One-dimensional Kerr-nonlinear Photonic Crystals

8/9/2019 One-dimensional Kerr-nonlinear Photonic Crystals

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applications in integrated optical devices. The 1-D

NLPC, which consists of Kerr-nonlinear materials

(e.g., semiconductors, glasses and polymers) [3],

has recently been applied to such nonlinear opticaldevices as low-threshold optical limiting[4], short

pulse compressors[5], nonlinear optical diodes[6]

and all-optical switching [7], etc. In these devices,

the dielectric constant is changed by a high-inten-

sity incident beam to dynamically control the

transmission of light. Under this condition, the op-

tical limiting is achieved by modifying the incident

intensity while the all-optical switching requires an

additional strong pump beam to control the

switching of a weak probe signal tuned to the band

edge region. The other nonlinear devices men-

tioned are based on related nonlinear phenomena

[3].

The dynamic control mechanism has previously

been analyzed by Scalora et al. [8], Tran [9] and

Scholz et al. [10]. They used nonlinear models

and calculated energy transmission spectra. The

complementary information about spectral prop-

erties can be obtained by analyzing the band struc-

ture of the 1-D NLPC. For example, Huttunen

and Torma[11]have reported a novel Fourier nu-

merical method which was applied to analyze band

structures in the 1-D NLPC. However, the finite-difference time-domain (FDTD) method can also

be applied for the same purpose. In the last de-

cades, much attention has been paid to model non-

linear media with this method and therefore its

adaptation would result in the powerful tool for

analyzing band structures in the 1-D NLPC. There

is a number of works dedicated to model nonlinear

media with the FDTD method. Merewether and

Radasky [12] were the first who used it in this

way. Various techniques to model Kerr-nonlinear

media and dispersive nonlinear effects were pre-sented by Zheng and Chen [13], Goorjian et al.

[14]and Joseph et al. [15]. With these techniques,

one can investigate the propagation and scattering

of femtosecond electromagnetic solitons. Alterna-

tive approaches such as the combined Maxwell

Bloch FDTD scheme and the Z-transform-based

FDTD scheme were later proposed by Forysiak

et al. [16,17], Ziolkowsky et al. [18] and Sullivan

[19,20]to model more complicated dispersive and

nonlinear materials. Particularly, the dynamic

nonlinear optical skin effect and self-induced trans-

parency effects were studied. Tran[21]was the first

who calculated band structures in Kerr-nonlinear

photonic crystals. The notable feature of his ap-proach is the calculation of the electric field from

a cubic equation derived from the constitutive

equation for a Kerr-nonlinear medium. We have

recently applied the Trans approach to calculate

band structures in Kerr-nonlinear photonic crystal

slabs[22].

In this paper, we present a new approach for

analyzing band structures in 1-D NLPC. It com-

bines the FDTD method, based on the numerical

simulation of oscillating dipole radiation, with

the Kerr-nonlinear model. Such FDTD method

was first proposed by Sakoda[23]and successfully

used to calculate band structures in metallic sys-

tems. With this method, the photonic crystals are

excited by an embedded oscillating dipole. The

Kerr-nonlinear model used is borrowed from

[21,24,25].

The paper is organized as follows. First, we dis-

cuss the theoretical considerations of the 1-D

NLPC. In the second section, we present the

FDTD method and the Kerr-nonlinear model,

which are used to analyze band structures. The

next section discusses the results. In it, the energyspectrum plots calculated for the 1-D LPC and

1-D NLPC are identified and the band structures

in the 1-D LPC calculated with the FDTD and

the plane wave expansion methods are compared.

We then go on to discuss the band structures in the

1-D NLPC for different amplitudes of the oscillat-

ing dipole. We demonstrate the validity of our ap-

proach by comparing the results with analogical

results. Finally, we discuss the physical origins of

the red-shift of the band structures.

