Linewidth of gain-coupled distributed feedback lasers

A simple expression for the linewidth of vertical cavity, periodic gain, laser diodes is reported. Quantum wells, spaced half a wavelength apart are assumed to be driven by time-independent electrical currents J _k, and to possess distinct phase amplitude coupling factors α_k. The usual linewidth formula is found to be applicable if the photon lifetime is defined as the round trip time divided by a factor of 6, and the averaging of the α_k is made with J_k as a weighting factor. Furthermore, half the spatial variance of α must be subtracted. The expression involves cross terms of the form α_iα_j, i≠j