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Igor V. Dzedolik Vol. 24, No. 10 /October 2007 /J. Opt. Soc. Am. B 2741

One-dimensional controllable photonic crystal

Igor V. Dzedolik

Taurida V.Vernadsky National University, 4, Vernadsky Avenue, Simferopol, 95007, Ukrainedzedolik@crimea.edu

Received May 23, 2007; accepted July 27, 2007;posted August 9, 2007 (Doc. ID 83384); published September 26, 2007

The nonlinear one-dimensional model of controllable photonic crystals is considered. A photonic crystal isformed in a dielectric film on which the electromagnetic pump and signal waves fall. It is shown that the shapeof the superlattice is controlled by the variation of the external electrostatic field imposed on the film. Thus itleads to changing boundaries of the allowed and forbidden zones of the signal wave spectrum. © 2007 OpticalSociety of America

OCIS codes: 190.0190, 130.0130, 200.4740.

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. INTRODUCTIONne-, two- or three-dimensional nonlinear macroscopic

tructures arise in the media with a continuous inflow ofnergy from an exterior source, with the reflux and dissi-ation of the energy [1]. Such macroscopic structures arehe solitary and periodic stationary spatial waves, the op-ical vortices, the spiral waves, etc., that appear in the di-lectric medium by passing the electromagnetic field. Thehape-incipient macroscopic object depends on propertiesf the dielectric medium and parameters of the externallectromagnetic field.

Photonic crystals are the dielectric periodic structuresith forbidden regions (bandgaps) in an optical spectrum,nd they are recently actively explored [1,2]. The photonicrystals were made so that even one of the sizes of theireriodic structure was about the length of a light wave.hus they can function in both the linear and the nonlin-ar regime; the intensity of the electromagnetic field andhe structure period determine the spectrum profile [3].

At the interaction of the electromagnetic field with theedium, the quasi-particles named polaritons are gener-

ted in the nonlinear crystals and in the nonlinear-morphous dielectric media. In case the dielectric me-ium has the boundaries, the polaritons in one- and two-imensional models form the Bose–Einstein quasi-ondensate at a cross section of the resonator longitudinalxis [4]. The polaritons are at the ground state in theuasi-condensate when they propagate strictly along thexis of the dielectric resonator, and they have the zero-ross impulse components. The excitations of the quasi-ondensate form the stationary spatial waves with theero of amplitude on the resonator transverse axis [5]. Inhe areas around the wave maxima the refractive index ofhe medium increases and around the wave minima prac-ically does not vary [6]. Thus in the dielectric mediumhe superlattice is formed. The shape of the superlatticean be controlled by changing parameters of the electro-agnetic field.If the signal electromagnetic wave falls on the dielec-

ric medium with the superlattice, then it will be scat-ered on the inhomogeneities of the medium, i.e. on the

0740-3224/07/102741-5/$15.00 © 2

eriodic potential. For the signal wave there will be thereas of the allowed and the forbidden frequencies withhe changeable boundaries by the varying of the externallectrostatic field. Thus, in the dielectric medium the con-rollable photonic crystal is formed.

. BOSON HAMILTONIANe shall consider the quantum processes of interaction

etween the electromagnetic wave and the nonlinearmorphous dielectric medium in the volumetric unclosedesonator. Let us write the equation for the classical elec-ric field E in medium �2E−���E�− 1 � c2 �2� �t2�E=0,here �=�1+4��3E*E is the nonlinear permittivity, �1 is

he linear permittivity, and �3 is the cubic susceptibility ofedium. We shall use the electromagnetic field potentials

n Hamilton gauge �=0, E=−c−1�A /�t, and we obtain thequation for the vector potential:

�2A − ���A� −1

c2� �

�t�

�

�t�A = 0. �1�

he last term in Eq. (1) we shall rewrite as

1

c2� �

�t�

�

�t�A =1

c2

�

�t��1

�A

�t � +4��3

c4 � �2A*

�t2

�A

�t

�A

�t

+ 2�A*

�t

�2A

�t2

�A

�t � ,

here the linear dispersion �1=�1��� is taken into ac-ount, and the nonlinear response of medium �3=const. ismall. For the quasi-monochromatic field ���� we shallrite the linear response like the Brillouin operator [7]

