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Journal of the Optical Society of America B

Fresnel coefficients for a phase conjugator

Henk F. Arnoldus, Thomas F. George

Published Jan 1, 1989• 1 citation
Publisher

Abstract

Optical phase conjugation by four-wave mixing is examined in detail. The Fresnel coefficients for reflection and transmission of a plane wave irradiating the surface of the phase conjugator are calculated. It appears that for normal incidence and a weak nonlinear interaction the device produces the phase-conjugated beam with respect to the incoming beam. For finite angles of incidence or stronger nonlinearities in the crystal, the generated wave deviates from the ideal conjugated wave, and also a second, specularly reflected, wave appears.

DOI
10.1364/josab.6.000030

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Optical phase conjugation by four-wave mixing is examined in detail. The Fresnel coefficients for reflection and transmission of a plane wave irradiating the surface of the phase conjugator are calculated. It appears that for normal incidence and a weak nonlinear interaction the device produces the phase-conjugated beam with respect to the incoming beam. For finite angles of incidence or stronger nonlinearities in the crystal, the generated wave deviates from the ideal conjugated wave, and also a second, specularly reflected, wave appears.
Reflection of travelling and evanescent plane waves by a four-wave mixing phase-conjugator is studied in detail. No restrictions are imposed on the nonlinear interaction strength, the angle of incidence or the frequency mismatch between the pump-beams and the incoming waves. We only assume that the incident field is weak compared to the pump fields, which justifies a classical field description of the pumps. The wave-vectors, amplitudes and phases for the various waves are evaluated, without the slowly-varying amplitude approximation. Familiar phase-matching resonances for certain values of the interaction length are recovered, and in addition strong resonances are found if the angle of incidence is finite and the incident light is not in perfect resonance with the pumps. The latter resonances appear at the transitions from a travelling to an evanescent wave. The significance of finite angles of incidence and evanescent waves for spectroscopic applications is pointed out.
Optical phase conjugation by four-wave mixing in a Kerr medium is considered. Two strong counterpropagating lasers pump a nonlinear medium in which they excite the third-order polarization. Mixing with a weak incident field then generates the phase-conjugated image of that field. It is shown how the polarization of the pumps and the tensorial nature of the nonlinear interaction can be accounted for by a geometrical polarization tensor. The electric field is shown to obey two coupled wave equations, which couple a positive- and negative-frequency component of the field. These wave equations allow plane-wave solutions, and we derive the dispersion relation for the wave vectors of these modes. By matching the field of such a mode across the boundaries of the medium to an external incident field, we are able to obtain analytically the Fresnel coefficients for reflection and transmission. Our expressions reduce to earlier results in the appropriate limits.
Abstract Reflection of travelling and evanescent plane waves by a four-wave mixing phase-conjugator is studied in detail. No restrictions are imposed on the nonlinear interaction strength, the angle of incidence or the frequency mismatch between the pump-beams and the incoming waves. We only assume that the incident field is weak compared to the pump fields, which justifies a classical field description of the pumps. The wave-vectors, amplitudes and phases for the various waves are evaluated, without the slowly-varying amplitude approximation. Familiar phase-matching resonances for certain values of the interaction length are recovered, and in addition strong resonances are found if the angle of incidence is finite and the incident light is not in perfect resonance with the pumps. The latter resonances appear at the transitions from a travelling to an evanescent wave. The significance of finite angles of incidence and evanescent waves for spectroscopic applications is pointed out.
We present a plane-wave analysis of a recently demonstrated self-pumped phase conjugator. This device uses four-wave mixing to produce the phase-conjugate replica of an incident optical wave. All the waves are derived from the single incident wave: there are no externally supplied pumping beams. We consider the case of four-wave mixing in two interaction regions coupled by simple reflection. We calculate the phase-conjugate reflectivity as a function of coupling strength, taking into account imperfect coupling between the two interaction regions, and show that there is a threshold coupling strength below which the reflectivity is zero and above which the reflectivity is multiple valued. We also compute the coupling strength per unit length for a photorefractive crystal of barium titanate.
We describe an all-optical method to transform a weak incident optical beam with constant amplitude and pulsated phase jumps into an amplified optical beam with amplitude modulation. The proposed method uses either the reflected or the transmitted beams by a phase-conjugate mirror. Our analysis assumes that the operation of optical phase conjugation is due to nearly degenerate four-wave mixing where two monochromatic pump fields are mixed in a Kerr-like material with the incident field, producing a transmitted field and a "phase-conjugate" reflected field. Numerical results show that the depth of amplitude modulation decreases with the gain of the phase-conjugate mirror, while the duration of the amplitude pulses increases along with that gain. An original configuration with a single output beam is proposed. The equivalence of the device with coherent detection schemes is discussed.
Optical phase conjugation is concerned with the use of nonlinear optical effects to precisely reverse the direction of propagation of each plane wave in an arbitrary beam of light. This causes the return beam to exactly retrace the path of the incident beam, leading to many applications of automatic pointing and tracking (even in the presence of an inhomogeneous medium). The main goals of this course are familiarization with how each of the many nonlinear optical effects gives rise to a phase conjugate signal, and understanding and appreciation of the many potential applications for optical phase conjugation. Conjugation techniques involving Kerr-like degenerate four-wave mixing, resonant degenerate four-wave mixing, stimulated Brillouin scattering, photorefractive effects, surface phenomena, and others will be carefully reviewed. Applications of particular importance to this course will include distortion correction, pointing, tracking, laser-resonator design, optical information processing, etc. Throughout the course, key references will be identified for those wishing to pursue any topic more vigorously.