BTech Project Report on Surface Plasmon Polariton Waves

IIInnnssstttiiitttuuuttteee ooofff RRRaaadddiiioooppphhyyysssiiicccsss &&& EEEllleeeccctttrrrooonnniiicccsss h

r

d

s

i

TTThhheee UUUnnniiivvveeerrrsssiiitttyyy ooofff CCCaaalllcccuuuttttttaaa

TTThhheeeooorrreeetttiiicccaaalll ssstttuuudddyyy ooofff SSSuuurrfffaaaccceee PPPlllaaasssmmmooonnn PPPooolllaaarrriiitttooonnn WWWaaavvveee

RRReeepppooorrrttteeedd bbbyyy

MMMooouuullliiinnnaaattthhh RRRaaayyy &&&

SSSuuubbbhhhaaa CCChhhaaakkkrrraaabbbooorrrtttyyy

333 rrrddd... YYYeeeaaarrr BBB... TTTeeeccchhh,,, IIInnnsssttt iiitttuuuttteee ooofff RRRaaadddiiioooppphhhyyysssiiicccsss &&& EEEllleeeccctttrrrooonnniiicccsss,,,

UUUnnniiivvveeerrrssiiitttyyy ooofff CCCaaalllcccuuuttttttaaa

SSSuuupppeeerrrvvviissseeeddd aaannnddd aaadddvvviiissseeeddd bbbyyy

PPPrrrooofff... DDDrrr... PPP... KKK... SSSaaahhhaaa

CCCooonnnttteennnttt::: e

t nd por

PPPaaagggeee CCChhhaaapppttteeerrr 111::: IIInnntttrrroooddduuucccttooorrryyy wwwooorrrdddsss::: AAAiiimmm,,, ssscccooopppeee aaanndd ssstttrrruuuccctttuuurrreee ooofff ttthhheee rrreeeppoorrttt 111---555 Introduction: What & Why 2 Historical background 3 Objective of the project 3 Structure of the report 4 CCChhhaaapppttteeerrr 222::: BBBaaasssiiiccc EEEllleeeccctttrrrooommmaaagggnnneeettt iiisssmmm aaannnddd lll iiiggghhhttt aaasss eeellleeeccctttrrrooommmaaagggnnneeettt iiiccc wwwaaavvveee 666---111999 2.1 : Introduction 7 2.2 : Maxwell’s equation 7 2.3 : Boundary conditions 8 2.4 : Propagation through matter 9 2.5 : Power of electromagnetic wave 13 2.6 : Propagation in bounded and unbounded media 14 1. Unbounded medium 14 2. Bounded medium 15 2.7 : Discussions 19 CCChhhaaapppttteeerrr 333::: RRRaaayyy ooopppttt iiiccc aaapppppprrroooaaaccchhh fffooorrr ggguuuiiidddeeeddd wwwaaavvveeesss ttthhhrrrooouuuggghhh dddiiieeellleeecccttt rrr iiiccc ssslllaaabbb wwwaavvveeegguuuiiidddeeesss 222000---444000 a g 3.1 : Introduction 21 3.2 :The structure of the thin layer dielectric waveguide 21 3.3 : Basic optical laws in dielectric waveguides 22 3.4 : Phase shift in total internal reflection at the dielectric interfaces 24 1. TE wave 24 2. TM wave 27 3.5 : Goos-Hanchen Shift and field penetration on the basis of ray-optics 31 3.6 : Ray optical explanation of SWG modes : Discrete nature of phase constant 37 3.7 : Discussions 40 CCChhhaaapppttteeerrr 444::: MMMooodddaaalll aaannnaaalllyyysssiiisss ooofff dddiiieeellleeecccttt rrr iiiccc ssslllaaabbb wwwaaavvveeeggguuuiiidddeeesss 444111---777222

e

4.1 : Introduction 42 4.2 : Wave equation 42 4.3 : Trapped modes in asymmetric 3 layer dielectric waveguide 44 1. TE modes field distribution and eigenvalue equation 44 Eigenvalue equation in normalized form 46 Modal cut-off 48 Mode numbers 49 Normalization in terms of power flow 50 Effective width 51 Confinement factor 52 2. TM modes field distribution and eigenvalue equation 55 Eigenvalue equation in normalized form 58 Modal cut-off 59 Mode numbers 60 Normalization in terms of power flow 60 Effective width 62 Confinement factor 63 4.4 : Symmetric SWG 66 1. Odd and Even TE modes distribution 66 2. Modal cut-off 66 4.5 : Weakly guiding symmetric SWG 71 4.6 : Discussions 72 CCChhhaaapppttteeerrr 555::: SSSWWWGGG wwwiii ttthhh mmmeetttaaalll ::: BBBaaasssiiiccc ppprrrooopppeeerrrttt iiieeesss ooofff ttthhheee mmmooodddeeesss 777333---777999 5.1 : Introduction 74 5.2 : Dielectric property of metal : Drude Theory of dielectric constant 74 5.3 : Investigated results for several metals 77

5.4 : General discussion on modes under the effect of metal layer 78 5.5 : Discussions 79 CCChhhaaapppttteeerrr 666::: SSSuuurrrfffaaaccceee ppplllaaasssmmmooonnn wwwaaavvveee iiinnn sssiiinnngggllleee iiinnnttteeerrrfffaaaccceee SSSWWWGGG 888000---888666

e

rie i

ie i

::

e

6.1 : Introduction 81 6.2 : Geometry of the structure 81 6.3 : Modal analysis 82 1. TE modes 82 2. TM modes 83 6.4 : Discussions 86 CCChhhaaapppttteeerrr 777::: SSSuuurrrfffaaaccceee ppplllaaasssmmmooonnn pppooolllaaarrr iii tttooonnn wwwaaavvveee iiinnn aaasssyyymmmmmmeeettt rrr iiiccc mmmeeetttaaalll ccclllaaaddddddeeddd SSSWWWGGG 888777---111000777 7.1 : Introduction 88 7.2 : Geometry of the SWG 88 7.3 : The characteristic equation 88 7.4 : Modal cut off 90 7.5 : Maximum value of b : Dispersion profile 92 TE Modes 92 TM Modes 92 7.6 : Field distribution : Complete set of field equations : 97 7.7 : Propagation loss and propagating length 104 7.8 : Effective width of the optical beam 106 7.9 : Discussions 107 CCChhhaaapppttteeerrr 888::: SSSuuurrfffaaaccceee ppplllaaasssmmmooonnn pppooolllaaarrr iii tttooonnn wwwaaavvveee iiinnn SSSWWWGGG wwwiii ttthhh mmmeeetttaaalll lll iiiccc ggguuuiiidddeee sssaaannndddwwwiiiccchhheeeddd bbbeeetttwwweeeeeennn tttwwwooo dddiieellleeecccttt rrr iiccc lllaaayyyeeerrrsss 111000888---111222999 8.1 : Introduction 109 8.2 : Characteristic equation 110 8.3 : Maximum value of B : Decoupled and coupled Fane mode 112 8.4 : Field distribution : Complete set of field equations 116 8.5 : Cut off 119 8.6 : Mode spot size 121 8.7 : Propagation loss : Long Range and Short Range SPP 122 8.8 : Symmetric guide : Even and Odd symmetric modes 124 8.9 : Discussions 129 CCChhhaaapppttteeerrr 999::: MMMooodddaaalll aaannnaaalllyyysssiiisss ooofff ttthhhiiinnn mmmeeetttaaalll sssttt rrr iiippp ooofff fff iiinnniii ttteee wwwiiidddttthhh aaasss ooopppttt iiicccaaalll wwwaaavvveeeggguuuiiidddeee::: EEEfff fffeeecccttt iiivvveee dddiieellleeecccttt rrr iiccc cccooonnnssstttaaannnttt mmmeeettthhhoooddd 111333000---111444888 9.1 : Introduction 131 9.2 : Geometry of the structure 131 9.3 : General discussion on modes in 3D waveguides 131 9.4 : Modal analysis using effective dielectric constant method 133 9.5 : Numerical results using the effective dielectric constant method 137 9.6 : Modified application of the EDC method 141 Confirmation of improvement of results 144 9.7 : Discussion 148 CCChhhaaapppttteeerrr 111000::: CCCooonnncccllluuusssiiivvveee wwwooorrrdddsss: FFFuuutttuuurrreee ssscccooopppeee ooofff WWWooorrrkkk 111444999---111555333 AAAppppppeeennndddiiixxx AAA111---AAA666222 Newton Raphson method to solve algebraic equations A1 Solving the characteristic equation in Newton Raphson method A2 Programs for the plots A4 RRReeefffeeerrreennnccceeesss RRR111---RRR333

CCChhhaaapppttteeerrr (((iii)))

IIInnntttrrroooddduuuccctttooorrryyy wwwooorrrdddsss::: AAAiiimmm,,, ssscccooopppeee aaannnddd ssstttrrruuuccctttuuurrreee ooofff ttthhheee rrreeepppooorrrttt

--- 111 ---


IIInnntttrrroooddduuuccctttiiiooonnn::: WWWhhhaaattt aaannnddd WWWhhhyyy Surface plasmon polaritons are surface electromagnetic waves that exist at the interface of a dielectric and a metal. These modes are very much surface bound in nature and decay evanescently from the interface on both sides of the interface. The existence of these modes at the metal-dielectric interfaces is attributed to the peculiar dielectric property of certain metals especially in the optical frequencies starting from the far infrared region to the ultraviolet region. Certain conducting metals like Gold, Silver, Aluminum, Copper etc. have frequency dependant complex dielectric constants in this range with a large negative real part and a small imaginary part. The surface electromagnetic waves propagating through the interface of a material with positive dielectric constant and one with negative dielectric constant is generally termed as Surface Polariton waves. When metal is used as the negative dielectric constant material the surface polariton wave is termed as a SSSurface PPP lasmon PPPolariton wave. These waves are in general lossy in nature. It is the imaginary part of the dielectric constant of the metals that gives rise to loss. The surface Plasmon Polariton wave existing at the interface between a lossless dielectric and a lossless metal is a idealized solution and is termed as a Fano wave. So Surface Plasmon Polaritons are basically Fano waves with losses incorporated. In the entire scope of this project we would study the theoretical background of propagation characteristics and field distributions of Surface Plasmon waves. Surface plasmon polaritons have received much attention for their ability to guide electromagnetic energy. Unlike dielectric waveguides, which confine volume electromagnetic waves to an optically dense core, these surface electromagnetic waves are localized at interfaces between dielectric materials and metals or ionic solids that support charge density oscillations. This surface localization has led researchers to explore the potential for transporting information via guided polariton modes with smaller spatial extents than can be achieved with diffraction-limited dielectric waveguides. In days of bulk optics, surface plasmons were just another curious physical effect of largely academic interest. Theoretical study reveals that they are tricky to excite, occupy a very small area, and can’t travel far. So for a long time they had very limited practical interest beyond certain sensing application. This has changed with the development of techniques for fabricating micro and nanoscale structures that alter the interaction of light with electrons. These structures can control optical interaction on a sub-wavelength scale for application including sensing, nonlinear device optical storage and signal processing. So the once obscure field has become a hot research area now. Surface plasmons have been recognized for decades but they were not initially identified as responsible for their first application, surface enhanced Raman scattering. Electrochemist Martin Fleischmann, now better known for his cold fusion research, discovered it in the early 1970s when he began studying the Raman spectra of electrode surface. He tested a variety of materials, and found that using roughened silver surfaces enhanced the Raman signal by as much as a million fold. Only later was the effect linked to surface plasmons, which concentrated the electromagnetic field to multiply the intensity of nonlinear Raman signal. SPR can be used as the basis for a sensor which is capable of sensitive and quantitative measurement of a broad spectrum of chemical and biological entities. It offers a number of important practical advantages over current analytical techniques. Sensing application based on resonances of surface plasmon became well established over the past two decades. They can be used for real time detection of specific material such as bio-molecules that are present at low concentrations. A recent trend in plasmon resonance sensing is the development of high resolution microscopy and measurement. One recent example is profiling a strongly focused beam with sub-micrometer resolution by using it to excite surface plasmon on a thin gold film.

--- 222 ---


The surface plasmon modes are strong at the interface and decay evanescently from the interface. With proper choice of guide structure they can be made narrow enough even to the sub-wavelength dimensions. This makes these modes so special relative to the other modes. In integrated optical devices the normal optical modes spread over the entire thickness and therefore reflection from the interfaces are inescapable. This puts a limit to the miniaturization of the optical devices to the sub-wavelength scale. But miniaturization of the integrated devices is a need for technological revolutions. Surface plasmon modes being narrow enough do not penetrate inside by much and hence reflection from the layer boundaries is evaded. Therefore it is one of the prime aspects that are being and going to be investigated regarding the integrated optical device fabrications. And it can be anticipated that these waves will definitely bring revolutions to the integrated device fabrication technologies. The new interest in surface plasmons owes much to the development of photonic band-gap materials. Besides, operating in the sub-wavelength scale surface plasmons are opening new door in optical devices, effect and integration. They offer way to circumvent the important limitation imposed by conventional optics in areas like beam divergence and density of component integration. It is although too early to say what will prove practical, but interesting developments are certain. HHHiiissstttooorrriiicccaaalll bbbaaaccckkkgggrrrooouuunnnddd:::

Since 1950s, there were many people who begun to investigate the waveguide theory and different kinds of analytic solutions and the choice of methods had expanded greatly. Sommerfeld and his student Zenneck in 1907 were the earliest investigators of surface electromagnetic waves phenomenon. Their theories of propagation of surface electromagnetic waves were over imperfect conductors. The inhomogeneous plane wave that exists at the interface of one media with absorption and the other one non-absorbing has since then known as Zenneck Wave. Gaubau was the first person who applied the theoretical analysis of surface electromagnetic wave into practice by using a transmission wire as surface waveguide. It was an outstanding breakthrough and since then open boundary waveguide had been studied extensively both experimentally and theoretically. In the late 1970s, H. Reather had begun his investigation in surface plasmon and according to him surface plasmons exist at the boundary of a solid (metal or semiconductor) whose electrons behave like those of a quasi-free electron gas. However, the research was not greatly investigated until recent years. OOObbbjjjeeeccctttiiivvveee ooofff ttthhheee ppprrrooojjjeeecccttt::: In the project our main objective is to analyze the various properties of surface plasmons in different waveguide structures. The area of this project begins with a good understanding of basic electromagnetic theory. Electromagnetic field theory is a discipline concerned with the study of charges, at rest and in motion, that produce currents and electric-magnetic fields. Maxwell’s equations relate the electric field and magnetic field together and they can be applied to all the situations discussed in this project. The electric and magnetic field vector are solutions of the inhomogeneous vector wave equations. Therefore electric and magnetic fields for a given boundary value problem can be obtained either as solutions to Maxwell’s or the wave equation. For doing so, rigorous electromagnetic solution of Maxwell’s equations had been necessary. So our report starts with fundamental electromagnetic laws for guided and unguided waves. We also studied in details the electromagnetism of the dielectric two-dimensional slab waveguides and extended the analysis for the case of metallic layers. We tried to emphasize on the surface plasmon waves at the interface of metal with dielectrics. We considered both single interface and double interface rectangular structures for modal analysis. We considered various situations regarding the position of the metal layer in the two dimensional structures. Finally we discussed the various modes present in the thin finite width metal strip as

--- 333 ---


an optical channel embedded into asymmetrically placed dielectric covers. The method adopted for such three dimensional structure was the effective dielectric constant method. Rigorous analytical solutions cannot be obtained in such structures and therefore approximate methods like the effective dielectric constant method has been adopted. Therefore the main objective of our project is to provide a theoretical background to the Surface Plasmon waves in such finite width metal strip structures in asymmetrically placed semi infinite dielectric medium. The pertinent background needed for such analysis have also been studied. The excitation of the modes in such guides, the application part and other experimental details for sensing the modes are beyond the scope of this project.

SPP

SSStttrrruuuccctttuuurrreee ooofff ttthhheee rrreeepppooorrrttt::: The report deals with the electromagnetism associated with the electromagnetic waves in some dielectric and metallic waveguides. The structure of the report has been attempted to be in compliance with the gradual approach towards the metallic structures, starting from the basic electromagnetic theory of guided waves. Chapter 2 of the report has been devoted to the development of the basic electromagnetism. We start with Maxwell’s equations and the boundary conditions and solutions for bounded and unbounded waves in general. In this chapter we discussed about phase and group velocity, guided and unguided wavelengths, power flow associated with the waves etc. In Chapter 3 we tried to pay attention to the ray nature of optical waves, their fundamental laws, reflection and refraction phenomena, guiding through multiple reflections, penetration into non-propagating media under the condition of total internal reflection, and the ray optic analog of the modal analysis in such optical waveguide structures. Chapter 4 starts with the modal analysis of dielectric slab waveguides. The various supported modes have been analyzed in this chapter with recourse of the wave equations. The TE and TM modes have been analyzed separately with fair degree of entirety. The modal dispersion, modal cut-off, field penetration, etc. are discussed on the basis of the characteristic equation for the structure which we derive in the chapter starting from the boundary conditions for the electric and magnetic field components. In Chapter 5 we introduce to the dielectric behavior of metals in the optical region of frequencies. Some metals exhibit negative dielectric constant in this region which makes them extremely handy for propagation of the surface Plasmon modes. In this chapter the elementary theory of this dielectric behavior of those metals has been discussed in brevity. In Chapter 6 we started analysis of the surface modes of optical waves in a single interface structure containing metal at one side of the structure and dielectric at the other side. We solved Maxwell’s equations for such structures for bothTE and modes and found that only one TM solution can lead to the existence of the surface mode in such structure but TE modes are not supported.

TM

Chapter 7 starts with the extension of the theory of dielectric slab waveguides derived in chapter 5 to the case of metal covered slab waveguides. In this chapter we studied the theoretical aspects of both TE and

modes and sought for the surface Plasmon Polariton wave in the structure. The distinct features of this mode to the other volume modes in this structure are also pointed out as far as possible. The field amplitude distribution for the different volume modes and that of the mode is probably the most interesting score of this chapter.

TM

SPP

--- 444 ---


In Chapter 8 we discuss about the modes in slab waveguide with a thin metal strip sandwiched between two dielectric layers of semi-infinite dimension. We start from the same equations developed in the chapter 4 for the general and modify it to be useful for the structure. The even and the odd modes are of much importance in this structure.

SPP

SWG

The Chapter 9 starts with the modal analysis of finite three dimensional structures. We considered the case of thin metal strip of finite width embedded in asymmetrically placed dielectric covers. Since exact analytical solution the finite structure is next to impossible as it would lead to numerous number of boundary conditions we use one of the most widely used approximate methods called the Effective Dielectric Constant ( ) method for the modal analysis. Wave solutions for the structure have been looked for to agree with the reported results. The method was further modified for the improvement of the derived results.

EDC

The Chapter 10 is the conclusive chapter where we pointed out a few drawbacks of the method for analyzing the present structure like, why it did not give the best solution. Further modifications to the approach for the structure using the technique has also been suggested which we could not carry on due to shortage of time.

EDC

EDC

--- 555 ---

CCChhhaaapppttteeerrr (((iiiiii)))

BBBaaasssiiiccc EEEllleeeccctttrrrooommmaaagggnnneeetttiiisssmmm aaannnddd llliiiggghhhtt aaasss t

eeellleeeccctttrrrooommmaaagggnnneeetttiiiccc wwwaaavvveee

--- 666 ---


222...111 IIInnntttrrroooddduuuccctttiiiooonnn::: In optical communication, the light wave is routed through specific bounded media. The wave is essentially not a plane wave in this case as it is in case of transmission through unbounded media, where the wave front is infinite in extent in the direction perpendicular to the direction of propagation. The wave essentially satisfies some boundary conditions at the interfaces. Therefore we need to solve the wave equation in such guide and apply the boundary conditions to analyze the transmission characteristics in such a waveguide. 222...222 MMMaaaxxxwwweeellllll ’’’sss eeeqqquuuaaatttiiiooonnnss::: s The four Maxwell’s equations are

0

.ερΕ =∇

rr (2.2.1)

0. =∇ Βrr

(2.2.2)

t∂

∂−=×∇

ΒΕr

rr (2.2.3)

t

J 000 ∂∂

+=×∇ΕεµµΒr

rrr (2.2.4)

where Εr

( ) & t,rr Βr

( ) are the electric & magnetic field intensities associated with the wave, t,rr

ρ ( ) & t,rr Jr

( ) are the total charge density & total current density at the field point t,rr rr , at any time instant t , 0ε & 0µ are permittivity & permeability of free space given by 0ε = 2212 m.N/C10854.8 −×

m/H104 70

−×= πµ

In terms of free charge & current densities fρ ( t,rr ) & fJr

( t,rr ) the above set of equations can be rewritten as

fD. ρ=∇rr

(2.2.5)

0H. =∇rr

(2.2.6)

t∂

∂−=×∇

ΒΕr

rr 2.2.7)

tDJH f ∂∂

+=×∇r

rrr (2.2.8)

where ( ) is the magnetic induction field and Hr

t,rr Dr

( t,rr ) is the electric displacement vector respectively given by

EDrr

ε= (2.2.9)

Βµrr 1H = (2.2.10)

µ & ε are the permeability & permittivity of the medium respectively. In a source free region where

--- 777 ---


0J f

rr=

& 0f =ρ

the above equations can be written as 0. =∇ Ε

rr (2.2.5 )

0H. =∇

rr (2.2.6)

t

H∂∂

−=×∇r

rrµΕ (2.2.7)

t

H∂∂

=×∇Εεr

rr (2.2.8)

222...333 BBBooouuunnndddaaarrryyy cccooonnndddiiitttiiiooonnnsss::: When an electromagnetic wave propagates through a medium and falls on the interface with another medium, some boundary conditions are satisfied at the interface. If 1H

r& 2H

r be the magnetic induction

vectors and 1Εr

& 2Εr

the corresponding electric fields just on the two sides of the interface, the boundary conditions can be written as (2.3.1) 0t

2t1 =− ΕΕ

(2.3.2) fn22

n11 σΕεΕε =−

nKHH ft2

t1 ×=−

r (2.3.3)

(2.3.4) 0HH n22

n11 =− µµ

where the superscripts t & denote the components tangential & normal to the interface respectively and

n1µ , 2µ , 1ε , 2ε are respective permeability & permittivity values of the two media on the two sides of

the interface. The terms fσ & are the free surface charge density on the interface & free current per unit length on the interface, is unit normal on the interface. In particular if there is no free charge or free current on the interface we can write

fKr

n

(2.3.5) 0t

2t1 =− ΕΕ

(2.3.6) 0n22

n11 =− ΕεΕε

(2.3.7) 0HH t2

t1 =−

(2.3.8 a) 0HH n22

n11 =− µµ

that is in words the tangential components of the electric field intensity & the magnetic induction vector and the normal components of the displacement vector & the magnetic field intensity are continuous across such a source free interface.

--- 888 ---


In the discussion we will be mainly concerned about propagation through dielectric & metallic slabs and their interfaces which are assumed to be hardly magnetic in nature. So for all the slabs we can write

021 µµµµ === (2.3.9)

Therefore equation (2.3.8 a) can be written as

(2.3.8 b) 0HH n2

n1 =−

222...444 PPPrrrooopppaaagggaaatttiiiooonnn ttthhhrrrooouuuggghhh mmmaaatttttteeerrr::: Let us consider an arbitrary uniform medium whose cross sectional dimensions and the parameters do not vary in the direction of propagation. We consider the medium to be source free i.e. it contains no free charges or active media. If it has some finite conductivity σ the free charge density can be related to the electric field as

fJr

Εσrr

=fJ (2.4.1) Thus for such homogeneous isotropic dielectric source free medium Maxwell’s equations can be written from equations (2.2.5 - 2.2.8) as

0. =∇ Εrr

(2.4.2) 0H. =∇

rr (2.4.3)

t∂

∂−=×∇

ΒΕr

rr (2.4.4)

t

H∂∂

+=×∇ΕεΕσr

rrr (2.4.5)

Let us write the electric field vector in the form )r.kt(jexp)t,r( 0

rrrrr−= ωΕΕ (2.4.6)

where k

r is the propagation vector in the arbitrary direction of propagation, defined as , and r

r ω is the angular frequency of the wave defined as

λ

πω cf2 == (2.4.7)

where is the frequency of the wave,f λ the corresponding wave-length and is the speed at which the wave propagates in the direction

crr , given by

µε1c = (2.4.8)

The refractive index of the medium is defined as

cc

n 0= (2.4.9)

--- 999 ---


or, n00εµ

µε= (2.4.10)

or, rn ε≈ (2.4.11)

using equation (2.3.9), where is the speed of the electromagnetic wave in free space given as 0c

00

01cεµ

= (2.4.12)

and rε is the relative permittivity of the medium defined as r0εεε = (2.4.13) Now using equation (2.4.6) in equations (2.4.2 - 2.4.5) we get

0. =∇ Ε

rr (2.4.13)

0H. =∇rr

(2.4.14) Hj 0

rrrωµΕ −=×∇ (2.4.15)

Εεωεvrr

0crjH =×∇ (2.4.16)

where is the effective complex relative permittivity, given by c

rε

0

rcr j

ωεσεε −= (2.4.17)

Using these four equations (2.4.13-2.4.16) and using vector identities we get the following wave equations for Ε

r field & field respectively, H

r

0)k( 22rr

=+∇ Ε (2.4.18 a) 0H)k( 22

rr=+∇ (2.4.18 b)

where is the wave number in an unbounded medium defined as k

λπ2kk ==

r (2.4.19)

and

2

2

2

2

2

22

zyx ∂∂

+∂∂

+∂∂

≡∇

in Cartesian coordinate system defined by

zzyyxxr ++=r

(2.4.20) For any one of the orthogonal components of the Ε

r & H

rfields in any coordinate system, we can write in

general

--- 111000 ---


(2.4.21) 0)k( 22 =+∇ ψ

Using the separation of variable technique equation (2.4.21) can be solved by setting )z(Z)y(Y)x(X)z,y,x( =ψ (2.4.22) Then the solutions in Cartesian coordinate system are

)zjkexp(~)z(Z

)yjkexp(~)y(Y

)xjkexp(~)x(X

z

y

x

−

−

−

i.e. [ ])zjkyjkxjk(exp~ zyx ++−ψ (2.4.23) where

2

22

z2

y2

x2 4kkkk

λπ

=++= (2.4.24)

zyx kzkykxk ++=r

(2.4.25) If the z direction is considered to be the direction of propagation we can write the solution the electric field and the magnetic field can be written from equation (2.4.23) as )}zkt(jexp{)z,y,x(E)t,z,y,x( z0 −= ωΕ (2.4.26 a) )}zkt(jexp{)z,y,x(H)t,z,y,x(H z0 −= ω (2.4.26 b) and therefore equations (2.4.21) take the form of what is called the Helmholtz’s equation (2.4.27) 0)k( 2

c2

t =+∇ ψ

where 2

2

2

22

t yx ∂∂

+∂∂

≡∇

and (2.4.28) 2y

2x

2z

22c kkkkk +=−=

Now if γ be the propagation constant in the direction of propagation ( z - direction) given by βαγ j+= (2.4.29) where α is the attenuation constant and β is the phase constant in the direction of propagation inside the guide the field variation in z direction is written as )zexp( γ , so we can write from equation (2.4.26)

--- 111111 ---


zjk=γ (2.4.30 a) αβ jkz −= (2.4.30 b) i.e. from equation (2.4.28) we have the relation (2.4.31) 222

c kk γ+= If the propagation is lossless through the guide we set 0=α and then β=zk (2.4.32) Now let us write the components of E

r and H

rfields along z,y,x directions as and

respectively i.e. zyx E,E,E

zyx H,H,H zyx0 EzEyEx)z,y,x(E ++= (2.4.33 a) and zyx0 HzHyHx)z,y,x(H ++= (2.4.33 b) Then using equation (2.4.33) in equation (2.4.26) and putting them in equation (2.4.15, 2.4.16) we get on simplification the four equations relating yxyx H,H,,E Ε to the components & as zE zH

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−=x

Ey

Hk

jE zz02

cx βωµ (2.4.34 a)

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=y

Ex

Hk

jE zz02

cy βωµ (2.4.34 b)

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=x

Hy

Ek

jH zzcr02

cx βεωε (2.4.34 c)

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−=y

Hx

Ek

jH zzcr02

cy βεωε (2.4.34 d)

If the medium is perfectly dielectric in nature 0=σ and we can use the same relations by setting

. The effectiveness of these equations is that if we know the field pattern of & all the field components can be found out exclusively. For example for T

rcr εε = zE zH

E w.r.t. z mode direction we set and solve the wave equation (2.4.27) for and all the other components can be found out. But

these equations cannot be applied directly to TEM waves as for them and we will have to solve the Maxwell’s equations separately. But since we are concerned about the guided waves which cannot in general be for the existence of the wave at all, we keep our attention restricted into the present context of equations.

0Ez = zH

0k 2c =

TEM

--- 111222 ---


222...555 PPPooowwweeerrr oooff eeellleeeccctttrrrooommmaaagggnnneeetttiiiccc wwwaaavvveee::: f According to Poynting Theorem the energy flux per unit area of an electromagnetic wave per unit time is given by the instantaneous Poynting vector HES

rrr×= (2.5.1)

If we now express the electric and magnetic fields by complex notations we have to take the real parts in calculating the vector S

r. Thus

)HRe()ERe(S

rrr×= (2.5.2)

where denotes the real part. Now if Re ∗E

r and ∗H

rdenote the complex conjugate of the complex

fields Er

and Hv

respectively we may write

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+

+=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∗

∗

HH

EE

21

H

ERe

rr

rr

r

r

so that we can write equation (2.5.2) as

)HH()EE(41S ∗∗ +×+=

rrrrr

[ ])HE()HE()HE()HE(41 rrrrrrrr

×+×+×+×= ∗∗∗∗ (2.5.3)

The time averaged energy flow per unit time per unit area is obtained by averaging the right hand side of the equation over entire time period of the fields. In notations we write

∗∗∗∗ ×+×+×+×= HE41HE

41HE

41HE

41S

rrrrrrrrr (2.5.4)

Now if we write the fields as

)tjexp()r(HH

)tjexp()r(EE

ω

ωrrr

rrr

=

=

so that the period of the fields is ωπ2T = we get

∗∗ ×==×=× ∫ HEdt)tj2exp()r(H)r(ET1HE

T

0

rrrrrrrrΟω

--- 111333 ---


and )r(H)r(Edt)r(H)r(ET1HE

T

0

rrrrrrrrrr×=×=× ∗∗∗ ∫

)r(H)r(Edt)r(H)r(ET1HE

T

0

rrrrrrrrrr ∗∗∗ ×=×=× ∫

Therefore from equation (2.5.4) we get that the time averaged energy flow per unit time per unit area is

)HEHE(41S

rrrrr×+×= ∗∗

)HERe(21)HERe(

21S

rrrrr×=×= ∗∗ (2.5.5)

222...666 PPPrrrooopppaaagggaaatttiiiooonnn iiinnn bbbooouuunnndddeeeddd aaannnddd uuunnnbbbooouuunnndddeeeddd mmmeeedddiiiaaa::: 111... UUUnnnbbbooouuunnndddeeeddd mmmeeedddiiiuuummm ::: For propagation through an unbounded perfectly dielectric medium of relative permittivity

rε there is no

boundary condition for the fields to satisfy, because an unbounded medium is one which has no discontinuous interface for the wave to pass through. Therefore if z direction is considered to be the direction of propagation there is no field variation in transverse direction. That is to say

0yx

2t ≡∇≡

∂∂

≡∂∂

(2.6.1.1) 0k 2c =

β== zkk (2.6.1.2)

Under such conditions the electromagnetic wave travels in the z direction with propagation constant β . The wave front is flat and extends indefinitely in the transverse direction i.e. the wave is in nature. For such waves we may symbolically write (

TEMkHErrr

⊥⊥ ) as the Er

& Hr

fields are contained in the plane of the wave front which is perpendicular to the direction of propagation. Now the phase velocity of a wave is defined as the velocity at which the energy of the wave is propagated. Since for wave the flow of energy is in the direction of TEM k

r itself the phase velocity of an unbounded

plane wave is given by

)(

1nc

ck

0p µεβ

ωωυ ===== (2.6.1.3)

i.e. the phase velocity of the wave is the velocity of light in the medium. Thus a perfectly dielectric ( ) unbounded medium in non-dispersive, because the phase velocity in independent of frequency. r

cr εε =

--- 111444 ---


The term group velocity comes in case when a number of electromagnetic waves having different frequencies propagate through a medium .As the refractive index of a medium is dependent on frequency, different waves travel at different phase velocities and they superimpose with each other to form what is called the wave packet which actually carries the energy. The group velocity is defined as the velocity of the wave packets and is given by

βωυ

dd

g = (2.6.1.4)

Since for unbounded waves ckc == βω

)(

1nc

c 0pg µε

υυ ====

222... BBBooouuunnndddeeeddd mmmeeedddiiiuuummm ::: In bounded medium the wave is subjected to the boundary conditions at the interfaces and the wave cannot be TEM in nature. This is because if we set & both equal to zero, we get from equations (2.4.34)

all the components of zE zH

Εr

& fields vanish which means there is no wave at all! The modes in such bounded media may be either T

Hr

E or TM or neither of these. Since bounded waves are influenced by the interfaces of the medium with other medium, complete set of solutions can be found out by solving equation (2.4.27) and applying the boundary equations (2.3.5-2.3.8). A detailed study of these is done in chapter 4 for multilayer dielectric slab waveguides. For the present context let us assume a guiding medium between two perfectly dielectric identical medium where the interface is along y~x plane as shown in figure 2.6.1.

Fig. 2.6.1 : Guided wave propagation through a bounded medium, depicted as ray The wave front propagates through multiple reflections at the interfaces as depicted by the arrow. Now since β is the effective propagation constant in the direction of power flow the phase velocity is given by

βπ

βωυ f2

p == (2.6.2.1)

Again we can define the guide wavelength gλ as the wavelength of the wave measured in the guide which is the effective wavelength in the direction of power flow with phase velocity pυ . So

--- 111555 ---


fp

g

υλ = (2.6.2.2)

or gp fλυ =

g

2λπβ = (2.6.2.3)

Again the wavelength λ associated with the direction of k

r is the wavelength in an unbounded media.

Thus

λω fk

c == (2.6.2.4)

or λπ2k = (2.6.2.5)

Finally, for bounded waves in a lossless dielectric medium we have from equation (2.4.31) (2.6.2.6) 2

c22 kk =− β

Putting expressions of & k β we get

2

2c

2g

2 4k11πλλ

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

For dimensional match we write

c

c2kλπ

= (2.6.2.7)

which yields the relations

2c

2g

2

111λλλ

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

or2

c

g)(1 λλ

λλ−

= (2.6.2.8)

2

c

p)(1

c

λλυ

−= (2.6.2.9)

and 2

c )(1k λλβ −= (2.6.2.10)

For guided waves in presence of boundary conditions and hence which means 02t ≠∇ 0k 2

c ≠β≠k , gλλ ≠ , cp ≠υ . Now for cλλ = we have 0=β which means the wave ceases to flow

for wavelength equal to cλ . Thus cλ is the upper cut-off of wavelength that can be supported by the guide. In figure 2.6.2 plots of cg λλ , cpυ and kβ against cλλ are shown for bounded and unbounded

waves. This shows λλ =g for unbounded waves as and0k 2c = λλ >g . The plots are generated

using MATLAB 5.3 and the corresponding program is written in appendix (Prog. 1).

--- 111666 ---


(a) (b)

(c) Fig. 2.6.2 : Variation of k,, pg βυλ with cλλ for unbounded and bounded waves

The group velocity is given by

βωυ

dd

g =

or gυ 2c )(1c λλ−= (2.6.2.11)

and 2

02gp n

cc ⎟

⎠

⎞⎜⎝

⎛==υυ

--- 111777 ---


In figure 2.6.3 we have drawn phase velocity & group velocity for bounded waves in the same graph to show a comparison between them. Now from figure 2.6.2 & figure 2.6.3 the following characteristics of the bounded waves are obtained: 1. When cλλ << , cp →υ and λλ →g the nature of propagation tends to that in an unbounded medium.

Fig. 2.6.3 : Variation of group vel. And phase vel. of bounded wave with cλλ

2. As cλλ → , ∞→gλ , 0→β and ∞→pυ . For a finite frequency, infinite velocity means infinite wavelength, zero time to travel finite distances. So the propagation ceases as cλλ → . 3. At cλλ > , β becomes imaginary, so the wave cannot propagate through the guide and attenuates with distance. So cλ is essentially the cut-off wavelength of the waveguide. The corresponding cut-off frequency is defined through cf

ccr

0 fc

c λε

== (2.6.2.12)

that is, cλ is the wavelength measured in an unbounded medium corresponding to a frequency at which propagation ceases on a guiding line.

cf

4. As seen from the equation (2.6.2.9) pυ is a function of wavelength λ and therefore the medium is dispersive in nature .This means waves of different frequency will travel at different speed through it and chromatic dispersion will take place among them. 5. From figure 2.6.3 we see that cp ≥υ in a bounded medium. But the energy is carried at the group velocity along the guide and from figure 2.6.3 cg ≤υ , that is the wave propagates in a bounded medium at a slower speed than in unbounded medium.

--- 111888 ---


222...777 DDDiiissscccuuussssssiiiooonnnsss:::

In this chapter we had tried to introduce the fundamental laws of electromagnetism obeyed by electromagnetic waves in unbounded and bounded media. We have seen that the guided waves propagate within the waveguides with phase velocity different from their free velocities unlike the case of unbounded waves. They have a cut-off wavelength that must not be exceeded in order that the wave propagates at all. As the wave is guided through the guiding layer the electromagnetic energy is confined within the guide and it decays exponentially in the subsidiary portions. In the next chapters the electromagnetism of different optical guide models will be discusses. The boundary conditions at the interfaces are the starting tools for analyzing such models. In the next chapters these boundary value problems will be solved in order to obtain the propagation characteristics of the waves in various structures. The four basic equations relating the four transverse field components with the two components in the direction of propagation will be used frequently to find the field distributions. The equation for power flow in the section (2.5) will also be utilized in order to normalize the field amplitudes to proportionately correct values.

--- 111999 ---

CCChhhaaapppttteeerrr (((iiiiii iii)))

RRRaaayyy oooppptttiiiccc aaapppppprrrooaaaccchhh ffooorrr ggguuuiiidddeeeddd wwwaaavvveeesss o f

t tthhhrrrooouuuggghhh dddiiieeellleeeccctttrrriiiccc ssslllaaabbb wwwaaavvveeeggguuuiiidddeeesss

--- 222000 ---


333...111 IIInnntttrrroooddduuuccctttiiiooonnn::: Since light is one form of electromagnetic wave, the properties of light wave can be explained in terms of Maxwell’s equations. Therefore rigorous way to calculate the intensity and phase of a light wave means to derive the wave equations and solve them directly subject to the boundary conditions. But as long as the wavefront of the light is much larger than the dimensions of the guiding medium, the concept of ‘light ray’ and ray equations are convenient to use for analysis. That is why for transmission of light through waveguides, refraction and total internal reflection at the guide boundaries etc. are useful and accurate tools for analysis as long as the plane wavefront assumption is justified. Under these situations the guided wave can be approximated as a plane wave without much loss of accuracy although strictly speaking they are not so as the wave is subject to satisfy the boundary conditions which truncates the wave front and bend them. In this chapter emphasis will be given on deriving the properties of transmission of light through dielectric slab waveguides on the basis of refraction and total internal reflection phenomena, using the simple process of ray tracing. Also Maxwell’s equation will be taken recourse of to evaluate the phase shifts associated with the refraction and total internal reflection phenomena. 33..22 TThhee ssttrruuccttuurree ooff tthhee tthhiinn llaayyeerr ddiieelleeccttrriicc wwaavveegguuiiddee:: The basic thin layer dielectric waveguide consists of three layers. 1. The waveguide, 2. The cladding and 3. The substrate. In figure 3.2.1 the structure of a 3-layer step-index dielectric waveguide is shown. With reference to the coordinate system considered the wave is guided through the middle layer along the z axis. The refractive indices of the three layers must satisfy the inequality csf nnn >>

Fig. 3.2.1 : Physical structure of 3-layer slab waveguide for the wave to be bounded in the middle layer. If the interfaces are infinite in extent in the y direction light confinement takes place only in the direction. Such waveguides are not practicable but only of theoretical importance. Such waveguides are called the

xD2 waveguides or slab waveguides. In such waveguides the

guided-light width expands in the y direction due to diffraction during propagation. So additional light confinement is achieved by truncating the waveguide dimensions in the y direction also, to achieve efficient guiding. The structures in which the light is confined in andx y directions both, are called the D3

--- 222111 ---


waveguides or channel waveguides. However we shall mainly concentrate on D2 waveguides in this chapter. 333...333 BBBaaasssiiiccc oooppptttiiicccaaalll lllaaawwwsss iiinnn dddiiieeellleeeccctttrrriiiccc wwwaaavvveeeggguuuiiidddeeesss::: Let us consider a ray of light traveling through a homogeneous isotropic medium (1) of refractive index be incident on the interface between medium (1) and another homogeneous isotropic medium (2) of refractive index as shown in

1n

2n

Fig. 3.3.1 : Light ray at the interface of two dielectric media figure 3.3.1. The interface is along the z~y plane whose cross-section through plane is shown. Then some part of the incident wave intensity will be refracted into medium (2) according to Snell’s equation

0x =

r2i1 sinnsinn θθ = (3.3.1) where iθ & rθ are the angle of incidence and angle of refraction as shown. If we have then there exists a cut-off angle of incidence

21 nn >

cθ called the critical angle such that if the ray is incident at an angle greater than cθ the rays is totally reflected back into the medium (1). The critical angle is obtained from equation (3.3.1) putting which yields o

r 90=θ

⎟⎟⎠

⎞⎜⎜⎝

⎛= −

1

21critical n

nsinθ (3.3.2)

So under this condition the energy associated with the wave is totally bound in medium (1). This criterion can be utilized to explain guiding of optical waves through 3 layer step index dielectric waveguide. Let the refractive indices of the substrate, waveguide and cladding layers be respectively. Then the cfs n,n,n

--- 222222 ---


critical angles for the cladding interface and the substrate interface be sc ,θθ respectively. Then we have from equation (3.3.2)

⎟⎟⎠

⎞⎜⎜⎝

⎛= −

f

c1c n

nsinθ (3.3.3 a)

⎟⎟⎠

⎞⎜⎜⎝

⎛= −

f

s1s n

nsinθ (3.3.3 b)

In order to assure propagation through the waveguide the required condition is that the incidence of the ray at the cladding interface should be greater than cθ and that at the substrate interface should be greater than sθ so that light can be internally reflected at the two interfaces to be bounded inside the waveguide layer. Therefore we must have and . As in practice the three layers are made such that

and hence cf nn > sf nn >

csf nnn >> cs θθ > .Therefore three conditions may arise. 1. o

is 90<< θθ In that case the ray will be reflected back from both the interfaces and thus the wave will be strongly bound to the guiding layer. This mode is the desired mode and called the guided mode. 2. sic θθθ << In that situation the ray will be reflected from the cladding layer but will transmit into the substrate. Therefore the amplitude of the wave variables will decrease along the propagation direction significantly. This mode is called the substrate radiation mode. 3. o

io 900 << θ

In this situation the ray gets refracted into both, the substrate and the cladding layer. This mode is called the substrate-clad radiation mode. In this mode the wave amplitude decays even faster along the direction of propagation. Now from wave vector point of view, with reference to the notations used in previous chapter we may write for the guiding medium ( ) fn if0fx cosnkk θ= (3.3.4 a) βθ == if0fz sinnkk (3.3.4 b) where the subscript denotes the guiding layer as usual. For propagation of the wave as a whole in thef z direction the z component of the wave vector should be continuous across the interfaces, i.e. β=== czszfz kkk We define effective index of refraction as effn

--- 222333 ---


⎟⎟⎠

⎞⎜⎜⎝

⎛==

0ifeff k

sinnn βθ (3.3.5)

such that β may be redefined as the propagation constant in an unbounded medium of refractive index . The range of for the three modes in the waveguide are correspondingly effn effn 1. for supporting guided modes in the middle layer. feffs nnn <<

2. for substrate radiation mode. seffc nnn <<

3. for substrate-clad radiation mode. effc nn < Obviously the discussion is limited to the condition . If or be greater than the waveguide supports leaky modes whose energy leaks to the covering regions (cladding & substrate).

csf nnn >> sn cn fn

333...444 PPPhhhaaassseee ssshhhiiifffttt iiinnn tttoootttaaalll iiinnnttteeerrrnnnaaalll rrreeefffllleeeccctttiiiooonnn aaattt ttthhheee dddiiieeellleeeccctttrrriiiccc iiinnnttteeerrrfffaaaccceeesss::: We now consider the reflection coefficient and phase change that take place while a ray of light is incident on the interface of two dielectric media of refractive indices and ( ) at . The interface is infinite in extent in the

1n 2n 21 nn > 0x =y direction and the wave as a whole propagates in the z direction.

Maxwell’s equations are inevitable tools to do so. Since there is no variation of field pattern in the y direction we can set 0y ≡∂∂ in the set of equations (2.4.34). Let us consider the case of two special cases. 111... TTTEEE wwwaaavvveee::: For this wave the component of the electric field in the z direction is zero, i.e. 0z =Ε . Using this along with 0yy 22 ≡∂∂≡∂∂ in equations (2.4.34) we get

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

xH

kjH z

2c

xβ (3.4.6 a)

0H y = (3.4.6 b) 0E x = (3.4.6 c)

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛=

xH

kjE z

2c

0y

ωµ (3.4.6 d)

Therefore the non-zero tangential electric field is . yEAlso since we start the wave equation from equation (2.4.27) 0Ez =

0H)k( z2

c2

t =+∇

or 0HkxH

z2

c2z

2

=+∂∂

(3.4.7)

whose solution can be written in general as

--- 222444 ---


)xjkexp(AH xz −= (3.4.8) where as and therefore from equation (3.4.6) we get introducing the propagation term xc kk = 0k y =

)zkt(jexp z−ω

)zkt(jexp)xjkexp(Akk

E zxx2c

0y −−⎟

⎟⎠

⎞⎜⎜⎝

⎛= ω

ωµ

)zkt(jexp)xjkexp(k

Azx

x

0 −−= ωωµ

)zkt(jexp)xjkexp(E zxo −−= ω say (3.4.9) Therefore the tangential electric and magnetic fields associated with the incident, transmitted and the reflected ray can be written as

⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

o

1xiozi

z1x

ioyi

kEH

)zkt(jexp)xjkexp()E(E

ωµ

ω

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

o

1xroyr

z1x

royr

kEH

)zkt(jexp)xjkexp()E(E

ωµ

ω

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−=

o

2xtoyt

z2x

toyt

kEH

)zkt(jexp)xjkexp(

)E(E

ωµ

ω

1xk , and satisfy the following ray equations 2xk zk

(3.4.10)

⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪

⎬

⎫

==

=

=

=+

=+

220110z

2202x

1101x

22

20

2z

22x

21

20

2z

21x

sinnksinnkk

cosnkk

cosnkk

nkkk

nkkk

θθ

θ

θ

--- 222555 ---


Now applying the boundary conditions for tangential components of the electric & magnetic field as given in equation (2.3.5) & (2.3.7) we get at the interface 0x = toroio EEE =+ (3.4.11) and t02xro1xio1x EkEkEk =− (3.4.12) Eliminating from these equations we get tA )EE(k)EE(k oroi2xoroi1x +=− or or2x1xoi2x1x E)kk(E)kk( +=−

or )kk()kk(

EE2x1x

2x1xioro +

−= (3.4.13)

Therefore the reflection coefficient is given by

)kk()kk(

EE

r2x1x

2x1x

io

roE +

−== (3.4.14)

As long as both and are real there is partial reflection 1xk 2xk )1r( E < . As 1θ increases increases and & decrease. For a certain value of

zk

1xk 2xk 1θ , say criticalθ , 2θ becomes and . At this condition from equations (3.4.10) we get using the expressions for ,

090 0k 2x =

zk

1

2critical n

nsin =θ

or ⎟⎟⎠

⎞⎜⎜⎝

⎛= −

1

21critical n

nsinθ

which confirms equation (3.4.2). Now for critical1 θθ > , becomes imaginary and 2xk 2x2 jk=γ becomes real. Then we can write the reflection coefficient from equation (3.4.14) Er

TEE21x

21xE 2r

)jk()jk(

r Φγγ

∠=−+

= say.