� �

�t�1

�

�t� = f�� + ���.

hen we expand the operator f that acts at the quasi-onochromatic field A�t ,r�= A�t ,r�e−i�t in the Taylor se-

ies:

007 Optical Society of America

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Tw=

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3Lddwtti

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it

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w

WgwEtAsde�aHzt

2742 J. Opt. Soc. Am. B/Vol. 24, No. 10 /October 2007 Igor V. Dzedolik

f�� + ���A � �f���A + i�df���/�d��� � A/�texp�− i�t�,

here f���=−�2�1. If we suppose that the field is changinglowly enough T0 � ln A /�t �1 for the period T0, then therst approximation gives the system of equations forlowly varying amplitudes of the potential A= ��1 ,�2 ,�3�:

i��j

�t= − 0�2�j + 0�j��

j�

�j��j�� − 1�j − 2��j�

�j�* �j���j,

�2�

here 0=c2�d��2�1� /d��−1, 1=�1�2�d��2�1� /d��−1, 24��3c−2�4�d��2�1� /d��−1 are coefficients [8], and j1,2,3. The Lagrangian of the system is

L =�

c2�j�i�j � �j

*/�t + 0���j − 1j�j�

�j��j�� � �j* −

1

2�j

*�j

−2

16��j

*�j + �j�j*�2 + c.c. .

We shall write the Hamiltonian of the system

H =� dV� �j,=0,1,2,3

��j � L \ ��j*

+ �j* � L \ ��j

� − L y the field operators for the amorphous medium in theesonator with the volume V0:

�j�t,r� =c��

�2�V0�

kajke−i��+���t exp�ikr�,

�j*�t,r� =

c��

�2�V0�

kajk

+ ei��+���t exp�− ikr�.

hen we obtain the Hamiltonian in the boson operatorith the relations �ajk ,aj�k�

+ �=�jj��kk�, �ajk ,aj�k���ajk

+ ,aj�k�+ �=0:

H = ��j,k��w�k� + �

j�,k��ajk�

+ ajk� +1

2� �ajk+ ajk +

1

2�−

0

2kj��

j�k�

kj��ajk+ ajk� + ajkajk�

+ ��� , �3�

here ��k�=�+��+1+0k2 , =�c22 /4�V0. The bosonamiltonian in form of Eq. (3) is well known in the quan-

um field theory [9]. In this case the Hamiltonian (3) de-cribes the bound states of photons in the amorphous di-lectric medium, i.e., the polaritons.

. POLARITON QUASI-CONDENSATEet us consider Chiao’s model “electromagnetic field—ielectric medium in the resonator” [4]. The dielectric me-ium is placed in the unclosed Fabry–Perot resonatorith the flat mirrors. The electromagnetic wave falls on

he input mirror of the resonator and then leaves throughhe output mirror. As the resonator has such a high qual-ty factor the polaritons can stay in the resonator long

nough to establish the statistical equilibrium. The quad-ate of the wave number of �th resonator mode is equal to

�2=k��

2 +k3�2 , where k3�= �� /L is a propagation constant

f a mode along the longitudinal axis 3→z of the resona-or, L is the length of the resonator, and �= ±1, ±2, . . ..he cross-impulse components of the polariton thatropagate strictly along the z axis are equal to zero �k1�

�k2�=0. We can calculate the energy system changes byhe introduction of the effective chemical potential , be-ause the average number of quasi-particles is stable inhe resonator. Chemical potential is introduced as theagrange factor in one- or two-dimensional models. Thehange of the wave vector k= �k1 ,k2 , �� /L� of the polar-ton impulse p=�k� is taken into account only in a crosslane. The Hamiltonian is H→H�=H−N, where N ishe complete number of polaritons,

N = �j

NJ0 + �0 �j,k�0

ajk+ ajk0�, = �E0/�N

s the chemical potential, and E0= �0 H 0� is the energy ofhe ground state.

