2005 Two-dimensional Kerr-nonlinear Photonic Crystals

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Band structure calculation in two-dimensionalKerr-nonlinear photonic crystals

I.S. Maksymov, L.F. Marsal *, M.A. Ustyantsev, J. Pallare `sDepartament d Enginyeria Electro ` nica, Ele ctrica i Automa` tica, Universitat Rovira i Virgili, Campus Sescelades,

Avda. Pa sos Catalans 26, 43007 Tarragona, Spain

Received 17 September 2004; received in revised form 26 November 2004; accepted 9 December 2004

Abstract

Using the nite-difference time-domain method, based on the numerical simulation of oscillating dipole radiation,we analyze band structures in two-dimensional Kerr-nonlinear photonic crystals. This method is more thorough at cal-culating band structures in two-dimensional Kerr-nonlinear photonic crystals than approaches proposed earlier. Wend that the band structures calculated for both TE and TM polarizations are dynamically red-shifted with regardto the linear case. For a positive Kerr coefficient, this red-shift increases as the intensity of the oscillating dipole

increases. 2004 Elsevier B.V. All rights reserved.

PACS: 42.65.Pc; 42.65; 42.70.QsKeywords: Two-dimensional nonlinear photonic crystal; Finite-difference time-domain method; Oscillating dipole; Band structure;Kerr coefficient; Dynamic red-shift

1. Introduction

Nonlinear photonic crystals [1] have been of par-

ticular interest to researchers because of such prom-ising applications in photonic devices as opticallimiting [2], short pulse compressors [3], nonlinear

optical diodes [4] and all-optical switching [57].The behaviour of these devices depends heavily onwhether the working frequency is within the band

gap of the constituent photonic crystal or whetherit is tuned to the band edge region. It is well-knownthat the position of the band gap depends on theintensity of the input signal. Consequently, in orderto design such devices, it is essential to know theband structure of the constituent nonlinearphotonic crystals based, e.g., on Kerr-nonlinearmaterials. Since nonlinear photonic crystals have

0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.optcom.2004.12.022

* Corresponding author. Tel.: +34 977 559 625; fax: +34 977559 605.

E-mail addresses: [email protected], [email protected](L.F. Marsal).

Optics Communications 248 (2005) 469477

www.elsevier.com/locate/optcom
mailto:[email protected]:[email protected]


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a large refractive index contrast and the elec-tromagnetic elds inside them obey the Maxwellequations and additionally increase this contrast,

it is more difficult to calculate their band structures.Therefore, special numerical approaches should beused.

A considerable amount of literature, startingwith Tran [8], deals with the calculation of bandstructures in photonic crystals, which consist of Kerr-nonlinear materials (see e.g. [811]). Recently,we presented a novel approach for analyzing bandstructures in one-dimensional Kerr-nonlinear pho-tonic crystals. In this approach, we combined thenite-difference time-domain (FDTD) method,based on the numerical simulation of oscillatingdipole radiation [12], with a Kerr-nonlinear model[13].

In this paper, we extend our approach and ana-lyze band structures in two-dimensional nonlinearphotonic crystals (2-D NLPC) consisting of asquare lattice of circular air rods collocated in aKerr-nonlinear material. We calculate them forboth TM and TE polarizations and, in order todemonstrate the validity of the approach, comparethem with those presented in [8]. We estimate thefeasible intensities of the electric and magnetic di-

poles that are needed to induce the nonlinearity.Finally, we study the convergence of the FDTDmethod used to calculate the band structures.

