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    Photonics and Optical Communication, Spring 2007, Dr. D. Knipp

    1Waveguides

    Photonics and Optical Communication

    (Course Number 300352)

    Spring 2007

    Waveguides

    Dr. Dietmar Knipp

    Assistant Professor of Electrical Engineering

    http://www.faculty.iu-bremen.de/dknipp/

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    Photonics and Optical Communication

    3 Waveguides3.1 Introduction

    3.2 Reflection and Refraction at the Boundary between two Media3.3 Total internal reflection

    3.3.1 Light propagation in an optical fiber

    3.3.2 Acceptance angle

    3.4 Planar Waveguide

    3.4.1 Planar Mirror Waveguide3.4.2 Planar Dielectric Waveguide

    3.5 Modes in Waveguides

    3.5.1 Transverse Electric Waves

    3.5.2 Transverse Magnetic Waves

    3.5.3 Transverse Electro Magnetic Waves3.5.4 Calculating Modes in a planar wave guide

    3.5.5 The effective refractive index

    3.5.6 The Mode chart

    3.5.7 Designing a planar wave guide

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    3.5.8 TE versus TM Modes

    3.5.9 Types of Modes

    3.5.10 Numbering of modes

    3.6 Coupling between Waveguide

    References

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    3.1 Introduction

    Light can be confined by an optical waveguide. The waveguide is formed by a

    medium which is embedded by an another medium of lower refractive index.

    The medium of higher refractive index acts as a light trap. Light is confined in

    the waveguide by multiple total internal reflections. By doing so light can betransported from one location to another location. Waveguides can be

    distinguished in terms of slabs, strips and fibers. The most widely applied

    waveguide structure is the optical fiber, which is made out of two concentrically

    cylinders of low-loss glass with slightly different refractive index.

    Waveguides can be distinguished in terms of a slab, a strip or a fiber.

    Ref: Saleh & Teich, Fundamentals of Photonics

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    3.1 Introduction

    If a lot of optical component like wave guides, light sources and light receivers

    are integrated together on a substrate (chip) we speak about integrated

    optics. The goal is to miniaturize optics like electronics to improve performance

    and reduce cost.

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    3.1 Introduction

    The optical fiber in its existing form (the fiber consists of a core and a cladding)

    was invented 40 years ago. The first fibers were used in the near infrared

    wavelength region at around 800nm-900nm. As technology of fibers and light

    sources evolved the optical transmission window was shifted to 1310nm in themid 1980s and 1550nm in the 1990s.

    Internal reflection is a requirement for the guidance or confinement of waves in

    a waveguide. Total internal reflection can only be achieved if the refractive

    index of the core is larger than the refractive index of the cladding. In the

    following, we will briefly repeat the related ray optics.

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    3.2 Reflection and Refraction at the Boundary between two Media

    The reflection and refraction of light at an interface can be described by Snells

    law. The angle of incidence is given by 1 which is related to the angle ofrefraction 2.

    Reflection of rays at an interface. (a) From a high to a low refractive medium, (b)

    The critical angle, (c) Total internal reflection.

    Ref: J.M. Senior, Optical Fiber Communication

    2211 sinsin = nn

    Snells law.

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    3.3 Total internal reflection

    With increasing angle of incidence 1 the angle of refraction 2 also increases.

    If n1 > n2, there comes a point when 2 =/2 radians. This happens when1=sin

    -1(n2 / n1). For larger values of 1, there is no refracted ray, and all the

    energy from the incident ray is reflected. This phenomena is called totalinternal reflection. The smallest angle for which we get total internal reflection

    is called the critical angle and 2equals /2 radians.

    The total internal reflection is an requirement for the guidance of light in an

    optical fiber.

    1

    2sinn

    nc = Critical angle

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    3.3.1 Light propagation in an optical fiber

    An optical fiber can be described by an cylindrical core surrounded by a

    cladding. Usually (at least for optical communication) the fiber core and the

    cladding are made of silica (SiO2). The refractive index of the core is slightly

    higher than the refractive index of the cladding so that the light is guided in thefiber.

