VOLUME 82, NUMBER 4 PHYSICAL REVIEW LETTERS 25 JANUARY 1999 Photonic Crystal Optics and Homogenization of 2D Periodic Composites P. Halevi Instituto de Astrofı ´sica, Optica y Electrónica, Apartado Postal 51, Puebla, Puebla 72000, México A. A. Krokhin and J. Arriaga Instituto de Fı ´sica, Universidad Autónoma de Puebla, Apartado Postal J-48, 72570, Puebla, México (Received 4 September 1998) We study the long-wavelength limit for an arbitrary photonic crystal (PC) of 2D periodicity. Light propagation is not restricted to the plane of periodicity. We proved that 2D PC’s are uniaxial or biaxial and derived compact, explicit formulas for the effective (“principal”) dielectric constants; these are plotted for silicon-air composites. This could facilitate the custom design of optical components for diverse spectral regions and applications. Our method of “homogenization” is not limited to optical properties, but is also valid for electrostatics, magnetostatics, dc conductivity, thermal conductivity, etc. Thus our results are applicable to inhomogeneous media where exact, explicit formulas are scarce. Our numerical method yields results with unprecedented accuracy, even for very large dielectric contrasts and filling fractions. [S0031-9007(98)08154-X] PACS numbers: 42.70.Qs, 41.20.Jb, 42.25.Lc Photonic crystals (PC’s) are arrays of dielectric ma- terials with one-, two-, or three-dimensional periodicity. Since the suggestion [1] that PC’s may be useful for controlling light emission, their properties have been re- searched intensively [2,3]. Recently, it was proposed that PC’s could advance photonic information technology [4– 6]. These ideas rely on the existence of a photonic band gap — a frequency region in which light propagation is for- bidden. The region well below the gap received much less attention [7–12]. Here the wavelength is much greater than the lattice period; hence light “sees” a homogeneous medium. This situation is analogous to light propagation in natural crystals, whose optical properties like birefrin- gence are described in crystal optics [13]. We studied analytically, for the first time, propagation in an arbitrary direction in space for a 2D PC. In general, a PC supports two distinct propagation modes. Independently of the existence of a band gap, for sufficiently low frequencies v, the dispersion relations v versus the vector of propagation k are linear for the two modes. The slopes vyk define two effective dielec- tric constants e eff › sck yvd 2 [7–12]. Thus, in this long- wavelength limit the composite may be treated as if it were homogeneous. At the same time e eff does depend on the direction of propagation skyk d, implying that the effective medium is anisotropic [7,9,10,12]. An electrostatic calcu- lation sv › 0d must yield the same values for the e eff as the previously described quasistatic approach sv ! 0d. In fact, the homogenization of composites [14] has been stud- ied for many years by means of both the static [15 – 23] and the quasistatic [7,24,25] methods. The homogenization of 1D structures has been accomplished a long time ago [23]. Solution for 3D photonic crystals was given in Ref. [7]. The object of our investigation is a periodic, 2D array of infinitely long cylinders. The cross section of a cylinder can have an arbitrary shape, and the unit cell is in general a parallelogram. These rods are assumed to be made of a homogeneous material (dielectric constant e a ), as is the interstitial material (e b ). The cylinders occupy a fraction f of space. For propagation of light parallel to the plane of periodicity there are two independent modes: the EsHd mode has its electric (magnetic) field parallel to the cylinders [2 – 4,26 – 28]. In a preliminary work we have examined the in-plane behavior of these modes in the low- frequency limit [12]. We also note that homogenization has been performed for the E mode in the case of metallic cylinders modeled with e a › 1 2v 2 p yv 2 [29]. In this Letter we achieve a homogenization of our photonic crystal for an arbitrary direction of propagation in space; namely, we take a 3D approach to composites of 2D periodicity. We are aware only of a single, numerical study (p. 66 of Ref. [2]) of out-of-plane propagation. Crystal optics [13] is the product of homogenization of the periodic atomic structure. In the same way photonic crystal optics is the result of homogenizing a periodic com- posite with macroscopic inhomogeneities. Hence it should be possible to give a complete description of this structure in terms of a dielectric tensor, which becomes diagonal in the principal set of axes, embedded in the crystal. In this system of coordinates the dielectric response is simply D i › e i E i si › 1, 2, 3d, the e i being the principal dielec- tric constants of the composite. Now there is a simple, but crucial, consideration that for the (in-plane) E mode the displacement vector D must be parallel to the cylin- ders at every point. Then the coordinate axis parallel to the cylinders (say, z ) must be a principal axis. Moreover, it is well known [14] that if E is parallel to all the dielec- tric interfaces, then e eff is equal to the weighted average of the dielectric constants of the constituents. This gives immediately one of the principal dielectric constants, e 3 › e › e a f 1e b s1 2 f d . (1) 0031-9007y 99 y 82(4) y 719(4)$15.00 © 1999 The American Physical Society 719