Then we consider the mode with polarization along theransverse axis. In the Hamiltonian (3) we shall allocatehe terms with zero cross impulse p�=0, taking into ac-ount only the pair interactions of the polaritons k�1�k�2�=k��1�+k��2� with the opposite wave vectors k andk:

H = H0 + H1 + H2, �4�

here

H0 = �����0��a0+a0 +

1

2� + �a0+a0 +

1

2�2 ,

H1 = ��k�0

���k��ak+ak +

1

2� ,

H2 = ��k�0

�k��0

�ak+ak +

1

2��ak�+ ak� +

1

2� ,

���k� = � + �� + 1 + 0�k22 + k3

2�.

e shall assume that the majority of polaritons are in theround state with zero cross impulse p�=0. In the stateith zero impulse the actual bosons form the Bose–instein condensate, but in one- or two-dimensional sys-

ems the Bose–Einstein condensation is not possible [10].s the polaritons are the quasi-particles, it is possible topeak about the quasi-condensation in one- or two-imensional systems like the unclosed resonator, consid-ring that the number of quasi-particles is very largeN0�1�. Then the eigenvalues of the operators a0 and a0

+

re approximately equal ��N0+1��N0�. We obtain theamiltonian by exchanging in Eq. (4) the operators with

ero cross impulse by their eigenvalues [9], within theerms of the second order of smallness:

w=

H

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Te

a

Tm

4PWw

Lsb�c

�

fs

w=

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=

wt

=lia

=q��

Igor V. Dzedolik Vol. 24, No. 10 /October 2007 /J. Opt. Soc. Am. B 2743

H = E�0� + ��k�0

� �k��ak+ak + a−k

+ a−k + 1�

+ N0�ak+a−k

+ + aka−k� , �5�

here E�0�=�����k�=0�+ �N0+1/2���N0+1/2�, �k����k�+ �N0+1/2�.We shall have the Hamiltonian (5) in the diagonal form

ˆ = E�0�−� �k�0

�k��bk+bk+b−k

+ b−k+1� by the Bogolubov

–v-exchanging, in letting the boson operators bk=ukakvka−k

+ and bk+=ukak

+−vka−k, where uk, vk are the c num-ers that are symmetric at k and satisfy the requirement

k2−vk

2=1. We find the numbers uk and vk by equating theiagonalized Hamiltonian and the Hamiltonian (5):

ukvk = −N0

2 �k�, uk

2 =1

2� �k�

�k�+ 1�, vk

2 =1

2� �k�

�k�− 1� .

hen we obtain the spectrum of Bogolubov type weaklyxcitations above the ground state:

�k� = �����2 + 2���N0 + 1/2�

+ 2�N0 + 1/4� 1/2

,

nd the energy of ground state

E0 = E�0� + ��k�0

�k�.

he chemical potential is ��E0 /�N0�����k�=0� in theodel.

. GROSS–PITAEVSKII EQUATION FOROLARITON QUASI-CONDENSATEe shall write the nonlinear Schrodinger equation for theave function of the polariton quasi-condensate:

i����t,r�

�t= − � �2

2mef�2 + ���t,r�

+ �� �*�t,r����t,r��U�r − r��dV� ��t,r�.

�6�

et us assume that the density of the quasi-condensatelowly varies at the wave length of the electromagneticeam. Then in Eq. (6) it is possible to put mef�� /c2,*�t ,r���t ,r��U�r−r��dV����*�t ,r���t ,r� and to re-eive the Gross–Pitaevskii equation [9]:

i����t,r�

�t= − � �2

2mef�2 + ���t,r� + ��*�t,r���t,r���t,r�.

�7�

We shall present a wave function of excitations as�t ,r�= ��r�exp�−i t�, and we obtain the equation for the

unction ��r� that depends only on the coordinates by theubstituting wave function in Eq. (7):

�2� + g1� − g2�*�� = 0, �8�

here the coefficients are g1=2 �� +� /�c2 and g2

2 /c2.