    Transmission of a light ray in a perfect optical fiber.

    Ref: J.M. Senior, Optical Fiber Communication

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    3.3.2 Acceptance angleTotal internal reflection is required to guide light in an optical fiber. We know

    that only light under sufficient shallow angles (angle greater than the critical

    angle) can propagate in the fiber. The question is now under what angle a ray

    can enter a fiber? It is clear that not all rays entering the fiber core will continue

    to be propagated along the fiber. Only rays that enter the fiber within aacceptance cone (acceptance angle) will propagate along the fiber, whereas

    rays outside of the cone will not be guided.

    Coupling of a ray into a fiber. The ray can only be coupled into the fiber

    when the angle of incident is within the acceptance cone.

    Ref: J.M. Senior, Optical Fiber Communication

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    3.3.2 Acceptance angle

    In the following we will derive an expression for the the acceptance angle

    from the refractive indices of the three media involved, namely the core of the

    fiber (n1), the cladding of the fiber (n2) and the air (n0).

    In order to enter the fiber Snells law has to be fulfilled.

    The angle 2can now be

    described by

    So that the Snells law can

    be modified to

    2110 sinsin = nn

    =2

    2

    cossin 110 = nn

    Coupling of a ray into a fiber. The ray can onlybe coupled in the fiber when the angle of

    incident is within the acceptance cone.

    Ref: J.M. Senior, Optical Fiber

    Communication

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    3.3.2 Acceptance angleIf we consider now the trigonometrically relationship

    The expression can be modified to

    Now the equation can be combined with the equation for the critical angle

    Leading to the relationship for the numerical aperture

    The acceptance angle can now be calculated by

    ( ) ( ) 1cossin 22 =+

    2110 sin1sin = nn

    ( )121sin nn=

    22

    2110 sin nnnNA == Numerical aperture

    Acceptance angle

    =<

    0

    22

    211

    1 sinn

    nna

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    3.4.1 Planar Mirror waveguideA planar mirror waveguide is shown on this

    slide. The cladding of the waveguide is formed

    by a conducting material, which can be a

    mirror.

    As a consequence of the conducting cladding

    the tangential components of the electric and

    the magnetic field is zero at the interface

    between the core and the cladding.

    Therefore, the waves can not extend in the

    cladding of the waveguide. Subsequently the

    modes of propagation are defined by thedimensions of the core of the waveguide.

    Cross section of a mirror

    waveguide

    Ref: Back to Basics in Optical

    Communications, Tutorial

    Agilent Technologies

    0=TE 0=TB

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    15Waveguides

    3.4.2 Planar Dielectric waveguideA planar dielectric waveguide is shown

    on this slide. The cladding of the

    waveguide is formed by a dielectric

    medium of lower refractive index.

    The waves extend in the cladding of

    the waveguides. The wave propagating

    the waveguide can be described by

    The complex amplitude of the wave

    corresponds to the transverse standing

    wave perpendicular to the direction of

    propagation. Due to the fact that wavesextend in the cladding the wavelengths

    that can propagate are larger than 2

    times the diameter of the core. is apropagation constant.

    Cross section of a dielectric

    waveguide.

    Ref: Back to Basics in Optical

    Communications, Tutorial Agilent

    Technologies

    ( )( )ztjrEtrE = exp,

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    3.5 Modes in Waveguides

    The planar waveguide is the simplest form of an optical waveguide. The

    waveguide can be realized by a simple sandwich structure which consists of a

    slab embedded between two regions of lower refractive index.

    The optical ray within the waveguide can be described by a transverseelectromagnetic wave, which can be a TE, TM or TEM wave.

    Propagation of a wave in a planar

    waveguide. We can impose self-consistency condition which

    requires that the wave reproduces

    itself. Fields that satisfy this

    conditions are called Eigenmodes

    (modes) of the waveguide.

