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arXiv:physics/9902014

Theoretical Study of Phase Conjugation in Mesoscopic Interaction Volumes

Torsten Andersen

Published Feb 8, 1999• 1 citation
arXivHTMLPDF

Abstract

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).

DOI
10.48550/arxiv.physics/9902014

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•Jan 1, 1982•0 citations
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).
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.
The phase conjugated response from nonmagnetic multilevel metallic quantum wells is analyzed and an essentially complete analytical solution is presented and discussed. The description is based on a semiclassical local-field theory for degenerate four-wave mixing in mesoscopic interaction volumes of condensed media developed by the present authors [T. Andersen and O. Keller, Phys. Scr. 58, 132 (1998)]. The analytical solution is supplemented by a numerical analysis of the phase conjugated response from a two-level quantum well in the case where one level is below the Fermi level and the other level is above. This is the simplest configuration of a quantum-well phase conjugator in which the light-matter interaction can be tuned to resonance. The phase conjugated response is examined in the case where all the scattering takes place in one plane, and linearly polarized light is used in the mixing. In the numerical work we study a two-monolayer thick copper quantum well using the infinite barrier model potential. Our results show that the phase conjugated response from such a quantum-well system is highly dependent on the spatial dispersion of the matter response. The resonances showing up in the numerical results are analytically identified from the expressions for the linear and nonlinear response tensors. In addition to the general discussion of the phase conjugated response with varying frequency and parallel component of the wave vector, we present the phase conjugated response in the special case where the light is in resonance with the interband transition.
We present a calculation of the phase conjugated response from a single-level metallic quantum well. The description builds upon a recently developed local-field theory for degenerate four-wave mixing in mesoscopic interaction volumes of condensed media. The single-level quantum well represents the simplest possible configuration of a quantum well phase conjugator. Furthermore, the single-level quantum well is an interesting object, since the optical response contains no dipole terms. The discussion of the response is based on the use of linearly polarized light to excite the process. We demonstrate that the phase conjugation process is extremely efficient in the evanescent regime of the wave-vector spectrum. We also address the problem of plane-wave excitation of the high wave-number end of the evanescent regime. We end our discussion by suggesting the use of a broadband source to excite the process. One such broad angular band source is a quantum wire, and the phase conjugated angular spectrum from a quantum wire is presented and discussed.
We present a theoretical investigation of optical phase conjugation by use of nearly degenerate four-wave mixing in a two-level atomic system. The description is based on the semiclassical dressed-state approach, which offers a very transparent picture of the physics involved in the nonlinear process. The theory provides an explanation in terms of a quantum interference between two channels for the strong reduction of the optical phase-conjugate yield when the pump fields are tuned to the atomic resonance. In addition, the model predicts an efficient generation of the optical phase conjugate if the system is prepared in a pure dressed state.
The phase-conjugate field fidelity in backward-going four-wave mixing, based on the diffraction model solvable analytically, is investigated. The method assumes interaction of the non-depleted infinite plane pump waves with the spatially limited probe and signal waves inside the nonlinear medium. Solutions are obtained applying the Fourier transformation and the Green function technique under various boundary conditions for coupled wave equations. The probe and its phase-conjugate waves are defined by using a complex phase formulation including both the phase variation and amplitude modulation. It is shown that the analysed approach can involve assumptions leading to the ideal or distorted phase conjugation.
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.