Planar Leaky Light-Guides and Couplers

A quantitative theory of light propagation in a dielectric slab guide with general cladding media is presented. It is based on a plane wave which bounces in a zigzag fashion along the guide as a result of total or partial reflections at the two surfaces of the film. Two mechanisms are considered which contribute to the attenuation of the guide: losses due to absorption in the slab and cladding materials, and radiation losses if the guide is a leaky one. We point out the significance of the Goos-Hgmchen effect for all questions relating to the power flow in the slab guide. The theory is illustrated by discussing dispersion and attenuation of guides with various low-index and high-index claddings, operating above and below cutoff. The low-index leaky guide is considered particularly in detail. Its high attenuation by leakage can be reduced to practically acceptable values (< ldB/cm) by increasing the film thickness to > 402. One application of this guide is in the leaky wave coupler. This coupler may be viewed as a prism-film coupler simplified by omission of the gap. It offers a new approach to the problem of broad-band coupling to thin-film light guides. Index Headings: Integrated optics Thin films Goos-H~nchen effect The planar dielectric slab guide has become a model light guide for many experimental and theoretical studies relating to integrated optics [1-5]. It consists of a thin transparent, high-index film, sandwiched between two transparent low-index claddings. Propagation of light in this guide can be explained by assuming that a plane wave in the film undergoes total reflections alternatingly at the upper and lower surfaces of the film, and thus bouncing in a zigzag fashion along the guide [2, 5] (Fig. 1). This "classical" guide is a special case of a more general class of slab guides in which the relative magnitudes of the indices and of the losses in the three regions are all arbitrary. A discussion of this general class of guides is the subject of the present paper. The existence of freely propagating modes is not restricted to guides based on total reflection of the wave at the interfaces between the slab and the claddings. Partial reflection can produce guided modes, too. However, their propagation along the guide is attenuated. Although this is basically undesirable, such guides are of interest because their attenuation can be reduced to practically acceptable levels by sufficiently increasing the thickness of the film. A reasonable discrimination against higher modes is also possible in such "thick-film" guides because their higher modes have considerably higher attenuations. Two reasons for nontotal reflections will be considered here: absorption of light in the cladding materials, and refraction of light into the claddings. In the former case the guide is lossy; in the latter case it is leaky. A type of leaky guide that we shall discuss particularly is the planar equivalent of the hollow optical fiber: a low-index dielectric film sandwiched between two high-index claddings. The theoretical results on the attenuation of this guide have been confirmed experimentally. Other types of leaky guides are known in different contexts, e. g. as Lummer-Gehrcke plate or as prism-film coupler. Here they will be considered from the viewpoint of light-guiding. The low-index gap in the prism-film coupler, though advantageous for very thin guides, is not an absolute necessity. A gapless coupler can give the same 56 R. Ulrich and W. Prettl coupling efficiency, provided the film thickness can be adjusted to a suitable thick value (typically 3 gin). Gapless couplers may be useful for measuring the attenuation of low-loss guides. Freely Propagating Modes in a Slab For a quantitative description of light-guiding in a slab we make the following rather general assumptions: the slab of thickness W consists of a homogeneous material 1 of complex refractive index h l = n l + i t r 1. The "cladding" regions 0 and 2, adjacent to the slab (Fig. 1), are homogeneous in the plane parallel to the slab, but they may be of arbitrary composition in the normal direction within an interval Izl < D of an arbitrary, but finite width 2D, containing the slab. Outside this interval, the cladding regions must be homogeneous also in the normal direction. We characterize the regions 0 and 2 in the following only by their amplitude reflection coefficients qo and r12 at the boundaries to material 1. When we discuss specific types of slab guides, we have to insert specific functions for r,o and r~2, depending on the nature of regions 0 and 2. Let us take the mean direction of propagation as the x axis of our coordinate system. The z axis is normal to the slab. We consider only waves that are independent of the y direction, i.e. O/Oy = O. In the interior of the slab, the field is the superposition of two plane waves A and B (see Fig. 1). Their spatial amplitude distribution is written as V(x, z) = A exp {ik(f lx + ~z)} (1) + Bexp {ik([3x ~z)}. From the symmetry of the system it follows that we have two polarization eigenstates, T E and T M . In both cases, the amplitudes A, B, and V denote the field component transverse to the x z plane, which is the electrical field E =E~ for T E waves, or the magnetic field H--Hy for T M polarized waves 9 The remaining field components can then be derived by Maxwell's equations from V(x, z). A time dependence exp(-icot) of all fields with a constant frequency co is tacitly assumed throughout this paper. By k = co/c we denote the propagation constant of light in free space, so that fl and ~ = (n~ fiz)~/2 are the relative propagation constants along the slab and perpendicular to it, respectively 9 As fl is common to waves A and B and also [6] to the fields in regions 0 and 2, it describes the propagation of the entire field along the guide 9 Therefore, fl is the principal quantity characterizing the propagation along the guide 9 We allow for attenuation of the guided wave in the direction of propagation so that fl may be complex fi = N + i K . (2) Consequently, the waves A and B have plane phase fronts, but their amplitudes decay in x direction. The real part N of fi is related by vph = c /N to the phase velocity along the guide, i.e. N is the effective refractive index [4, 7] of the guide. The imaginary part K is the attenuation constant of the wave amplitude. The power decays as I(x) / I(O)= e x p ( 2 k K x ) , and the attenuation constant ~ in practical units is given by ~--8.686 kK decibels per unit length 9 Our goal now is to find the possible values of N and K. We obtain a condition for the free propagation of light along the slab guide from the requirement that, after completing one zigzag, the plane wave (A' in Fig. 1) must be identical with the original wave (A in Fig. 1) so that the field is a unique function of space. We write A' = FA, where the factor F accounts for the reflections at the two film surfaces and for two traversals of the film thickness 9 We find the condition