    Ref: Saleh and Teich,

    Fundamentals of Photonics

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    3.5 Modes in a Waveguides

    As a consequence of the superposition of planar waves we get an interference

    pattern, which is formed in the waveguide (the z-direction is the propagation

    direction of the wave). If the total phase change upon two successive reflections

    is equal to 2m constructive interference is observed, where m is a positive

    integer. The phase shift has two contributions. The reflection of the plane wave atthe interface leads to a phase shift, which depends on the angle and thedistance traveled. Later on we will derive an expression for the modes of

    propagation in such a structure.Formation of modes in a planar

    dielectric waveguide. (a) planewave propagation in a waveguide

    and corresponding electric field

    distribution in the optical fiber.

    The interference of the plane

    waves in the waveguide isforming the lowest order mode

    (m=0).

    Ref: J.M. Senior, Optical Fiber

    Communication

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    3.5 Modes in a Waveguides

    In the figure on the previous slide the lowest order mode (m=0) is shown. A mode

    of propagation is only observed when the angle between the propagation vector

    and the interface (boundary of the cladding and the core) has particular values.

    For all modes of propagation a standing wave is formed in the waveguide.

    Depending on the mode of propagation an electric field distribution is formed. Forthe lowest order mode the electric field is maximized in the center of the core.

    The electric field decays towards the boundaries. For all modes of propagation

    the self-consistency condition has to be satisfied which means that the wave in

    the waveguide reproducing itself.

    Before discussing the modes of propagation mathematically we will discuss the

    propagation of waves in a waveguide phenomenologically .

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    19Waveguides

    3.5 Modes in WaveguidesAgain we assume a plane wave which

    propagates in the z-direction. We

    observe constructive interference

    across the waveguide as a

    consequence of the superposition of the

    propagating waves. In the examples

    shown on this slide the propagation

    modes are m=1, 2 and 3. The number

    of modes corresponds to the number of

    zeros in the transverse electric field

    pattern.

    How do we determine the self-

    consistency conditions. In order to

    achieve total internal reflection theangle of incidence has to be smaller

    than

    Propagation of waves in a waveguide

    and the corresponding transverseelectric (TE) field pattern of three

    lower order models m=1, 2, 3.

    Ref: J.M. Senior, Optical Fiber

    Communication

    =

    <

    1

    21

    1

    21 cossin

    2 n

    n

    n

    nC

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    For self-consistency the wave

    reproduces itself and the phase

    shift between the two waves has

    to be zero or a multiple of 2.We can assume that the field in

    the slab is in the form of a

    monochromatic plane wave

    bouncing back and forth at an

    angle smaller than the criticalangle C. A round trip can bedescribed by:

    For self-consistency the phaseshift between the two waves has

    to be zero or a multiple of 2.

    Planar dielectric waveguide.

    Ref: Saleh and Teich, Fundamentals of

    Photonics

    3.5 Modes in Waveguides

    sin2dABAC =

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    3.5 Modes inWaveguides

    There is a phase r introduced by each internal reflection at the boundary. Thereflection phase is a function of the angle and it depends on the polarization ofthe incidence wave (TE, TM or TEM wave), which is described by the complex

    reflection and transmission coefficients (see Review of optics). The complex

    reflection and transmission coefficients can be separated in an amplitude and a

    phase, where the phase depends on the angle of incidence.

    ,2,1,02sin22

    == mformd r

    Mode Equation

    Reflection coefficient and phase shift on

    reflection for a transverse electric wave

    as a function of the angle of incidencefor a glass/air interface.

    Ref: J.M. Senior, Optical Fiber

    Communication

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    3.5.1 Transverse Electric WavesIn the case of a transverse electric field (TE mode) the electric field is

    perpendicular to the direction of propagation of the wave (z-direction). As we

    are dealing with electro-magnetic waves each wave consists of a periodically

    varying electric and magnetic field which is again perpendicular to each other.