2. Theory

Here, we consider the FDTD method based on

the numerical simulation of oscillating dipole radi-

ation and the Kerr-nonlinear model. The oscillat-

ing dipole is embedded in the 1-D NLPC, whose

general geometry is illustrated inFig. 1. It consists

of alternating layers of materials with high (e1) and

214 I.S. Maksymov et al. / Optics Communications 239 (2004) 213222


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low (e2) dielectric constants. The optical thickness

of the layers with a high dielectric constant isband

the optical thickness of the layers with a low di-

electric constant is a b, where a is the period.The layers with a high dielectric constant are

doped with a Kerr-nonlinear material and are also

characterized by the Kerr coefficient v(3).

It is apparent that the 1-D NLPC is invariant

when we change y toye; v3 x; y e; v3 x;y : 1

Therefore only the region where 06y6C is con-

sidered for calculation and in the y-direction the

1-D NLPC is truncated by the symmetric bound-

ary condition (BC). The periodic BC truncates

the structure in the x-direction. In the z-direction,

the structure is assumed to be infinite. The oscillat-

ing dipole is situated at ~r0 x0;y0. The parame-tersnxandnydenote the number of subcells, which

divide the unit cell (the period a). The parameters

Dx and Dy characterize the dimensions of the sin-gle subcell. All these parameters will be defined in

the following section.

For a description of the FDTD method, we

start with Maxwells equations (Gaussian units)

r ~E~r; t 1

c

o

ot~H~r; t 4p~PM~r; t

; 2

r ~H~r; t 1

c

o

ot~D~r; t 4p~PE~r; t

; 3

where~E~r; t and ~H~r; t are the electric and mag-netic fields; ~D~r; t is the electric displacement;~PE~r; t and ~PM~r; t are the polarization fields 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.

(2) and (3), we need so-called constitutive equation

that relates ~D~r; t to ~E~r; t. For the linearmedium, it is simply

~D~r; t e~r ~E~r; t : 4

In the nonlinear case, we will have a different con-

stitutive equation. For the Kerr-nonlinear medium

(the layers with a high dielectric constant), the di-

electric constant depends on the electric field ~E~r; tand the Kerr coefficient v(3) [21,24,25]

e~r e1 v3 ~E~r; t 2: 5

Now Eq.(5)can be substituted into Eq.(4), which

is resulted in the constitutive equation for the

Kerr-nonlinear medium

~D~r; t e1 v3 ~E~r; t 2 ~E~r; t : 6

The polarization fields of the electric and mag-netic dipoles can be expressed in the explicit form

as:

~PE~r; t ~eld~r~r0 eixt; 7

~PE~r; t ~hld~r~r0 eixt; 8

where ~el and~hl are the amplitudes of the electricand magnetic dipoles, ~r0 indicates the position ofthe dipoles within the photonic crystal and x is

the angular frequency of the oscillation; i refers

to the imaginary unit andd~r~r0 denotes the Di-rac delta function.

The electromagnetic energy density W emitted

per unit time by the oscillating dipole at ~r0 canbe calculated by using the following expression

[26]:

W 1

8p~E~r ~D~r ~H~r

2h i: 9In accordance with the Bloch theorem[23], the pe-

riodic BC can be written as

Fig. 1. Schematic representation of the geometry of the 1-D

NLPC. The oscillating dipole is situated at ~r0 x0;y0.

I.S. Maksymov et al. / Optics Communications 239 (2004) 213222 215


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~U~r~a; t ~U~r; t ei~k~a; 10

where~U is any field component (~Eor ~H),~a is theperiod and~k is the wave vector in the first Brill-ouin zone. The symmetric BC can be written as

[27]

~U x; 0 ~U x;Dy ; 11

~U x;C ~U x;C Dy : 12

Restricting ourselves to the H-polarization, Eqs.

(2) and (3) in two dimensions reduce to

1

c

o

ot Hz 4phlzd x x0 d yy0 e

ixt

oEy

ox

oEx

oy ; 13

1

c

o

ot Dx 4pelxd x x0 d yy0 e

ixt

oHz

oy ;

14

1

c

o

ot Dy 4pelyd x x0 d yy0 e

ixth i

oHz

ox :

15

From Eq.(6), we get wanting electric field compo-nents Ex and Ey

Ex Dx

e1 v3 Exj j2; 16

Ey Dy

e1 v3 Ey 2: 17

In the denominator of Eqs.(16) and (17)values of

the electric field components are assumed to be

known from the previous time step (see the follow-ing section).

