Many-mode stability analysis in the mean-field limit for absorptive bistability

The eigenvalue equation for absorptive bistability in a ring cavity is analyzed in the mean-field limit. Perturbative solutions are obtained in two distinct limits: $\kappa \ll {\gamma }_{\parallel },{\gamma }_{\perp }$ and $\kappa \sim {\gamma }_{\parallel },{\gamma }_{\perp }$, where $\kappa$ and ${\gamma }_{\parallel }$,${\gamma }_{\perp }$ are field and atomic damping rates, respectively. For $\kappa \sim {\gamma }_{\parallel },{\gamma }_{\perp }$, the eigenvalues satisfy a cubic equation, derived previously in the theory of self-pulsing in absorptive bistability. For $\kappa \ll {\gamma }_{\parallel },{\gamma }_{\perp }$, while the "dressed-field-mode" eigenvalues are solutions to this cubic equation, the "dressed-atomic-mode" eigenvalues are not.