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Journal of the Optical Society of America

Nonideal phase-conjugate resonators—a canonical operator analysis

Moshe Nazarathy, Joseph Shamir, Amos A. Hardy

Published May 1, 1983• 9 citations
Publisher

Abstract

An unfolded equivalent model is constructed for the round-trip transfer operator of a phase-conjugate mirror (PCM) resonator with a cavity described by complex ABCD ray matrix elements, taking into account hard-edge mirror apertures and the possible frequency shift of the wave reflected by the PCM. The treatment makes use of recently developed canonical operator techniques and extends these methods further to incorporate wavelength variation and the effect of apertures.

DOI
10.1364/josa.73.000587

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An unfolded equivalent model is constructed for the round-trip transfer operator of a phase-conjugate mirror (PCM) resonator with a cavity described by complex ABCD ray matrix elements, taking into account hard-edge mirror apertures and the possible frequency shift of the wave reflected by the PCM. The treatment makes use of recently developed canonical operator techniques and extends these methods further to incorporate wavelength variation and the effect of apertures.
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.
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.
We have developed a theory for nondegenerate oscillations in optical resonators containing an intracavity phase-conjugate element. The phase-conjugate element consists of a nonlinear transparent medium that is pumped externally by a pair of counterpropagating laser beams of the same frequency and intensity. Phase conjugation of an input beam of slightly different frequency occurs because of nondegenerate four-wave mixing. The theory takes into account linear absorption (or gain) in the medium and is applied to study the threshold behavior of phase-conjugate oscillators. For the special case of no conventional mirrors, the phase-conjugate oscillator reduces to an ordinary phase-conjugate mirror, and our general formulation yields the results of previous studies. Our analysis shows that the parametric gain required for oscillation increases (or decreases) as a result of linear absorption (or gain) in the medium, and oscillation can occur at a frequency different from that of the pump beams in the presence of large linear gain (or loss). The effects of linear absorption (or gain) on the filter operation are also examined.
The fidelity of the image beam produced by a phase conjugate mirror (PCM) is less when pump beams are not plane waves. On the other hand, high fidelity is a desirable property for most proposed applications, while energy or power should also be economized. We show that PCM fidelity may be characterized in terms of eigenfunctions—and values—of the reflection operator, and optimization may be achieved through use of an appropriate phase filter.1,2 In the present view, the filter has the role of transforming the incident object field into a (near) eigenfunction of the reflectivity operator. An equivalence theorem is presented which demonstrates parallels between the PCM and a certain kind of optical cavity. Perturbation theory applied in the small distortion case leads to an alternative approach to characterization and calculation technique.
The response of a phase-conjugate mirror (PCM) based on nondegenerate four-wave mixing to an incident photon wave packet has been calculated. In the monochromatic limit the phase-compensating property of the PCM, which makes the response independent of optical path length, is recovered. However, for a photon wave packet of nonzero bandwidth, the response of the PCM varies in a fairly complicated manner with the wave-packet length. In particular, the length of the pulse reflected from the PCM becomes independent of the length of the incident pulse for sufficiently short pulses.