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Progress of Theoretical Physics

Theory of Nonlinear Susceptibility

Tatuo Kawasaki

Published Feb 1, 1968• 6 citations
Publisher

Abstract

A diagrammatic approach to the theory of nonlinear susceptiblities is developed. Including damping, the steady state nonlinear response functions to several simultaneously applied monochromatic electromagnetic fields are obtained in terms of linear response functions in the weak coupling system. Saturation phenomena in microwave resonance and in parametric processes can be understood as lower order responses in the system with higher spin temperature. The condition for the Raman emission is discussed in some details. Results are compared with previous theories, in particular with that of Bloembergen and Shen. The theory is extended to the case where the spacially inhomogeneous perturbation is imposed.

DOI
10.1143/ptp.39.331

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Cambridge University Press eBooks•Jul 27, 1990•0 citations
A diagrammatic approach to the theory of nonlinear susceptiblities is developed. Including damping, the steady state nonlinear response functions to several simultaneously applied monochromatic electromagnetic fields are obtained in terms of linear response functions in the weak coupling system. Saturation phenomena in microwave resonance and in parametric processes can be understood as lower order responses in the system with higher spin temperature. The condition for the Raman emission is discussed in some details. Results are compared with previous theories, in particular with that of Bloembergen and Shen. The theory is extended to the case where the spacially inhomogeneous perturbation is imposed.
The steady-state response of a nonlinear medium to several simultaneously applied monochromatic electromagnetic fields is described by the density matrix method. A Fourier series expansion in terms of ascending powers of the amplitudes of the applied fields is especially useful to describe the parametric response in spectral regions, where the absorption is small. As the resonances of the material system are approached, the general formalism exhibits the mixture and interference between parametric processes and single- and multiple-photon absorption and emission processes. previously discussed examples of the two-level and three-level system are generalized. The reaction of the general nonlinear medium on the electromagnetic fields is incorporated.
We apply a diagrammatic density matrix perturbation technique to analyze third order nonlinear optical processes. The diagrammatic approach is shown to yield a significant simplification in the computation of both steady state and transient response while including the effects of damping and linewidth. The topology of the diagrams is stressed in identifying and interpreting their associated perturbation terms as well as predicting frequency resonance and transient behavior. We demonstrate the practical utility of the technique by presenting a detailed analysis and discussion of the processes of stimulated Raman scattering, hot luminescence, and pressure induced resonances in four wave mixing.
Diagrammatic methods for the calculation of the nonlinear response of a system that is dressed by resonant external fields are developed. The double-sided diagrams used earlier by Prior [IEEE J. Quantum Electron. QE-20, 37 (1984)] for the calculation of nonlinear susceptibilities in the presence of weak fields are generalized. Additional types of double-sided diagrams are given that incorporate explicitly (a) the relaxation-induced population transfers among the dressed levels and (b) the effects of the nonvanishing of the diagonal elements of the dipole operator in the dressed-state basis. The rules for the diagrams are worked out from the analytical structure of the nonlinear response. Applications of the diagrammatic techniques to several intense-field phenomena are discussed.
The equivalence between the Hamiltonians representing nonlinear optical phenomena and N two-level atoms interacting via the electromagnetic field is exploited to find a solution to a parametric amplifier, a frequency converter and to the problems of Brillouin and Raman scattering. The n-photon probability and average number of photons as a function of time are obtained for all these cases. The results of the parametric amplifier can be readily applied to the Brillouin and Raman effects. In particular the results obtained for spontaneous fluorescence are applicable to the Stokes line of the stimulated Raman effect. Other nonlinear effects are discussed in the context of the present theory.
A nonlinear response theory of a quantum-mechanical system undergoing arbitrary relaxation and interacting with fields, some of which may be strong enough to saturate optical transitions, is developed. Explicit expressions for second-order and third-order susceptibilities are obtained. If the fields are weak, then these expressions show the existence of additional resonant contributions to ${\ensuremath{\chi}}^{(n)}$ which arise due to inelastic collisions. Various applications of these ${\ensuremath{\chi}}^{(n)}$'s to modulation spectroscopy, four-wave mixing, and pump-probe experiments are discussed. The general structure suggests how the additional resonances can be used to determine inelastic rates in a Doppler-broadened medium. For saturating fields, ${\ensuremath{\chi}}^{(n)}$'s become dependent on the intensity of such fields and can be formally obtained from weak-field ${\ensuremath{\chi}}^{(n)}$'s, provided proper identification of unperturbed eigenfunctions, eigenvalues, and relaxation times is made. Such intensity-dependent ${\ensuremath{\chi}}^{(n)}$'s have resonant denominators which lead to resonances at Rabi frequencies and submultiples of these frequencies, the widths of which are also dependent on the strength of the field. In the special cases intensity-dependent ${\ensuremath{\chi}}^{(1)}$ agrees with the work of Cohen-Tannoudji and co-workers.
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).