Angular photonic band gap

Materials and structures that strongly discriminate electromagnetic radiation based on one, or more of its properties (e.g. polarization, frequency) play an enabling role for a wide range of physical phenomena. For example, polarizers can selectively transmit light based on its polarization [1] over a wide range of frequencies; photonic crystals [2] (PhCs) can reflect light of certain frequencies irrespective of the angle of incidence and irrespective of the polarization. A material system that could transmit light based primarily on the angle of incidence might also enable a variety of novel physical phenomena. Light incident at a prescribed range of angles would be nearly perfectly transmitted, while light from other angles of incidence would be nearly perfectly reflected [Fig. 1(a)]. Ideally, such an angular selectivity would apply independent of the incoming polarization and over a broad range of frequencies. Structures with such strong angular selectivity do not currently exist. For example, PhCs exhibit some angular discrimination of light, but this discrimination is always strongly dependent on frequency, as illustrated in Fig. 1(b). In this Brief Report, we present a material system that opens the desired angular gaps, as shown in Fig. 1(c). For example, using realistic constituent material parameters, we present numerical calculations demonstrating an angular photonic band gap (PBG) material system in which light close to normal incidence is nearly perfectly transmitted for a wide range of frequencies, independent of the polarization. In contrast, light of angles further from the normal (e.g., 22.5◦–90◦) can be nearly perfectly reflected over >100% fractional frequency band gap. The key to these interesting angular PBG material systems lies in exploring PhCs whose constituents have anisotropic permittivity and/or permeability. To demonstrate the fundamental physics principle at work here, we first show in Fig. 2(a) an angular PBG material system that opens an angular gap for the TM (electric field is in the plane of incidence) polarization and a certain frequency range only. In this example, we take layer A to have an anisotropic effective permittivity εA = (1.23,1.23,2.43) whereas layer B has isotropic permitivity εB = 1.23. To accomplish the required anisotropy, one could either use naturally anisotropic materials [3] such as TiO2, or explore metamaterial approaches. An example of a metamaterial system that in the long-wavelength limit possesses an effective ε of (1.23,1.23,2.43) is shown to the left of the inset of Fig. 2(a) and consists of a two-dimensional (2D) periodic square lattice of dielectric rods having radius r = 0.2d where d (the in-plane period) is given by d = 0.1a (a is the thickness of each bilayer [4]), and made out of an isotropic material with εrods = 12.25 [5]. To the right of the inset of Fig. 2(a), we show a schematic diagram of normally incident TM(in blue) and TE(in red) polarized light incident on the above-described multilayer. Since E lies in the xy plane, both polarizations experience nA = nB = √ 1.23, so because of the absence of any contrast in the refractive index, there is no photonic band gap and normally incident light of all frequencies and both polarizations gets transmitted, apart from the small reflections at boundaries between the structure and air. As seen in Fig. 2(b), the situation changes with oblique incidence: TM-polarized light now has Ez = 0, and thus experiences an index contrast (nTM A = nTM B = √ 1.23); therefore, a photonic band gap opens, with strong reflections for TM light. In contrast, TE-polarized light incident at oblique angle still has Ez = 0 and thus experiences no index contrast as shown [Fig. 2(b): nA = nB = √ 1.23]; therefore, it gets transmitted for all frequencies. Thereby, TM light is transmitted for small θinc and reflected for large θinc. This structure works only for a certain frequency range [9.3% in the case of Fig. 2(b)]. Later in the Brief Report, we show how to generalize our approach to both polarizations and wide frequency ranges. It is useful to look at analytical expressions for the effective refractive index nA of the anisotropic layer. A simple calculation starting from Maxwell’s equations [6] yields the following refractive indices (n = c/vphase) experienced by TE and TM light, respectively, in the anisotropic layer A: