The spontaneous emission factor for lasers with gain induced waveguiding

The expression for the spontaneous emission factor for lasers with gain induced waveguiding has a factor
K
, called by Petermann "the astigmatism parameter." This factor has been invoked to explain spectral and dynamic characteristics of this class of lasers. We contend that the widely accepted form of the
K
factor is based on a derivation which is not appropriate for the typical laser situation where the spontaneous emission factor is much smaller than unity. An alternative derivation is presented which leads to a different form for the
K
factor, The new expression predicts much smaller values under conditions where the previous theory gave values large compared to unity. Petermann's form for the
K
factor is shown to be relevant to large gain linear amplifiers where the power output is amplified spontaneous emission noise. The expression for the power output has Petermann's value of
K
as a factor. The difference in the two situations is that in the laser oscillator the typical atom of interest couples a small portion of its incoherent spontaneous emission into the dominant mode, whereas in the amplifier only the atoms at the input end are important as sources and their output is converted to a greater degree into the dominant mode through the propagation process. In our analysis we use a classical model of radiating point dipoles in a continuous medium characterized by a complex permittivity. Since uncritical use of this model will lead to infinite radiation resistance we address the problem of its self-consistency.