Frequency dependence of a ring laser with backscattering

The coupled third-order equations of motion of the two-mode ring laser with spontaneous emission noise are investigated in the presence of detuning and backscattering. The steady-state solution is obtained without linearizing approximations, and it is shown to depend on backscattering only through the sum $R_1$+$R_2^{\mathrm{*}}$, where $R_1$ and $R_2$ are complex backscattering coefficients. Above threshold the effect of backscattering is found to be equivalent to incrementing the pump parameter of each laser mode by (1/2)‖$R_1$+$R_2^{\mathrm{*}}$‖. The variations of the mean light intensities of the two modes with detuning are calculated for several different pump parameters, from well below to well above threshold. The results are compared with direct measurements of a He:Ne ring laser, and good agreement is obtained over a wide range, with the adjustment of just a single scale constant.