Then we have 1rE =

and ⎟⎟⎠

⎞⎜⎜⎝

⎛= −

1x

21TE k

tanγ

Φ (3.4.15)

Thus in total internal reflection a phase shift takes place. In figure (3.4.1) a plots of and |r| E TEΦ are shown with and . The corresponding program is in Appendix (Prog. 2)

5.1n1 = 4.1,3.1,2.1,1.1,0.1n2 =

--- 222666 ---


Figure 3.4.1(a) : Plot of Er against iθ for dielectric interface

222... TTTMMM wwwaaavvveee::: For wave the component of the magnetic field in the direction of propagation is zero i.e. we set

in equations (2.3.34) which yield the other field components as TM

0H z =

Figure 3.4.1(b) : Plot of TEΦ against iθ for dielectric interface

--- 222777 ---


0H x = (3.4.16 a)

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

xE

kjH z

2c

r0y

εωε (3.4.16 b)

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

xE

kjE z

2c

xβ (3.4.16 c)

0E y = (3.4.16 d) The non-zero tangential component of magnetic field is . Now we write the wave equation in this case as

yH

0E)k( z

2c

2t =+∇

or 0EkxE

z2

c2z

2

=+∂∂

whose solution can be written as )xjkexp(EE x0z −= where as and therefore as done previously xc kk = 0k y =

)zkt(jexp)xjkexp(Ekk

H zx0x2c

r0y −−⎟

⎟⎠

⎞⎜⎜⎝

⎛= ω

εωε

or )zkt(jexp)xjkexp(Ek

nH zx0

x

20

y −−⎟⎟⎠

⎞⎜⎜⎝

⎛= ω

ωε (3.4.17)

as . Therefore the tangential electric and magnetic fields associated with the incident, transmitted and the reflected ray can be written as

2r n≈ε

)E(E

)zkt(jexp)xjkexp(k

nEH

i0zi

z1x

1x

21i0

0yi

−−=⎟⎟⎠

⎞⎜⎜⎝

⎛

ω

ωµ

)E(E

)zkt(jexp)xjkexp(k

nEH

r0yr

z1x

1x

21r0

0yr

−=⎟⎟⎠

⎞⎜⎜⎝

⎛−

ω

ωµ

--- 222888 ---


)E(E

)zkt(jexp)xjkexp(k

nEH

t0zt

z2x

2x

22t0

0yt

−−=⎟⎟⎠

⎞⎜⎜⎝

⎛

ω

ωµ

2x1x k,k & satisfy the same equations as in equation (3.4.10). Now again applying the boundary conditions at the interface we get

zk0x =

t0r0i0 EEE =+ (3.4.18)

t02x

22

r01x

21

i01x

21 E

kn

Ekn

Ekn

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛ (3.4.19)

Eliminating from these equations we get t0E

)EE(kn

)EE(kn

r0i02x

22

r0i01x

21 +⎟

⎟⎠

⎞⎜⎜⎝

⎛=−⎟

⎟⎠

⎞⎜⎜⎝

⎛

or r01x

222x

21i01x

222x

21 E)knkn(E)knkn( +=−

or i02x

211x

22

2x2

11x2

2r0 E

)knkn()knkn(

E+

−= (3.4.20)

Therefore the reflection coefficient in this case can be written as

)knkn()knkn(

EE

r2x

211x

22

2x2

11x2

2

i0

r0H

+

−== (3.4.21)

With and both real and different it is possible for be equal to zero unlike , for a particular value of angle of incidence,

1xk 2xk Hr Er

Βθ called the Brewster’s angle. Under the condition 2

12x2

21x nknk = known as the Brewster’s condition. At this condition using equations (3.4.10) we get r1B2 cosncosn θθ = which along with equation (3.4.1) can be used to find the relation rB 2sin2sin θθ = or 0)cos()sin( rBrB =+− θθθθ which for ri θθ ≠ yields

--- 222999 ---


2

)( rBπθθ =+ (3.4.22)

and 2

1B n

ntan =θ (3.4.23)

Figure 3.4.2(a) : Plot of Hr against iθ for dielectric interface

Figure 3.4.2(b) : Plot of TMΦ against iθ for dielectric interface

--- 333000 ---


Thus the refracted and the reflected rays are at right angles to each other when the ray is incident at the Brewster’s angle. The electric field vector of the incident ray points directly along the beam direction for the reflected beam and therefore cannot couple in that direction. This justifies why 0rH = under this condition. In the case of wave also, there exists a critical angle of incidence TM criticalθ for which transmission in the second medium ceases as . For 0k 2x = criticali θθ > , becomes imaginary and total internal reflection takes place. Therefore the reflection coefficient can be written in the form as in the case of T

2xkE

wave as

TMH2x

211x

22

2x2

11x2

2H 2r

)knkn()knkn(

r Φ∠=+

−= say.

Then we can find 1rH =

and ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛= −

1x

22

2

11TM kn

ntan

γΦ (3.4.24)

Thus in total internal reflection a phase shift takes place for TM wave also. In figure (3.4.2) a plots of

|and r| H TMΦ are shown with and 5.1n1 = 4.1,3.1,2.1,1.1,0.1n2 = . The corresponding program is Prog. 3. From the above discussions two points clearly come out: 1. There is always some phase difference between the incident and the reflected ray during total internal reflection. 2. Since during total internal reflection becomes imaginary the electric field associated with the transmitted ray is of the form

2xk

)zkt(jexp)xexp(~E z2yt −− ωγ where 2γ is real positive. Thus the transmitted ray exponentially decays away from the interface in the

direction and it is termed as ‘evanescent’. The field thus penetrates second medium and dies out, so it transports energy in the x

z direction but not in the direction. x 333...555 GGGoooooosss---HHHaaannncchhheeennn SSShhhiiifffttt aaannnddd fffiiieeelllddd ppeeennneeetttrrraaatttiiiooonnn ooonnn tthhheee bbbaaasssiiisss ooofff rrraaayyy---oooppptttiiicccsss::: c p t Thus far we have seen that during total internal reflection at dielectric surfaces the field associated with the optical ray undergoes a phase-shift which is a function of the refractive indices of the media across the interface of which the ray is incident and reflected back. Also a field exists in the medium of lower refractive index i.e. beyond the apparent plane of reflection. The field is evanescent in nature under the condition of total internal reflection and decays off with penetration. This is the outcome of solution of Maxwell’s equations at the interface and from ray optical approach we must have a physical model to explain it. One such model exists which is based on the physical observation that a beam of finite width is laterally shifted on total reflection. A beam can be decomposed into a collection of uniform plane waves with slightly different wave vectors and on incidence on the reflecting interface they have slightly different angles, and each of these components may be reflected differently. The beam displacement is the result of various small

--- 333111 ---


but different phase changes in the constituent plane waves of the incident beam—the sum of these slightly phase-shifted waves is a laterally displaced reflected beam. Physically, there is a flow of wave energy parallel to the shift direction. This lateral shift was first observed by Goos and Hanchen and was termed as Goos-Hanchen shift after them. To accommodate this lateral shift the ray optic model of the shift demands that the ray is reflected not from the physical boundary of the medium of higher refractive index but from a distance inside the medium of lower refractive index. This is also in conformity with the penetration of the field inside the medium of lower refractive index.

Fig. 3.5.1 : Ray incident and reflected at interface of two dielectric media To have an estimate of GH Shift we may first assume the spectrum of the incident beam to be consisting of just two plane waves with slightly different angle of incidence. With our conventional assumption we take the positive direction of z axis as the direction of propagation and the interface is along the z~y plane. Then for an arbitrary angle of incidence θ we may write

and (3.5.1) ⎪⎭

⎪⎬⎫

==

=

βθ

θ

sinnkk

cosnkk

10z

10x

Therefore the associated incident and the reflected field at the field point can be written as )y,x(

and [ ][ ])sinzcosx(jexp)tjexp(E)y,x(E

)sinzcosx(jexp)tjexp(E)y,x(E

0r

0i

θθω

θθω

+−=

+−−=

the amplitude being same as in internal total reflection the modulus of the reflection coefficient reaches the unity value. The

0E)tjexp( ω term is neglected with the assumption that it is always there with .

Therefore for the beam of finite cross-section incident on the interface if the two plane waves have angles of incidence

0E

)( θ∆θ + and )( θ∆θ − where θθ∆ << the total incident field can be written as

[ ]

[ ])}sin(z)cos(x{njkexpE

})sin(z)cos(x{njkexpEE

100

100i

θ∆θθ∆θ

θ∆θθ∆θ

−+−−−+

+++−−=

--- 333222 ---


We use the approximations θθ∆ << as θθ∆θ cos)cos( ≈± and θ∆θθθ∆θ ⋅±≈± cossin)sin( and the expression above reduces to [ ] )cosnkcos()sinzcosx(njkexpE2E 10100i θ∆θθθ ⋅+−−≈ (3.5.2) Let )( θ∆θφ ± be the phase shift of the two waves on total internal reflection. With θθ∆ << we may write

θ∆θφθφθ∆θφ ⎟⎠⎞

⎜⎝⎛∂∂

±=± )()(

and the reflected wave can be written as

[ ] [ ]

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−⋅⋅

×

+−≈

θφ

θθ∆θ

θφθθ

)cosnk(1zcosnkcos

)(jexp)sinzcosx(njkexpE2E

1010

100r

(3.5.3)

Comparing equations (3.5.2) and (3.5.3) we can conclude that the beam shifts laterally in the z direction by an amount given by Sz2

S10

S z)cosnk(

1z2 ∆θφ

θ=⎟

⎠⎞

⎜⎝⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛−= (3.5.4)

Again as the axial propagation constant is given by θβ sinnk 10= equation (3.5.4) can be simplified in terms of β as

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=βφ

21zS (3.5.5)

Now for TE and TM modes we already derived the expressions for the phase shifts associated with total internal reflection in subsection 3.4. The phase shift for TE modes is given in equation (3.4.15) as

⎟⎟⎠

⎞⎜⎜⎝

⎛= −

1x

21TE k

tanγ

φ

and ⎟⎟⎠

⎞⎜⎜⎝

⎛== −

1x

21TE k

tan22γ

φφ

Here

θβ cosnk)nk(k 1022

12

01x =−=

--- 333333 ---


and )nsinn(k)nk( 22

2210

22

20

22 −=−= θβγ

Using these relations we can write

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −= −

θθ

φcos

)nn(sintan

21

22

21

TE

and hence on detailed derivation w.r.t. θ we get

⎟⎟⎠

⎞⎜⎜⎝

⎛

−⋅

⎟⎠⎞⎜

⎝⎛ −+

=∂∂

)nn(kcosnk

cos

)nn(sinsincossin22

22

12

0

221

20

2

21

22

22

TE θθ

θθθθ

θφ

or )nn(

n)nn(sin)nn1(sin

22

21

21

21

22

2

21

22TE

−⋅

−

−⋅=

∂∂

θθ

θφ

)nn(sin

sin2

12

22 −

=θ

θ

and therefore from equation (3.5.4)

)nsinn(k

tan2)TE(z22

222

10

S−

=θ

θ

or )TE(ztan2)TE(z2 S2S ∆γθ == (3.5.6)

Fig. 3.5.2 : Goos Hanchen shift from ray optic approach The depth of penetration is the distance inside the medium of lower refractive index from where the ray is reflected and it is simply given by θcotz)TE(d SP = or 2P 1)TE(d γ= (3.5.7) For the TM modes also similar results can be obtained starting from the expression for the phase shift in equation (3.4.24)

--- 333444 ---


⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛= −

1x

22

2

211

TM knn

tanγ

φ

and ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛== −

1x

22

2

211

TM knn

tan22γ

φφ

Using the expressions for 2γ and we may write 1xk

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −= −

θ

θφ

cos)nn()nn(sin

tan 21

22

21

22

21

TM

and hence

)TM(z)cosnsinn()nsinn(k

tann2)TM(z2 S

222

221

22

2210

22

S ∆θθθ

θ=

⎟⎠⎞⎜

⎝⎛ −⋅−

= (3.5.8)

We define a term especially for TM waves

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛= θθ 22

2

2

1 cossinnn

q (3.5.9)

Then equation (3.5.8) can be written as

)TM(z)nsinn(qk

tan2)TM(z2 S22

2210

S ∆θ

θ=

−= (3.5.10)

And the field penetration depth )q(1cotz)TM(d 2SP γθ == (3.5.11) Thus we get the relation between the GH Shifts and the field penetration depth for the TE and modes as

TM

)TE(zq1)TM(z SS ∆∆ ⋅⎟⎟⎠

⎞⎜⎜⎝

⎛= (3.5.12)

and )TE(dq1)TM(d PP ⋅⎟⎟⎠

⎞⎜⎜⎝

⎛= (3.5.13)

The results are very much significant when we come to examine propagation in planer dielectric guides or thin film between two media of lower index since under these conditions it is perfectly possible for the penetration depth to be compatible in order of magnitude to the guide thickness itself or even greater. Under these conditions the wave spends more time outside the guiding layer than inside it. The essential point to infer from these is that the wave is not completely bound inside the core layer and the properties of the cladding layer, especially it’s refractive index profile and thickness are as important as the properties of the guiding layer. In the figure 3.5.1 the variation of the penetration depths for TE and TM modes with the angle of incidence are shown for different index differences. The corresponding program is in appendix (Prog. 4). The graphs show sharp cut-off at the critical angles of incidences and it increases with the decrease in index difference for a given angle of incidence. Also it is notable that for typical dielectric interface and for conditions (θ exceeding criticalθ substantially) the penetration is small enough. But for guide thickness of the order of magnitudes in mµ the penetration is a significant consideration.

--- 333555 ---


From the above discussion we can infer that when a ray of light is guided through a medium of refractive index higher than the surrounding layers via total internal reflection they suffer a penetration into the surrounding layers as if they are reflected from a little deep inside from the surrounding layers. So the wave propagates thought the guide with effective width that is equal to

⎟⎟⎠

⎞⎜⎜⎝

⎛++=

ccss q1

q1d2w

γγ (3.5.14)

Fig. 3.5.1 (a)

Figure 3.5 Penetration depths for TE and TM modes

Fig. 3.5.1 (b)

--- 333666 ---


where

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛= θθ 22

2

s

fs cossin

nn

q for TM modes

for TE modes 1=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛= θθ 22

2

c

fc cossin

nn

q for TM modes

for TE modes 1=

Fig. 3.5.2 : Effective width of a beam guided through a slab waveguide 333...666 RRRaaayyy oooppptttiiicccaaalll eeexxxppplllaaannnaaatttiiiooonnn ooofff SSSWWWGGG mmmooodddeeesss::: DDDiiissscccrrreeettteee nnnaaatttuuurrreee oofff ppphhhaaassseee cccooonnnssstttaaannnttt::: o

Fig. 3.6.1 : Ray guiding via multiple reflection in 2D SWG

--- 333777 ---


The ray theory appears to allow rays at angle of incidence greater than the critical angle. In the figure the rays are shown to be guided via multiple reflections at the interfaces of different layers in the dielectric

. Referred to the figure the allowable value of the angle SWG ϕ is criticalϕϕ < where criticalϕ is given by the complement to the critical angle of incidence i.e. )nn(sin2 fs

1critical

−−= πϕ (3.6.1) Obviously have once again considered the assumption csf nnn ≥>Now when the phase of the plane wave associated with the ray is taken into account, it is seen that only rays at certain discrete angle greater than the critical angle i.e. less than criticalϕ are capable of propagating on the guide. As the plane wave associated with any trapped ray travels, it undergoes a phase change ∆ . The plane wave front is assumed to be infinite, or at least larger than the cross section of the guide that that is intercepted. Otherwise they would not fit to the definition of a plane wave which requires a constant phase over the plane. Thus there is much overlapping of the waves as they travel in the zigzag path. The phase shift ∆ is given by

sn2snk ff0 λπ∆ == (3.6.2)

where is the distance traveled along the ray path by the wave. There are additional phase changes at the interface due to reflection. These phase changes have been calculated previously in this chapter. In order for the wave to propagate the phase of the doubly reflected ray must be same as the incident ray, that is, the waves must interfere constructively with itself. If the phase condition is not satisfied the wave would interfere destructively and die out.

s

From the figure we see that the ray CD suffers two internal reflections at the boundaries as it travels from one phase front A to another phase front .Hence the phase changes suffered by the rays and B CDAB must differ by integral multiple of π2 .

Let and . Then between the two phase fronts one ray travels a distance and the other ray a distance . The ray along

1sAB = 2sCD = 1s

2s AB suffers no phase shift due to reflection between these two phase fronts and the other ray faces two internal reflections and associated phase changes. Therefore the coherent phase condition for constructive interference may be written as πφφ∆∆ N222)( fcfsABCD =−−− (3.6.3) or πφφ N222)ss(nk fcfs12f0 =−−− (3.6.4) Here N is an integer and fs2φ and fc2φ are the phase shifts associated with the internal reflections at the core-substrate and core-cladding layers. These terms had been derived in this chapter previously separately for the TE and TM modes. For TE modes we can write

--- 333888 ---


⎪⎪

⎭

⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

−

−

f

c1TMfc

f

s1TMfs

hh

tan22

hh

tan22

φ

φ

(3.6.5)

and for the TM modes

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

−

−

f

c

2

c

f1TMfc

f

s

2

s

f1TMfs

hh

nn

tan22

hh

nn

tan22

φ

φ

(3.6.6)

as derived in the section 3.4. Here the parameters

212

c2

02

c

212s

20

2s

)nk(h

)nk(h

−=

−=

β

β

and 2122f

20f )nk(h β−=

Now we find from the figure

⎟⎟⎠

⎞⎜⎜⎝

⎛=

ϕsind2s2 (3.6.7)

ϕϕϕϕϕ cos)tand2coss(cos)BECE(cosCBs 21 −=−==

or )sin(cossin

d2cos)cossin(cossin

d2s 2221 ϕϕ

ϕϕϕϕϕ

ϕ−⎟⎟

⎠

⎞⎜⎜⎝

⎛=−⎟⎟

⎠

⎞⎜⎜⎝

⎛= (3.6.8)

Therefore

ϕϕϕϕ

sindnk4)sincos1(sin

d2nk)ss(nk f022

f012f0 =+−⎟⎟⎠

⎞⎜⎜⎝

⎛=− (3.6.9)

Again we know that the propagation constant in the core region in the direction of propagation ϕsinnkh f0f = Hence dh4)ss(nk f12f0 =− (3.6.10) Therefore we can write from equation (3.6.4) fcfsf 22N2dh4 φφπ ++= or fcfsf 2Ndh2 φφπ ++= (3.6.11) Therefore forTE modes the coherent phase condition is

--- 333999 ---


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+= −−

f

c1

f

s1f h

htan

hh

tanNdh2 π (3.6.12)

and for TM modes

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+= −−

f

c

2

c

f1

f

s

2

s

f1f h

hnn

tanhh

nn

tanNdh2 π (3.6.13)

These equations can alternatively be written as

)hhh(

)hh(h)Ndh2tan(

cs2

f

csff

−

+=− π for TE modes

and

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=−

2c

c2

s

s

2

2f

f

2c

c2

s

s2

f

f

f

nh

nh

n

h

nh

nh

n

h

)Ndh2tan( π for TM modes

333...777 DDDiiissscccuuussssssiiiooonnnsss::: In this chapter we developed the ray optic theory of dielectric slab structures. We discussed the reflection phenomena at the interfaces between two dielectrics both in single interface and double interface structures. It must be carefully noted that the ray model of electromagnetic waves is valid only when the guide dimensions are large compared to the feasible wavefront of the wave. In case of the double interface structure as the thickness decreases the wave ceases to be reflected from the interfaces and the wave stops propagating. These phenomena will be discussed in detail in relation to mode cut off in the next chapter where the pure electromagnetic solutions to the structure will be sought. We shall see that the same coherent phase condition will be derived as a solution of the boundary value problem for the structure and this will give rise to several modes in the structure. The GH shift phenomena will also come out as a consequence of satisfaction of the electromagnetic boundary conditions and the law of conservation of energy. However despite that the ray model can explain the modes in a purely dielectric slab waveguide it cannot reflect any light on the modes existing in slab waveguides with metal. This is because as we will see, certain metals posses complex dielectric constants with real negative parts and correspondingly the refractive index also comes out complex with a predominant imaginary part. This makes the critical angle of incidence complex which can be dealt with in mathematics but cannot be depicted in ray model. In fact the interesting modes in waveguides with metal are chiefly the surface waves that do not exactly follow the multiple reflection process to propagate. In fact these modes are bound near one interface and often do not have significant amplitude at the other interface. This means these modes do not reach the other interface and hence the idea of internal reflection is not valid there.

--- 444000 ---

CCChhhaaapppttteeerrr (((iiivvv)))

MMMooodddaaalll aaannnaaalllyyysssiiisss oofff dddiiieeellleeeccctttrrriiiccc ssslllaaabbb wwwaaavvveeeggguuuiiidddeeesss o

--- 444111 ---


444...111 IIInnntttrrroooddduuuccctttiiiooonnn::: In this chapter detailed modal analysis of the dielectric slab waveguides will be carried out. We shall mainly concentrate on the 3 layer slab waveguide structure which is infinite in the direction parallel to the interface and perpendicular to the direction of propagation. Thus we shall be restricted to the case of step-index

D2 dielectric case only. The convention of notations will be borrowed from the previous chapter, that is the refractive indices will be taken as for the respective layers and the coordinate system will be taken in such a way that

scf n,n,nz direction is the direction of propagation and the y direction has no boundary,

the confinement takes place only in the direction. We shall also consider the case of metal cladding and try to formulate the losses in that case.

x

Fig. 4.1 : 2D Slab Waveguide structure 444...222 WWWaaavvveee eeeqqquuuaaatttiiiooonnn::: We first consider the case of TE modes. For this mode of propagation 0Ez = . Also since there is no field

variation in the y direction we have as before 0yy 22 ≡∂∂≡∂∂ and therefore the field equations (2.4.34) reduce to the equations for (3.3.6) which shows the only non-zero component of the electric field is .Therefore we can write the wave equation (2.4.27) for as yE yE 0E)k( y

2c

2t =+∇

or 0Ekx y

2c2

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

(4.2.1)

where (4.2.2) )nk()k()(k 222

022222

c βββµεω −=−=−= Therefore the wave equations for the three layers can be written as

--- 444222 ---


0E)nk(Ex y

22j

20y2

2

=−+∂∂ β (4.2.3)

where for ⎪⎩

⎪⎨

⎧=

s

c

f

j

nnn

nsubstratecladding

layerguiding

Similarly for TM waves, it can be shown from equations (3.3.16) that the nonzero magnetic field component satisfies the same equation yH

0H)nk(Hx y

22j

20y2

2

=−+∂∂ β (4.2.4)

The nature of the solutions will be exponential or sinusoidal depending on the sign of the term

. For guidance through the middle layer, ( ) the solution should be sinusoidal in the middle layer and decay exponentially in the other two layers. Therefore we need the guide to satisfy the inequality.

)nk( 22j

20 β− fn

f0 nk<β and s0c0 nk,nk>β Usually for most cases the following condition is imposed. cs0f nnkn >>> β Therefore the different conditions may be discussed in brief about the field distributions as a function of β which reproduces the results in chapter 3.3 for various possible modes. 1. f0 nk>β In that case the solution is exponential in all three regions. Such field distribution is unrealizable in practice since in the outer layer the field grows exponentially outward. However these solutions can be achieved with use of metals. These are the modes we are going to concentrate later on. 2. f0s0c0 nknk,nk << β The field is sinusoidal in the middle layer and decays in the other two layers. So this mode is completely trapped or bound and this mode is called the guided mode. 3. s0c0 nknk << β The field is not completely bound, rather it decays in the cladding layer only but not in the substrate. This mode is called the substrate radiation mode. 4. c0 nk<β In this case the field varies sinusoidally in the clad and substrate layers both. This mode is called the clad- substrate radiation mode. The wave continuously loses energy through the cladding and substrate and tends to die out as it propagates.

--- 444333 ---


444...333 TTTrrraaapppppeeeddd mmmooodddeeesss iiinnn aaasssyymmmmmmeeetttrrriiiccc 333 lllaaayyyeeerrr dddiiieeellleeeccctttrrriiiccc wwwaaavvveeeggguuuiiidddeee::: p y

v 111... TTTEEE mmmooodddeeesss fff iiieeelllddd dddiiissstttrrr iiibbbuuuttt iiiooonnn aaannnddd eeeiiigggeeennnvvaaallluuueee eeeqqquuuaaattt iiiooonnn For modes we start with equations (4.2.3). Since the desired solution of the equations (4.2.3) are sinusoidal in the middle layer and evanescent in the other two layers we write the solutions as

TE

)xh()]xd(hexp[AsinB)xhcos(A

)]xd(hexp[A)x(E f

ss

fff

cc

y

++−

= , for, (4.3.1.1) dxd|x|

dx

−≤≤≥

where the transverse wave numbers and are given by cf h,h sh

(4.3.1.2)

⎪⎪

⎭

⎪⎪

⎬

⎫

>−=

>−=

>−=

0nkh

0nkh

0nkh

2c

20

22s

2c

20

22c

22f

20

2f

β

β

β

Again using equation (3.3.7) we get on differentiation of both sides of equation (3.3.6 d)

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛=

xEj)x(H y

0z ωµ

(4.3.1.3 a)

The other non-zero field component is given by equation (3.4.6) as xH

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

xH

kjH z

2c

xβ

or y0

x EH ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

ωµβ

(4.3.1.3 b)

Equations (4.3.1.1) and (4.3.1.3) give the tangential electric and magnetic fields for the TE mode and for continuity they have to be continuous across the interfaces dx ±= . Equating s at and at

separately we get 'E y dx =

dx −= )dhsin(B)dhcos(AA ffffc += and )dhsin(B)dhcos(AA ffffs −= from which we eliminate and as fA fB

--- 444444 ---


⎪⎪

⎭

⎪⎪

⎬

⎫

−=

+=

)dhsin(2)AA(

B

)dhcos(2)AA(

A

f

scf

f

scf

(4.3.1.4)

Again equating s (i.e. the derivatives of ) at zH yE dx = and at dx −= separately we get )dhcos(hB)dhsin(hAhA ffffffcc +−=−

and )dhcos(hB)dhsin(hAhA ffffffss += and thus replacing and from equations (4.3.1.4) we get the two homogeneous equations in and

fA fB sA

cA 0)]dhcot(hh[A)]dhcot(hh[A ffssffcc =−−++ 0)]dhtan(hh[A)]dhtan(hh[A ffssffcc =+−++− For non-trivial solution i.e. for arbitrary and we set the determinant of the coefficient matrix equal to zero i.e.

cA sA

0)]dhtan(hh[)]dhtan(hh[)]dhcot(hh[)]dhcot(hh[

ffsffc

ffsffc =+−+−−−++

(4.3.1.5)

which on simplification yields 0)dhtan()hhh(2]1)dh()[tanhh(h fsc

2ff

2scf =−+−+

or )hhh(

)hh(h)dh(tan1

)dhtan(2

sc2

f

scf

f2

f

−

+=

−

or )hhh(

)hh(h)dh2tan(

sc2

f

scff

−

+=

or )hhh(

)hh(h)dhtan(

sc2

f

scf0f

−

+= (4.3.1.6)

This is the characteristics or eigenvalue equation for the trapped TE modes in the waveguide. Another way of writing the field equations given by equations (4.3.1.1) are

)]xd(hexp[)dhcos(

)xhcos()]xd(hexp[)dhcos(

C)x(E

sf

f

cf

y

++−

−−=

φφ

φ for, (4.3.1.7)

dxd|x|

dx

−≤≤≥

--- 444555 ---


where we have already matched the electric field at yE dx ±= . The beauty of the field equations (4.3.1.7) is that it contains only one constant in the amplitude Now applying the boundary conditions for

(i.e. the derivative of ) at the boundaries we get respectively at zH yE dx = )dhsin(h)dhcos(h fffc φφ −−=−−

or f

cf h

h)dhtan( =−φ (4.3.1.8 a)

and at dx −= )dhsin(h)dhcos(h fffs φφ +=+

or f

sf h

h)dhtan( =+ φ (4.3.1.8 b)

The constant φ can be easily evaluated from equations (4.3.1.8) as

)dhtan()dhtan(1

)dhtan()dhtan()2tan(

ff

ff

φφφφ

φ−++

−−+=

or )hhh(

h)hh()2tan( 2

fsc

fcs

+

−=φ (4.3.1.9)

Similarly adding the equations (4.3.1.8) we get back the eigenvalue equation as

)dhtan()dhtan(1

)dhtan()dhtan()dh2tan(

ff

fff φφ

φφ−+−

−++=

or, )dh2tan( f )hhh()hh(h

sc2

f

scf

−

+=

There are number of solutions to the characteristic equation for and thus there are various Tdh f E modes

supported by the . For the SWG thN mode ( ) the eigenvalue equation can be written as NTE

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

++= −

)hhh(

)hh(htanNdh

sc2

f

scf10f π (4.3.1.10)

or ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+= −−

f

s1

f

c10f h

htan

hh

tanNdh π

or ⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−+= −−

s

f1

c

f10f h

htan

hh

tan)1N(dh π (4.3.1.11)

where corresponds to the lowest ( ) mode. 0N = 0TE Eigenvalue equation in normalized form The equation (4.3.1.11) contains a lot of parameters to define a specific mode e.g. SWG β,n,n,n scf etc. We can normalize it by defining the following terms

--- 444666 ---


)nn(dkv 2s

2f0 −= (4.3.1.12)

)nk(ddhu 22f

20f β−== (4.3.1.13)

and )nn(k)nk(

vu1b 2

s2

f2

0

2s

20

2

2

2

−

−=−=

β (4.3.1.14)

The term v is called the normalized frequency (spatial) and the term is called the normalized propagation constant. We also define asymmetry factor

bη as

)nn(

)nn(2

s2

f

2c

2f

−

−=η (4.3.1.15)

such that for symmetrical as we set SWG sc nn = we have 1=η . In general since we have for asymmetric ,

cs nn >SWG 1>η . Then in terms of these parameters the equation (4.3.1.11) becomes

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−+= −−

d)uv(

)du(tan)dbv()du(tan)1N(u

22

11

ηπ

or ⎟⎟⎠

⎞⎜⎜⎝

⎛

−+−

−⎟⎟⎠

⎞⎜⎜⎝

⎛ −−+=− −−

)1b()b1(tan

b)b1(tan)1N()b1(v2 11

ηπ (4.3.1.16)

This is the normalized eigenvalue equation for TE modes in asymmetric . On the basis of this equation we can plot the normalized dispersion curve or the curve for the TE modes for different values of asymmetry factor

SWGv~b

η .This curve is shown in the figure (4.3.1.1) and the corresponding program is Prog. 5.

Fig. 4.3.1.1 : Universal b~v plot for TE modes for various asymmetry

--- 444777 ---


The beauty of these curves is that if one knows the parameters etc. the field profile for the different T

d,n,n,n scf

E mode can be obtained graphically. The curves corresponding to the condition ∞→η are of special important for us. This condition means

from equation (4.3.1.15) that if , i.e. the refractive index of the cladding layer is complex and the imaginary part of exceeds the real part by a large amount. This condition is satisfied by metals in the optical range of frequency and these metal-cladding give rise to surface plasmon polariton waves. The modal analysis of such waves under metal cladding will be discussed in the next chapter.

sf nn ≠ −∞→2cn

cn

Modal cut-off At cut-off the field just ceases to decay exponentially outside the guiding layer and as a result the wave is no longer bound to the waveguide as such. This happens when β becomes so small that either

or become zero. Since according to our convention , for all realch sh cs nn > β , and therefore cut-off starts in the substrate layer and the corresponding condition is given by

sc hh >

0hs = or s0 nk=β This condition for cut-off is identical to the cut-off condition stated in the last chapter. The ray optic equivalence of the cut-off is that the ray just ceases to be internally reflected from the interface and the corresponding angle of incidence can be obtained from the ray optic expressions of β and which yields the critical angle as given in equation (3.3.3 a) Then we have the relations )nn(k)nk(h 2

s2

f022

f2

0f −=−= β (4.3.1.17)

and )nn(k)nk(h 2c

2s0

2c

20

2c −=−= β (4.3.1.18)

Then we get from the characteristic equation (4.3.1.10) putting 0hc =

)nn()nn(

N)nn(dk2tan 2s

2f

2c

2s2

s2

f0−

−=⎟

⎠⎞⎜

⎝⎛ −− π

This is the cut-off condition for the thN TE mode in a generally asymmetric . We once again take recourse of the normalized parameters and in terms of them the cut-off condition can be obtained as

SWG

)1()Nv2tan( −=− ηπ

or )1(tan21

2Nvv 1

c −+== − ηπ (4.3.1.19)

cv is the cut-off value of v at which 0hs = . Also from equation (4.3.1.16) we see that when 0b =

--- 444888 ---


Fig. 4.3.1.2 : variation of normalized cut-off frequency with asymmetry factor

For the special case ∞→η we have

22

1Nvcπ

⎟⎠⎞

⎜⎝⎛ +=

)1(

1tan2

)1N(v2 1

−−−+= −

ηππ

or c1 v)1(tan

21

2Nv =−+= − ηπ

Thus the curves in the normalized dispersion profile start at for each mode. We see that is a direct function of the asymmetry factor

v~b cv cvη . A plot of variation of against cv η is shown in the figure

(4.3.1.2). The corresponding figure is Prog. 6 of Appendix. We note from equation (4.3.1.19) that for large value of η , almost converges tocv 2)1N( π+ . Also as 1=η for symmetric the cut-of frequency for symmetric is

SWGSWG 2N π .Thus varies between cv 2N π and 2)1N( π+ .

Mode numbers Since for the thN TE mode the cut-off frequency is given by equation (4.3.1.19) all the lower modes,

…..,TE,TE,TE 210 1NTE − and NTE will be supported by the waveguide, if . Thus for a given cvv >v the number of guided modes is given by

{ }int

1 )1(tanv21M ⎥⎦⎤

⎢⎣⎡ −−= − ηπ

(4.3.1.20)

--- 444999 ---


where the subscript ‘int’ indicates the next largest integer and the mode up to which TE waves are supported for a given value of v is 1MTE − for which cvv = . Normalization in terms of power flow The field equations for the TE modes are all written in equation (4.3.1.7) in terms of a single constant amplitude term .We can find an expression of C in terms of the total power flow in the and thus we can normalize the expressions of by properly adjusting C in such a way that represents a power flow of per unit length in the

C)x(E y )x(E y

1 W y direction. According to Poynting theorem the time averaged power flow through the entire cross-section is given by the

Pz component of the Poynting vector, i.e. in usual notation

dx)HERe(21P

z∫∞

∞−

∗×=rr

Since for the TE modes this reduces, for the 0HE yx == thN mode to

dxHE21P

Nx

NyN ∫

∞

∞−

∗−=

Now combining equations (3.3.6 a) and (3.3.6 d) we have for the thN mode

Ny

0

NNx EH ⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

ωµβ

and therefore dxE2

P2N

y0

NN ∫

∞

∞−⎟⎟⎠

⎞⎜⎜⎝

⎛=

ωµβ

or ⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛= ∫∫∫

∞

−

−

∞−

dxEdxEdxE2

Pd

2Ny

d

d

2Ny

d 2Ny

0

NN ωµ

β

The power flow through the cladding layer is

dxE2

Pd

2Ny

0

Nc ∫

∞

⎟⎟⎠

⎞⎜⎜⎝

⎛=

ωµβ

(4.3.1.21)

or [ ]dx)xd(h2exp)dh(cosC2

P cd

f22

0

Nc −−⎟⎟

⎠

⎞⎜⎜⎝

⎛= ∫

∞

φωµβ

)hh(

)hh(2

C2 2

c2

f

c2

f2

0

N

+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

ωµβ

(4.3.1.22)

where we used the equation (4.3.1.8 a). Similarly the power flow in the substrate can be evaluated using equation (4.3.1.8 b) to be

dxE2

Pd 2N

y0

Ns ∫

−

∞−⎟⎟⎠

⎞⎜⎜⎝

⎛=

ωµβ

)hh(

)hh(2

C2 2

s2

f

s2

f2

0

N

+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

ωµβ

(4.3.1.23)

--- 555000 ---


at last the power flow in the guiding layer

dxE2

Pd

d

2Ny

0

Nf ∫

−⎟⎟⎠

⎞⎜⎜⎝

⎛=

ωµβ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

++

++⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

)hh(h

)hh(h

d22

C2 2

c2

f

c2

s2

f

s2

0

N

ωµβ

(4.3.1.24)

and therefore the total power flow is given by scfN PPPP ++=

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

cs

2

0

NN h

1h1d2

2C

2P

ωµβ

(4.3.1.25)

Therefore the normalizing constant for theC thN mode is given as

2

1

cs

21

N

0NN h

1h1d2P2C

−

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛=

βωµ

(4.3.1.26)

For symmetric this can further be written as SWG

2

1

s

21

N

0NN h

1d2

P2C−

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

βωµ

(4.3.1.27)

Effective width We note from the expression of that the term in the last parenthesis is of the dimension o length where

the terms and have been added to the actual width of the guide. From the field equations

(4.3.1.1) it can be easily found that is the distance inside the cladding (substrate) region at which the field intensity has fallen to

NC1

ch − 1sh − d2

)h(h 1s

1c

−−

e1 of the value at the cladding (substrate) interface. Therefore the term

⎟⎟⎠

⎞⎜⎜⎝

⎛++=

scTE h

1h1d2W (4.3.1.28)

is the effective width for theTE modes in the waveguide. This same expression had been derived in the last chapter from the ray-optic approach using the phase-shifts in total internal reflection. For symmetrical this result simplifies to SWG

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

cTE h

1d2W

Now we can again normalize in terms of by writing the normalized effective width as TEW v,b

)nn(kWw 2s

2f0TETE −= (4.3.1.29)

Then we get in terms of we get v,b

--- 555111 ---


)nk(

)dv(

)nk(

)dv(v2w2

c2

022

s2

02

TE−

+−

+=ββ

{ }22 )du()dv(

)dv(b

1v2−

++=η

or ⎟⎟⎠

⎞⎜⎜⎝

⎛

−+++=

)1b(1

b1v2wTE η

(4.3.1.30)

where we used the expressions for and b,v,u η . For symmetrical we set SWG 1=η which reduces it to

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

b1v2wTE

A plot can be derived to show the variation of as a function of TEw v for different degrees of asymmetry. This plot is shown in figure (4.3.1.3). Program is in appendix (Prog. 7).

Fig. 4.3.1.3 : Variation of normalized effective width with normalized frequency

We see from this set of curves that for large v the field penetration depth into the substrate and cladding layers tend to the asymptotic expression )1v2( + . But for values of v close to cut-off of each mode the field spreads substantially into the two layers and. Confinement factor We get from the expressions of and in equations (4.3.1.22), (4.2.23), (4.2.24) and (4.2.25) that the fraction of the total power that is confined into the guiding layer is given by

cfs P,P,P NP

--- 555222 ---


⎟⎟⎠

⎞⎜⎜⎝

⎛++

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

++

++

=

sc

2f

2s

s2

f2

c

c

N

f

h1

h1d2

)hh(h

)hh(h

d2

PP

(4.3.1.31)

This is called the field confinement factor. Similarly the fraction 0f the power contained outside the guiding layer is

⎟⎟⎠

⎞⎜⎜⎝

⎛++

+=

+

sc

2f

2s

s2

f

N

sc

h1

h1d2

)hh(

hh

P)PP(

(4.3.1.32)

Now we one again express them in terms of the normalized parameters and b,v,u η and the confinement factor turns into

TE

2c

2f0

2s

20

2

2c

2f

20

2c

20

22c

2f0

N

f

w

)nn(k

)nk()nn(k

)nk)(nn(kv2

PP ⎟⎟

⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−+

−

−−+

=

ββ

TEw

b)1b(

v2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+

−++

=η

η

(4.3.1.33)

By similar calculations we can also show that

TE

22

2

2

2

f

sc

w)du()dv()dv(

)du()dv(b)du(

P)PP( ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

−+

=+ ηη

TEw

)1b()b1(

b)b1(

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+−

+−

=ηη

(4.3.1.34)

For symmetrical the respective results simplify to SWG

TEN

f

w)bv(2

PP +

=

and bw

)b1(2P

)PP(

TEN

sc −=

+

We plot the variation of these two fractions to have an idea of the degree of confinement of a particular TE mode. The plots are shown in figure (4.3.1.4) and (4.3.1.5) respectively. The corresponding programs are in Prog. 7 in appendix. The two functions that are plotted are complementary to each other i.e. at any value of v their sum is unity. This confirms the conservation of energy in the total system. From these curves it is to

--- 555333 ---


be noted hat the confinement is high in case of large values of v for all modes. Near cut-off the confinement sharply falls to negligible value as the wave no longer remains bound o the guiding layer and the total power is radiated into the substrate and cladding layers. We also see from the graphs that for higher values of v , mode is more confined than the other modes present in the guide. 0TE

Fig. 4.3.1.4 : Plot of confinement factor with the normalized guide thickness

Fig. 4.3.1.5 : Plot of radiation factor with the normalized guide thickness

--- 555444 ---


Fig. 4.3.1.6 : Field distribution of the TE mode in asymmetric dielectric SWG In the figure 4.3.1.6 we plot the field amplitude distribution corresponding to the first two TE waves for

a set of guide specification atyE

0.1n&2.2n,9.2n 2c

2c

2f === m633.0 µλ = . The plot is

generated in the program Prog. 8 of Appendix. Here we see that the fields vary sinusoidally in the guiding layer and decay evanescently in the substrate and cladding layers. From the figure we see the asymmetry of the distribution in the substrate and the cladding layers. We also see that the field has one maxima whereas the mode has two maxima on opposite sides of the

0TE

1TE 0x = axis. 222... TTTMMM mmmooodddeeesss fff iiieeelllddd dddiiissstttrrriiibbbuuuttt iiiooonnn aaannnddd eeeiiigggeeennnvvvaaallluuueee eeeqqquuuaaattt iiiooonnn For waves we have and from equations (3.3.16) we see that the only non-zero tangential component of the magnetic field is .Then the guided mode solutions for from equation (4.2.4) in the three layers can be written in the similar way as in equations (4.3.1.1).

TM 0H z =

yH yH

yE

)xh()]xd(hexp[PsinQ)xhcos(P

)]xd(hexp[P)x(H f

ss

fff

cc

y

++−

= for, (4.3.2.1) dxd|x|

dx

−≤≤≥

and the non-zero tangential electric field is which can be evaluated by differentiating equation (3.4.16 b) and using equation (4.2.4)

zE

zr0y Ej

xH

εωε=∂

∂

or ⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛−=

xH

njE y

20

z ωε (4.3.2.2 a)

The other non-zero component of the TM mode field is given by xE

--- 555555 ---


y20

z2

cx H

nxE

kjE ⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

ωεββ (4.3.2.2 b)

where we have used equations (3.4.16) for evaluation. Therefore applying the boundary conditions at the interfaces dx ±= for both and separately we get

zE yH

)dhsin(Q)dhcos(PP ffffc += )dhsin(Q)dhcos(PP ffffs −=

)dhcos(n

hP)dhsin(

n

hP

nh

P f2f

fff2

f

ff2

c

cc +−=−

)dhcos(nh

Q)dhsin(n

hP

nh

P ff

fff2

f

ff2

s

ss +=

which on eliminating and results in two simultaneous homogeneous equations in and as fP fQ cP sP

0)dhcot(n

h

nh

P)dhcot(n

h

nh

P f2f

f2

s

ssf2

f

f2

c

cc =

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−−+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+

0)dhtan(n

h

nh

P)dhtan(n

h

nh

P f2f

f2

s

ssf2

f

f2

c

cc =

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+−+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+−

These for non-trivial solutions results in

0

)dhtan(n

h

nh

)dhtan(n

h

nh

)dhcot(n

h

nh

)dhcot(n

h

nh

f2f

f2

c

sf2

f

f2

c

c

f2f

f2

s

sf2

f

f2

c

c

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛++

On simplification

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

==

2s

s2

c

c4

f

2f

2s

s2

c

c2

f

f

0ff

nh

nh

n

h

nh

nh

n

h

)dhtan()dh2tan( (4.3.2.3)

This is the eigenvalue equation for TM modes in the . The detailed calculations are similar to the case of T

SWGE modes. Then for the thN TM mode that can be supported by the we can write NTM SWG

--- 555666 ---


⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+= −

2s

s2

c

c4

f

2f

2s

s2

c

c2

f

f

10f

nh

nh

n

h

nh

nh

n

h

tanNdh π (4.3.2.4)

or ⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+= −−

f

s

2

s

f1

f

c

2

c

f10f h

hnn

tanhh

nn

tanNdh π

or ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−+= −−

s

f

2

f

s1

c

f

2

f

c10f h

hnn

tanhh

nn

tan)1N(dh π (4.3.2.5)

Thus far the results for TM modes are identical in nature to the results for the modes, only difference has been the additional term underneath . The lowest mode corresponds to . From this analogy in the expressions we define two unitless terms namely

TE2

jn jh 0TM 0N =

fsη and fcη as

⎪⎩

⎪⎨

⎧

=2

sf

fs

)nn(

1η for mode

TM

TE

and

⎪⎩

⎪⎨

⎧

=2

cf

fc

)nn(

1η for mode

TM

TE

Then in terms of these parameters the eigenvalue equation for both TE and TM modes can be unified to

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+= −−

f

sfs

1

f

cfc

10f h

htan

hh

tanNdh ηηπ

or ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−+= −−

s

f

fs

1

c

f

fc

10f h

h1tanhh1tan)1N(dh

ηηπ

In case of symmetrical we have SWG fcfs ηη = and cs hh = as cs nn = . Therefore in that case we have the eigenvalue equation as

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−+= −−

f

sfs

1

s

f

fs

10f h

htan2N

hh1tan2)1N(dh ηπ

ηπ

Similar to the TE modes the field equations for the TM modes can also be written in the form

--- 555777 ---


)]xd(hexp[)dhcos(

)xhcos()]xd(hexp[)dhcos(

D)x(H

sf

f

cf

y

++−

−−=

ϕϕ

ϕ for, (4.3.2.6)

dxd|x|

dx

−≤≤≥

where the boundary conditions for the tangential magnetic field has already been applied. Now applying the boundary conditions for the tangential electric field at the interfaces dx ±= we get ϕ as

)dhsin(n

h)dhcos(

nh

f2f

ff2

c

c ϕϕ −−=−−

or ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=−

f

cfc

f

c

2

c

ff h

hhh

nn

)dhtan( ηϕ (4.3.2.7 a)

and )dhsin(n

h)dhcos(

nh

f2f

ff2

s

s ϕϕ +=+

or ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=+

f

sfs

f

s

2

s

ff h

hhh

nn

)dhtan( ηϕ (4.3.2.7 b)

which can be used to reproduce the eigenvalue equation once again by eliminating ϕ from these two equations just like the TE case. Eigenvalue equation in normalized form In order to normalize the results we take recourse of the previously considered normalized parameters and the asymmetry factor which had been defined in equation (4.3.1.12) through equation (4.3.1.15). Then in terms of these parameters it follows from the analysis for the TE case the eigenvalue equation for the

modes can be written as TM

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −⎟⎟⎠

⎞⎜⎜⎝

⎛−+=− −−

)1b()b1(1tan

b)b1(1tan)1N()b1(v2

fc

1

fs

1

ηηηπ (4.3.2.8)

or ⎥⎦

⎤⎢⎣

⎡

−−+

+⎥⎦

⎤⎢⎣

⎡

−+=− −−

)b1()1b(tan

)b1(btanN)b1(v2 fc

1fs

1 ηηηπ (4.3.2.9)

In case of symmetrical we set SWG fcfs ηη = and 1=η and the equation reduces to

⎥⎦

⎤⎢⎣

⎡

−+=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −⎟⎟⎠

⎞⎜⎜⎝

⎛−+=− −−

)b1(btan2N

b)b1(1tan2)1N()b1(v2 fs

1

fs

1 ηπη

π

So we see that unlike the case of TE modes this normalized eigenvalue equation still contains the individual refractive indices of the different layers and hence cannot be numerically solved for b as a function of v unless the refractive indices of the three layers are known definitely. However in the figure (4.3.2.1) we plot the curves for different TM modes for some specific typical values of

and hence for specific v~b

csf n,n,n η . The corresponding program is Prog. 9 in appendix. In this chapter we have considered the case of pure dielectric layers for which the refractive indices are real positive. The curves for in fig (same) show clear cut-offs like the modes but the curves intersect as we increase the TE

--- 555888 ---


asymmetryη . Another important point to note is that in plotting the curves we set and fixed but asymmetry has been developed by varying Infinite asymmetry corresponds to the condition which makes the denominator of

fn cn

sn fs nn →η almost zero.

