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Physical Review A

Quantum images and critical fluctuations in the optical parametric oscillator below threshold

A. Gatti, Luigi A. Lugiato

Published Aug 1, 1995• 84 citations
Publisher

Abstract

In the first part of the paper we define the spatial spectrum of squeezing following an operational procedure, and we show in general how the spatial structure of squeezed states can be described in terms of correlation functions of quadrature components of the electric field. In the second part we formulate an appropriate quantum model for a degenerate optical parametric oscillator (OPO) with plane mirrors, and we analyze it extensively below the threshold for signal generation. The correlation length diverges when the OPO approaches threshold; this picture provides an ideal completion of the analogy with critical phenomena in second-order phase transition at equilibrium. The spatial configuration of the correlation function exhibits the phenomenon of ``quantum image:'' the signal field, which is purely generated by quantum noise, shows an ordered spatial structure which is observable via the spatial correlation function.

DOI
10.1103/physreva.52.1675

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In the first part of the paper we define the spatial spectrum of squeezing following an operational procedure, and we show in general how the spatial structure of squeezed states can be described in terms of correlation functions of quadrature components of the electric field. In the second part we formulate an appropriate quantum model for a degenerate optical parametric oscillator (OPO) with plane mirrors, and we analyze it extensively below the threshold for signal generation. The correlation length diverges when the OPO approaches threshold; this picture provides an ideal completion of the analogy with critical phenomena in second-order phase transition at equilibrium. The spatial configuration of the correlation function exhibits the phenomenon of ``quantum image:'' the signal field, which is purely generated by quantum noise, shows an ordered spatial structure which is observable via the spatial correlation function.
We illustrate the spatial quantum features that emerge in a degenerate optical parametric oscillator below threshold, where the signal field is purely generated by quantum fluctuations. We analyze the behaviour of the spatial correlation function of a generic quadrature component of the signal field in both cases of a cavity with plane mirrors and a cavity with spherical mirrors. The correlation length diverges when the semiclassical threshold for signal field generation is approached. We focus on the novel concept of quantum image, which is analyzed both in the near field and in the far field.
We consider a quantum model for a degenerate optical parametric oscillator below threshold in a cavity with spherical mirrors. The spatial correlation function of a generic quadrature component of the signal field is calculated analytically in terms of an expansion over the Gauss-Laguerre modes. The critical behavior and the quantum image phenomenon, which emerge when threshold is approached, are described in detail. We calculate also the spatial correlation function of the intensity fluctuations, which can be more easily measured by direct detection and exhibits the same qualitative features.
Quantum images, that is inhomogeneous field distributions purely generated by quantum fluctuations, persist when passing from the degenerate to the non-degenerate case of optical parametric oscillators (OPO). Below the threshold for parametric oscillation where the near-field distributions are homogeneous both in intensity and phase, appropriate spatial correlation functions anticipate the transverse spatial pattern that appears above threshold. In particular, the angular dependence of the far field spatial correlation function is able to reveal the travelling-wave nature of the phase pattern above threshold typical of nondegenerate OPOs. Cross-correlation functions between signal and idler intensities show clear evidence of the non-classical nature of the output light.
A Langevin model is introduced to study quantum fluctuations below the threshold of pattern formation for optical parametric oscillators (OPO's). In particular we compare analytical and numerical results for the OPO with one and two transverse spatial dimensions and in the presence of either plane or spherical cavity mirrors. The far-field structure and the correlation functions of the fluctuating signal field anticipate the onset of a transverse spatial pattern which arises classically even in the presence of Gaussian input beams. Correlation functions also reveal the squeezed nature of the OPO field. Numerical simulations of the Langevin model describe the result of short time measurements and show that close to threshold the near-field signal is a noisy spot pattern evolving on a time scale longer than the inverse decay rate of the resonator. This and other far-field features should be experimentally accessible.
We illustrate the concept of quantum image in the context of degenerate parametric oscillators. In the case of a cavity with planar mirrors, we show the gradual transformation of a quantum image into a classical image as the threshold region is scanned. Thus the quantum image behaves as a precursor of the roll pattern which appears above threshold. In the far field, the configuration of the spatial correlation function of the intensity and of the quadrature components shows unambiguously the quantum nature of fluctuations that generate the quantum image, leading to effects of quantum noise reduction below the shot noise level, and to the formulation of an Einstein–Podolsky–Rosen paradox.
We demonstrate that the optical parametric oscillator is an ideal system for studying spatial pattern formation and quantum effects in spatial structures. In the first part of the article we analyze a semiclassical model which includes diffraction in the paraxial approximation, and we show the formation of rolls, zig-zag patterns, dislocations, filamentation, and optical chaos. In the second part of the paper we describe the spatial structure of the squeezed vacuum state emitted by the optical parametric oscillator below the threshold for signal generation.