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Pure and Applied Optics Journal of the European Optical Society Part A

Vectorial electromagnetic theory of degenerate four-wave mixing in isotropic Kerr-like media

Zdeněk Bouchal, Jan Peřina

Published Sep 1, 1998• 1 citation
Publisher

Abstract

The polarization properties of the phase-conjugate wave generated by means of degenerate four-wave mixing in isotropic Kerr-like media are analysed for monochromatic plane waves of arbitrary polarizations using the Laplace transformation to solve the coupled wave equations. The model presented enables us to find the reflection and transmission matrices defined for the electric and magnetic strength vectors and to discuss the efficiency and fidelity of the phase conjugation.

DOI
10.1088/0963-9659/7/5/013

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Physica Scripta•Aug 1, 1998•4 citations
The polarization properties of the phase-conjugate wave generated by means of degenerate four-wave mixing in isotropic Kerr-like media are analysed for monochromatic plane waves of arbitrary polarizations using the Laplace transformation to solve the coupled wave equations. The model presented enables us to find the reflection and transmission matrices defined for the electric and magnetic strength vectors and to discuss the efficiency and fidelity of the phase conjugation.
Optical phase conjugation by four-wave mixing in a Kerr medium is considered. Two strong counterpropagating lasers pump a nonlinear medium in which they excite the third-order polarization. Mixing with a weak incident field then generates the phase-conjugated image of that field. It is shown how the polarization of the pumps and the tensorial nature of the nonlinear interaction can be accounted for by a geometrical polarization tensor. The electric field is shown to obey two coupled wave equations, which couple a positive- and negative-frequency component of the field. These wave equations allow plane-wave solutions, and we derive the dispersion relation for the wave vectors of these modes. By matching the field of such a mode across the boundaries of the medium to an external incident field, we are able to obtain analytically the Fresnel coefficients for reflection and transmission. Our expressions reduce to earlier results in the appropriate limits.
This thesis describes the theoretical solution and experimental verification of phase conjugation via nondegenerate four-wave mixing in resonant media. The theoretical work models the resonant medium as a two-level atomic system with the lower state of the system being the ground state of the atom. Working initially with an ensemble of stationary atoms, the density matrix equations are solved by third-order perturbation theory in the presence of the four applied electro-magnetic fields which are assumed to be nearly resonant with the atomic transition. Two of the applied fields are assumed to be non-depleted counterpropagating pump waves while the third wave is an incident signal wave. The fourth wave is the phase conjugate wave which is generated by the interaction of the three previous waves with the nonlinear medium. The solution of the density matrix equations gives the local polarization of the atom. The polarization is used in Maxwell's equations as a source term to solve for the propagation and generation of the signal wave and phase conjugate wave through the nonlinear medium. Studying the dependence of the phase conjugate signal on the various parameters such as frequency, we show how an ultrahigh-Q isotropically sensitive optical filter can be constructed using the phase conjugation process. In many cases the pump waves may saturate the resonant medium so we also present another solution to the density matrix equations which is correct to all orders in the amplitude of the pump waves since the third-order solution is correct only to first-order in each of the field amplitudes. In the saturated regime, we predict several new phenomena associated with degenerate four-wave mixing and also describe the ac Stark effect and how it modifies the frequency response of the filtering process. We also show how a narrow bandwidth optical filter with an efficiency greater than unity can be constructed. In many atomic systems the atoms are moving at significant velocities such that the Doppler linewidth of the system is larger than the homogeneous linewidth. The latter linewidth dominates the response of the ensemble of stationary atoms. To better understand this case the density matrix equations are solved to third-order by perturbation theory for an atom of velocity v. The solution for the polarization is then integrated over the velocity distribution of the macroscopic system which is assumed to be a gaussian distribution of velocities since that is an excellent model of many real systems. Using the Doppler broadened system, we explain how a tunable optical filter can be constructed whose bandwidth is limited by the homogeneous linewidth of the atom while the tuning range of the filter extends over the entire Doppler profile. Since it is a resonant system, sodium vapor is used as the nonlinear medium in our experiments. The relevant properties of sodium are discussed in great detail. In particular, the wavefunctions of the 3S and 3P states are analyzed and a discussion of how the 3S-3P transition models a two-level system is given. Using sodium as the nonlinear medium we demonstrate an ultrahigh-Q optical filter using phase conjugation via nondegenerate four-wave mixing as the filtering process. The filter has a FWHM bandwidth of 41 MHz and a maximum efficiency of 4 x 10-3. However, our theoretical work and other experimental work with sodium suggest that an efficient filter with both gain and a narrower bandwidth should be quite feasible.
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.