Fig. 4.3.2.1 : b~v curve for TM modes for different asymmetry Modal cut-off As in the case of TE modes the cut-off frequency can be defined as the value of v for which the

mode just ceases to propagate through the guide. This condition occurs when TM s0 nk=β i.e. 0b = which when put in the eigenvalue equation (4.3.2.7) gives the cut-off frequency { })1(tanNv2 fc

1c −+= − ηηπ

or { })1(tan21

2Nv fc

1c −+= − ηηπ

(4.3.2.10)

Thus the cut-of frequency depends on the asymmetry factor cv η and also on and . For the symmetric case we put

cn fn1=η which yields

2

Nvcπ

=

like the TE case. However for ∞→η , there may be two conditions for . cv ∞→η corresponds to

either or . For the first condition sf nn → −∞→2cn

22

1N)(vcπη ⎟⎠⎞

⎜⎝⎛ +=∞→

--- 555999 ---


as the term terminates to 1tan− 2π . This corresponds to weak guidance for the TM modes as and will be discussed in the second next subsection of this chapter. For the second condition

which corresponds to the special case we get sf nn →

−∞→2

cnLt

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛− )1(nn

tan2

c

f1 η = −∞→2

cnLt

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−⎟⎟⎠

⎞⎜⎜⎝

⎛−

)nn()nn(

nn

tan 2s

2f

2c

2s

2

c

f1 0=

and hence

2

Nvcπ

=

Thus for TM modes the variation of with cv η will be strongly dependent on the nature and value of . cn Mode numbers Since for the thN TM mode the cut-off frequency is given by equation (4.3.2.8) all the lower modes,

….. and will be supported by the waveguide, if . Thus for a given

,TM,TM,TM 210 1NTM − NTM cvv >v the number of guided modes is given by

{ }[ ]int

fc1 )1(tanv21M −−= − ηη

π (4.3.2.11)

where the subscript ‘int’ indicates the next largest integer and the mode up to which TM waves are supported for a given value of v is for which 1MTM −

cvv = . Normalization in terms of power flow The constant amplitude part D in the field equations (4.3.2.6) can be evaluated in terms of the power flow though the total waveguide. Using Poynting theorem the time averaged power flow in the z direction is given by, as done in the case of TE modes

dx)HERe(21P

z∫∞

∞−

∗×=rr

Since for the TM modes and are both zero for the xH yE thN TM mode we may write

dxHE21P N

yN

xN ∫∞

∞−

∗=

Now as for TM modes

Ny2

0

NNx H

nE ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

ωεβ

dxHn2

P2N

y2j0

NN ∫

∞

∞−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=

ωεβ

--- 666000 ---


or ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛= ∫∫∫

∞−

−

−

∞−

dxHn1dxH

n1dxH

n1

2P

d

2Ny2

c

d

d

2Ny2

f

d 2Ny2

s0

NN ωε

β

The power flow through the waveguide layer )dx( ±= the flow of power is

dx)xh(cosDn

12

Pd

df

222

f0

Nf ∫

−

−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛= ϕ

ωεβ

or

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

4f

2f

4c

2c

2c

c2

f

4f

2f

4s

2s

2s

s2

f2

2f0

Nf

n

h

nh

nh

n1

n

h

nh

nh

n1

d22

Dn

12

Pωεβ

(4.3.2.12)

or ⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

++⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

)hh(

h

)hh(

hd2

2D

n1

2P 2

f2

c2

fc

cfc2

f2

s2

fs

sfs2

2f0

Nf η

η

η

ηωεβ

The power flow in the cladding layer

[ ]dx)xd(h2exp)dh(cosDn1

2P

dcf

222

c0

Nc ∫

∞

−−⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛= ϕ

ωεβ

or

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

4c

2c

4f

2f

c

4f

2f

2

2c0

Nc

nh

n

hh

n

h

2D

n1

2P

ωεβ

(4.3.2.13)

or ⎥⎥⎦

⎤

⎢⎢⎣

⎡

+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

)hh(

)hh(2

Dn

12

P 2s

2fc

2f

c2

ffc2

2f0

Nc η

ηωεβ

and the power in the substrate is

[ ]dx)xd(h2exp)dh(cosDn1

2P

d

sf22

2s0

Ns ∫

−

∞−

++⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛= ϕ

ωεβ

or

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

4s

2s

4f

2f

2s

s

4f

2f

2

2s0

Ns

nh

n

h

nh

n

h

2D

n1

2P

ωεβ

(4.3.2.14)

or ⎥⎥⎦

⎤

⎢⎢⎣

⎡

+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

)hh(

)hh(2

Dn

12

P 2s

2fs

2f

s2

ffs2

2f0

Ns η

ηωεβ

--- 666111 ---


where we used the relations obtained in equation (4.3.2.7 a) and (4.3.2.7 b) in deriving the results. Therefore the total time averaged power flow through the whole waveguide structure in the z direction is given by csfN PPPP ++=

or ⎥⎥⎦

⎤

⎢⎢⎣

⎡

+

++

+

++⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

)hh(h

)hh(

)hh(h

)hh(d2

2D

n1

2P 2

c2

fc2

fc

2c

2ffc

2s

2fs

2fs

2s

2ffs

2

2f0

NN η

η

η

ηωεβ

(4.3.2.15)

Therefore the normalizing constant D can be written in terms of the power as NP

2

1

2c

2fc

2fc

2c

2ffc

2s

2fs

2fs

2s

2ffs

21

NN

2f0

N )hh(h

)hh(

)hh(h

)hh(d2P

n2D

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+

++

+

++

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=

η

η

η

ηβ

ωε (4.3.2.16)

In case of symmetrical this reduces to SWG

21

2s

2fs

2fs

2s

2ffs

21

NN

2f0

N )hh(h

)hh(dP

2n

2D−

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+

++

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=

η

ηβ

ωε

Effective width From the expression of we see that similar to the TND E case we can define effective width of the waveguide as the total term containing the physical thickness having a power of d2 )21(− , i.e. the effective thickness for TM modes is

)hh(h

)hh(

)hh(h

)hh(d2W 2

c2

fc2

fc

2c

2ffc

2s

2fs

2fs

2s

2ffs

TM η

η

η

η

+

++

+

++= (4.3.2.17)

and for the symmetric SWG

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

++=

)hh(h

)hh(d2W 2

s2

fs2

fs

2s

2ffs

TM η

η

We can once again take recourse of the normalized parameters and define the normalized effective thickness as

)nn(kWw 2s

2f0TMTM −=

[ ][ ]

[ ][ ]2222

fc222

2222fc

22fs

2

22fs

d)uv()du(d)uv(

)dv(d)uv()du(

)dv(b)du(b)dv(

)dv(b)du(v2

−+−

−++

+

++=

ηηη

ηη

η

η

or TMw [ ] [ ])1b()b1()1b(b)b1(bv2 2

fc

fc2

fs

fs

−++−−++

+−+=

ηηη

ηη

η

η (4.3.2.18)

In figure (4.3.2.2) we plot the variation of against normalized frequency TMw v for five lowest TM modes for some specific values of the refractive indices of the different layer. The program is in Prog. 10 of Appendix.

--- 666222 ---


Fig. 4.3.2.2 : Variation of normalized effective width with normalized frequency In case of modes also we see that the field penetration into the substrate and cladding layers are substantially high. Penetration is significant up to the points of inflection of the curves. The penetration profile is more sharply defined for higher asymmetry as we proceed from the red curves towards the black curves. A comparison between the T

TM

E and TM modes field penetration depths will be tried in the third next subsection. But we see here, comparing the graphs of the TE and TM modes, that the points of inflection for the TM modes correspond to higher value of v which means TE modes are more suitable for propagation in the in general. SWG Confinement factor The confinement factor is given from the expressions for and as fP NP

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

++

+

++

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

++

++

=

)hh(h

)hh(

)hh(h

)hh(d2

)hh(

h

)hh(

hd2

PP

2c

2fc

2fc

2c

2ffc

2s

2fs

2fs

2s

2ffs

2f

2c

2fc

cfc2

f2

s2

fs

sfs

N

f

η

η

η

η

η

η

η

η

or TM

2f

2c

2fc

cfc2

f2

s2

fs

sfs

N

f

W

)hh(

h

)hh(

hd2

PP ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

++

++

=η

η

η

η

--- 666333 ---


In terms of the normalize parameters we may write it, by multiplying both the numerator and the

denominator by )nn(k 2s

2f0 −

TM

22222fc

22fc

222fs

2fs

N

f

w

)du(d)uv(

d)uv()dv(

)du(b)dv(

b)dv(v2

PP ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

+−

−+

++

=ηη

ηη

η

η

TM

2fc

fc2

fs

fs

w

)b1()1b(

)1b(

b)b1(

bv2

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−+−+

−++

+−+

=ηη

ηη

η

η

(4.3.2.19)

For symmetrical this simplifies to SWG

TM

2fs

fs

N

f

w

b)b1(

bv

2PP ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

+−+

=η

η

Also the fraction of power radiated in the substrate and cladding layers in

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+

++

+

++

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

++

+=

+=−

)hh(h

)hh(

)hh(h

)hh(d2

)hh(

)hh(

)hh(

)hh(

P)PP(

PP

1

2c

2fc

2fc

2c

2ffc

2s

2fs

2fs

2s

2ffs

2c

2fc

2f

c2

ffc2

s2

fs2

f

s2

ffs

N

cs

N

f

η

η

η

η

η

η

η

η

In normalized parameters

{ }

TM

22222fc

222fc

222fs

2fs

N

cs

w

)du(d)uv(

)du()dv()du(

)du(b)dv(

)dbv()du(

)dv(P

PP⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+−

−+

+=

+ ηη

ηη

η

η

TM

2fc

fc2

fs

fs

w

)b1()1b(

)1b()b1(

b)b1(

b)b1(⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−+−+

−+−+

+−

−

=ηη

ηη

η

η

(4.3.2.20)

In figure (4.3.2.3) and figure (4.3.2.4) we plot the variations of these two factors with v for specific values of the refractive indices of the three layers. The plots are complementary to each other as expectedvalue of

as at any v they should sum up to unity. The corresponding programs are in Prog. 10 of Appendix.

--- 666444 ---


Fig. 4.3.2.3 : Variation of confinement factor with normalized frequency

Fig. 4.3.2.4 : Variation of radiation factor with normalized frequency

--- 666555 ---


444...444 SSSyyymmmmmmeeetttrrriiiccc SSSWWWGGG:::

111... OOOdddddd aaannnddd EEEvvveeennn TTTEEE mmmooodddeeesss dddiiissstttrrr iiibbbuuuttt iiiooonnn For TE field distribution (equation (4.3.7)) we find that at ,0x = 0E y = if 0cos =φ i.e. 2πφ = which putting in equation (4.3.8) gives

f

s

f

cf h

hhh

)dhcot( −=−=

This essentially needs and therefore sc hh = sc nn = . Thus we consider the case of symmetric for which we have identical clad and substrate layers and therefore we set

SWG

cs nn = which yields cs hh = . Then the situation effectively means the existence of a plane of symmetry at

which simulates a conducting or short circuit or an electric symmetry plane for the T0)0x(E y ==

0x = E modes. For symmetric the SWG TE modes which satisfy the condition

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

f

cf h

h)dhcot(

or ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

c

ff h

h)dhtan( (4.4.1)

are called the odd TE as the field pattern on the two sides of the plane of symmetry has odd symmetry. Similarly for the field to be maximum at the plane of symmetry we must have from equation (4.3.7) 1cos =φ i.e. 0=φ which yields from equation (4.3.8) the relation

f

cf h

h)dhtan( = (4.4.2)

The modes satisfying this relation are called the even modes as the field distribution on the two sides of the plane of symmetry has even symmetry. This condition effectively simulates an open circuit or magnetic symmetry plane at .

TE TE

0x = 222... MMMooodddaaalll cccuuuttt ---ooffffff o At cut-off the field in the covering layers just ceases to decay exponentially and as a result the wave is no longer bound to the waveguide as such. Thus at cut-of we have 0hs = or s0 nk=β This condition for cut-off is identical to the condition stated in the last chapter. The ray optic equivalence of the cut-off is that the ray just ceases to be internally reflected from the interface and the corresponding

--- 666666 ---


angle of incidence can be obtained from the ray optic expressions of β and which yields the critical angle as given in equation (3.3.3 a) Then we have the relation

)nn(k)nk(h 2s

2f0

22f

20f −=−= β (4.4.3)

Then for odd and even TE modes we get from equation (4.4.1) and equation (4.4.2) For even TE : πυ'dh f = ,.....2,1,0' =υ For odd TE : )2(''dh f πυ= ,....5,3,1'' =υ )egersintodd( Thus in general we can write theTE cut-off condition by combining the two as υπ=0f dh ,...3,2,1,0=υ (4.4.4)

or υπ=− )nn(dk 2s

2f00 (4.4.5)

where the even values of υ corresponds to even modes and odd values to odd modes. Since for guidance

for confinement of the mode we must have 0hc >thυ

)nn(dk

)nn(nsf

220

22

sf+

>−=πυ∆

or )nn(d4

nsf

20

20

2

+>

λυ∆ (4.4.6)

This condition gives which modes can be supported by the symmetric for a given SWG n∆ and thickness . We see from the condition (4.4.6) the lowest mode 0d 0TE )0( =υ has no cut-off i.e. even if

0n →∆ or 0)d( 00 →λ the mode remains bounded in the waveguide. In figure (4.4.1) and (4.4.2) we show the variation of the required thickness for supporting specific modes while n∆ varies and also variation of the maximum supported mode number with n∆ for given )d( 00 λ . Both these graphs have been drawn for a particular value of . The programs for these graphs are given in appendix (Prog. 11 & 12).

fn

A more general plot can be generated by considering normalized term

v)nn(dkdh 2s

2f0f =−= (4.4.7)

Then the cut-off condition becomes

int2

v πυ⎟⎠⎞

⎜⎝⎛= (4.4.8)

where the subscript ‘int’ denotes truncation to the integral value just below the value of the right hand side of the expression in equation (4.4.8).

--- 666777 ---


From the plot of υ against v one can easily find the maximum number of modes supported for any given set of values of sf00 n,n,d,λ .This plot is shown in figure (4.4.3). Prog. 13 is for reference.

Fig. 4.4.1 : Variation of required thickness with index difference for supporting specific mode

Fig. 4.4.2 : Variation of supported mode number with index difference for different guide thicknesses

--- 666888 ---


Now we have from the definition of and ch fh

2f

2s

hh

)nk()nk(

22s

20

2s

20

2

ββ

−

−=

2f

2f

2s

2f

20

h

]h)nn(k[ −−=

22

f

22f

2s

2f

220

dh

]dh)nn(dk[ −−=

Fig. 4.4.3 : Variation of mode number with normalized frequency So from equation (4.4.1) and (4.4.2) we can write for guided even and odd TE modes the characteristics equations as

dh

]dh)nn(dk[

)(

)(

)dhtan(

)dhcot(

:even

:odd

f

22f

2s

2f

220

f

f −−

+

−= (4.4.9)

This forms a set of transcendental equations in as function of for even and odd Tdh f dk0 E modes for symmetric . Once again from equation (4.3.6) we get for symmetrical SWG SWG

)hh(

hh2)dh2tan( 2

s2

f

sff

−=

--- 666999 ---


or )hh(

hh2)]dh(tan1[

)dhtan(22

s2

f

sf

f2

f

−=

−

On simplification this gives

0hh

)dhtan(hh

)dhtan(s

ff

f

sf =⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛−

which shows the only TE modes present in the symmetrical are those for which SWG

f

sf h

h)dtan(h =

i.e. the evenTE mode or

s

ff h

h)dtan(h −=

i.e. the odd TE mode. Therefore equations (4.4.9) give in general the eigenvalue equations for a symmetrical . SWG

Fig. 4.4.4 : Field distribution for TE modes in symmetric SWG In the figure 4.4.4 we plot the field distribution of the field component for a few low order TyE E modes in

a symmetric guide with and the guide thickness 3.2n,5.2n 2s

2f == m2d2 µ= . We see that the

even modes are distributed symmetrically about the plane of symmetry whereas the odd mode is anti symmetrically distributed. The program corresponding to the figure is Prog. 14 of Appendix. From similar steps we can also find the even and odd TM modes satisfying the eigenvalue equations

f

sfsf h

h)dtan(h η=

--- 777000 ---


and s

f

fsf h

h1)dtan(h ⎟⎟⎠

⎞⎜⎜⎝

⎛−=η

444...555 WWWeeeaaakkklllyyy ggguuuiiidddiiinnnggg sssyyymmmmmmeeetttrrriiiccc SSSWWWGGG:::

Weakly guiding are those in which the cladding and the substrate layers are identical and the index difference between the guiding layer and the substrate layers is small. Symbolically this means

SWG

cs nn = and fsf n)nn(n <<−=∆ Then under these conditions we have

1nn 2

s

ffs ≈⎟⎟

⎠

⎞⎜⎜⎝

⎛=η

1nn 2

c

ffc ≈⎟⎟

⎠

⎞⎜⎜⎝

⎛=η

and 1=η Then the eigenvalue equation for the modes can approximately be written from equation (4.3.2.8) as TM

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−+=− −

b)b1(tan2)1N()b1(v2 1π

And that for theTE modes also reduce to

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−+=− −

b)b1(tan2)1N()b1(v2 1π

Thus both the modes of waves satisfy the same eigenvalue equation and therefore in these s theTE andTM modes become approximately degenerate. This simplifies the tools of analysis as for these s one has to deal with the solutions of one type only. The variation of normalized parameters and (effective width) and that of the confinement factor and fractional radiation terms against

SWG

SWGb w

v for both,TE andTM modes are similar to the ‘red’ curves plotted for theTE modes because the red curves are corresponding to the condition 1=η .The original eigenvalue equations in terms of the actual transverse propagation constants can also be written in the unified form for TE and TM modes as

⎟⎟⎠

⎞⎜⎜⎝

⎛−+= −

s

f1f h

htan2)1N(dh π

or ⎟⎟⎠

⎞⎜⎜⎝

⎛+= −

f

s1f h

htan2Ndh π

or )hh(

hh2)Ndh2tan( 2

s2

f

sff

−=− π (4.5.1)

--- 777111 ---


The normalized cut-off frequency and the total number of modes supported by the are found from equations (4.3.1.19) and (4.3.1.20) as

SWG

⎟⎠⎞

⎜⎝⎛=

2Nvcπ

(4.5.2)

and int

v2M ⎟⎠⎞

⎜⎝⎛=π

(4.5.3)

where the subscript ‘int’ signifies the next integer value. In these the normalized parameter has particularly simple interpretation. is defined as SWG b b

)nn(k)nk(

b 2s

2f

20

2s

20

2

−

−=

β

)nn)(nn(k)nk)(nk(

sfsf2

0

s0s0

−+

−+=

ββ

Since and sf nn ≈ s0f0 nknk >> β )nk()nk()nn(k)nn(k s0s0sf0sf0 −>>+≈+<<− ββ

and hence )nn(k

)nk(b

sf0

s0

−−

≈β

(4.5.4)

or )nn(bnnk sfseff

0

−+≈=β

(4.5.5)

With the help of this equation the curves are simply interpreted in terms of effective index vs. wave number for any given set of waveguide parameters and .These curves are identical to the curves developed for the T

v~b effnk sn,d2 fn

E modes in a general ( without weakly guiding approximation ) . SWG 444...777 DDDiiissscccuuussssssiiiooonnnsss::: In this chapter we tried to discuss the important modal features of a general D2 rectangular structure, the modal dispersions, modal cut-off characteristics their penetration into the subsidiary layers for bothTE and

modes. The eigenvalue equation for the TTM E and modes will be used in the further chapters while dealing with the structures containing metals at one or more supporting layers. We will see that these same equations can be slightly modified in format to satisfy the existence of the modes which are somewhat different from the modes we discussed in this chapter. The modes in this chapter are of the general behavior that they are sinusoidal in the main guiding layer but evanescent in the cladding and substrate layer. But metals have the unique feature of supporting surface modes which are evanescent even in the guiding layer. These modes can be dealt with appropriately with the concepts of the volume modes in the dielectric waveguides without considering the same boundary conditions once again separately and solving for the characteristic equations once again. The eigenvalue equation developed here is also self satisfactory to discard the existence of T

TM

E modes in some special cases we will deal with in the subsequent chapters while justifying that only TM modes can be the surface modes in the structures. The field penetrations in the supporting layers also draw much interest in the case of metallic layers.

--- 777222 ---

CCChhhaaapppttteeerrr (((vvv)))

SSWWWGGG wwwiiittthhh mmmeeetttaalll ::: BBBaaasssiiiccc ppprrrooopppeeerrrtttiiieeesss ooofff ttthhheee S a mmmooodddeeesss

--- 777333 ---


555...111 IIInnntttrrroooddduuuccctttiiiooonnn::: In this chapter we shall extend the discussion on modes to the case of metallic layer. Up to the previous chapter emphasis was given on purely dielectric s which were almost lossless in nature as the dielectric constants of the three layers were implicitly assumed to be pure real quantities. But in case of metals the dielectric constant is essentially a frequency dependent complex quantity. And since the refractive index is entirely dependent on the dielectric properties of the material it is also a frequency dependent complex quantity. The results for the TE and modes in the dielectric can be extended for the case of metal used as one of the layers in the only by replacing the refractive index of that layer by the complex refractive index of metals. In fact, since all the results previously obtained contain the square of the refractive index it is customary to use the complex dielectric constant of the metal in stead of the square of it’s complex refractive index. However it will be tried to be established how the modes of such metallic optical waveguide are affected by the complex nature of the dielectric constant of the metal. In photonics, metals are not usually thought of as being very useful, except perhaps as mirrors. In most cases, metals are strong absorbers of light, a consequence of their large free-electron density. However, in the miniaturization of photonic circuits, it is now being realized that metallic structures can provide unique ways of manipulating light at length scales smaller than the wavelength.

SWGSWG

TM SWGSWG

555...222 DDDiiieeellleeeccctttrrriiiccc ppprrrooopppeeerrrtttyyy ooofff mmmeeetttaaalll::: DDrrruuudddeee TTThhheeeooorrryyy ooofff dddiiieellleeeccctttrrriiiccc cccooonnnssstttaaannnttt::: D e It was obvious from equation (2.4.17) that in general for any material with certain conductivity the dielectric constant is a complex quantity in general. For dielectrics with negligible conductivity the imaginary part of

is negligible and it becomes a pure real positive quantity. But for conductors the imaginary part is very much significant for obvious reason. Thus equation (2.4.17) suggests that the imaginary part of dielectric constant is in general a negative quantity under the convention we adopted, that, the time variation of the fields is given as

crε

)tjexp( ω and the. It is also to quote here before justifying that the real part of the dielectric constant of most metals is also a negative quantity over the entire spectral range of frequency varying from the deep infrared region to the ultraviolet region covering the entire range used in optical communication. An estimate of the variation and change of sign of the dielectric constant of metals can be obtained from Drude Theory. The existence of plasma oscillation in metal surfaces is the reason for complex permittivity of metals. Plasma is the state of matter where the electrons are not bound to the atoms like in the case of dielectrics. Plasma is a neutrally charged group of positively and negatively charged particles (approximately the same number of each). A plasma oscillation is said to take place when one group of charges moves relative to the other, such as in a metal with electrons moving with respect to the positive ions. The quantum of the plasma oscillation is a quasi-particle referred to as a plasmon. It is a boson, so it follows Bose-Einstein statistics and has integer spin. However we will not look at it in that much detail in waveguide analysis. In 1900 P. K. Drude published a theory of electrical conduction. (Ann. Physik 1(1900) 566). He assumed that metals contained electrons which were free to move through the lattice. These free electrons were taken to be particles obeying Maxwell-Boltmann statistics and Newton’s laws of motion. The periodic movement of ions or electrons close to the surface of a solid (typically an ionic crystal or a metal) can create an interface polarization. This displacement leads to the formation of a time-dependent polarization or magnetization

and the creation of an associated time-dependent electromagnetic field close to the interface. As a consequence, ions or free charge carriers in the solid are subject to restoring Coulomb forces that lead to an acceleration of charge and hence to an oscillatory motion.

Pr

Mr

--- 777444 ---


The Drude model yields an expression for the metal permittivity under the assumption that all the conduction electrons can freely move around inside the metal to form a gas of non interacting particles. In this simple model, the electron gas acts as a kind of to account for the binding of the rigid, positively charged metal ions. In this picture, a free electron of mass and charge subject to an arbitrary electric field

glue

0m e Er

obeys Newton's equation of motion

Eevmdtvdm 00

rrr

−=⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

τ (5.2.1)

where dtrdv rr

= denotes the electron velocity and τ is the electron momentum relaxation time. This time phenomenologically describes the interaction between a conduction electron and the resting, positively charged metal ions. For this reason, the second term in the upper equation is interpreted as a damping term.

Fig. 5.2.1 : Schematic of the SPP electromagnetic field The electron collision frequency which describes the scattering of the conduction electrons with lattice vibrations (phonons) and lattice impurities is then given by

⎟⎠⎞

⎜⎝⎛=

πτ21vT (5.2.2)

If the electric field and the displacement take the form )r.ktjexp(E)t(E 0

rrrr−= ω

)r.ktjexp(r)t(r 0rrrr

−= ω for the forced electronic motion this leads to an expression for the specific AC conductivityσ of a metal because the electron velocity is related tov

rσ via Ohm’s law

EvNeJ

rrrσ=−= (5.2.3)

These three equations in conjunction give the results for the conductivity

)j1()j1(

1m

Ne)( 0

0

2

ωτσ

ωττωσ

+=

+⎟⎟⎠

⎞⎜⎜⎝

⎛= (5.2.4)

--- 777555 ---


Here, N is the particle density of the electron gas. For 1<<ωτ ,σ reduces to 0σ , the specific DC

conductivity. If we identify the polarization Pr

inside the metal with the forced oscillation of the free electrons, )t(rNe)t(P rr

−= (5.2.5) and use that EPED 0

rrrrεε =+= (5.2.6)

we obtain together with Ohm's law the relation between permittivity ε and specific conductivityσ as

⎥⎦

⎤⎢⎣

⎡−=

0

)(j1)(ωεωσωε (5.2.7)

By defining the plasma frequency

2

1

00

2

P mNe

⎟⎟⎠

⎞⎜⎜⎝

⎛=

εω (5.2.8)

we can write

)j(

1)(t

2

2P

ωωωω

ωε+

−= (5.2.9)

or ⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−=

)1(j

)(1)( 2

t2

t

2P

2t

2

2P

ωωωωω

ωωω

ωε (5.2.10)

where tω is the electron collision rate . Now for typical highly conducting metals 1−τ N is of the order of magnitude in the range to . Putting ,

and we get from equation (5.2.8)

328 m10 − 329 m10 − kg101.9m 310 ×≈

212120 mNC1085.8 −−−×=ε C106.1e 19−×= Pω is in the

ranges and the corresponding frequency in is in the tens of ranges. On the other hand relaxation time

116 sec10 − Hertz HzPetaτ is of the order of which gives sec10 13−

tω in the range of . So 113 sec10 −

Pω is at least thousand times as great as tω and therefore we may neglect tω terms in the expression for ε and get

⎟⎟⎠

⎞⎜⎜⎝

⎛−= 2

2P1)(

ωω

ωε (5.2.11)

This means that as long as the frequency is greater than Pω the dielectric constant is positive but for frequency less than Pω it is negative. We also saw that for most metals like gold and silver, this “plasma reflectivity edge” is usually in the ultraviolet (UV) part of the spectrum. So in the optical range used for communication the metal exhibits negative dielectric constant. Above the plasma frequency the metal becomes effectively transparent while below all incident radiation is reflected. Thus metals are optically active in the optical frequencies. We also see that )Im()Re( εε >> which conforms to the proven results for dielectric constants for several metals.

--- 777666 ---


555...333 IIInnnvvveeessstttiiigggaaattteeeddd rrreeesssuuullltttsss fffoorrr ssseevvveeerrraaall mmmeeetttaaalllsss::: o e l It has been observed that at wavelength nm650≥λ Drude model gives a good description of ε , but at shorter wavelengths there is a significant discrepancy. In the table below we list the values of the plasma frequency and the collision rates for some investigated metals. This table is taken from Reference r1 of the list at the end of the report. TTTaaabbbllleee 111

Metals

1P secin −ω 1

t secin −ω

Au 16101.37 × 131005.4 ×

Ag 16101.37 × 131073.2 ×

Pt 1510 7.82 × 131005.1 × In table 2 we also site he dielectric constants of some of the investigated metals at specific wavelengths. This table is derived from the table of refractive indices of metals in the reference b1. TTTaaabbbllleee 222

Metals m633.0 µλ = m6.10 µλ =

Au j0.55-16.3- j790.3-2796.8

Ag 5414.0j32.16 −− j1476.6-4646.5

Cu 52.0j996.15 −− j1612.8-3937.2

Cr 42.14j07.5 − j501.6-398.84

Al 8.16j56.47 −− j3350-3864

Figure 2.2: Variation of the real and imaginary part of the relative dielectric constant of metals withλ

--- 777777 ---


From these tables we see that tP ωω >> and also at the wavelengths specified )Im()Re( εε >> . Special attention given to , it has been find that the assumption of the Drude model gives a good agreement with reported values for its permittivity. The huge change of the permittivity with frequency is remarkable for . While at wavelength

Au

Au m633.0 µλ = , we have 5.0j16)Au( −−≈ε , at m300µλ = , i.e. 54 1023.6j1062.8)Au( ×−×−≈ε )Im()Re( εε < . Again at m55.1 µλ = , 65.12j95.131)Au( −−≈ε . Furthermore, for frequencies higher than optical

frequencies the Drude model fails to account for the experimental data. Among the metals, the noble category shows the highest value of )Im()Re( εε with and presenting the highest. The variation of that ratio with wavelength for and is shown in figure 5.3.1. From the above graph it is shown that the ratio between

Ag AuAu Ag

)Re( ε and )Im( ε becomes maximum at m9.0 µ hence the metal approaches ideal conditions. From to 4.0 m9.0 µ this ratio increases, and

decreases for longer wavelengths beyond m9.0 µ . It is also shown that silver exhibits higher ratios than gold, hence providing less losses and higher reflectivity. However these factors will be discussed in the subsequent chapters. However an important exception is for which the real part comes to be negative and also the real part is much smaller than the imaginary part so far as the magnitude is concerned.

Cr

555...444 GGGeeennneeerrraaalll dddiiissscccuuussssssiiiooonnn ooonnn mmmooodddeeesss uuunnndddeeerrr ttthhheee eeeffffffeeecccttt oofff mmmeeetttaaalll lllaaayyyeeerrr::: o In this subsection the general effect of metallic layer on the modal characteristics of the guided waves will be discussed. In the previous subsection it was found that the modulus of the real part of the dielectric constant is much greater than the modulus of the imaginary part. A lossless metal is that for which the imaginary part totally vanishes. Under this condition the wave propagation constant in the direction of propagation is the phase constant β and the attenuation constant is zero. However this is totally a idealized situation as the imaginary part if the dielectric constant actually exists however small. This makes α to exist giving rise to attenuation of the wave amplitude in the direction of propagation. This practical situation can be thought of as the idealistic situation with perturbation as it is small enough. If we therefore consider the ideal situation |'| mm εε −= the transverse propagation constant in the metallic layer will accordingly be modified. If cladding layer is made metallic becomes ch

|)'|k(h m2

02

c εβ += Thus the field dies out faster in the cladding layer in case when metal is used than in the case when the cladding was dielectric. So the field is confined in the guiding layer more strongly and the penetration in the cladding layer is negligible. On the other hand the guiding layer made metallic is modified as fh

)|'|k(jh 2m

20f βε +=

--- 777888 ---


i.e. the field no longer remains sinusoidal in the transverse direction in the guiding layer but decays away from interfaces. This makes the modes in such guide surface-bound waves as the transverse wave amplitude decays in both sides from the interfaces, in the dielectric cladding or substrate as well as the guiding layer. These surface waves present at the metal-dielectric interface are called ‘Fano waves’. The surface waves in case of real metal are termed as ‘Surface Plasmon Polariton waves’. The name comes because of the plasma contribution to the negative dielectric constant as discussed in the last subsection. So a Surface Plasmon Polariton wave is basically a Fano wave with losses incorporated. Although analytically difficult, Surface Plasmon Polariton waves can be analyzes by taking into account the imaginary part of the dielectric constant of the metal layer as a small perturbation. 555...555 DDDiiissscccuuussssssiiiooonnnsss::: In this chapter we discussed the background, why metals show negative dielectric constants in the optical regions of frequencies. This feature of metals makes them useful for propagation of optical frequency surface modes which we will discuss in the next few chapters. However, Drude theory is a classical model assuming that the valence electrons of atoms can move freely and independently in a solid. One of the striking conclusions of Drude theory is the ultraviolet transparency of metals i.e. for frequencies greater than plasma frequency metals become effectively transparent. This is in contradiction with the observed reddish color of copper or the yellowish color of gold, for example. The reason for this behavior is that for high frequencies the metal band structure can no longer be neglected and additional effects such as inter band absorption arise. Nevertheless, for frequencies ranging in the optical region the Drude model for metals and doped or intrinsic semiconductors holds good and gives generally a good agreement with experimental observations. For other cases the model is modified accordingly. But the discussion of these modifications is beyond the scope of the report as we are concerned only about the metals in the optical frequencies and their negative dielectric constants in this region.

--- 777999 ---

CCChhhaaapppttteeerrr (((vvviii)))

SSSuuurrrfffaaaccceee ppplllaaasssmmmooonnn wwwaaavvveee iiinnn sssiiinnngggllleee iiinnnttteeerrrfffaaaccceee SSSWWWGGG

--- 888000 ---


666...111 IIInnntttrrroooddduuuccctttiiiooonnn::: This chapter will be dedicated to electromagnetic behavior of single structure and the modes that are supported. The reason we discuss it here after introducing 3 layers is that we shall seek out the possibilities of existence of surface plasmon waves in this chapter. We shall see that modes that are bound at the interface and die out in both directions away from the interface can exist only in case when one of the two layers have negative dielectric constant and the other positive. Thus it is the best-fit situation to analyze the modal solutions of the Maxwell’s equations after introducing the negative nature of the dielectric constant of metals. We shall derive the behavior of both T

SWG

E and TM modes in this structure. 666...222 GGGeeeooommmeeetttrrryyy ooofff ttthhheee ssstttrrruuuccctttuuurrreee:::

The structure we consider in this chapter contains single plane interface between two optical media of different refractive indices. The interface is flat and infinitely extended in the y direction. The direction of propagation is considered to be the z direction and the direction is the transverse direction. The interfacial plane is the plane . We consider the refractive index of the region of negative value of

to be and that of positive value of to be and we consider the condition . The dielectric constants of the respective layers are and .

x0x =

x 1n x 2n 21 nn >2

11 n=ε 222 n=ε

Fig. 6.2.1 : Dielectric constant profile of single interface structure with metal Now as discussed earlier if for a mode 102 n)k(n << β the mode will be periodic in the negative

direction in the region of r.i. but will be evanescent in the other region. Since there is no other interface to confine the mode the wave energy will thus flow in the direction of negative in the first medium. In order to confine the modal energy around the interface the condition

x 1nx

210 n,n)k( >β must be satisfied so that the field amplitude falls exponentially in both directions from the interface. We shall seek this result.

--- 888111 ---


666...333 MMMooodddaaalll aaannnaaalllyyysssiiisss::: 111... TTTEEE MMMooodddeeesss We shall first consider the TE waves in this structure. Since as stated in the earlier chapter there is no variation in the y direction in this structure the only field components for the TE modes that can exist are

and given by )x(H),x(E zy )x(H x

y0

x EH ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

ωµβ

(6.3.1.1)

and ⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛=

xEjH y

0z ωµ

(6.3.1.2)

where the tangential electric field satisfies the wave equation yE

0x

0xfor

0Ehx

0Ehx

y2

22

2

y2

12

2

≥

≤

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂

(6.3.1.3)

where the ’s are transverse propagation constants given by h 2,1j;)nk(h 2

j2

022

j =−= β Obviously real values of both and gives the surface bound solutions for the tangential electric field. Imaginary value of any of these propagation constants gives propagative solution in the direction. The solutions to the equations can be written as

1h 2hx

(6.3.1.4) 0x

0xfor

)xhexp(A

)xhexp(A)x(E

22

11

y

≥

≤

−=

where we have already adopted a realistic assumption that the field amplitude cannot increase away from the interfaces as this will lead to infinite field amplitude at large distances. We expect realistic solutions which go to zero at large distances. So we take both and to be positive. 1h 2h Now continuity conditions demand the tangential electric field must be continuous at the interface. Also the tangential magnetic field should be continuous across the interface. These two conditions respectively mean and

yE

zH

yE )xE( y ∂∂ must be continuous across the interfaces. Equating at from both sides we get

yE0x =

21 AA = and equating )xE( y ∂∂ at from both sides we get 0x =

--- 888222 ---


2211 AhAh −= or 0hh 21 =+ (6.3.1.5) Since modes bound at the interface need to have both 1 and 2 positive, otherwise the field amplitudes will increase going away from the interface in one side leading to radiation of the field energy. So for such bound modes equation (6.3.1.5) cannot be satisfied. Thus T

h h

E surface bound modes are not possible in case of single interface . Conceptually, a SWG TE wave, whose electric field vector is parallel to the interface, merely creates a motion of charge parallel to the interface. For an ideal gas of charge carriers, however, no restoring force can build up and consequently no propagating wave can be supported. 222... TTTMMM MMMooodddeeesss In case of TM modes the only non-zero field components are and and they are related as

)x(E),x(E zx )x(H y

y0

2j

x Hn

E⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=

εωβ

(6.3.2.1)

and ⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−=

xH

njE y

02

jz εω

(6.3.2.2)

where the tangential magnetic field satisfies the wave equation )x(H y

0x

0xfor

0Hhx

0Hhx

y2

22

2

y2

12

2

≥

≤

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂

(6.3.2.3)

The solutions for the surface bound tangential magnetic fields can once again be written as

(6.3.2.4) 0x

0xfor

)xhexp(B

)xhexp(B)x(H

22

11

y

≥

≤

−=

We now apply the boundary conditions. The continuity of the tangential magnetic field at the interface demands

)x(H y

0x = 21 BB = Again the continuity of the normal electric field at the interface demands )x(E x

--- 888333 ---


+=−=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛

0x

y2

20x

y2

1 xH

n1

xH

n1

i.e. 22

21

1

1 hBhB⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛εε

or 0hh

2

2

1

1 =⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛εε

(6.3.2.5)

Since both and are positive this equation can be satisfied if and only if 1h 2h 1ε and 2ε have opposite signs. This requires one of the layers must have negative value of dielectric constant. In the last chapter we just introduced that metal in the entire optical range of frequencies exhibit negative dielectric constant. Thus surface bound TM mode exists in metal-dielectric single interface structure. For the TM waves the magnetic field vector is parallel to the interface. This results in a charge accumulation at the interface itself because now the electric field vector has a non-vanishing component perpendicular to the interface. Since the charge carriers are in fact trapped inside the solid, they cannot escape and consequently a restoring force will build up. Hence, the surface plasmon polariton can be regarded as a longitudinal wave in which the charge carriers move in a direction parallel to the propagation direction. The corresponding value of the propagation constant β is given by, using the definition of in equation (6.3.2.5) jh

0)k()k( 22

02

112

02

2 =−+− εβεεβε

or )(k 21

21

2

0 εεεεβ+

=⎟⎟⎠

⎞⎜⎜⎝

⎛ (6.3.2.6)

Now if we have lossless metal with real negative value of dielectric constant The propagation constant is real quantity. We take || 11 εε −= , || 22 εε = and |||| 21 εε > and get

|)||(|

||||k 21

21

2

0 εεεεβ

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ (6.3.2.7)

Thus )k( 0β is greater than 2ε and since 1ε . This makes and both positive. Therefore the field is confined at the metal-dielectric interface and falls away evanescently as we go away from the interface in both directions. This surface bound wave that exists at the interface between lossless metal and lossless dielectric is termed as Fano Waves.

1h 2h

Practically, of course, there are always some losses in the material i.e. the imaginary part of the dielectric constant is non-zero. The surface modes still exist but suffer propagation loss in the direction of propagation. The surface modes that exist at the interface between one lossy metal and one lossless or lossy dielectric are termed as Surface plasmon waves. In this case we can no longer consider the phase constant as the axial propagation constant , rather the attenuation constant comes into significance and in all the results previously obtained β has to be replaced by given by zk αβ jkz −=

--- 888444 ---


Thus equation (6.3.2.6) changes to

)(k

k

21

21

2

0

z

εεεε+

=⎟⎟⎠

⎞⎜⎜⎝

⎛ (6.3.2.8)

We now write the dielectric constant of the dielectric and the metal in most general form, considering the losses as

|''|j|'|dielectric

|''|j|'|metal

222

111

εεεεεε

−=−−=

with |''||'|,|'||'| 1121 εεεε >>> and |''||'2| 2εε >> as the losses are in general very small. Then from equation (6.3.2.8) we get on simplification

221

12

222

1

21

21

2

0

z

|)'||'(||)''||'||''||'(|j

|)'||'(||'||'|

kk

εεεεεε

εεεε

−+

−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛ (6.3.2.9)

where higher order terms in |''| 1ε and |''| 2ε and their product terms are neglected for brevity. Thus we get

|)'||'(|

|'||'|k 21

21

2

0 εεεεβ

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ (6.3.2.10)

and 221

12

222

12

0 |)'||'(||)''||'||''||'(|

k2

εεεεεεαβ

−+

=⎟⎟⎠

⎞⎜⎜⎝

⎛ (6.3.2.11)

These equations exclusively characterize the propagation of the Surface plasmon mode in the single interface . We note that if the double – primed imaginary parts for doth the layers are neglected we retrieve the Fano wave solution given in equation (6.3.2.7) with attenuation constant

SWG0=α . However most

of the dielectrics used are practically lossless and we can neglect |''| 2ε . Also since |''| 1ε is very small equation (6.2.3.11) suggests that attenuation constant α is very small in most metals. So the transverse propagation constants ’s are also predominantly real positive with very little imaginary part. Thus the field amplitudes exponentially decay going away from the interface. This happens with most metals. If on the other hand the imaginary parts of the dielectric constants are not negligible with respect to the real parts the results for

jh

β and α come out without any approximation

⎥⎦

⎤⎢⎣

⎡−+

−⎥⎦

⎤⎢⎣

⎡++−

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛2

21

12

222

12

212

21

221

212

2

0

z

|)'||'(||)''||'||''||'(|

j|)''||''(||)'||'(|

)|||'||||'(|kk

εεεεεε

εεεεεεεε

(6.3.2.12)

Thus now α is not any more negligible and so the transverse propagation constants have substantial imaginary part. If we write )hIm(j)hRe(h jjj += the field variation is given by |)x|)hIm(jexp(|)x|)hRe(exp(~ jj −− the second term in the product being oscillatory in the transverse direction. Thus in this case the field tries to propagate with a

--- 888555 ---


spatial transverse frequency but the amplitude exponentially falls at the rate as we go away from the interface. In the figure (6.3.2.1) the transverse distribution of the tangential magnetic field is shown for various metals where the other layer is taken to be lossless dielectric of dielectric

constant

)hIm( j )hRe( j

)x(H y

25.22 =ε . The corresponding program is written in Prog. 15. The wavelength chosen is the He-Ne Laser wavelength m633.0 µλ = . The complex dielectric constants of the various metals are taken from the table 2 of the previous chapter. In the plots we see there is clear peculiarity of the field for Cr . It looks like the heavily damped oscillatory field distribution owing to substantial imaginary part in . This is because as we look in the table the real part of

2h)Cr(ε is positive and the imaginary part is larger in

magnitude than the real part. So Cr behaves like a heavily lossy dielectric at this frequency.

Fig. 6.3.2.1 : Distribution of the Hy (x) field for various metals in single interface SWG However we leave the discussion on single interface structure right here and from the next chapter we go into the detailed study of double interface SWG 666...444 DDDiiissscccuuussssssiiiooonnnsss::: In this chapter we studied the Fano modes in single interface structures. We saw that there may exist

field in this structure that decays evanescently from the interface on both sides. However when this interface is brought near another interface the boundary conditions are altered. Accordingly the Fano mode couples with the TM modes in the other interface and thus modal dispersion result with separation between the two interfaces. These give rise to Surface Plasmon Polariton Waves in double interface

s with at least one layer metallic. The detailed modal analysis of various kinds of such structures is discussed in the subsequent chapters.

TM

SWG

--- 888666 ---

CCChhhaaapppttteeerrr (((vvviiiiii)))

SSSuuurrrfffaaaccceee ppplllaaasssmmmooonnn pppooolllaarrriiitttooonnn wwwaaavvveee iiinnn a

aaasssyyymmmmmmeeetttrrriiiccc mmmeeetttaaalll ccclllaaaddddddeeeddd SSSWWWGGG

--- 888777 ---


777...111 IIInnntttrrroooddduuuccctttiiiooonnn::: In this chapter we shall consider the modal analysis of asymmetric metal cladded . We shall try to develop the detailed theoretical approach to the modes existing in such waveguide. We shall try to establish the existence of the surface plasmon polariton mode in this structure. Our objective will be to analyze the theoretical aspects of all the supported modes and also the surface plasmon polariton mode and to put emphasis on the special properties this unique mode holds.

