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Physical Review A

Singular perturbation analysis of the mean-field limit of semiclassical optics

Robert R. Snapp, William C. Schieve

Published Jan 1, 1990• 1 citation
Publisher

Abstract

A linear stability analysis of a plane-wave model of a unidirectional ring cavity, which allows the slowly varying field amplitudes to vary along the length of the atomic medium, yields a characteristic equation having singularities coincident with estimates of the atomic-dominant eigenvalues derived under the mean-field limit. This result calls into question the validity of previous estimates of these eigenvalues. Using this general characteristic equation, we estimate the cavity- and atom-dominant eigenvalues by applying a perturbation expansion in sequential powers of the uniform-field parameter \ensuremath{\eta}=2 ln\ensuremath{\Vert}scr\ifmmode \bar{E}\else \={E}\fi{}(L)/scr\ifmmode \bar{E}\else \={E}\fi{}(0)\ensuremath{\Vert}. The resulting eigenvalue estimates appear as asymptotic expansions with limited domains of validity. The cavity-dominant eigenvalues, which are responsible for the well-known ``self-pulsing'' instability in unidirectional ring lasers and optically bistable resonators, are shown to be consistent with estimates derived in previous studies. However, the new expressions for the atom-dominant eigenvalues exhibit a more intricate distribution in the complex plane.

DOI
10.1103/physreva.41.421

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A linear stability analysis of a plane-wave model of a unidirectional ring cavity, which allows the slowly varying field amplitudes to vary along the length of the atomic medium, yields a characteristic equation having singularities coincident with estimates of the atomic-dominant eigenvalues derived under the mean-field limit. This result calls into question the validity of previous estimates of these eigenvalues. Using this general characteristic equation, we estimate the cavity- and atom-dominant eigenvalues by applying a perturbation expansion in sequential powers of the uniform-field parameter \ensuremath{\eta}=2 ln\ensuremath{\Vert}scr\ifmmode \bar{E}\else ={E}\fi{}(L)/scr\ifmmode \bar{E}\else ={E}\fi{}(0)\ensuremath{\Vert}. The resulting eigenvalue estimates appear as asymptotic expansions with limited domains of validity. The cavity-dominant eigenvalues, which are responsible for the well-known ``self-pulsing'' instability in unidirectional ring lasers and optically bistable resonators, are shown to be consistent with estimates derived in previous studies. However, the new expressions for the atom-dominant eigenvalues exhibit a more intricate distribution in the complex plane.
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: $\ensuremath{\kappa}\ensuremath{\ll}{\ensuremath{\gamma}}_{\ensuremath{\parallel}},{\ensuremath{\gamma}}_{\ensuremath{\perp}}$ and $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}_{\ensuremath{\parallel}},{\ensuremath{\gamma}}_{\ensuremath{\perp}}$, where $\ensuremath{\kappa}$ and ${\ensuremath{\gamma}}_{\ensuremath{\parallel}}$,${\ensuremath{\gamma}}_{\ensuremath{\perp}}$ are field and atomic damping rates, respectively. For $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}_{\ensuremath{\parallel}},{\ensuremath{\gamma}}_{\ensuremath{\perp}}$, the eigenvalues satisfy a cubic equation, derived previously in the theory of self-pulsing in absorptive bistability. For $\ensuremath{\kappa}\ensuremath{\ll}{\ensuremath{\gamma}}_{\ensuremath{\parallel}},{\ensuremath{\gamma}}_{\ensuremath{\perp}}$, while the "dressed-field-mode" eigenvalues are solutions to this cubic equation, the "dressed-atomic-mode" eigenvalues are not.
A single-mode dynamic equation is obtained from the Maxwell equation with the slowly-varying amplitude approximation, and the boundary conditions of the cavity. Here the mean-field limit and the good-quality cavity case have been used. This equation is useful in describing the dynamic behavior of intrinsic optical bistable systems and/or lasers with injected signals for any kinds of materials, including two-level atoms. Furthermore, according to the linear stability analysis, a generalized instability criterion is derived.
Analytical traveling-wave self-pulsing solutions of a homogeneously broadened ring laser are presented in the limit that the dipole relaxation rate is much greater than the atomic inversion relaxation rate. The phase velocity v(\ensuremath{\Lambda}) as a function of the normalized pump \ensuremath{\Lambda} is given explicitly. This function has a simple relation with the upper boundary ${\ensuremath{\alpha}}_{\mathrm{max}(\mathrm{\ensuremath{\Lambda}}}$) of the Risken-Nummedal-Graham-Haken (RNGH) instability domain in the (\ensuremath{\alpha},\ensuremath{\Lambda}) plane, where \ensuremath{\alpha} is the wave vector of the perturbations. The self-pulsing may appear in different ways, depending on how the stationary solution becomes unstable. If the instability of the stationary solution is caused by an unstable mode touching upon the upper boundary of the RNGH-instability domain in the (\ensuremath{\alpha},\ensuremath{\Lambda}) plane, the self-pulsing is supercritical; if the unstable mode lies on the lower boundary, a subcritical self-pulsing arises, and the system is bistable. This rule provides new signatures of the self-pulsing phenomena and will be helpful to experimental identification of the self-pulsing arising from the RNGH instability. The linear stability analysis reveals two kinds of instabilities for these self-pulsing solutions. One kind of instability is rooted in the RNGH instability of the stationary solution and may occur even when the amplitude of the self-pulsing solution approaches zero. The other kind of instability occurs when the oscillating amplitude of the self-pulsing solution becomes large. Above this large-amplitude instability threshold no traveling-wave self-pulsing can be stable any longer. The results for the corresponding Lorenz model are also presented.
The coupling of an electric field to an atomic beam prepared in a phase-locked coherent superposition of energy eigenstates is examined for a ring-laser cavity. The active medium is assumed to be a collection of noninteracting homogeneously broadened two-level atoms whose transition is electric-dipole allowed. The atom-field interaction is treated semiclassically and the atomic coherence injected into the cavity is assumed to be phase matched to the unidirectional plane-wave cavity mode. The polarization is derived for the case in which the electric-field-mode envelope has at most linear phase variations in time and is applied to the problem of frequency locking in a cavity. The presence of the atomic coherence causes the active medium to possess an inherent polarization, i.e., a field-independent contribution. This enables an optical cavity to support laser oscillation over the entire range of relative excitations, in particular, even for strongly absorbing media. A simple linear stability analysis is performed on the electric-field envelope by describing the medium with a slowly varying polarization valid when the time scales of the atomic response are much smaller than those of the variations in the envelope. The stability of frequency-locked oscillation depends on the detuning of the cavity from line center and the strength of the injected coherence relative to that of the injected population difference.