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Annual Meeting Optical Society of America

Giant four-wave mixing amplification in Kerr nonlinear medium with dispersion

C. T. Law, A. E. Kaplan

Published Jan 1, 1989• 0 citations

Abstract

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.

DOI: 10.1364/oam.1989.mmm1

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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.

This paper presents an exact solution to the slowly varying envelope equations for the probe and signal fluxes in degenerate four-wave mixing in Kerr media. The analysis is performed for a noncollinear geometry, and all beams have the same linear polarization in the nonlinear medium. It is assumed that the angle separating the directions of the counterpropagating pumps and probe and signal beams is large enough so that cross-coupled waves do not contribute to the envelope equations describing the spatial evolution of the pump, probe, and signal beams. Multiple values of the transmitted probe are obtained for equal pump fluxes above a threshold determined by the nonlinear phase shift in the medium. Significant pump depletion occurs in this regime, but it can also occur without the appearance of multiple solutions. Since exact optical phase conjugation occurs only in the limit where the pumps are undepleted in Kerr media, this result implies that there are fundamental upper limits on the pump intensity that can be used to increase signal reflectivities in optical phase conjugation.

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 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.

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.

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].

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.