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Journal of the Optical Society of America B

Transverse dynamics of a phase-conjugate resonator I: sluggish nonlinear medium

G. Reiner, Pierre Meystre, Ewan M. Wright

Published May 1, 1987• 11 citations
Publisher

Abstract

We present a fully nonlinear theory of a phase-conjugate resonator in which the phase-conjugate mirror is a diffusive Kerr medium with an arbitrary ratio of cavity round-trip time to medium response time. The model is restricted to the Raman–Nath scattering regime but includes pump depletion and transverse coupling, which is provided by diffraction inside the cavity and diffusion inside the phase-conjugate mirror. A transverse linear-stability analysis of steady-state solutions is performed. The case of a sluggish medium is studied numerically in both low- and high-diffraction geometries.

DOI
10.1364/josab.4.000675

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We present a fully nonlinear theory of a phase-conjugate resonator in which the phase-conjugate mirror is a diffusive Kerr medium with an arbitrary ratio of cavity round-trip time to medium response time. The model is restricted to the Raman–Nath scattering regime but includes pump depletion and transverse coupling, which is provided by diffraction inside the cavity and diffusion inside the phase-conjugate mirror. A transverse linear-stability analysis of steady-state solutions is performed. The case of a sluggish medium is studied numerically in both low- and high-diffraction geometries.
The exact theory of degenerate four-wave mixing in a Kerr-like medium in the Bragg scattering regime is used to study the reflectivity of phase-conjugate mirrors pumped by intense laser beams. The analysis takes into account pump depletion, nonlinear phase modulation and the relaxation time of the nonlinearity. The steady-state response exhibits multivalued reflectivity, hysteresis and potential bistability; however, the transient response shows that at most one value of the phase-conjugate reflectivity is stable. Periodic or chaotic pulsations are predicted beyond threshold values of pump intensity. The onset of a natural oscillation in the Kerr-like medium is also predicted in the absence of a continuous probe beam; the resulting field amplitude is stable and weakly sensitive upon the presence of external noise.
The phase-conjugate reflectivity produced by degenerate four-wave mixing in a Kerr medium in the Bragg scattering regime is shown to be unstable when the intensity of the external pump beams is greater than a threshold value that can be predicted analytically. Below this threshold, a single branch of the reflectivity curve is stable, eliminating any possibility for bistability. A numerical simulation indicates that the unstable regime is characterized by fast oscillations with a period related to the transit time in the nonlinear medium. A finite relaxation time of the nonlinearity eliminates the high-frequency oscillations but can increase the amplitude of the low-frequency components. The insertion of a phase-conjugate mirror into an optical cavity leads to an instability whose threshold can be evaluated analytically. The temporal evolution of the field intensity in the cavity is shown to change from a stable regime to periodic oscillations and then to quasi-periodic oscillations as the pump intensity is increased.
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.
A system for phase conjugation of Stokes radiation by a four-wave interaction involving intracavity stimulated Raman scattering (STRS) in a Raman-active medium with a nonresonant nonlinearity. This method can be used in construction of high-power sources of highly directional Stokes radiation on the basis of round-trip STRS amplifiers coupled to a phase-conjugating element. It is shown that an STRS converter utilizing organic liquids and subjected to laser pumping of less than 100 MW/cm2 intensity inside a resonator can ensure a highly efficient phase conjugation with a peak reflection coefficient in excess of unity. Depending on the ratio of the Raman and nonresonant susceptibilities of a medium, there is an optimal value of the coefficient representing the degree of conversion of laser pump radiation into STRS radiation, which ensures the maximum phase conjugation efficiency.
Transverse effects are known to play a central role in nonlinear-optical systems.Since experiments are generally performed with lasers with finite transverse dimensions, the transverse nature of the field must be considered.Experimental evidence, for example, indicates that mechanisms in the form of diffusive and diffractive coupling within the medium, free-space diffraction, or some combination of these, can create transverse structures which are important in optical bistability and self-focusing/defocusing.In particular, transverse coupling is attributed to such phenomena as radial dependence of the switch up times of bistable loops, spatial hysteresis, and the formation of solitary waves in passive systems.The dynamical effects of free-space diffraction in a unidirectional ring cavity containing a thin resonant absorber driven by an external coherent signal is considered.Minimal focusing effects exist in this system by incorporating a thin medium whose width is small compared to the cavity length and Raleigh range.There is no restriction on the medium response time relative to the round trip time; i.e. the time-dependent polarization is not adiabatically eliminated from the equations of motion.Free-space diffraction effects are isolated and diffraction and diffusive coupling within the medium is negligible.In this model where free-space diffraction is the dominant mechanism for changing the field profile in the system, shifting, deformation, and radial variation of the bistable loop and spatial hysteresis is attributed to an inherent dispersion induced by the free-space diffraction.