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Electronics and Communications in Japan (Part II Electronics)

Angular dependence of phase conjugate intensity in forward four-wave mixing with Gaussian beam pumping

Kei Hagiwara, Atsushi Okamoto, Hiroyuki Kaneko, Kojiro Koyanagi, Kunihiro Sato

Published Apr 1, 1997• 0 citations

Abstract

With progress in phase conjugate optics, studies on forward four-wave mixing have been actively carried out. From the practical optical system design point of view, it is important to understand the angular dependence of phase conjugate intensity when pumping beams with limited spot size are used, especially with beam separation taken into consideration. In this paper, the angular dependence of phase conjugate intensity distribution is theoretically analyzed for the first time, in forward four-wave mixing when an optical Kerr medium is used. First, the phase conjugate intensity is calculated for a nonlinear medium thinner than the overlapping region of two pumping beams, where the intensity is relatively low. The results are used to calculate the output angle of the phase conjugate beam against the z-axis and the azimuth angle in the x-y plane. Then, a similar analysis is conducted for a medium sufficiently thick to cover the interaction region of two pumping beams. The differences between the above two cases are described and comparison with plane-wave pumping is made. Finally, the effects of spot size and curvature variation with Gaussian beam pumping on conjugate beam intensity are discussed for the case of a thick medium. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 2, 80(4): 67–75, 1997

DOI: 10.1002/(sici)1520-6432(199704)80:4<67::aid-ecjb8>3.0.co;2-3

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Electronics and Communications in Japan (Part II Electronics)•Jun 1, 2000•0 citations

With progress in phase conjugate optics, studies on forward four-wave mixing have been actively carried out. From the practical optical system design point of view, it is important to understand the angular dependence of phase conjugate intensity when pumping beams with limited spot size are used, especially with beam separation taken into consideration. In this paper, the angular dependence of phase conjugate intensity distribution is theoretically analyzed for the first time, in forward four-wave mixing when an optical Kerr medium is used. First, the phase conjugate intensity is calculated for a nonlinear medium thinner than the overlapping region of two pumping beams, where the intensity is relatively low. The results are used to calculate the output angle of the phase conjugate beam against the z-axis and the azimuth angle in the x-y plane. Then, a similar analysis is conducted for a medium sufficiently thick to cover the interaction region of two pumping beams. The differences between the above two cases are described and comparison with plane-wave pumping is made. Finally, the effects of spot size and curvature variation with Gaussian beam pumping on conjugate beam intensity are discussed for the case of a thick medium. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 2, 80(4): 67–75, 1997

We present in this paper a theoretical study of the angular dependence of phase-conjugate emission via resonant degenerate four-wave mixing (FWM) in Doppler-broadened gas media. The nonlinear optical polarization is calculated through an expansion of the atomic density matrix up to the third order in the incident fields. One finds that, when the incident frequency is resonant for a single-photon transition, both strength and lineshape of the phase-conjugate emission strongly depend on the angle 0 between standing pump wave and probe wave. The emission intensity, maximum at grazing incidences, decreases rapidly by several orders of magnitude when 0 increases. Simultaneously the emission linewidth increases from homogeneous to Doppler width. Similar effects are predicted for nearly-degenerate FWM (different pump and probe frequencies). In particular, one shows that the bandwidth of the equivalent optical reflection filter undergoes a Doppler-broadening linearly increasing with sin θ. On the opposite, none of the above effects is predicted in two-photon-resonant FWM which does not exhibit any angular dependence.

The effects of side diffractions, generated from the interaction of a weak probe beam and its phase conjugate with two strong counterpropagating pump waves, and the effect of various intensity-dependent self- and mutual-phase modulations on the intensities of the probe beam and its phase conjugate are analytically studied in the nonpump-depletion limit. The results show that the conditions for the existence of, and threshold pump intensity parameters for, the divergence in the intensities of the probe and other generated beams are highly dependent on the wave-mixing angles and are different from the usual four-wave mixing phase conjugation process where the diffractions are not taken into account.>

The phase-conjugate mirror realized by means of degenerate four-wave mixing in a Kerr-like medium is investigated using the angular spectrum representation of interacting fields. The analysed model is based on the nonlinear mixing of two infinite plane pump waves with the spatially limited probe and signal waves. Relations among harmonic components of the probe and signal fields are derived assuming non-depleted pumping and neglecting the evanescent waves. The influence of the injected signal field on the frequency spectrum of the generated phase-conjugate field is also examined.

Mutually pumped phase conjugation, a process by which two incident laser beams can mutually pump each other to generate the phase-conjugate replica of the other beam, is studied in a transparent Kerr medium with an electrostrictive nonlinearity. The nonlinear medium acts as a phase-conjugating mirror to both of the incident beams and generates their Stokes-shifted phase-conjugated outputs by the nearly degenerate four-wave mixing process. Our results show that this process will occur only if the coupling strength, defined by the product of the Kerr amplitude gain coefficient, the total input intensity, and the interaction length, is above the threshold value of 2. If the intensity of one of the input beams exceeds the threshold value and the second input beam is weaker than the first, then the weak beam is phase conjugated with a large reflectivity that has a maximum value of the intensity ratio of the strong to the weak input beam. For negligible absorption in the medium, a complete transfer of energy of both the incident beams into the phase-conjugate beams can occur for finite interaction lengths that are only slightly larger than the threshold value. Seeding the oscillation beams tends to lower the phase-conjugate power reflectivities, probably because of increased contribution from the competing two-wave mixing process.

Phase-conjugate beam fidelity is studied in degenerate four-wave mixing with spatially varying pump beams. The theory includes long-cell-length (or high-gain) effects, diffraction, and nonlinear phase variation, focusing of the probe beam, and, finally, losses. Relatively simple algebraic expressions for the reflectivity are given that cover the cases of collinear or near-collinear beam geometries. An easily calculated parameter is defined that characterizes phase-conjugate fidelity and is applied to study focusing effects. It is shown that focusing effects may be useful in greatly reducing the principal distortion of the phase-conjugate beam over a considerable range of variation of beam parameters.

When the nonlinear effect of the crystal is strong in four-wave mixing using a photorefractive crystal, the generated phase conjugate light takes multiple values. In this paper, a new method of controlling the bistable values of the phase conjugate reflectivity, that is, the ratio of the phase conjugate beam to the probe beam, via the erase beam is proposed for use in four-wave mixing using the fanning beam generated in the crystal in place of the forward pump beam. It is extremely difficult to solve analytically the problem of four-wave mixing including the fanning beam. By means of a model in which a part of the probe beam is used for fanning and acts as the forward pump beam in the interaction region, the previous method of analysis for four-wave mixing can be used. First, a steady-state analysis of the phase conjugate reflectivity is carried out. It is shown that the phase conjugate reflectivity has bistable values as function of the intensity of the erase beam. Next, by time response analysis, the intensity of the erase beam is varied and the bistable values of the phase conjugate reflectivity are controlled. Finally, the theory is compared with experiments. © 2000 Scripta Technica, Electron Comm Jpn Pt 2, 83(6): 8–14, 2000