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Physical Review A

Dynamics of frequency conversion of an optical pulse in a microcavity

J. C. Martinez

Published Jan 5, 2005• 6 citations
Publisher

Abstract

Although it is known that the nonlinear Schr\"odinger equation describes the propagation of light in self-focusing media, the quantization conditions that emerge from its corresponding semiclassical version have not been sufficiently appreciated. For the propagation of an optical pulse in a Kerr-effect microcavity, we find that the transverse energy, intensity, number of particles, and frequency detuning are not arbitrary but are interrelated by quantization conditions. We identify the binding energy of the collective bound state of particles with the energy loss of the pulse and indicate how the energy distribution of photons can be studied from the point of view of a master-equation approach to a heat reservoir interacting with the photons.

DOI
10.1103/physreva.71.015801

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Although it is known that the nonlinear Schr"odinger equation describes the propagation of light in self-focusing media, the quantization conditions that emerge from its corresponding semiclassical version have not been sufficiently appreciated. For the propagation of an optical pulse in a Kerr-effect microcavity, we find that the transverse energy, intensity, number of particles, and frequency detuning are not arbitrary but are interrelated by quantization conditions. We identify the binding energy of the collective bound state of particles with the energy loss of the pulse and indicate how the energy distribution of photons can be studied from the point of view of a master-equation approach to a heat reservoir interacting with the photons.
We obtain two stationary solutions for an intense optical pulse propagating in a Kerr-effect nonlinear ring cavity using elliptic functions associated with a parameter mmm. The solutions may be associated with distinct phases of the system, the transition from one to the other being dictated by an energy ratio. Semiclassical quantization reveals excitations that are quantized rotational states for the spatially constant solution and NNN-particle bound states for the soliton solution. On invoking periodic boundary conditions, a constraint relating the number of particles, wavelength, and mmm emerges. This constraint fundamentally modifies the binding energy of the soliton for a large range of mmm. At the phase transition point, the energy undergoes a discontinuity in both value and derivative, a conclusion that differs markedly from results of a mean-field calculation.
We obtain stationary solution for optical solitons propagating in a Kerr-effect nonlinear cavity using elliptic functions and quantize them semiclassically. On invoking box boundary conditions, a constraint relating the number of particles, wavelength, and a parameter associated with the elliptic function emerges. This constraint fundamentally modifies the binding energy of the soliton and lends the system a rich plethora of solution types with diverse behavior as a function of excitation number. We also speculate on how the bright soliton can thermalize through a path of frequency conversion.
We study the quantum fluctuations about a dark soliton generated in a photon fluid confined in a Kerr-effect nonlinear microcavity. Semiclassical quantization imposes quantization conditions that lend a particle interpretation to the fluctuations. A Bogoliubov–de Gennes analysis complements this view by yielding the single-particle energies and wave functions. The zero modes are found to correspond to the background while the scattering solutions to the notch. We compute the mass associated with the soliton.
Analytical and numerical methods are used to study the transverse reshaping that occurs when ultrashort optical pulses propagate through media resonant to the pulse-carrier frequency. The field is described by the scalar quasi-optic equation, and the material system by the full Bloch equations describing a distribution of homogeneously broadened two-level molecules. Transverse derivatives in the field equation provide the coupling leading to the development of radially dependent frequency variations and phase curvature. We have studied the origin and evolution of transverse energy flow and investigated the conditions under which this leads to significant coherent self-focusing. The physical mechanisms responsible for the onset and development of the focusing phenomenon are discussed and related to other types of self-confinement of optical pulses.
In this chapter we review a number of quantum features of optical fields associated with propagation of intense light through nonlinear, isotropic media. We begin with the classical description of field propagation, introducing the nonlinear polarization of the medium which enters as the source term into the approximate field equations obtained from the Maxwell equations in the slowly varying amplitude approximation. This establishes the classical background for further quantum considerations. Next, we quantize the field and construct the effective Hamiltonian from which we get equations of motion for the quantum fields. We discuss a number of quantum effects such as photon antibunching, squeezing, the formation of Schrodinger cats and kittens, changes in field polarization due to the quantum nature of the field, as well as the quantum description of the field phase. The characteristic feature of quantum evolution — the periodicity — is strongly affected by dissipation. We give exact analytical formulae describing the quantum evolution of the field, including dissipation. We have collected results illustrating various aspects of quantum evolution and we believe that this review, although far from complete, will be a useful source of information on the subject.
Ever since the advent of nonlinear optics, the generation of light by frequency-conversion is drawing continued attention, and leading to emerging applications such as supercontinuum sources for ultra stable clocks and advanced microscopy. A modern approach to frequency-conversion is to switch light confined in micro- and nanocavity resonances to enable on-chip operation. Supposedly, nonlinear frequency conversion in such confined media differs from traditional non-linear optics in three key features regarding output spectrum, frequency shift, and critical time scale. Therefore, we switch GaAs-AlAs microcavities by the electronic Kerr effect, and study a range of quality factors to bridge the confined and the traditional non-linear regimes. We uncover the key role of the density of vacuum fluctuations, i.e., the local density of optical states (LDOS) for newly generated frequencies, a concept inspired by cavity quantum electrodynamics (cQED). As a result, we succeed to establish a framework, which not only describes nonlinear optics both in traditional bulk and in confined media but also opens a new control dimension in changing the color of light.