2. Theory

For a description of the FDTD method, westart with Maxwell s equations (Gaussian units)

r ~ E ~r ; t 1c

o

o t ~ H ~r ; t 4p ~ P M ~r ; t

; 1

r ~ H ~r ; t 1c

o

o t ~ D~r ; t 4p ~ P E ~r ; t ; 2

where ~ E ~r ; t and ~ H ~r ; t are the electric and mag-netic elds; ~ D~r ; t is the electric displacement;~ P E ~r ; t and ~ P M ~r ; t are the polarization elds of the electric and magnetic dipoles; c is the speedof light in vacuum and e~r is the position-depen-dent dielectric constant. In order to solve Eqs.(1) and (2) , we need a so-called constitutive equa-

tion that relates ~ D~r ; t to ~ E ~r ; t . For the Kerr-nonlinear medium, the dielectric constant dependson the electric eld ~ E ~r ; t and the Kerr coefficient

v(3)

[13]e~r e v3 ~ E ~r ; t

2; 3

where e is an intensity independent dielectric con-stant. The constitutive equation for the Kerr-nonlinear medium can be expressed as

~ D~r ; t e v3 ~ E ~r ; t 2

~ E ~r ; t : 4The polarization elds of the electric and mag-

netic dipoles can be expressed in the explicit form

as~ P E ~r ; t ~el d~r ~r 0e ix t ; 5

~ P M ~r ; t ~hl d~r ~r 0e ix t ; 6

where ~el and ~hl are the amplitudes of the electricand magnetic dipoles, ~r 0 indicates the position of the dipoles within the photonic crystal and x isthe angular frequency of the oscillation; i refersto the imaginary unit and d~r ~r 0 denotes theDirac delta function.

The electromagnetic energy density W emittedper unit time by the oscillating dipole at ~r 0 canbe calculated by using the following expression[14]

W 18p

~ E ~r ~ D~r ~ H ~r 2

h i: 7In accordance with the Bloch theorem [12], the

periodic boundary condition (BC) can be writtenas

~U~r ~a ; t ~U~r ; t ei~k ~a ; 8

where ~U is any eld component

~ E or

~ H , ~a is theperiod and ~k is the wave vector in the rst Brillouin

zone (BZ).In the case of the TE-polarization, for which the

magnetic eld is parallel to the z axis, Eqs. (1) and(2) in two dimensions reduce to

1c

o

o t H z 4p hl z d x x0d y y 0e

ix t

o E y o x

o E xo y

; 9

470 I.S. Maksymov et al. / Optics Communications 248 (2005) 469477


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1c

o D xo t

o H z o y

; 10

1c

o D y

o t o H

z o x : 11

From Eq. (4), we obtain the electric eld compo-nents E x and E y

E x D x

e v3j E xj2 ; 12

E y D y

e v3j E y j2 : 13

In the case of the TM-polarization, for which theelectric eld is parallel to the z axis, Eqs. (1) and(2) in two dimensions reduce to

1c

o

o t D z 4p el z d x x0d y y 0e

ix t

o H y o x

o H xo y

; 14

1c

o H y o t

o D z o x

; 15

1c

o H xo t

o D z o y

: 16

From Eq. (4), we obtain the electric eld compo-nent E z

E z D z

e v3j E z j2 ; 17

In the denominator of Eqs. (12), (13) and (17) ,the values of the electric eld component are as-sumed to be known from the previous time step[13].

3. Numerical method

In order to numerically solve Eqs. (9)(17) , weuse the nite-difference scheme proposed by K.Yee [15], which is the standard for solving Max-well s equations. The nite-difference expressionsfor Eqs. (9)(13) are

H p 12

z l 12

; m 12 H p 12 z l 12 ; m 12

cD t E p y l 1; m 12

E p y l; m 12

D x" E p x l 12 ;m 1 E p x l

12 ;m D y

4p ix hl z D t

D xD y dll 0 dmm0 exp ix p D t ; 18

D p 1 x l 12

; m D p x l 12 ; m

cD t D y

H p 12

z l 12

; m 12 H p 12 z l 12 ; m 12 ;

19

D p 1 y l; m 12 D p y l; m 12

cD t D x

H p 12

z l 12

; m 12 H p 12 z l 12 ; m 12 ;

