Research Article Band Structure Engineering in 2D Photonic ...downloads.hindawi.com/journals/aot/2014/780142.pdf · Research Article Band Structure Engineering in 2D Photonic Crystal

Research ArticleBand Structure Engineering in 2D Photonic CrystalWaveguide with Rhombic Cross-Section Elements

Abdolrasoul Gharaati1 and Sayed Hasan Zahraei2

1 Department of Physics Payame Noor University PO Box 19395-3697 Tehran Iran2Nanotechnology Research Institute of Salman Farsi University PO Box 73196-73544 Kazerun Iran

Correspondence should be addressed to Abdolrasoul Gharaati agharaatipnuacir

Received 6 February 2014 Revised 14 April 2014 Accepted 15 April 2014 Published 12 May 2014

Academic Editor Jose Luıs Santos

Copyright copy 2014 A Gharaati and S H ZahraeiThis is an open access article distributed under theCreativeCommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

Two-dimensional photonic crystal (2D PhC) waveguides with square lattice composed of dielectric rhombic cross-section elementsin air background by using plane wave expansion (PWE) method are investigated In order to study the change of photonic bandgap (PBG) by changing of elongation of elements the band structure of the used structure is plotted We observe that the size ofthe PBG changes by variation of elongation of elements but there is no any change in the magnitude of defect modes However theused structure does not have any TE defect modes but it has TM defect mode for any angle of elongation So the used structurecan be used as optical polarizer

1 Introduction

PhCs are class of media represented by natural or artificialstructures with periodic modulation of the refractive index[1ndash3] Such optical media have some peculiar propertieswhich gives an opportunity for a number of applications tobe implemented on their basis In 2D PhCs the periodicmodulation of the refractive index occurs in two directionswhile in one other direction structure is uniform Whenthe refractive index contrast between elements of the PhCand background is high enough a range of frequenciesexists for which propagation is forbidden in the PhC andcalled photonic band gap (PBG)The PBG depends upon thearrangement and shape of elements of the PhC fill factorand dielectric contrast of the two mediums used in formingPhCThemost important feature of PhCs is ability to supportspatially electromagnetic localized modes when a perfectlyperiodic PhC has spatial defects [4ndash6] In recent years a lot ofresearches are devoted to study 2D PhC with circular squareand elliptic cross-section elements [7 8] However less workwas devoted to study of PhC with rhombic cross-sectionelements In this paper we study band structure for 2D PhCwaveguide with dielectric rhombic cross-section elements

with a square lattice and how band structure is affected byelongating of elements

2 PWE Method and Numerical Analysis

We consider 2D PhC waveguide as shown in Figure 1(a)consisting of a square lattice of GaAs rhombic cross-sectionelements in air background having a lattice constant of 119886 =

815 nmThe rhombuses have 04 119886 side and a refractive indexof 119899119886= 337 [8]Thewaveguide core is formed by substitution

of a row of rhombuses with a row of different rhombuses withrefractive index 119899

119889= 1 and 04 119886 side along the 119910 direction

Figure 1(b) shows the unit cell for the structure used which iscomposed of the elements as shown in Figure 1(c) [1]

To obtain the band structure of the considered 2D PhCwaveguide the PWEmethod has been employed [1 5] Basedon the symmetry considerations the general form of themagnetic field vector of a TE-polarizedmode and the electricfield vector of a TM-polarized mode expanded into planewave vector with respect to the 2D reciprocal lattice vector labeled with a Bloch wave number 119896

119910 which is given by

[1]

Hindawi Publishing CorporationAdvances in Optical TechnologiesVolume 2014 Article ID 780142 5 pageshttpdxdoiorg1011552014780142

2 Advances in Optical Technologies

a

y

z x

(a) (b)

y

x

120579

(c)

Figure 1 (a) 2D PhC waveguide (b) the unit cell of the 2D PhC waveguide and (c) the element of the unit cell

For TE-polarized mode

= (0 0119867119911119896119910

(119909 119910))

119867119911119896119910

( 119903) = sum

119866

int

120587

119886

minus

120587

119886

119889119896119909119867119911( + ) exp (119894 ( + ) sdot 119903)

