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Journal of the Optical Society of America B

Dynamics of Fabry–Perot resonators with a phase-conjugate mirror

Ari T. Friberg, Martti Olavi Kauranen, Rainer Salomaa

Published Dec 1, 1986• 9 citations
Publisher

Abstract

We study the dynamics of a stable Fabry–Perot resonator in which one mirror is a phase-conjugate mirror (PCM) with instantaneous response. The temporal variations in the model are caused by round-trip effects. An exact solution is given in a form of an infinite series. The step-function response is summed exactly. Numerical studies with Gaussian input pulses and PCM reflectivity variations are presented, and their behavior is explained. Approximate formulas for responses to slowly varying pulses are derived. Noise effects are simulated with a model in which the phases of the probe and the amplitude reflection coefficient of the PCM make uncorrelated jumps with Poissonian statistics. The theoretical model is aimed for optimizing experiments on phase-conjugate resonators involving transients and fluctuating fields.

DOI
10.1364/josab.3.001656

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Phase conjugate resonators (PCR) are optical resonators in which (at least) one of the mirrors is a phase-conjugate one (PCM), e.g. a four-wave mixing cell. These systems are of considerable experimental and theoretical interest, specially in view of their potential to eliminate phase distorsions within the resonator1. In recent work2, we discussed a PCR consisting of a Fabry-Perot resonator with one end mirror replaced by a lossless Kerr medium of infinitely fast response time pumped by two counterpropagating beams of same frequency. We described the PCM within the Raman-Nath approximation3, but including fully the effects of pump depletion. The normal mirror was positioned such as to couple the first order scattered field back into the PCM. We found that this system can break into self-oscillations corresponding to the half-axial modes predicted by linear theories1,4. For higher pump intensities, this is followed by a sequence of bifurcations to chaos, with in general the coexistence of several basins of attraction. We also showed5 that the first of this sequence of bifurcations is in general already sufficient to induce a quasi-periodic motion of the intracavity field phase, leading in practice to its complete indetermination.

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We study the dynamics of a stable Fabry–Perot resonator in which one mirror is a phase-conjugate mirror (PCM) with instantaneous response. The temporal variations in the model are caused by round-trip effects. An exact solution is given in a form of an infinite series. The step-function response is summed exactly. Numerical studies with Gaussian input pulses and PCM reflectivity variations are presented, and their behavior is explained. Approximate formulas for responses to slowly varying pulses are derived. Noise effects are simulated with a model in which the phases of the probe and the amplitude reflection coefficient of the PCM make uncorrelated jumps with Poissonian statistics. The theoretical model is aimed for optimizing experiments on phase-conjugate resonators involving transients and fluctuating fields.
This paper is the second in a series describing propagation of electromagnetic radiation in phase-conjugate oscillators. We apply the formulation developed in the first paper to study the phase-conjugate resonator (PCR) and the Fabry–Perot cavity with an intracavity phase-conjugate mirror (PCM). Our results show that nondegenerate oscillation occurs in the PCR for large parametric gains and fixed separation between the conventional reflector and the PCM. The bandwidth is considerably reduced from that of a PCM alone. In the phase-conjugate oscillator, oscillation will occur at the pump frequency only if there is no separation between the conventional mirrors and the phase-conjugate element and if the nonlinear medium fills the entire Fabry–Perot cavity.
An analytical method is derived for the evaluation of the average conjugate and specular reflections of a probe field from a phase-conjugate resonator. Statistical fluctuations described by Poissonian sudden jumps may simultaneously occur in the amplitude and phase of both the probe field and the complex reflectivity of the phase-conjugate mirror. In the possible noise scenarios any mixture of phase and amplitude jump probabilities can be included. The resonator is assumed to stay in the stable regime. It is shown that phase fluctuations have the strongest influence on the resonator performance. Amplitude noise is important only if jumps are infrequent and if the resonator operates close to the instability threshold or if the phase-conjugate-mirror reflectivity is approximately unity.
The spectra, coherence properties, and intensity fluctuations of waves generated by reflections of a noisy probe field from a phase-conjugate Fabry-Perot resonator are discussed. The emphasis of the analysis is on the spectral response of the resonator. The results obtained are in fill agreement with those derived using specific noise models in the time domain. Advantages of the frequency-domain analyses include full tractability of general probe-wave noise models and fast convergence of numerical computations. The reflected-wave coherence properties reveal an interesting periodic structure due to multiple intracavity reflections. The intensity fluctuation levels, which depend on the statistical models, are calculated explicitly for pure Markovian phase noise and Gaussian amplitude noise of the probe. Combinations of these two, yielding simple laser models, and other extensions of noise effects in phase-conjugate resonators are briefly discussed.>
An analytical method is derived for the evaluation of the average conjugate and specular reflections of a probe field from a phase-conjugate resonator. Statistical fluctuations described by Poissonian sudden jumps may simultaneously occur in the amplitude and phase of both the probe field and the complex reflectivity of the phase-conjugate mirror. In the possible noise scenarios any mixture of phase and amplitude jump probabilities can be included. The resonator is assumed to stay in the stable regime. It is shown that phase fluctuations have the strongest influence on the resonator performance. Amplitude noise is important only if jumps are infrequent and if the resonator operates close to the instability threshold or if the phase-conjugate-mirror reflectivity is approximately unity.
Phase conjugate resonators (PCR) are optical resonators in which (at least) one of the mirrors is a phase-conjugate one (PCM), e.g. a four-wave mixing cell. These systems are of considerable experimental and theoretical interest, specially in view of their potential to eliminate phase distorsions within the resonator1. In recent work2, we discussed a PCR consisting of a Fabry-Perot resonator with one end mirror replaced by a lossless Kerr medium of infinitely fast response time pumped by two counterpropagating beams of same frequency. We described the PCM within the Raman-Nath approximation3, but including fully the effects of pump depletion. The normal mirror was positioned such as to couple the first order scattered field back into the PCM. We found that this system can break into self-oscillations corresponding to the half-axial modes predicted by linear theories1,4. For higher pump intensities, this is followed by a sequence of bifurcations to chaos, with in general the coexistence of several basins of attraction. We also showed5 that the first of this sequence of bifurcations is in general already sufficient to induce a quasi-periodic motion of the intracavity field phase, leading in practice to its complete indetermination.