20

E p 1 x l 12

; m D p 1 x l

12 ;m e v 3 E p x l 12 ;m

2 ; 21

E p 1 y l; m 12 D

p 1 y l; m 12 e v3 E p y l; m 12 2 ; 22

The nite-difference expressions for Eqs. (14)(17)are D p 1 z l ;m

D p z l ;m cD t H

p 12 y l 12 ;m H

p 12 y l 12 ;m D x"

H p 12 x l; m 12

H

p 12 x l;m 12

D y # 4p ix el z D t D xD y

dll 0 dmm0 exp ix p D t ; 23

H p 12

x l; m 12 H p 12 x l; m 12

cD t D y

E p z l ; m 1 E p z l ; m ;24

I.S. Maksymov et al. / Optics Communications 248 (2005) 469477 471


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H p 12

y l 12

; m H p 12 y l 12 ; m

cD t D x E

p

z l 1; m E p

z l ; m ;25 E p 1 z l ; m

D p 1 z l ; me v3 E p z l ; m

2 : 26

Here index p refers to a grid point in time and indi-ces l and m denote x and y, respectively. Theexpressions dll 0 and dmm0 denote the position of the dipole on the spatial grid. The spatial stepsD x and D y are calculated as 2 p a /nx and 2 p a /ny,

where nx and ny are the number of subcells inthe unit cell in the x- and y-directions, respectively.The temporal step D t is calculated as 2 p /(x Nt ),where Nt is the total number of temporal steps.The Courant stability condition [15] is satised

with Nt P ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinx2 ny 2q =x a. All the compo-nents of the electromagnetic eld are set to be zerofor all points of the calculation space before thecalculation process.

The energy density is now calculated from theelds known from Eqs. (18)(26) . The following -

nite-difference expressions are obtained from Eq.(7) by spatial discretization. We have

W 18p X

nx;ny

l ;m0 E x l

12

; m D x l 12 ; m E y l; m

12 D y l; m 12

H 2 z l 12

; m 12 27

for TE polarization and

W 18p X

nx;ny

l ;m0 E z l ; m D z l ; m H 2 x l; m

12

H 2 y l 12

; m 28for TM polarization, respectively. Fig. 1 shows theow chart of the calculation process for a compo-nent of the wave vector ~k (so-called k -point). Inthis gure, x denotes the angular frequency, whichvaries between x 1 and x 2 with the step Dx ; i c de-

notes the number of the current oscillating cycle,which varies between zero and the total numberof oscillating cycles N c. The grey blocks corre-

spond to the ow chart of the sequence of opera-tions within the main body of the FDTDalgorithm for the TE polarization. If the polariza-tion were TM, the eld components D x , E x , D y, E yand H z would be substituted for H x , H y, Dz andE z, correspondingly.

4. Results and discussion

In this section, we consider the two-dimensionalnonlinear photonic crystal (2-D NLPC) whoseunit cell is shown schematically in Fig. 2 . In orderto demonstrate the validity of the numerical method,we consider a square lattice of air holes with thesame parameters that were used in [8]. Theseparameters are: a = 1, r = 0.5 a , ea = 1, eb = 5.0,

Fig. 1. Flow chart of the calculation process.



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v3a 0; v3b 0:005 ;

x 1a2p c 0:01;

x 2a2p c 1:0 and

Nt = 600060 (depending on the frequency). Theangular frequencies 0 6 x < x 1 are not calculated

because of an enormous increase in Nt [13]. Forthe two-dimensional linear photonic crystal (2-DLPC) the parameters are the same, but both Kerrcoefficients are zero. In the case of the TE polari-zation, the magnetic dipole is used for excitationwhile the electric one is used in the case of theTM polarization. For all calculations, 50 oscillat-ing cycles are used. The unit cell is divided into50 50 subcells.