(1)

For TM-polarized mode

= (0 0 119864119911119896119910

(119909 119910))

119864119911119896119910

( 119903) = sum

119866

int

120587119886

minus120587119886

119889119896119909119864119911( + ) exp (119894 ( + ) sdot 119903)

(2)

where and are magnetic field vector electric fieldvector 2D reciprocal lattice vector and plane wave vectorrespectively The sum and integral are taken over the firstBrillouin zone of the 2D PhC waveguide used [1 5]

SolvingMaxwellrsquos equations in CGS unit for themagneticand electric fields leads to the following vector wave equa-tions

1205962

= nabla times (

1

120576 ( 119903)

nabla times ) (3)

1205962

=

1

120576 ( 119903)

nabla times (nabla times ) (4)

where 120576( 119903) is the dielectric function of the unit cellSubstituting (1) in the vector wave (3) and (2) in the vector

wave (4) we get two eigenvalue problems for the square offrequency 120596 for each polarized mode


1205962

(119896119910)119867119911( + )

= minussum

119866

int

120587119886

minus120587119886

1198891198961015840

119909120581 ( + minus

1015840

minus 1015840

)

times [(1015840

+ 1015840

) sdot ( + minus 1015840

minus 1015840

)]119867119911(1015840

+ 1015840

)

(5)


1205962

(119896119910) 119864119911( + )

= minussum

119866

int

120587119886

minus120587119886

1198891198961015840

119909120581 ( + minus

1015840

minus 1015840

)

times

10038161003816100381610038161003816(1015840

+ 1015840

)

10038161003816100381610038161003816

2

119864119911(1015840

+ 1015840

)

(6)

That 120581( 119896+) is the Fourier expansion of the inverse dielectric

function of 2D PhC waveguide that is written as

120581 ( + ) =

1

Sunit cellint

unit cell119889 119903

1

120576 ( 119903)

exp (minus119894 ( + ) sdot 119903)

(7)

That integral is taken over the unit cell in Figure 1(b) Fora given value of a Bloch wave number 119896

119910as propagation

constant 120573 (5) and (6) constitute two eigenvalue problemswith respect to the square of frequency 120596(119896

119910) Finally using

a trapezoidal approximation of the 1D integral 1198961015840119909and the

numerical solutions for (5) and (6) we get the band structureof the structure used [1] The computation method used forimplementation of PWE method for 2D PhC waveguide issimilar to the one which is used for the computation ofthe band structure of strictly periodic PhC There is someessential difference in the structure parameters definitions[1 2] First in 2D PhC waveguide the unit cell consists ofseveral PhC elements rather than one The defect of periodicstructure is also introduced to form the waveguide core Alsoin case of 2D PhC band structure computation we set the 119896-path to pass through all high symmetry points of the Brillouinzone However as we have considered in this section com-putation of the 2D PhC waveguide band structure requirestransversal wave vector consideration only The longitudinalcomponent stays in this case for the propagation constant andthe propagation constant is limited by the boundaries of theBrillouin zone One more difference from strictly periodicPhC is the definition of the reciprocal lattice vectors set [1ndash3]

3 Elongation of the Rhombuses

According to Figure 2 we can change the elongation angle 120579

that it makes with 119909 axis for transformation of rhombuses

Advances in Optical Technologies 3

y

x

120579

Figure 2 Schematic elongation angle 120579 that it makes with 119909 axis in the unit cell

0 05 1 15 2 25 3 350

01

02

03

04

05

times106

Freq

uenc

y120596a2120587c

Propagation constant 120573 (mminus1)

(a)

0 05 1 15 2 25 3 350

01

02

03

04

05

times106

Freq

uenc

y120596a2120587c


(b)

Figure 3 Band structures for elongation angle 120579 = 1205874 rad for (a) TE-polarized mode and (b) TM-polarized mode

By changing of the elongation angle 120579 when the definition ofthe unit cell is being made in discrete form by setting valuesof inversed dielectric function to mesh nodes we define theborders of rhombuses in each element of the unit cell asfunction of the elongation angle 120579 as

