Assume the electric field is a linear combination of the ideal modes (with no grating

perturbation), such that

where *a _{i}^{(+)}* and

*a*are the slowly varying amplitudes of ith mode traveling in the +z

_{i}^{(-)}and –z directions. β

_{i}and is the propagation constant and modal field of the ith

mode.

The above electric field is used as trial solution in the Maxwell’s equation. The

following Coupled mode Equations (CMEs) can be derived by using the properties of

waveguide modes,

The coupling coefficient between modes *k* and *i* is given by:

where is the periodic refractive index perturbation of the grating,

and *n _{0}(x, y)* is the index profile of waveguide.

*n(x, y, z)*is the grating index profile.

In OptiGrating, the coupled mode equations are based on non-orthogonal coupled

mode theory. Both the waveguide nature coupling and grating coupling are

considered. In order to formulate the coupled mode equations, waveguide modal

constants, fields, and coupling coefficients are calculated based on waveguide and

grating profiles. The coupled mode equations are then solved by the two mode or by

the multi-mode coupling formulation.