4.1. Two-dimensional linear photonic crystal

In this section, we present the band structures(TM and TE polarizations) in the 2-D LPC calcu-lated with our FDTD method and compare themwith etalons. The etalons we used were the bandstructures calculated in the same 2-D LPC forthe TM and TE polarizations but with the planewave expansion method [12]. In Fig. 3 (a) and(b), the band structures calculated with our methodare represented by open circles and the etalonband structures are represented by solid lines. Ascan be seen, our FDTD method provides the accu-

rate result for both TE and TM polarizations.

4.2. Two-dimensional nonlinear photonic crystal

Fig. 4 (a) and (b) show the band structures in

the 2-D LPC (solid line) calculated with theplane wave method and in the 2-D NLPC (opencircle) calculated with our FDTD method forTM (a) and TE (b) polarizations. The solid cir-cles in Fig. 4 (a) show the result for the TMpolarization borrowed from [8] (only the rstfour bands were calculated there and the TEpolarization was not considered). We can seethat our FDTD method provides the same resultas in [8]. Furthermore, we can see the third (anti-symmetric) band in the CX and CM direction. In[8] Tran did not calculate it and, therefore, didnot show it. The reason why he did not can befound, e.g., in [16]. If the initial eld is incidentalong the high-symmetry axes of the rst BZ(e.g., CX -direction), it cannot excite the modeswith odd parity (antisymmetric modes) with re-spect to the axes. This is because the initial eldis even (symmetric) with respect to the axes. Ingeneral, if we put the oscillating dipole at a pointthat does not coincide with the high-symmetryaxes of the rst BZ, we do not encounter anyproblem with the excitation of all modes. So

we conclude that our numerical approach ismore thorough.

The amplitudes of the electric and magneticdipole used to calculate the band structuresfor the TM and TE polarizations were el z 1500 arb : units and hl z 225 arb : units, respec-tively. We selected them so that we could obtainthe same red-shift of the band structure as Tran[8] did. This can be seen from Fig. 4 (a) and (b),where the bands of the band structures are red-shifted with regard to the linear case. The forbid-

den band gap that takes place in the bandstructure for TM polarization is also red-shifteddue to the shift of the rst and the second bandswhich make it up. The band structure for the TEpolarization has no forbidden band gap for therange of frequencies considered. The shifting canbe qualied by the following argument. In the fre-quency domain, the band structure is determinedby the difference between the dielectric constantsof the materials which make up the photonic crys-tal. From Eq. (3) we can express this difference as

Fig. 2. Schematic representation of the unit cell of the 2-DNLPC. The oscillating dipole is situated at ~r 0 x0; y 0.



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D e eb v3 ~ E ~r ; t 2

h iea : 29Theoretically, the value of De increases as the

intensity increases if v(3) > 0 and decreases if v(3) < 0. As the oscillating dipole excites the struc-ture, the value of De changes and the bands of theband structure dynamically shift. The results show

that for the positive Kerr coefficient the value of D eincreases and the bands dynamically red-shift.This red-shift increases as the intensity of the oscil-lating dipole increases. This process is the basis forintensity-driven optical limiting and it is extremelyimportant that it be understood if all-opticalswitching devices are to be modelled.

Fig. 3. The comparison between the band structures for the TM (a) and TE (b) polarizations in the 2-D LPC calculated with ourFDTD method (open circles) and the plane wave expansion method (solid line).



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4.3. Field intensity estimation, unit cell discretization and convergence

In all our calculations, we give both the mag-netic and the electric elds in arbitrary units

(arb. units). To estimate the feasible intensitiesneeded to induce the nonlinearity which shiftsthe bands, we draw a parallel between the arb.units and the Gaussian units. We obtain that after50 oscillating cycles the magnetic dipole with an

Fig. 4. The band structures in the 2-D NLPC and 2-D LPC calculated with our FDTD method (open circles) and the plane waveexpansion method (solid line) for the TM (a) and TE (b) polarizations. The solid circles correspond to the band structure for the TMpolarization borrowed from [8]. The amplitudes of the electric and magnetic dipoles were el z 1500 and hl z 225 arb : units,respectively.



