Analytic description of a homogeneously broadened injection laser

Abstract

Analytic expressions for modeling the steady-state spectral output of homogeneously broadened lasers are presented. The expressions define the laser output in terms of the pumping rate, resonator reflectivities, arbitrary gain, spontaneous emission, and saturation profiles for each mode. To derive the analytic expressions the laser is idealized as a homogeneously broadened amplifier contained by a plane-parallel Fabry-Perot resonator. Rate equations are used to describe the interaction of the laser-mode intensities and population inversion over an infinitesimal section of the amplifier and differential equations describing the amplification of the modes as they pass through the gain section are deduced. Exact analytic solutions to these differential equations are found. These solutions, along with the boundary conditions imposed by the resonator, are used to specify completely the output of the laser. The model predicts a single-pass intensity gain which increases exponentially with pumping below threshold and saturates above threshold. An interesting result is that for a laser which is symmetric, an analysis which takes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I^{+} = I^{-} = I</tex> , whereI <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">±</sup> are intensities traveling in the ± direction, gives results identical to an analysis which explicitly considers both <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I^{+}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I^{-}</tex> , irrespective of the resonator mirror reflectivities. The predictions of the model are compared to the spectral output and single-pass intensity gain of a GaAs diode laser and found to be in good agreement. The laser model may be generalized in a straightforward manner to include feedback, compound cavities, and optical amplifiers.