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Symmetrical Analysis of Complex Two-Dimensional Hexagonal Photonic Crystals

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Symmetrical Analysis of Complex Two-Dimensional Hexagonal Photonic Crystals PHYSICAL REVIEW B 67, 125203 ͑2003͒ Symmetrical analysis of complex two-dimensional hexagonal photonic crystals N. Malkova, S. Kim, T. DiLazaro, and V. Gopalan Materials 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 2003͒ We study complex hexagonal photonic crystals with unit cells that include different dielectric cylinders. A general symmetrical perturbation approach for a hexagonal lattice with up to three basis rods is presented that systematically develops other structural derivatives including comblike structures. We show how the band spectrum 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 number͑s͒: 71.15.Dx, 42.70.Qs, 71.20.Nr I. INTRODUCTION semiconductors in Refs. 11 and 12. In Ref. 10 the model has been applied to complex square photonic crystals in which, In recent years, experimental and theoretical studies of perpendicular to the selected diagonal, the layers with differ- artificially manufactured dielectric media, the so-called pho- ent rods alternated. When studying the band spectrum of the tonic band-gap materials or photonic crystals, have attracted complex crystals, we start from the band spectrum of the considerable attention.1 The photonic crystals may be di- prophase, suggested by simple symmetry analysis. Introduc- vided by their application either as photonic insulators or as tion and identification of the appropriate prophase is a crucial photonic conductors. In the first case, the most important step in this model, which differentiates this approach from property of the photonic structures is a band gap, where the nearly free electron model for the electronic band propagating modes for any magnitude and direction of the structure13 or with the symmetrical model of the photonic wave vector are forbidden for either specific or all crystals developed in Ref. 4 in which the starting point is the polarizations.2 The most important feature of the photonic band spectrum of the free electrons or free photons. In the conductors, such as those exhibiting the superprism effect, is present approach, the plane-wave functions for the prophase the possibility of tuning the opening of the band gap at some states are used as the basis. The band spectrum of the pertur- low-symmetry points of the Brillouin zone.3 Thus, under- bative phase is then obtained as a perturbation of the plane- standing the behavior of defect-free photonic crystals and wave spectrum of the prophase. designing devices based on them require a knowledge of the Two important classes of periodic lattice symmetries are wave propagation properties in the crystals.4 In the same way studied in photonic structures: the class of square lattices and as electron properties of electronic crystals are governed by their complex derivatives ͑containing fourfold or twofold ro- the solid’s band structure, the information about the photon tational symmetry axes͒,14,15 and the class of hexagonal lat- propagation properties is contained in the band structure and tices and their complex derivatives ͑containing sixfold- or eigenmodes of the dielectric periodic structure. threefold rotational symmetry axes͒.16,17 Reference 10 dealt A variety of methods have been used to calculate the pho- with the first class of square lattices. This paper deals with tonic band structure. All these approaches may be divided the second important class of hexagonal lattices and their into two groups. The first group comprises the so-called ab complex derivatives. The motivation of this work is two- initio models, implementation of which does not demand any step. First is that since the model depends critically on the empirical parameters. They are the plane-wave technique,5 presence and selection of a suitable prophase lattice, demon- transfer-matrix methods,6 and different types of numerical strating the presence of a suitable prophase for the complex schemes to solve Maxwell’s equations.7 A disadvantage of square lattices ͑as in Ref. 10͒ does not automatically imply the ab initio calculations is that they are very time and or naturally suggest a similar suitable prophase for the entire memory consuming. Another group of the photonic band- class of complex hexagonal lattices. In this paper, we show structure models consists of empirical models. For periodic that indeed such a prophase exists. The second goals is to dielectric structures, this group includes the tight-binding understand the common scheme of the evolution of the band model, first adopted for photonic crystals in Ref. 8. This spectrum for the whole class of the complex crystals