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amplitude hl z 1 arb : unit corresponds to anintensity of about (1 2) 108 erg/s/cm 2 [13]. Afterthe same number of oscillating cycles, the electric

dipole with an amplitude el z 1 arb : unit corre-sponds to an intensity of about (0.15 0.3) 108 erg/s/cm 2. Conversion into SI units resultsin intensities of about 0.010.02 and 0.0015 0.003 kW/cm 2, respectively.

To our knowledge, the physical interpretationof such a difference in amplitudes of the electricand magnetic dipoles has not yet been claried.It could be explained as follows: in both linearand nonlinear photonic crystals, the Purcell effect[17,18] has to be taken into account. It has beenshown that for the linear photonic crystals andangular frequencies x a2p c 6 1:0, electric and magneticdipoles behave like in a linear homogeneous med-ium [19,20]. In our explanation, the behaviour of the dipoles in the nonlinear case is assumed to besimilar to that in the linear one, i.e., in the nonlin-ear photonic crystal the dipoles behave like in anonlinear homogeneous medium.

In the homogeneous Kerr-nonlinear medium,there is a growth of the transverse eld compo-

nents at the cost of the longitudinal one. Let usanalyse the Maxwell equation sets for the TEand TM polarizations. In the TE case, we focus

our attention on the transverse electric eld com-ponents because they play an important role ininducing the nonlinearity (Eqs. (12) and (13) ).The amplitudes of these eld components increaseat the cost of the magnetic dipole radiation. As aresult of this process, the induced nonlinearity in-creases as the amplitudes of the electric eld com-ponents increases. In the TM case, however, thetransverse electric eld is absent by denition.Consequently, the energy transfer mechanismfrom the electric dipole to the transverse magneticeld components does not facilitate the inductionof the nonlinearity and therefore, the larger ampli-tudes of the electric dipole should be applied to in-duce the same nonlinearity as with the smalleramplitudes of the magnetic dipole.

Fig. 5 shows the convergence of the FDTDmethod we used to calculate the band structures.The abscissa represents the number of mesh pointsN in the unit cell. The ordinate corresponds to theangular frequency. The curves marked by , . , }

Fig. 5. The convergence behavior of our FDTD method (TM polarization). The parameters are the same as in Fig. 4 (a). The curvesmarked by , . , } , and h correspond to the rst, second, third and fourth bands of the band structure for TM polarization ( CX direction).



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and h correspond to the rst, second, third andfourth bands of the band structure for TM polar-ization ( CX direction). The values of the angular

frequency are taken for the X point of the rstBZ. The parameters are the same as in Fig. 4 (a).As can be seen, the method is convergent for

N P 40. Any decrease in this number will giveinaccurate results. In other directions of the BZ,the same behaviour of the bands was observedfor both TM and TE polarizations. To play safe,we use 50 mesh points.

5. Conclusions

Using the nite-difference time-domain method,based on the numerical simulation of oscillatingdipole radiation, we analyzed band structures intwo-dimensional Kerr-nonlinear photonic crystals.We considered a square lattice of circular air rods.The calculations revealed that a photonic bandgap exists for the TM polarization. We found thatthe band structures are dynamically red-shiftedwith regard to the linear case. The method we usedwas more thorough at calculating band structuresin two-dimensional Kerr-nonlinear photonic crys-

tals than approaches proposed earlier. It is usefulfor understanding such phenomena as intensity-driven optical limiting and all-optical switchingwith Kerr-nonlinear photonic crystals.

Acknowledgments

This work has been supported by the SpanishCommission of Science and Technology (Ci-CYT), Project No. TIC2002-04184-C02. M.A.U.

acknowledges the scholarship from Generalitatde Catalunya.

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2005 Two-dimensional Kerr-nonlinear Photonic Crystals