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PHYSICAL REVIEW B 67, 125203 ~2003!

Symmetrical analysis of complex two-dimensional hexagonal photonic crystals

N. Malkova, S. Kim, T. DiLazaro, and V. GopalanMaterials Research Institute and Department of Materials Science and Engineering, Pennsylvania State University,

University Park, Pennsylvania 16802~Received 29 August 2002; revised manuscript received 6 January 2003; published 12 March 2003!

We study complex hexagonal photonic crystals with unit cells that include different dielectric cylinders. Ageneral symmetrical perturbation approach for a hexagonal lattice with up to three basis rods is presented thatsystematically develops other structural derivatives including comblike structures. We show how the bandspectrum of these complex structures evolves from the most symmetrical prophase. The results are in agree-ment with the plane-wave calculations of the band spectrum.

DOI: 10.1103/PhysRevB.67.125203 PACS number~s!: 71.15.Dx, 42.70.Qs, 71.20.Nr

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I. INTRODUCTION

In recent years, experimental and theoretical studiesartificially manufactured dielectric media, the so-called phtonic band-gap materials or photonic crystals, have attraconsiderable attention.1 The photonic crystals may be dvided by their application either as photonic insulators orphotonic conductors. In the first case, the most importproperty of the photonic structures is a band gap, whpropagating modes for any magnitude and direction ofwave vector are forbidden for either specific orpolarizations.2 The most important feature of the photonconductors, such as those exhibiting the superprism effecthe possibility of tuning the opening of the band gap at solow-symmetry points of the Brillouin zone.3 Thus, under-standing the behavior of defect-free photonic crystalsdesigning devices based on them require a knowledge owave propagation properties in the crystals.4 In the same wayas electron properties of electronic crystals are governedthe solid’s band structure, the information about the phopropagation properties is contained in the band structureeigenmodes of the dielectric periodic structure.

A variety of methods have been used to calculate the ptonic band structure. All these approaches may be diviinto two groups. The first group comprises the so-calledabinitio models, implementation of which does not demand aempirical parameters. They are the plane-wave techniq5

transfer-matrix methods,6 and different types of numericaschemes to solve Maxwell’s equations.7 A disadvantage ofthe ab initio calculations is that they are very time anmemory consuming. Another group of the photonic banstructure models consists of empirical models. For periodielectric structures, this group includes the tight-bindimodel, first adopted for photonic crystals in Ref. 8. Thmodel contains some empirical parameters, coupled maelements, which have to be fitted toab initio or experimentalresults. Tight-binding calculations of the band structurevery simple in their numerical implementation, giving aanalytical solution. Recently, this scheme was also succfully used for various photonic structures.9

In our previous paper10 a symmetrical model for theanalysis of the band structure of the complex photonic strtures has been developed. A similar theoretical approachbeen suggested for electronic band structures of the com

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semiconductors in Refs. 11 and 12. In Ref. 10 the modelbeen applied to complex square photonic crystals in whperpendicular to the selected diagonal, the layers with difent rods alternated. When studying the band spectrum ofcomplex crystals, we start from the band spectrum ofprophase, suggested by simple symmetry analysis. Introtion and identification of the appropriate prophase is a crustep in this model, which differentiates this approach frothe nearly free electron model for the electronic bastructure13 or with the symmetrical model of the photoncrystals developed in Ref. 4 in which the starting point is tband spectrum of the free electrons or free photons. Inpresent approach, the plane-wave functions for the prophstates are used as the basis. The band spectrum of the pbative phase is then obtained as a perturbation of the plwave spectrum of the prophase.

