Silicon Photonics: Fundamentals and Devices

11.2 Model for Mode Coupling

Electromagnetic energy propagating in one mode may be coupled into another mode with the introduction of a small perturbation in the RI. The purpose of this section is to develop a mathematical model for the mechanisms for the coupling [1, 6, 9].

Let us first consider a slab waveguide, the geometry of which is shown in Figure 11.1, in which an EM wave of angular frequency propagates along the z-direction. The RI distribution of the guide along the x-direction is denoted by n(x). The general solution to the wave equation in the structure is a superposition of different modes with propagation constants β_m and an electric field profile f_m(x), given as

(11.1)

The field profiles are assumed orthogonal, such that

(11.2)

where δ_mn is the Kronecker delta. In Eq. (11.1)E_m(z) is the z-dependent complex amplitude of the electric field of mode m.

Until now the different modes denoted by m have represented different independent solutions to the wave equation and there has been no interaction between these modes. Should there be any coupling between different modes, there must be present a perturbation of the RI distribution 9 of the following form:

(11.3)

where