1003816100381610038161003816119910 + tan 120579119909

1003816100381610038161003816= 04119886 (sin 120579)

1003816100381610038161003816119910 minus tan 120579119909

1003816100381610038161003816= 04119886 (sin 120579)

(8)

And we change the elongation angle 120579 and get the bandstructure for any angle 120579

4 Band Structures

First we plot the band structure for the 2D PhC waveguidecomposed of square lattice of GaAs rhombic cross-sectionelements with side 119889 = 326 nm and refractive index 119899

119886= 335

in air background with a row of line defects for both TE- andTM-polarized modes The results are shown in Figure 3 forthe elongation angle 120579 = 1205874 rad The filled areas in Figure 3are the continuum of states of the perfectly periodic 2D PhCwhich the 2D PhC waveguide is made from All radiations

with frequencies which hit these areas (with red color) will beable to propagate inside the PhC surrounding the waveguidecore But the radiations with frequencies which lie in the PBG(with white color) do not leak into the surrounding periodicmedia so that radiations are guided through the waveguidecore and are called defect modes [1ndash4]

In order to study how band structure is affected byelongating of elements we change the angle 120579 and plot theband structure for a few important angles of elongationsFigure 4 shows the band structures for the elongation angles120579 (1205876amp1205873 rad) for both TE- and TM-polarized modesFromnumerical results in Figures 3 and 4 it is evident that byincreasing the elongation angle magnitude of defect modeswill be constant but the PBG width increases Although forthe case TE there is no defectmode the structure can be usedas optical polarizer waveguide (OPW) which has TM defectmode and does not have TE defect mode So the structuretransmits one state of polarization and blocks TE defectmode[7ndash13] Calculationswere performed for two important anglesof elongation and all our computational results for any angleconfirm these results


0 05 1 15 2 25 3 350

01

02

03

04

05

times106

Freq

uenc

y120596a2120587c


(a)

0 05 1 15 2 25 3 350

01

02

03

04

05

Freq

uenc

y120596a2120587c

times106Propagation constant 120573 (mminus1)

(b)

0 05 1 15 2 25 3 350

01

02

03

04

05

times106

Freq

uenc

y120596a2120587c


(c)

0 05 1 15 2 25 3 350

01

02

03

04

05Fr

eque

ncy120596a2120587c


(d)

Figure 4 Band structure for 120579 = 1205876 rad in (a) TE mode and (b) TMmode and for 120579 = 1205873 rad in (c) TE mode and (d) TMmode

5 Conclusion

Using PWE method we have studied band structure for2D PhC waveguide with dielectric rhombic cross-sectionelements in air background Less works were devoted tostudy of PhC with rhombic cross-section elements So weconsidered variations of the elements elongation for theused structure Numerical results show that by increasing inthe elongation of elements magnitude of the defect modesremains constant but the size of PBG increases Also theused 2DPhCwaveguide blocksTEdefectmode and transmitsTM modes So this kind of structure can be used as opticalpolarizer waveguide

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work has been financially supported by Payame NoorUniversity (PNU) I R of Iran

References

[1] M Skorobogatiy and J Yang Fundamentals of Photonic CrystalGuiding Cambridge University Press 2009

[2] K Sakoda Optical Properties of Photonic Crystals SpringerBerlin Germany 2001

[3] J D Joannopoulos S G Johnson J N Winn and R DMeade Photonic Crystals Molding the Flow of Light PrincetonUniversity Press 2008

[4] B E A Saleh andMC Teich Fundamentals of PhotonicWiley-Interscience New York NY USA 2007

[5] Y Kalra and R K Sinha ldquoPhotonic band gap engineering in 2Dphotonic crystalsrdquo PramanamdashJournal of Physics vol 67 no 6pp 1155ndash1164 2006


[6] S Robinson andRNakkeeran ldquoPCRRbased band pass filter forC and L+U bands of ITU-T G6942 CWDM systemsrdquo Opticaland Photonic Journal vol 1 no 3 pp 142ndash149 2011

[7] R Stopper H J W M Hoekstra R M De Ridder E VanGroesen and F P H Van Beckum ldquoNumerical studies of 2Dphotonic crystals waveguides coupling between waveguidesand filtersrdquo Optical and Quantum Electronics vol 32 no 6 pp947ndash961 2000