Two important classes of periodic lattice symmetriesstudied in photonic structures: the class of square latticestheir complex derivatives~containing fourfold or twofold ro-tational symmetry axes!,14,15 and the class of hexagonal latices and their complex derivatives~containing sixfold- orthreefold rotational symmetry axes!.16,17 Reference 10 dealwith the first class of square lattices. This paper deals wthe second important class of hexagonal lattices and tcomplex derivatives. The motivation of this work is twostep. First is that since the model depends critically onpresence and selection of a suitable prophase lattice, demstrating the presence of a suitable prophase for the comsquare lattices~as in Ref. 10! does not automatically implyor naturally suggest a similar suitable prophase for the enclass of complex hexagonal lattices. In this paper, we shthat indeed such a prophase exists. The second goalsunderstand the common scheme of the evolution of the bspectrum for the whole class of the complex crystals baon hexagonal and square symmetries within the framewof one model. We compare and contrast these two imporcrystal classes.

In particular, we consider a hexagonal lattice with altnating layers of the dielectric rods perpendicular to a selechexagon diagonal. The selection of this one direction incrystal will be shown to be of utmost importance in all baproperties of these structures. We start from a simple hegon lattice with one rod in the basis, when all the layersequivalent. Then we simulate complex lattices by introdu

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N. MALKOVA, S. KIM, T. DiLAZARO, AND V. GOPALAN PHYSICAL REVIEW B 67, 125203 ~2003!

ing two additional rods on the selected diagonal of the inilattice. As a limit case when the radius of one of the adtional rods is equal to zero, we can derive the hexagocomblike lattice considered first in Refs. 16 and 17. Thusshow that in the framework of the developed model we csystematically study the band structure of the entire clascomplex lattices, including square, hexagonal, and thcomplex derivatives such as comblike lattices. Fromsymmetrical analysis, the opening of the band gap is shoto be predicted for the class of the layered photonic strtures. We state that these layered photonic crystals preswide class of tunable photonic structures.

The outline of this paper is as follows. A theoretical dvelopment of the model for the case of the complex hexanal lattices is presented in Sec. II. In Sec. III we comparetheoretical predictions with the plane-wave calculationsthe photonic crystals in question. In Sec. IV we give concsions, where discussion of the evolution of the band sptrum for complex crystals with square and hexagonal symetries is presented.

II. SYMMETRICAL MODEL FOR COMPLEXHEXAGONAL LATTICES

We consider a two-dimensional dielectric system periocal in thex-y plane and homogeneous along thez axis. Thedielectric constant for the periodical system is a positionpendent and periodic function of the vectorr in the x-yplane, satisfying the relatione(r1Rl)5e(r ), where for anyintegersl 1,2, Rl5 l 1a11 l 2a2 defines a two-dimensional Bravais lattice with the unit cell constructed on the primitivtranslation vectorsa1 and a2 . The magnitudeA5ua13a2ugives the area of a primitive unit cell of this lattice.

In the framework of the plane-wave method, the banstructure problem for the two-dimensional lattice is reducto the eigenvalue problem4

detS H2v2

c2 D 50, ~1!

where

H~G, G8!5uk1Guuk1G8uh~G2G8!

for the E polarization~electric field is parallel to thez axis!and

H~G,G8!5~k1G!•~k1G8!h~G2G8!

for the H polarization ~magnetic field is parallel to thezaxis!. The functionh(G) is defined as Fourier transforms othe inverse dielectric constant

h~G!51

AEuc

1

e~r !e2 iG•rdr , ~2!

where the integral is taken over the unit cell with an areaA.Herek is the two-dimensional reciprocal vector lying insidthe Brillouin zone and the translation vectors of the recipcal latticeGn5n1b11n2b2 , wheren1,2 are integers, andb1,2are the primitive reciprocal vectors.

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We study the complex hexagonal lattice, shown in F1~a!. This lattice is obtained from the simple hexagonal ltice by placing equidistantly, two different rods on one of tdiagonals of hexagon. We first note that this is a layer lattwith alternating layers of the different rods perpendicularthe selected diagonal. The unit cell is defined by the veca15a(1,0) anda25a(1/2,A3/2), shown in Fig. 1~a! by boldlines. The point group symmetry of this lattice isC3 . Themost important point is that this lattice can be representethree embedded identical hexagonal sublattices, determby the set of same primitive vectorsa1 anda2 , but which areshifted with respect to each other by the vectort5a/2(1,1/A3). These sublattices are shown in Fig. 1~a! bydashed-dotted lines. The inverse dielectric constant of sulattice can be expressed as

1

e~r !5(

G@h1~G!eiG•r1h2~G!eiG•(r1t)

1h3~G!eiG•(r12t)#. ~3!

Here G5n1b11n2b2 (n1,2 are integers! are the vectors ofthe corresponding reciprocal lattice, with the primitive vetors b15(2p/a)(1,21/A3) and b25(2p/a)(0,2/A3);h1,2,3(G) are the Fourier transforms of the inverse dielect

FIG. 1. Hexagonal lattice with two rods in basis~a! and itsBrillouin zone~b!. The unit cells of the simple hexagonal prophaand perturbative phase are shown by dashed and bold lines, retively.

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SYMMETRICAL ANALYSIS OF THE COMPLEX TWO- . . . PHYSICAL REVIEW B 67, 125203 ~2003!

constants for the three selected sublattices. The Brillozone of this lattice is a hexagon shown as a shaded areFig. 1~b!.

We note thatG•t52p(n11n2)/3. So the inverse dielectric constant can be expressed as a sum of the three pote

1

e~r !5V0~r !1V1~r !1V2~r !. ~4!

The potentialV0 is represented as

V0~r !5(G0

@h1~G0!1h2~G0!1h3~G0!#eiG0•r. ~5!

Here the sum runs over the reciprocal vectors of the simhexagonal latticeG05(2p/a)@n11n2 ,A3(n12n2)#, withthe primitive vectorsb1,2

0 52p/a(1,6A3). The correspond-ing real space unit cell is defined by the primitive vectoa1,2

0 5(a/2)(1,61/A3). The potentialV0 may be consideredas a potential of the simple hexagonal prophase obtafrom the lattice in question if all the cylinders were identicIt has to be characterized byC6v symmetry of the simplehexagonal lattice, satisfying the translation relation

V0~r1t!5V0~r !.

The unit cell and Brillouin zone of the simple hexagonprophase are shown in Fig. 1 by dashed lines.

Two potentialsV1,2 have the form

V1,2~r !5(G0

@h1~G07Q!2h2~G07Q!e7 ip/3

2h3~G07Q!e6 ip/3#ei (G07Q)•r. ~6!

Here summing is over the reciprocal vectors of the simhexagon latticeG0 again, a new reciprocal vectorQ5(4pA3/3a)(0,1) being selected.

We note that the potentialsV1,2 increase the translatioscaling of the lattice, satisfying the translation relation:

V1,2~r1t!52V1,2~r !e7 ip/3.

In general case, when all the rods in the basis are diffe@this is the case in Fig. 1~a!#, the symmetry of the lattice islowered toC3v . But in the particular case, when two rodsthe basis are equivalent, that is,V15V2 , the lattice keeps thesymmetryC6v of the prophase. As seen from Fig. 1, the arof the unit cell for the complex hexagon lattice is greathan three times the area of the simple hexagon lattice ofprophase, while the areas of the reciprocal unit cells, namthe Brillouin zones, are characterized by the inverse relatship.

The potentialsV1,2 can be treated as perturbations charterized by the difference in properties of the dielectric rodsthe basis. Inclusion of the perturbation potentials resultstripling the area of the direct lattice and in complete nestof the Brillouin zone of the simple hexagonal prophase ithe Brillouin zone of perturbative phase. This is governedthe vectorQ shown in Fig. 1~b!. So that the pointsG andJ0of the prophase will be combined by the vectorQ into oneG

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point of the perturbed phase, all the pointsJ0 of the prophasebeing combined into one point of the prophase. The poX0 of the prophase become the pointsX in the perturbativephase. But the vectorQ picks up one of the diagonals of thhexagonal Brillouin zone of the prophase in such a way ttwo of the X points of the perturbative phase are obtainfrom the the middle points of theGJ0 branches of theprophase and four others—from theX0 points of theprophase.

