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Journal of Modern Optics

Quantum properties of a spatially-broadband traveling-wave phase-sensitive optical parametric amplifier

Michael Vasilyev, Muthiah Annamalai, Nikolai M. Stelmakh, Prem Kumar

Published Jul 15, 2010• 24 citations
Publisher

Abstract

We present the quantum theory of a spatially-multimode traveling-wave phase-sensitive optical parametric amplifier (OPA) pumped by a beam with arbitrary spatial profile. By using Green's functions of the classical OPA, we derive the normally-ordered quadrature correlators at the OPA output, which provide complete quantum description of the phase-sensitive OPA and enable determination of its independently-squeezed eigenmodes. Two analytically treatable examples of plane-wave pump and infinite spatial bandwidth of the crystal are discussed in detail.

DOI
10.1080/09500340.2010.495463

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We present the quantum theory of a spatially-multimode traveling-wave phase-sensitive optical parametric amplifier (OPA) pumped by a beam with arbitrary spatial profile. By using Green's functions of the classical OPA, we derive the normally-ordered quadrature correlators at the OPA output, which provide complete quantum description of the phase-sensitive OPA and enable determination of its independently-squeezed eigenmodes. Two analytically treatable examples of plane-wave pump and infinite spatial bandwidth of the crystal are discussed in detail.
Optical parametric amplifiers (OPAs) configured as phase-sensitive amplifiers (PSAs) can be used for noiseless optical image amplification, generation of non-classical states of light, and, in particular, for multimode squeezed light generation. For all these applications and for effective use of the quantum properties of the multimode PSA, we need to know its independently squeezed (or amplified) eigenmodes. First we present the quantum theory of a spatially multimode traveling-wave PSA pumped by a high-power pump beam with arbitrary spatial profile. To quantitatively identify the Green's function of the PSA, we have developed a semi-analytical coupled-mode-theory of the PSA using 2D Hermite-Gaussian mode expansions of the signal- and pump-beam spatial distributions. By using Green's functions of the classical OPA, we calculate the normallyordered quadrature correlators at its output, which provide complete quantum description of the phase-sensitive OPA and enable determination of its independently squeezed eigenmodes. We find the number of the supported eigenmodes and their shapes for a spatially broadband frequency-degenerate optical parametric amplifier with elliptical Gaussian pump. We conclude by discussing our recent extensions of the coupled-mode-theory to study higher-order pump modes, compact representation of the PSA eigenmodes (especially convenient for experiments), and effect of non-zero phase mismatch on the PSA eigenmodes. We expect that the results from our model can be used for optimum mode matching in phase-sensitive image amplification and multimode squeezing generation.
We investigate experimentally the eigenmodes of a Gaussian-beam-pumped traveling-wave phase-sensitive optical parametric amplifier (PSA). By varying the waist of an input LG(00) signal mode, we show that PSA performance improves with increasing spatial overlap between the input and the theoretically predicted fundamental eigenmode. For optimum waist, we report amplification and deamplification markedly higher than those observed for the traditional case of signal waist=√2× (pump waist). Lastly, we demonstrate the generation and verification of the PSA fundamental eigenmode.
We experimentally investigate the gain response of a frequency-degenerate but polarization-nondegenerate traveling-wave optical parametric amplifier that consists of a type II phase-matched potassium titanyl phosphate crystal pumped by a frequency-doubled Q-switched mode-locked Nd:YAG laser. Both the optical phase-sensitive and phase-insensitive configurations of the parametric amplifier are studied. Experimental results are in excellent agreement with the theory of an optical parametric amplifier when the Gaussian-beam nature of the various fields is taken into account. In the phase-sensitive configuration a gain of >100 (20 dB) could be easily obtained in the amplified quadrature, which is limited only by the available pump power. Because of gain-induced diffraction and phase fluctuations, however, maximum deamplification in the orthogonal quadrature is limited to <0.5(-3 dB).
We analyze the spatial-frequency dependence of the gain of phase-insensitive and phase-sensitive optical parametric amplifiers with plane-wave pumping. We discuss the dependence of their spatial bandwidths on pump power and crystal length, L, and observe that the well-known (k s/L)1/2 approximation of spatial bandwidth is not very accurate (here k s is the signal beam's wavevector magnitude). We derive an alternative approximation that is highly accurate at large gains and moderately accurate at low gains. The differences between the phase-insensitive and phase-sensitive amplifier bandwidths are shown to be insignificant for gains above several dB. Maximum phase-sensitive gain and bandwidth are realized by imposing an optimum phase profile onto the input signal's spatial spectrum, which has nearly parabolic shape with added series of π/2 phase discontinuities at spatial frequencies outside the main bandwidth of the amplifier. We show that, apart from the discontinuities, such a profile can be closely approximated by placing the image plane into the middle of the nonlinear crystal (converging input beam). The phase discontinuities contribute an additional narrow component to the point-spread function of the phase-sensitive amplifier.
We compute the eigenmodes of a spatially-broadband optical parametric amplifier with elliptical Gaussian pump and show that the well-amplified eigenmodes can be compactly represented by a low-dimensional subspace of the first few Laguerre- or Hermite-Gaussian (LG or HG, respectively) modes of an appropriate waist size. We also show that the first few eigenmodes are well matched to single LG or HG modes. For sufficiently large pump waists, the optimum waist size of the compact basis is in the vicinity of the geometric average of the pump waist size and the inverse spatial bandwidth of the nonlinear crystal in the parametric amplifier. The use of such compact representation can greatly simplify numerical computation of the spatial eigenmodes of the amplifier and thus lead to improving the experiments on traveling-wave image amplification and spatially-broadband vacuum squeezing.