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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Quantum images in optical parametric oscillators with spherical mirrors and Gaussian pump

K.I. Petsas, A. Gatti, Luigi A. Lugiato

Published Dec 1, 1998• 6 citations
Publisher

Abstract

We perform a study of the quantum image phenomenon appearing below threshold in the optical parametric oscillator for the near and far field, for the case of a cavity with spherical mirrors and Gaussian pump profile. Because of the finite size of the pump, the whole set of Gauss-Laguerre cavity modes are coupled and therefore an analytical solution cannot be found. We propose a numerical treatment taking into account the mode couplings up to a finite order, and using an input-output formalism we evaluate the correlation functions for an arbitrary quadrature component of the signal field in the frequency domain. Major quantum spatial features of the plane-wave pump case are shown to persist when the pump waist is progressively decreased to realistic experimental values.

DOI
10.1088/1355-5111/10/6/011

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We perform a study of the quantum image phenomenon appearing below threshold in the optical parametric oscillator for the near and far field, for the case of a cavity with spherical mirrors and Gaussian pump profile. Because of the finite size of the pump, the whole set of Gauss-Laguerre cavity modes are coupled and therefore an analytical solution cannot be found. We propose a numerical treatment taking into account the mode couplings up to a finite order, and using an input-output formalism we evaluate the correlation functions for an arbitrary quadrature component of the signal field in the frequency domain. Major quantum spatial features of the plane-wave pump case are shown to persist when the pump waist is progressively decreased to realistic experimental values.
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.
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.
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.
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 consider here the problem of quantum noise in light produced by an optical parametric oscillator operating below threshold inside a single-sided cavity. By employing the correct boundary conditions on quantum fields at the output mirror, a procedure that is valid for arbitrary mirror transmission, we derive exact expressions for quantum-noise reduction for arbitrary pump phase and output-mirror coupling. We find that for a given output-mirror power reflectance R, squeezing of quantum noise in the intracavity field increases to a final fractional value equal to R/(1+R) as the pump intensity is increased and the oscillation threshold is approached. The output field, on the other hand, exhibits perfect squeezing of noise in one quadrature of its central Fourier component at threshold regardless of the output-mirror coupling. The present approach, which is based on boundary conditions, is quite general and applicable to any matter-field interaction problem inside a cavity.
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.