[8] A V Dyogtyev I A Sukhoivanov and R M De La RueldquoPhotonic band-gap maps for different two dimensionallyperiodic photonic crystal structuresrdquo Journal of Applied Physicsvol 107 no 1 Article ID 013108 7 pages 2010

[9] R K Sinha and Y Kalra ldquoDesign of optical waveguide polarizerusing photonic band gaprdquo Optics Express vol 14 no 22 pp10790ndash10794 2006

[10] I Guryev I A Sukhoivanov S Alejandro-Izquierdo et alldquoAnalysis of integrated optics elements based on photoniccrystalsrdquo Revista Mexicana de Fisica vol 52 no 5 pp 453ndash4582006

[11] T Liu A R Zakharian M Fallahi J V Moloney and MMansuripur ldquoDesign of a compact photonic-crystal-basedpolarizing beam splitterrdquo IEEEPhotonics Technology Letters vol17 no 7 pp 1435ndash1437 2005

[12] M Bayindir E Cubukcu I Bulu T Tut E Ozbay andC M Soukoulis ldquoPhotonic band gaps defect characteristicsand waveguiding in two-dimensional disordered dielectric andmetallic photonic crystalsrdquo Physical Review B vol 64 no 19Article ID 195113 7 pages 2001

[13] M J A DeDood E Snoeks AMoroz and A Polman ldquoDesignand optimization of 2D photonic crystal waveguides based onsiliconrdquo Optical and Quantum Electronics vol 34 no 1ndash3 pp145ndash159 2002

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Navigation and Observation



DistributedSensor Networks



a

y

z x

(a) (b)

y

x

120579

(c)

Figure 1 (a) 2D PhC waveguide (b) the unit cell of the 2D PhC waveguide and (c) the element of the unit cell


= (0 0119867119911119896119910

(119909 119910))

119867119911119896119910

( 119903) = sum

119866

int

120587

119886

minus

120587

119886

119889119896119909119867119911( + ) exp (119894 ( + ) sdot 119903)

(1)


= (0 0 119864119911119896119910

(119909 119910))

119864119911119896119910

( 119903) = sum

119866

int

120587119886

minus120587119886

119889119896119909119864119911( + ) exp (119894 ( + ) sdot 119903)

(2)

where and are magnetic field vector electric fieldvector 2D reciprocal lattice vector and plane wave vectorrespectively The sum and integral are taken over the firstBrillouin zone of the 2D PhC waveguide used [1 5]

SolvingMaxwellrsquos equations in CGS unit for themagneticand electric fields leads to the following vector wave equa-tions

1205962

= nabla times (

1

120576 ( 119903)

nabla times ) (3)

1205962

=

1

120576 ( 119903)

nabla times (nabla times ) (4)

where 120576( 119903) is the dielectric function of the unit cellSubstituting (1) in the vector wave (3) and (2) in the vector

wave (4) we get two eigenvalue problems for the square offrequency 120596 for each polarized mode


1205962

(119896119910)119867119911( + )

= minussum

119866

int

120587119886

minus120587119886

1198891198961015840

119909120581 ( + minus

1015840

minus 1015840

)

times [(1015840

+ 1015840

) sdot ( + minus 1015840

minus 1015840

)]119867119911(1015840

+ 1015840

)

(5)


1205962

(119896119910) 119864119911( + )

= minussum

119866

int

120587119886

minus120587119886

1198891198961015840

119909120581 ( + minus

1015840

minus 1015840

)

times

10038161003816100381610038161003816(1015840

+ 1015840

)

10038161003816100381610038161003816

2

119864119911(1015840

+ 1015840

)

(6)

That 120581( 119896+) is the Fourier expansion of the inverse dielectric

function of 2D PhC waveguide that is written as

120581 ( + ) =

1

Sunit cellint

unit cell119889 119903

1

120576 ( 119903)

exp (minus119894 ( + ) sdot 119903)

(7)

That integral is taken over the unit cell in Figure 1(b) Fora given value of a Bloch wave number 119896

