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Annual Meeting Optical Society of America

Instabilities due to quantum multiwave mixing

Murray Sargent

Published Jan 1, 1986• 0 citations
Publisher

Abstract

Given one oscillating mode in a laser or an optical bistability cavity, can other modes build up, that is, is single-mode operation unstable? The answer depends on an interplay of spontaneous emission, saturated gain, and four-wave mixing processes that permit one or more sidemodes both to be resonant with cavity modes and to have more gain than loss. We discuss this problem for cavities containing one- or two-photon two-level media subjected to a classical wave of arbitrary intensity and one or two weak quantized sidemodes. While a single sidemode may have gain, two sidemodes symmetrically placed about a strong pump mode can have greater gain due to four-wave mixing contributions sometimes called combination tones. The theory is valid for the buildup of sidemodes from quantum noise until the sidemodes begin to saturate. Subsequent dynamics including chaos must be handled by a theory that allows all modes to saturate. The same physical mechanisms are important in saturation spectroscopy and control the generation of squeezed states in a cavity. However while the instabilities depend on sidemode gain and spontaneous emission, the generation of squeezed states of the vacuum avoids these phenomena, depending solely on quantum combination tones.

DOI
10.1364/oam.1986.mf2

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•Jan 1, 1988•0 citations
Given one oscillating mode in a laser or an optical bistability cavity, can other modes build up, that is, is single-mode operation unstable? The answer depends on an interplay of spontaneous emission, saturated gain, and four-wave mixing processes that permit one or more sidemodes both to be resonant with cavity modes and to have more gain than loss. We discuss this problem for cavities containing one- or two-photon two-level media subjected to a classical wave of arbitrary intensity and one or two weak quantized sidemodes. While a single sidemode may have gain, two sidemodes symmetrically placed about a strong pump mode can have greater gain due to four-wave mixing contributions sometimes called combination tones. The theory is valid for the buildup of sidemodes from quantum noise until the sidemodes begin to saturate. Subsequent dynamics including chaos must be handled by a theory that allows all modes to saturate. The same physical mechanisms are important in saturation spectroscopy and control the generation of squeezed states in a cavity. However while the instabilities depend on sidemode gain and spontaneous emission, the generation of squeezed states of the vacuum avoids these phenomena, depending solely on quantum combination tones.
Over the past three years we have developed a quantum theory of multiwave mixing designed to determine effects of quantum noise on laser and optical bistability instabilities, pump/probe saturation spectroscopy, AM and FM modulation spectroscopy, and phase conjugation. The theory treats a classical pump field and one or two quantum-mechanical sidemode fields interacting with one- or two-photon, two-level media. The theory1 produces a master equation for the reduced sidemode density operator able to describe all of the above phenomena in a uniform way closely related to the corresponding semiclassical theories. It has led to a number of significant predictions about the effects of spontaneous emission on laser spectroscopy and optical instabilities.
In a recent paper1, a quantum theory of multiwave interactions with a two-level medium is presented. In this paper we apply this theory to sidemode instabilities pertinent to cavity con figurations in optical bistability and ring lasers. We assume the cavity to oscillate on a single mode, and we evaluate the gain and spontaneous emission at other frequencies. Because of the saturating presence of the strong, laser mode, two symmetrically situated sidemodes can grow together, inducing populations pulsations that influence the other sidemode. The instability must therefore be treated as a double sidemode problem.
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.
We use the semiclassical strong-signal theory of the laser to predict and explain the onset of side-mode buildup in lasers with one oscillating mode. Two general categories are considered: one for which the side modes and the oscillating mode all have the same wavelength and the other for which they have different wavelengths. The treatments include an arbitrary amount of inhomogeneous broadening. Our approach unifies the treatments of the side-mode instabilities presented earlier and extends them to handle standing waves in addition to the previously treated running waves. We write the field and the population matrix elements as Fourier series in the adjacent-mode beat frequency. This approach has been used extensively in both multimode laser theory and saturation spectroscopy. This technique coincides with linear stability analyses used by others, provided that our beat frequency includes a contribution that is proportional to the complex side-mode gain. We give a solution that allows for detuned operation along with its simpler, centrally tuned special case. The connection with saturation spectroscopy clearly reveals that the side-mode instabilities require side-mode gain. For the single-wavelength case, nonlinear anomalous dispersion is also required. The side-mode gain and dispersion result from both inhomogeneous broadening and population pulsations. The lowest instability thresholds occur when both of these mechanisms play a role. The approach can also be used to treat instabilities in optical bistability by substituting the appropriate equation of state for the strong-mode intensity and by changing the sign of the absorption coefficient. In homogeneously broadened, standing-wave lasers, we show that multiwavelength instabilities depend strongly on the position of the medium in the cavity. We illustrate the theory by giving numerical results for the output pulsation frequency and for the instability threshold by using parameters that are appropriate for the He–Xe laser. These results correlate well with experimental observations.
This dissertation presents theories of multiwave mixing in two- and three-level media. The first part of the dissertation treats the semiclassical theories in two-level media. Chapter 2 gives the simple semiclassical theory of four-wave mixing when the two pump frequences differ by more than the reciprocal of the population-difference lifetime. This difference washes out the pump spatial holes as well as one of the two reflection gratings. We compare the results to the degenerate treatment of Abrams and Lind and find significant differences in the reflection coefficient spectra. Chapter 3 presents the semiclassical theory of multiwave in a squeezed vacuum characterized by unequal in-phase and in-quadrature dipole decay times. For a highly squeezed vacuum, we find sharp resonances in both probe absorption and reflection coefficients, which provide sensitive ways to measure the amount of squeezing in the vacuum. The second part of the dissertation treats the quantum theories in two- and three-level media. Chapter 4 develops the fourth-order quantum theory of multiwave mixing to describe the effects of sidemode saturation in two-level media. We derive explicit formulas for the fourth-order quantum coefficients and show that the fourth-order quantum theory reproduces the third-order semiclassical coefficient obtained by truncating a continued fraction. We apply the results to cavity problems and find significant differences in the sideband spectra given by the second- and fourth-order treatments, particularly as the sidemode approaches the laser threshold. The final chapter presents a quantum theory of multiwave mixing in three-level cascades with a two-photon pump. The explicit formulas for the resonance fluorescence spectrum and the quantum combination-tone source term are derived. The theory is applied to the generation of squeezed states of light. We find almost perfect squeezing for some strong pump intensities and good broad-band squeezing for low pump intensities.