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Journal of Modern Optics

Local Gauge Invariance of the Two-beam Coupling Equations

François Sanchez

Published Oct 1, 1993• 0 citations
Publisher

Abstract

An analytical comparison between the exact model of two-wave mixing in thin nonlinear local response absorbing media and a simplified theory is presented. The approach adopted here consists of studying the dynamical interaction of four waves: the two incident beams and the two main diffracted orders. The simplified model assumes energy conservation in a weakly absorbing medium. It is demonstrated that the two models give exactly the same results for any absorption coefficient. It is shown that the simplified model is correct in weakly or highly absorbing media. This is due to a local gauge invariance of the coupled equations.

DOI
10.1080/09500349314552041

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An analytical comparison between the exact model of two-wave mixing in thin nonlinear local response absorbing media and a simplified theory is presented. The approach adopted here consists of studying the dynamical interaction of four waves: the two incident beams and the two main diffracted orders. The simplified model assumes energy conservation in a weakly absorbing medium. It is demonstrated that the two models give exactly the same results for any absorption coefficient. It is shown that the simplified model is correct in weakly or highly absorbing media. This is due to a local gauge invariance of the coupled equations.
A systematic study of degenerate two-wave mixing in the stationary regime in thin nonlinear local-response media is presented. The model is based on the coherent interaction of four waves, the two incident beams, and the two main diffracted waves. Analytical results are obtained in all cases investigated under the assumption of weak-beam amplification. It is demonstrated that the energy transfer arises only from the forward degenerate four-wave mixing terms. In particular, the theoretical results show two new features: (1) the gain factor of the interaction is substantially reduced because of the phase change associated with the two-wave mixing terms and (2) a phase mismatch can compensate for this phase change, leading to an appreciable gain enhancement. Moreover, the general case is solved numerically. Particular attention is given to the influence of separate two-wave and forward degenerate four-wave mixing terms. Self-diffraction effects and the gain dependence versus the ratio of the pump-beam intensity to the signal-beam intensity are investigated.
A general theory of two-wave mixing (TWM) in nonlinear media is presented. Assuming a gain (or absorption) grating and a refractive index grating are generated because of the nonlinear process in a nonlinear medium, the coupled-wave equations of TWM are derived based on Maxwell's wave equation. The coupled-wave equations can be decoupled as coupled equations for the intensity and coupled equations for the phase of both beams, and these two sets of coupled equations can be solved analytically by using average total intensity in the medium instead of using the total intensity. Compared with the previous theory of TWM, the theory presented here is more general, and the applications of the theory to photorefractive materials, Kerr media, and semiconductor broad-area amplifiers are described.
We study the self-action of an amplitude-modulated beam in a two-level saturable absorbing medium. We also consider the radial quadratic dependence of the linear refraction index to apply the results to doped waveguides. As the modulation period approaches the relaxation times, the medium response is no more instantaneous, so that one should solve the full set of Maxwell-Bloch equations. We propose a second-order scheme with the Gauss-Laguerre transformation of the transverse field pattern. A simplified approach based on the synchronous interaction approximation is used for thin layers. We analyze the transient behavior of the medium response and its manifestation in the modulation of the beam diffracting after a thin saturable absorber. Nonlinear distortions of the modulation signal passed through a doped waveguide appear to be unexpectedly small compared with those of the local polarization and population difference.
A local-field theory describing optical phase conjugation in condensed media in the special case of degenerate four wave mixing is established. The aim of the theory is to form the framework for microscopic studies of optical phase conjugation (i) of evanescent fields in the context of near-field optics, (ii) in mesoscopic films and quantum wells, (iii) in small particles, and (iv) in lossy media where the field penetration depth is comparable to or (substantially) less than the vacuum wavelength of the driving field. The aforementioned goal makes it necessary to abandon both the slowly varying envelope- and the electric dipole approximations usually adopted in phase conjugation studies where spatially slowly decaying or modulated fields are mixed. By keeping in the interaction Hamiltonian the term of second order in the vector potential and in the current-density operator the term of first order in the vector potential new microscopic field-matter interaction processes of particular importance in the present context are included. The physics of the various nonlinear microscopic processes is analysed, and systematised by presenting in diagrammatic form the nonlocal electrodynamics hidden in the nonlinear constitutive relation. A new nonlocal conductivity tensor, enabling one to describe the degenerate four wave mixing process among the prevailing local fields, is presented and the eigensymmetries of its various parts are analysed. Starting from the general local-field theory a degenerate four wave mixing response tensor of relevance for media exhibiting two-dimensional translational invariance is established and discussed. In the last part of the paper an integral equation allowing one to obtain the phase conjugated local field inside and outside the nonlinear medium is established, discussed and formally solved, and it is pointed out that this equation for systems with two-dimensional translational invariance often can be analysed analytically and numerically using methods previously developed in theoretical studies of the linear local-field electrodynamics of mesoscopic films.
We show that four-wave mixing in optically dense media is very different than in an optically thin medium. In time-resolved degenerate four-wave mixing we experimentally observe ``negative'' time delay response, fast decay rates at short delay times, and broad shoulders at long delays. The observations are explained very well in terms of pulse propagation effects, and a theoretical framework is presented for the analysis of the nonlinear interaction. We treat both the incident and generated fields self-consistently, and solve for the interaction of short pulses with a resonantly absorbing medium. The implications for the extraction of relaxation rates from four-wave-mixing experiments involving ultrashort pulses in optically dense matter are discussed.