. DIELECTRIC FILM AS FABRY–PEROTESONATORe shall consider a process of quantum spatial-wave for-ation at the polariton quasi-condensate in a thin amor-

hous dielectric film. The dielectric film with light reflect-ng cross edges represents the one-dimensional Fabry–erot resonator (Fig. 1). The electromagnetic beam fallst the input “mirror” and leaves through the output “mir-or” of the resonator. The electrostatic field E0 is posi-ioned along the transverse axis of the resonator in thelm plane. The electrostatic field E0 allows us to controlhe permittivity of the medium. In such a resonator theolaritons are generated as a result of photon interactionith the dielectric medium. As the resonator has the highuality factor, the polaritons should stay in the resonatorong enough to establish the statistical equilibrium.

For one-dimensional model k=1xkx+1z�� /L we takento account the changing of the polariton impulse p�k� only along the transverse axis x. The quadrate of

he wave vector of �th mode of the resonator is equal to

�2=kx�

2 +kz�2 , where kx� is the cross component of the wave

ector, kz�= �� /L is the propagation constant of the modelong the longitudinal axis, and L is the length of theesonator, �= ±1, ±2, . . .. The cross component of the po-ariton impulse is equal to zero ��kx�=0� if the polariton isropagating along the z axis. The permittivity of the films �=1+4���1���+�2E0�, where �1���is the linear suscep-ibility of medium, �2 is the square-law susceptibility ofedium that describes the linear electro-optical Pockels

ffect.We receive the equation for the wave function �

�0f�X� of the 1D model from Eq. (8):

d2f

dX2 + f − f3 = 0, �9�

here �0=�g1 /g2, X=x�g1, and g1=g1−�2�2 /L2. We ob-ain the solution of Eq. (9) as spatial cnoidal wave f

b2 sn�b1��2X , k� [6], where sn�X , k�� �0,K�k�� is the el-iptic sine of Jacobi, k=b2 /b1 is the module of the ellipticntegral of the first order K�k�, 0� k2�1, 0� f�b2�b1,

02= �df /dX�0

2− f04 /2+ f0

2, 2a02=b1

2b22, and b1

2+b22=2.

We shall estimate the distance between zeros �x0

1/�g1� of the spatial density wave f2 in the polaritonuasi-condensate. For the laser radiation with frequency�1015 s−1 in the medium with parameters �1=1.5, �310−20 cm3/erg we obtain value x0�10−5 cm. The fre-

Fig. 1. Fabry–Perot resonator based on the dielectric film.

qlttdo

6CWtiwdo

ta

si

−fi+ss

w

I

−rErpttE

w�t

pt

�pps

Aetoa

w

ato

2744 J. Opt. Soc. Am. B/Vol. 24, No. 10 /October 2007 Igor V. Dzedolik

uency of radiation, dispersion properties of medium, theength of resonator L, and the intensity of exterior elec-rostatic field E0 determine the distance between zeros ofhe spatial density wave in the quasi-condensate. Thisistance can be as small as the principle of indeterminacyf Heisenberg �x�1/�kx allows.

. ONE-DIMENSIONAL PHOTONICRYSTALe shall consider the propagation process of signal elec-

romagnetic wave in the film in which the spatial polar-ton wave is exited by the powerful electromagnetic pumpave. As the result of Kerr nonlinearity the refractive in-ex of dielectric medium in the areas around the maximaf the polariton wave,

�1 =� g�

g2b2 sn�b1�g�

�2x,k�exp�− i t + i �

�

Lz� ,

hat is excited by pump wave, will be higher than in thereas around the minima [6]. Thus, the one-dimensional

uperlattice forms by the potential �sn2�b1�g���2x , k�n the thin dielectric film, i.e., the tapers with width �1

tanh2��g���2x� and the period �=2K�k� appear in thelm. The polariton signal wave �2�x�exp�−i�2ti� ���L �z� scatters at the superlattice. The superlatticehape can be controlled by varying the exterior electro-tatic field E0.

