Silicon Photonics: Fundamentals and Devices

10.5 Wave Theory of Light Guides

We now wish to develop the theory of optical waveguides and of the modes by using the EM wave equation.

10.5.1 Wave Equation in a Dielectric

Like all other EM phenomena, the propagation of EM waves along a dielectric waveguide is governed by Maxwell's equations, given by Eqs. (4.1a)—(4.1d). For most of the guiding materials, M = 0; further, the current density and free charge densities are assumed to be absent. The electric polarization is related to the electric field through a susceptibility χ, which is taken to be constant in most of the situations.

By taking curl of Eq. (4.1a) and using (4.1b)–(4.1d), the wave equation is obtained in the following form:

(10.25)

where is the relative permittivity of the medium. Now assuming that E varies sinusoidally as , we may write

(10.26)

where is the velocity of light in free space. In general, ε is complex and its real and imaginary parts are related to the refractive index n and absorption coefficient α (see ...