SWG

777...222 GGGeeeooommmeeetttrrryyy ooofff ttthhheee SSSWWWGGG:::

In this structure the cladding layer is taken to be metallic. The refractive index of the middle layer is positive real, so also is that of the substrate layer. Thus the guide consists of dielectric guiding layer supported by a dielectric substrate with lower refractive index and a metallic cladding layer. In the last chapter we saw that in the optical range of frequencies the dielectric constant of the metal is a complex quantity with real part being large negative. The complete schematic r.i. profile of the guide in different layers is shown in figure (7.2.1)

Fig. 7.2.1 : Dielectric constant profile of metal cladded SWG The z direction is once again the direction of propagation and the guide is infinitely spread in the y direction. The direction is the transverse direction in which the transverse fields are distributed across

the interfaces. The two boundaries are perpendicular to the axis. x

x 777...333 TTThhheee ccchhhaaarrraaacccttteeerrriiissstttiiiccc eeeqqquuuaaatttiiiooonnn::: In this structure the cladding layer is taken to be metallic. With such configuration the guiding condition within the middle layer, that, 2

0 )k( β lies within a maximum value of and a minimum value of

with is still satisfied as the real part of is negative which keeps . Therefore the dielectric constant of the guiding layer may be still arbitrary and there will be guiding action. The eigenvalue equation for the

2fn 2

sn2

c2

s nn > 2cn 2

c2

s nn >

TE and TM modes in combined form can be modified using the notation of the dielectric constant of the metal used as the cladding layer

--- 888888 ---


(7.3.1) 22cccc )jKn(n''j' −==+= εεε

in stead of using the notation , as cn

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛+= −−

f

cfc

1

f

sfs

1f h

htan

hh

tanNdh2 ηηπ (7.3.2)

where

⎪⎩

⎪⎨

⎧

=2

sf

fs

)nn(

1η for mode

TM

TE

and (7.3.3)

⎪⎩

⎪⎨

⎧

=)n(

1

c2

f

fc

εη for mode

TM

TE

and

)''k'kk()kk(h c2

0c2

02

zc2

02

zc εεε −−=−= (7.3.4 a)

)nkk(h 2s

20

2zs −= (7.3.4 b)

)knk(h 2z

2f

20f −= (7.3.4 c)

where is now the axial propagation constant in the zk z direction which takes into account the loss and is defined in equation (2.4.30 b) as )0(jkz >−= ααβ Again as mentioned in the previous chapter the dielectric constant of highly reflecting metals in the optical range of frequency that we are considering, has both the real and imaginary parts negative. Thus we can write the metal cladding dielectric constant as |''|j|'| ccc εεε −−= with )1|,''(||'| cc εε >> (7.3.5) In normalized form also we may write the eigenvalue equation as before

⎥⎦

⎤⎢⎣

⎡

−−+

+⎥⎦

⎤⎢⎣

⎡

−+=− −−

)b1()1b(tan

)b1(btanN)b1(v2 fc

1fs

1 ηηηπ (7.3.6)

where the asymmetry factor is

)nn(

|''|j

)nn(

|)'|n(

)nn(

)n(2

s2

f

c2

s2

f

c2

f2

s2

f

c2

f

−+

−

+=

−

−=

εεεη (7.3.7)

i.e. the real and the imaginary part of the asymmetry factor are

--- 888999 ---


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

−=

−

+=

)nn(|''|

)nn(

|)'|n(

2s

2f

ci

2s

2f

c2

fr

εη

εη

(7.3.8)

If we additionally impose the weak guidance condition as 2

f2

s2

f n)nn( <<− with the condition (7.3.5) we may therefore infer that )1,(|| ir ηηη >>≈ The normalized parameters and are redefined as v,b u

)nn(kv 2s

2f0 −=

)knk(d''ju'uu 2z

2f

20 −=+=

and )nn(k)nkk(

vu1''jb'bb 2

f2

f2

0

2s

20

2z

2

−

−=⎟

⎠⎞

⎜⎝⎛−=+=

The single and the double primed quantities denote the real and imaginary parts respectively. Now we shall evaluate the cut-off value of the normalized frequency v and the final value of as b ∞→v 777...444 MMMooodddaaalll cccuuuttt oooffffff::: The cut-off value of v is once again defined as the condition when 0b = . From equation (7.3.6) we may define it as

( )[ ])1(tanN21v fc

1c −+= − ηηπ

Since for TE modes 1fc =η and 1|| >>η the arc tan function on the right hand side can be approximated to

( )2

)1(tan fc1 πηη ≈−− (7.4.1)

and hence for the TE modes the cut-off value of the normalized frequency is

22

1Nvcπ⎟⎠⎞

⎜⎝⎛ +≈ (7.4..2)

--- 999000 ---


Here we site an example to justify the approximation. If we have silver ( ) cladding with Ag 32.16c −=ε

at a wavelength of 5414.0j− m633.0 µλ = and we have 25.2n,3.2n 2s

2f ==

83.10j40.372 +=η and ( ) 51897.1)1(tan fc

1 =−− ηη

whereas

57079.12=

π

For modes however the factor TM fcη is complex as cε is so. As |''||'| cc εε >> we can approximate

fcη to |)'|n( c2

f ε− and therefore for TM modes the real value of the arc tan function is negative and so

2

Nvcπ

< (7.4.3)

An important factor coming out of this is that for the mode as 0TM 0N = , is negative. But cv v being a

realistically physical parameter defined as )nn(d2 2f

2f −

λπ

cannot be negative. So the significant

feature of the mode is that it has no cut-off at all. The wave propagates even if , for which for the mode. For all other TM modes as the thickness tends towards the cut-off thickness

the modes cease to propagate. We once again take recourse of the same example of Silver cladding for which for mode

0TM 0dk0 →0b > 0TM

TMN th

( ) 00603.0j21794.1)1(tan fc1 +−=−− ηη

and

( )00603.0j21794.1N21vc +−= π

i.e. for mode the cut-off normalized frequency is 0TM 60897.0)vRe( 0c −= The value of b as or for the mode is obtained from the eigenvalue equation (7.3.6) by putting . This gives equal and oppositely signed values of the arguments of the two arc tan functions. This makes the right hand side of the equation vanish. Thus we get

0dk0 → 0v → 0TM0v =

)b1(

)1b()b1(

bfcfs −

−+−=

−ηηη

or )1b(nb 4s

2c −+= ηε

or )nn(

)n(nb)n( 2

s2

f

c2

s4s

4s

2c

−

−⋅=−

εε

--- 999111 ---


or )n)(nn(

nb 2

sc2

s2

f

4s

+−−=

ε (7.4.4)

Using the modified definition of in terms of we therefore have b zk

)n(

nn

kk

c2

s

4s2

s

2

0

z

ε+−=⎟⎟

⎠

⎞⎜⎜⎝

⎛

or )n(

nkk

c2

s

c2

s2

0

z

εε+

=⎟⎟⎠

⎞⎜⎜⎝

⎛ (7.4.5)

Splitting intozk β & α and using the condition (7.3.5) we get

)n|'(|

|'|nk 2

sc

c2

s2

0 −=⎟⎟

⎠

⎞⎜⎜⎝

⎛

εεβ

(7.4.6 a)

and

22

sc

c4

s2

0 )n|'(||''|n

k2

−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

εεαβ (7.4.6 b)

777...555 MMMaaaxxxiiimmmuuummm vvvaaallluuueee ooofff bbb ::: DDDiiissspppeeerrrsssiiiooonnn ppprrrooofffiii llleee:::

The final value of is attained as b ∞→v . In order to allow ∞→v we write the eigenvalue equation (7.3.6) in the form

⎥⎦

⎤⎢⎣

⎡

−−+

−+⎥

⎦

⎤⎢⎣

⎡

−−+

−= −−

)b1()1b(tan

)b1(1

)b1(btan

)b1(1

)b1(Nv2 fc

1fs

1 ηηηπ (7.5.1)

TTTEEE MMooodddeeesss M Now as ∞→v the right hand side of the equation (7.5.1) must also tend to ∞ . Now for TE modes as

fsη and fcη are unity, one feasible solution is as 1b → ∞→v . Under this condition, for TE modes all the three terms on the right hand side go to ∞ . Even for mode the first term on the right hand side vanishes but the other two terms both tend towards

0TE∞ . So for all TE modes the feasible final value of is

. In figure (7.5.1) we have plotted the variation of the real value of against b

0j1b += b v for the TE modes. The value of for different values of b v is obtained by iterating the eigenvalue equation (7.3.6). In the figure (7.5.2) and (7.5.3) we have also shown the modulation of the effective index profile, i.e. the variation of 0kβ and 0kα against . The curves confirm that the value cannot exceed unity for any of the T

00 dk bE modes as we allow ∞→v i.e. the variation of 0kβ is limited within and . The

plots are generated by MATLAB in Prog. 16 of Appendix. sn fn

TTTMMM MMMoodddeeesss o For the modes however the term TM fcη can be approximated to |)'|n( c

2f ε− . Thus as the

first arc tan function tends to

1b →)2( π+ whereas the second arc tan term tends to )2( π− . Thus these

--- 999222 ---


two phase terms add to zero as so that the second and third term on the right hand side add to zero. But if the right hand side of the equation (7.5.1) still goes to

1b →0N > ∞ as . Therefore for the

modes etc. the feasible solution for eigenvalue equation (7.5.1) as 1b →

,.....TM,TM 21 ∞→v is . 0j1b +=

Fig. 7.5.1 : Variation of normalized propagation const. with normalized frequency for metal cladded SWG

Fig. 7.5.2 : Effective index profile for TE modes

--- 999333 ---


But for mode the first term on the right hand side itself vanishes and therefore the right hand side cannot go to as . Therefore for the mode the final value of must differ from unity so that the sum of the second term and the third term on the right hand side of equation (7.5.1) go to

0TM∞ 1b → 0TM b

∞ as ∞→v . This means for the mode we must have for 0TM ∞→v .

Fig. 7.5.3 : Loss profile for TE modes

∞=⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡

−−+

+⎥⎦

⎤⎢⎣

⎡

−−−−

)b1()1b(tan

)b1(btan

)b1(1

fc1

fs1 ηηη (7.5.2)

To mathematically manipulate this equation and allowing b exceeding unity value, we convert the arc tan functions into arc tan hyperbolic functions as )jz(tanhjztan 11 −− −= so that the equation (7.5.2) reduces with 0b ≠ to

−∞=⎥⎦

⎤⎢⎣

⎡

−−+

+⎥⎦

⎤⎢⎣

⎡

−−−

)1b()1b(tanh

)1b(btanh fc

1fs

1 ηηη (7.5.3)

Writing the two arguments inside the arc tanh as and respectively and adding the two arc tanh functions we get

sz cz

−∞=⎟⎟⎠

⎞⎜⎜⎝

⎛++−

cs

cs1

zz1zz

tanh

This would require the argument of the arc tanh function to be unity which leads to

--- 999444 ---


)zz1()zz( cscs +−=+ 0)z1)(z1( cs =++ Therefore the solutions are 1z;1z cs −=−= For we have 1zs −=

1)1b(

bfs −=

−η

or )1(

1b 2fs −

−=η

which leads to

2

02

f2

s

2f

2s

2

0

z

k)nn(

nnkk

⎟⎟⎠

⎞⎜⎜⎝

⎛=

+=⎟⎟

⎠

⎞⎜⎜⎝

⎛ β (7.5.4)

Thus ( ) s0 nk <β which leads to the unguided solution. The other solution 1zc −= leads to

1)1b(

)1b(fc −=

−−+ηη

or )1(

1b 2fc

2fc

η

ηη

−+=

or [ ]

)n)(nn(nn)nn(

bc

2f

2s

2f

2s

2fc

2s

2f

εε

+−

−−=

and therefore

[ ]

)n(

nn)nn(n

kk

c2

f

2s

2fc

2s

2f2

s

2

0

z

ε

ε

+

−−+=⎟⎟

⎠

⎞⎜⎜⎝

⎛

or )n(

nkk

c2

f

c2

f2

0

z

ε

ε

+=⎟⎟

⎠

⎞⎜⎜⎝

⎛ (7.5.5)

Equating the real and imaginary parts on both sides we get

)n|'(|

|'|nk 2

fc

c2

f2

0 −=⎟⎟

⎠

⎞⎜⎜⎝

⎛

ε

εβ (7.5.6)

and 22

fc

c4

f2

0 )n|'(|

|''|n

k2

−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

ε

εαβ (7.5.7)

--- 999555 ---


Fig. 7.5.4 : Variation of normalized propagation const. with normalized frequency for metal cladded SWG

Fig. 7.5.5 : Effective index profile for TM modes

--- 999666 ---


Fig. 7.5.6 : Loss profile for TM modes

Since |''||'| cc εε >> , we can infer that the final value of 0kα is much smaller than the final value of 0kβ . In figure (7.5.4) we have plotted the variation of the real value of against b v for the TM modes and in the figure (7.5.5) and (7.5.6) we have also shown the variation of 0kβ and 0kα against . The curves confirm that the value cannot exceed unity for any of the TM modes except mode as we allow

00 dkb 0TM

∞→v i.e. the variation of 0kβ is limited within and for etc. modes. But for the mode the value of exceeds unity and

sn fn ,....TM,TM 21

0TM b 0kβ exceeds . Also as there still exists finite positive value of for mode which means there is no cut-off for the mode. Since for all the T

fn 0v →b 0TM 0TM

E and TM modes except mode the final value of b is 0TM 0j1 + the loss factor α decreases to 0 as we increase v . But for the mode the value of 0TM 0kα reaches up to the value governed by the equation (7.5.7). The programs for the three figures for the TM modes is written in Prog. 17 of Appendix. 777...666 FFFiiieeelllddd dddiiissstttrrriiibbuuutttiiiooonnn::: CCCooommmpppllleeettteee ssseeettt ooofff fff iiieeelllddd eeeqqquuuaaatttiiioonnnsss::: b o The results in equations (7.5.6) and (7.5.7) are very important in this context because they impose a very peculiar property on the mode that, for this mode b can exceed the value i.e.0TM 1 β can exceed the value . If we recall the definition of , we see that the imaginary value of is large with real part being negligible. If we approximate

f0 nk fh fh0|''| ≈ε and 0k0 ≈α we have

--- 999777 ---


)n|'(|

nkjh 2

fc

2f0

f−

=ε

Thus the nature of variation of the transverse magnetic field will be exponentially decaying in the guiding layer from the metal-dielectric interface. It will be customary to define the transverse propagation constant as

yH

ff jhH −= so that the field decays at a rate inside the guiding layer. The mode will thus be strongly confined in that interface unlike the other modes. In a nutshell, the solutions lead to justification for the so-called Surface plasmon wave as long as the guide thickness is large.

fH 0TM

For theTE modes the other non-zero field components are the transverse magnetic field and the longitudinal magnetic field . Therefore the complete set of field equations for the three layers for the

xH

zHTE modes is given by

(7.6.1 a) ⎪⎩

⎪⎨

⎧

−≤≤≥

++−

−−=

dxd|x|

dxfor

)]xd(hexp[)dhcos()xhcos(

)]xd(hexp[)dhcos(C)x(E

sf

f

cf

y

φφ

φ

⎪⎩

⎪⎨

⎧

−≤++≤−≥−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

dx)]xd(hexp[)dhcos(d|x|for)xhcos(

dx)]xd(hexp[)dhcos(C)x(H

sf

f

cf

0x

φφφ

ωµβ (7.6.1 b)

⎪⎩

⎪⎨

⎧

++−−

−−−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

dx)]xd(hexp[)dhcos(hd|x|for)xhsin()h(dx)]xd(hexp[)dhcos()h(

Cj)x(H

sfs

ff

cfc

0z

≤≤≥

φφφ

ωµ (7.6.1

c) where the continuity of the tangential component of the magnetic field, at the two interfaces gives a definition of

zHφ as

⎪⎪

⎭

⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛=+

⎟⎟⎠

⎞⎜⎜⎝

⎛=−

f

sf

f

cf

hh

)dhtan(

hh

)dhtan(

φ

φ

(7.6.2)

For the modes the field components that exist are and as discussed in chapter (IV). The complete set of field equations is

TM yz H,E xE

--- 999888 ---


(7.6.3 a) ⎪⎩

⎪⎨

⎧

−≤≤≥

++−

−−=

dxd|x|

dxfor

)]xd(hexp[)dhcos()xhcos(

)]xd(hexp[)dhcos(D)x(H

sf

f

cf

y

φφ

φ

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

−≤++⎟⎟⎠

⎞⎜⎜⎝

⎛

≤−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

≥−−⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=

dx)]xd(hexp[)dhcos(n1

d|x|for)xhcos(n

1

dx)]xd(hexp[)dhcos(1

D)x(E

sf2s

f2f

cfc

0x

φ

φ

φε

ωεβ

(7.6.3 b)

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

≤++⎟⎟⎠

⎞⎜⎜⎝

⎛

≤−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

≥−−⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−=

dx)]xd(hexp[)dhcos(nh

d|x|for)xhsin(n

h

dx)]xd(hexp[)dhcos(h

Dj)x(E

sf2s

s

f2f

f

cfc

c

0z

φ

φ

φε

ωε (7.6.3 c)

where the continuity of the tangential electric field defines zE φ as

⎪⎪

⎭

⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛=+

⎟⎟⎠

⎞⎜⎜⎝

⎛=−

f

sfsf

f

cfcf

hh

)dhtan(

hh

)dhtan(

ηφ

ηφ

(7.6.4)

However since for the mode as we discussed, is imaginary for this particular mode the more realistic field equations can be given the form

0TM fh

(7.6.5 a) ⎪⎩

⎪⎨

⎧

−≤≤≥

++−

−−=

dxd|x|

dxfor

)]xd(hexp[)dHcosh()xHcosh(

)]xd(hexp[)dHcosh(D)x(H

sf

f

cf

y

ϕϕ

ϕ

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

−≤++⎟⎟⎠

⎞⎜⎜⎝

⎛

≤−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

≥−−⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=

dx)]xd(hexp[)dHcosh(n1

d|x|for)xHcosh(n

1

dx)]xd(hexp[)dHcosh(1

D)x(E

sf2s

f2f

cfc

0x

ϕ

ϕ

ϕε

ωεβ

(7.6.5 b)

--- 999999 ---


⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

≤++⎟⎟⎠

⎞⎜⎜⎝

⎛

≤−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

≥−−⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−=

dx)]xd(hexp[)dHcosh(nh

d|x|for)xHsinh(n

H

dx)]xd(hexp[)dHcosh(h

Dj)x(E

sf2s

s

f2f

f

cfc

c

0z

ϕ

ϕ

ϕε

ωε (7.6.5 c)

where

⎪⎪

⎭

⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛=+

⎟⎟⎠

⎞⎜⎜⎝

⎛−=−

f

sfsf

f

cfcf

Hh

)dHtanh(

Hh

)dHtanh(

ηϕ

ηϕ

(7.6.6)

In the figure (7.6.1) through (7.6.3) we show the field distribution of the different non-zero TE field components in this structure corresponding to the same power level of . Similarly in figures (7.6.4) through (7.6.6) the non-zero field components for the modes are plotted. The guide thickness and the refractive indices of the different layers are written on the plots. The guide supports the TE and mode. The graphs show that the distribution of the TE modes are quite similar to the case of dielectric s. Figure (7.6.1) assures that the tangential electric field continues across the interfaces. Also from figure (7.6.2) we find that tangential magnetic field continues across the interfaces. Moreover because of same permeabilities of the three layers the normal component of the magnetic field

mW5.0TM

01 ,0 TMTE,

1TMSWG yE

zH

)B(H xx µ= is also continuous across the interfaces. The MATLAB programs for the three TE distributions is written in Prog. 18 of Appendix and those for the TM modes in Prog. 19.

Fig.7.6.1 : Field distribution of TE modes in metal cladded SWG In fact the continuity of the fields demand

--- 111000000 ---


(7.6.7 a) ⎪⎭

⎪⎬

⎫

===

−==−=

+−

+−

)dx(E)dx(E

)dx(E)dx(E

yy

yy

Fig.7.6.2 : Field distribution of TE modes in metal cladded SWG

Fig.7.6.3 : Field distribution of TE modes in metal cladded SWG

--- 111000111 ---


(7.6.7 b) ⎪⎭

⎪⎬

⎫

===

−==−=

+−

+−

)dx(H)dx(H

)dx(H)dx(H

xx

xx

and (7.6.7 c) ⎪⎭

⎪⎬

⎫

===

−==−=

+−

+−

)dx(H)dx(H

)dx(H)dx(H

zz

zz

For the TE fields we see that the electric fields merely penetrate the metallic cladding layer. For the

modes however the penetration is larger inside the metallic cladding layer. In the plots for the modes we see the surface nature of the mode. From the plots we see that the continuity of the

tangential magnetic field is satisfied at the two interfaces so also is of the tangential electric field . However due to difference in the permittivity of the three layers the normal electric field component

does not continue across the interfaces. In a nutshell the continuity equations demand

TMTM 0TM

yH zE

xE

(7.6.8 a) ⎪⎭

⎪⎬

⎫

===

−==−=

+−

+−

)dx(H)dx(H

)dx(H)dx(H

yy

yy

Fig. 7.6.4 : Field distribution of TM modes in metal cladded SWG

--- 111000222 ---


Fig.7.6.5 : Field distribution of TM modes in metal cladded SWG

Fig.7.6.6 : Field distribution of TM modes in metal cladded SWG

(7.6.8 b)

⎪⎪⎭

⎪⎪⎬

⎫

===

−==−=

+−

+−

)dx(E)dx(En

)dx(En)dx(En

xcx2

f

x2

fx2

s

ε

--- 111000333 ---


(7.6.8 c) ⎪⎭

⎪⎬

⎫

===

−==−=

+−

+−

)dx(E)dx(E

)dx(E)dx(E

zz

zz

777...777 PPPrrrooopppaaagggaaatttiiiooonnn lllooossssss aaannnddd ppprrrooopppaaagggaaatttiinnnggg llleeennngggttthhh::: i From these plots we see that the TE mode hardly penetrates the metal layer. The penetration for the TM modes are rather substantial. For the mode i.e. the plasmon wave the field penetrates most into the metal layer and subsequently the loss of the field energy is maximum for this mode. From the plots of

0TM

0kα in figure (7.5.3) and (7.5.6) also we see that the attenuation in the direction of propagation for the mode is almost one order of magnitude higher than the other modes. 0TM

Since α is itself the measure of the attenuation of the field amplitude in the direction of propagation the energy carried with the field in the longitudinal direction dies away at a rate of α2 . Therefore we define the propagation loss in dB as )lengthunitdB(68.8)e(log20Loss 10 αα ≈−= − A rough estimate of the propagation loss for the mode can be dawn from the curve (7.5.6). We may roughly select a value of

0TM3

0 105)k( −×≡α for the mode and with 0TM m633.0 µλ = this gives mNeper05.0 µα ≈ and hence the propagation loss is nearly mdB43.0 µ . The loss is strongly

dependent on the index difference between the guiding layer and the substrate layer and a plot may be derived for the variation of the loss with this index difference n∆ as shown in the figure (7.7.1). This plot is generated using Prog. 20 in Appendix. The plots show that for mode the loss is maximum of the lot and for any TM mode the loss is almost one order of magnitude higher than the loss for the corresponding

0TM

TE mode. The propagating length is defined as the distance the wave travels in the direction of propagation before it’s power content becomes e1 times that of it’s initial value. This length is therefore given by

α21LPR = (7.7.1)

Therefore with the previous configuration we find that the propagation length for the mode is nearly 0TM

m20µ . That is the surface plasmon wave can effectively travel a distance of only m20µ before getting highly attenuated. For all the other modes this length is larger than that. This means propagation loss is a problem for the surface plasmon mode for propagation. 0TM However the propagation loss will be different for different metals and also will depend on the wavelength used because for different wavelengths the refractive index or the permittivity of the same metal differs substantially. In the figure (7.7.2) we plotted the propagation losses for the mode at two special frequencies used for optical communication, (1)

0TMm633.0 µλ = , the Helium-Neon laser frequency and

(2) m6.10 µλ = , the Carbon di oxide laser frequency. We see that the loss is as much as two order of

--- 111000444 ---


Fig. 7.7.1 : Variation of propagation loss with index difference magnitude lower in case of the higher wavelength m6.10 µλ = . It is nearly mdB01.0 µ at that frequency for the metals used and the corresponding propagating length is nearly m500µ . Thus at higher wavelength the guide is enhanced in it’s range. The corresponding program is given in Prog. 21 of Appendix.

Fig . 7.7.2 : Propagation loss for different metals in metal cladded SWG

--- 111000555 ---


777...888 EEEffffffeeeccctttiiivvveee wwwiiidddttthhh ooofff ttthhhee oooppptttiiicccaaalll bbbeeeaaammm::: e From the field distribution curves for the TE and modes in figure (7.6.1) and (7.6.2) we see that all the modes but the mode exists over entire width of the guiding layer and decays outside it . The effective width of the beam can therefore be defined as before for these modes as in equation (4.3.1.28) and (4.3.2.17). The mode exists only on the interface between the cladding layer and the guiding layer and falls on both side of the interface. This brings forward he importance to define the effective width of the

mode differently. The distance within the guiding layer where the field falls substantially is

TM0TM

0TM

0TM 0TM yH

fH1 and that in the cladding layer is ch1 . Therefore the effective width of the beam is

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

cf0TM h

1H1W (7.8.1)

In figure (7.8.1) we show the variation of effective width for a few lower order TE and TM modes with the normalized width of the guiding layer. The plots for all the modes other than the show typical behavior as in the case of dielectric . The mode show peculiar behavior as it has very small width. The curve for this mode is insignificant at lower values of . We see from the curve (7.5.4) that the real

value of attains unity value at . The corresponding value of with and

is nearly . For less than this value

0TMSWG 0TM

dk2 0

b 75.0v ≈ dk2 0 523.2n 2f =

1n 2s = 2.1 dk2 0 1b < and hence 2

f2

0 n)k( <β . Thus the field does not behave as surface plasmon wave and is distributed over the entire width of the guiding layer just like the other modes. Therefore a critical value of guide thickness is essential for supporting the surface plasmon wave. For the present configuration with m633.0 µλ = this value is )2/(2.1d λπ=

. The corresponding program is in Prog. 22 of Appendix. nm60≈

Fig. 7.8.1 : Effective beam width of TE and TM modes in metal cladded SWG

--- 111000666 ---


To make the mode bound to the surface for all values of (d ∞→0 ) we must have such that sn

f

0d20

nk

≥⎟⎟⎠

⎞⎜⎜⎝

⎛

→

β

i.e. 2f2

sc

c2

s n)n|'(|

|'|n≤

−εε

or )n|'(|

|'|nn 2

fc

c2

f2s

+≤

ε

ε (7.4.7)

Under this condition the modal nature remains evanescent in both sides of the metal-guiding layer interface. 777...999 DDDiiissscccuuussssssiiiooonnn::: In this chapter we discussed about the Surface plasmon modes in asymmetrically metal cladded . We found that the structure supports both T

SWGE and modes but only the mode exhibits surface

bound nature. This particular mode has no cut-off, meaning that the mode exists even for diminishing thickness of the guide. But there is a critical value for the thickness which must be exceeded in order for the mode being a surface bound mode. At thicknesses lower than that all the other modes cease to exist but this particular mode continues with volume wave nature. Under this situation this guide may be used as a single mode structure but the mode radiates away in the transverse direction.

TM 0TM

Similar to the case of asymmetrically metal cladded structure the symmetrically metal cladded s also support modes. Since there are two metal-dielectric interface the structure supports two

modes. As the thickness of the dielectric guiding layer remains small these two modes couple to each other and to the other non modes that are present in the structure. As the thickness goes high the modes decouple and the result is generation of two Fano modes at two independent metal-dielectric single interface structures.

SWGSPP

SPP SPPSPP

--- 111000777 ---

CCChhhaaapppttteeerrr (((vvviiiiii iii)))

SSSuuurrrfffaaaccceee ppplllaaasssmmmooonnn pppooolllaarrriiitttooonnn wwwaaavvveee iiinnn a

s

SSSWWWGGG wwwiiittthhh mmmeeetttaaalllllliiiccc ggguuuiiidddeee ssaaannndddwwwiiiccchhheeeddd bbbeeetttwwweeeeeennn tttwwwooo dddiiieeellleeeccctttrrriiiccc lllaaayyyeeerrrsss

--- 111000888 ---


888...111 IIInnntttrrroooddduuuccctttiiiooonnn::: In this chapter we shall turn our attention to the modal analysis of with metallic layer sandwiched between two semi-infinite dielectric layers. We shall try to develop the detailed theoretical approach to the modes existing in such waveguide. We shall first consider the general asymmetric structure and then go to the symmetrically sandwiched structure. The dielectric constant of the metallic middle layer is complex with real part being large negative quantity and the dielectric constants of the other two layers are real positive as they are pure dielectric as per assumption.

SWG

Fig. 8.1.1 : Dielectric constant profile of metallic guide SWG With this kind of configuration also, we will see that the modal solution leads to existence of surface plasmon wave. Since the middle layer is metallic we can write the refractive index of the guiding layer as fn

|''|j|'|n fff2

f εεε −−== with |''||'| ff εε >> If we consider lossless condition neglecting the imaginary part we see the condition for guidance through the middle layer 2

s2

02

f n)k(n >> β is not being satisfied as is negative. Therefore conventional bound rays are not supported by this kind of waveguide. On the basis of ray picture it can be assessed that the condition of total internal reflection is never achieved in this waveguide and every incidence on the interfaces from the guiding layer will result in radiation of wave energy into the substrate and cladding layers. The only mode that can be supported by the structure, if it exists, is the mode for which the effective index exceeds the refractive index of the supporting layers because only then the wave amplitude dies away evanescently in the supporting layers going in the transverse direction. Now as discussed in the subsection (5.3) the transverse propagation constant in the guiding layer with lossless metal is large imaginary quantity. Therefore the only feasible mode that can exist in this structure is a mode for which the wave amplitude falls exponentially both ways from the interface. Thus the mode that can exist is essentially a surface bound wave i.e. Fano wave which hardly penetrates the metallic guide. Also it can be intuitively inferred that the mode must be TM mode. The reason for saying this is that for modes the only transverse field that exists is and since on ideal metallic

2fn

TE yE

--- 111000999 ---


surfaces the electric field comes to zero value, there is no sense in talking about TE surface bound modes which decays exponentially from the interface. We will see that there exists such TM surface wave solution in this configuration that can be guided along the interface. 888...222 CCChhhaaarrraaacccttteeerrriiissstttiiiccc eeeqqquuuaaatttiiiooonnn::: We once again start with the general eigenvalue equation

⎢⎣

⎡

−−+

+⎥⎦

⎤⎢⎣

⎡

−+=− −−

b1(b(tan

)b1(btanN)b1(v2 fc

1fs

1 ηηηπ ⎥⎦

⎤

))1 (8.2.1)

where )n(dkv 2sf0 −= ε

and )n(k)nkk(

b 2sf

20

2s

20

2z

−

−=

ε

with ηηη ,, fcfs redefined as

⎪⎩

⎪⎨

⎧

=)n(

1

2sf

fs

εη for mode

TM

TE

and

⎪⎩

⎪⎨

⎧

=)n(

1

2cf

fc

εη for mode

TM

TE

and )n|''|j|'(|

)n|''|j|'(|

)n(

)n(2

sff

2cff

2sf

2cf

++

++=

−

−=

εε

εε

ε

εη

Now if we consider |''||'| ff εε >> the definition of v leads to imaginary value. So we define a new quantity V as

jv)n(dkV f2

s0 −=−= ε (8.2.2) which, when approximated to the case of lossless metal reduces to a positive real quantity. Again according to the definition of b , leads to the cut off condition 0b = 2

s2

0z n)kk( = , and leads to the

condition

1b =|'|n)2kk( f

2f0z ε−== . This defines a non propagative wave in the z direction as, for

it β comes imaginary. Therefore for the mode to be supported in the metal guide, if we seek for a solution

s0 n)k( >β we must have as the denominator in the expression of is negative. So we define another normalized parameter

0b < b

--- 111111000 ---


b)n(k)nkk(

Bf

2s

20

2s

20

2z −=

−

−=

ε (8.2.3)

such that for the supported mode we must have with 0B ≥ 0B = as the cut off condition. Then in terms of the new parameters the equation (8.2.1) can be rewritten as

⎥⎦

⎤⎢⎣

⎡

+−+

+⎥⎦

⎤⎢⎣

⎡

++=+ −−

)b1()1B(jtan

)B1(BjtanN)B1(jV2 fc

1fs

1 ηηηπ

or ⎥⎦

⎤⎢⎣

⎡

+−+

−⎥⎦

⎤⎢⎣

⎡

+−−=+ −−

)B1()1B(jtanj

)B1(BjtanjjN)B1(V2 fc

1fs

1 ηηηπ

or ⎥⎦

⎤⎢⎣

⎡

+−+

−⎥⎦

⎤⎢⎣

⎡

+−−=+ −−

)B1()1B(tanh

)B1(BtanhjN)B1(V2 fc

1fs

1 ηηηπ (8.2.4)

Because of non-periodicity of the hyperbolic functions this equation does not any more lead to non-degenerate harmonic solutions as before and we need to absorb the term πjN inside the tan hyperbolic functions. We use the identity )xjNtanh()xtanh( += π to get the characteristic equation as

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡

+−+

+⎥⎦

⎤⎢⎣

⎡

+−=+ −−

)B1()1B(tanh

)B1(Btanh)B1(V2 fc

1fs

1 ηηη (8.2.5)

In terms of these normalized parameters the transverse propagation constants in the three layers can be rewritten as

BV)nkk(ddh 2s

20

2zs =−=

)1B(V)nkk(ddh 2c

20

2zc η−+=−=

and )B1(V)kk(ddjhdH f2

02

zff +=−=−= ε Then in terms of them the eigenvalue equation for theTM modes can be rewritten as

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛−= −−

f

cfc

1

f

sfs

1f H

htanh

Hh

tanhdH2 ηη (8.2.6)

--- 111111111 ---


or

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−= −

2c

c2

s

s

2

2f

f

2c

c2

s

s2

f

f

1f

nh

nhH

nh

nhH

tanhdH2

ε

ε

or 0nh

nhH

)dH2tanh(nh

nhH

2c

c2

s

s

f

ff2

c

c2

s

s

2

f

f =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

εε (8.2.7)

888...333 MMMaaaxxxiiimmmuuummm vvvaaalluuueee ooofff BBB::: DDDeeecccooouuupppllleeeddd aaannnddd cccooouuupppllleeeddd FFFaaannnooo mmmooodddeeesss::: l To find the value of as we rewrite the equation (8.2.5) as B ∞→V

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡

+−+

+⎥⎦

⎤⎢⎣

⎡

++−= −−

)B1()1B(tanh

)B1(Btanh

)B1(1V2 fc

1fs

1 ηηη (8.3.1)

As the right hand side of the equation (8.3.1) should also tend towards ∞→V ∞+ . This with 1B −≠ for desired mode leads to

−∞=⎥⎦

⎤⎢⎣

⎡

+−+

+⎥⎦

⎤⎢⎣

⎡

+−−

)B1()1B(tanh

)B1(Btanh fc

1fs

1 ηηη

Writing the arguments of the two arc tanh functions as and we get sZ cZ

−∞=⎟⎟⎠

⎞⎜⎜⎝

⎛++−

cs

cs1

ZZ1ZZ

tanh

or 1ZZ1ZZ

cs

cs −=⎟⎟⎠

⎞⎜⎜⎝

⎛++

or )ZZ1()ZZ( cscs +−=+ or 0)1Z)(1Z( cs =++ This leads to two solutions. For we get 1Z s −=

1)B1(

Bfs −=

+η

or 2fsB

11 η=⎟⎠⎞

⎜⎝⎛ +

or )1(

1B 2fs −

=η

(8.3.2)

--- 111111222 ---


For the TE modes if we put 1fs =η we get ∞→B i.e. ∞→20 )k( β which leads to a non

propagative solution as for this the phase velocity 0p =υ . For the TM modes we put =fsη )n( 2sfε

and get

)n)(n(

n)n(k)nkk( 2

sf2

sf

4sf

2s

202

s2

02

z+−

−=−

εε

ε

or )n(

nn)kk( 2

sf

4s2

s2

0z+

−=ε

or )n(

n)kk(

f2

s

f2

s20z ε

ε

+= (8.3.3)

Equating the real and imaginary parts on both sides and with the assumption |''||'| ff εε >> we get

)n|'(|

|'|n)k( 2

sf

f2

s20

−=

ε

εβ (8.3.4)

and 22

sf

f4

s2

0 )n|'(|

|''|n

k2

−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

ε

εαβ (8.3.5)

The other solution 1Zc −= leads in the similar way to

1)1(

B 2fc

2fc −−

=η

ηη (8.3.6)

For the TE modes this once again leads to ∞→20 )k( β i.e. non-propagative solution. For the

modes we get from equation (8.3.1) TM

)n(

n)kk(

f2

c

f2

c20z ε

ε

+= (8.3.7)

and hence

)n|'(|

|'|n)k( 2

cf

f2

c20

−=

ε

εβ (8.3.8)

22

cf

f4

c2

0 )n|'(|

|''|n

k2

−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

ε

εαβ (8.3.9)

Thus only modes are supported in this structure as predicted before. We see that two such modes can exist as . As i.e. the distance between the two interfaces the solution in equation (8.3.4) shows

TM∞→V ∞→V ∞→d2

2s

20 n)k( >β and that in equation (8.3.8) shows 2

c2

0 n)k( >β . These two solutions lead to two separate Fano waves. In figure (8.3.1) we show the variation of transverse magnetic field for these two quasi-Fano modes in a metal-guide . The plots are generated by the program Prog. 23 of Appendix. The red curve is for the mode (1) corresponding to equation (8.3.8) and the blue curve is for the mode (2) corresponding to equation (8.3.4). These field profiles show that the mode (1) is confined at the metal-cladding interface and the mode (2) is confined at the metal-substrate interface.

yH SWG

--- 111111333 ---


Both these modes are so sharply defined that they do not penetrate up to the other interface for even small finite thickness of the guiding layer. Thus these two Fano modes are completely decoupled of each other even for finite thickness of the guiding layer.

Fig. 8.3.1 : Decoupled Fano modes at the two interfaces of metal guiding SWG

Now as the thickness of the guiding layer decreases the two modes tend to couple to each other. Since there exist two final values of as we infer that there are two supported modes existing in the guide even for finite thickness of the guide. The effective index for the coupled modes is different from the two Fano modes and can be found by solving the characteristic equation numerically. From the field profile we see that coupling occurs when the thickness of the guide is less than the sum of the penetration of the two modes into the guiding layer. Numerical data about coupling is of admirable interest. We find that for the mode (1)

B 0V →

0.01455j6768.2)kk( 20z −=

met/Neper)105.9986j104.3267(H 57f ×+×=

and for the mode (2) j0.010546-2.27893)kk( 2

0z =

met/Neper)106.10848j104.2812(H 57f ×+×=

Adding the inverse of these two transverse propagation constants the effective penetration into the guide is obtained as 46nmm464650.0nPenetratio ≈= µ Thus a guide thickness of m1.0d2 µ= or even less will give rise to decoupled Fano waves existing at the two interfaces. As the thickness is made smaller than the penetration, coupling between the two modes occur and the effective index values for the two modes are altered due to interaction with the other interface whose presence is now significant. In figure (8.3.2) the normalized index profiles for the two modes are shown. The blue curve once again represents the mode (2) and the red curve corresponds to the mode (1). For increasing value ofV the values of for the two coupled modes reach the values given in equations B

--- 111111444 ---


(8.3.4) and (8.3.9) respectively. These steady values of represent the achievement of the decoupled modes as at this situation the effective index profile does not change with variation of or V . From the figure (8.3.2) we see constant values of for the two modes are reached at

Bd

B 5.2|V| ≈ which leads to the critical thickness m117.0d2 µ≈ . As the thickness or V is made smaller than this critical value, for the mode (2) increases and that for the mode (1) decreases to zero value.

B

Fig. 8.3.2 : B~V curves for the coupled modes in metallic guide SWG

Fig . 8.3.3 : Effective index profile for metallic guide SWG

The corresponding values of β and α are also plotted in the figure (8.3.3) and (8.3.4) respectively. The three plots are generated in Prog. 24 of Appendix. We see from the curves that with increase of thickness of the guiding layer the )k( 0β and the )k( 0α values for the mode (1) increase and that for the mode (2)

--- 111111555 ---


decrease to the asymptotic values governed by equations (8.3.4), (8.3.5), (8.3.8) and (8.3.9). The difference between the asymptotic values decreases as the asymmetry of the index of refraction between the cladding layer and the substrate layer decreases and coincide for symmetrically cladded guide. Symmetrically cladded metallic guide will be discussed in more detail a little bit later in this chapter. SWG

Fig . 8.3.4 : Loss profile for metallic guide SWG

888...444 FFFiiieeelllddd dddiiissstttrrriiibbbuuutttiiiooonnn::: CCCooommmpppllleeettteee ssseeettt ooofff fff iiieeelllddd eeeqqquuuaaatttiiiooonnnsss::: Since the field distribution of the transverse magnetic field inside the guiding layer is governed by propagation constant given by the field variation inside the guiding layer is given by the term

and it follows exponentially decaying nature unlike the case of dielectric s. Since the modes in this structure are TM to the direction of propagation the other two non-zero components of the field distribution are and given by equations (3.4.16).

)x(H y

fH )jh( f−

)xHcosh( f SWG

xE zE Therefore the field equations for the modes can be written taking recourse of equations (4.3.2.6) as

⎪⎩

⎪⎨

⎧

−≤≤≥

++−

−−=

dxd|x|

dxfor

)]xd(hexp[)dHcosh()xHcosh(

)]xd(hexp[)dHcosh(D)x(H

sf

f

cf

y

ϕϕ

ϕ (8.4.1 a)

--- 111111666 ---


⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

−≤++⎟⎟⎠

⎞⎜⎜⎝

⎛

≤−⎟⎟⎠

⎞⎜⎜⎝

⎛

≥−−⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=


d|x|for)xHcosh(1


D)x(E

sf2s

ff

cf2c

0x

ϕ

ϕε

ϕ

ωεβ (8.4.1 b)

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

≤++⎟⎟⎠

⎞⎜⎜⎝

⎛

≤−⎟⎟⎠

⎞⎜⎜⎝

⎛

≥−−⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=


d|x|for)xHsinh(H


Dj)x(E

sf2s

s

ff

f

cf2c

c

0z

ϕ

ϕε

ϕ

ωε (8.4.1 c)

The continuity of the magnetic field has already been applied across the interfaces. Application of the continuity of the tangential electric field at the interfaces yield zE

⎪⎪

⎭

⎪⎪

⎬

⎫

⎟⎟⎠

⎞⎜⎜⎝

⎛=+

⎟⎟⎠

⎞⎜⎜⎝

⎛−=−

f

sfsf

f

cfcf

Hh

)dHtanh(

Hh

)dHtanh(

ηϕ

ηϕ

(8.4.2)

In figure (8.4.1) the field variation of for the two coupled modes are shown. The corresponding variations of and are also shown in figures (8.4.2) and (8.4.3) respectively. The corresponding program to generate these plots is Prog. 25 of Appendix. The red curve is for the mode (1) and the blue curve for the mode (2) once again. In order to achieve coupling the guide thickness is made as little as

. We can see the dissimilarity between the field distributions of the two modes. The red curve has two peaks at the two interfaces which have the same sign. The difference between the two peaks is owing to the asymmetry of the guide. If the guide were symmetric the two peaks would have been of same amplitude. Thus the red curve leads to the Even mode as we approach symmetry. Also as the guide thickness increases this mode leads to the single interface mode existing on the metal-cladding interface (red curve in figure (8.3.1)). The other mode represented by the blue curve leads to the Odd symmetric mode as we approach symmetry because the peaks have opposite signs. And this is the mode that leads to the single interface mode existing on the metal-substrate interface (blue curve in figure (8.3.1)) as the guide thickness is increased. The Even and Odd modes will be discussed in a little more detail in the subsequent proceeding in context of symmetric guide.

)x(H y

xE zE

nm80

The field profiles reveal that the tangential magnetic field yH )B( y µ is continuous across the interfaces, so also is the tangential (To the direction of propagation) electric field . But the transverse electric field is discontinuous across the interfaces. In fact for this component we must have according to the conditions of continuity

zE

xE

--- 111111777 ---


Fig . 8.4.1 : Field distributions of the TM modes in metallic guide SWG


)dx(E)dx(En xfx

2s +− −==−= ε

and )dx(E)dx(En xfx2

c −+ === ε

--- 111111888 ---



888...555 CCCuuuttt oooffffff:::

We see from the normalized profile that the real value of decreases to 0 for the mode (1) as the normalized thickness V decreases to zero. Therefore a critical value of V must be exceeded in order to support the mode (1) otherwise the effective index will be less than and the wave will radiate out. This cut value of V for the mode (1) can be found out putting

B

sn0B = eigenvalue equation (8.3.1) and we get

{ })1(tanh21V fc

1c ηη −−= − (8.5.1)

Since for lossless metal fcη is pure real negative quantity and 1<η is also pure real

{ })1(tanh fc1 ηη −− is negative real, giving feasible real positive cut-off value of . However in

practical case, is complex with positive real part and the imaginary part being insignificantly small. This means unlike the case of metal cladded there exists a finite cut off thickness of the guiding layer which must be exceeded for propagation of the plasmon wave. A numerical example can be cited. With the configuration taken in the figures (8.3.3) through (8.3.4) we have

cV

cVSWG

j0.000160.9946 +=η 0.24609j -7.41818 -=fcη and hence

0.00719j +0.30466=cV which yields the guide thickness

--- 111111999 ---


m)101.42261j101.4226(d2 -10-8c µ×+×=

that is the guide thickness must exceed in order to support the mode (1). For guide thicknesses lower than this, the guide ceases to support the mode (1). The mode (2) is still supported, but it has a high loss.

nm15

The condition of guidance imposes a lower limit on the cladding layer refractive index. We see that for the mode (1) given by equation (8.3.8) 2

c2

0 n)k( >β is confirmed but it does not confirm 2

s2

0 n)k( >β unless exceeds a lower limit, given by the condition cn

2s

f2

c

f2

c n)n(

n≥

+ ε

ε

or )n(

nn 2

sf

f2

s2c

−≥

ε

ε (8.5.2)

or for lossless metal

)n|(|

||nn 2

sf

f2

s2c

+≥

ε

ε (8.5.3)

If this condition is not satisfied field energy corresponding to the mode (1) flows out through the substrate layer. But for large guide thickness as the mode becomes decoupled of the substrate-metal interface and hardly reaches the metal-substrate interface, it’s radiating out through that remote layer is meaningless. However it cannot radiate through the metal-cladding interface as 2

c2

0 n)k( >β is confirmed. This is the condition to be satisfied for large guide thickness i.e. for the decoupled modes. For the coupled modes the value of )k( 0β is even less than the value given in equation (8.3.8) which leads to larger value that should exceed as we decrease the guide thickness. A plot of variation of real cn )k( 0β with index of refraction of the cladding layer is shown in the figure (8.5.1) for different guide thicknesses. These plots are generated by the program Prog. 26 of Appendix. Here we have taken 3.2ns = and 5414.0j32.16f −−=ε i.e. at the free space wavelength Ag

m633.0 µλ = . The dotted red curve represents the allowable limit of the effective index for being a surface mode. As long as the mode becomes cut-off if sc nn ≤ s0 n)k( =β . But if we allow the condition of cut-off turns into

sc nn >

c0 n)k( =β because then, if )k( 0β comes below the modal energy radiates through the cladding layer. The red dotted curve represents as long as and represents as . Therefore the allowed region of values of

cn

sn sc nn ≤

cn sc nn > )k( 0β is the region above the red dotted curve. From the curves we see that there exists a critical lower limit of so that the surface plasmon mode exists and as the thickness is increased a larger asymmetry is needed to cut - off the mode.

cn

--- 111222000 ---


Fig. 8.5.1 : Variation of effective index for the LRSPP with r.i. of cladding layer 888...666 MMMooodddeee ssspppooottt sssiiizzzeee::: The effective thickness up to which the field exists in the transverse direction penetrating the dielectric layers is defined as the given, as discussed earlier, as size spotMode

Fig. 8.6.1 : Mode spot size vs. r.i. of cladding layer at different guide thicknesses

--- 111222111 ---


⎟⎟⎠

⎞⎜⎜⎝

⎛++=

cs h1

h1d2W

The variation of the with the refractive index of the clad is shown in the figure (8.6.1). The refractive index of the clad has been allowed to exceed once again and all other components are kept identical to the case of figure (8.5.1). Both these plots are generated in the same program, Prog. 26 of Appendix.

MSSsn

We see that the goes to a large value as we approach towards both the cut-off values. It is because for lower value of cut-off occurs as

MSScn s0 n)k( =β which leads ∞→sh and in the region

cut-off occurs as sc nn > c0 n)k( =β which leads ∞→ch . At the point of cut-off the mode is infinitely large into the higher index media and we consider the mode cut-off. It is essentially a plane wave traveling in that medium under such situations. Therefore the mode is only of a practical size for a limited range of . cn 888...666 PPPrrrooopppaaagggaaatttiiiooonnn lllooossssss ::: LLLooonnnggg RRRaaannngggeee aaannnddd SSShhhooorrrttt RRRaaannngggeee SSSPPPPPP:::

We see from the figure (8.3.4) that for the mode (2) the attenuation constant increases as we decrease the guide thickness whereas for the mode (1) the reverse happens. Thus if the guide thickness is kept sufficiently small the mode (2) becomes very much lossy in the direction of propagation but for the mode (1) attenuation constant becomes extremely small. This gives very small propagation loss and hence very large propagating length for the mode (1) compared to the mode (2). That is why the mode (2) is called the

and the mode (1) the . An estimate of the loss can be done roughly from the graphs for

SPPRange Short SPPRange Long )k( 0α . With a guide thickness as small as the value of nm16 )k( 0α

Fig . 8.7.1 : Propagation loss for LRSPP in metallic guide SWG

--- 111222222 ---


is of the order of for the which with 510 − SPPRange Long m633.0 µλ = gives a loss of the order of mmdB1 to mmdB10 whereas that for the with typical value SPPRange Short

10 10)k( −=α comes out to be mdB10 µ . Even for a standard metal cladded the loss was

typically below SWG

mdB1 µ . Thus a thin strip of metal between two dielectric layers allows for which the loss is significantly small compared to the other cases. In figure (8.7.1)

the variation of propagation loss with the guide thickness is shown for the in case of various metals with the configuration of the other two layers being kept unchanged. The corresponding program is Prog. 27 of Appendix.