In accordance with the original algorithm

[23], for H-polarization, for which the magnetic

field is parallel to z-axis, we use the magnetic di-

pole to excite the structure. For one-dimensional

photonic crystals in the in-plane case of the E-

polarization, for which the electric field is parallel

to the z-axis, the band structure is the same as in

the case of the H-polarization, but the electric

dipole will be used instead of the magnetic one.

3. Numerical method

In order to numerically solve Eqs. (13)(17),

one of the existing finite-difference schemes [28]can be used to approximate both derivatives in

space and time. We will use the scheme proposed

by K. Yee, which is the standard for solving Max-

wells equations. The finite-difference expressions

corresponding to Eqs. (13)(17)are

Hp12

z l 1

2;m

1

2

Hp12

z l 1

2;m

1

2

Dt

Epy l 1;m 12 Epy l;m 12 Dx

"Epx l

12;m 1

Epx l

12;m

Dy

#

4pixhlzDt

DxDy dll0dmm0 exp ixpDt ; 18

Dp1x l 1

2;m

Dpx l 1

2

;m DtDy

Hp12

z l 1

2

;m 1

2

Hp12

z l 1

2;m

1

2

4pixelxDt

DxDy dll0dmm0

exp ixpDt ; 19

Dp1y l;m 1

2

Dpy l;m 1

2

Dt

Dx Hp

12

z l 1

2;m

1

2

Hp12

z l 1

2

;m 1

2

4pixelyDt

DxDy dll0dmm0 exp ixpDt ; 20

Ep1x l 1

2;m

Dp1x l 12;m

e1 v3 E

px l

12;m

2; 21

Ep1y l;m 1

2

Dp1y l;m 12

e1 v3 E

py l;m

12

2; 22



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where indexp refers to a grid point in time and in-

dicesland m denotex and y, respectively. The ex-

pressions dll0 and dmm0 denote the position of the

dipole on the spatial grid. We normalize the angu-lar frequency and the wave vector as xa/2pc and

ka/2p, respectively. Because of this normalization,

the values ofDxand Dyare now calculated as 2pa/

nx and 2pa/ny, where nx and ny are the number of

the subcells in the unit cell in the x- and y-direc-

tions, respectively. If the symmetric boundary con-

dition is set in the y-direction, Dyis defined as pa/

ny. The temporal step Dtis calculated as 2p/(xNt),

where Nt is the total number of temporal steps.

The Courant stability condition [28] is satisfied

with NtPffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi

nx

2

2 ny

2q =x a. All compo-nents of the electromagnetic field are set to be zero

for all points of the calculation space before the

calculation process.

The energy density is now calculated from the

fields known from Eqs.(18)(22). The following fi-

nite-difference expression is obtained from Eq.(9)

by spatial discretization.

W 1

8p

Xnx;nyl;m0

Ex l 1

2;m

Dx l

1

2;m

Ey l;m

1

2

Dy l;m

1

2

H2z l 1

2;m

1

2

: 23

The flow chart representation of the calculation

process for a component of the wave vector ~k(so-called k-point) is shown inFig. 2. In this figure, x

denotes the angular frequency, which varies be-

tween x1 and x2 with the step Dx; ic denotes the

number of the current oscillating cycle, which var-

ies between zero and the total number of oscillating

cyclesNc. In the first step, we set up the k-point. Allfurther calculations are carried out only for this k-

point. In the second step, we set up the current an-

gular frequency x for which the radiated energy

will be calculated. In the third step, we calculate

Dtand zerorize the counter of the oscillating cycles

ic. The fourth step consists ofNc of dipoles oscillat-

ing cycles. In the fifth step, we calculate the radiated

energy for the current angular frequency x, which

was set up in the second step. We repeat the steps

IIIV for all angular frequencies. After this process

has finished for allk-points (it is enough to take in-

to account about 1015 k-points for a wave vector),

the accumulated electromagnetic energy density

distribution Wis a function of the angular frequency

for the specified wave vector. The band structure is

obtained from the resonance peaks of this electro-

magnetic energy density distribution.

