Surface EM waves on 1D Photonic Crystals

Photonic crystals are artificial low-loss dielectric structures with periodic modulation of refractive index, which have attracted considerable attention in the last two decades. Due to Bragg reflection, electromagnetic (optical) waves cannot propagate through such structures in certain directions, at certain frequencies. Hence, photonic crystals can control the propagation of electromagnetic waves in novel ways, with obvious application to dielectric mirrors, dielectric waveguides, and dielectric laser cavities. As a way to efficiently inject light into a photonic crystal (PC) waveguide, it has recently been proposed to use surface electromagnetic waves (SEW)[1, 2]. In those papers, the photonic crystal was a two dimensional array of rods, of infinite length normal to the plane of incidence. Instead, we have studied SEW on a semi-infinite one-dimensional (1D) photonic crystal sketched in Fig. 1. While retaining all useful properties of 2D and 3D photonic crystals, a 1D dielectric periodic structure with high refractive index contrast is more attractive from a technological point of view. The usual theoretical methods for wave propagation in 1D photonic crystals, including SEW , are the FloquetBloch modal formalism, coupled wave theory, and the transfer matrix method. Among these three, the coupled wave approach[3, 4, 5] offers superior physical insight and gives simple analytical results in limiting cases. Unfortunately, the conventional coupled wave theory of Kogelnik fails in the case of high refractive index contrast, which is essential for a functional 1D photonic crystal. In this paper, we apply our recently developed semiclassical version of coupled wave theory[6, 7, 8] to SEW on 1D photonic crystals. The method is analytically almost as simple as conventional coupled wave theory, and is practically exact for the achievable ratios (e.g. 1.5:4.6) of the indices of refraction of the materials available to build devices. We present here a unified description of TE and TM SEW. A detailed account of the properties of the TE surface modes has recently been given by us in Ref. [9]; here we complement these findings with those for TM modes, which are slightly more complex due to the presence of Brewster points in the bandgaps. As a result, we thoroughly clarify the systematics of solutions for surface EM waves in semi-infinite 1D photonic crystals. Our method is formally quite different from that recently presented in Ref. [10], or those in Ref. [11], so in Section II we provide a short summary of the transfer matrix approach, in the notation of our previous work[7]. In Section III we rederive the exact equations for SEW of TM modes and obtain from them various analytic approximations for a semi-infinite crystal. The analogous equations for TE modes were given in Ref. [9]. With these in hand, we discuss systematics of SEW. In Section IV we apply the semiclassical approximations of Refs. [7] and [8] to surface waves, and show that the second approximation is very accurate both for the dispersion relation and the bandgap boundaries.