The periodicity of theV0 potential implies that it can mixonly the states differing by the reciprocal vectorsG0 of thesimple hexagon lattice while the translation properties ofV1,2 potentials result in mixing of the prophase states serated by the vectors6Q. Perturbative analysis similar to thapresented in Ref. 10 shows that the matrix HamiltonianH~1! is 333 block matrix:

HG0 ,G085S Hk Vk,k1Q Vk,k2Q

Vk1Q,k Hk1Q Vk1Q,k2Q

Vk2Q,k Vk2Q,k1Q Hk2Q

D . ~7!

Here the diagonal blocks give the band spectrum ofprophase in the pointsk, k1Q andk2Q. They interact witheach other through the block matricesVk,k8 characterized bythe matrix elements of the potentialsV1 and V2 . In theplane-wave basis these matrix elements, for the case of tEpolarization, are defined as10

vk,k85ukuuk8uhk2k82 , ~8!

where h25h12(h21h3)cos(p/3)2 i (h22h3)sin(p/3).We can interpret the effect of including theV1,2 potentials astriple folding of the simple hexagon Brillouin zone. So ththe spectrum of the perturbative phase along any direcM1M2 will be a sum of the three prophase branches:M1M2 ,(M11Q)(M21Q) and (M12Q)(M22Q). Here we define(Mi6Q) as a point inside the Brillouin zone with the coodinate (2p/3a)(xi ,yi)6(2p/3a)(0,2A3), assuming that thecoordinates of the pointM are (2p/3a)(xi ,yi). We concludethat the folding of the Brillouin zone of the simple hexagonlattice is governed by the vectorQ, resulting in a strongmixing of the states separated by this vector.

III. PLANE-WAVE CALCULATION

We apply this symmetrical analysis to study the planwave band spectrum of the complex hexagonal lattishown in Fig. 1. It is obvious10 that in the region of the lightwavelength much larger than the radii of the rods, the effof varying the radii of the rods versus varying their dielectconstants on the band spectrum should be similar in the ptonic crystals studied. We have performed numerical callations for wavelengths larger than the lattice period. Thefore, in our numerical analysis, since the radius of tdielectric rods is a more flexible parameter, we have mopaid attention to the case of varying the radii of the rods, akeeping the dielectric constants of the rods unchanged.the results presented can be easily extended for the casvarying the dielectric constants of the rods.

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FIG. 2. The band spectrum forE polarization of the simple hexagonal lattice with one rod (r 150.2a) in the basis constructed in thBrillouin zone of the prophase~a! and in the Brillouin zone of the perturbative phase~b!; and the band spectrum of the complex hexagolattices withr 15r 250.2a, r 350.19a ~c!; r 350.17a ~d!. Discussed points are labeled by circles.

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A system of dielectric rods in the air has been considerwith the dielectric constants of the rodsea511.9~Si! and aireb51. To calculate the band spectrum for the photonic crtals with complex basis we used the technique developeRef. 18. The results that follow were obtained using 5plane waves for the simple hexagonal lattice with one rodbasis and 961 plane waves for the complex lattices. Themerical data were tested using 1221 plane waves showthat the accuracy of the results is better than 1%.

At first we study the generation of the band spectrumthe complex crystals from the band spectrum of the prophunder small perturbation. Figure 2 shows the spectrumE-polarization of the simple hexagonal lattice with one r(r 150.2a) in the basis constructed in the Brillouin zonethe prophase~a! and in the Brillouin zone of the perturbativphase~b!; and the band spectrum of the complex latticwith r 15r 250.2a, r 350.19a ~c! andr 350.17a ~d!. This isthe simplest case of the hexagonal photonic lattice withrods in the basis when the radius of one of the rods is kunchanged and only the radius of the second rod changethis case, the matrix elementsh1 and h2 are equal to each

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other, and as follows from Eq.~6!, the perturbative potentialsV1,2 are complex conjugate.

