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Journal of Modern Optics

Squeezing in Second-harmonic Generation

S. Kielich, R. Tanaś, Roman Zawodny

Published Jun 1, 1987• 17 citations
Publisher

Abstract

The production of squeezed states in second-harmonic generation is discussed theoretically from the viewpoint of the structure of the nonlinear medium, phase-matching conditions, and the polarization state of the fields. The nonlinear coupling constants are derived in explicit form for all 102 magnetic symmetry groups for crystalline as well as electrically polarized isotropic media. The phase-matching conditions are discussed in detail. Phase mismatch is shown to accumulate along the optical path, and is calculated in full detail for various media and geometries. We refer to this as the mismatch accumulation effect. General formulae for normally ordered variances of the second-harmonic and fundamental fields are derived by perturbative procedure, and are specified for some special cases of practical interest.

DOI
10.1080/09500348714550881

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•May 16, 2003•0 citations
The production of squeezed states in second-harmonic generation is discussed theoretically from the viewpoint of the structure of the nonlinear medium, phase-matching conditions, and the polarization state of the fields. The nonlinear coupling constants are derived in explicit form for all 102 magnetic symmetry groups for crystalline as well as electrically polarized isotropic media. The phase-matching conditions are discussed in detail. Phase mismatch is shown to accumulate along the optical path, and is calculated in full detail for various media and geometries. We refer to this as the mismatch accumulation effect. General formulae for normally ordered variances of the second-harmonic and fundamental fields are derived by perturbative procedure, and are specified for some special cases of practical interest.
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.
Combinations of quadratic nonlinear crystals with different phase mismatches are studied numerically in terms of amplitude squeezing in the second-harmonic generation. It is predicted that an appropriate choice of phase mismatches can lead to large amplitude squeezing in the fundamental field with much larger stability than in the phase-matched case. It is also predicted that another choice of phase mismatches can yield amplitude squeezing in the harmonic field beyond the theoretical limit of the phase-matched case.
We have analyzed squeezing of the fundamental field by means of traveling-wave, type-II phase-matched, secondharmonic generation, taking into account depletion of the fundamental field and phase mismatch between the fundamental and the harmonic fields. For the phase-matched case we show that the generated squeezing is S = 1 - gamma, where gamma is the harmonic conversion efficiency. In the case of a large phase-mismatch we have identified a new mechanism that generates squeezing. It is the nonlinear phase shift of the fundamental field that is due to the cascaded chi((2)) nonlinearity.
Squeezing can be generated by means of traveling-wave (TW) second-harmonic generation (S1 1C), even when there is a large phase mismatch between the fundamental and the harmonic fields. We show, in particular, that in the type-II geometry squeezed vacuum is created in the mode that is polarized orthogonal lo the fundamental beam. In the large phase-mismatch limit, where the fundamental field experiences little depletion and negligible second harmonic is generated, nonlinear phase shift of the fundamental wave introduced during the 5HG process is responsible for the creation of squeezing.
Abstract Squeezed states of light in a three-channel nonlinear coupler with second-order nonlinearity using the phase space method and the analytical perturbative method is reported in this paper. The system is studied under the frequency mismatch condition, where the frequencies of the pump mode are not common. The effect of frequency mismatch on the generated squeezed states is investigated. We have found that the frequency mismatch does significantly affect the behavior of the generated squeezed states only at higher values of linear coupling between the waveguides. By increasing the frequency mismatch, the squeezing pattern also evolves from a collapses-revivals-like squeezing into a constant oscillation.
Cascading of quadratic nonlinearity has been attracting great interests for its potential application to parametric devices, such as phase conjugators or effective Kerr media. In most configurations, the device consists of a single nonlinear element with uniform phase-mismatch. Cascading of several elements with different phase-mismatch has been theoretically investigated and predicted to improve performance of the device for classical applications. In this paper, amplitude squeezing in second-harmonic generation using cascaded quadratic nonlinear elements is numerically analyzed. The analyses are based on linearization of nonlinear coupling equations, where interacting fields are approximated as plane waves. Phase-mismatch of each element is varied independently and tolerance of squeezing performance to the fluctuation of the phase-mismatch is also investigated. It is predicted that the performance as a squeezing device can be also superior to that of a single element device, if a proper combination of the phase-mismatch of each element is chosen. For the fundamental wave, the tolerance to the fluctuation of the phase-mismatch will improve by nearly tenfold compared with the single element case. For the harmonic wave, squeezing beyond the limit of perfect phase-matched case (3dB) will be available, though the tolerance to the fluctuation of the phase-mismatch is quite small. These improvements can be attributed to the nonlinear phase rotation that keeps squeezed axis coincided with the amplitude phase in a stable manner.