based model contains some empirical parameters, coupled matrix on hexagonal and square symmetries within the framework elements, which have to be fitted to ab initio or experimental of one model. We compare and contrast these two important results. Tight-binding calculations of the band structure are crystal classes. very simple in their numerical implementation, giving an In particular, we consider a hexagonal lattice with alter- analytical solution. Recently, this scheme was also success- nating layers of the dielectric rods perpendicular to a selected fully used for various photonic structures.9 hexagon diagonal. The selection of this one direction in the In our previous paper10 a symmetrical model for the crystal will be shown to be of utmost importance in all band analysis of the band structure of the complex photonic struc- properties of these structures. We start from a simple hexa- tures has been developed. A similar theoretical approach has gon lattice with one rod in the basis, when all the layers are been suggested for electronic band structures of the complex equivalent. Then we simulate complex lattices by introduc- 0163-1829/2003/67͑12͒/125203͑9͒/$20.0067 125203-1 ©2003 The American Physical Society N. MALKOVA, S. KIM, T. DiLAZARO, AND V. GOPALAN PHYSICAL REVIEW B 67, 125203 ͑2003͒ ing two additional rods on the selected diagonal of the initial lattice. As a limit case when the radius of one of the addi- tional rods is equal to zero, we can derive the hexagonal comblike lattice considered first in Refs. 16 and 17. Thus we show that in the framework of the developed model we can systematically study the band structure of the entire class of complex lattices, including square, hexagonal, and their complex derivatives such as comblike lattices. From the symmetrical analysis, the opening of the band gap is shown to be predicted for the class of the layered photonic struc- tures. We state that these layered photonic crystals present a wide class of tunable photonic structures. The outline of this paper is as follows. A theoretical de- velopment of the model for the case of the complex hexago- nal lattices is presented in Sec. II. In Sec. III we compare our theoretical predictions with the plane-wave calculations of the photonic crystals in question. In Sec. IV we give conclu- sions, where discussion of the evolution of the band spec- trum for complex crystals with square and hexagonal sym- metries is presented. II. SYMMETRICAL MODEL FOR COMPLEX HEXAGONAL LATTICES We consider a two-dimensional dielectric system periodi- cal in the x-y plane and homogeneous along the z axis. The dielectric constant for the periodical system is a position de- pendent and periodic function of the vector r in the x-y ͑ ͒ plane, satisfying the relation ⑀(rϩR )ϭ⑀(r), where for any FIG. 1. Hexagonal lattice with two rods in basis a and its l Brillouin zone ͑b͒. The unit cells of the simple hexagonal prophase integers l , R ϭl a ϩl a defines a two-dimensional Bra- 1,2 l 1 1 2 2 and perturbative phase are shown by dashed and bold lines, respec- vais lattice with the unit cell constructed on the primitive ϭ͉ ϫ ͉ tively. translation vectors a1 and a2 . The magnitude A a1 a2 gives the area of a primitive unit cell of this lattice. In the framework of the plane-wave method, the band- We study the complex hexagonal lattice, shown in Fig. structure problem for the two-dimensional lattice is reduced 1͑a͒. This lattice is obtained from the simple hexagonal lat- to the eigenvalue problem4 tice by placing equidistantly, two different rods on one of the diagonals of hexagon. We first note that this is a layer lattice ␻2 with alternating layers of the different rods perpendicular to detͩ HϪ ͪ ϭ0, ͑1͒ 2 the selected diagonal. The unit cell is defined by the vectors c ϭ ϭ ͱ ͑ ͒ a1 a(1,0) and a2 a(1/2, 3/2), shown in Fig. 1 a by bold where lines. The point group symmetry of this lattice is C3 . The most important point is that this lattice can be represented as H͑G, GЈ͒ϭ͉kϩG͉͉kϩGЈ͉␩͑GϪGЈ͒ three embedded identical hexagonal sublattices, determined by the set of same primitive vectors a and a , but which are for the E polarization ͑electric field is parallel to the z axis͒ 1 2 shifted with respect to each other by the vector ␶ and ϭa/2(1,1/ͱ3). These sublattices are shown in Fig. 1͑a͒ by H͑G,GЈ͒ϭ͑kϩG͒ ͑kϩGЈ͒␩͑GϪGЈ͒ dashed-dotted lines. The inverse dielectric constant of such a • lattice can be expressed as for the H polarization ͑magnetic field is parallel to the z axis͒. The function ␩(G) is defined as Fourier transforms of the inverse dielectric constant 1 ϩ␶ ϭ͚ ͓␩1͑G͒eiG•rϩ␩2͑G͒eiG•(r ) ⑀͑r͒ G 1 1 Ϫ ␩͑ ͒ϭ ͵ iG•r ͑ ͒ G e dr, 2 3 iG (rϩ2␶) A uc⑀͑r͒ ϩ␩ ͑G͒e • ͔. ͑3͒ where the integral is taken over the unit cell with an area A.
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