In Fig. 3, we present the flow chart of the se-

quence of operations inside the main body of theFDTD algorithm. Here jdenotes the number of

the current temporal step and Nt is the total num-

Fig. 2. Flow chart of the calculation process.

Fig. 3. Flow chart of the FDTD calculation.



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ber of temporal steps. We only notice that despite

the fact that the magnetic dipole is used to excite

the structure, the electric field is got from the iter-

ation process and induces the nonlinearity.

4. Results and discussion

In this section, we consider the 1-D NLPC

shown schematically inFig. 1. The parameters that

we will use for all further calculations are as fol-

lows: a = 1, b =0.2a, e1=13.0, e2= 1, (x1a/

2pc)=0.01, (x2a/2pc)=1.0, (Dxa/2pc)=0.001 and

Nt= 6000 to 60 (depending on the frequency).

The behavior of the band structure at x=0 iswell-known. Therefore, the angular frequencies

06x


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depends on the number of oscillating cycles. For

example, a resonance peak is observed at xa/

2pc =0.189 for 10 cycles while it is at xa/

2pc =0.185 for 50 cycles. The last number of theoscillating cycles is suitable to accurately calculate

band structures.

Fig. 5shows the normalized accumulated elec-

tromagnetic energy density distribution calculated

for the 1-D LPC (a) and 1-D NLPC (b) as a func-

tion of the angular frequency for ka/2p=0.5. The

amplitudes of the magnetic dipoles used for the cal-

culation were hlz 1:0 and hlz 100 arb: units,respectively. There were 50 oscillating cycles. As

can be seen from Fig. 5(a), in the linear case the

electromagnetic energy density distribution is very

clean. InFig. 5(b), the electromagnetic energy den-

sity distribution is not clean because a noise is pro-

duced by the nonlinear calculation. Therefore, each

peak in the energy distribution cannot be taken as

an individual mode. Tran [21] reported the same

problem. He stated that in the nonlinear case the

energy distribution is very similar to that of the lin-

ear case, which indicates that the Kerr nonlinearity,

apart from shifting its angular frequency, does not

drastically affect the linear solution. Following this

statement, in Fig. 5(b) only the peaks indicated

by an arrow are meaningful for plotting bandstructures.

4.2. One-dimensional linear photonic crystal

In this section, we plot the band structure in the

1-D LPC calculated with the FDTD method and

compare it with an etalon. The etalon we used

was the band structure calculated for the same

1-D LPC but with the plane wave expansion meth-

od. InFig. 6, this band structure is represented by

open circles and the etalon band structure is repre-sented by a solid line. As can be seen, the FDTD

method provides the accurate result for the 1-D

LPC.

4.3. One-dimensional nonlinear photonic crystal

Fig. 7shows the bands of the band structures in

the 1-D LPC (dashed line) and 1-D NLPC (solid

line) calculated with the proposed approach. In

the nonlinear case, the results are presented for

the amplitudes of the magnetic dipole hlz 100and hlz 450 arb: units. The open circles corre-spond to the band structure calculated by Huttunen

and Torma [11], who presented band structures in

an analogical 1-D NLPC. Therefore, we use their

results to validate our approach. They calculated

the band structure with the nonlinear Fourier

method and, in contrast to the geometry shown

in Fig. 1, their 1-D NLPC was surrounded by

the metal. Another difference is that the additional

pump wave was used to control the switching.

Fig. 5. The normalized accumulated electromagnetic energy

density distribution for the 1-D LPC (a) and the 1-D NLPC (b).

The arrows indicate the peaks, which correspond to the

individual modes. The amplitudes of the magnetic dipole are

hlz 1:0 andhlz 100 arb: units, respectively. The number ofoscillation cycles is 50.



