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Physical Review A

Effects of spatial transverse correlations in second-harmonic generation

Diney S. Ether, P. H. Souto Ribeiro, Carlos Henrique Monken, R. L. de Matos Filho

Published May 23, 2006• 7 citations
Publisher

Abstract

Second-harmonic generation is studied for the case where the fundamental field is light produced in a spontaneous parametric down-conversion process. We show that second-harmonic generation is sensitive to the transverse correlations between signal and idler fields. In particular, when the fundamental is prepared in a state exhibitting spatial antibunching, the second-harmonic intensity may be zero, independent of the intensity of the fundamental field.

DOI
10.1103/physreva.73.053819

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Second-harmonic generation is studied for the case where the fundamental field is light produced in a spontaneous parametric down-conversion process. We show that second-harmonic generation is sensitive to the transverse correlations between signal and idler fields. In particular, when the fundamental is prepared in a state exhibitting spatial antibunching, the second-harmonic intensity may be zero, independent of the intensity of the fundamental field.
We investigate theoretically the spatial quantum correlations of the light produced by spontaneous emission in type-I second-harmonic generation in the traveling-wave configuration. It is first shown that spontaneous emission occurs at both fundamental and second-harmonic frequency. Considering the quantum fluctuations of the intensities collected in symmetrical parts of the far-field plane, nonclassical correlations below the shot noise are predicted not only at fundamental frequency, but also at second-harmonic frequency. The latter cannot be traced back to any twin-photon emission mechanism, but are generated by a secondary process acting on twin photons. This mechanism also creates correlations between fundamental and second-harmonic field, at a given transverse wave number, and at opposite wave numbers. The analysis of a simplified few-mode model, on a quantum level, provides a good qualitative understanding of these correlations.
We experimentally demonstrate how the two-photon field generated by spontaneous parametric down-conversion contains an intriguing fine structure associated with the positional spread within the photon pair. The obtained results provide a three-dimensional picture of the near-field correlations, which are determined by the phase-matching conditions. These correlations are compared with previous results on second-harmonic generation, spatial antibunching, and transverse entanglement in parametric down-conversion.
The quantum state for spontaneous parametric down-conversion was evaluated to include information about the transverse spatial characteristic of the down-conversion fields. This state was used to calculate the angular spectrum of down-conversion in the case of both single-channel and coincidence measurements, while retaining the characteristic nonclassical correlations. Actual transverse profiles for typical experimental parameters were evaluated numerically and measurements were performed with results that compared well with the numerical calculation. This theoretical spatial model was then applied nonrigorously to the case of induced coherence without induced emission. The theoretical results implied that there was a reduction in the degree of induced coherence due strictly to the spatial distribution of the down-conversion beams. This was interpreted as a result of increased distinguishability of the overlapping idler photons. Measurements were conducted which support these results qualitatively. In addition, the present theory explained an unexpected quantitative discrepancy between second- and fourth-order visibilities in previous induced-coherence experiments. This discrepancy was interpreted phenomenologically as a difference between intrinsic and nonintrinsic indistinguishability, and some implications towards quantum measurement theory and the role of the observer were suggested.
We investigate the spatial quantum noise properties of the one-dimensional transverse pattern formation instability in intracavity second-harmonic generation. The Q representation of a quasi-probability distribution is implemented in terms of nonlinear stochastic Langevin equations. We study these equations through extensive numerical simulations and analytically in the linearized limit. Our study, made below and above the threshold of pattern formation, is guided by a microscopic scheme of photon interaction underlying pattern formation in second-harmonic generation. Close to the threshold for pattern formation, beams with opposite direction of the off-axis critical wave numbers are shown to be highly correlated. This is observed for the fundamental field, for the second-harmonic field, and also for the cross-correlation between the two fields. Nonlinear correlations involving the homogeneous transverse wave number, which are not identified in a linearized analysis, are also described. The intensity differences between opposite points of the far fields are shown to exhibit sub-Poissonian statistics, revealing the quantum nature of the correlations. We observe twin beam correlations in both the fundamental and second-harmonic fields, and also nonclassical correlations between them.
In experiments with spontaneous parametric down conversion the near-field plane is usually defined as the output facet of the generating crystal. Experimentally, however, the spatial correlations are accessed via an imaging system and observed in a region around the image of the output facet. We show that the imaged two-photon field has a very rich transverse and longitudinal structure, which is determined by the phase-matching conditions. We observe many intriguing three-dimensional structures that demonstrate the presence of spatial antibunching, an extreme localization of twin photons, and spatial correlations that resemble Bessel beams in propagation. We link these observations with previous results in second harmonic generation and predict the presence of fourth-order phase singularities. Both experiments and theory are presented, yielding further insight into the nature of the two-photon field.