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Journal of the Optical Society of America B

Degenerate four-wave mixing and beam amplification in a Kerr medium

T. H. Liu, I. C. Khoo

Published Feb 1, 1987• 0 citations
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Abstract

According to the theory of two-beam coupling in a thick Kerr medium, which is based on the firstorder coupled-wave method,1 the energy exchange between the two incident beams is forbidden because the phase shift between the intensity grating and refractive-index grating is zero in the Kerr media. The energy exchange must be achieved by artificially creating a phase shift, e.g., by moving the intensity grating with respect to the medium at a speed of the order of one fringe per response time of the medium.

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According to the theory of two-beam coupling in a thick Kerr medium, which is based on the firstorder coupled-wave method,1 the energy exchange between the two incident beams is forbidden because the phase shift between the intensity grating and refractive-index grating is zero in the Kerr media. The energy exchange must be achieved by artificially creating a phase shift, e.g., by moving the intensity grating with respect to the medium at a speed of the order of one fringe per response time of the medium.
The theory of probe beam amplification via degenerate optical wave mixing in a Kerr medium is presented. This theory includes consideration of the self-phase modulation, two-wave mixing, and multiwave mixing terms in the coupled-wave equations. Both analytical solutions, under the assumptions of a nondepleted pump beam and a transparent Kerr medium, and numerical solutions with these assumptions removed are presented. Our solutions show that under certain conditions, phase matching can be satisfied and a high probe beam gain is possible, even for a zero phase shift between the refractive index and the intensity gratings.
We have shown previously that two strong laser beams (with the same frequency) counterpropagating in a Kerr nonlinear medium with linear dispersion become un stable when their intensities exceed a certain threshold.1 We demonstrate here that the system exhibits large (theoretically unlimited) amplification of a probe beam when the two pumping beams have their intensities slightly below the threshold. This is in contrast to the conventional phase conjugation due to four-wave mixing in a nondispersive Kerr nonlinear medium. The mechanism of amplification in our system can be explained in terms of positive distributed feedback from the Kerr nonlinear index grating formed by two strong pumping laser beams in the presence of dispersion giving rise to a light-induced distributed-feedback resonator. This feedback is attributed to the fact that the weak beams with both signal frequency, ω + δ, and conjugate frequency, ω − δ, propagate with phase speed different (due to dispersion) from that of a pumping wave with frequency ω, which triggers the exponentially increasing energy exchange between weak and strong beams at the nonlinear index grating (this energy exchange is prohibited in nondispersive media). When the probe frequency approaches a frequency of one of the longitudinal modes of the light-induced distributed resonator, and the pumping intensity approaches the threshold of instability, the resonant amplification increases infinitely; for presently available laser intensity stabilities, amplification as large as 104 is feasible.
We consider degenerate four-wave mixing in a Kerr medium with a standard noncollinear geometry and the polarization directions of all the beams parallel to one another. The two pump beams, the probe beam, and the phase-conjugate beam are denoted by the subscripts 1,2,4 and 3, respectively. The nonlinear medium is placed between z = 0 and z = 1; the values of the intensities of the beams at z = 0 and 1 are denoted by Ii0 andIi1, where i = 1,2,3,4. Beams 2 and 4 are incident at z = 1; and beam 1 is incident at z = 0. We have determined the dependence of the phase-conjugate reflectivity R = I31/I41 and the pump transmission T = I11/I10 on the following parameters: the probe ratio, q=I41/(I10 + I21); the pump ratio, r = I21/I 1 0; the absorption coefficient–path length product, al; and the coupling constant of the medium, q. In the absence of nonlinear refraction, pump depletion, and linear absorption in the medium, R = tan2q. When all of these effects are taken into account, the maximum value of R is given by Rmax = exp (-2α1)/(q(1 + r)). The minimum value that T can reach is proportional to q for q < 1. Even for small values of q, multistable solutions for R and T are obtained when q is greater than a threshold value, which we denote by qth. We have determined the dependence of qth on the parameters q and r.
Nonlinear optics normally involves the change in the susceptibility of a material due to the presence of optical radiation. In many cases, the light-matter interaction produces a change in the refractive index of the medium, as occurs in photorefractive media, as well as in Kerr-like media, Brillouin-active media and absorbing media with thermallyinduced refractive index changes. Such nonlinear refractive index variation corresponds to a change in the real part of the optical susceptibility %. In resonant materials, the optical radiation induces stimulated emission (or absorption) causing transfer of population between different energy states of the medium. This can lead to saturation of the amplification (or absorption) corresponding to changes in the imaginary part of the susceptibility x= i2odk where the amplitude gain coefficient a(I) is a saturable function of the intensity /, and k is the magnitude of the wavevector of the optical radiation. The interference of two (or more) coherent beams I(x) in the laser amplifier can cause a modulation of the population inversion N(x). This modulation of the inversion is commonly known as spatial hole burning in lasers and leads to formation of a gain-grating a(x) = 1/2GN(X) where G is the stimulated emission cross-section. The gain grating can act as a very efficient diffractive optical element exhibiting high diffraction efficiency that may be even greater than unity [16] and also very high phase conjugate reflectivity using four-wave mixing (FWM) geometries [5-7].