Effective-index matrix analysis of distributed feedback semiconductor lasers

A method is developed for calculating the threshold gain in distributed feedback (DFB) semiconductor lasers consisting of a four-layer waveguide. Each corrugation region is expressed by a transfer matrix comprising the complex effective index, and the threshold calculation is reduced to the matrix multiplication. It is found that a periodic variation of the effective gain and loss induced by the modulation of the optical confinement factors gives rise to the threshold gain difference between the two adjacent modes on both sides of the Bragg frequency in standard DFB lasers without facet reflections. The absorption loss in the guiding layer increases the amplitude of the gain modulation, resulting in the increase of the threshold gain difference