SPPRange Long SPPRange Long

The propagating length being defined as is typically in the sub-millimeter to millimeter range for the

and has a negatively sloped variation with increasing guide thickness.

1)2( −αSPPRange Long

Fig. 8.7.2 : Variation of loss of LRSPP with r.i. of cladding layer The propagation loss for the with the refractive index of the cladding layer at different guide thicknesses are plotted in figure (8.7.2). Here we allowed exceeding so that the condition of cut -off once again becomes

SPPRange Long

cn sn

sc0 norn)k( =β whichever is larger. The other data are identical to the data taken for the figure (8.5.1). This curve is also an outcome of the program, Prog. 26. We see that the loss increases with the increase of the guide thickness. At the cut-off points, the attenuation goes to zero as all of the power resides in one of the dielectric layers, and is not absorbed by the metal. The two points of cut-off broaden up as we increase the guide thickness.

--- 111222333 ---


888...888 SSSyyymmmmmmeeetttrrriiiccc ggguuiiidddeee::: EEEvvveeenn aaannnddd OOOdddddd sssyyymmmmmmeeetttrrriiiccc mmmooodddeeesss::: u n The symmetrically cladded metallic guide is defined by the condition SWG cs nn = . This makes the previous equations look simpler as under this symmetric configuration fcfs ηη = , and cs hh = 1=η . These lead the eigenvalue equation (8.3.1) to the simpler form of as

0nhH

2)dH2tanh(nhH

2s

s

f

ff

2

2s

s

2

f

f =⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

εε (8.8.1)

or [ ]⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

=+ 2

2s

s

2

f

f

2s

s

f

f

f2

f

nhH

nhH

)dH(tanh1)dHtanh(

ε

ε

This on simplification leads to

0h

H1)dHcoth(hH1)dHtanh(

s

f

fsf

s

f

fsf =

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

ηη

Thus in a symmetrical guide also there exist twoTM solutions namely

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=

s

f

fsf h

H1)dHtanh(η

(8.8.2 a)

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

f

sfsf H

h)dHtanh( η (8.8.2 b)

In normalized form also we may write the equations as

[ ]B

)B1(1)B1(Vtanhfs

+⎟⎟⎠

⎞⎜⎜⎝

⎛−=+η

(8.8.2 a)

[ ])B1(

B)B1(Vtanh fs +−=+ η (8.8.2 b)

In figure (8.8.1) the variation of with V for the two modes are shown for a specific guide with guiding layer made of covered by dielectric of refractive index

BAg 3.2ns = . The corresponding variations of

)k( 0β and )k( 0α with guide thickness are also plotted in the figures (8.8.2) and (8.8.3) respectively. These three plots are generated by the program Prog. 28 of Appendix. The red curves and blue curves once again represent the mode (1) and mode (2) as in case of the general asymmetric guide. The difference between the dispersion profiles for the symmetric guide with the asymmetric guide is that as the guide thickness increases the values of )k( 0β and )k( 0α merge with each other leading to two identical single interface solutions corresponding to two separate metal-dielectric

--- 111222444 ---


interfaces. Another important point about the symmetrical guide is that there is no cut off condition for the guide to support the modes. For the mode (2) is always positive and for the mode (1) reaches zero value at . This can also be justified analytically if we recall equation (8.3.1) and put

B B0V = 1=η owing to

symmetry.

Fig. 8.8.1 : B~V plot for symmetric guide with metallic guiding layer

Fig. 8.8.2 : Effective index variation in a symmetric metallic guide SWG

This gives . This means as we let the guide thickness the modes still continue to propagate. However since the value of for the mode (2) at

0Vc = 0d2 →B 0V = is not still determined we try to

estimate it now. Since the left hand side of equation (8.8.2 a) is zero unless . For any 0V = ∞→B

--- 111222555 ---


positive finite value of the right hand side cannot go to zero. Therefore the only possible solution that can match the both sides of the equation (8.8.2 a) is

B∞→B . This leads to the limiting value condition that

must be satisfied by the mode (2)

⎟⎟⎠

⎞⎜⎜⎝

⎛−=+ −

∞→→

fs

1

B0V

1tanh)B1(V2Limη

(8.8.3)

Fig. 8.8.3 : Variation of attenuation const. in a symmetric metallic guide SW

Fig. 8.8.4 : Distribution of transverse magnetic field in the symmetric guide

--- 111222666 ---


For example at with0001.0V = m633.0 µλ = , and 3.2n 2s = )5414.0j32.16(f −−=ε

computation yields and hence 45 1038.31002.5B ×−×≈ 4.58)k( 0 ≈α i.e. the propagation loss is nearly nmdB5 .

Fig. 8.8.5 : Variation of propagation loss with guide thickness for Even symmetric mode

Fig. 8.8.6 : Variation of propagation loss with guide thickness for Odd symmetric mode

--- 111222777 ---


Thus at small guide thickness there is huge propagation loss for the mode (2) as seen from it’s loss profile. As long as the guide thickness is kept small enough to achieve coupling, the mode (2) is the

and the mode (1) the one. In figure (8.8.4) the transverse magnetic field in the guide for the two modes are shown. These field distributions are generated using the program Prog. 29 of Appendix. The mode (1) (red) shows the even symmetric distribution in the transverse direction and the mode (2) (blue) shows the odd symmetric distribution. So the mode (1) is also called the symmetric or Even symmetric mode and the mode (2) the anti-symmetric or Odd symmetric mode.

SPPRange Short Range Long

It is worth mentioning that from the plots of losses in figure (8.8.3) we find the losses for the anti-symmetric mode is larger than the symmetric mode. In figure (8.8.5) and (8.8.6) we plot the variation of the loss in dB against the guide thickness for both the modes in case of a few metals at two specific wavelengths. The corresponding program is given in Prog. 30 of Appendix. We see that in case of both these modes the loss enormously decreases as we use higher wavelengths. With a typical guide thickness of the loss for the even mode at the He-Ne laser frequency

nm10d2 =m633.0 µλ = the loss is of the order of a few

hundreds of dB per centimeter whereas for the Carbon di oxide laser frequency m6.10 µλ = it is in the range of less than cmdB1 . In case of the odd symmetric mode however the loss is enormous, as high as

cmdB10 5 for the lower wavelength and cmdB100 for the higher wavelength. The corresponding propagating length for the two modes are respectively of the order of a few millimeters and a few microns at the lower wavelengths and a few tens of centimeters and a few hundred microns for the higher wavelength used. So from this point of view the even mode is the ‘supported’ mode in the symmetrical guide with metallic middle layer. However the loss can also be controlled by choosing suitable refractive

Fig. 8.8.7 : Variation of propagation loss with clad r.i. for symmetric mode

index of the symmetrical supporting layers. This is because the decoupled values of attenuation constant depend on the refractive index of these layers. In figure (8.8.7) the variation of loss with the refractive index of the cladding layers for the symmetric mode is plotted for different guide thicknesses at m633.0 µλ = in case of . We see that the loss decreases as we decrease the optical density i.e. refractive index of the Ag

--- 111222888 ---


cladding layers placed symmetrically around the guide. The plot is generated by the program Prog. 31 of Appendix.

Ag

888...999 DDDiiissscccuuussssssiiiooonnnsss:::

This chapter deals with perhaps the most efficient structures using metals. We see that this structure does not support T

SWGD2E modes. Furthermore it supports only two TM modes which are both

Surface plasmon modes in nature. These two modes show different characteristics. While for the antisymmetric mode the attenuation increases sharply with diminishing thickness of the metal strip the even mode shows decreasing loss. Therefore the thickness of the metal strip can be made just above the cut-off of the symmetric mode to support it while the other mode ceases to propagate due to large longitudinal loss. Therefore this structure effectively can give rise to single mode behavior at small thicknesses. Also compared to the case of the metal cladded guide discussed in the previous structure the metal strip thickness required is much smaller, in the nanometer scale. The metals used in the structure are all precious metals and therefore the requirements for such metals need to be minimized. This structure ensures minimal use of guiding metal. The metal thickness can further be made even smaller for the symmetrically dielectric covered structure where the symmetric mode does not have any cut-off.

--- 111222999 ---

CCChhhaaapppttteeerrr (((iiixxx)))

MMMooodddaaalll aaannnaaalllyyysssiiisss ooofff ttthhhiiinnn mmmeeetttaaalll ssstttrrriiippp ooofff fffiiinnniiittteee wwwiiidddttthhh aaasss oooppptttiiicccaaalll wwwaaavvveeeggguuuiiidddeee ::: EEEffffffeeeccctttiiivvveee

dddiiieeellleeeccctttrrriiiccc cccooonnnssstttaaannnttt mmmeeettthhhoooddd

--- 111333000 ---


999...111 IIInnntttrrroooddduuuccctttiiiooonnn::: Up to this point we discussed all about D2 slab waveguides. The guides were assumed to be extending infinitely in the y direction and the interface planes were perpendicular to the transverse direction. Light confinement in such structures takes place in the direction only. But all practical optical waveguides are designed to achieve light confinement in the

xx

y direction also in addition to confinement along the depth. Such waveguides are from this perspective. Such geometrical structures impose additional boundary conditions on the modes and in general the Maxwell’s equations cannot be solved analytically as such for these waveguide structures unless some approximate tools are applied. In this chapter we will try the effective dielectric constant method for analyzing the modes in a very special kind of like waveguides, the thin metallic strip embedded into dielectric surroundings substrate.

D3

D3

999...222 GGGeeeooommmeeetttrrryyy oooff ttthhheee ssstttrrruuuccctttuuurrreee::: f In the figure we schematically draw the structure of an asymmetric metal strip waveguide.

Fig. 9.2.1 : geometry of the thin metal strip structure Here we see that the structure is finite in the direction as well as x y . The thickness of the metal strip of dielectric constant fε is and the width in the t2 y direction is no more infinite but where W2 Wt < . The substrate is usually of the highest dielectric constant sε and the dielectric constant of the cover region is cε i.e. cs εε > whereas fε is usually complex with large negative real part with small imaginary contribution. The origin of the co-ordinate system for the sake of convenience is taken at the centre of the metal strip. Here the thin metal strip is assumed to be embedded into the layer of lower dielectric constant. In the other cases the strip can be in the substrate. Here the structure is symmetric about the axis but asymmetric about the

xy axis. Therefore it is one of the asymmetric metal strip structures. However

symmetric metal structures can be considered also in which the thin metal strip is assumed to be embedded into a dielectric medium surrounding it from all directions.

999...333 GGGeeennneeerrraaalll dddiiissscccuuussssssiiiooonnn ooonnn mmmooodddeeesss iiinnn 333DDD wwwaaavvveeeggguuuiiidddeeesss::: In general four of the six field components for an arbitrary electromagnetic mode can be represented in terms of the rest two, which are the components in the direction of propagation, namely and as shown in equations (2.4.34). They are once again recalled in this section.

zE zH

--- 111333111 ---


⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−=x

Ey

Hk

jE zz02

cx βωµ (9.3.1 a)

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=y

Ex

Hk

jE zz02

cy βωµ (9.3.1 b)

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=x

Hy

Ek

jH zzcr02

cx βεωε (9.3.1 c)

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−=y

Hx

Ek

jH zzcr02

cy βεωε (9.3.1 d)

In case of D2 waveguides the guide dimensions were supposed to be infinite in extent in the y direction. This imposed a condition on the modes that there were no field variations in the y direction. Therefore in case of D2 waveguides the equations reduce in size as we set 0)x()x( 22 ≡∂∂≡∂∂ . For the TE modes we additionally set which ultimately leaves only three non-zero components,

and . Thus only electric field that exists is . For the TM modes we set which leaves the non-zero field components and . The transverse (to the direction of propagation) electric field component is the field and it is the dominant mode.

0Ez =)x(H),x(H xz )x(E y )x(E y

0H z = )x(E),x(E xz )x(H y

)x(E x

However, in case of waveguides or alternatively called channel waveguides we do not have the liberty to set

D30)x()x( 22 ≡∂∂≡∂∂ as there is confinement from the direction of y axis also and the

equations (9.3.1) cannot be reduced to any degree of brevity in general. Microwave rectangular waveguides with perfectly conducting metallic walls can support pure TE or TM modes. On the contrary in optical

waveguides surrounded by different dielectric materials, pure TD3 E or TM modes are not supported and two families of hybrid modes exist. These modes are principally polarized in the plane perpendicular to the direction of propagation, i.e. in the or x y direction. These guided modes are therefore classified depending on whether the main electric field component lies in the or in the x y direction. The mode having the main electric field component resemble the TM modes in xE D2 slab waveguides and consequently they are called the TM like modes or the modes. Similarly the are those which have the main electric field component in the

xmnE y

mnEy direction, i.e. . They are the TyE E like modes in the slab

waveguides. The propagation constant in the case of waveguides can in general be written as D3 2

z2

y2

x2 kkkk ++=

or (9.3.2) 2y

2x

20

2z kkkk −−= ε

In case of lossless media we therefore can write the propagation constant in the direction of propagation

)kkk(k 2y

2x

20z −−== εβ (9.3.3)

--- 111333222 ---


whereas for lossy medium we can write )j(kz αβ −= In case of the D2 slab waveguides the direction y was not bounded and this led to the condition

, but in the present case we cannot set 0k y = 0k y = as there is boundary in the y direction and consequently there is field variation in the y direction. In general the boundary value problem cannot be solved accurately for the waveguides analytically. However there are approximate solutions that enable one to obtain analytical solutions in closed form. But these approximate solutions are effective far from the cut-off condition and as long as the aspect ratio

D3

)tW( is larger than unity. We shall be considering one of the most widely used approximate methods the Effective Dielectric Constant Method for our analysis. 999...444 MMMooodddaaalll aaannnaaalllyyysssiiisss uuusssiiinnnggg eeefffffeeeccctttiiivvveee dddiiieeelleeeccctttrrriiiccc cccooonnnssstttaaannnttt mmmeeettthhhoooddd::: f l The Effective dielectric constant (EDC) method, alternatively called the Effective index method was first developed by Knox and Toulis for image guides. It has subsequently been applied to other channel structures also. This approximate method will also be applied to the structure we are concerned about in the present context. We recall the equation (9.3.3) once again. Had we have 0k y = we know, in that case the propagation constant in the direction of propagation would have been

)kk(k 2x

20zs −= ε

or the effective refractive index for the wave in the medium of dielectric constant ε would be 2

0x0zseffeff )kk()kk( −=== εεη (9.4.1) If we put this in equation (9.3.3) once again we get the propagation constant in the z direction )kk(k 2

y2

eff2

0z −= η

)kk( 2yeff

20 −= ε (9.4.2)

This method is the heart of effective dielectric constant method for modal analysis of waveguides. The way we first neglect for the evaluation of the effective dielectric constant

D3yk effε effectively simulates the

D2 slab structures we considered in the previous chapters because only semi-infinite extent in the y direction gives zero variation in the y direction and hence 0k y = . Then again using the effective

dielectric constant effε in the equation (9.4.2) effectively means that we have another D2 slab structures having boundaries parallel to the y axis and the central layer is of dielectric constant effε .

--- 111333333 ---


Therefore we now outline the method of using the method to the structure of the present context. EDC 1. First we divide the guide cross section into constituent regions. Since the guide is symmetric about the y axis we divide the guide cross section in the three regions as shown in the figure (9.4.1 (a)). The region

containing the metallic strip is assumed to extend to infinity in the y direction as shown in the figure (9.4.1 (b)). This results in a rectangular D2 slab waveguide with thickness having boundary only in the

direction. t2

x If for this virtual planer structure the propagation constant in the direction of propagation be zsβ the decay constants in the dielectric regions sε and cε and the same in the metal region are given by

Fig. 9.4.1 Analytical models for the thin metallic strip structure for EDC method

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

−=

−=

−=

21f

20

2zsfx

21c

20

2zscx

21s

20

2zssx

)k(h

)k(h

)k(h

εβ

εβ

εβ

(9.4.3)

where ''j' fff εεε −−=

--- 111333444 ---


The structure we derived is identical to the structure we analyzed in chapter 8. We know that this structure does not support TE solutions. Only two TM solutions are supported by this structure. The characteristic equation for the hypothetical structure can therefore be written as in the chapter 8 as

0hhh

)th2tanh(hhh

c

cx

s

sx

f

fxfx

c

cx

s

sx

2

f

fx =⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

εεεεεε (9.4.4)

This equation can be solved for the hypothetical propagation constant in the direction of propagation zsβ . There are two possible solutions for zsβ which give rise to two surface Plasmon modes, one called the even or symmetric mode and the other the odd or antisymmetric mode. They have been discussed in detail in the previous chapter. For small thicknesses the two modes are coupled together and dispersion occurs. As the thickness increases the two modes get decoupled and lead to two separate Fano waves with

[ ])(

kfs

fs0zs εε

εεβ

+=∞

and [ ])(

kfc

fc0zs εε

εεβ

+=∞

respectively for the odd and even mode. The field components for these virtual modes are and with dominating. )x(E),x(E xz )x(H y )x(E x

2. Now we replace the region I of the actual structure in figure (9.4.1 (a)) by the effective dielectric constant 2

0zseff )k()t2( βε = (9.4.5) Thus we get another equivalent planer structure as shown in the figure (9.4.1 (c)). This equivalent guide is now bounded in the y direction. The dielectric constant of the middle layer is written as )t2(effε as it is a function of the guide thickness . Since the metal strip is embedded in the dielectric with lower dielectric constant the surrounding dielectric constant of the equivalent guide is

t2cε and the structure is symmetric

about the axis. x The virtual guide of figure (9.4.1 (b)) supports only two Plasmon modes for each of which the effective dielectric constant is different. Therefore for each of the Plasmon modes in the previous structure we consider separate effective dielectric constants for the guiding layer and these effective dielectric constants are implicit functions of the guide thickness. The effective index for the middle layer in the equivalent structure is now a complex quantity with the real part positive and the imaginary part signifying propagation loss. If the metal in the strip had been lossless it would have led to purely real effective index. For the modes in the structure (9.4.1 (b)) above cut-off the effective refractive index of the middle layer comes out greater than the surrounding mediums. Therefore in the equivalent guide of figure (9.4.1 (c)) the middle layer is of higher refractive index than the symmetrically placed cover. This leads to the conclusion that for both cases of the Plasmon modes (even or odd), the equivalent guide behaves like symmetric dielectric D2 slab waveguide discussed in the chapter 4. This structure supports both TE and TM modes which are volume waves in nature spreading into the entire middle layer with sinusoidal field distribution in the transverse direction ( y direction). For the TE case the

--- 111333555 ---


main electric field component is in the direction now while for the TM case it is in the x y direction. But the Plasmon modes in the structure (9.4.1 (b)) had it’s main electric field component in the direction. So for the equivalent structure in figure (9.4.1 (c)) we shall consider only the T

xE modes.

Therefore in total, the modes that exist in the structure have all it’s field components in general with predominant polarization in the direction. Therefore the modes are hybrid and as per notation they are

like modes or the modes.

D3x

TM xmnE

At this point we can further introduce to another special class of notation for the modes. Let us consider the equivalent guide with the effective dielectric constant that corresponds to the even mode in the previous structure (Fig. 9.4.1 (b)). Since the main transverse electric field for the even mode is symmetric-like about the y axis, we put a letter ‘s’ to the modes existing in the equivalent guide. The TE modes in the corresponding symmetric dielectric in figure (9.4.1 (c)) may be classified as symmetric or antisymmetric depending on the symmetry of the main transverse electric field component about the axis. Therefore another letter ‘s’ or ‘a’ may be used prior to the earlier letter to designate them. Therefore the modes in the complete structure may be designated as etc. Similarly in case of the odd mode in the structure of figure (9.4.1 (b)) the modes in the guide may be denoted as

etc. The subscript denotes whether the modes are bound modes or leaky, here bound. The superscripts represent the order of the T

SWGx

1b

0b

1b

0b as,as,ss,ss

D31b

0b

1b

0b aa,aa,sa,sa b

E modes in the equivalent guide structure in figure (9.4.1 (c)). The fundamental modes are the and the modes. 0

b0b

0b as,sa,ss 0

baa Now the propagation constant in the direction of z axis, β , is given by the solution of the eigenvalue equation for TE modes in the equivalent guide structure as

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

cy

fyfy h

h)Whtan( (9.4.6)

for the odd TE modes and

⎟⎟⎠

⎞⎜⎜⎝

⎛=

fy

cyfy h

h)Whtan( (9.4.7)

for the even TE modes. Here the propagation constant in the middle layer is

2eff

20fy )t2(kh βε −= (9.4.8)

and the decay constant in the region surrounding it is

c2

02

cy kh εβ −= (9.4.9) For a specific guide thickness the effective dielectric constant t2 effε can be found out by solving the equation (9.4.4) iteratively and using the value of effε in the equation (9.4.6) or (9.4.7) the actual propagation constant β in the direction of propagation for the odd or even TE respectively can be found out once again by iterative technique.

--- 111333666 ---


999...555 NNNuuummmeeerrriiicccaaalll rrreeesssuullltttsss uuusssiiinnnggg ttthhheee eeeffffffeeeccctttiiivvveee dddiiieeellleeeccctttrrriiiccc cccooonnnssstttaaannnttt mmmeeettthhhoooddd::: u For numerical evaluation we use as usual the Newton-Raphson iteration technique of solving transcendental equations. In the present case we consider the metal strip width to be m1W2 µ= . The substrate layer is of real dielectric constant 0.4s =ε and the cladding layer is with 91.3c =ε . The metal used is with

Ag53.0j19f −−=ε and the free space wavelength of the optical wave is m633.0 µλ = . Here the

dielectric constant of at Ag m633.0 µλ = has been taken to be different from the value previously used. This is because we are going to see whether the results obtained by the method comes in good agreement with the reliable results reported. The results (reference paper 11 of the list at the end of this report) we are taking as precursor are those, reported by Pierre Berini and in these reports the dielectric constant of at

EDC

Ag m633.0 µλ = has been taken to be 53.0j19f −−=ε . The other value of )Ag(ε we used elsewhere was 32.16f −=ε 5414.0j− which was taken from the reference paper

8. The variation of the normalized propagation constant )k( 0β is determined by varying the thickness of the strip. For each value of the effective dielectric constant of the middle layer is determined using the equation (9.4.4). For both the symmetric and the antisymmetric modes the effective dielectric constant is determined separately. Using these values of

t2

effε in equation (9.4.6) and (9.4.7) the actual propagation constant β is determined. In the figure (9.5.1 (a)) the dispersion profile for the four fundamental modes are plotted according to the effective dielectric constant method. The corresponding loss profiles are also plotted in the same figure 9.5.1 (b). Both these set of plots are generated using the program Prog. 32 of Appendix. The Black curves represent the even and odd modes in the corresponding having the same thickness but width infinite. These modes can be denoted as the and the mode respectively which are the Plasmon modes discussed in the previous chapter. The Blue curves represent the and the modes and the Red curves represent the and the modes respectively. For the and the modes the guiding layer dielectric constant is the effective dielectric constant corresponding to the antisymmetric mode in the corresponding infinite width structure and let this be denoted as

SWGD2

bs ba

0bsa 0

bss 0bas 0

baa0bsa 0

baa

)t2(oddε . Then for the equivalent secondary structure the normalized propagation constant

)k( 0β for all the pure TE modes lies within cε and oddε . The mode is the fundamental

mode in the symmetric

0bsa

D2 structure in the equivalent configuration and the mode is the next higher mode for a given width of the equivalent guides (here

0baa

m1W2 µ= ). The normalized propagation constant )k( 0β is always higher for mode. On the other hand if 0

bsa )t2(evenε be the effective dielectric constant corresponding to the symmetric structure with infinite width the normalized propagation constant

)k( 0β for the and the mode always lie within the range between0bss 0

bas cε and evenε . And for

the fixed width m1W2 µ= the )k( 0β value corresponding to the mode is higher than the mode due to the same reason.

0bss 0

bas

In the figure (9.5.1 (b)) the loss corresponding to the four fundamental modes are plotted. The curves for the

and modes coincide with each other on the loss profile corresponding to the mode in the equivalent structure with infinite width. Similarly the loss curves for the and modes coincide with each other on the loss profile corresponding to the mode in the equivalent structure with infinite width. Thus according to the assumption there is no prominent effect of shortening the width on the loss

0bas 0

bss bs0bsa 0

baa

baEDC

--- 111333777 ---


profile for the structure. Only effect the finite width produces is that, it gives rise to multiple numbers of modes, all of nature and almost degenerate due to little difference of their phase constant values. SPP

Fig. 9.5.1 (a)

Fig. 9.5.1 (b) Fig. 9.5.1 : Dispersion profile and loss profile for finite width thin metal strip structure using EDC method However, the actual results are quite different from the results we obtained. In fact, not too many results are reported regarding this type of structures. This is chiefly because of the complexity that arises while trying to analyze the structure.

--- 111333888 ---


Fig. 9.5.2 (a)

Fig. 9.5.2 (b) Fig. 9.5.2 : Reported dispersion and loss profile for the finite width thin metal strip structure

--- 111333999 ---


So we rely on the reports provided by one of the pioneers in this subject, Pierre Berini. His reports are taken as the trailblazer for these kinds of complex structures. The proper dispersion profile and the loss profile for the structure of our discussion are given in the figure (9.5.2) according to these reports. The curves are obtained from the reference paper 11 listed later. The results are supposed to be obtained using the Method of Lines (MoL) as par report and are supposed to fit to experimental observations. According to the results we observe that for sufficiently large thickness (about ) the and modes are much like the corresponding modes in a symmetric structure except that the fields are localized near the cover. As the strip thickness decreases the modes evolve from a symmetric-like mode to asymmetric-like mode having field localized along the substrate-metal interface. Since the substrate dielectric constant is larger than the cladding dielectric constant the modes seem to be ‘pulled’ from a symmetric-like mode to an asymmetric-like mode as the thickness goes smaller. This behavior of these two fundamental modes is quite unexpected. In the asymmetric structure both modes exhibit increasing attenuation with decreasing thickness and the mode does not have a cut-off thickness. The dotted Blue curve represents the mode in case of a symmetric structure with the cladding layer dielectric constant being also

nm100 0bas 0

bss

0bas 0

bss0.4c =ε ,

identical to the substrate layer. The curve in figure (9.5.3) may also be taken for comparison for the symmetric structure. This curve is obtained from the reference 14. The behavior of the modes in the asymmetric structure with the decreasing film thickness is quite unexpected, differing completely from the behavior in the corresponding symmetric structure. In the symmetric structure, we see, as the film thickness decreases the mode shows decreasing attenuation. So the mode in the symmetric structure is the main long ranging mode and the mode has a cut-off. But in the asymmetric structure both and

modes exhibit increasing attenuation with decreasing thickness and the mode does not have a cut-off thickness. However in both the cases the and the modes show degeneracy especially at the larger thicknesses. As the thickness decreases the modes become slightly apart i.e. non-degenerate.

0bss 0

bss0bas 0

bss0bas 0

bas0bsa 0

baa

Fig. 9.5.3 : Dispersion profile for thin metal strip embedded in a dielectric medium We shall not go tot the details of these modes any further because our approach to the structure could not explain the mode switch over from the symmetric like mode to the antisymmetric like mode. Also the

--- 111444000 ---


normalized phase constant values corresponding to the fundamental modes are different from the reported values by plenty. For example the )k( 0β value for the mode at large thicknesses are reported to be as high as whereas the results obtained from the method are below . Therefore modifications must be adopted to the modeling scheme of the structure.

0bss

6.2~ EDC 1.2

999...666 MMMooodddiiifffiiieeeddd aaappppppllliiicccaaatttiiiooonnn ooofff ttthhheee EEEDDDCCC mmmeeettthhhoooddd::: In order to explain the higher values of the normalized phase constant obtained from the reports of Pierre Berini we adopt to the following modification. We know that the thin metal strip of infinite width structure can support two pure TM modes, one the symmetric-like mode and the other the antisymmetric-like mode. The symmetric-like mode, has field maxima at the metal dielectric interface with the dielectric of lower dielectric constant and the antisymmetric-like mode has field maxima at the metal dielectric interface with the dielectric of higher dielectric constant. Therefore the phase constant of the mode is always higher than that of the mode. As long as the thickness of the metal strip is large these two modes remain decoupled with respect to each other. The symmetric mode spreads evanescently in the cladding dielectric and penetrates merely into the metal. Similarly the antisymmetric mode spreads evanescently in the substrate layer and penetrates negligibly into the metal. Therefore the effective dielectric constant of the cladding region faced by the supermodes in the finite width structure may be assumed to be the

bs

ba

ba

bs

)k( 0β value corresponding to the symmetric mode in the infinite width structure for a give value of the thickness. Similarly the effective dielectric constant of the substrate region faced by the supermodes in the finite width structure may be assumed to be the )k( 0β value corresponding to the anti-symmetric mode in the infinite width structure for the give value of the thickness. As the thickness of the infinite width structure is reduced the two modes seem to couple to each other resulting in variation of the respective effective dielectric constant values according to the variation of the )k( 0β values with thickness for the two modes in the infinite width structure. The thickness dependant effective indices of the cladding and the substrate layers are already discussed in the chapter 8. Now with this idea in mind the effective dielectric constant model for the finite width metal strip structure can be modified in the following way. 1. First we derive at the two effective dielectric constants or effective refractive index values to replace the substrate and the cladding layer in the infinite width structure. This step involves nothing but reproduction of the )k( 0β values, or more specifically the )kk( 0z values for the infinite width structure as done in the previous chapter. Let these two effective indices be represented as )t2(evenη and )t2(oddη . The bracketed terms signify that the values vary with the change of the strip thickness. Obviously )t2(evenη and )t2(oddη are the solutions to the equation (9.4.4) for varying thickness. These effective indices can be found by first solving for the two possible values of the hypothetical propagation constant zsβ and replacing zsβ for symmetric like solution by )t2(k even0η and for anti-symmetric like solution by

)t2(k odd0η . 2. We then replace the cladding layer by the material of hypothetical effective index of refraction

)t2(evenη and the substrate layer by the material of hypothetical effective index )t2(oddη . This makes the structure modified for the supermodes as shown in the figure (9.6.1). This effectively means that the

--- 111444111 ---


finite width metal strip is surrounded by the dielectrics of dielectric constants and

)t2(2oddη

)t2(2evenη

Fig. 9.6.1 replacing sε and cε respectively. These hypothetical dielectrics are lossy even though the actual dielectrics were not. This is because of the imaginary part of the dielectric constant of the metal which gives rise to loss for the and modes in the infinite width structure. bs ba 3. We now apply the effective index method to the actual structure in the similar way done in the section 9.4 of this chapter. Obviously the infinite width structure considered in doing so is the structure in the figure (9.6.1). This infinite width structure can support two TM modes which are obtained by solving the characteristic equation (9.4.4) where we replace sε by and )t2(2

oddη cε by . The two solutions are the two Surface modes, one being the symmetric-like mode and the other the anti-symmetric-like mode. The variation of the corresponding propagation constants with thickness for the modes in this equivalent infinite width structure depends on the thickness dependence of the effective dielectric constants

and . Let the normalized phase constants in the derived infinite structure of figure (9.6.1) be respectively

)t2(2evenη

)t2(2oddη )t2(2

evenη)t2(symη and )t2(asymη . As the thickness tends towards infinity the effective

dielectric constant saturates to the value corresponding to the phase constant of the Fano mode existing at the metal cladding

)t2(2evenη

)( cε interface and

)(

)(fc

fc2even εε

εεη

+=∞ (9.6.1)

Thus the modified cladding layer dielectric constant varies from sε to given in equation (9.6.1) as the thickness varies from 0 to ∞ . Similarly

)(2even ∞η

oddη being the phase constant corresponding to the antisymmetric mode in the unmodified infinite structure the value of oddη varies from ∞ to the value

)(

)(fs

fs2odd εε

εεη

+=∞ (9.6.2)

--- 111444222 ---


Fig. 9.6.2 : complete description of the modified approach using EDC method to the thin finite width metal strip structure as the thickness increases from to ∞ . Therefore the effective dielectric constant of the modified cladding layer always remains lower than that of the modified substrate layer and hence in the modified infinite width structure the symmetric like mode exists at the interface and the antisymmetric like

mode at the interface.

0

)t2(~ 2evenf ηε

)t2(~ 2oddf ηε

Now as the thickness tends towards cut-off thickness for the unmodified structure the effective dielectric constant of the modified cladding layer tends towards and hence the phase

constant for the symmetric like mode tends towards . As the thickness tends towards

2evenη s

2even )0( εη =

s2

even )0( εη = ∞

the effective dielectric constant tends towards given by equation (9.6.1) and the phase constant for the symmetric like mode tends towards

2evenη )(2

even ∞η

{ }))(

)()(

f2

even

f2

even2sym εη

εηη

+∞

∞=∞

)2( cf

fc

εεεε+

= (9.6.4)

The phase constant for the symmetric like mode in the modified infinite width structure therefore varies from

sε to )(sym ∞η given by equation as the thickness increases from to 0 ∞ .

--- 111444333 ---


Similarly, for the antisymmetric mode as the thickness tends towards zero the phase constant tends towards according to it’s nature discussed in the previous chapter. As the thickness tends towards the phase

constant tends towards ∞ ∞

{ }))(

)()(

f2

odd

f2

odd2asym εη

εηη

+∞

∞=∞

)2( sf

fs

εεεε+

= (9.6.5) 4. For the symmetric-like mode we now replace the middle layer by a hypothetical dielectric of dielectric constant and for the antisymmetric-like mode we replace the middle layer by the dielectric constant

)t2(2symη

)t2(asymη . Thus we get the equivalent dielectric slab waveguide structure, symmetric about the axis as in the figure (9.4.1 (c)). For the mode symmetric-like about the x y axis the middle layer is

replaced by the effective dielectric constant . Therefore the structure now becomes purely symmetric about the axis with the core being a lossy dielectric of dielectric constant varying from

)t2(2symη

x sε to

as the thickness varies from 0 to )(2sym ∞η ∞ while the other two layers are lossless dielectrics of

dielectric constant cε . Similarly for the mode antisymmetric-like about the y axis the middle layer is

replaced by the effective dielectric constant which varies from )t2(2asymη ∞ to as the

thickness varies from 0 to ∞ . The complete pictorial description of the process is depicted in the figure (9.6.2). The dielectric thus formed supports multimode. With in the middle layer the

modes that are supported are the and the modes whereas with in the middle layer

the modes supported are the and the modes.

)(2asym ∞η

SWG )t2(2symη

nbss n

bas )t2(2asymη

nbsa n

baa Now the modes are TE like in dielectric s symmetric about the axis. Therefore the values of the phase constants for the modes lie between the refractive indices of the core layer and the supporting layers under the guiding condition. Thus for the and the modes he phase constant lie between

SWG x

nbss n

bas cε

and )t2(symη . Similarly for the and the modes the phase constant lie between nbsa n

baa cε and )t2(asymη . For sufficiently large width (like m1W2 µ= in the present structure) the phase constant for

the fundamental modes almost reach to the values )t2(symη and therefore at large thicknesses

( ) the propagation constant for the and the modes are almost equal to nm100t2 ≥ 0bss 0

bas )(sym ∞η

and for the and the modes, almost equal to 0bsa 0

baa )(sym ∞η Confirmation of improvement of results If we consider a lossless metal with dielectric constant mf εε −=

--- 111444444 ---


Then we get from equations (9.6.4) and (9.6.5) the values of the propagation constants for the and the modes, for large width and thickness is

0bss

0bas

)2(

)(cm

mcsym εε

εεη

−=∞ (9.6.6)

And that for the and the modes is 0bsa 0

baa

)2(

)(sm

msasym εε

εεη

−=∞ (9.6.7)

Without this modification the maximum value would have been, as we saw in the section 9.5

)(

)(cm

mceven εε

εεη

−=∞ (9.6.8)

and

)(

)(sm

msodd εε

εεη

−=∞ (9.6.9)

It is obvious that

oddasym

evensym

ηη

ηη

>

>

Thus we see the values of the phase constants increase from the values predicted by the conventional

method towards the reported results in the papers of Pierre Berini. In particular if we take the example EDC

0.4s =ε , 91.3c =ε and 53.0j19f −−=ε at m633.0 µλ = , for sufficiently large guide thickness and large width )nm100t2( ≥ )m1W2( µ=

(9.6.10)

⎪⎪

⎭

⎪⎪

⎬

⎫

−≈

−≈

−≈

−≈

021.0j41.2

027.0j63.2007.0j11.2008.0j25.2

sym

asym

even

odd

η

ηηη

and therefore the normalized propagation constants for the four fundamental modes at these large thickness and large width are 021.0j41.2)kk()kk( 0

b0b as0zss0z −≤=

and 027.0j63.2)kk()kk( 0b

0b aa0zsa0z −≤=

If we investigate the results in the plots (9.5.2) wee see that the normalized phase constants for the four fundamental modes asymptotically saturate to the values in the set of values in (9.6.10). Obviously these results agree very well with the results reported in the paper of reference 11. Thus this modification may be useful at the higher thickness and higher widths of the metal strip.

--- 111444555 ---


But at lower film thicknesses the modal switch over of for the and the modes from the symmetric like mode to antisymmetric like mode cannot be explained in this modified model. A set of plots for the results according to the modified application of the method to the structure are shown in the figure (9.6.3 (a) and (b)). The corresponding MATLAB program is given in Prog. 33 of Appendix.

0bss 0

bas

EDC

Fig. 9.6.3 (a)

Fig. 9.6.3 (b) Fig. 9.6.3 : Dispersion and loss profile of finite width metal strip structure according to the modified model

--- 111444666 ---


The red curves represent the symmetric modes in the equivalent guides both for the symmetric-like and asymmetric-like solutions for effective dielectric constants. Similarly the blue curves correspond to the respective antisymmetric modes in the equivalent structure. Obviously these curves fit better to the curves provided in the report of reference 11 at the larger thicknesses. But the and the modes do not seem to switch over. In fact below nearly

0bss 0

basnm5045 − the modes are cut-off according to the model. The

)t2(symη value falls below the effective index )t2(oddη for thickness below . Therefore this

model is not explicable for the very thin structure. Also the and the modes do not seem to be degenerate although their propagation constant profiles are almost similar.

nm45~0baa 0

bsa

The loss profiles however satisfy the fact that all the modes in the asymmetric structure are short ranging for decreasing thicknesses. The and the modes are expectedly short ranging. The and the

also show increasing attenuation with decreasing thickness at small ranges of thicknesses. But the loss values for the and the modes and those of the and the modes are so similar that they cannot be distinguished in the figure (9.6.3 (b)). Also, although the short ranging behavior of the four fundamental modes are obvious from the modified structure the values of the corresponding attenuation values do not agree vary well and so also not do the nature of the plots.

0baa 0

bsa 0bss

0bas

0bss 0

bas 0baa 0

bsa

We can also find the dispersion profiles at other values of width. In the report of reference 11, the dispersion profile at the width m5.0W2 µ= has also been derived. The plot in the figure (9.6.4) shows the variation of the normalized propagation constant for the structure with strip width m5.0W2 µ= where the constituents are the same as before and the wavelength used is also the same. The plot in figure (9.6.4) has

Fig. 9.6.4 : Normalized propagation constant at thickness 0.5 µm

--- 111444777 ---


been taken from the reference 11 with some modification in ‘Paints’ (Windows). The fundamental modes have only been shown with the higher modes rubbed away. The green curves show the corresponding and modes. On the other hand the figure (9.6.5) shows the same profile using the method. This set of curves has been generated using the same program Prog. 33 in Appendix, just by changing the width from

bs

ba EDC

m1W2 µ= to m5.0W2 µ= . The similarities and the dissimilarities are obvious from the figures.

Fig. 9.6.5 : Normalized propagation constant at thickness 0.5 µm according to the EDC method adopted 999...777 DDDiiissscccuuussssssiiiooonnnsss::: The purely bound optical modes supported by a thin lossy metal film of finite width, supported by a semi-infinite substrate and covered by a semi-infinite superstrate dielectric have been characterized and described in this chapter in brief. We see that the modes differ significantly from the modes in the corresponding slab structures. In addition to the four fundamental modes there are numerous higher modes existing in this structure. The dispersion of the modes with the film thickness have been tried to assess in this chapter. There are not many reported results to our knowledge about this kind of structure. So we adopted the effective dielectric constant method. We have found from the reports of Pierre Berini that the fundamental modes show unexpected behavior at the ranges of thickness near cut-off. The method is supposed to be useful even near cut-off as compared to Marcatili’s method. But the basic assumption behind the successful application of this method is that the geometrical discontinuities must not perturb the modal behaviors. However this assumption is probably not a justified one in the case of metal strips. This is chiefly because the modes are surface sensitive and as we shall discuss in brief in the next chapter, there exist corner modes due to sharp discontinuity of dielectric behavior at the edges of the rectangular cross section of the metal strip. More dedicated methods probably will be more useful to analyze this kind of structure.

EDC

--- 111444888 ---

CCChhhaaapppttteeerrr (((xxx)))

CCCooonnncccllluuussiiivvveee wwwooorrrdddsss::: FFFuuutttuuurrreee ssscccooopppeee oooff wwwooorrkkk s f r

--- 111444999 ---


Then entire scope of the project was to analyze the theoretical existence of the Surface Plasmon Polariton waves in metallic waveguides. In the initial chapters the generalized electromagnetism of D2 slab waveguides were discussed in order to have knowledge about the modes in general guiding structures and the approach needed for such analysis. We applied Maxwell’s equations to the modes with proper boundary conditions to develop the relevant wave equations and the characteristic equations that give rise to evolution of the modes in such hypothetical semi-infinite structures. Then specific layers in the structures were replaced by metals and we analyzed the modes under the influence of metals. For this, pertinent idea about the dielectric behavior of metals was introduced such that the inclusion of metal in place of pure dielectric in the structures just makes the dielectric constant of the corresponding layer negative and in general complex. The same characteristic equations were adopted for the structures with slight modification in their layout. This is the way we analyzed the structure with metallic strip of small thickness between symmetrically or asymmetrically placed dielectric cover. The structure was still hypothetical as we considered that there is only one directional boundary for the structure and the other directions were unbounded. With the knowledge about the modes in such structures we finally tried to analyze the modes in a thin metal strip of finite width. The effective index method was assumed to be the best fit to the structure. The effective index method led to two virtual D2 configurations that were known to us in the previous chapters, one is the structure with thin metal strip of infinite width sandwiched between two unlike dielectrics and the second one, a symmetrical dielectric slab waveguide structure with the middle layer having highest permittivity. These two hypothetical structures were D2 in nature as is obvious in the previous chapter. Here lies the utility of analyzing the hypothetical semi infinite structures in the previous chapters. But in the last chapter we saw that the effective index method produced results for the finite structures that do not agree with the practical values or with the results that come out of more dedicated analytical or numerical techniques. The evolution of the supermodes was not explicable in the orthodox approach of effective index method. So modifications were made to the approach of using the method of effective index. The details of the mathematical and structural assumption have been discussed in the previous chapter. The modified results gave better results but not still the accurate ones. The higher order modes were not of problem as these modes do not show practical evidence of evolution into supermodes and these modes can be explained adequately in the effective index approach. But the fundamental modes do not fit to the derived results to the best degree of agreement. Especially in the ranges of smaller thicknesses the results are drastically differed. In the larger thicknesses the results agree fairly well. Therefore the effective index method was of moderate utility to the finite structure we considered. But it can be foreseen and presumed that the effective index method can be prudently modified to explain the modes in such structures exclusively. This is firstly because the effective index method can explain properly the existence of the multiple modes in the structure with their symmetry classification, namely etc. and their higher eigenvalues. And it is needed the astute approach of using the effective index method for closest fit to the actual situation. One such modification was done in our project which gave good results in the large guide thicknesses but at small thicknesses this proved invalid. A proper theoretical explanation to this can be obtained by considering the background of evolution of the supermodes.

1b

0b

1b

0b aa,aa,sa,sa

The supermodes in the finite metal strip structure evolve due to coupling between “edge” and “corner” modes supported by each metal dielectric interface defining the structure. As the metal strip thickness decreases, the coupling between these interface modes intensifies leading to dispersion and possibly evolution of the supermodes. In the asymmetric structure like this the interface modes at the dissimilar interfaces with the metal also couple to create the supermodes. For instance a mode with three extrema along the bottom interface may couple to a mode having one field extrema along the top interface and so on. The selection criteria determining which modes will couple to each other to create the supermodes are a similarity in the value of their propagation constants and a shared field symmetry or asymmetry with respect to the central vertical axis.

--- 111555000 ---


But in the effective dielectric constant method the basic assumption behind it’s validity was that the geometrical discontinuities at the strip edges are negligible. This in other words means that the fields at the corner regions of the structure are negligible. This may be a good approximation as long as the strip is thick enough. For small thicknesses this assumption is invalid as the discontinuity at the corners is sharp and the corner modes are strong enough to couple to the straight interface modes. In the metal strip structures the thickness of the strip is of the order of few tens of nanometers whereas the width is as large as m1µ i.e. the width is al least ten times higher in magnitude. So the coupling is expected to be prominent and the effective dielectric constant method is not applicable as such, unless modified to take care of these corner modes. In the short duration of the project we had not got enough scope to go into the various modifications. It may be proposed to the successors to proceed to consider different sorts of symmetry in the structure including the corners. One such analysis needs solution of the Maxwell’s equations at semi infinite structures with proper boundary conditions adopted at the corners. The various structures that can be useful for the complete study may be as shown in the figure (10.1).

Fig. 10.1 (a) Fig. 10.1 (b) Fig. 10.1 : Schematic of several structures containing the corners The structure in figure (10.1 (a)) can be useful if possible to analyze, for the interface between the metal and the dielectric of lower dielectric constant in the thin finite width metal strip in our present context. The structure in figure (10.1 (b)) may be useful for the interface with higher dielectric constant as there the corner consists of regions with three dissimilar dielectric constants. The structure in the figure of part (a) may also be considered as a special case of the structure in the figure of part (b) where we set 32 εε = . The exact solutions to the wave equations in these structures may be instinctively presumed to be decaying in both andx y directions with extrema at the interfaces. Therefore the modes may have sharp peak at the corners which intrinsically may give rise to the corner mode solutions. Now, properly analyzed, these two structures can be combined using the effective index method for constructing either halves of the finite width thin metal strip structure we discussed in the last chapter while divided vertically along the axis in the figure (9.2.1). These two halves can once again be combined symmetrically along the axis to form the original structure. The schematic of the proposed method is depicted below in the figure 10.2.

xx

This entire process of combining may be considered to be a modified approach to the effective dielectric constant method which we think might be the best fit for the structure. So this needs starting with the Maxwell’s equations and the boundary conditions for the structures in the figure (10.1). The boundary conditions are to be properly stated at the corners for correct results. So this topic probably leaves scope of

--- 111555111 ---


Fig. 10.2 : Schematic of the proposed combination of the structures in the figure 10.1 future work behind the modal analysis of the thin finite width metal strip waveguides. More complex structures may also be a matter of interest for application of the effective dielectric constant method with properly chosen equivalent structures. We may also add at this point some of the other important factors regarding . One of them is their excitation in the metallic guides. It is a fact that the dispersion relation of freely propagating electromagnetic waves and s do not intersect at any frequency. This signifies that one cannot generate s by simply irradiating a metal surface with light, for example. Instead, coupling between propagating waves and surface evanescent waves requires additional technique. All these techniques have in common that they can be used for coupling free electromagnetic radiation with surface evanescent waves (incoupling) or vice versa (outcoupling). This symmetry is in fact a direct consequence of the invariance of Maxwell's equations under time reversal. These techniques might also be important aspects of study.

SPP

SPP SPP

Fig. 10.3 : Several metal strips on a dielectric substrate The next objective of study regarding this topic may be the analysis of the structure shown in the figure 10.3. In this structure several identical or on-identical metal strips are deposited on a dielectric substance. The

--- 111555222 ---


analysis essentially requires the successful analysis of the single finite width strips of metal as discussed in the previous chapter. The analysis may be carried out for each single strip without considering the effect of the existence of the other. A model of coupling of the modes in each single structure may be aimed at to develop the total modal analysis. The structure cross section may be divided along the plane of symmetry to start the modal analysis of the total structure. However, we could not attempt to go that far because of shortage of time during the course. It will be delightful if the study on these topics goes on and successful results are explored.

