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Physical Review A

Effects of χ(3) nonlinearities in traveling-wave second-harmonic generation

Vladimir I. Kruglov, M. K. Olsen

Published Oct 1, 2001• 10 citations
Publisher

Abstract

We investigate the effects of self-phase and cross-phase ${\ensuremath{\chi}}^{(3)}$ nonlinearities, in the process of traveling-wave second-harmonic generation. We derive a semiclassical analytical solution for the field intensities, comparing this with the numerically obtained fully quantum solutions. We also investigate the effects of the cross-phase modulation on the quantum statistical properties of the fields. We find that, as the ${\ensuremath{\chi}}^{(3)}$ components increase, there are qualitative changes to both the field intensities and the quantum statistics.

DOI
10.1103/physreva.64.053802

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We investigate the effects of self-phase and cross-phase \ensuremathχ(3){\ensuremath{\chi}}^{(3)}\ensuremathχ(3) nonlinearities, in the process of traveling-wave second-harmonic generation. We derive a semiclassical analytical solution for the field intensities, comparing this with the numerically obtained fully quantum solutions. We also investigate the effects of the cross-phase modulation on the quantum statistical properties of the fields. We find that, as the \ensuremathχ(3){\ensuremath{\chi}}^{(3)}\ensuremathχ(3) components increase, there are qualitative changes to both the field intensities and the quantum statistics.
We investigate the effects of higher-order, ${\ensuremath{\chi}}^{(3)},$ nonlinearities on the process of second-harmonic generation. In the traveling-wave case we find substantive differences in the macroscopic behavior of the fields when the ${\ensuremath{\chi}}^{(3)}$ components are present. In the intracavity case, which has been investigated before using a linearized analysis, we investigate regions where these analyses may not be valid, comparing and contrasting the full quantum simulations with previous results.
We analyze squeezing of both the fundamental and harmonic fields undergoing traveling-wave, second-harmonic generation (SHG) in second-order (${\mathrm{\ensuremath{\chi}}}^{(2)}$) nonlinear media. We take into account depletion of the fundamental field as well as the phase mismatch between the fundamental and harmonic fields. The behavior of the quantum noises on the propagating fields is studied by linearizing the nonlinear operator equations around the mean-field values. We first consider the degenerate case that is applicable to type-I phase-matching geometries, obtaining expressions for squeezing in both the fundamental and harmonic fields under the conditions of perfect phase matching and large phase mismatch. We show that in the case of a large phase mismatch, the intensity-dependent self-phase shift of the fundamental field, arising due to cascading of the ${\mathrm{\ensuremath{\chi}}}^{(2)}$ nonlinearity, is responsible for the squeezing generation. We also numerically solve the linearized quadrature-operator equations together with the nonlinear mean-field equations. We find that in the case of a finite phase mismatch the harmonic field can be highly squeezed. This is in contrast to the perfectly phase-matched case where the maximum squeezing is limited to 50%. Finally, we analyze the nondegenerate case that applies to type-II phase-matching geometries. Here we show that the commonly used type-II phase-matched SHG process with the input harmonic field in the vacuum state is equivalent to a type-I SHG process in parallel with a degenerate optical-parametric process. The latter causes squeezing in the mode that is polarized orthogonal to the fundamental beam. In the perfect phase-matching case, the squeezing in the orthogonally polarized mode follows the simple expression S=1-\ensuremath{\gamma}, where \ensuremath{\gamma} is the harmonic conversion frequency.
We analyze the evolution of quantum noise in both the fundamental and the harmonic fields that are undergoing traveling-wave interaction in a second-order [χ(2)] nonlinear medium. Assuming perfect phase matching between the fundamental and the harmonic fields and arbitrary input boundary conditions, the behavior of quantum noise in the propagating fields is studied by linearization of the nonlinear-operator equations around the mean-field values. We first consider the degenerate case that is applicable to type-I phase-matching geometries, obtaining expressions for squeezing in both the fundamental and the harmonic fields. We then analyze the polarization-nondegenerate case that applies to type-II phase-matching geometries. In the special case, when the two orthogonally polarized fundamental inputs are of equal amplitude, we obtain analytical results and show that the type-II phase-matched second-harmonic-generation process can be configured to generate sub-Poissonian light in both polarization components of the fundamental field. Finally, we numerically solve the linearized quadrature-operator equations along with the nonlinear mean-field equations for the general case of type-II phase matching. For both type-I and type-II processes, we find that whenever the fundamental field experiences deamplification it is associated with amplitude squeezing and phase desqueezing. If, in contrast, the fundamental field experiences amplification, then it is accompanied by amplitude desqueezing, but with squeezing in the phase quadrature. The harmonic field is amplitude squeezed if the input boundary condition leads to harmonic conversion and is phase squeezed if the input boundary condition leads to parametric amplification.
Using the full nonlinear equations of motion, we calculate the quantum-nondemolition (QND) correlations for the traveling-wave second-harmonic generation. We find that, after a short interaction length, these are qualitatively different from results calculated previously using a linearized fluctuation analysis. We demonstrate that, although individual QND criteria can be very good in certain regions, there is no region where all three of the standard criteria are perfect, as has previously been claimed. We also show that only the amplitude quadrature of the output field can be considered as a QND quantity, with the phase quadrature not satisfying all the criteria.
With the use of a quantum theory of optical propagation, a set of nonlinear stochastic partial differential equations may be derived which describes the quantum-statistical properties of traveling waves due to self-phase modulation in a nonlinear medium. We calculate exact moments for the field, which exhibit classical self-phase modulation in the short-interaction limit and periodic quantum evolution through field-superposition states in the long-interaction limit. Reversible and irreversible behaviors of the stochastic description are reviewed. The relation of the present work to the corresponding single-mode nonlinear oscillator is discussed.
We calculate the spectrum of third-harmonic generation (THG) for multimode fundamental input fields. We consider input fields from (1) a laser with gain on several cavity modes, (2) pulses with short temporal duration, and (3) frequency-modulated input fields. The effects of phase modulation in the nonlinear medium on the spectrum of the THG field are considered. Using the estimates by Miles and Harris [IEEE J. Quantum Electron. QE-9, 470 (1973)] for the nonlinear susceptibilities ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-3\ensuremath{\omega};\ensuremath{\omega},\ensuremath{\omega},\ensuremath{\omega}) and ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-\ensuremath{\omega};\ensuremath{\omega},-\ensuremath{\omega},\ensuremath{\omega}) in alkali-metal vapors, we find that phase modulation can profoundly affect the efficiency and the spectrum of the THG. Generally, the self-phase modulation of the fundamental will tend to reduce the phase matching of an intense fundamental field and the third harmonic and thereby lower the conversion efficiency. New frequencies not contained in the set of frequencies ${\mathrm{\ensuremath{\omega}}}_{1\mathit{i}}$+${\mathrm{\ensuremath{\omega}}}_{1\mathit{j}}$+${\mathrm{\ensuremath{\omega}}}_{1\mathit{k}}$, where {${\mathrm{\ensuremath{\omega}}}_{1\mathit{i}}$} are frequencies in the fundamental laser spectrum, are created even at relatively low conversion due to phase modulation. Effects of cross phase modulation are also studied.