    In the case of a TE transverse wave the electric is perpendicular to thedirection of propagation (Ez=0) and the magnetic field has a (small) component

    to the direction of propagation. This is due to the fact that the traveling wave is

    not propagating in a straight line in the wave guide, meaning the ray is

    propagation on a zigzag path.

    Propagation of a TE wave in a slab

    waveguide.

    Ref: J.C. Palais, Fiber Optic

    Communication

    z

    x

    B

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    3.5.2 Transverse Magnetic WavesIn the case of transverse TM modes the magnetic field is perpendicular to the

    direction of propagation and the electric field has a (small) component to the z-

    direction of propagation. Again the traveling wave is propagating on a zigzag path

    rather than a straight line in the wave guide.

    Propagation of a TM wave in a slab

    waveguide.

    Ref: J.C. Palais, Fiber Optic

    Communication

    z

    x

    E

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    3.5.3 Transverse Electro Magnetic Waves

    In the case of a TEM transverse wave (TEM modes) both the electric and the

    magnetic field are perpendicular to the direction of propagation, which means

    that the rays propagate straight in the fiber. Such cases occurs only for single

    mode fibers.

    3.5.4 Calculating Modes in a planar wave guideThe mode equation and the equation for the phase shift have to be merged,

    which leads to a transcendental equation. The transcendental equation has to

    be solved to get the modes which propagate in a given waveguide structure. A

    detailed mathematical description is given by Saleh and Teich in their book

    Fundamentals of Photonics.

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    3.5.5 The effective refractive indexThe effective refractive index is defined by:

    The effective refractive index is a key parameter for waveguides like the

    refractive index is a key parameter for the free space propagation of waves.

    The effective refractive index changes the wavelength in the same way that abulk refractive index does. The idea of the effective refractive index gets clear

    by simply looking at the structure of a waveguide. The effective refractive

    index is a corrected refractive index which simply assumes that the wave

    propagates in a straight line the media (in our case in the core of the

    waveguide structure.)

    sin1nneff =

    Plane wave propagating in a

    waveguide. The effective refractive

    index considers that the plane wavepropagates by following a zigzag path.

    Ref: J.C. Palais, Fiber Optic

    Communicationz

    x

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    26Waveguides

    3.5.6 The Mode chartThe mode chart for a waveguide structure is an absolutely essential graph to

    study the propagation of modes in a given waveguide structure. The

    thickness/diameter (d) of the core of the waveguide is usually normalized by

    the wavelength of the incident light. The different modes of propagation are

    plotted for the propagation angle and the effective refractive index.

    Mode chart for a

    symmetric slab. The

    following refractive indices

    were assumed for the core

    n1=3.6 and the cladding

    n2

    =3.55 (AlGaAs

    structure).

    Ref: J.C. Palais, Fiber

    Optic Communication

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    3.5.6 The Mode chartFor the particular mode chart shown on the previous slide an AlGaAs waveguide

    structure (used for a laser diode) is assumed The critical angle for the structure

    is

    Therefore, the range of angles for which the ray is trapped in the waveguide is

    then 80.4 90. As a consequence the effective refractive index ranges from

    3.55 to 3.6.

    From the mode chart we can draw the following conclusions:

    When the core thickness is very small in comparison to the wavelength of the

    propagating light the wave travels very close to the critical angle and the

    effective index is close to the refractive index of the refractive index of the

    cladding. The wave penetrates deeply into the outer layers, because the rays are

    near the critical angle.With increasing thickness of the core the ray travels at larger angles. The ray

    travels more parallel to the waveguide axis. For thick films (thickness is large in

    comparison with the wavelength of the propagating light) the effective index is

    very close to the refractive index of the film itself.