--- 111555333 ---

AAAppppppeeennndddiiixxx

AAAppppppeeennndddiiixxx ::: NNNeeewwwtttooonnn RRRaaappphhhssooonnn mmmeeettthhhoooddd tttooo sssooolllvvveee aaalllgggeeebbbrrraaaiiiccc eeeqqquuuaaatttiiiooonnnsss s Let us have a well defined function in independent variable . The numerical solution to the equation

)x(f x

0)x(f = (A.1) can be found by Newton Raphson method. It is an iterative technique widely used to solve transcendental equations where we cannot solve it analytically for that satisfies the equation (A.1). x Let be an approximate solution to the equation (A.1) and the exact solution. Then 0x 1x 0)x(f 1 = (A.2) and the error in the solution, if is taken as the solution is 0x )xx(h 01 −= (A.3) i.e. )hx(x 01 += (A.4) Then from (A.2) we get 0)hx(f 0 =+ (A.5) Expanding in Taylor series

0....xdx

fd!2

hxdx

dfh)x(f0

2

22

00 =+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛+ (A.6)

Since is small we neglect and subsequent higher powers of and get using (A.3) h 2h h

0xdx

df)xx()x(f0

010 =⎟⎠⎞

⎜⎝⎛−+

or 0

01 x)dxdf()x(fxx ⎥⎦

⎤⎢⎣

⎡−= (A.8)

Similarly starting with the new value of a still better approximation to the solution may be achieved as where

1x,x

2x

1

12 x)dxdf()x(fxx ⎥⎦

⎤⎢⎣

⎡−=

In general this iteration can me carried any arbitrary number of times to get gradually improved solution

1n

1nn x)dxdf()x(fxx

−− ⎥

⎦

⎤⎢⎣

⎡−= (A.9)

AAA 111


This is known as Newton Raphson formula or the Newton iteration formula. This iteration can be carried on until the error i.e. the difference between and becomes smaller than the permissible error. Under this condition the solution to the equation can be achieved correct to the desired degree of accuracy.

nx 1nx −

SSSooolllvvviiinnnggg ttthhheee ccchhhaaarrraaacccttteeerrriiisssttiiiccc eeeqqquuuaaatttiiiooonnn iiinnn NNNeeewwwtttooonnn RRRaaappphhhsssooonnn mmmeeettthhhoooddd t In the entire scope of the project we faced the characteristic equation in numerous occasions. The equation relates the propagation constant for a guided wave in a D2 slab waveguide to the thickness of the core layer of the . The equation can be symbolically written as SWG 0)N,k,d( z =ζ (A.10) where the thickness of the core region is and is the phase constant for the d2 zk thN mode in the structure. We are to find the variation of the propagation constant with the variation of thickness. From the real part of we find the phase constant

zk

zk β and from the imaginary part we find the attenuation constant α related as )j(kz αβ −= (A.11) The normalized version of the characteristic equation is also often used which can be written symbolically as 0)N,b,v( =ξ (A.12) where the normalized thickness of the guide’s core region is and the normalized propagation constant

for thev2

b thN mode. To find the solution we set N to the integral value corresponding to the mode number. Then we set a particular value of the independent variable in the equation A.10 (A.12) so that the function )v(d

)(ξζ reduces to a single variable equation in . This equation can be now solved for the solution for at that particular value of by Newton Raphson technique. Then the parameter can be increased or decreased from it’s previous value and the solution for can again be obtained by iteration at that value of . In this way the complete solution of the characteristic equation for a specific range of values of the thickness can be obtained.

)b(kz

)b(kz )v(d )v(d)b(kz

)v(d

A value of can be preset to launch the iteration. For example if we start from , the cut-off thickness, the initial value of can be taken as where the symbols are of their usual meaning. Similarly going with the normalized equation if we start with

)b(kz cutoffdd =

zk s0 nk

cutoffvv = corresponding to the normalized propagation constant can be initiated at

cutoffdd =

0b = which corresponds to . Similarly if we start with large value of or, far away of cut-off, and then decrease it up to cut-off the value of can be initiated at or correspondingly

=zk s0 nkd zk

f0 nk 1b = . The following flowchart may be depicted to clarify the scheme.

AAA 222


Start

Know the dielectric constants of the different layers

Select the number of modes to be solved for n

Select a value of d or v to start iteration

Select a suitable value of kz or b to initiate iteration

Set mode number N = 0 for the 0th mode

Set a counter to count the number of iteration steps k=m

Find ζ(kz) or ξ(b)

Find dζ/dkz or dξ/db

Find corrected value of kz or b by

kz ← kz – ζ/(dζ/dkz) or b ← b – ξ/(dξ/db)

Decrement k

Is k=0?

No

Yes

Has d or v reached the final value ?

No

Increase N for the next mode

Yes

Is N =n

End

Yes

Plot or store kz or b against d or v

Increase or decrease d

or v by small amount

No

AAA 333


PPPrrrooogggrraaammmsss fffooorrr ttthhheee pppllloootttsss r

% For fig. 2.6.2 & 2.6.3 l_c=5e-6; l=1e-8:1e-8:1e-5; l_g1=l./(1-(l./l_c).^2).^0.5; l_g2=l; y=linspace(0,2,80); x=linspace(0.1,0.25,100); figure(1); plot(l./l_c,l_g1./l_c,'r',l./l_c,l_g2./l_c,'b',1,y,'k',x,1.8,'b',x,1.6,'r'); xlabel('\lambda / \lambda_c'); ylabel('\lambda_g / \lambda_c'); text(0.3,1.8,'Unbounded'); text(0.3,1.6,'Bounded'); axis([0,2,0,2]); c=2e8; v_p1=c./(1-(l./l_c).^2).^0.5; v_p2=c; y=linspace(0,4,80); x=linspace(0.1,0.25,100); figure(2); plot(l./l_c,v_p1./c,'r',l./l_c,v_p2./c,'b',1,y,'k',x,3.6,'b',x,3.2,'r'); xlabel('\lambda / \lambda_c'); ylabel('v_p / c'); text(0.3,3.6,'Unbounded'); text(0.3,3.2,'Bounded'); axis([0,2,0,4]); k=2*pi./l; beta_1=k.*(1-(l./l_c).^2).^0.5; beta_2=k; y=linspace(0,1.2,80); x=linspace(0.1,0.25,100); figure(3); plot(l./l_c,beta_1./k,'r',l./l_c,beta_2./k,'b',1,y,'k',x,0.4,'b',x,0.3,'r'); xlabel('\lambda / \lambda_c'); ylabel('\beta /k'); text(0.3,0.4,'Unbounded'); text(0.3,0.3,'Bounded'); axis([0,2,0,1.2]); v_g=c.*(1-(l./l_c).^2).^0.5; figure(4); y=linspace(0,2,80); x=linspace(0.1,0.25,100); plot(l./l_c,v_p1./c,'r',l./l_c,v_g./c,'b',1,y,'k',y,1,'k',x,0.45,'b',x,0.3,'r'); text(0.3,0.45,'v_g'); text(0.3,0.3,'v_p');

AAA 444


axis([0,2,0,2]); xlabel('\lambda / \lambda_c'); ylabel('v_p / c and v_g / c ');

% For fig. 3.4.1 (a) and (b) n1=1.5; for(n2=1:0.1:1.4) theta_i_in_degree=0:0.001:90; theta_i=theta_i_in_degree.*pi/180; theta_r=asin(n1.*sin(theta_i)./n2); kx1_by_k0=n1.*cos(theta_i); kx2_by_k0=n2.*cos(theta_r); r_E=abs((kx1_by_k0-kx2_by_k0)./(kx1_by_k0+kx2_by_k0)); figure(1); plot(theta_i_in_degree,r_E); hold on; grid on; gamma_c_by_k0=j*(kx2_by_k0); phi_E=atan(gamma_c_by_k0./kx1_by_k0); phi_E_in_degree=phi_E.*180/pi; figure(2); plot(theta_i_in_degree,phi_E_in_degree); hold on; grid on; end; figure(1); text(31,0.45,'n_2=1.0'); text(41,0.55,'n_2=1.1'); text(48,0.65,'n_2=1.2'); text(56,0.75,'n_2=1.3'); text(65,0.85,'n_2=1.4'); text(21,0.95,'n_1=1.5'); xlabel('angle of incidence \theta_i in degree'); ylabel('|r_E|'); title('variation of reflection coefficient with angle of incidence for TE wave'); figure(2); xlabel('angle of incidence \theta_i in degree'); ylabel('phase shift \Phi_T_E in degree'); title('variation of phase shift with angle of incidence for TE wave'); grid on; text(38,5,'n_2=1.0'); text(44,15,'n_2=1.1'); text(52,25,'n_2=1.2'); text(61,36,'n_2=1.3'); text(71,46,'n_2=1.4'); text(41,84,'n_1=1.5');

AAA 555


% For fig. 3.4.2 (a) and (b) n1=1.5; for(n2=1:0.1:1.4) theta_i_in_degree=0:0.001:90; theta_i=theta_i_in_degree.*pi/180; theta_r=asin(n1.*sin(theta_i)./n2); kx1_by_k0=n1.*cos(theta_i); kx2_by_k0=n2.*cos(theta_r); r_E=abs(((n2.^2.*kx1_by_k0-n1.^2.*kx2_by_k0)./(n2.^2.*kx1_by_k0+n1.^2.*kx2_by_k0))); figure(1); plot(theta_i_in_degree,r_E); hold on; grid on; gamma_c_by_k0=j*kx2_by_k0; phi_TM=180/pi.*atan((n1./n2).^2.*gamma_c_by_k0./kx1_by_k0); figure(2); plot(theta_i_in_degree,phi_TM); hold on; grid on; end; figure(1); xlabel('angle of incidence \theta_i in degree'); ylabel('|r_H|'); title('variation of reflection coefficient with angle of incidence for TM wave'); grid on; text(38,0.55,'n_2=1.0'); text(44,0.65,'n_2=1.1'); text(49,0.75,'n_2=1.2'); text(56,0.85,'n_2=1.3'); text(65,0.95,'n_2=1.4'); text(21,0.95,'n_1=1.5'); figure(2); xlabel('angle of incidence \theta_i in degree'); ylabel('phase shift \Phi_T_M in degree'); title('variation of phase shift with angle of incidence for TM wave'); text(47,55,'n_2=1.0'); text(50,45,'n_2=1.1'); text(53,35,'n_2=1.2'); text(58,25,'n_2=1.3'); text(66,15,'n_2=1.4'); text(21,84,'n_1=1.5');

AAA 666


% For fig. 3.5.1 (a) and (b) n1=1.5; for(n2=1:0.1:1.4) theta_critical=asin(n2./n1); theta=linspace(theta_critical+0.000001,pi./2,10000) lambda=1.3e-6; k_0=2.*pi./lambda; gamma=k_0.*(n1.^2.*(sin(theta)).^2-n2.^2).^0.5; dP_TE=gamma.^(-1).*10^6; theta_in_degree=180.*theta./pi; figure(1); plot(theta_in_degree,dP_TE); hold on; figure(2); q=(n1./n2).^2.*(sin(theta)).^2-(cos(theta)).^2; k_0=2.*pi./lambda; gamma=k_0.*(n1.^2.*(sin(theta)).^2-n2.^2).^0.5; dP_TE=(q.*gamma).^(-1).*10^6; theta_in_degree=180.*theta./pi; plot(theta_in_degree,dP_TE); hold on; end; figure(1); axis([0,90,0,4]); grid on; title('plot of depth of penetration vs. angle of incidence for TE modes'); xlabel('angle of incidence \theta in degree'); ylabel('depth of penetration in \mum'); text(5,0.9,'\lambda = 1.3 \mum'); text(5,0.6,'n_1 = 1.5'); text(39,1,'n_2 = 1.0'); text(44,1.5,'n_2 = 1.1'); text(50,2,'n_2 = 1.2'); text(56,2.5,'n_2 = 1.3'); text(65,3,'n_2 = 1.4'); figure(2); axis([0,90,0,4]); grid on; title('plot of depth of penetration vs. angle of incidence for TM modes'); xlabel('angle of incidence \theta in degree'); ylabel('depth of penetration in \mum'); text(5,0.9,'\lambda = 1.3 \mum'); text(5,0.6,'n_1 = 1.5'); text(39,1,'n_2 = 1.0'); text(44,1.5,'n_2 = 1.1'); text(50,2,'n_2 = 1.2'); text(56,2.5,'n_2 = 1.3'); text(65,3,'n_2 = 1.4');

AAA 777


% For fig. 4.3.1.1 for(N=0:1:4) b=0.001; eta=1; cut_off=(N.*pi./2)+0.5.*atan((eta-1).^0.5); v=linspace(cut_off,8,2000); for(i=0:1:100) func=2.*v.*(1-b).^0.5-(N+1).*pi+atan(((1-b)./b).^0.5)+atan(((1-b)./(eta+b-1)).^0.5); diff=-(1-b).^(-0.5).*(v+0.5.*b.^(-0.5)+0.5.*(eta+b-1).^(-0.5)); b=b-func./diff; end; plot(v,b,'r'); hold on; hold on; end; for(N=0:1:4) b=0.001; eta=2; cut_off=(N.*pi./2)+0.5.*atan((eta-1).^0.5); v=linspace(cut_off,8,2000); for(i=0:1:100) func=2.*v.*(1-b).^0.5-(N+1).*pi+atan(((1-b)./b).^0.5)+atan(((1-b)./(eta+b-1)).^0.5); diff=-(1-b).^(-0.5).*(v+0.5.*b.^(-0.5)+0.5.*(eta+b-1).^(-0.5)); b=b-func./diff; end; plot(v,b,'b'); hold on; hold on; end; for(N=0:1:4) b=0.001; eta=10; cut_off=(N.*pi./2)+0.5.*atan((eta-1).^0.5); v=linspace(cut_off,8,2000); for(i=0:1:100) func=2.*v.*(1-b).^0.5-(N+1).*pi+atan(((1-b)./b).^0.5)+atan(((1-b)./(eta+b-1)).^0.5); diff=-(1-b).^(-0.5).*(v+0.5.*b.^(-0.5)+0.5.*(eta+b-1).^(-0.5)); b=b-func./diff; end; plot(v,b,'g'); hold on; hold on; end; for(N=0:1:4) b=0.001; eta=1e20; cut_off=(N.*pi./2)+0.5.*atan((eta-1).^0.5); v=linspace(cut_off,8,2000);

AAA 888


for(i=0:1:100) func=2.*v.*(1-b).^0.5-(N+1).*pi+atan(((1-b)./b).^0.5)+atan(((1-b)./(eta+b-1)).^0.5); diff=-(1-b).^(-0.5).*(v+0.5.*b.^(-0.5)+0.5.*(eta+b-1).^(-0.5)); b=b-func./diff; end; plot(v,b,'k'); hold on; hold on; end; grid on; xlabel('normalized frequency v'); ylabel('normalized propagation constant b'); title('b~v curve for for TE modes in asymmetrical SWG'); text(0.5,.1,'TE_0'); text(2.1,.1,'TE_1'); text(3.8,.1,'TE_2'); text(5.5,.1,'TE_3'); text(7.2,.1,'TE_4'); text(1,.9,'\eta=1'); text(1,.85,'\eta=2'); text(1,.8,'\eta=10'); text(1,.75,'\eta=\alpha'); x=linspace(.2,.8,2000); y=0.9; plot(x,y,'r'); hold on; y=0.85; plot(x,y,'b'); hold on; y=0.8; plot(x,y,'g'); hold on; y=0.75; plot(x,y,'k'); hold on;

% for fig. 4.3.1.2 for(N=0:1:4) eta=1:0.001:10; v_c=N.*pi./2+atan((eta-1).^0.5); plot(eta,v_c); hold on; end; xlabel('asymmetry factor \eta'); ylabel('cut-off value of normalized frequency v_c'); title('variation of cut-off frquency with asymmetry factor'); grid on; text(5,pi./2-.3,'TE_0');

AAA 999


text(5,2.*pi./2-.3,'TE_1'); text(5,3.*pi./2-.3,'TE_2'); text(5,4.*pi./2-.3,'TE_3'); text(5,5.*pi./2-.3,'TE_4');

% For fig 4.3.1.3, 4.3.1.4 and 4.3.1.5 for(N=0:1:4) b=0.001; eta=1; cut_off=(N.*pi./2)+0.5.*atan((eta-1).^0.5); v=linspace(cut_off,8,2000); for(i=0:1:100) func=2.*v.*(1-b).^0.5-(N+1).*pi+atan(((1-b)./b).^0.5)+atan(((1-b)./(eta+b-1)).^0.5); diff=-(1-b).^(-0.5).*(v+0.5.*b.^(-0.5)+0.5.*(eta+b-1).^(-0.5)); b=b-func./diff; end; w=(2.*v+b.^(-0.5)+(eta+b-1).^(-0.5)); confinement=(2.*v+b.^(0.5)+(eta+b-1).^0.5./eta)./w; deconfinement=(1-b).*(b.^(-0.5)+(eta.*(eta+b-1).^0.5).^(-1))./w figure(1); plot(v,w,'r'); hold on; figure(2); plot(v,confinement,'r'); hold on; figure(3); plot(v,deconfinement,'r'); hold on; end; for(N=0:1:4) b=0.001; eta=2; cut_off=(N.*pi./2)+0.5.*atan((eta-1).^0.5); v=linspace(cut_off,8,2000); for(i=0:1:100) func=2.*v.*(1-b).^0.5-(N+1).*pi+atan(((1-b)./b).^0.5)+atan(((1-b)./(eta+b-1)).^0.5); diff=-(1-b).^(-0.5).*(v+0.5.*b.^(-0.5)+0.5.*(eta+b-1).^(-0.5)); b=b-func./diff; end; w=(2.*v+b.^(-0.5)+(eta+b-1).^(-0.5)); confinement=(2.*v+b.^(0.5)+(eta+b-1).^0.5./eta)./w; deconfinement=(1-b).*(b.^(-0.5)+(eta.*(eta+b-1).^0.5).^(-1))./w figure(1); plot(v,w,'b'); hold on; figure(2); plot(v,confinement,'b');

AAA 111000


hold on; figure(3); plot(v,deconfinement,'b'); hold on; end; for(N=0:1:4) b=0.001; eta=10; cut_off=(N.*pi./2)+0.5.*atan((eta-1).^0.5); v=linspace(cut_off,8,2000); for(i=0:1:100) func=2.*v.*(1-b).^0.5-(N+1).*pi+atan(((1-b)./b).^0.5)+atan(((1-b)./(eta+b-1)).^0.5); diff=-(1-b).^(-0.5).*(v+0.5.*b.^(-0.5)+0.5.*(eta.^(0.5)+b-1).^(-0.5)); b=b-func./diff; end; w=(2.*v+b.^(-0.5)+(eta+b-1).^(-0.5)); confinement=(2.*v+b.^(0.5)+(eta+b-1).^0.5./eta)./w; deconfinement=(1-b).*(b.^(-0.5)+(eta.*(eta+b-1).^0.5).^(-1))./w figure(1); plot(v,w,'g'); hold on; figure(2); plot(v,confinement,'g'); hold on; figure(3); plot(v,deconfinement,'g'); hold on; end; for(N=0:1:4) b=0.001; eta=10000000000000000000; cut_off=(N.*pi./2)+0.5.*atan((eta-1).^0.5); v=linspace(cut_off,8,2000); for(i=0:1:100) func=2.*v.*(1-b).^0.5-(N+1).*pi+atan(((1-b)./b).^0.5)+atan(((1-b)./(eta+b-1)).^0.5); diff=-(1-b).^(-0.5).*(v+0.5.*b.^(-0.5)+0.5.*(eta+b-1).^(-0.5)); b=b-func./diff; end; w=(2.*v+b.^(-0.5)+(eta+b-1).^(-0.5)); confinement=(2.*v+b.^(0.5)+(eta+b-1).^0.5./eta)./w; deconfinement=(1-b).*(b.^(-0.5)+(eta.*(eta+b-1).^0.5).^(-1))./w; figure(1); plot(v,w,'k'); hold on; figure(2); plot(v,confinement,'k'); hold on; figure(3); plot(v,deconfinement,'k'); hold on; end;

AAA 111111


figure(1); axis([0,8,0,30]); grid on; xlabel('normalized frequency v'); ylabel('normalized effective thickness w_T_E'); title('variation of normalized effective thickness vs. normalized frequency for TE modes in asymmetrical SWG'); text(0.5,20,'TE_0'); text(2,20,'TE_1'); text(3.6,20,'TE_2'); text(5,20,'TE_3'); text(6.9,20,'TE_4'); text(6,10,'\eta=1'); text(6,8,'\eta=2'); text(6,6,'\eta=10'); text(6,4,'\eta=\alpha'); x=linspace(6.8,7.3,2000); y=10; plot(x,y,'r'); hold on; y=8; plot(x,y,'b'); hold on; y=6; plot(x,y,'g'); hold on; y=4; plot(x,y,'k'); hold on; figure(2); grid on; xlabel('normalized frequency v'); ylabel('normalized confinement factor (P_f/P_N)'); title('variation of normalized confinement factor vs. normalized frequency for TE modes in asymmetrical SWG'); text(0.4,.1,'TE_0'); text(1.9,.1,'TE_1'); text(3.5,.1,'TE_2'); text(5.1,.1,'TE_3'); text(6.7,.1,'TE_4'); text(0.5,.95,'\eta=1'); text(0.5,.9,'\eta=2'); text(0.5,.85,'\eta=10'); text(0.5,.8,'\eta=\alpha'); x=linspace(.05,.3,2000); y=0.95; plot(x,y,'r'); hold on; y=0.9; plot(x,y,'b'); hold on; y=0.85; plot(x,y,'g');

AAA 111222


hold on; y=0.8; plot(x,y,'k'); hold on; figure(3); grid on; xlabel('normalized frequency v'); ylabel('fraction of power in the cladding and substrate layer (P_c+P_s)/P_N'); title('variation of fraction power radiated vs. normalized frequency for TE modes in asymmetrical SWG'); text(0.4,.9,'TE_0'); text(1.9,.9,'TE_1'); text(3.5,.9,'TE_2'); text(5.1,.9,'TE_3'); text(6.7,.9,'TE_4'); text(0.5,.2,'\eta=1'); text(0.5,.15,'\eta=2'); text(0.5,.1,'\eta=10'); text(0.5,.05,'\eta=\alpha'); x=linspace(.05,.5,2000); y=0.2; plot(x,y,'r'); hold on; y=0.15; plot(x,y,'b'); hold on; y=0.1; plot(x,y,'g'); hold on; y=0.05; plot(x,y,'k'); hold on;

% For fig. 4.3.1.6 % The texts, labels and the dotted lines ae drawn in the figure window manually d=1e-6; lambda=0.633e-6; k_0=2.*pi./lambda; omega=2.*pi.*3e8./lambda; mu_0=4*pi*1e-7; for(N=0:1:1) b=0.01; n_s=(2.2-0*j).^0.5; n_f=(2.9-0*j).^0.5; n_c=(1-0*j).^0.5; eta_fs=1; eta_fc=1; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=real((N.*pi./2)+0.5.*atan((eta-1).^0.5.*eta_fc));

AAA 111333


if(cut_off<0) cut_off=0.01; end; a=k_0.*d.*(n_f.^2-n_s.^2).^0.5; for(v=linspace(cut_off,a,100)) n=0; while(n<20) n=n+1; func=2.*v-(1-b).^(-0.5).*(N.*pi+atan(eta_fs.*(b./(1-b)).^0.5)+atan(eta_fc.*((eta+b-1)./(1-b)).^0.5)); diff=(2.*(1-b)).^(-1).*(func-2.*v-eta_fs./((1-b+eta_fs.^2.*b).*b.^0.5)-eta_fc.*eta./((1-b+eta_fc.^2.*(eta+b-1)).*(eta+b-1).^0.5)); b=b-func./diff; end; end; w=2.*v+(eta_fs./(b.^0.5.*(1-b+eta_fs.^2.*b))+eta_fc.*eta./((eta+b-1).^0.5.*((1-b)+eta_fc.^2.*(eta+b-1)))); W=w.*d./v; beta=k_0.*(b.*(n_f.^2-n_s.^2)+n_s.^2).^0.5 C_N=2.*(omega.*mu_0./beta./50000).^0.5.*W.^(-0.5); phi=0.5.*atan((eta_fs.*b.^0.5./(1-b).^0.5-eta_fc.*(eta+b-1).^0.5./(1-b).^0.5)./(1+eta_fs.*eta_fc.*(b.*(eta+b-1)).^0.5./(1-b)))+N.*pi./2; h_s=(beta.^2-n_s.^2.*k_0.^2).^0.5; h_c=(beta.^2-n_c.^2.*k_0.^2).^0.5; h_f=(n_f.^2.*k_0.^2-beta.^2).^0.5; for(x=-2.*d:1e-9:2.*d) if(x<-d) Ey=(((C_N.*cos(v.*(1-b).^0.5+phi).*exp(v.*b.^0.5./d.*(d+x))))); elseif((x>=-d)&(x<=d)) Ey=(((C_N.*cos((v.*(1-b).^0.5.*x./d-phi))))); elseif(x>d) Ey=(((C_N.*cos(v.*(1-b).^0.5-phi).*exp(v.*(eta+b-1).^0.5./d.*(d-x))))); end; Hx=-beta./omega./mu_0.*Ey; if(N==0) plot(x*1e6,(Ey),'r'); hold on; elseif(N==1) plot(x*1e6,(Ey),'b'); hold on; end; hold on; end; hold on; end; hold on; axis([-2,2,-100,100]); xlabel('x in \mum'); ylabel('E_y ( x )')

AAA 111444


%For fig. 4.3.2.1 for(N=0:1:4) b=0.01; n_f=3.5; n_s=2.0; n_c=2.0; eta_sf=(n_f./n_s).^2; eta_cf=(n_f./n_c).^2; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=(N.*pi./2)+0.5.*atan((eta-1).^0.5.*eta_cf); v=linspace(cut_off,8,2000); for(i=0:1:100) func=2.*v.*(1-b).^0.5-(N+1).*pi+atan(eta_sf.^(-1).*((1-b)./b).^0.5)+atan(eta_cf.^(-1).*((1-b)./(eta+b-1)).^0. 5); diff=-(1-b).^(-0.5).*(v+0.5.*eta_sf.^(-1).*b.^(-0.5)./(b+eta_sf.^(-2).*(1-b))+eta.*eta_cf.^(-1).*(eta+b-1).^(-0. 5)./(eta+b-1+eta_cf.^(-2).*(eta+b-1))); b=b-func./diff; end; plot(v,real(b),'r'); hold on; end; for(N=0:1:4) b=0.01; n_f=3.5; n_s=2.35; n_c=2.0; eta_sf=(n_f./n_s).^2; eta_cf=(n_f./n_c).^2; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=(N.*pi./2)+0.5.*atan((eta-1).^0.5.*eta_cf); v=linspace(cut_off,8,2000); for(i=0:1:100) func=2.*v.*(1-b).^0.5-(N+1).*pi+atan(eta_sf.^(-1).*((1-b)./b).^0.5)+atan(eta_cf.^(-1).*((1-b)./(eta+b-1)).^0. 5); diff=-(1-b).^(-0.5).*(v+0.5.*eta_sf.^(-1).*b.^(-0.5)./(b+eta_sf.^(-2).*(1-b))+eta.*eta_cf.^(-1).*(eta+b-1).^(-0. 5)./(eta+b-1+eta_cf.^(-2).*(eta+b-1))); b=b-func./diff; end; plot(v,real(b),'b'); hold on; end; for(N=0:1:4) b=0.01; n_f=3.5; n_s=2.9; n_c=2.0; eta_sf=(n_f./n_s).^2; eta_cf=(n_f./n_c).^2;

AAA 111555


eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=(N.*pi./2)+0.5.*atan((eta-1).^0.5.*eta_cf); v=linspace(cut_off,8,2000); for(i=0:1:100) func=2.*v.*(1-b).^0.5-(N+1).*pi+atan(eta_sf.^(-1).*((1-b)./b).^0.5)+atan(eta_cf.^(-1).*((1-b)./(eta+b-1)).^0. 5); diff=-(1-b).^(-0.5).*(v+0.5.*eta_sf.^(-1).*b.^(-0.5)./(b+eta_sf.^(-2).*(1-b))+eta.*eta_cf.^(-1).*(eta+b-1).^(-0. 5)./(eta+b-1+eta_cf.^(-2).*(eta+b-1))); b=b-func./diff; end; plot(v,real(b),'g'); hold on; end; for(N=0:1:4) b=0.01; n_f=3.5; n_s=3.49; n_c=2.0; eta_sf=(n_f./n_s).^2; eta_cf=(n_f./n_c).^2; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=(N.*pi./2)+0.5.*atan((eta-1).^0.5.*eta_cf); v=linspace(cut_off,8,2000); for(i=0:1:100) func=2.*v.*(1-b).^0.5-(N+1).*pi+atan(eta_sf.^(-1).*((1-b)./b).^0.5)+atan(eta_cf.^(-1).*((1-b)./(eta+b-1)).^0. 5); diff=-(1-b).^(-0.5).*(v+0.5.*eta_sf.^(-1).*b.^(-0.5)./(b+eta_sf.^(-2).*(1-b))+eta.*eta_cf.^(-1).*(eta+b-1).^(-0. 5)./(eta+b-1+eta_cf.^(-2).*(eta+b-1))); b=b-func./diff; end; plot(v,real(b),'k'); hold on; end; grid on; xlabel('normalized frequency v'); ylabel('normalized propagation constant b'); title('b~v curve for TM modes in asymmetrical SWG'); text(0.5,.1,'TM_0'); text(2.1,.1,'TM_1'); text(3.8,.1,'TM_2'); text(5.5,.1,'TM_3'); text(7.2,.1,'TM_4'); text(0.65,.9,'n_f=3.5,n_s=2.0,n_c=2.0'); text(0.65,.85,'n_f=3.5,n_s=2.35,n_c=2.0'); text(0.65,.8,'n_f=3.5,n_s=2.9,n_c=2.0'); text(0.65,.75,'n_f=3.5,n_s=3.49,n_c=2.0'); x=linspace(.2,.6,2000); y=0.9; plot(x,y,'r'); hold on; y=0.85; plot(x,y,'b');

AAA 111666


hold on; y=0.8; plot(x,y,'g'); hold on; y=0.75; plot(x,y,'k'); hold on;

% For fig. 4.3.2.2, 4.3.2.3 and 4.3.2.4 for(N=0:1:4) b=0.01; n_f=3.5; n_s=2.0; n_c=2.0; eta_sf=(n_f./n_s).^2; eta_cf=(n_f./n_c).^2; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=(N.*pi./2)+0.5.*atan((eta-1).^0.5.*eta_cf); v=linspace(cut_off,8,2000); for(i=0:1:100) func=2.*v.*(1-b).^0.5-(N+1).*pi+atan(eta_sf.^(-1).*((1-b)./b).^0.5)+atan(eta_cf.^(-1).*((1-b)./(eta+b-1)).^0. 5); diff=-(1-b).^(-0.5).*(v+0.5.*eta_sf.^(-1).*b.^(-0.5)./(b+eta_sf.^(-2).*(1-b))+eta.*eta_cf.^(-1).*(eta+b-1).^(-0. 5)./(eta+b-1+eta_cf.^(-2).*(eta+b-1))); b=b-func./diff; end; w=2.*v+(eta_sf./(b.^0.5.*(1-b+eta_sf.^2.*b))+eta_cf.*eta./((eta+b-1).^0.5.*((1-b)+eta_cf.^2.*(eta+b-1)))); confinement=(2.*v+eta_sf.*b.^0.5./((1-b)+eta_sf.^2.*b)+eta_cf.*(eta+b-1).^0.5./((1-b)+eta_cf.^2.*(eta+b-1)) )./w; deconfinement=(eta_sf.*(1-b)./(b.^0.5.*(eta_sf.^2.*b+(1-b)))+(eta_cf.*(1-b)./((eta+b-1).^0.5.*((1-b)+eta_cf.^ 2.*(eta+b-1)))))./w; figure(1); plot(v,real(w),'r'); hold on; figure(2); plot(v,confinement,'r'); hold on; figure(3); plot(v,deconfinement,'r'); hold on; end; for(N=0:1:4) b=0.01; n_f=3.5; n_s=2.35; n_c=2.0; eta_sf=(n_f./n_s).^2;

AAA 111777


eta_cf=(n_f./n_c).^2; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=(N.*pi./2)+0.5.*atan((eta-1).^0.5.*eta_cf); v=linspace(cut_off,8,2000); for(i=0:1:100) func=2.*v.*(1-b).^0.5-(N+1).*pi+atan(eta_sf.^(-1).*((1-b)./b).^0.5)+atan(eta_cf.^(-1).*((1-b)./(eta+b-1)).^0. 5); diff=-(1-b).^(-0.5).*(v+0.5.*eta_sf.^(-1).*b.^(-0.5)./(b+eta_sf.^(-2).*(1-b))+eta.*eta_cf.^(-1).*(eta+b-1).^(-0. 5)./(eta+b-1+eta_cf.^(-2).*(eta+b-1))); b=b-func./diff; end; w=2.*v+(eta_sf./(b.^0.5.*(1-b+eta_sf.^2.*b))+eta_cf.*eta./((eta+b-1).^0.5.*((1-b)+eta_cf.^2.*(eta+b-1)))); confinement=(2.*v+eta_sf.*b.^0.5./((1-b)+eta_sf.^2.*b)+eta_cf.*(eta+b-1).^0.5./((1-b)+eta_cf.^2.*(eta+b-1)) )./w; deconfinement=(eta_sf.*(1-b)./(b.^0.5.*(eta_sf.^2.*b+(1-b)))+(eta_cf.*(1-b)./((eta+b-1).^0.5.*((1-b)+eta_cf.^ 2.*(eta+b-1)))))./w; figure(1); plot(v,real(w),'b'); hold on; figure(2); plot(v,confinement,'b'); hold on; figure(3); plot(v,deconfinement,'b'); hold on; end; for(N=0:1:4) b=0.01; n_f=3.5; n_s=2.9; n_c=2.0; eta_sf=(n_f./n_s).^2; eta_cf=(n_f./n_c).^2; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=(N.*pi./2)+0.5.*atan((eta-1).^0.5.*eta_cf); v=linspace(cut_off,8,2000); for(i=0:1:100) func=2.*v.*(1-b).^0.5-(N+1).*pi+atan(eta_sf.^(-1).*((1-b)./b).^0.5)+atan(eta_cf.^(-1).*((1-b)./(eta+b-1)).^0. 5); diff=-(1-b).^(-0.5).*(v+0.5.*eta_sf.^(-1).*b.^(-0.5)./(b+eta_sf.^(-2).*(1-b))+eta.*eta_cf.^(-1).*(eta+b-1).^(-0. 5)./(eta+b-1+eta_cf.^(-2).*(eta+b-1))); b=b-func./diff; end; w=2.*v+(eta_sf./(b.^0.5.*(1-b+eta_sf.^2.*b))+eta_cf.*eta./((eta+b-1).^0.5.*((1-b)+eta_cf.^2.*(eta+b-1)))); confinement=(2.*v+eta_sf.*b.^0.5./((1-b)+eta_sf.^2.*b)+eta_cf.*(eta+b-1).^0.5./((1-b)+eta_cf.^2.*(eta+b-1)) )./w; deconfinement=(eta_sf.*(1-b)./(b.^0.5.*(eta_sf.^2.*b+(1-b)))+(eta_cf.*(1-b)./((eta+b-1).^0.5.*((1-b)+eta_cf.^ 2.*(eta+b-1)))))./w; figure(1); plot(v,real(w),'g'); hold on;

AAA 111888


figure(2); plot(v,confinement,'g'); hold on; figure(3); plot(v,deconfinement,'g'); hold on; end; for(N=0:1:4) b=0.01; n_f=3.5; n_s=3.49; n_c=2.0; eta_sf=(n_f./n_s).^2; eta_cf=(n_f./n_c).^2; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=(N.*pi./2)+0.5.*atan((eta-1).^0.5.*eta_cf); v=linspace(cut_off,8,2000); for(i=0:1:100) func=2.*v.*(1-b).^0.5-(N+1).*pi+atan(eta_sf.^(-1).*((1-b)./b).^0.5)+atan(eta_cf.^(-1).*((1-b)./(eta+b-1)).^0. 5); diff=-(1-b).^(-0.5).*(v+0.5.*eta_sf.^(-1).*b.^(-0.5)./(b+eta_sf.^(-2).*(1-b))+eta.*eta_cf.^(-1).*(eta+b-1).^(-0. 5)./(eta+b-1+eta_cf.^(-2).*(eta+b-1))); b=b-func./diff; end; w=2.*v+(eta_sf./(b.^0.5.*(1-b+eta_sf.^2.*b))+eta_cf.*eta./((eta+b-1).^0.5.*((1-b)+eta_cf.^2.*(eta+b-1)))); confinement=(2.*v+eta_sf.*b.^0.5./((1-b)+eta_sf.^2.*b)+eta_cf.*(eta+b-1).^0.5./((1-b)+eta_cf.^2.*(eta+b-1)) )./w; deconfinement=(eta_sf.*(1-b)./(b.^0.5.*(eta_sf.^2.*b+(1-b)))+(eta_cf.*(1-b)./((eta+b-1).^0.5.*((1-b)+eta_cf.^ 2.*(eta+b-1)))))./w; figure(1); plot(v,real(w),'k'); hold on; figure(2); plot(v,confinement,'k'); hold on; figure(3); plot(v,deconfinement,'k'); hold on; end; figure(1); grid on; axis([0,8,0,30]); xlabel('normalized frequency v'); ylabel('normalized effective width w_T_M'); title('plot of variation of normalized effective width with normalized frequency for TM modes in asymmetrical SWG'); text(0.7,20,'TM_0'); text(2.2,20,'TM_1'); text(3.8,20,'TM_2'); text(5.3,20,'TM_3'); text(7.3,20,'TM_4');

AAA 111999


text(5.3,6.5,'n_f=3.5,n_s=2.0,n_c=2.0'); text(5.3,5,'n_f=3.5,n_s=2.35,n_c=2.0'); text(5.3,3.5,'n_f=3.5,n_s=2.9,n_c=2.0'); text(5.3,2,'n_f=3.5,n_s=3.49,n_c=2.0'); x=linspace(7.4,7.9,2000); y=6.5; plot(x,y,'r'); hold on; y=5; plot(x,y,'b'); hold on; y=3.5; plot(x,y,'g'); hold on; y=2; plot(x,y,'k'); hold on; figure(2); grid on; xlabel('normalized frequency v'); ylabel('confinement factor ( P_f / P_N)'); title('variation of confinement factor against normalized frequency v for TM modes in asymmetrical SWG'); text(0.5,.1,'TM_0'); text(2.1,.1,'TM_1'); text(3.8,.1,'TM_2'); text(5.1,.1,'TM_3'); text(6.6,.1,'TM_4'); text(0.65,.9,'n_f=3.5,n_s=2.0,n_c=2.0'); text(0.65,.85,'n_f=3.5,n_s=2.35,n_c=2.0'); text(0.65,.8,'n_f=3.5,n_s=2.9,n_c=2.0'); text(0.65,.75,'n_f=3.5,n_s=3.49,n_c=2.0'); x=linspace(.2,.6,2000); y=0.9; plot(x,y,'r'); hold on; y=0.85; plot(x,y,'b'); hold on; y=0.8; plot(x,y,'g'); hold on; y=0.75; plot(x,y,'k'); hold on; figure(3); grid on; xlabel('normalized frequency v'); ylabel('fraction of power radiated ( P_c + P_s ) / P_N'); title('plot of variation fractional radiation with normalized frequency for TM modes in asymmetrical SWG'); text(0.5,.8,'TM_0'); text(2.1,.8,'TM_1'); text(3.8,.8,'TM_2'); text(5.3,.8,'TM_3');

AAA 222000


text(6.7,.8,'TM_4'); text(0.5,.3,'n_f=3.5,n_s=2.0,n_c=2.0'); text(0.5,.23,'n_f=3.5,n_s=2.35,n_c=2.0'); text(0.5,.15,'n_f=3.5,n_s=2.9,n_c=2.0'); text(0.5,.07,'n_f=3.5,n_s=3.49,n_c=2.0'); x=linspace(.1,.4,2000); y=0.3; plot(x,y,'r'); hold on; y=0.23; plot(x,y,'b'); hold on; y=0.15; plot(x,y,'g'); hold on; y=0.07; plot(x,y,'k'); hold on;

% For fig. 4.4.1 n_f=3.0; lambda_0=.9; for(mode=0:1:5) mode=mode+1; n_c=1.0:0.00001:(n_f-0.1); delta_n=n_f-n_c; d_0=mode.*lambda_0./(2.*(n_f.^2-n_c.^2).^0.5); plot(delta_n,d_0); hold on; end; grid on; xlabel('\Deltan'); ylabel('required thickness d_0 in \mum'); title('variation of required guide thickness vs. \Deltan for supporting different TE modes'); text(0.02,0.37,'TE_0'); text(0.02,1.14,'TE_1'); text(0.02,1.74,'TE_2'); text(0.02,2.3,'TE_3'); text(0.02,2.85,'TE_4'); text(0.02,3.35,'TE_5'); text(1.02,3.85,'n_f=3.0'); text(1.02,3.6,'\lambda_0=0.9\mum');

AAA 222111


% For fig. 4.4.2 n_f=3.0; lambda_0=0.9; for(d_0=1:1:5) n_c=1.0:0.00001:(n_f); delta_n=n_f-n_c; mode=round((n_f.^2-n_c.^2).^0.5.*2.*d_0./lambda_0); plot(delta_n,mode); hold on; end; grid on; text(1.6,4,'d_0=1\mum'); text(1.6,10,'d_0=2\mum'); text(1.6,16,'d_0=2\mum'); text(1.6,22,'d_0=3\mum'); text(1.6,28,'d_0=4\mum'); xlabel('\Deltan'); ylabel('number of supported modes\nu'); title('variation of the number of modes supported vs \Deltan for given d_0'); text(1,34,'n_f=3.0'); text(1,32,'\lambda_0=0.9\mum');

% For fig. 4.4.3 v=0:0.001:30; nu=floor(2.*v./pi); plot(v,nu); grid on; xlabel('normalized frequency v'); ylabel('mode number \nu'); title('variation ofnormlised frequency vs. maximum number of TE modes supported');

% For fig. 4.4.4 % The texts, labels and the dotted lines ae drawn in the figure window manually d=1e-6; lambda=0.633e-6; k_0=2.*pi./lambda; omega=2.*pi.*3e8./lambda; mu_0=4*pi*1e-7; for(N=0:1:2)

AAA 222222


b=0.01; n_s=(2.3-0*j).^0.5; n_f=(2.5-0*j).^0.5; n_c=(2.3-0*j).^0.5; eta_fs=1; eta_fc=1; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=real((N.*pi./2)+0.5.*atan((eta-1).^0.5.*eta_fc)); if(cut_off<0) cut_off=0.01; end; a=k_0.*d.*(n_f.^2-n_s.^2).^0.5; for(v=linspace(cut_off,a,100)) n=0; while(n<20) n=n+1; func=2.*v-(1-b).^(-0.5).*(N.*pi+atan(eta_fs.*(b./(1-b)).^0.5)+atan(eta_fc.*((eta+b-1)./(1-b)).^0.5)); diff=(2.*(1-b)).^(-1).*(func-2.*v-eta_fs./((1-b+eta_fs.^2.*b).*b.^0.5)-eta_fc.*eta./((1-b+eta_fc.^2.*(eta+b-1)).*(eta+b-1).^0.5)); b=b-func./diff; end; end; w=2.*v+(eta_fs./(b.^0.5.*(1-b+eta_fs.^2.*b))+eta_fc.*eta./((eta+b-1).^0.5.*((1-b)+eta_fc.^2.*(eta+b-1)))); W=w.*d./v; beta=k_0.*(b.*(n_f.^2-n_s.^2)+n_s.^2).^0.5 C_N=2.*(omega.*mu_0./beta./50000).^0.5.*W.^(-0.5); phi=0.5.*atan((eta_fs.*b.^0.5./(1-b).^0.5-eta_fc.*(eta+b-1).^0.5./(1-b).^0.5)./(1+eta_fs.*eta_fc.*(b.*(eta+b-1)).^0.5./(1-b)))+N.*pi./2; h_s=(beta.^2-n_s.^2.*k_0.^2).^0.5; h_c=(beta.^2-n_c.^2.*k_0.^2).^0.5; h_f=(n_f.^2.*k_0.^2-beta.^2).^0.5; for(x=-2.*d:1e-9:2.*d) if(x<-d) Ey=(((C_N.*cos(v.*(1-b).^0.5+phi).*exp(v.*b.^0.5./d.*(d+x))))); elseif((x>=-d)&(x<=d)) Ey=(((C_N.*cos((v.*(1-b).^0.5.*x./d-phi))))); elseif(x>d) Ey=(((C_N.*cos(v.*(1-b).^0.5-phi).*exp(v.*(eta+b-1).^0.5./d.*(d-x))))); end; Hx=-beta./omega./mu_0.*Ey; if(N==0) plot(x*1e6,(Ey),'r'); hold on; elseif(N==1) plot(x*1e6,(Ey),'b'); hold on; elseif(N==2) plot(x*1e6,(Ey),'g'); hold on; end; hold on;

AAA 222333


end; hold on; end; hold on; axis([-2,2,-100,100]); xlabel('x in \mum'); ylabel('E_y ( x )')

% For Fig. 6.3.2.1 for(n=0:1:3) if(n==0) e_1=(1.2-7*j)^2;% Al elseif(n==1) e_1=(3.19-2.26*j)^2;% Cr elseif(n==2) e_1=(0.15-3.2*j)^2;% Au elseif(n==3) e_1=(0.065-4*j)^2;% Ag end; e_2=1.5.^2; beta_k_0=(e_1.*e_2./(e_1+e_2)).^0.5; lambda=0.633e-6; k_0=2*pi./lambda; beta=beta_k_0.*k_0; h_1=(beta.^2-k_0.^2.*e_1).^0.5; h_2=(beta.^2-k_0.^2.*e_2).^0.5; for(x=linspace(-0.5e-6,1.5e-6,6000)) if(x<=0) Hy=1.*exp(h_1.*x); else Hy=1.*exp(-h_2.*x); end; if(n==0) plot(x.*1e6,Hy,'r');% Al elseif(n==1) plot(x.*1e6,Hy,'b');% Cr elseif(n==2) plot(x.*1e6,Hy,'g');% Au elseif(n==3) plot(x.*1e6,Hy,'m');% Ag end; hold on; end; end; xlabel('distance x in \mum'); ylabel('H_y ( x )'); title('H_y field for different metals in single interface SWG');

AAA 222444


x=linspace(-0.2,1.2,100) plot(0,x,'k'); hold on; text(0.55,1,'\lambda = 0.633 \mum'); text(0.75,0.9,'Al'); text(0.75,0.8,'Cr'); text(0.75,0.7,'Au'); text(0.75,0.6,'Ag'); x=linspace(0.6,0.7,50); plot(x,0.9,'r',x,0.8,'b',x,0.7,'g',x,0.6,'m'); hold on; text(-0.4,0.4,'metal'); text(0.5,0.4,'dielectric ( \epsilon_2 = 2.25 )'); x=linspace(-0.5,1.5,100); plot(x,0,'k');

% For fig. 7.5.1, 7.5.2 and 7.5.3 for(N=0:1:3) b=0.01; n_f=(2.523-0*i).^0.5; n_s=(1-0*i); n_c=(-16.32-0.5414.*i).^0.5; eta_fs=1; eta_fc=1; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=real((N.*pi./2)+0.5.*atan((eta-1).^0.5.*eta_fc)); for(v=logspace(log10(cut_off),log10(80),5000)) n=0; while(n<20) n=n+1; func=2.*v-(1-b).^(-0.5).*(N.*pi+atan(eta_fs.*(b./(1-b)).^0.5)+atan(eta_fc.*((eta+b-1)./(1-b)).^0.5)); diff=(2.*(1-b)).^(-1).*(func-2.*v-eta_fs./((1-b+eta_fs.^2.*b).*b.^0.5)-eta_fc.*eta./((1-b+eta_fc.^2.*(eta+b-1)).*(eta+b-1).^0.5)); b=b-func./diff; end; beta_by_k0=(b.*(n_f.^2-n_s.^2)+n_s.^2).^0.5; k0_into_2d=real(2.*v./(n_f.^2-n_s.^2).^0.5); if(N==0) figure(1); plot(v,b,'r'); hold on; figure(2); semilogx(k0_into_2d,real(beta_by_k0-n_s),'r'); hold on; figure(3); loglog(k0_into_2d,imag(-beta_by_k0),'r'); hold on;