A key point in our symmetrical analysis is demonstratby comparing Fig. 2~a! and Fig. 2~b!. By doing this, we canfollow the folding of the band spectrum of the prophase inthe Brillouin zone of the perturbative phase. From the preous analysis, we know that in the perturbative phase,band spectrum alongGX direction will be a sum of the threeprophase branchesGX, GX6Q. Simple symmetrical analy-sis shows that the band spectrum alongGX branch in theperturbative phase must be composed from the three dient branches of the prophase:GX5(GX)pr1(J0X)pr

1(J0X0)pr, where index pr means the states of the prophaThe dashed line in Fig. 2~a! separates the first two brancheIn order to get the band spectrum of the perturbative ph@Fig. 2~b!#, we have to fold the band spectrum of thprophase first alongJ0 line, and then along the dashed lineas to overlap the states separated by the vectorQ shown inFig. 2~a!. The branches involved in this mapping are markby one dash in Fig. 1~b!. All these branches are different ithe prophase. They have to show splitting at the points of

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degeneracy when perturbation is included. This is a casethe degeneracy point between seventh and eighth bmarked by the circle in Figs. 2~b, c, d!.

The band spectrum alongGJ direction of the perturbativephase is composed from the three branches of the propagain:GJ5(GJ)pr1(J0J)1

pr1(J0J)2pr . They are mapped in

Fig. 1~b! as cuts with double dashes marker. To get the bspectrum of the perturbative phase@Fig. 2~b!# from theprophase spectrum@Fig. 2~a!# we have to take the states frothe point G up to the dash-dotted line in Fig. 2~a!, whichmarks the pointJ of the perturbative phase, and then overlthe (J0J)pr branches. But these latter branches do not liethe symmetrical directions of the hexagonal [email protected]~b!#, and so they are not present in the spectrum ofprophase@Fig. 2~a!#. From Fig. 1~b! we note that thebranches (J0J)1,2

pr are equivalent in the simple hexagonal latice, and therefore, they completely overlap in the proph@the second and third bands in Fig. 2~b!, for example#. Whenincluding the perturbationsV1,2, the scaling of the latticeincreases@Fig. 1~a!# and these branches split. This processclearly shown in Figs. 2~c, d!. We can also follow the opening of the band gaps at the points of the degeneracy betwthe seventh and the eighth bands on theGJ branch, markedby the circle in Figs. 2~b, c!. Figure 2~d! shows that thesplitting of the seventh and eighth bands at the degenepoints on theGJ and GX directions results in opening thcomplete band gap forE bands, bordered by two dashelines.

An obvious reason for the opening of the band gaps atband intersection points created by the Brillouin zone folddescribed above, is that these bands are related to thedifferent rods in the basis of the lattice. In support of thstatement, we present in Fig. 3, the spatial distribution ofPoynting vector for the seventh@v5va/(2pc)50.6056#~a! and the [email protected]# ~b! bands at the labeled poinon the GJ direction of the photonic crystal withr 15r 250.2a and r 350.19a @Fig. 2~c!#. Figure 3 shows that themaximum intensity of the energy for these states is locaon different rods in the basis. In the prophase, all the rodsidentical and the two bands have the same energy. Howein the perturbative phase, the greater the difference betwdielectric rods, the greater the energy splitting between thtwo bands@Figs. 2~c, d!#.

As was noted above, this lattice has the same group smetryC6v as a hexagonal lattice of the prophase. That is wthe symmetry of the bands at the symmetrical points ofBrillouin zone should not change. Data in Figs. 2~c, d! showthat the double degeneracy, permitted by theC6v , C3v , andC2v point groups at theG, J, and X points, respectively,survives.

