Analytic description of a homogeneously broadened injection laser: Extension to include a waveguide scattering/absorption loss

-An approximate analytic model of a homogeneously broadened injection laser is presented. The model is based on solutions to differential equations which describe the steady-state amplification of the modes for unidirectional propagation through a homogeneously broadened gain medium. The gain medium is characterized by independently adjustable gain, saturation, and spontaneous- emission profiles for each mode and a waveguide scattering/absorption loss a which is constant for all modes. The approximate nature of the laser model is due to the neglect of the counter propagating intensity. This beam had to be neglected in order to find analytic solutions to the coupled, non-linear differential equations. In the limit of
\alpha = 0
, the model is exact, thus one expects that the model is accurate for small values of α. To verify these expectations, the predictions of the analytic model are compared with results obtained from a numerical integration of the equations for bidirectional propagation through the amplifier. It is found that for
\alpha l \leq 4
(
l
is the length of the laser) the results of the analytic model agree with the results of the numerical integration to within ≃10 percent. There exist several qualities which make the material presented in this paper interesting. First, the analytic model is solved quickly and efficiently for numerical values which are accurate approximations to the true values. Second, the form of coupling of the spontaneous light into the modes for bidirectional propagation through the gain medium may be deduced from the analytic solutions to the unidirectional case. This means that the functional dependence of the spontaneous noise on α and the parameters which characterize the active medium and its effect on the division of the output power amongst the modes is determined and given in analytic form. Third, solutions for unidirectional propagation through a homogeneously broadened amplifying medium are given. These solutions should find application in the study of ring lasers, optical amplifiers, or any other structure where the backward propagating intensity is negligible in comparison to the forward traveling beam. Fourth, it is shown that results predicted by an
\alpha = 0
calculation may be scaled to be similar to results calculated for
\alpha l \lsim 4
.