AAA 222555


elseif(N==1) figure(1); plot(v,b,'b'); hold on; figure(2); semilogx(k0_into_2d,real(beta_by_k0-n_s),'b'); hold on; figure(3); loglog(k0_into_2d,imag(-beta_by_k0),'b'); hold on; elseif(N==2) figure(1); plot(v,b,'g'); hold on; figure(2); semilogx(k0_into_2d,real(beta_by_k0-n_s),'g'); hold on; figure(3); loglog(k0_into_2d,imag(-beta_by_k0),'g'); hold on; elseif(N==3) figure(1); plot(v,b,'k'); hold on; figure(2); semilogx(k0_into_2d,real(beta_by_k0-n_s),'k'); hold on; figure(3); loglog(k0_into_2d,imag(-beta_by_k0),'k'); hold on; end; hold on; end; end; hold on; figure(1); grid on; axis([0,8,0,1]); title('plot of variation of b against v for TE modes in metal cladded SWG'); xlabel('normalized frequency v'); ylabel('normalized propagation constant b'); text(0.5,0.9,'n_f = ( 2.523 -0 j )^0^.^5 ( Dielectric polymer film )'); text(0.5,0.82,'n_s = ( 1.0 - 0 j )^0^.^5 ( Air )'); text(0.5,0.74,'n_c= ( -16.32 - 0.5414 j )^0^.^5 ( Silver )'); text(0.5,0.66,'\lambda = 0.6328 \mum'); text(0.6,0.1,'TE_0'); text(2.4,0.1,'TE_1'); text(4.0,0.1,'TE_2'); text(5.7,0.1,'TE_3'); figure(2); grid on;

AAA 222666


axis([0.1,100,0,0.8]); title('\beta / k_0 vs. normalized thickness 2k_0d for TE modes in Silver cladded SWG'); xlabel('2k_0d'); ylabel('\beta / k_0 '); text(0.13,0.77,'n_f = ( 2.523 -0 j )^0^.^5 ( Dielectric polymer film )'); text(0.13,0.675,'n_s = ( 1.0 - 0 j )^0^.^5 ( Air )'); text(0.13,0.725,'n_c= ( -16.32 - 0.5414 j )^0^.^5 ( Silver )'); text(0.13,0.64,'\lambda = 0.6328 \mum'); text(4,0.525,'TE_0'); text(6,0.425,'TE_1'); text(8.0,0.3,'TE_2'); text(9,0.2,'TE_3'); figure(3); grid on; axis([0.1,100,1e-8,1e-1]); title('\alpha / k_0 vs. normalized thickness 2k_0d for TE modes in Silver cladded SWG'); xlabel('2k_0d'); ylabel('\alpha / k_0 '); text(0.12,4e-7,'n_f = ( 2.523 -0 j )^0^.^5 ( Dielectric polymer film )'); text(0.12,1.5e-7,'n_s = ( 1.0 - 0 j )^0^.^5 ( Air )'); text(0.12,6e-8,'n_c = ( -16.32 - 0.5414 j )^0^.^5 ( Silver )'); text(0.12,2.5e-8,'\lambda = 0.6328 \mum'); text(1.2,2e-3,'TE_0'); text(3.5,1.5e-3,'TE_1'); text(6,1e-3,'TE_2'); text(8.5,0.8e-3,'TE_3');

% For fig. 7.5.4, 7.5.5 and 7.5.6 for(N=0:1:3) b=0.01; n_f=(2.523-0*i).^0.5; n_s=(1-0*i); n_c=(-16.32-0.5414.*i).^0.5; eta_fs=(n_f./n_s).^2; eta_fc=(n_f./n_c).^2; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=real((N.*pi./2)+0.5.*atan((eta-1).^0.5.*eta_fc)); for(v=linspace(cut_off,80,4000)) n=0; while(n<20) n=n+1; func=2.*v-(1-b).^(-0.5).*(N.*pi+atan(eta_fs.*(b./(1-b)).^0.5)+atan(eta_fc.*((eta+b-1)./(1-b)).^0.5)); diff=(2.*(1-b)).^(-1).*(func-2.*v-eta_fs./((1-b+eta_fs.^2.*b).*b.^0.5)-eta_fc.*eta./((1-b+eta_fc.^2.*(eta+b-1)).*(eta+b-1).^0.5)); b=b-func./diff; if((b>=1.30)&(N==0)) b=1.302935;

AAA 222777


n=21; end; end; beta_by_k0=(b.*(n_f.^2-n_s.^2)+n_s.^2).^0.5; k0_into_2d=real(2.*v./(n_f.^2-n_s.^2).^0.5); if(N==0) figure(1); plot(v,b,'r'); hold on; figure(2); semilogx(k0_into_2d,real(beta_by_k0-n_s),'r'); hold on; figure(3); loglog(k0_into_2d,imag(-beta_by_k0),'r'); hold on; elseif(N==1) figure(1); plot(v,b,'b'); hold on; figure(2); semilogx(k0_into_2d,real(beta_by_k0-n_s),'b'); hold on; figure(3); loglog(k0_into_2d,imag(-beta_by_k0),'b'); hold on; elseif(N==2) figure(1); plot(v,b,'g'); hold on; figure(2); semilogx(k0_into_2d,real(beta_by_k0-n_s),'g'); hold on; figure(3); loglog(k0_into_2d,imag(-beta_by_k0),'g'); hold on; elseif(N==3) figure(1); plot(v,b,'k'); hold on; figure(2); semilogx(k0_into_2d,real(beta_by_k0-n_s),'k'); hold on; figure(3); loglog(k0_into_2d,imag(-beta_by_k0),'k'); hold on; end; hold on; end; end; figure(1); axis([0,8,0,1.4]);

AAA 222888


grid on; title('plot of variation of b against v for TM modes in metal cladded SWG'); xlabel('normalized frequency v'); ylabel('normalized propagation constant b'); text(3,1.22,'n_f = ( 2.523 -0 j )^0^.^5 ( Dielectric polymer film )'); text(3,1.14,'n_s = ( 1.0 - 0 j )^0^.^5 ( Air )'); text(3,1.06,'n_c= ( -16.32 - 0.5414 j )^0^.^5 ( Silver )'); text(3,0.98,'\lambda = 0.6328 \ mum'); text(0.2,0.5,'TM_0'); text(2.3,0.5,'TM_1'); text(4.7,0.5,'TM_2'); text(7.2,0.5,'TM_3'); figure(2); grid on; axis([0.1,100,0,0.8]); title('effective refractive index \beta / k_0 vs. normalized thickness 2k_0d for TM modes in Silver cladded SWG'); xlabel('2k_0d'); ylabel('( \beta / k_0 )- n_s'); text(0.12,0.76,'n_f = ( 2.523 -0 j )^0^.^5 ( Dielectric polymer film )'); text(0.12,0.68,'n_s = ( 1.0 - 0 j )^0^.^5 ( Air )'); text(0.12,0.72,'n_c= ( -16.32 - 0.5414 j )^0^.^5 ( Silver )'); text(0.12,0.64,'\lambda = 0.6328 \ mum'); text(0.9,0.5,'TM_0'); text(4,0.4,'TM_1'); text(6.5,0.32,'TM_2'); text(9,0.22,'TM_3'); figure(3); axis([0.1,100,1e-8,1e-1]); title(' \alpha / k_0 vs. normalized thickness 2k_0d for TM modes in Silver cladded SWG'); xlabel('2k_0d'); ylabel('\alpha / k_0'); text(0.12,4e-7,'n_f = ( 2.523 -0 j )^0^.^5 ( Dielectric polymer film )'); text(0.12,1.5e-7,'n_s = ( 1.0 - 0 j )^0^.^5 ( Air )'); text(0.12,6e-8,'n_c = ( -16.32 - 0.5414 j )^0^.^5 ( Silver )'); text(0.12,2.5e-8,'\lambda = 0.6328 \mum'); text(0.9,0.005,'TM_0'); text(3,2.5e-3,'TM_1'); text(5.7,1.5e-3,'TM_2'); text(10,1e-3,'TM_3');

% For fig. 7.6.1, 7.6.2 and 7.6.3 d=1e-6; lambda=0.633e-6; k_0=2.*pi./lambda; omega=2.*pi.*3e8./lambda; mu_0=4*pi*1e-7;

AAA 222999


for(N=0:1:1) b=0.01; n_f=(2.3-0*j).^0.5; n_s=(1-0*j).^0.5; n_c=(-16.32-0.5414.*j).^0.5; eta_fs=1; eta_fc=1; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=real((N.*pi./2)+0.5.*atan((eta-1).^0.5.*eta_fc)); if(cut_off<0) cut_off=0.01; end; a=k_0.*d.*(n_f.^2-n_s.^2).^0.5; for(v=linspace(cut_off,a,100)) n=0; while(n<20) n=n+1; func=2.*v-(1-b).^(-0.5).*(N.*pi+atan(eta_fs.*(b./(1-b)).^0.5)+atan(eta_fc.*((eta+b-1)./(1-b)).^0.5)); diff=(2.*(1-b)).^(-1).*(func-2.*v-eta_fs./((1-b+eta_fs.^2.*b).*b.^0.5)-eta_fc.*eta./((1-b+eta_fc.^2.*(eta+b-1)).*(eta+b-1).^0.5)); b=b-func./diff; end; end; w=2.*v+(eta_fs./(b.^0.5.*(1-b+eta_fs.^2.*b))+eta_fc.*eta./((eta+b-1).^0.5.*((1-b)+eta_fc.^2.*(eta+b-1)))); W=w.*d./v; beta=k_0.*(b.*(n_f.^2-n_s.^2)+n_s.^2).^0.5; C_N=2.*(omega.*mu_0./beta./2000).^0.5.*W.^(-0.5); phi=0.5.*atan((eta_fs.*b.^0.5./(1-b).^0.5-eta_fc.*(eta+b-1).^0.5./(1-b).^0.5)./(1+eta_fs.*eta_fc.*(b.*(eta+b-1)).^0.5./(1-b)))+N.*pi./2; h_s=(beta.^2-n_s.^2.*k_0.^2).^0.5; h_c=(beta.^2-n_c.^2.*k_0.^2).^0.5; h_f=(n_f.^2.*k_0.^2-beta.^2).^0.5; for(x=-2.*d:1e-9:2.*d) if(x<-d) Ey=(((C_N.*cos(v.*(1-b).^0.5+phi).*exp(v.*b.^0.5./d.*(d+x))))); Hz=-j/(omega.*mu_0).*h_s.*Ey; elseif((x>=-d)&(x<=d)) Ey=(((C_N.*cos((v.*(1-b).^0.5.*x./d-phi))))); Hz=j/(omega.*mu_0).*h_f.*C_N.*sin(h_f.*x-phi); elseif(x>d) Ey=(((C_N.*cos(v.*(1-b).^0.5-phi).*exp(v.*(eta+b-1).^0.5./d.*(d-x))))); Hz=j/(omega.*mu_0).*h_c.*Ey; end; Hx=-beta./omega./mu_0.*Ey; if(N==0) figure(1); plot(x*1e6,abs(Ey),'r'); hold on; figure(2); plot(x*1e6,abs(Hx),'r'); hold on;

AAA 333000


figure(3); plot(x*1e6,abs(Hz),'r'); hold on; elseif(N==1) figure(1); plot(x*1e6,abs(Ey),'b'); hold on; figure(2); plot(x*1e6,abs(Hx),'b'); hold on; figure(3); plot(x*1e6,abs(Hz),'b'); hold on; end; hold on; end; hold on; end; figure(1); axis([-2,2,0,800]); figure(2); axis([-2,2,0,3]); figure(3); axis([-2,2,0,0.6]);

% For fig. 7.6.4, 7.6.5 and 7.6.6 d=1e-6; lambda=0.633e-6; k_0=2.*pi./lambda; omega=2.*pi.*3e8./lambda; epsilon_0=8.854e-12; for(N=0:1:1) b=0.01; n_f=(2.523-0*j).^0.5; n_s=(1-0*j).^0.5; n_c=(-16.32-0.5414.*j).^0.5; eta_fs=(n_f./n_s).^2; eta_fc=(n_f./n_c).^2; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=real((N.*pi./2)+0.5.*atan((eta-1).^0.5.*eta_fc)); if(cut_off<0) cut_off=0.01; end; a=k_0.*d.*(n_f.^2-n_s.^2).^0.5; for(v=linspace(cut_off,a,40)) n=0; while(n<20)

AAA 333111


n=n+1; func=2.*v-(1-b).^(-0.5).*(N.*pi+atan(eta_fs.*(b./(1-b)).^0.5)+atan(eta_fc.*((eta+b-1)./(1-b)).^0.5)); diff=(2.*(1-b)).^(-1).*(func-2.*v-eta_fs./((1-b+eta_fs.^2.*b).*b.^0.5)-eta_fc.*eta./((1-b+eta_fc.^2.*(eta+b-1)).*(eta+b-1).^0.5)); b=b-func./diff; end; end; w=2.*v+(eta_fs./(b.^0.5.*(1-b+eta_fs.^2.*b))+eta_fc.*eta./((eta+b-1).^0.5.*((1-b)+eta_fc.^2.*(eta+b-1)))); W=w.*d./v; beta=k_0.*(b.*(n_f.^2-n_s.^2)+n_s.^2).^0.5; C_N=2.*(omega.*epsilon_0.*n_f.^2./beta./2000).^0.5.*W.^(-0.5); phi=0.5.*atan((eta_fs.*b.^0.5./(1-b).^0.5-eta_fc.*(eta+b-1).^0.5./(1-b).^0.5)./(1+eta_fs.*eta_fc.*(b.*(eta+b-1)).^0.5./(1-b))); h_s=(beta.^2-n_s.^2.*k_0.^2).^0.5; h_c=(beta.^2-n_c.^2.*k_0.^2).^0.5; h_f=(n_f.^2.*k_0.^2-beta.^2).^0.5; for(x=-2.*d:1e-10:2.*d) if(x<-d) Hy=(abs((C_N.*cos(v.*(1-b).^0.5+phi).*exp(v.*b.^0.5./d.*(d+x))))); Ex=beta./(n_s.^2.*epsilon_0.*omega).*Hy; Ez=j./(n_s.^2.*epsilon_0.*omega).*Hy.*h_s; elseif((x>=-d)&(x<=d)) Hy=(abs((C_N.*cos((v.*(1-b).^0.5.*x./d-phi))))); Ex=beta./(n_f.^2.*epsilon_0.*omega).*Hy; Ez=-j./(n_f.^2.*epsilon_0.*omega).*h_f.*(C_N.*sin(h_f.*x-phi)); elseif(x>d) Hy=(abs((C_N.*cos(v.*(1-b).^0.5-phi).*exp(v.*(eta+b-1).^0.5./d.*(d-x))))); Ex=beta./(n_c.^2.*epsilon_0.*omega).*Hy; Ez=-j./(n_c.^2.*epsilon_0.*omega).*h_c.*Hy; end; if(N==0) figure(1); plot(x*1e6,Hy,'r'); hold on; figure(2); plot(x*1e6,abs(Ex),'r'); hold on; figure(3); plot(x*1e6,abs(Ez),'r'); hold on; elseif(N==1) figure(1); plot(x*1e6,Hy,'b'); hold on; figure(2); plot(x*1e6,abs(Ex),'b'); hold on; figure(3); plot(x*1e6,abs(Ez),'b'); hold on; end;

AAA 333222


hold on; end; hold on; end; figure(1); axis([-2,2,0,8]); figure(2); axis([-2,2,0,2000]); figure(3); axis([-2,2,0,700]);

% For fig. 7.7.1 for(f=0:1:1) for(N=0:1:3) lambda=0.633*1e-6; k_0=2*pi/lambda; d=1e-6; n_f=(2.523-0*i).^0.5; for(n_s=1.0:0.001:n_f-0.01) n_c=(-16.32-0.5414.*i).^0.5; if(f==0) eta_fs=(n_f./n_s).^2; eta_fc=(n_f./n_c).^2; else eta_fs=1; eta_fc=1; end; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=real(N.*pi./2+0.5.*atan((eta-1).^0.5.*eta_fc)); b=0.01; v_0=k_0*d*(n_f.^2-n_s.^2).^0.5; for(v=linspace(cut_off,v_0,100)) n=0; while(n<10) n=n+1; func=2.*v-(1-b).^(-0.5).*(N.*pi+atan(eta_fs.*(b./(1-b)).^0.5)+atan(eta_fc.*((eta+b-1)./(1-b)).^0.5)); diff=(2.*(1-b)).^(-1).*(func-2.*v-eta_fs./((1-b+eta_fs.^2.*b).*b.^0.5)-eta_fc.*eta./((1-b+eta_fc.^2.*(eta +b-1)).*(eta+b-1).^0.5)); b=b-func./diff; end; end; d_n=n_f-n_s; beta_by_k0=(b.*(n_f.^2-n_s.^2)+n_s.^2).^0.5; alpha=-imag(beta_by_k0.*k_0); loss_dB_mum=8.68.*alpha./1e6; if(N==0) semilogy(d_n,loss_dB_mum,'r');

AAA 333333


hold on; elseif(N==1) semilogy(d_n,loss_dB_mum,'b'); hold on; elseif(N==2) semilogy(d_n,loss_dB_mum,'g'); hold on; elseif(N==3) semilogy(d_n,loss_dB_mum,'k'); hold on; end; end; hold on; end; hold on; end; hold on; x=linspace(0,1,5) for(axis_count=0:1:5) y=x.^0.*10^(-axis_count); loglog(x,y,'k:'); hold on; end; y=linspace(1e-5,1,5) for(axis_count=0.1:0.1:1) x=y.^0.*axis_count; loglog(x,y,'k:'); hold on; end; text(0.02,5*1e-1,'TM_0'); text(0.01,1*1e-3,'TM_1'); text(0.02,3*1e-3,'TM_2'); text(0.07,8*1e-3,'TM_3'); text(0.02,1*1e-4,'TE_0'); text(0.02,3*1e-4,'TE_1'); text(0.05,1*1e-3,'TE_2'); text(0.11,2*1e-3,'TE_3'); axis([0,0.6,1e-5,1]); xlabel('\Delta n'); ylabel('Propagation loss in dB / \mum'); title('Variation of propagation loss with index difference for variour TE and TM modes in Metal cladded SWG') text(0.4,3*1e-1,'n_f^2 = 2.523'); text(0.4,1.5e-1,'d = 1 \mum'); text(0.4,8*1e-2,'\epsilon_r ( Ag ) = - 16.32 - 0.5414 i'); text(0.4,5e-2,'\lambda = 0.633 \mum');

AAA 333444


% For fig. 7.7.2 n_f=2.523^0.5; n_s=1; for(n0_of_metal=1:1:4) for(no_of_wavelength=1:1:2) if(no_of_wavelength==1) lambda=0.633e-6; if(n0_of_metal==1) n_c=1.2-7*i;% Al elseif(n0_of_metal==2) n_c=0.148-3.211*i; % Au elseif(n0_of_metal==3) n_c=0.15-3.2*i;% Cu else n_c=0.065-4*i;% Ag end; else lambda=10.6e-6; if(n0_of_metal==1) n_c=25-67*i;% Al elseif(n0_of_metal==2) n_c=12.6-64.3*i; % Au elseif(n0_of_metal==3) n_c=7.4-53.4*i;% Cu else n_c=10.7-69*i;% Ag end; end; k0=2*pi./lambda; eta_fs=(n_f./n_s).^2; eta_fc=(n_f./n_c).^2; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=real(0.5.*atan((eta-1).^0.5.*eta_fc)); if(cut_off<0) cut_off=0.01; end; b=1e-4; d_0=3e-6; v_0=k0.*d_0.*(n_f.^2-n_s.^2).^0.5; for(v=logspace(-2,log(v_0),500)) k=0; while(k<20) k=k+1; func=2.*v-(1-b).^(-0.5).*(atan(eta_fs.*(b./(1-b)).^0.5)+atan(eta_fc.*((eta+b-1)./(1-b)).^0.5)); diff=(2.*(1-b)).^(-1).*(func-2.*v-eta_fs./((1-b+eta_fs.^2.*b).*b.^0.5)-eta_fc.*eta./((1-b+eta_fc.^2.*(eta+ b-1)).*(eta+b-1).^0.5)); b=b-func./diff; end;

AAA 333555


beta_by_k0=(b.*(n_f.^2-n_s.^2)+n_s.^2).^0.5; k0_into_2d=real(2.*v./(n_f.^2-n_s.^2).^0.5); d_into_2=k0_into_2d./k0; alpha=(8.68.*imag(-beta_by_k0*k0))./1e6; if(n0_of_metal==1) loglog(d_into_2*1e6,alpha,'r'); hold on; elseif(n0_of_metal==2) loglog(d_into_2*1e6,alpha,'b'); hold on; elseif(n0_of_metal==3) loglog(d_into_2*1e6,alpha,'g'); hold on; elseif(n0_of_metal==4) loglog(d_into_2*1e6,alpha,'k'); hold on; end; hold on; end; hold on; end; hold on; end; hold on; axis([0,10,1e-4,1e1]); xlabel('Thickness of the guide 2d in \mum'); ylabel('Propagation loss in dB / \mum'); title('Propagation loss for some metals at different wavelengths'); text(3,2,'\lambda = 0.633 \mum'); text(3,4e-2,'\lambda = 10.6 \mum'); x=linspace(3,3.25,100); y=2e-3; plot(x,y,'r'); y=1e-3; plot(x,y,'b'); y=5e-4; plot(x,y,'g'); y=2.3e-4; plot(x,y,'k'); text(3.4,2e-3,'Al'); text(3.4,1e-3,'Au'); text(3.4,5e-4,'Cu'); text(3.4,2.3e-4,'Ag'); grid on; x=linspace(0.01,100,50) for(axis_count=0:1:8) y=x.^0.*10^(-axis_count); loglog(x,y,'k:'); hold on; end;

AAA 333666


%For fig. 7.8.1 for(f=0:1:1) lambda=0.633*1e-6; k_0=2.*pi./lambda; for(N=0:1:3) b=0.01; n_f=(2.523-0*i).^0.5; n_s=(1-0*i); n_c=(-16.32-0.5414.*i).^0.5; if(f==0) eta_fs=(n_f./n_s).^2; eta_fc=(n_f./n_c).^2; else eta_fs=1; eta_fc=1; end; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=real((N.*pi./2)+0.5.*atan((eta-1).^0.5.*eta_fc)); if(cut_off<0) cut_off=0.75; end; for(v=linspace(cut_off,20,10000)) n=0; while(n<20) n=n+1; func=2.*v-(1-b).^(-0.5).*(N.*pi+atan(eta_fs.*(b./(1-b)).^0.5)+atan(eta_fc.*((eta+b-1)./(1-b)).^0.5)); diff=(2.*(1-b)).^(-1).*(func-2.*v-eta_fs./((1-b+eta_fs.^2.*b).*b.^0.5)-eta_fc.*eta./((1-b+eta_fc.^2.*(eta+ b-1)).*(eta+b-1).^0.5)); b=b-func./diff; if(b>1.30) b=1.302945; end; end; d=v./(k_0.*(n_f.^2-n_s.^2).^0.5); beta_by_k0=(b.*(n_f.^2-n_s.^2)+n_s.^2).^0.5; beta=beta_by_k0.*k_0; if((N==0)&(f==0)) H_f=(beta.^2-k_0.^2.*n_f.^2).^0.5; h_c=(beta.^2-k_0.^2.*n_c.^2).^0.5; W=(H_f.^(-1)+h_c.^(-1))*1e6; else w=2.*v+(eta_fs./(b.^0.5.*(1-b+eta_fs.^2.*b))+eta_fc.*eta./((eta+b-1).^0.5.*((1-b)+eta_fc.^2.*(eta+b-1)) )); W=w.*d./v.*1e6; end; k0_into_2d=real(2.*v./(n_f.^2-n_s.^2).^0.5); if(N==0) plot(k0_into_2d,real(W),'r');

AAA 333777


hold on; elseif(N==1) plot(k0_into_2d,real(W),'b'); hold on; elseif(N==2) plot(k0_into_2d,real(W),'g'); hold on; elseif(N==3) plot(k0_into_2d,real(W),'k'); hold on; end; hold on; end; hold on; end; end; hold on; grid on; axis([0,20,0,5]); text(1.2,1,'TE_0'); text(3.6,1.5,'TE_1'); text(6.2,2,'TE_2'); text(8.8,2.5,'TE_3'); text(2,0.25,'TM_0'); text(3,0.75,'TM_1'); text(5,1,'TM_2'); text(8,1.25,'TM_3'); xlabel('2k_0d'); ylabel('Effective width in \mum'); title('Effective beam width vs. normalized physical width for TE and TM modes in metal cladded SWG'); text(12,4.5,'n_f^2 = 2.523'); text(12,4.2,'n_s^2 = 1'); text(12,3.9,'\epsilon_r ( Ag ) = -1 6.32 - 0.5414 i'); text(12,3.6,'d = 1 \mum'); text(12,3.3,'\lambda = 0.633 \mum');

% For fig. 8.3.1 p=0; while(p<2) p=p+1; d=1e-6; lambda=0.633e-6; k_0=2.*pi./lambda; omega=2.*pi.*3e8./lambda; epsilon_0=8.854e-12; n_s=2.3.^0.5; n_c=2.2.^0.5;

AAA 333888


n_f=(-16.32-0.5414*i).^0.5; eta_fs=(n_f./n_s).^2; eta_fc=(n_f./n_c).^2; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); if(p==1) beta_k_0=(n_f.^2.*n_c.^2./(n_f.^2+n_c.^2)).^0.5; else beta_k_0=(n_f.^2.*n_s.^2./(n_f.^2+n_s.^2)).^0.5; end; beta=k_0.*beta_k_0; h_f=(-beta.^2+k_0.^2.*n_f.^2).^0.5; h_s=(beta.^2-k_0.^2.*n_s.^2).^0.5; h_c=(beta.^2-k_0.^2.*n_c.^2).^0.5; C_N=0.5e-21; if(p==1) phase=0; else phase=pi./2; end; phi=0.5.*atan((eta_fs.*h_s./h_f-eta_fc.*h_c./h_f)./(1+h_c.*h_s.*eta_fc.*eta_fs./h_f.^2))+phase; for(x=linspace(-5.*d,5.*d,1e4)) if(x<-d) Hy=(abs((C_N.*cos(h_f.*d+phi).*exp(h_s.*(d+x))))); elseif((x>=-d)&(x<=d)) Hy=(abs((C_N.*cos(h_f.*x-phi)))); elseif(x>d) Hy=(abs((C_N.*cos(h_f.*d-phi).*exp(h_c.*(d-x))))); end; if(p==1) plot(x*1e6,Hy,'r'); else plot(x*1e6,Hy,'b'); end; hold on; end; hold on; end; line1=linspace(0,8,7); x1=line1.^0; x2=-line1.^0; plot(x1,line1,'k:',x2,line1,'k:'); ylabel('|H_y(x)|'); xlabel('x in \mum'); title('Field amplitude distribution of |H_y(x)| for TM modes in Ag guide SWG'); text(-3.8,4,'Substrate'); text(-0.9,4,'Silver guide '); text(2.4,4,'Cladding'); text(-4.5,7.6,'Guide thickness 2d = 2 \mum'); text(-4.5,7.1,'n_s^2 = 2.3'); text(-4.5,6.5,'n_c^2 = 2.2'); text(-4.5,6.0,'\epsilon_r( Ag )= - 16.32 - 0.5415i');

AAA 333999


text(-4.5,5.5,'\lambda = 0.633 \mum');

% For fig. 8.3.2, 8.3.3 and 8.3.4 n_f=(-16.32-0.5414*i)^0.5; n_s=2.3^0.5; n_c=2.2^0.5; eta_fs=(n_f./n_s).^2; eta_fc=(n_f./n_c).^2; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); n_1_eff=n_c; lambda=0.633e-6; k_0=2.*pi./lambda; beta_k0=(n_s.^2.*n_f.^2./(n_s.^2+n_f.^2)).^0.5; beta=beta_k0.*k_0; for(d=linspace(200e-9,1e-9,2001)) for(k=0:1:20) h_f=(beta.^2-k_0.^2.*n_f.^2).^0.5; h_s=(beta.^2-k_0.^2.*n_s.^2).^0.5; h_c=(beta.^2-k_0.^2.*n_c.^2).^0.5; func=(h_f.^2+eta_fs.*eta_fc.*h_s.*h_c).*tanh(2.*h_f.*d)+h_f.*(eta_fs.*h_s+eta_fc.*h_c); diff=(2+eta_fs.*eta_fc.*(h_s./h_c+h_c./h_s)).*beta.*tanh(2.*h_f.*d)+2.*d.*beta./h_f.*(h_f.^2+eta_fs.*eta_fc.*h_s.*h_c).*(sech(2.*h_f.*d)).^2+eta_fs.*beta.*(h_f./h_s+h_s./h_f)+eta_fc.*beta.*(h_f./h_c+h_c./h_f); beta=beta-func./diff; end; beta_k0=beta./k_0; figure(1); plot(2.*k_0.*d,beta_k0-n_s,'b'); hold on; figure(2); semilogy(2.*k_0.*d,imag(-beta_k0),'b'); hold on; b=(beta_k0.^2-n_s.^2)./(n_s.^2-n_f.^2); v=k_0.*d.*(n_s.^2-n_f.^2).^0.5; figure(3); plot(v,b,'b'); hold on; end; beta_k0=(n_c.^2.*n_f.^2./(n_c.^2+n_f.^2)).^0.5; beta=beta_k0.*k_0; for(d=linspace(200e-9,5e-9,2000)) for(k=0:1:20) h_f=(beta.^2-k_0.^2.*n_f.^2).^0.5; h_s=(beta.^2-k_0.^2.*n_s.^2).^0.5; h_c=(beta.^2-k_0.^2.*n_c.^2).^0.5; func=(h_f.^2+eta_fs.*eta_fc.*h_s.*h_c).*tanh(2.*h_f.*d)+h_f.*(eta_fs.*h_s+eta_fc.*h_c);

AAA 444000


diff=(2+eta_fs.*eta_fc.*(h_s./h_c+h_c./h_s)).*beta.*tanh(2.*h_f.*d)+2.*d.*beta./h_f.*(h_f.^2+eta_fs.*eta_fc.*h_s.*h_c).*(sech(2.*h_f.*d)).^2+eta_fs.*beta.*(h_f./h_s+h_s./h_f)+eta_fc.*beta.*(h_f./h_c+h_c./h_f); beta=beta-func./diff; end; beta_k0=beta./k_0; figure(1); plot(2.*k_0.*d,beta_k0-n_s,'r'); hold on; figure(2); semilogy(2.*k_0.*d,imag(-beta_k0),'r'); hold on; b=(beta_k0.^2-n_s.^2)./(n_s.^2-n_f.^2); v=k_0.*d.*(n_s.^2-n_f.^2).^0.5; figure(3); plot(v,b,'r'); hold on; end; figure(1); axis([0,4,0,0.3]); grid on; xlabel('2k_0d'); ylabel('\beta / k_0 - n_s'); title('Effective index variation against normalized guide thickness for metallic guide SWG'); text(2.5,0.25,'n_s^2 = 2.3'); text(2.5,0.225,'n_c^2 = 2.2'); text(2.5,0.2,'\epsilon _r ( Ag ) = - 16.32 - i 0.5414'); text(2.5,0.175,'\lambda = 0.633 \mu m'); figure(2); axis([0,4,1e-4,1]); grid on; xlabel('2k_0d'); ylabel('\alpha / k_0 - n_s'); title('Variation of \alpha / k_0 against normalized guide thickness for metallic guide SWG'); text(2.5,2e-1,'n_s^2 = 2.3'); text(2.5,1e-1,'n_c^2 = 2.2'); text(2.5,5.5e-2,'\epsilon _r ( Ag ) = - 16.32 - i 0.5414'); text(2.5,2.5e-2,'\lambda = 0.633 \mu m'); figure(3); axis([0,8,0,0.06]); grid on; xlabel('V'); ylabel('B'); title('Variation of normalized prop. const. with normalized guide thickness for metallic guide SWG'); text(5,0.05,'n_s^2 = 2.3'); text(5,0.045,'n_c^2 = 2.2'); text(5,0.04,'\epsilon _r ( Ag ) = - 16.32 - i 0.5414'); text(5,0.035,'\lambda = 0.633 \mu m');

AAA 444111


% For fig. 8.4.1, 8.4.2 and 8.4.3 d=0.04*1e-6; lambda=0.633e-6; k_0=2.*pi./lambda; epsilon_0=8.854e-12; omega=2*pi*3e8./lambda; for(N=0:1:1) n_f=(-16.32-0.5414*i)^0.5; n_s=2.3^0.5; n_c=2.2^0.5; eta_fs=(n_f./n_s).^2; eta_fc=(n_f./n_c).^2; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); v_0=k_0.*d.*(n_s.^2-n_f.^2).^0.5; cut_off=4; if(N==0) b=eta.*eta_fc.^2./(eta_fc.^2-1)-1; else b=1./(eta_fs.^2-1); end; for(v=cut_off:-0.01:v_0) for(k=0:1:20) func=2.*v+(1+b).^(-0.5).*(atanh(eta_fs.*(b./(1+b)).^0.5)+atanh(eta_fc.*((b+1-eta)./(1+b)).^0.5)); diff=(2.*(1+b)).^(-0.5).*(func-2.*v+eta_fs./(b.^0.5.*(1+b-eta_fs.^2.*b))+eta.*eta_fc./((b+1-eta).^0.5.*(1+b-eta_fc.^2.*(1+b-eta)))); b=b-func./diff; end; hold on; end; beta_by_k0=(b.*(n_s.^2-n_f.^2)+n_s.^2).^0.5; beta=k_0.*beta_by_k0; h_f=(-beta.^2+k_0.^2.*n_f.^2).^0.5; h_s=(beta.^2-k_0.^2.*n_s.^2).^0.5; h_c=(beta.^2-k_0.^2.*n_c.^2).^0.5; w=2.*d+eta_fs.*(h_f.^2+h_s.^2)./(h_s.*(h_f.^2+eta_fs.^2.*h_s.^2))+eta_fc.*(h_f.^2+h_c.^2)./(h_c.*(h_f.^2+eta_fc.^2.*h_c.^2)) C_N=n_f./((beta.*w).^(0.5)); if(N==0) phase=0; else phase=pi./2; end; phi=0.5.*atan((eta_fs.*h_s./h_f-eta_fc.*h_c./h_f)./(1+h_c.*h_s.*eta_fc.*eta_fs./h_f.^2))+phase; for(x=linspace(-5.*d,5.*d,2e3)) if(x<-d) Hy=(C_N.*cos(h_f.*d+phi).*exp(h_s.*(d+x))); Ex=beta./(n_s.^2.*epsilon_0.*omega).*Hy;

AAA 444222


Ez=j./(n_s.^2.*epsilon_0.*omega).*Hy.*h_s; elseif((x>=-d)&(x<=d)) Hy=(C_N.*cos(h_f.*x-phi)); Ex=beta./(n_f.^2.*epsilon_0.*omega).*Hy; Ez=-j./(n_f.^2.*epsilon_0.*omega).*h_f.*(C_N.*sin(h_f.*x-phi)); elseif(x>d) Hy=(C_N.*cos(h_f.*d-phi).*exp(h_c.*(d-x))); Ex=beta./(n_c.^2.*epsilon_0.*omega).*Hy; Ez=-j./(n_c.^2.*epsilon_0.*omega).*h_c.*Hy; end; if(N==0) figure(1); plot(x*1e9,Hy,'r'); hold on; figure(2); plot(x*1e9,(Ex),'r'); hold on; figure(3); plot(x*1e9,(Ez),'r'); hold on; else figure(1); plot(x*1e9,Hy,'b'); hold on; figure(2); plot(x*1e9,(Ex),'b'); hold on; figure(3); plot(x*1e9,(Ez),'b'); hold on; end; hold on; end; hold on; end; figure(1); ylabel('H_y(x)'); xlabel('x in nm'); title('Field amplitude distribution of H_y(x) for TM modes in Ag guide SWG'); x=linspace(-200,200,100); plot(x,0,'k'); hold on; y=linspace(-0.4,1,100); plot(-40,y,'k',40,y,'k'); text(-150,0.8,'Substrate'); text(100,0.8,'Cladding'); text(-10,0.8,'Guide'); text(-175,-0.05,'n_s^2 = 2.3'); text(-175,-0.15,'n_c^2 = 2.2'); text(-175,-0.25,'\epsilon_r ( Ag ) = - 16.32 - 0.5414 i'); text(-175,-0.35,'\lambda = 0.633 \mum');

AAA 444333


text(-175,0.05,'2d = 80 nm'); box on; figure(2); ylabel('E_X(x)'); xlabel('x in nm'); title('Field amplitude distribution of E_x(x) for TM modes in Ag guide SWG'); x=linspace(-200,200,100); plot(x,0,'k'); hold on; y=linspace(-100,300,100); plot(-40,y,'k',40,y,'k'); text(-150,250,'Substrate'); text(100,250,'Cladding'); text(-10,250,'Guide'); text(-175,-5,'n_s^2 = 2.3'); text(-175,-25,'n_c^2 = 2.2'); text(-175,-45,'\epsilon_r ( Ag ) = - 16.32 - 0.5414 i'); text(-175,-65,'\lambda = 0.633 \mum'); text(-175,-85,'2d = 80 nm'); figure(3); ylabel('E_Z(x)'); xlabel('x in nm'); title('Field amplitude distribution of E_z(x) for TM modes in Ag guide SWG'); x=linspace(-200,200,100); plot(x,0,'k'); hold on; y=linspace(-2,3,100); plot(-40,y,'k',40,y,'k'); text(-150,2.50,'Substrate'); text(100,2.50,'Cladding'); text(-10,2.50,'Guide'); text(-175,-.1,'n_s^2 = 2.3'); text(-175,-.5,'n_c^2 = 2.2'); text(-175,-.9,'\epsilon_r ( Ag ) = - 16.32 - 0.5414 i'); text(-175,-1.3,'\lambda = 0.633 \mum'); text(-175,-1.7,'2d = 80 nm');

% For fig. 8.5.1, 8.6.1 and 8.7.2 n_f=(-16.32-0.5414*i)^0.5; n_s=2.3.^0.5; lambda=0.633e-6; k_0=2*pi/lambda; for(d=0.005e-6:0.005e-6:0.03e-6) for(n_c=linspace(1.4,1.65,5000)) eta_fs=(n_f./n_s).^2; eta_fc=(n_f./n_c).^2; b=0.001;

AAA 444444


if(n_c<n_s) eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=real(-0.5.*atanh((1-eta).^0.5.*eta_fc)); v_0=k_0.*d.*(n_s.^2-n_f.^2).^0.5; for(v=linspace(cut_off,v_0,20)) for(k=0:1:10) func=2.*v+(1+b).^(-0.5).*(atanh(eta_fs.*(b./(1+b)).^0.5)+atanh(eta_fc.*((b+1-eta)./(1+b)).^0.5)); diff=(2.*(1+b)).^(-0.5).*(func-2.*v+eta_fs./(b.^0.5.*(1+b-eta_fs.^2.*b))+eta.*eta_fc./((b+1-eta).^0.5.*( 1+b-eta_fc.^2.*(1+b-eta)))); b=b-func./diff; end; end; beta_k0=(b.*(n_s.^2-n_f.^2)+n_s.^2).^0.5; else eta=(n_f.^2-n_s.^2)./(n_f.^2-n_c.^2); cut_off=real(-0.5.*atanh((1-eta).^0.5.*eta_fs)); v_0=k_0.*d.*(n_c.^2-n_f.^2).^0.5; for(v=linspace(cut_off,v_0,20)) for(k=0:1:10) func=2.*v+(1+b).^(-0.5).*(atanh(eta_fc.*(b./(1+b)).^0.5)+atanh(eta_fs.*((b+1-eta)./(1+b)).^0.5)); diff=(2.*(1+b)).^(-0.5).*(func-2.*v+eta_fc./(b.^0.5.*(1+b-eta_fc.^2.*b))+eta.*eta_fs./((b+1-eta).^0.5.*( 1+b-eta_fs.^2.*(1+b-eta)))); b=b-func./diff; end; end; beta_k0=(b.*(n_c.^2-n_f.^2)+n_c.^2).^0.5; end; MPA=8.68.*imag(-beta_k0).*k_0./1e3; h_s=real((beta_k0.^2-n_s.^2).^0.5.*k_0); h_c=real((beta_k0.^2-n_c.^2).^0.5.*k_0); MSS=(2.*d+1./h_s+1./h_c).*1e6; if(beta_k0>=n_s&beta_k0>=n_c) if(d==0.005e-6) figure(1); plot(n_c,MSS,'c'); hold on; figure(2); plot(n_c,beta_k0,'c'); hold on; figure(3); semilogy(n_c,MPA,'c'); hold on; elseif(d==0.01e-6) figure(1); plot(n_c,MSS,'b'); hold on; figure(2); plot(n_c,beta_k0,'b'); hold on; figure(3); semilogy(n_c,MPA,'b');

AAA 444555


hold on; elseif(d==0.03e-6) figure(1); plot(n_c,MSS,'r'); hold on; figure(2); plot(n_c,beta_k0,'r'); hold on; figure(3); semilogy(n_c,MPA,'r'); hold on; elseif(d==0.02e-6) figure(1); plot(n_c,MSS,'k'); hold on; figure(2); plot(n_c,beta_k0,'k'); hold on; figure(3); semilogy(n_c,MPA,'k'); hold on; elseif(d==0.025e-6) figure(1); plot(n_c,MSS,'m'); hold on; figure(2); plot(n_c,beta_k0,'m'); hold on; figure(3); semilogy(n_c,MPA,'m'); hold on; else figure(1); plot(n_c,MSS,'g'); hold on; figure(2); plot(n_c,beta_k0,'g'); hold on; figure(3); semilogy(n_c,MPA,'g'); hold on; end; hold on; end; hold on; end; hold on; end; figure(1); axis([1.4,1.65,0,2]); grid on;

AAA 444666


xlabel('n_c'); ylabel('Mode Spot size in \mu m'); text(1.62,0.7,'2d=30 nm'); text(1.62,1.8,'2d=25 nm'); text(1.435,1.8,'2d=20 nm'); text(1.575,1.8,'2d=15 nm'); text(1.47,1.7,'2d=10 nm'); text(1.52,1.8,'2d=05 nm'); text(1.575,0.4,'n_s^2 = 2.3'); text(1.555,0.25,'\epsilon _r ( Ag ) = ( -16.32 - j 0.5414 )'); text(1.575,0.1,'\lambda = 0.633 \mu m'); title('Plot of Mode spot size n_c for different guide thickness'); figure(2); n_c=linspace(1.4,n_s,60); x=n_s.*n_c.^0; plot(n_c,x,'r:'); hold on; n_c=linspace(n_s,1.65,80); x=n_c; plot(n_c,x,'r:'); hold on; axis([1.4,1.65,1.51,1.65]); grid on; xlabel('n_c'); ylabel('\beta / k_0'); text(1.525,1.62,'2d=30 nm'); text(1.525,1.6,'2d=25 nm'); text(1.515,1.58,'2d=20 nm'); text(1.51,1.555,'2d=15 nm'); text(1.5,1.535,'2d=10 nm'); text(1.5,1.525,'2d=05 nm'); text(1.45,1.64,'n_s^2 = 2.3'); text(1.45,1.63,'\epsilon _r ( Ag ) = ( -16.32 - j 0.5414 )'); title('Plot of \beta / k_0 vs. n_c for different guide thickness'); figure(3); axis([1.4,1.65,1e-1,1e3]); grid on; xlabel('n_c'); ylabel('\Propagation loss in dB / mm'); text(1.45,3e2,'2d=30 nm'); text(1.5,1.3e2,'2d=25 nm'); text(1.55,1e2,'2d=20 nm'); text(1.5,5e1,'2d=15 nm'); text(1.55,2e1,'2d=10 nm'); text(1.5,2e0,'2d=05 nm'); text(1.55,3,'n_s^2 = 2.3'); text(1.55,1.5,'\epsilon _r ( Ag ) = ( -16.32 - j 0.5414 )'); text(1.55,0.7,'\lambda = 0.633 \mu m'); title('Plot of propagation vs. n_c for different guide thickness'); x=[1.4:0.01:1.65]; for(count=0:1:2)

AAA 444777


plot(x,10.^count.*x.^0,'k:'); hold on; end; box on;

% For fig. 8.7.1 n_s=2.3^0.5; n_c=2.2^0.5; for(no_of_metal=1:1:3) lambda=0.633e-6; if(no_of_metal==1) n_f=0.144-3.221*i; % Au elseif(no_of_metal==2) n_f=0.15-3.2*i;% Cu else n_f=0.065-4*i;% Ag end; k0=2*pi./lambda; eta_fs=(n_f./n_s).^2; eta_fc=(n_f./n_c).^2; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); cut_off=real(-0.5.*atanh((1-eta).^0.5.*eta_fc)); b=eta.*eta_fc.^2./(eta_fc.^2-1)-1; for(v=linspace(2,cut_off,10000)) k=0; while(k<50) k=k+1; func=2.*v+(1+b).^(-0.5).*(atanh(eta_fs.*(b./(1+b)).^0.5)+atanh(eta_fc.*((b+1-eta)./(1+b)).^0.5)); diff=(2.*(1+b)).^(-0.5).*(func-2.*v+eta_fs./(b.^0.5.*(1+b-eta_fs.^2.*b))+eta.*eta_fc./((b+1-eta).^0.5.*(1+b -eta_fc.^2.*(1+b-eta)))); b=b-func./diff; end; beta_by_k0=(b.*(n_s.^2-n_f.^2)+n_s.^2).^0.5; k0_into_2d=real(2.*v./(n_s.^2-n_f.^2).^0.5); d_into_2=k0_into_2d./k0; alpha=(8.68.*imag(-beta_by_k0*k0))./1e3; if(no_of_metal==1) semilogy(d_into_2*1e9,alpha,'r'); hold on; elseif(no_of_metal==2) semilogy(d_into_2*1e9,alpha,'b'); hold on; elseif(no_of_metal==3) semilogy(d_into_2*1e9,alpha,'g'); hold on; end; hold on;

AAA 444888


end; hold on; end; hold on; axis([10,30,1e-1,1e3]); xlabel('Thickness of the guide 2d in nm'); ylabel('Propagation loss in dB / mm'); title('Propagation loss for some metals'); grid on; x=linspace(27,29,100); y=2e1; plot(x,y,'r'); y=1e1; plot(x,y,'b'); y=5e0; plot(x,y,'g'); text(26,2e1,'Au'); text(26,1e1,'Cu'); text(26,5e0,'Ag'); text(26,2e0,'n_s^2 = 2.3'); text(26,1e0,'n_c^2 = 2.2'); text(26,0.5e0,'\lambda = 0.633 \mum');

% For fig. 8.8.1, 8.8.2 and 8.8.3 n_f=(-16.32-0.5414*j)^0.5; n_s=2.3.^0.5; eta_fs=(n_f./n_s).^2; for(N=0:1:1) b=1./(eta_fs.^2-1); for(v=8:-0.001:0.001) if(N==0) for(k=0:1:20) func=tanh(v.*(1+b).^0.5)+(eta_fs.^(-1).*((1+b)./b).^0.5); diff=(2.*(1+b)).^(-0.5).*(v.*(sech(v.*(1+b).^0.5)).^2+eta_fs.^(-1).*b.^0.5.*(b.^2-1)./b.^2); b=b-func./diff; end; else for(k=0:1:20) func=tanh(v.*(1+b).^0.5)+eta_fs.*(b./(1+b)).^0.5; diff=(2.*(1+b)).^(-0.5).*(v.*(sech(v.*(1+b).^0.5)).^2+eta_fs./(b.^0.5.*(1+b))); b=b-func./diff; end; end; beta_k0=(b.*(n_s.^2-n_f.^2)+n_s.^2).^0.5; k_0_2d=2.*v./(n_s.^2-n_f.^2).^0.5; if(N==0) figure(1);