The above analysis provides evidence for a qualitaagreement between symmetrical model and exact plawave calculation of the band spectrum of the complex hagonal crystals. In the case of the small difference betwthe rods, the perturbative approach can be used to quantively solve the eigenvalue problem with Hamiltonian~7!.For the points of degeneracy, the problem can be reducethe three-band approximation10

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U ~Ek0!22v2 vk,k1Q vk,k2Q

vk1Q,k ~Ek1Q0 !22v2 vk1Q,k2Q

vk2Q,k vk2Q,k1Q ~Ek2Q0 !22v2

U50. ~9!

Here Ek0 is a normalized frequency for the treated bands

the pointk of the prophase,vk,k8 are the corresponding matrix element of the block matricesVk,k8 ~10!, and v5va/(2pc) is an unknown normalized frequency.

We perform the perturbative calculations for the secoand third split bands at theG point for the photonic crystashown in Fig. 2~c!. This is the lowest labeled circle on theGaxis in Fig. 2~c!. We note that in this case the problem canreduced to a 232 matrix, because for these bandsk50 andall the matrix elementsvk,k1Q ~8! are equal to zero. Thenonzero matrix element is

vk1Q,k2Q5uQu2h2Q2 , ~10!

FIG. 3. The space distribution of the Poynting vector for t

seventh (v50.6056) ~a! and the eighth (v50.6196) ~b! bands atthe labeled point on theGJ direction for the photonic crystal withr 15r 250.2a; r 350.19a @Fig. 2~c!#. The unit cell is superimposed

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where

h2Q2 5~r 12r 2!S 1

ea2

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ebD 2pJ1~2QR!

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Here J1 is the first-order Bessel function andA5a2/A3 isthe area of the prophase unit cell. In this approximationimmediately arrive at

DEk;vk1Q,k2Q

Ek0

. ~11!

Taking into account that for these bandsEk050.37 @Fig.

2~b!#, we getDE050.0093. The exact plane-wave solutiogives the splittingDE50.372720.368050.0047, which isthe same order of magnitude asDE0 .

We apply the same approach for the fourth and fifth sG bands marked by the second circle onG axis in Fig. 2~c!.In this case as well, the problem allows the two-bandproximation. The nonzero matrix element is

vk1G02Q,k1G081Q5uk1G02Quuk1G081QuhG02G

0822Q2

,

~12!

whereG052p/a(1,A3) andG0852p/a(1,2A3). Using theapproximation formula~11! with Ek

050.53, we have gotDE50.0189. Within the limits of the accuracy of the planwave calculations, this is in good agreement with the exsolution for these bandsDE50.540520.526550.014.

We note that the above estimations have been perforwithin the very rough two-band approximation in compason with the more accurate plane-wave calculations us;1000 plane waves~bands!. A reasonable agreement btween perturbative andab initio data, without any fitting pro-cedure, demonstrates the power of even this simple two-bmodel for the prediction of the band splitting for small peturbations. However, for large perturbations compared wthe distance from far bands in the prophase, one needconsider the many-band model of the perturbative Hamtonian in order to get a good agreement with theab initiocalculations. In this case, the perturbative approach losesimplicity and the standard plane-wave calculations are mreasonable.

Next we study the band spectrum of the complex hexanal lattice when perturbation potential is increased, buth1

5h2. Figure 4 presents the plane-wave band spectrumthe E andH bands in the casesr 15r 250.2a, r 350.1a ~a!,r 350.05a ~b! andr 350a ~c!. First we note that thecomblikelatticesare included in the family of these lattices as a limiting case when the radius of one of the rods tends to z@Fig. 4~c!#. These lattices have been carefully studiedRefs. 16 and 17. Here, we shall only emphasize thatcomblike lattices can be considered within the frameworkour model. However, in this case, the structures do not althe perturbative approach, because both the perturbativetentialsV1,2 do not have limit transitionV1,2→0.

