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IEEE Journal of Quantum Electronics

Transient four-wave mixing and real time holography in atomic systems

Avigail Yariv, J. Auyeung

Published Apr 1, 1979• 16 citations
Publisher

Abstract

The problem of transient four-wave mixing with noncoincident optical pulses is analyzed using the formalism of the time evolution and the density matrix operators. The results are relevant to problems involving real time holography and wave conjugation. The treatment establishes a bridge between the conventional formalisms of nonlinear optics and of photon echoes.

DOI
10.1109/jqe.1979.1069986

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The problem of transient four-wave mixing with noncoincident optical pulses is analyzed using the formalism of the time evolution and the density matrix operators. The results are relevant to problems involving real time holography and wave conjugation. The treatment establishes a bridge between the conventional formalisms of nonlinear optics and of photon echoes.
The general formalism of transient four-wave mixing is developed using the diagrammatic technique of Yee and Gustafson, and applied systematically to two-, three-, and four-level systems. The results predict all possible photon-echo phenomena including free-induction decay in the low-intensity limit. Connections to transient saturation spectroscopy and other coherent transient effects can also be made. The treatment can be readily extended to the more general problem of coherent transient optical effects with nnn waves.
Summary form only given. In this tallk I will try to supplement what I learned in the field. Here are the topics to be touched on. The order is not historical, but serves a tutorial purpose. 1. Time reversibility (TR) of optical wave propagation. 2. Mathematical description of TR in terms of complex amplitudes and complex conjugate amplitudes. 3. Static holography by real and conjugate images. 4.Static holography: illumination by counterpropagating reference waves at the recording and reconstruction stages. 5. Dynamic holography/four-wave mixing for phase conjugation (R. Hellwarth, A. Yariv, Yariv-Pepper instability. 6. Delayed and hidden holograms, photon echoes, electro-acoustic echoes. 7. Two-photon coherence, fast nonlinearities and three-wave mixing via X(2) nonlinearity. 8. Yet unrealized: Bragg (periodic domain). Time-domain phase conjugation: reversal of chirp and compensation of the dispersion spread in fibers. 10. Why a laser may (and may not) oscillate at a single transverse mode: discrimination of gain/loss. 11. Phase conjugation via stimulated Brillouin scattering (V.V. Ragulsky; V.V. Shkunov). 12. Brillouin-enhanced four-wave mixing for phase conjugation, ten to the tenth power amplification of the image (G.A. Pasmanik). 13. Photorefractive crystals: nonlinear response at low intensity at a cost of long build-up time. 14. Double phase conjugation 15. Feinberg's cat and cat conjugator.
Optical phase conjugation is an important nonlinear process, with many applications in the areas of optical communications and optical processing. This is as a result of the wave-front correction properties that phase conjugation offers and this has generated much interest in the area. A common method of producing phase conjugated wavefronts is the four-wave mixing interaction and many publications on the steady state solution to this problem have appeared over the last decade. More recently, however, interest has been focussed upon the temporal behaviour of four-wave mixing systems, with instabilities and chaos being both demonstrated by Gauthier et al [1] and predicted by Królikowski et al [2]. To date, the analysis for the temporal behaviour of anisotropic four wave mixing, in the transmission grating regime, has involved the direct numerical integration of the equations describing four wave mixing. This analysis however, does not utilise the symmetries of the four wave mixing process, and thus results in a more complex problem. Through exploitation of these symmetries the complexity of the problem has been reduced from one containing four complex variables, to one containing three real variables. This has been accomplished through the use of the Special Unitary Group 2, a group providing a two dimensional matrix, whose elements are functions of three real variables. The multiplication of this matrix, together with one containing the boundary conditions for the problem, thus enables the four complex beam amplitudes to be reexpressed in terms of three real quantities. Using this technique, the temporal nature of anisotropic four wave mixing has been studied, and shown under certain conditions, to exhibit chaotic behaviour when an electric field is present. The effect of any material absorption on the chaotic nature is also demonstrated.
An efficient method has been developed for the calculation of third-order time- and frequency-resolved optical signals. To obtain the general four-wave mixing signal, seven auxiliary density matrices have to be propagated in time. For the special cases of two-pulse photon-echo and transient-grating signals, two or three density matrices, respectively, are required. The method is limited to weak laser fields (it is thus valid within the third-order perturbation theory) but allows for any pulse durations and automatically accounts for pulse-overlap effects. To illustrate the method, we present the explicit derivation of the three-pulse photon-echo signal. Any other third-order optical signal can be calculated in the same manner. As an example, two- and three-pulse photon-echo and transient-grating signals for a weakly damped displaced harmonic oscillator have been calculated.