Ray optics determination of the DFB coupling coefficient for trapezoidal gratings

A ray optics technique is used to derive a closed-form expression for the first-order coupling coefficient of trapezoidal gratings in a generic five-layer separate confinement heterostructure distributed feedback (DFB) laser configuration for transverse electric (TE) mode. Expressions for a rectangular grating with the duty cycle of 0.5 and a symmetric triangular grating are also obtained as special cases of our general expressions. Based on the comparison of the coupling coefficient calculated by the ray optics technique with those calculated by the extended additional layer method, we show that the ray optics technique is very useful to calculate the coupling coefficient for most cases of realistic grating depths and profiles used in multilayer DFB devices without using any numerical techniques