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Physical Review A

Quantum theory of multiwave mixing. III. Averages over inhomogeneous broadening, spatial hole burning, and Gaussian beams

David A. Holm, Murray Sargent, Lois M. Hoffer

Published Aug 1, 1985• 26 citations
Publisher

Abstract

The effects of inhomogeneous broadening, spatial hole burning, and Gaussian beams are calculated for the four coefficients derived in previous papers in this series [Phys. Rev. A 31, 3112 (1985); 31, 3124 (1985)] on the quantum theory of multiwave mixing. These coefficients uniformly describe how one or two strong classical-field modes and one or two weak quantum-field modes interact in a two-level medium. The phenomena for which this theory is applicable includes resonance fluorescence, Rayleigh scattering, laser and/or optical bistability instabilities, and phase conjugation. The averaged formulas are illustrated by resonance-fluorescence spectra. We find the sidepeaks can be ``washed out'' and that asymmetries can occur.

DOI
10.1103/physreva.32.963

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Annual Meeting Optical Society of America•Jan 1, 1987•1 citation
The effects of inhomogeneous broadening, spatial hole burning, and Gaussian beams are calculated for the four coefficients derived in previous papers in this series [Phys. Rev. A 31, 3112 (1985); 31, 3124 (1985)] on the quantum theory of multiwave mixing. These coefficients uniformly describe how one or two strong classical-field modes and one or two weak quantum-field modes interact in a two-level medium. The phenomena for which this theory is applicable includes resonance fluorescence, Rayleigh scattering, laser and/or optical bistability instabilities, and phase conjugation. The averaged formulas are illustrated by resonance-fluorescence spectra. We find the sidepeaks can be ``washed out'' and that asymmetries can occur.
This dissertation formulates and applies a theory describing how one or two strong classical waves and one or two weak quantum mechanical waves interact in a two-level medium. The theory unifies many topics in quantum optics, such as resonance fluorescence, saturation spectroscopy, modulation spectroscopy, the build up of laser and optical bistability instabilities, and phase conjugation. The theory is based on a quantum population pulsation approach that resembles the semiclassical theories, but is substantially more detailed. Calculations are performed to include the effects of inhomogeneous broadening, spatial hole burning, and Gaussian transverse variations. The resonance fluorescence spectrum in a high finesse optical cavity is analyzed in detail, demonstrating how stimulated emission and multiwave processes alter the spectrum from the usual three peaks. The effects of quantum noise during the propagation of weak signal and conjugate fields in phase conjugation and modulation spectroscopy are studied. Our analysis demonstrates that quantum noise affects not only the intensities of the signal and conjugate, but also their relative phase, and in particular we determine a quantum limit to the semiclassical theory of FM modulation spectroscopy. Finally, we derive the corresponding theory for the two-photon, two-level medium. This yields the first calculation of the two-photon resonance fluorescence spectrum. Because of the greater number of possible interactions in the two-photon two-level model, the theoretical formalism is considerably more complex, and many effects arise that are absent in the one-photon problem. We discuss the role of the Stark shifts on the emission spectrum and show how the Rayleigh scattering is markedly different.
We present a theory that describes how one strong classical wave and one or two weak quantum-mechanical waves interact in a nonlinear two-level medium. The analysis is applicable to several popular problems with and without cavities. In particular, the theory treats laser and optical bistability instabilities, predicting when the instabilities grow from spontaneous emission. The theory is a multimode extension of Scully-Lamb theory that derives the equations that describe population pulsations, combination tones, mode locking, resonance fluorescence, Rayleigh scattering, and phase conjugation with quantum-mechanical fields. Hence the theory both presents new results on instabilities and phase conjugation, and also unifies the treatment of a variety of phenomena in the context of Lamb theory. The present paper (first of the series) presents the basic formalism, leaving most applications to subsequent papers in this series. The following paper presents equivalent derivations based on a purely operator formalism.
This paper treats the interaction of one strong classical field mode and one or two weak quantal modes with two-level systems. The analysis is carried out in a purely operator formalism that complements the method presented in the preceding paper and produces the same results. The present approach automatically collects all contributions to second order in the weak-field interactions, and can be easier to extend to more complicated cases. The analysis allows one to treat problems such as predicting the growth of sidemodes in laser- and optical-bistability instabilities, the resonance fluorescence spectrum, Rayleigh scattering, and quantum noise in small-signal phase conjugation.
We extend the quantum theory of multiwave mixing developed in previous papers in this series to include side-mode saturation. Specifically, the earlier papers calculate the absorption and resonance-fluorescence coefficients to second order in the weak-field interaction for an arbitrarily intense pump field. This paper extends those calculations to fourth order in the weak-field interaction. We find significant differences in the amplified resonance fluorescence spectra given by the second- and fourth-order treatments, particularly as the side mode approaches the laser threshold. In particular, while the second-order spectra diverge at this threshold, the fourth-order spectra remain finite. Although fourth- and higher-order theories are very desirable for applications to larger fields and optical instabilities, the present analysis reveals that their calculation is very complex. This fact poses an important limitation to the generalization of the theory to still higher-order accuracies.
We extend our theoretical formalism describing how one strong classical wave and one or two weak quantum-mechanical waves interact in a nonlinear medium to the ``two-photon two-level'' model. In this model, field transitions between two levels not connected by an electric dipole occur by means of a two-photon transition using off-resonant intermediate levels. This model has many similarities with, as well as significant differences from, the corresponding one-photon two-level model that has been treated in the previous papers in this series. The theory yields the first detailed calculation of the two-photon resonance fluorescence spectrum, as well as the complete set of four coefficients that describe the quantum theory of two-, three-, and four-wave mixing.
Our semiclassical theory of two-photon multiwave mixing1 is a generalization of the degenerate four-wave mixing case2 to treat nondegenerate pump/signal/conjugate frequencies. We apply that theory to four-wave mixing in two-photon media. We present explicit formulas for the nondegenerate signal/conjugate absorption and coupling coefficients for cases for which the two-photon Stark shifts are sufficiently small compared to the homogeneous linewidths. More general cases require that the spatial hole burning averages be performed numerically. We find intricate dependencies of the transmission and reflection spectra on the pump intensity and on dynamic Stark shifts. In particular, the reflection coefficients do not necessarily bleach to zero as the pump intensity is increased indefinitely, and interferences occur between pure index and mixed absorption/index contributions