Figure 4 shows that the common character of the chanof theH bands is similar to theE bands discussed above. Thperturbation potential is the reason for splitting of the seco

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FIG. 4. The plane-wave band spectrum for theE andH bands inthe casesr 15r 250.2a, r 350.1a ~a!, r 350.05a ~b!, andr 350.0a~c!. The E andH bands are shown by dots and stars, respectivThe E-band gaps are shown by dashed lines, the complete bgaps are shown by the dashed stripes. Discussed points are laby circles.

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and the thirdH bands, which overlap in the prophase~notshown!. We also note that the splitting of the second athird E bands that had appeared with a small perturbationFigs. 2~c, d! results, in the case of a large perturbation, infirst band gap, shown by two dashed lines in Fig. 4. Tband gap increases with decreasing the radius of the serod, reaching the maximum for a comblike lattice atr 350.

It is worth mentioning that in the prophase at the pointG,the fourth and fifthE bands@Fig. 2~b!# and the second anthird H bands are degenerate. Moreover, accidentally forcrystal studied, the energy of theE bands overlaps with theenergy of theH bands. The degeneracy is removed in tperturbative phase. It is remarkable here that the splittingtheE andH bands at theG point, labeled by circles in Fig. 4has approximately the same magnitude for all the crysshown in Fig. 4. The explanation of this fact is the followinFirst, the relatively large distance of the splitH bands fromthe far bands results in a small effect of the far bands andpossibility of using a two-band approximation even in tcase of a large perturbation. The splitting of the secondthird H-bands is determined by the matrix element~10!. Adirect estimation by formula~11!, for the crystal shown inFig. 4~a!, gives the magnitudeDE50.05 that is close to theexact value 0.06. As seen from Fig. 4, the two-band apprmation for theE bands is hardly applicable. But in a rougapproximation we may still use formula~12! to estimate thesplitting of the fourth and fifthE bands. Since the energiesthe unperturbative phaseEk

0 for theE andH bands are equaeach other, Eq.~11! estimates that the splitting of theH andE bands is approximately equal. The binding of theE andHbands explains the fact that the lowest complete band gathese structures may be opened only between the sixth~sev-enth! E band and the fourthH band. The complete band gais shown by the shaded stripes in Fig. 4. We note the dmatic increase in the first complete band gap for the comlike crystal @Fig. 4~c!#.

As the last step, we study the band spectrum of the cplex hexagonal crystals when all the rods are different. Fure 5 shows the spectrum of theE bands and theH bands ofthe hexagonal lattice withr 150.2a,r 250.21a, r 350.05a~a!; r 150.2a,r 250.24a,r 350.05a ~b!; and r 150.2a,r 250.3a,r 350.05a ~c!.

First, we note that in this case both the potentialsV1 andV2 ~6! are different, and the group symmetry of the latticelowered toC3v . The different magnitudes of the potentiaV1,2 differentiate the two rods in the basis. This results inseparation of the states at the pointJ into two groups relatedto two different rods. This is the reason for the removingthe degeneracy of the first and second as well as of the foand fifth E bands at the pointJ and for the opening of thesband gaps between theseE bands. These states are markby circles in Figs. 5~a, b!. We can follow, in Fig. 5, theopening and increasing of these band gaps~bordered by twodashed lines! when the difference between two rods in tbasis increases. The splitting of the second and third asas fourth and fifthH bands~marked by dashed circles! isagain caused by the difference in theV1 andV2 potentials. Itis also important to note that theC3v point group symmetryof the pointsG and J allows the existence of the doubl

12520

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FIG. 5. The plane-wave spectrum of theE andH bands of thehexagonal lattice with r 150.2a,r 250.21a,r 350.05a ~a!; r 1

50.2a,r 250.24a,r 350.05a ~b!; and r 150.2a,r 250.3a,r 3

50.05a ~c!. The E and H bands are shown by dots and stars,spectively. TheE band gaps are shown by dashed lines, the coplete band gaps are shown by the dashed stripes. Discussed pare labeled by circles.