119910as propagation

constant 120573 (5) and (6) constitute two eigenvalue problemswith respect to the square of frequency 120596(119896

119910) Finally using

a trapezoidal approximation of the 1D integral 1198961015840119909and the

numerical solutions for (5) and (6) we get the band structureof the structure used [1] The computation method used forimplementation of PWE method for 2D PhC waveguide issimilar to the one which is used for the computation ofthe band structure of strictly periodic PhC There is someessential difference in the structure parameters definitions[1 2] First in 2D PhC waveguide the unit cell consists ofseveral PhC elements rather than one The defect of periodicstructure is also introduced to form the waveguide core Alsoin case of 2D PhC band structure computation we set the 119896-path to pass through all high symmetry points of the Brillouinzone However as we have considered in this section com-putation of the 2D PhC waveguide band structure requirestransversal wave vector consideration only The longitudinalcomponent stays in this case for the propagation constant andthe propagation constant is limited by the boundaries of theBrillouin zone One more difference from strictly periodicPhC is the definition of the reciprocal lattice vectors set [1ndash3]

3 Elongation of the Rhombuses

According to Figure 2 we can change the elongation angle 120579

that it makes with 119909 axis for transformation of rhombuses


y

x

120579


0 05 1 15 2 25 3 350

01

02

03

04

05

times106

Freq

uenc

y120596a2120587c


(a)

0 05 1 15 2 25 3 350

01

02

03

04

05

times106

Freq

uenc

y120596a2120587c


(b)



1003816100381610038161003816119910 + tan 120579119909

1003816100381610038161003816= 04119886 (sin 120579)

1003816100381610038161003816119910 minus tan 120579119909

1003816100381610038161003816= 04119886 (sin 120579)

(8)


4 Band Structures


119886= 335





0 05 1 15 2 25 3 350

01

02

03

04

05

times106

Freq

uenc

y120596a2120587c


(a)

0 05 1 15 2 25 3 350

01

02

03

04

05

Freq

uenc

y120596a2120587c


(b)

0 05 1 15 2 25 3 350

01

02

03

04

05

times106

Freq

uenc

y120596a2120587c


(c)

0 05 1 15 2 25 3 350

01

02

03

04

05Fr

eque

ncy120596a2120587c


(d)


5 Conclusion




Acknowledgment


References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










y

x

120579


0 05 1 15 2 25 3 350

01

02

03

04

05

times106

Freq

uenc

y120596a2120587c


(a)

0 05 1 15 2 25 3 350

01

02

03

04

05

times106

Freq

uenc

y120596a2120587c


(b)



1003816100381610038161003816119910 + tan 120579119909

1003816100381610038161003816= 04119886 (sin 120579)

1003816100381610038161003816119910 minus tan 120579119909

1003816100381610038161003816= 04119886 (sin 120579)

(8)


4 Band Structures


119886= 335





0 05 1 15 2 25 3 350

01

02

03

04

05

times106

Freq

uenc

y120596a2120587c


(a)

0 05 1 15 2 25 3 350

01

02

03

04

05

Freq

uenc

y120596a2120587c


(b)

0 05 1 15 2 25 3 350

01

02

03

04

05

times106

Freq

uenc

y120596a2120587c


(c)

0 05 1 15 2 25 3 350

01

02

03

04

05Fr

eque

ncy120596a2120587c


(d)


5 Conclusion




Acknowledgment


References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 05 1 15 2 25 3 350

01

02

03

04

05

times106

Freq

uenc

y120596a2120587c


(a)

0 05 1 15 2 25 3 350

01

02

03

04

05

Freq

uenc

y120596a2120587c


(b)

0 05 1 15 2 25 3 350

01

02

03

04

05

times106

Freq

uenc

y120596a2120587c


(c)

0 05 1 15 2 25 3 350

01

02

03

04

05Fr

eque

ncy120596a2120587c


(d)


5 Conclusion




Acknowledgment


References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Band Structure Engineering in 2D Photonic ...downloads.hindawi.com/journals/aot/2014/780142.pdf · Research Article Band Structure Engineering in 2D Photonic Crystal