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Journal of Physics B Atomic and Molecular Physics

Nonlinear atomic susceptibilities

Michael A. Yuratich, D.C. Hanna

Published Apr 1, 1976• 49 citations
Publisher

Abstract

Starting with well known formulae for nth-order atomic susceptibilities, irreducible spherical tensor techniques and Racah algebra are used to put these in a form suitable for the treatment of complex nonlinear processes, e.g. where fine and hyperfine interactions can be included and the interacting fields can have quite general polarizations. By way of illustration the susceptibilities are calculated in detail for third harmonic generation and Raman scattering in alkali vapours, taking into account the effect of spin-orbit coupling.

DOI
10.1088/0022-3700/9/5/017

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Starting with well known formulae for nth-order atomic susceptibilities, irreducible spherical tensor techniques and Racah algebra are used to put these in a form suitable for the treatment of complex nonlinear processes, e.g. where fine and hyperfine interactions can be included and the interacting fields can have quite general polarizations. By way of illustration the susceptibilities are calculated in detail for third harmonic generation and Raman scattering in alkali vapours, taking into account the effect of spin-orbit coupling.
In earlier chapters we have considered in detail the susceptibility formalism of nonlinear optics, which is perhaps the most familiar approach and has a wide range of application. Starting from the constitutive relations of Chapter 2, the susceptibility formalism is quite general. In many practical applications, a particular phenomenon can be described accurately by a single order of nonlinearity, and the susceptibility then provides a useful and convenient description. However, this is not always the case. Some of the most interesting phenomena in nonlinear optics involve close resonance with the transition frequencies of the medium, and perhaps also the use of very intense optical fields. As remarked in §4.5, in these circumstances the resonant susceptibilities display mathematical divergences which are clearly unphysical. Strictly, these divergences occur only because higher-order nonlinearities have been neglected. Successive orders of nonlinearity take into account such effects as saturation, power broadening and level shifts (optical Stark effect). For intense fields or very close resonance, the contributions from several orders of nonlinearity may be comparable in magnitude. Therefore, despite its generality, the susceptibility formalism does not necessarily provide the most practical approach for the description of resonant processes. This chapter is concerned with the problem of deriving alternative and more manageable descriptions of resonant nonlinear processes, illustrated by examples. For much of the chapter, the nonlinear medium is treated as a two-level system (described in §6.2).
A new propagator approach for the calculation of nonlinear susceptibilities is proposed. In this approach a higher order susceptibility is generated by the functional derivative of a lower order susceptibility. The procedure is illustrated through a calculation of the third-order susceptibility. A diagram technique is employed to select and calculate the important contributions. The results are applied to the resonance fluorescence from a two-level atom in a gas. The existing results of the impact theory are generalized by taking proper account of the lower state interaction and by calculating the higher order collision corrections.
A quantum theory of nonlinear phenomena is considered in which account is taken of all contributions from electric and magnetic multipoles, without any multipolar approximation. Exact closed analytical expressions for the multipoles are first obtained from quantum electrodynamics by applying a generalization of the Power-Zienau transformation. The standard procedure is invoked to calculate general electromagnetic susceptibility tensors ${\ensuremath{\chi}}_{\ensuremath{\mu}\ensuremath{\nu}{\ensuremath{\alpha}}_{1}{\ensuremath{\beta}}_{1}\ensuremath{\cdots}{\ensuremath{\alpha}}_{n}{\ensuremath{\beta}}_{n}}^{[n]}$ (${\ensuremath{\omega}}_{1}, \dots{}, {\ensuremath{\omega}}_{n}$). Such a tensor is a shorthand for the electric and magnetic susceptibilities as well as mixed susceptibilities. By considering a relativistic theory based on the Dirac equation, the spin susceptibilities are automatically included. The formalism in this way allows for the computation of a wide range of electro- and magneto-optical effects to any order of perturbation and to any order of multipolar approximation.
It is well known that a simple transformation of the electric dipole interaction provides a convenient means for ascertaining the effects of permanent dipoles on the optical behaviour of systems with a response dominated by two energy levels. By establishing the general validity of the procedure for parametric processes such as harmonic generation, it is shown how the detailed structure of the optical susceptibilities associated with arbitrary forms of optical nonlinearity can be ascertained by an algorithmic method, based on a novel interpretation of the relevant quantum electrodynamical Feynman diagrams. Application of the algorithm to second and third harmonic generation illustrates its usefulness and simplicity, whilst also providing new results and revealing features, related to the role of permanent dipoles, which have not hitherto been apparent.
This chapter uses all the concepts introduced and applied in Chapter 7 . Since these concepts are not iterated here, Chapter 8 frequently refers to Chapter 7 and is intentionally concise. It aims to highlight some specific aspects of the third-order interactions, including the coupled equations of complex amplitudes, intensities, and flux densities, the coefficients of the third-order electric susceptibility tensor whose sign can be reversed for quasi-phase-matching, the symmetry of the field tensor across different possible polarization configurations, and finally, the cases of third-harmonic generation in the pump depletion regime and third-order parametric down-conversion stimulated over two modes.
Part I of this thesis describes the theoretical and experimental aspects of a new area in coherent optics known as phase conjugate optics. By using nonlinear optical mixing techniques, the complex conjugate of an arbitrary incident monochromatic wave can be generated. In particular, the theory of the degenerate four-wave mixing technique is presented. Experimental observation of phase conjugate wave front generation in a liquid crystal material (MBBA) is described. The ability of the conjugate wave front to correct for phase aberration is demonstrated. The quantum mechanical origin of the nonlinear third order susceptibility responsible for the four-wave mixing process is illustrated in an atomic system. Because of a finite memory time of such an atomic system, the electromagnetic waves taking part in the mixing process need not occur simultaneously. The applications of the phase conjugation process to real-time information processing and the characteristics of a phase conjugate resonator are discussed in detail. The ability of an optical fiber to confine the propagation of electromagnetic waves over a long distance and within a small cross sectional area implies that large optical intensities can be maintained over a long " interaction length even at moderate incident optical power. Part II of this thesis describes some third order nonlinear optical phenomena that have been observed under this condition. Especially, phase conjugation by degenerate four-wave mixing in a fiber has been shown to require significantly less pumping power. The theoretical and experimental studies of four-wave mixing in a liquid-filled fiber are presented. Extensive experimental efforts have been devoted over the last few years to the study of stimulated Raman scattering and the oscillation of the Stokes radiation inside fibers. A detailed theoretical treatment will be given here to describe these effects. The analysis shows that pump depletion due to the nonlinear process is very important in describing the stimulated Raman scattering in low loss fibers and is the dominant saturation effect in a fiber Raman oscillator. Finally, the implications of nonlinear optical phenomena to fiber optic communication will be mentioned.