The equation for the polariton signal wave has the form

i��2�t,r�

�t= −

�

2m2ef�2�2�t,r� + �1

*�t,r��1�t,r��2�t,r�,

�10�

here m2ef���2 /c2. We shall rewrite Eq. (10) as

d2�2

dx2 +�2�22

c2 − �2�2

L2 −g��2b2

2

sn2�b1�g�

�2x,k� �2 = 0.

�11�

n Eq. (11) the functions U�X��= 2�22� c2 −�2��2�L2�

� g��2b22� � sn2�X� , k� (where X�= b1�g���2x) have the

eal period �=2K�k� : U�X�+n��=U�X��, n=0, ±1, ±2, . . ..quation (11) is the scattering-wave equation for the pe-

iodic potential. The exact solutions of Eq. (11) are theolynomials—Lame functions [11]. The physical modelhat is described by Eq. (11) is the one-dimensional pho-onic crystal with the variable width of superlattice.quation (11) has the form

H0�2 + �� − V��2 = 0, �12�

here H0=d2/dx2 is the unperturbed Hamiltonian,= 2�2

2� c2 −�2�2�L2 , and V�X��= g��2b22� sn2�x� , k� is

he perturbation.Let us assume that periodic potential V is the weak

erturbation [12]. At zero approximation if V=0 the solu-ions of the unperturbed equation are the plane waves

20�x�=exp�±ik2xx�. For the solution of Eq. (12) at zero ap-roximation we shall take a linear combination of thelane waves �20�x�=A exp�ik2xx�+B exp�−ik2xx� and wehall substitute it in Eq. (12):

A�− k2x2 + � − V�exp�ik2xx� + B�− k2x

2 + � − V�exp�− ik2xx�

= 0.

fter that we multiply the equation by exp�ik2xx�, then byxp�−ik2xx�, and then integrate by X� the received equa-ions, taking into account the periodicity of potential. Webtain the combined equations for the stationary values And B:

A�− k2x2 + � − V11� + BV12 = 0,

AV21 + B�− k2x2 + � − V22� = 0, �13�

here

V11 = V22 = �2

g�b22

�

−�/2

�/2

dX� sn2�X�,k�;

V12 =�2g�b2

2

�

−�/2

�/2

dX��exp�− i2�2k2x

b1�g�

X���sn2�X�,k�

�, V21 = V12*

re the matrix components of the perturbation operator inhe first Brillouin zone, � /2=K�k�. From the requirementf nontrivial solution of the linear combined Eqs. (13):

Fig. 2. Signal-wave spectrum at �2/b1�g�=1/2.

Fig. 3. Signal-wave spectrum at �2/b �g =1.

1 �

wt

w

F

ap

tn

w=qgIcaats=

ttcE

7Tpqitafiwwddtsop

R

1

1

1

Igor V. Dzedolik Vol. 24, No. 10 /October 2007 /J. Opt. Soc. Am. B 2745

��− k2x2 + � − V11� V12

V12* �− k2x

2 + � − V11�� = 0, �14�

e find the parameter � for the one-dimensional model athe first approximation:

� = k2x2 + V11 ± �V12�k2x�2, �15�

here

�V12�k2x�2

=�2g�b2

2

����−�/2

�/2

dX� sn�2�2k2x

b1�g�

X��sn2�X�,k��2

+ ��−�/2

�/2

dX� cos�2�2k2x

b1�g�

X��sn2�X�,k��2�1/2

.

