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Canadian Journal of Physics

Semiclassical behavior of the degenerate four-wave mixing without slowly varying amplitude approximation

Mahmood Hosseini Farzad, Mohammad Taghi Tavassoly

Published Aug 1, 2003• 0 citations
Publisher

Abstract

In this study, a semiclassical version of the degenerate four-wave mixing (DFWM) solution without slowly varying amplitude (SVA) approximation is investigated. The quadrature fluctuations of the output modes exhibit an oscillatory behavior with respect to that of the corresponding SVA modes. Furthermore, the variation of the two quadrature fluctuations of the combined mode, plotted against the normalized coupling constant κL, oscillates. Beyond a certain value of κL, the two quadratures oscillate in opposite directions, which implies that the squeezing effect of DFWM without the SVA approximation oscillates around that of the SVA. In addition, the squeezing effect in DFWM without SVA approximation depends on the refractive index of the medium and increases sharply upon the increase of the latter. This study also shows that the super-Poissonian behavior of the conjugate mode is preserved but is oscillatory. PACS No.: 42.65.Hw

DOI
10.1139/p02-127

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In this study, a semiclassical version of the degenerate four-wave mixing (DFWM) solution without slowly varying amplitude (SVA) approximation is investigated. The quadrature fluctuations of the output modes exhibit an oscillatory behavior with respect to that of the corresponding SVA modes. Furthermore, the variation of the two quadrature fluctuations of the combined mode, plotted against the normalized coupling constant κL, oscillates. Beyond a certain value of κL, the two quadratures oscillate in opposite directions, which implies that the squeezing effect of DFWM without the SVA approximation oscillates around that of the SVA. In addition, the squeezing effect in DFWM without SVA approximation depends on the refractive index of the medium and increases sharply upon the increase of the latter. This study also shows that the super-Poissonian behavior of the conjugate mode is preserved but is oscillatory. PACS No.: 42.65.Hw
The coupled wave equations describing the optical phase conjugation by degenerate four wave mixing DFWM is usually, solved in the so called slowly varying amplitude approximation (SVA), where only the first derivatives of the amplitudes of the probe and of conjugate waves are taken into account. Here, we solve the coupled equations exactly for nonresonance ]interactions. This leads to an oscillatory dependence of reflectivity and of transmittance on the normalized coupling constant, κL. Also the intensity distributions of the probe and conjugate waves show oscillatory beat-like behaviors around the corresponding SWA distributions. The latter can be interpreted as the diffraction of the conjugate (probe) waves from the super-position of the pump-pump and pump-probe gratings.
The coupled-wave equations describing the optical phase conjugation by degenerate four-wave mixing are usually solved in the so-called slowly varying amplitude approximation, wherein only the first derivatives of the amplitudes of the probe and the conjugate waves are taken into account. Here we solve the coupled-wave equations without this approximation for nonresonance interactions. This leads to an oscillatory dependence of both reflectivity and transmittance on the normalized coupling constant, κL. Also, the intensity distributions of the probe and the conjugate waves show oscillatory beatlike behaviors around the corresponding slowly varying amplitude approximation distributions. These behaviors can be interpreted as the diffraction of the conjugate (probe) wave from a grating resulting from the superposition of the pump–pump and the pump–probe gratings.
The suitability of degenerate four-wave mixing (DFWM) as a source of squeezed-state (two-photon coherent state) light is investigated, both theoretically and experimentally. In previous semiclassical theory, which neglected pump quantization and loss in the nonlinear medium, Yuen and Shapiro [Opt. Lett. 4, 334 (1979)] showed that such states would be generated by 50%-50% combination of the transmitted probe (TP) and the phase-conjugate (PC) output beams of DFWM. Here the effects of quantum fluctuations in the amplitudes and phases of the pump beams are calculated in a traveling-wave finite-interaction-length DFWM configuration. It is shown that in the limit of strong pump beams and weak nonlinear coupling, the classical-pumps assumption of Yuen and Shapiro is valid. With use of this assumption, loss in the nonlinear medium is incorporated into the model by coupling the interacting modes to a set of reservoir modes. It is shown that loss presents an absolute limit on the degree of squeezing that can be achieved with DFWM. Also, for comparison with experimental work, the photoelectron statistics of the TP, PC, and 50%-50% combination modes are calculated for lossy DFWM; significant deviations are found from the predictions for lossless operation. Quantum-noise measurements on the PC and TP beams generated in sodiumvapor DFWM are described. The mixer is pumped and probed by the nearly transform-limited pulses produced by a continuous-wave-oscillator-pulse-amplifier chain dye laser at 589 nm wave-length. The normalized second-factorial moment g2{g}_{2}g2​ of the PC and TP beams is determined from photoelectron-counting measurements. Inability to simultaneously achieve high DFWM reflectivity and acceptably low scattered-plus-fluorescent background, coupled with low photomultiplier quantum efficiency, restrict the experiment to a DFWM regime wherein theory predicts Poisson statistics. Experimental g2{g}_{2}g2​ values corroborate this prediction, indicating that systematic and excess noises have been eliminated. No attempt was made to measure the quantum statistics of the 50%-50% combination mode.
In order to study the possible phase conjugation of optical near-fields, it is necessary to go beyond the slowly varying envelope- and electric dipole approximations that are normally applied in phase conjugation studies where spatially non-decaying (or at least slowly decaying) modes are mixed. In the present dissertation, the minimal coupling Hamiltonian is used to create a microscopic theoretical description of degenerate four-wave mixing. It is a semiclassical description where the electromagnetic field is treated as a classical quantity and the active medium is treated quantum mechanically. Numerical results are given for a single-level quantum well (exclusively intraband contributions) and for a two-level quantum well (mainly interband contribution). Focusing of a phase conjugated field is also discussed. (Full-length abstracts in Danish and English included).
The authors present a quantum analysis of four-wave mixing in a non-resonant nonlinear medium inside a cavity which is driven by two driving fields. In this process a nonlinear medium couples two pump modes with a signal mode of the frequency equal to the half-sum of the two pump field frequencies. Three types of stable steady-state solutions of the deterministic problem, describing three regimes of oscillation, are found: one below threshold and two different above-threshold regimes. The spectra of fluctuations of the quadrature-phase amplitudes for all three modes are calculated. It is found that in the above-threshold regime each of three modes may display a squeezed noise reduction. Below threshold a squeezing is realized for the central mode.