Optical Waveguide Modes
The optical waveguide is the fundamental element that interconnects the various devices of an optical integrated circuit, just as a metallic strip does in an electrical integrated circuit. However, unlike electrical current that flows through a metal stri
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The optical waveguide is the fundamental element that interconnects the various devices of an optical integrated circuit, just as a metall ic strip does in an electrical integrated circui t. However, unlike electrical current that flows through a metal strip according to Ohm's law, optical waves tra vel in the waveguide in distinct optical modes. A mode, in this sense, is a spatial distribution of optical energy in one or more dimensions. In this chapter, the concept of optical modes in a waveguiding structure is discussed qualitatively, and key results of waveguide theory are presented with minimal proof to give the reader a general understanding of the nature of light propagation in an optical waveguide. Then, in Chap. 3, a mathematically sound development ofwaveguide theory is given.
2.1 Modes in a Planar Waveguide Structure As shown in Fig. 2.1, a planar waveguide is characterized by parallel planar boundaries with respect to one (x) direction, but is infinite in extent in the lateral directions (z and y). Of course, because it is infinite in two dimensions, it cannot be a practical waveguide for optical integrated circuits, but is forms the basis for the analysis of practical waveguides of reetangular cross section. It has therefore been treated by a number of
o Fig.2.1. Diagram of the basic three-Iayer planar waveguide structure. Three modes are shown, representing distributions of electric field in the x direction
R. G. Hunsperger, Integrated Optics: Theory and Technology © Springer-Verlag Berlin Heidelberg 1982
2.1 Modes in a Planar Waveguide Structure
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authors, including Me Whorter [2.1], M cKenna [2.2], Tien [2.3], Marcuse [2.4], Taylor and Yariv [2.5] and Kogelnik [2.6]. In Sect. 2.1.1 we follow the approach of Taylor and Yariv [2.5] to examine the possible modes in a planar waveguide, without fully solving the wave equation.
2.1.1 Theoretical Description of the Modes of a Three-Layer Planar Waveguide To begin the discussion of optical modes, consider the simple three-Iayer planar waveguiding structure of Fig. 2.1. The layers are all assumed to be infinite in extent in the y and z directions, and layers 1 and 3 are also assumed to be semi-infinite in the x direction. Light waves are assumed to be propagating in the z direction. It has been stated previously that a mode is a spatial distribution of optical energy in one or more dimensions. An equivalent mathematical definition of a mode is that it is an electromagnetic field which is a solution ofMaxwell's wave equation (2.1.1) where Eis the electric field vector, r is the radius vector, n is the index of refraction, and c is the speed of light in a vacuum. The solutions of (2.1.1) have the form
E(r, t) = E(r) exp {i [cot - tp(r)]} ,
(2.1.2)
where ta is the radian frequency and tp is a phase function. Substituting (2.1.2) into (2.1.1) we obtain (2.1.3) where k == co/c. If we assume, for convenience, a uniform plane wave propagating in the z direction, i.e. tp(r) = ßz (where ß is a propagation constant), and there is no explicit z dependence o
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