If one presents the periodic potential in the form of the

ourier series V�X��= �n=−�

�

Vn exp�inKxX��, where

Vn =g��2b2

2

�

−�/2

�/2

dX� sn2�X�,k�exp�− inKxX��

re the amplitudes of harmonics, Kx=2� /� ,n�0, it isossible to find the matrix components for the nth zone.We can find the frequency �2�0 from Eq. (15) and ob-

ain the spectrum of the polaritons that excited by the sig-al wave in the Brillouin zone k2x� �−� /�� ,� /���:

�2 =c2

4 �V11� ± �V12� �k2x�2 + ��V11� ± �V12� �k2x�2�2

+ �28�2

c2L2 +8

c2k2x2 �

12� ,

�16�

here V11� =V11/�2, �V12� �k2x�2=�V12�k2x�2 /�2, and ��

2�2K�k� /b1�g�. In the Brillouin zone we can see two fre-uency branches (Fig. 2; �2�1015 s−1, kx�105 cm−1). Theap in the signal spectrum takes place at �2/b1�g�=1/2.f the value of external electrostatic field E0 increases, theoefficients b1 and g� are incremented (since E0 enters b1nd g� through the coefficients 0, 1, 2) and, for ex-mple, we obtain the value �2/b1�g�=1 (Fig. 3). There ishe tendency to decrease in twice the spatial period of thepectrum along the Brillouin zone. The gap width ��

+ −

�2 −�2 is determined by “contrast” between the refrac-

ive index of the superlattice tapers. Thus we can controlhe polariton profile spectrum �2�k2x� in the photonicrystal, varying the value of external electrostatic field0.

. CONCLUSIONhus the polariton quasi-condensate arises at the amor-hous dielectric medium in the unclosed resonator. Theuantum interacting spatial cnoidal waves are generatedn the quasi-condensate. It is possible to create the quan-um spatial waves with zeros of the field at the transversexis of the resonator mirrors in the amorphous dielectriclm with reflecting edges. The electromagnetic pumpave produces the superlattice with the controllableidth by the external electrostatic field, i.e., the one-imensional controllable photonic crystal is formed in theielectric film. The one-dimensional photonic crystal withhe controllable breadth of the allowed and forbiddenpectral zones acts as perspective for the creation of theptical information transmission lines and for the opticalrocessing in the whole.

EFERENCES1. Y. S. Kivshar and G. P. Agraval, Optical Solutions: from

Fibers to Photonic Crystals (Academic, 2003).2. K. Sakoda, Optical Properties of Photonic Crystals

(Springer-Verlag, 2001).3. I. V. Guryev, O. V. Shulika, I. A. Sukhoivanov, and O. V.

Mashoshina, “Improvement of characterization accuracy ofthe nonlinear photonic crystals using finite elements-iterative method,” Appl. Phys. B: Photophys. Laser Chem.84, 83–87 (2006).

4. R. Chiao, “Bogolubov dispersion relation for a ‘photon fluid’:Is this a superfluid?” Opt. Commun. 179, 157–166 (2000).

5. I. V. Dzedolik, “Polariton Vortices in a TransparentMedium,” in Proceedings of International Conference onLaser and Fiber-Optical Networks Modeling (IEEE-LEOS,2006), pp. 96–99.

6. M. B. Vinogradova, O. V. Rudenko, and A. P. Sukhorukov,Wave Theory (Nauka, 1990) (in Russian).

7. L. D. Landau and E. M. Lifshitz, Theoretical Physics: VIII.Electrodynamics of Continuums (Nauka, 1982) (inRussian).

8. I. V. Dzedolik, “Spontaneous symmetry breaking in anelectromagnetic field—insulator system,” Tech. Phys. 51,932–937 (2006).

9. L. D. Landau and E. M. Lifshitz, Theoretical Physics: IXStatistical Physics. Part II (Nauka, 1978) (in Russian).

0. N. D. Mermin and H. Wagner, “Absence of ferromagnetismor antiferromagnetism in one- or two-dimensional isotropicHeisenberg models,” Phys. Rev. Lett. 17, 1133–1136 (1966).

1. E. T. Whittaker and G. N. Watson, A Course of ModernAnalysis (Cambridge U. Press, 1927).

2. L. D. Landau and E. M. Lifshitz, Theoretical Physics: III.Quantum Mechanics (Non-Relative Theory) (Nauka, 1989)

(in Russian).