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The amplitudes of the dipoles are not numerically

equivalent to the amplitudes of the control waves in

Huttunens paper. Their magnitudes were fitted

from the numerical experiment. In this experiment,

we calculated band structures for the amplitudes of

the magnetic dipole hlz 50 to 500 arb: units withthe step Dhlz 10 arb: units. As can be seen from

Fig. 7, for the selected amplitudes the bands of theband structures are red-shifted. This red-shift is in

good agreement with[11]. Consequently, we draw

the conclusion that they have the same effect on

the band structure. We note that because the metal

does not surround our 1-D NLPC, we do not ob-

serve the flat bands, which take place in the pres-

ence of the metal.

Here, we discuss the physical origins of the red-

shift of the bands of the band structures. For the

1-D NLPC, the red-shift can be explained with

Scaloras argumentation[8]. Because the dielectric

constant depends on the field intensity, the band

structure changes dynamically with the incident

field. This can be qualified by the following argu-

ment. In the frequency domain, the band structure

is determined by the difference between the dielec-

tric constants of the materials, which form the

photonic crystal. From Eq.(5), we can express this

difference as

De e1 v3 ~E~r; t

2h i

e2: 24

The value ofDe increases as the intensity increas-

es ifv(3)>0 and decreases ifv(3)


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creases. For example, as can be seen from Fig.

7(a), the intensity of the dipole just slightly shifts

the bands of the band structure while the more

than four-time increase in the amplitude of thedipole in fact significantly shifts them. The exist-

ing difference between the amplitudes (100 and

450 arb. units) is due to need to significantly

change the value ofDe and, consequently, signif-

icantly induce the degree of nonlinearity. It is not

yet studied, which rule obeys the red-shift. At

least, it is a nonlinear dependence, i.e., any uni-

form linear increase in the amplitude with the

step Dhlz does not mean that the bands of the

band structure would be red-shifted on the same

value ofxa/2pc with regard to the previous shift.

The study of the degree of the shifts nonlinear

dependence on the dipoles amplitude is one of

the possible directions for future investigations.

5. Conclusions

We have presented a new approach for analyz-

ing band structures in one-dimensional Kerr-non-

linear photonic crystals. This approach was used

to analyze the band structures in Kerr-nonlinear

one-dimensional photonic crystals as a functionof the intensity of the oscillating dipole. We found

that the bands of these band structures are dynam-

ically red-shifted with regard to the bands of band

structures in linear one-dimensional photonic crys-

tals. We have discussed the physical origins of this

red-shift and showed that the red-shift increases as

the incident intensity of the oscillating dipole in-

creases. The proposed approach was validated

and found to be useful for understanding such

phenomena as intensity-driven optical limiting

and all-optical switching with Kerr-nonlinear pho-tonic crystals.

Acknowledgements

This work has been supported by the Spanish

Commission of Science and Technology (CiCYT),

project No. TIC2002-04184-C02. The authors

thank Mr. M. Ustyantsev for his help with coding

and useful discussions.

Appendix A. Unit cell discretization and conver-

gence

Fig. A.1 shows the convergence of the FDTD

method used to calculate band structures. The ab-

scissa represents the number of mesh points N in

the unit cell in the x-direction. The ordinate corre-sponds to the angular frequency. The curves

marked by s, e and. correspond to the first, sec-

ond and third bands of the band structure of the 1-

D NLPC. The values of the angular frequency are

taken forka/2p=0.5. The parameters are the same

as inFig. 7(a).

As can be seen, the method is convergent for

NP 40. Any decrease in this number will give in-

accurate results. Due to the symmetry in the

y-direction, the number of parts into which the

unit cell is divided in this direction can be selectedas N/2.

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Fig. A.1. The convergence behavior of the FDTD method. Theparameters are the same as inFig. 7(a).The curves marked by

s, e and . correspond to the first, second and third bands of

the band structure of the 1-D NLPC.



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One-dimensional Kerr-nonlinear Photonic Crystals