    ( ) == 4.80sin 121 nnc

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    3.5.7 Designing a planar wave guide

    For any given propagation angle there is a set of film thicknesses that will allow

    rays to propagate. The following equation has to be satisfied for the higher

    modes, where m is a positive integer.

    In order to change the mode the normalized thickness has to change by:

    cos2 10 n

    mdd

    m+=

    ( )

    cos2

    1

    1n

    d =

    Table of TEm modes in a gallium arsenide waveguide.

    Ref: J.C. Palais, Fiber Optic Communication

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    3.5.7 Designing a planar wave guide

    The following equation can be applied to calculate the number of TE modes

    supported by the dielectric waveguide, where m is increased to the nearest

    integer.

    3.5.8 TE versus TM Modes

    So far we discussed only the propagation of TE modes. However, TM modes

    exhibit almost identical propagation behavior. This is why we will not distinguish

    between TM and TE modes. Therefore, the curve in the mode chart were labeled

    as both TE and TM modes. This is mostly true since the difference in the

    refractive index for the core and the cladding are very small (in the range of a fewpercent). Even with increasing difference the cutoff modes are identical. For each

    TE mode there will be always a TM mode. The number of total modes is

    therefore twice the number of TE modes. The electric field distribution for the

    different mode is shown in the following.

    ( )21

    22

    21

    0

    2

    nnNA

    NAd

    m

    =

    =

    Number of TE Modes

    Numerical Aperture

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    3.5.9 Types of ModesIn the case of a transverse TE (TE modes) wave the electric field is

    perpendicular to the direction of propagation and the magnetic field has a

    (small) component that is in the direction of the propagation.

    In the case of a transverse TM (TM modes) wave the magnetic field isperpendicular to the direction of propagation and the electric field has a (small)

    component that is in the direction of the propagation.

    In the case of a TEM transverse wave (TEM modes) both the electric and the

    magnetic field are perpendicular to the direction of propagation, which means

    that the rays propagate straight in the fiber. Such cases occurs only for single

    mode fibers.

    Furthermore, helical modes (HE or EH) modes exist. Under such conditions

    the ray travels in a circular path in the fiber and electric and magnetic field have

    components in the z-direction. These modes can be realized either as a HE or aEH mode depending on which field contributes most to the z-direction.

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    3.5.10 Numbering of Modes

    It turns out that the difference in refractive index between the cladding and the core

    is usually very small. The modes for TE, TM, HE and EH modes are very similar.

    Therefore, we can simplify the way we look at modes in waveguides and fibers. The

    listed modes can be summarized and explained using only a single set of LP (linear

    polarized) modes.The TE and the TM modes were numbered based on the number of zeros in their

    electric field pattern across the waveguide. Therefore a TE0 mode would be a

    continuous distribution with only a single maxima but no zeros. A TE00 mode would

    be a mode for a 2-dimensional waveguide structure and the electric field distribution

    would correspond to a single spot. Obliviously a waveguide structure does not haveto be symmetric. A TE21 would be now a pattern with 2 zeros in one direction and a

    third zero in the perpendicular direction.

    Electric field distribution for some

    symmetric and asymmetric slabwaveguides. The numbering for TE,

    TM and TEM mode is identical.

    Ref: H. Dutton, Understanding Optical

    Communication

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    3.5.10 Numbering of modes

    Numbering of linear polarized modes is different from numbering TE and TM

    modes, but LP numbers are only used for fibers (circular waveguides). LP

    modes are described by LPlm where m is the number of maxima along the

    radius of the fiber and l is half of the number of maxima around the

    circumference.

    Correspondence between the linear

    polarized modes and the traditional

    exact modes in a cylindrical fiber.

    Ref: J.M. Senior, Optical Fiber

    Communication

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    3.6 Coupling between waveguidesIf waveguides are sufficiently close to each other light can couple from one

    waveguide to the other. Coupling occurs if the electric fields of the two

    waveguides overlap. This effect can be used to build couplers and switches. The

    two waveguides are formed by two slabs of higher refractive index similar to single

    waveguide structures. Maxwell equations can be used to describe the coupling of

    modes from one waveguide to the other waveguide. The problem can be

    described by two coupled differential equations (Coupled-Mode Equations).

