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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

Asymmetric two-step GaAlAs quantum well for cascaded second-order processes

C. Kelaidis, David C. Hutchings, John M. Arnold

Published Jan 1, 1994• 0 citations
Publisher

Abstract

Recently there has been increasing interest in obtaining large nonlinear phase shifts by an effective intensity dependent refractive index n2 based on second-order nonlinearities. These phase shifts arise as down-conversion of the second harmonic, away from perfect phase-matching, to the fundamental, which in general is out of phase with the fundamental source field. This process has been termed cascaded,1,2 as it involves the consecutive application of second-order nonlinearities χ(2)(ω,ω):χ(2)(—ω,2ω).

DOI
10.1364/cleo_europe.1994.cwf39

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Recently there has been increasing interest in obtaining large nonlinear phase shifts by an effective intensity dependent refractive index n2 based on second-order nonlinearities. These phase shifts arise as down-conversion of the second harmonic, away from perfect phase-matching, to the fundamental, which in general is out of phase with the fundamental source field. This process has been termed cascaded,1,2 as it involves the consecutive application of second-order nonlinearities χ(2)(ω,ω):χ(2)(—ω,2ω).
It has recently been shown that the dipole matrix elements between subbands in semiconductor quantum wells are extremely large.1 Third-order nonlinear susceptibilities are proportional to the fourth power of these matrix elements. Large nonlinearities have thus been predicted for these systems, in particular an intensity-dependent index of refraction suitable for use in all-optical logic elements.1 Second-order nonlinearities vanish identically in symmetric wells, suggesting the use of asymmetrically graded quantum wells for quadratic interactions.2 External bias electric fields also break the inversion symmetry, resulting in Stark shifts and large quadratic susceptibilities whose sign depends on the direction of the applied field. The magnitude of the nonlinearity depends on the carrier density, the dc bias field, and the detuning of the fundamental and second harmonic frequencies from resonances.
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.