AAA 444999


plot(v,b,'b'); hold on; figure(2); plot(k_0_2d,real(beta_k0-n_s),'b'); hold on; figure(3); semilogy(k_0_2d,imag(-beta_k0),'b'); hold on; else figure(1); plot(v,b,'r'); hold on; figure(2); plot(k_0_2d,real(beta_k0-n_s),'r'); hold on; figure(3); semilogy(k_0_2d,imag(-beta_k0),'r'); hold on; end; hold on; end; hold on; end; hold on; figure(1); axis([0,8,0,0.1]); xlabel('V'); ylabel('B'); title('B~V plot for a symmetric metallic guide SWG'); text(5,0.08,'n_s^2=2.3'); text(5,0.073,'\epsilon ( Ag ) = - 16.32 - 0.5414 i'); text(5,0.066,'\lambda = 0.633 \mum'); grid on; figure(2); axis([0,4,0,0.5]); xlabel('2k_0d'); ylabel('\beta / k_0 - n_s'); title('Effective index variation with 2k_0d in a symmetric metallic guide SWG'); text(2.5,0.4,'n_s^2=2.3'); text(2.5,0.33,'\epsilon ( Ag ) = - 16.32 - 0.5414 i'); text(2.5,0.26,'\lambda = 0.633 \mum'); grid on; figure(3); axis([0,4,1e-6,1e-1]); grid on; title('Variation of \alpha / k_0 with 2k_0d in a symmetric metallic guide SWG'); xlabel('2k_0d');

AAA 555000


ylabel('\alpha / k_0'); text(2.5,1e-4,'n_s^2=2.3'); text(2.5,0.5e-4,'\epsilon ( Ag ) = - 16.32 - 0.5414 i'); text(2.5,0.2e-4,'\lambda = 0.633 \mum'); for(n=-5:1:-2) x=linspace(0,4,100); plot(x,10.^n); hold on; end;

% For fig. 8.8.4 d=0.04*1e-6; lambda=0.633e-6; k_0=2.*pi./lambda; epsilon_0=8.854e-12; for(N=0:1:1) n_f=(-16.32-0.5414*i)^0.5; n_s=2.3^0.5; eta_fs=(n_f./n_s).^2; v_0=k_0.*d.*(n_s.^2-n_f.^2).^0.5; cut_off=4; b=1./(eta_fs.^2-1); for(v=cut_off:-0.01:v_0) for(k=0:1:20) if(N==0) func=tanh(v.*(1+b).^0.5)+(eta_fs.^(-1).*((1+b)./b).^0.5); diff=(2.*(1+b)).^(-0.5).*(v.*(sech(v.*(1+b).^0.5)).^2+eta_fs.^(-1).*b.^0.5.*(b.^2-1)./b.^2); b=b-func./diff; else func=tanh(v.*(1+b).^0.5)+eta_fs.*(b./(1+b)).^0.5; diff=(2.*(1+b)).^(-0.5).*(v.*(sech(v.*(1+b).^0.5)).^2+eta_fs./(b.^0.5.*(1+b))); b=b-func./diff; end; end; end; beta_by_k0=(b.*(n_s.^2-n_f.^2)+n_s.^2).^0.5; beta=k_0.*beta_by_k0; h_f=(k_0.^2.*n_f.^2-beta.^2).^0.5; h_s=(beta.^2-k_0.^2.*n_s.^2).^0.5; w=2.*d+2*eta_fs.*(h_f.^2+h_s.^2)./(h_s.*(h_f.^2+eta_fs.^2.*h_s.^2)); C_N=n_f./((beta.*w).^(0.5)); if(N==0) phase=pi./2; else phase=0; end; for(x=linspace(-5.*d,5.*d,2e3))

AAA 555111


if(x<-d) Hy=(C_N.*cos(h_f.*d+phase).*exp(h_s.*(d+x))); elseif((x>=-d)&(x<=d)) Hy=(C_N.*cos(h_f.*x-phase)); elseif(x>d) Hy=(C_N.*cos(h_f.*d-phase).*exp(h_s.*(d-x))); end; if(N==0) plot(x*1e9,Hy,'b'); else plot(x*1e9,Hy,'r'); end; hold on; end; end; ylabel('H_y(x)'); xlabel('x in nm'); title('Field amplitude distribution of H_y(x) for TM modes in Ag guide SWG'); x=linspace(-200,200,100); plot(x,0,'k'); hold on; y=linspace(-0.8,0.8,100); plot(-d*1e9,y,'k',d*1e9,y,'k'); text(-160,-0.2,'n_s^2 = 2.3 = n_c^2'); text(-160,-0.3,'\epsilon_r ( Ag ) = - 16.32 - 0.5414 i'); text(-160,-0.4,'\lambda = 0.633 \mum'); text(-160,-0.5,'2d = 80 nm'); text(-150,0.6,'Substrate'); text(100,0.6,'Cladding'); text(-10,0.6,'Guide');

% For fig. 8.8.5 and 8.8.6 % The excess grid lines are removed in the Figure window for the figure 8.8.6 n_s=2.3^0.5; for(n0_of_metal=1:1:4) for(no_of_wavelength=1:1:2) if(no_of_wavelength==1) lambda=0.633e-6; if(n0_of_metal==1) n_f=1.2-7*i;% Al elseif(n0_of_metal==2) n_f=0.15-3.2*i; % Au elseif(n0_of_metal==3) n_f=0.145-3.211*i;% Cu else n_f=0.065-4*i;% Ag end;

AAA 555222


else lambda=10.6e-6; if(n0_of_metal==1) n_f=25-67*i;% Al elseif(n0_of_metal==2) n_f=12.6-64.3*i;% Au elseif(n0_of_metal==3) n_f=7.4-53.4*i;% Cu else n_f=10.7-69*i;% Ag end; end; k0=2*pi./lambda; eta_fs=(n_f./n_s).^2; b1=1./(eta_fs.^2-1); b2=1./(eta_fs.^2-1); for(v=logspace(1,-2,1000)) for(k=0:1:20) func1=tanh(v.*(1+b1).^0.5)+(eta_fs.^(-1).*((1+b1)./b1).^0.5); diff1=(2.*(1+b1)).^(-0.5).*(v.*(sech(v.*(1+b1).^0.5)).^2+eta_fs.^(-1).*b1.^0.5.*(b1.^2-1)./b1.^2); b1=b1-func1./diff1; func2=tanh(v.*(1+b2).^0.5)+eta_fs.*(b2./(1+b2)).^0.5; diff2=(2.*(1+b2)).^(-0.5).*(v.*(sech(v.*(1+b2).^0.5)).^2+eta_fs./(b2.^0.5.*(1+b2))); b2=b2-func2./diff2; end; beta_by_k01=(b1.*(n_s.^2-n_f.^2)+n_s.^2).^0.5; beta_by_k02=(b2.*(n_s.^2-n_f.^2)+n_s.^2).^0.5; k0_into_2d=real(2.*v./(n_s.^2-n_f.^2).^0.5); d_2=k0_into_2d./k0.*1e9; loss1=8.68.*imag(-beta_by_k01).*k0./1e2; loss2=8.68.*imag(-beta_by_k02).*k0./1e2; if(n0_of_metal==1) figure(1); loglog(d_2,loss1,'r'); hold on; figure(2); loglog(d_2,loss2,'r'); hold on; elseif(n0_of_metal==2) figure(1); loglog(d_2,loss1,'b'); hold on; figure(2); loglog(d_2,loss2,'b'); hold on; elseif(n0_of_metal==3) figure(1); loglog(d_2,loss1,'g'); hold on; figure(2); loglog(d_2,loss2,'g');

AAA 555333


hold on; elseif(n0_of_metal==4) figure(1); loglog(d_2,loss1,'k'); hold on; figure(2); loglog(d_2,loss2,'k'); hold on; end; hold on; end; hold on; end; hold on; end; hold on; figure(1); grid on; axis([0.1,1000,1e0,1e7]); text(0.5e2,0.3e7,'n_s^2 = 2.3'); text(1e0,1e4,'\lambda = 10.6 \mum'); text(1e0,1e6,'\lambda = 0.633 \mum'); for(n=1:1:6) x=logspace(-1,3,100); plot(x,10.^n); hold on; end; x=logspace(log10(3e-1),log10(6e-1),100); plot(x,1e3,'r',x,0.3e3,'b',x,0.1e3,'g',x,0.3e2,'k'); text(2e-1,1e3,'Al'); text(2e-1,0.3e3,'Au'); text(2e-1,0.1e3,'Cu'); text(2e-1,0.3e2,'Ag'); xlabel('Guide thickness in nm'); ylabel('Propagation loss in dB /cm'); title('Variation of propagation loss with guide thickness for anti symmetrical mode'); figure(2); grid on; axis([0.1,1000,1e-4,1e5]); text(3e1,0.3e-2,'n_s^2 = 2.3'); text(1e0,1e-1,'\lambda = 10.6 \mum'); text(1e0,1e2,'\lambda = 0.633 \mum'); for(n=-3:1:4) x=logspace(-1,3,100); plot(x,10.^n); hold on; end; x=logspace(2,1.5,100); plot(x,1e0,'r',x,0.3e0,'b',x,0.1e0,'g',x,0.3e-1,'k'); text(3e1,1e0,'Al'); text(3e1,0.3e0,'Au');

AAA 555444


text(3e1,0.1e0,'Cu'); text(3e1,0.3e-1,'Ag'); xlabel('Guide thickness in nm'); ylabel('Propagation loss in dB /cm'); title('Variation of propagation loss with guide thickness for symmetrical mode');

% For fig. 8.8.7 % The y axis grid is removed in the figure window manually for(d_2=10:15:70) d_into_2=d_2.*1e-9; d=d_into_2./2; for(no_of_wavelength=1:1:1) if(no_of_wavelength==1) lambda=0.633e-6; n_f=0.065-4*i; else lambda=10.6e-6; n_f=10.7-69*i; end; k0=2*pi./lambda; for(n_s=1.0:0.005:3) eta_fs=(n_f./n_s).^2; b=1./(eta_fs.^2-1); v_0=k0.*d.*(n_s.^2-n_f.^2).^0.5; for(v=linspace(4,v_0,20)) for(k=0:1:10) func=tanh(v.*(1+b).^0.5)+eta_fs.*(b./(1+b)).^0.5; diff=(2.*(1+b)).^(-0.5).*(v.*(sech(v.*(1+b).^0.5)).^2+eta_fs./(b.^0.5.*(1+b))); b=b-func./diff; end; end; beta_by_k0=(b.*(n_s.^2-n_f.^2)+n_s.^2).^0.5; loss=8.68.*imag(-beta_by_k0).*k0./1e2; if(d_2==10) semilogy(n_s,loss,'b'); elseif(d_2==25) semilogy(n_s,loss,'r'); elseif(d_2==40) semilogy(n_s,loss,'g'); elseif(d_2==55) semilogy(n_s,loss,'m'); elseif(d_2==70) semilogy(n_s,loss,'k'); end; hold on; end; hold on;

AAA 555555


end; hold on; end; grid on; xlabel('n_s'); ylabel('loss in dB / cm'); title('Variation of propagation loss with r.i. of the cladding layer'); text(2.6,1e3,'2d = 10 nm'); text(2.2,3e3,'2d = 25 nm'); text(1.8,3e3,'2d = 40 nm'); text(1.4,1.5e3,'2d = 55 nm'); text(1,1e3,'2d = 70 nm'); text(2.2,1e1,'\lambda = 0.633 \mum'); text(2.2,5e0,'\epsilon_r ( Ag ) = - 16.32 - 0.5414 i'); x=linspace(1,3,200); plot(x,1e1,'k',x,1e1,'k',x,1e2,'k',x,1e3,'k',x,1e4,'k',x,1e5,'k');

% For fig. 9.5.1 (a) and 9.5.1 (b) % The texts like ssbo, asb0 in the figures are introduced in the figure window separately n_f=(-19-0.53*j)^0.5; n_s=4^0.5; n_c=3.61.^0.5; lambda=0.633e-6; k_0=2.*pi./lambda; w_0=0.5e-6; n_odd=n_s.*n_f./(n_s.^2+n_f.^2).^0.5; n_odd_k0=n_odd.*k_0; n_even=n_c.*n_f./(n_c.^2+n_f.^2).^0.5; n_even_k0=n_even.*k_0; n_even_even=n_even-0.01; n_even_even_k0=n_even_even.*k_0; n_odd_even=n_odd-0.01; n_odd_even_k0=n_odd_even.*k_0; n_odd_odd=n_odd-0.01; n_odd_odd_k0=n_odd_odd.*k_0; n_even_odd=n_even-0.01; n_even_odd_k0=n_even_odd.*k_0; for(t=linspace(100e-9,10e-9,500)) eta_fs=(n_f./n_s).^2; eta_fc=(n_f./n_c).^2; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); k=0:1:10; h_f=(n_even_k0.^2-k_0.^2.*n_f.^2).^0.5; h_s=(n_even_k0.^2-k_0.^2.*n_s.^2).^0.5; h_c=(n_even_k0.^2-k_0.^2.*n_c.^2).^0.5; func=((h_f.^2+eta_fs.*eta_fc.*h_s.*h_c).*tanh(2.*h_f.*t)+h_f.*(eta_fs.*h_s+eta_fc.*h_c)); diff=((2+eta_fs.*eta_fc.*(h_s./h_c+h_c./h_s)).*n_even_k0.*tanh(2.*h_f.*t)+2.*t.*n_even_k0./h_f.*(h_f.^2+eta_

AAA 555666


fs.*eta_fc.*h_s.*h_c).*(sech(2.*h_f.*t)).^2+eta_fs.*n_even_k0.*(h_f./h_s+h_s./h_f)+eta_fc.*n_even_k0.*(h_f./h_c+h_c./h_f)); n_even_k0=n_even_k0-func./diff; n_even=n_even_k0./k_0; figure(1); plot(2.*t.*1e6,real(n_even),'k'); hold on; figure(2); semilogy(2.*t.*1e6,-imag(n_even),'k'); hold on; k=0:1:10; h_f=(n_odd_k0.^2-k_0.^2.*n_f.^2).^0.5; h_s=(n_odd_k0.^2-k_0.^2.*n_s.^2).^0.5; h_c=(n_odd_k0.^2-k_0.^2.*n_c.^2).^0.5; func=((h_f.^2+eta_fs.*eta_fc.*h_s.*h_c).*tanh(2.*h_f.*t)+h_f.*(eta_fs.*h_s+eta_fc.*h_c)); diff=((2+eta_fs.*eta_fc.*(h_s./h_c+h_c./h_s)).*n_odd_k0.*tanh(2.*h_f.*t)+2.*t.*n_odd_k0./h_f.*(h_f.^2+eta_fs.*eta_fc.*h_s.*h_c).*(sech(2.*h_f.*t)).^2+eta_fs.*n_odd_k0.*(h_f./h_s+h_s./h_f)+eta_fc.*n_odd_k0.*(h_f./h_c+h_c./h_f)); n_odd_k0=n_odd_k0-func./diff; n_odd=n_odd_k0./k_0; figure(1); plot(2.*t.*1e6,real(n_odd),'k'); hold on; figure(2); semilogy(2.*t.*1e6,-imag(n_odd),'k'); hold on; for(w=linspace(0.7e-6,0.5e-6,20)) n_odd_odd=n_odd-0.05; n_odd_odd_k0=n_odd_odd.*k_0; k=0:1:10; h_f=(k_0.^2.*n_odd.^2-n_odd_odd_k0.^2).^0.5; h_c=(n_odd_odd_k0.^2-k_0.^2.*n_c.^2).^0.5; func=h_c.*tan(h_f.*w)+h_f; diff=-n_odd_odd_k0.*(-tan(h_f.*w)./h_c+(sec(h_f.*w)).^2.*w.*h_c./h_f+1./h_f); n_odd_odd_k0=n_odd_odd_k0-func./diff; n_odd_odd=n_odd_odd_k0./k_0; end; figure(1); plot(2.*t.*1e6,real(n_odd_odd),'r'); hold on; figure(2); semilogy(2.*t.*1e6,-imag(n_odd_odd),'r'); hold on; n_odd_even=n_odd-0.01; n_odd_even_k0=n_odd_even.*k_0; for(w=linspace(0.7e-6,0.5e-6,20)) k=0:1:10; h_f=(k_0.^2.*n_odd.^2-n_odd_even_k0.^2).^0.5; h_c=(n_odd_even_k0.^2-k_0.^2.*n_c.^2).^0.5; func=h_f.*tan(h_f.*w)-h_c; diff=-n_odd_even_k0.*(tan(h_f.*w)./h_f+(sec(h_f.*w)).^2.*w+1./h_c);

AAA 555777


n_odd_even_k0=n_odd_even_k0-func./diff; n_odd_even=n_odd_even_k0./k_0; end; figure(1); plot(2.*t.*1e6,real(n_odd_even),'b'); hold on; figure(2); semilogy(2.*t.*1e6,-imag(n_odd_even),'b'); hold on; n_even_odd=n_even-0.1; n_even_odd_k0=n_even_odd.*k_0; for(w=linspace(0.7e-6,0.5e-6,20)) k=0:1:10; h_f=(k_0.^2.*n_even.^2-n_even_odd_k0.^2).^0.5; h_c=(n_even_odd_k0.^2-k_0.^2.*n_c.^2).^0.5; func=h_c.*tan(h_f.*w)+h_f; diff=-n_even_odd_k0.*(-tan(h_f.*w)./h_c+(sec(h_f.*w)).^2.*w.*h_c./h_f+1./h_f); n_even_odd_k0=n_even_odd_k0-func./diff; n_even_odd=n_even_odd_k0./k_0; end; figure(1); plot(2.*t.*1e6,real(n_even_odd),'r'); hold on; figure(2); semilogy(2.*t.*1e6,-imag(n_even_odd),'r'); hold on; n_even_even=n_even-0.01; n_even_even_k0=n_even_even.*k_0; for(w=linspace(0.7e-6,0.5e-6,20)) k=0:1:10; h_f=(k_0.^2.*n_even.^2-n_even_even_k0.^2).^0.5; h_c=(n_even_even_k0.^2-k_0.^2.*n_c.^2).^0.5; func=h_f.*tan(h_f.*w)-h_c; diff=-n_even_even_k0.*(tan(h_f.*w)./h_f+(sec(h_f.*w)).^2.*w+1./h_c); n_even_even_k0=n_even_even_k0-func./diff; n_even_even=n_even_even_k0./k_0; end; figure(1); plot(2.*t.*1e6,real(n_even_even),'b'); hold on; figure(2); semilogy(2.*t.*1e6,-imag(n_even_even),'b'); hold on; end; figure(1); grid on; xlabel('Metal strip thickness 2t in nm'); ylabel('Normalized propagation constant ( \beta / k_0 )'); axis([0,0.2,1.9,2.9]); figure(2); grid on;

AAA 555888


xlabel('Metal strip thickness 2t in nm'); ylabel('Normalized attenuation constant ( \alpha / k_0 )'); axis([0,0.2,1e-4,1e-1]);

% The texts like ssbo, asb0 in the figures are introduced in the figure window separately % For fig. 9.6.3 (a) and 9.6.3 (b) n_f=(-19-0.53*j)^0.5; n_s=4^0.5; n_c=3.61.^0.5; lambda=0.633e-6; k_0=2.*pi./lambda; w_0=0.5e-6; n_odd=n_s.*n_f./(n_s.^2+n_f.^2).^0.5; n_odd_k0=n_odd.*k_0; n_even=n_c.*n_f./(n_c.^2+n_f.^2).^0.5; n_even_k0=n_even.*k_0; n_asym=n_odd.*n_f./(n_odd.^2+n_f.^2).^0.5; n_asym_k0=n_asym.*k_0; n_sym=n_even.*n_f./(n_even.^2+n_f.^2).^0.5; n_sym_k0=n_sym.*k_0; n_ss=n_sym-0.01; n_ss_k0=n_ss.*k_0; n_sa=n_asym-0.01; n_sa_k0=n_asym.*k_0; n_aa=n_odd-0.01; n_aa_k0=n_aa.*k_0; n_as=n_sym-0.01; n_as_k0=n_as.*k_0; for(t=linspace(100e-9,7e-9,600)) eta_fs=(n_f./n_s).^2; eta_fc=(n_f./n_c).^2; eta=(n_f.^2-n_c.^2)./(n_f.^2-n_s.^2); for(k=0:1:10) h_f=(n_even_k0.^2-k_0.^2.*n_f.^2).^0.5; h_s=(n_even_k0.^2-k_0.^2.*n_s.^2).^0.5; h_c=(n_even_k0.^2-k_0.^2.*n_c.^2).^0.5; func=((h_f.^2+eta_fs.*eta_fc.*h_s.*h_c).*tanh(2.*h_f.*t)+h_f.*(eta_fs.*h_s+eta_fc.*h_c)); diff=((2+eta_fs.*eta_fc.*(h_s./h_c+h_c./h_s)).*n_even_k0.*tanh(2.*h_f.*t)+2.*t.*n_even_k0./h_f.*(h_f.^2+eta_fs.*eta_fc.*h_s.*h_c).*(sech(2.*h_f.*t)).^2+eta_fs.*n_even_k0.*(h_f./h_s+h_s./h_f)+eta_fc.*n_even_k0.*(h_f./h_c+h_c./h_f)); n_even_k0=n_even_k0-func./diff; n_even=n_even_k0./k_0; end; figure(1); plot(2.*t.*1e6,real(n_even),'k'); hold on;

AAA 555999


figure(2); semilogy(2.*t.*1e6,-imag(n_even),'k'); hold on; for(k=0:1:10) h_f=(n_odd_k0.^2-k_0.^2.*n_f.^2).^0.5; h_s=(n_odd_k0.^2-k_0.^2.*n_s.^2).^0.5; h_c=(n_odd_k0.^2-k_0.^2.*n_c.^2).^0.5; func=((h_f.^2+eta_fs.*eta_fc.*h_s.*h_c).*tanh(2.*h_f.*t)+h_f.*(eta_fs.*h_s+eta_fc.*h_c)); diff=((2+eta_fs.*eta_fc.*(h_s./h_c+h_c./h_s)).*n_odd_k0.*tanh(2.*h_f.*t)+2.*t.*n_odd_k0./h_f.*(h_f.^2+eta_fs.*eta_fc.*h_s.*h_c).*(sech(2.*h_f.*t)).^2+eta_fs.*n_odd_k0.*(h_f./h_s+h_s./h_f)+eta_fc.*n_odd_k0.*(h_f./h_c+h_c./h_f)); n_odd_k0=n_odd_k0-func./diff; n_odd=n_odd_k0./k_0; end; figure(1); plot(2.*t.*1e6,real(n_odd),'k'); hold on; figure(2); semilogy(2.*t.*1e6,-imag(n_odd),'k'); hold on; eta_fs=(n_f./n_odd).^2; eta_fc=(n_f./n_even).^2; eta=(n_f.^2-n_even.^2)./(n_f.^2-n_odd.^2); for(k=0:1:10) h_f=(n_sym_k0.^2-k_0.^2.*n_f.^2).^0.5; h_s=(n_sym_k0.^2-k_0.^2.*n_odd.^2).^0.5; h_c=(n_sym_k0.^2-k_0.^2.*n_even.^2).^0.5; func=((h_f.^2+eta_fs.*eta_fc.*h_s.*h_c).*tanh(2.*h_f.*t)+h_f.*(eta_fs.*h_s+eta_fc.*h_c)); diff=((2+eta_fs.*eta_fc.*(h_s./h_c+h_c./h_s)).*n_sym_k0.*tanh(2.*h_f.*t)+2.*t.*n_sym_k0./h_f.*(h_f.^2+eta_fs.*eta_fc.*h_s.*h_c).*(sech(2.*h_f.*t)).^2+eta_fs.*n_sym_k0.*(h_f./h_s+h_s./h_f)+eta_fc.*n_sym_k0.*(h_f./h_c+h_c./h_f)); n_sym_k0=n_sym_k0-func./diff; n_sym=n_sym_k0./k_0; end; eta_fs=(n_f./n_odd).^2; eta_fc=(n_f./n_even).^2; eta=(n_f.^2-n_even.^2)./(n_f.^2-n_odd.^2); for(k=0:1:10) h_f=(n_asym_k0.^2-k_0.^2.*n_f.^2).^0.5; h_s=(n_asym_k0.^2-k_0.^2.*n_odd.^2).^0.5; h_c=(n_asym_k0.^2-k_0.^2.*n_even.^2).^0.5; func=((h_f.^2+eta_fs.*eta_fc.*h_s.*h_c).*tanh(2.*h_f.*t)+h_f.*(eta_fs.*h_s+eta_fc.*h_c)); diff=((2+eta_fs.*eta_fc.*(h_s./h_c+h_c./h_s)).*n_asym_k0.*tanh(2.*h_f.*t)+2.*t.*n_asym_k0./h_f.*(h_f.^2+eta_fs.*eta_fc.*h_s.*h_c).*(sech(2.*h_f.*t)).^2+eta_fs.*n_asym_k0.*(h_f./h_s+h_s./h_f)+eta_fc.*n_asym_k0.*( h _f./h_c+h_c./h_f)); n_asym_k0=n_asym_k0-func./diff; n_asym=n_asym_k0./k_0; end;

AAA 666000


for(w=linspace(0.7e-6,0.5e-6,20)) n_aa=n_asym-0.05; n_aa_k0=n_aa.*k_0; k=0:1:10; h_f=(k_0.^2.*n_asym.^2-n_aa_k0.^2).^0.5; h_c=(n_aa_k0.^2-k_0.^2.*n_c.^2).^0.5; func=h_c.*tan(h_f.*w)+h_f; diff=-n_aa_k0.*(-tan(h_f.*w)./h_c+(sec(h_f.*w)).^2.*w.*h_c./h_f+1./h_f); n_aa_k0=n_aa_k0-func./diff; n_aa=n_aa_k0./k_0; end; figure(1); plot(2.*t.*1e6,real(n_aa),'b'); hold on; figure(2); semilogy(2.*t.*1e6,-imag(n_aa),'b'); hold on; for(w=linspace(0.7e-6,0.5e-6,20)) n_sa=n_asym-0.01; n_sa_k0=n_sa.*k_0; k=0:1:10; h_f=(k_0.^2.*n_asym.^2-n_sa_k0.^2).^0.5; h_c=(n_sa_k0.^2-k_0.^2.*n_c.^2).^0.5; func=h_f.*tan(h_f.*w)-h_c; diff=-n_sa_k0.*(tan(h_f.*w)./h_f+(sec(h_f.*w)).^2.*w+1./h_c); n_sa_k0=n_sa_k0-func./diff; n_sa=n_sa_k0./k_0; end; figure(1); plot(2.*t.*1e6,real(n_sa),'r'); hold on; figure(2); semilogy(2.*t.*1e6,-imag(n_sa),'r'); hold on; for(w=linspace(0.7e-6,0.5e-6,20)) n_as=n_sym-0.05; n_as_k0=n_as.*k_0; k=0:1:10; h_f=(k_0.^2.*n_sym.^2-n_as_k0.^2).^0.5; h_c=(n_as_k0.^2-k_0.^2.*n_c.^2).^0.5; func=h_c.*tan(h_f.*w)+h_f; diff=-n_as_k0.*(-tan(h_f.*w)./h_c+(sec(h_f.*w)).^2.*w.*h_c./h_f+1./h_f); n_as_k0=n_as_k0-func./diff; n_as=n_as_k0./k_0; end; figure(1); plot(2.*t.*1e6,real(n_as),'b'); hold on; figure(2); semilogy(2.*t.*1e6,-imag(n_as),'b'); hold on;

AAA 666111


for(w=linspace(0.7e-6,0.5e-6,20)) n_ss=n_sym-0.01; n_ss_k0=n_ss.*k_0; k=0:1:10; h_f=(k_0.^2.*n_sym.^2-n_ss_k0.^2).^0.5; h_c=(n_ss_k0.^2-k_0.^2.*n_c.^2).^0.5; func=h_f.*tan(h_f.*w)-h_c; diff=-n_ss_k0.*(tan(h_f.*w)./h_f+(sec(h_f.*w)).^2.*w+1./h_c); n_ss_k0=n_ss_k0-func./diff; n_ss=n_ss_k0./k_0; end; figure(1); plot(2.*t.*1e6,real(n_ss),'r'); hold on; figure(2); semilogy(2.*t.*1e6,-imag(n_ss),'r'); hold on; end; figure(1); grid on; xlabel('Metal strip thickness 2t in nm'); ylabel('Normalized propagation constant ( \beta / k_0 )'); axis([0,0.2,1.8,3.8]); figure(2); grid on; xlabel('Metal strip thickness 2t in nm'); ylabel('Normalized attenuation constant ( \alpha / k_0 )'); axis([0,0.2,1e-4,1e0]);

AAA 666222

RRReeefffeeerrreeennnccceee

RRReeefffeeerrreeennnccceeesss ::: 1. ‘‘SSuurrffaaccee PPllaassmmoonn PPoollaarriittoonnss aatt TTeerraahheerrttzz FFrreeqquueenncciieess oonn MMeettaall aanndd SSeemmiiccoonndduuccttoorr SSuurrffaacceess’’ by JJöörrgg SSaaxxlleerr, Institute of Semiconductor Electronics, (http://www.iht.rwth-aachen.de) at Aachen University of Technology, (http://www.rwth-aachen.de) Aachen, Germany.

2. ‘‘IInnttrroodduuccttiioonn ttoo LLoonngg--RRaannggee SSuurrffaaccee PPllaassmmoonn--PPoollaarriittoonnss ((LLRR--SSPPPP))’’ by HHssiiaaoo--KKuuaann YYuuaann and CChheenn--BBiinn HHuuaanngg, Purdue University School of Electrical and Computer Engineering. 3. ‘‘SSTTUUDDYY OOFF ((SSUURRFFAACCEE PPLLAASSMMOONN)) OOPPTTIICCAALL WWAAVVEESS FFOORR SSEENNSSIINNGG AAPPPPLLIICCAATTIIOONNSS’’ by AAggggeellooss TTaarrlliiss Department of Electronic and Electrical Engineering, University of Bath. 4. ‘‘SSuurrffaaccee PPllaassmmoonn--PPoollaarriittoonnss iinn TThhiinn MMeettaall SSttrriippss aanndd SSllaabbss:: WWaavveegguuiiddiinngg aanndd MMooddee CCuuttooffff’’ by IIaann GG.. BBrreeuukkeellaaaarr, B. Sc., B. A. Sc., Ottawa-Carleton Institute for Electrical and Computer Engineering, School of Information Technology and Engineering, Faculty of Engineering, University of Ottawa. 5. ‘‘GGooooss--HHäänncchheenn sshhiifftt iinn nneeggaattiivveellyy rreeffrraaccttiivvee mmeeddiiaa’’ by PP.. RR.. BBeerrmmaann, Michigan Center for Theoretical Physics, FOCUS Center and Physics Department, University of Michigan, Ann Arbor, Michigan. 6. ‘‘SSuurrffaaccee ppllaassmmoonn ppoollaarriittoonn pprrooppaaggaattiioonn aarroouunndd bbeennddss aatt aa mmeettaall--ddiieelleeccttrriicc iinntteerrffaaccee’’ by KKeeiissuukkee HHaasseeggaawwaa, JJeennss UU.. NNööcckkeell and MMiirriiaamm DDeeuuttsscchh, Oregon Center for Optics, 1274 University of Oregon. 7. ‘‘DDiieelleeccttrriicc wwaavveegguuiiddee mmooddeell ffoorr gguuiiddeedd ssuurrffaaccee ppoollaarriittoonnss’’ by RRaasshhiidd ZZiiaa, AAnnuu CChhaannddrraann, and MMaarrkk LL.. BBrroonnggeerrssmmaa, Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California December. 8. ‘‘MMeettaall--ccllaadd mmuullttiillaayyeerr ddiieelleeccttrriicc wwaavveegguuiiddee:: aaccccuurraattee ppeerrttuurrbbaattiioonn aannaallyyssiiss’’ by SShhoouu XXiiaann SShhee Dept. of Physics, Northern Jiaotong University, Beizing 100044, China. 9. ‘‘CChhaannnneell PPllaassmmoonn--PPoollaarriittoonn GGuuiiddiinngg bbyy SSuubbwwaavveelleennggtthh MMeettaall GGrroooovveess’’ by SSeerrggeeyy II.. BBoozzhheevvoollnnyyi, VVaalleennttyynn SS.. VVoollkkoovv, EEllooïïssee DDeevvaauuxx, and TThhoommaass WW.. EEbbbbeesseenn, Department of Physics and Nanotechnology, Aalborg University. 10. ‘‘PPllaassmmoonn--ppoollaarriittoonn wwaavveess gguuiiddeedd bbyy tthhiinn lloossssyy mmeettaall ffiillmmss ooff ffiinniittee wwiiddtthh:: BBoouunndd mmooddeess ooff aassyymmmmeettrriicc ssttrruuccttuurreess’’ by PPiieerrrree BBeerriinni, University of Ottawa, School of Information Technology and engineering, Canada. 11. ‘‘PPllaassmmoonn--ppoollaarriittoonn mmooddeess gguuiiddeedd bbyy aa mmeettaall ffiillmm ooff ffiinniittee wwiiddtthh bboouunnddeedd bbyy ddiiffffeerreenntt ddiieelleeccttrriiccss’’ by PPiieerrrree BBeerriinnii, University of Ottawa, School of Information Technology and engineering, Canada. 12. ‘‘AAnn EEffffeeccttiivvee aanndd AAccccuurraattee MMeetthhoodd ffoorr tthhee DDeessiiggnn ooff DDiirreeccttiioonnaall CCoouupplleerrss’’ by QQiiaann WWaanngg, SSaaiilliinngg HHee, Senior Member, IEEE, and Fangrong Chen. 13. ‘‘EEffffeeccttiivvee iinnddeexx mmeetthhoodd aanndd ccoouupplleedd mmooddee tthheeoorryy ffoorr aallmmoosstt ppeerriiooddiicc wwaavveegguuiiddee ggrraattiinnggss:: aa ccoommppaarriissoonn’’ by KKiimm AA.. WWiinniicckk. 14. ‘‘OOppttiiccaall PPrrooppeerrttiieess ooff NNaannoossttrruuccttuurreedd MMaatteerriiaallss SSpprriinngg 22000066 -- CCllaassss 1111’’, www.creol.ucf.edu.

RRR 111


15. ‘‘MMeettaall--ccllaadd ppllaannaarr ffoouurr--llaayyeerr ooppttiiccaall wwaavveegguuiiddee’’ by ZZii HHuuaa WWaanngg, Dept. of Electrical and computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-2991, and SS..RR.. SSeesshhaaddrrii, Dept. of Electrical and computer Engineering, University of Wisconsin-Madison, Wisconsin 53706-1691. 16. ‘‘EExxppeerriimmeennttaall oobbsseerrvvaattiioonn ooff ppllaassmmoonn--ppoollaarriittoonn wwaavveess ssuuppppoorrtteedd bbyy aa tthhiinn mmeettaall ffiillmm ooff ffiinniittee wwiiddtthh’’ by RRoobbeerrtt CChhaarrbboonnnneeaauu and PPiieerrrree BBeerriinnii, University of Ottawa, School of Information Technology and engineering, Canada, EEzziioo BBeerroolloo and LLiissiicckkaa SShhrrzzeekk, Communication Research Centre, Optical Network Technologies, Canada. 17. ‘‘AA CCoommpplleettee DDeessccrriippttiioonn ooff tthhee DDiissppeerrssiioonn RReellaattiioonn ffoorr TThhiinn MMeettaall FFiillmm PPllaassmmoonn--PPoollaarriittoonn’’ by FFrraasseerr aa.. BBuurrttoonn and SStteepphheenn AA.. CCaassssiiddyy.. 18. ‘‘AAnnaallyyssiiss OOff NNii--HHyyddrriiddee TThhiinn FFiillmm AAfftteerr SSuurrffaaccee PPllaassmmoonnss GGeenneerraattiioonn bbyy LLaasseerr TTeecchhnniiqquuee’’ by VV..VVIIOOLLAANNTTEE, EE.. CCAASSTTAAGGNNAA, CC.. SSIIBBIILLIIAA, SS.. PPAAOOLLOONNII.

19. ‘‘TThhee NNuummeerriiccaall MMeetthhoodd ooff LLiinneess ffoorr PPaarrttiiaall DDiiffffeerreennttiiaall EEqquuaattiioonnss’’ by MMiicchhaaeell BB.. CCuuttlliipp, UUnniivveerrssiittyy ooff CCoonnnneeccttiiccuutt aanndd MMoorrddeecchhaaii SShhaacchhaamm, Ben-Gurion University of the Negev. 20. ‘‘AA nneeww ggeenneerraalliizzeedd ddee--eemmbbeeddddiinngg mmeetthhoodd ffoorr nnuummeerriiccaall eelleeccttrroommaaggnneettiicc aannaallyyssiiss’ by YYuurrii OO. SShhlleeppnneevv,, Eagleware Corporation. 21. ‘‘SSuurrffaaccee ppllaassmmoonn--ppoollaarriittoonn mmeeddiiaatteedd lliigghhtt eemmiissssiioonn tthhrroouugghh tthhiinn mmeettaall ffiillmmss’’ by SStteepphheenn WWeeddggee aanndd WW.. LL.. BBaarrnneess,, Thin Films Photonics Group, School of Physics, Stocker Road, University of Exeter, Exeter. 22. ‘‘PPllaassmmoonn ppoollaarriittoonnss iinn mmeettaall nnaannoossttrruuccttuurreess:: tthhee ooppttooeelleeccttrroonniicc rroouuttee ttoo nnaannootteecchhnnoollooggyy’’, by MM.. SSAALLEERRNNOO,, JJ..RR.. KKRREENNNN,, BB.. LLAAMMPPRREECCHHTT,, GG.. SSCCHHIIDDEERR,, HH.. DDIITTLLBBAACCHHEERR,, NN.. FFÉÉLLIIDDJJ,, AA.. LLEEIITTNNEERR,, and FF..RR.. AAUUSSSSEENNEEGGGG. 23. ‘‘EEffffeeccttiivvee iinnddeexx mmeetthhoodd ffoorr hheetteerroossttrruuccttuurree--ssllaabb--wwaavveegguuiiddee--bbaasseedd ttwwoo--ddiimmeennssiioonnaall pphhoottoonniicc ccrryyssttaallss’’ by MMiinn QQiiuu, Laboratory of Optics, Photonics and Quantum Electronics, Department of Microelectronics and Information Technology, Royal Institute of Technology (KTH), Electrum 229. 24. ‘‘PPllaassmmoonniiccss:: ooppttiiccss aatt tthhee nnaannoossccaallee’’ by AAllbbeerrtt PPoollmmaann and HHaarrrryy AA.. AAttwwaatteerr. 25. ‘‘NNuummeerriiccaall ssiimmuullaattiioonnss ooff lloonngg--rraannggee ppllaassmmoonnss’’ by AAllooyyssee DDeeggiirroonn and DDaavviidd RR.. SSmmiitthh Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina. 26. ‘‘CCuurrvveedd lloonngg--rraannggee ssuurrffaaccee ppllaassmmoonn--ppoollaarriittoonn wwaavveegguuiiddeess’’ by PPiieerrrree BBeerriinnii,, University of Ottawa, School of Information Technology and engineering, Canada. 27. ‘‘NNoovveell ssuurrffaaccee ppllaassmmoonn wwaavveegguuiiddee ffoorr hhiigghh iinntteeggrraattiioonn’’ by LLiiuu LLiiuuaa,, Centre for Optical and Electromagnetic research, Zhejiang University, Joint Research Center of Photonics of the Royal Institute of Technology (Sweden) and Zhejiang University, Yu-Quan, Hangzhou, 310027, China, ZZhhaanngghhuuaa HHaannaa,, Laboratory of Photonics and Microwave Engineering, Department of Microelectronics and Information Technology, Royal Institute of Technology, Electrum 229, 16440 Kista, Sweden and SSaaiilliinngg HHeeaa Division of Electromagnetic Theory, Alfven Laboratory, Royal Institute of Technology, S-100 44 Stockholm, Sweden. 28. ‘‘FFIIRRSSTT BBRROOAADDBBAANNDD EEXXPPEERRIIMMEENNTTAALL SSTTUUDDYY OOFF PPLLAANNAARR TTHHZZ WWAAVVEEGGUUIIDDEESS’’ by RRaajjiinndd MMeennddiiss,, Bachelor of Science, University of Moratuwa, Moratuwa, Sri Lanka.

RRR 222


29. ‘‘LLeeaakkyy aanndd bboouunndd mmooddeess ooff ssuurrffaaccee ppllaassmmoonn wwaavveegguuiiddeess’’ by RRaasshhiidd ZZiiaa, MMaarrkk DD.. SSeellkkeerr and MMaarrkk LL.. BBrroonnggeerrssmmaa.. 30. ‘‘EEffffeeccttiivvee iinnddeexx mmeetthhoodd ffoorr ppllaannaarr lliigghhttwwaavvee cciirrccuuiittss ccoonnttaaiinniinngg ddiirreeccttiioonnaall ccoouupplleerrss’’ QQiiaann WWaanngg, GGeerraalldd FFaarrrreellll, TThhoommaass FFrreeiirr, Applied Optoelectronics Centre, School of Electronics and Communication Engineering, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland. 31. ‘‘IInntteeggrraatteedd OOppttiiccss aanndd OOppttiiccaall MMEEMMSS:: 11––TThheeoorryy ooff ooppttiiccaall wwaavveegguuiiddeess,, 11..33 EE--MM WWaavvee AApppprrooaacchh ttoo OOppttiiccaall WWaavveegguuiiddee TThheeoorryy,, 11..44 MMooddeess iinn RReeccttaanngguullaarr WWaavveegguuiiddeess’’ EEEE 553399AA UUnniivveerrssiittyy ooff WWaasshhiinnggttoonn. 32. ‘‘PPllaassmmoonniiccss -- AA RRoouuttee ttoo NNaannoossccaallee OOppttiiccaall DDeevviicceess’’ by SStteeffaann AA.. MMaaiieerr, MMaarrkk LL.. BBrroonnggeerrssmmaa, PPiieetteerr GG.. KKiikk, SShheeffffeerr MMeellttzzeerr, AArrii AA.. GG.. RReeqquuiicchhaa, and HHaarrrryy AA.. AAttwwaatteerr.. 33. ‘‘OObbsseerrvvaattiioonn ooff ccoouupplleedd ppllaassmmoonn--ppoollaarriittoonn mmooddeess iinn AAuu nnaannooppaarrttiiccllee cchhaaiinn wwaavveegguuiiddeess ooff ddiiffffeerreenntt eennggtthhss:: EEssttiimmaattiioonn ooff wwaavveegguuiiddee lloossss’’ by SStteeffaann AA.. MMaaiieerr, PPiieetteerr GG.. KKiikk, aanndd HHaarrrryy AA.. AAttwwaatteerr, Thomas J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California. 34. ‘‘DDeemmoonnssttrraattiioonn ooff iinntteeggrraatteedd ooppttiiccss eelleemmeennttss bbaasseedd oonn lloonngg--rraannggiinngg ssuurrffaaccee ppllaassmmoonn ppoollaarriittoonnss’’ by RRoobbeerrtt CChhaarrbboonnnneeaauu and NNaannccyy LLaahhoouudd,, Spectalis Corp., Nepean, Ontario, Canada, GGrreegg MMaattttiiuussssii, Ottawa, Ontario, Canada, PPiieerrrree BBeerriinnii University of Ottawa, School of Information Technology and engineering, Canada. 35. ‘‘TTeerraahheerrttzz ssuurrffaaccee ppllaassmmoonn ppoollaarriittoonnss’’, http://www-users.rwth-aachen.de/jaime.gomez/public.html, [email protected]. 36. ‘‘CCoohheerreenntt ffaarr--ffiieelldd eexxcciittaattiioonn ooff ssuurrffaaccee ppllaassmmoonnss uussiinngg rreessoonnaannttllyy ttuunneedd mmeettaall nnaannooppaarrttiiccllee aarrrraayyss’’ bbyy AAmmiittaabbhh GGhhoosshhaal, GGrraaddyy WWeebbbb--WWoooodd, CCllaarriissssee MMaazzuuiirr, College of Optics and Photonics, University of Central Florida, Orlando, USA and PPiieetteerr GG.. KKiikk Physics department, University of Central Florida, Orlando, USA. 37. ‘‘PPllaassmmoonnss iinn mmeettaall nnaannoossttrruuccttuurreess’’ by CCaarrsstteenn SSoonnnniicchhsseenn, physics department of the Ludwig-Maximilians-University of Munich. 38. ‘‘EEffffeeccttiivvee IInnddiicceess ooff HHiigghh CCoonnttrraasstt NNaannoo && MMiiccrroossttrruuccttuurreess’’ by AArriieell SScchhllaammmm Rochester Institute of Technology M. G. Moharam, REU Advisor. 39. ‘‘SSuurrffaaccee ppllaassmmoonn ppoollaarriizzaattiioonn ffiilltteerriinngg iinn aa ssiinnggllee mmooddee ddiieelleeccttrriicc wwaavveegguuiiddee’’ by PP.. SS.. DDaavviiddss, BB.. AA.. BBlloocckk, and KK.. CC.. CCaaddiieenn,, Components Research, Intel Corporation, Hillsboro. 40. ‘‘GGeeoommeettrriieess aanndd mmaatteerriiaallss ffoorr SSuubbwwaavveelleennggtthh ssuurrffaaccee ppllaassmmoonn mmooddeess’’ by RRaasshhiidd ZZiiaa, MMaarrkk DD.. SSeellkkeerr and MMaarrkk LL.. BBrroonnggeerrssmmaa.. 41. ‘‘AAnn IInnttrroodduuccttiioonn ttoo OOppttiiccaall WWaavveegguuiiddeess’’ by MM.. JJ.. AAddaammss Department of Electronics, The University, Southampton; John Willey & Sons.

42. ‘‘OOppttiiccaall IInntteeggrraatteedd cciirrccuuiittss’’ by HHiirroosshhii NNiisshhiihhaarraa, MMaassaammiittssuu HHaarruunnaa, TToosshhiiaakkii SSuuhhaarraa; McGraw-Hill Book Company. 43. ‘‘AAddvvaanncceedd EEnnggiinneeeerriinngg EElleeccttrroommaaggnneettiiccss’’ by CCoonnssttaannttiinnee AA.. BBaallaanniiss, Arizona State University, John Wiley & Sons.

RRR 333


44. ‘‘‘ MMMiiicccrrrooowwwaaavvveee DDDeeevvv iii ccceeesss aaannnddd CCiii rrrcccuuuii tttsss ’’’ by SSSaaammmuuueeelll YYY... LLL iiiaaaooo, Professor of Electrical Engineering, California State University, Fresno, Third Edition, PEARSON Education.

C i

G w

t

AB

45. ‘‘‘HHHiiiggghhheeerrr EEEnnngggiiinnneeeeeerrr iiinnnggg MMMaaattthhheeemmmaaattt iii cccsss ’’’ by DDDrrr ... BBB...SSS... GGrrreeewwaaalll and GGG...SSS... GGGrrreeewwwaaalll , Khanna Publishers. 46. ‘‘‘ IIInnntt rrroooddduuucccttt iiiooonnn tttooo eee llleeecccttt rrrooodddyyynnnaaammmiiicccsss ’’’ by DDDaaavvv iiiddd JJJ... GGGrrr iii fff fff iii ttthhhsss , 3rd Edition, Prentice Hall Private Limited. 47. ‘‘‘GGGeeettt ttt iiinnnggg ssstttaaarrr ttteeeddd iiinnn MMMAAATTTLLLAABB’’’ by RRRuuudddrrraaa PPPrrraaatttaaappp... 48. ‘‘‘PPPrrrooogggrrraaammmmmmiiinnnggg iiinnn MMMaaattt lllaaabbb’’’ by HHHeeerrrnnn iii ttteeerrr ...

RRR 444

Documents

BTech Project Report on Surface Plasmon Polariton Waves