Page 206 - Fiber Bragg Gratings

4.8 Grating simulation 183

so that the crossed state amplitude is

and the uncoupled amplitude,

Following the solutions for the codirectional coupled-mode Eqs. (4.4.9)
and (4.4.10), the T-matrix elements for thejth section are

Equations (4.8.18-21) complete the analysis for guided-mode interac-
tions.

Phase shifts within a grating

It is often useful and necessary to incorporate phase jumps within a
distributed grating structure. The phase jump opens up a bandgap within
the reflection bandwidth, creating a narrow transmission band. This pro-
cedure has been applied to distributed feedback (DFB) lasers to allow
stable single-mode operation [59]. A phase shift is accomplished in the T-
matrix by multiplying the reflectivity of the./th section by matrix elements
containing only phase terms. On this basis, the transfer matrix takes on
the form