    Coupling between two parallelplanar waveguides.

    At z1 most of the light is guided in

    waveguide 1,

    at z2 the light is equally divided

    amongst the two waveguides,at z3 most of the light is guided in

    waveguide 2.

    Ref.: B.E.A. Saleh, M.C Teich,

    Fundamentals of Photonics

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    3.6 Coupling between waveguidesLets assume that light propagates in waveguide 1. No light (at least at this

    point) propagates in waveguide 2. If the waveguides are close enough to each

    other and the electric fields of the waveguides overlap, the wave (in waveguide

    1) couples in the waveguide 2. Based on intuition it could be expected that half

    of the light is coupled in the waveguide 2. However, this is not the case. Almost

    all the light is coupled from waveguide 1 in waveguide 2. The length after which

    all the light is coupled form waveguide 1 to waveguide 2 is called the coupling

    length L0.

    If the two waveguides are close to each other for more than the coupling lengththe light starts to couple back in the waveguide 1. Depending on the length of

    the coupling structure the waves couple back and forth between the

    waveguides.

    Ph t i d O ti l C i ti S i 2007 D D K i

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    3.6 Coupling between waveguidesThis effect can be used to build switches and 3dB couplers. If the two

    waveguides are close to each other for a distance which is equal to the

    coupling length the entire optical power is coupled from one waveguide to the

    other waveguide. The structure can be used as switch. If the two waveguides

    are close to each other for only half of the coupling length the incoming opticalpower is divided into two equal intensities. Such a structure can be used as an

    3dB coupler. In discrete optics a beam splitter would be used to separate a

    beam into two equal beams.

    Waveguide based optical coupler, (left) switching of the power from one

    waveguide to the other, (right) a 3dB coupler.

    Ref.: B.E.A. Saleh, M.C Teich, Fundamentals of Photonics

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    3.6 Coupling between waveguidesSimilar coupling behavior is observed for single mode fibers. Two single-mode

    fibers have to be placed close and parallel to each other to accomplish

    coupling. In this case we are of course speaking about the cores of the single-

    mode fibers, which have to be placed close to each other. Like already

    described for the planar waveguides the electric field of the two fiber cores hasto overlap so that waves can couple from one fiber in the other. Again, we can

    define a coupling length. If the fiber cores are close for longer than the coupling

    length we observe an oscillation of the intensity from one fiber to the other fiber.

    Coupling between

    single mode fibers.

    Ref.: H. J.R. Dutton,

    Understanding optical

    communications

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    3.6 Coupling between waveguidesThe coupling length strongly dependent on the separation of the two single

    mode fiber cores. The further apart they are the greater the coupling length.

    Furthermore, the coupling length is strongly wavelength dependent. For

    different wavelengths the coupling length changes.

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    38Waveguides

    References:Stamatios V. Kartalopoulos, DWDM, Networks, Devices and Technology,

    IEEE press and Wiley Interscience, 2003.

    Eugene Hecht, Optics, Addison Wesly, 4th edition, 2002

    Fawwalz T. Ulaby, Fundamentals of Applied Electromagnetics, PrenticeHall, 2001.

    John M. Senior, Optical Fiber Communications, Prentice Hall Series in

    Optoelectonics, 2nd edition, 1992.

    Bahaa E.A. Saleh, Malvin Carl Teich, Fundamentals of Photonics,Wiley-Interscience (1991)

    Harry J. R. Dutton, Understanding Optical Communications,

    Prentice Hall Series in Networking, 1998. (Formerly freely available as a red

    book on the IBM red book server.

    Joseph C. Palais, Fiber Optic Communications,

    Prentice Hall Series, 1998. 4th edition.