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degenerateG and J bands, while all theX bands arenondegenerate.4,19 This explains the fact that the band gabetween first and second as well as fourth and fifthH bandscannot be opened. Figure 5 also shows that the compband gaps for these crystals, marked by shaded stripescrease with increasing the difference between two rods,at last disappear for the last crystal@Fig. 5~c!#. The overlapof the E- andH band gaps and, as a result, the appearingthe complete band gap, are described by the relative mment of the bands related to photon states with differentlarizations. This problem is beyond the developed mode

IV. CONCLUSION

We have developed in this paper the symmetrical mod10

for analyzing the band spectrum of the complex hexagophotonic crystals. We have shown that this symmetrimodel allows one to analyze the band spectrum and todict the opening of the band gaps for the whole classcomplex hexagonal photonic crystals.

Within the framework of the developed model, a qualitive understanding of the evolution of the band spectrumthe complex crystals can be obtained, starting from the bspectrum of the prophase and symmetry of the perturbapotential only, without any numerical calculations. The comon scheme for the analysis of the band spectrum ofcomplex crystals is to select the simplest prophase withmost symmetrical spectrum, and then to follow the genetion of the band spectrum of the complex photonic crystwhen including the perturbative potentials. The generationthe band spectrum of the complex crystals goes in two stIn the first step, the band spectrum of the prophase, goveby the corresponding vectorQ, is folded into the Brillouinzone of the perturbative phase. In the second step, the deeracy of the band spectrum is lifted by the perturbativetential. We emphasize that, without any fitting procedure,results of the perturbative analysis have shown a reasonagreement with the plane-wave calculations of the compphotonic crystals.

We now compare the complex hexagonal photonic crtals with complex square photonic lattices. It is importantnote that the evolution of the band spectrum of the compcrystals with hexagonal and square10 symmetries is similarwithin the the theoretical framework presented. In bocases, the qualitative understanding of the changes inband spectrum can be obtained from the band spectrumthe prophase and the symmetry of the prophase potenFor both symmetries, the band gap inside the Brillouin zo

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has been shown to open along the direction parallel tovector Q. This theoretical prediction of the symmetricmodel is supported by the numerical calculations. An imptant difference in the symmetrical analysis of the both symetries is the very lowC2 symmetry of the perturbativepotential of square lattice with three alternating layers, whin the case of the layered hexagonal lattice, the symmetrthe perturbative potential isC3v . It is generally easier to findcomplete band gaps in hexagonal lattices due to its hsymmetry,C6v . The perturbation of this lattice with aC3vpotential still maintains a high enough symmetry to maintan overlapping of the band gaps for both polarizations, tresulting in a complete band gap. This fact is an advantfor the complex hexagonal structures to be used as photisolators. In contrast, the low symmetry of the complsquare lattices results in a complete splitting of the degeate states at all symmetrical points, which is not the casethe complex hexagonal crystals. Therefore, by varyingparameters~radii or dielectric constants of the rods! of thecomplex square structures it is possible to create a bandfor a specific polarization of light. However, complete bagaps for such low-symmetry structures are difficultachieve. This fact is of great interest in possibly designincomplex square lattice based photonic insulator basedmaterials with low dielectric constant contrast.

The theoretical background of perturbative approachthis model was based on the plane-wave expansion forwave function of the prophase. We would like to point othat further theoretical development of this approach holdpossibility for the tight-binding formulation, based on usinthe Mai resonances as a basis for the prophase wfunctions.8 In this case the problem may be reduced toanalytical solution, having a simple numerical implemention.

To conclude, the developed symmetrical model has bshown to be useful for understanding and for the predictof the points in the Brillouin zone where the band gap canopened by some lattice perturbation. The hexagonal comcrystals considered here present an entire class of photstructures with predictable and tunable properties that caapplicable both for photonic insulator and photonic condtor devices.

ACKNOWLEDGMENTS

We would like to acknowledge the support from the Ceter for Collective Phenomena in Restricted Geometries~PennState MRSEC! under NSF Grant No. DMR-00800190, anthe National Science Foundation Grant No. ECS-998868

nd

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