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Physical Review A

Light scattering and nonlinear optical response near a critical point

Robert M. McGraw

Published Aug 1, 1990• 14 citations
Publisher

Abstract

The connection between light scattering and nonlinear optical response is examined quantitatively for Kerr media and is generalized to include isotropic materials with nonlocal interactions near a critical point. A number of important quantities, including the nonlinear refractive index (${\mathit{n}}_{2}$) and the four-wave-mixing coefficient, are shown to acquire increasing grating wave-vector (q) dependence as the critical point is approached. This dependence is identical to the q dependence obtained for light scattering in the Ornstein-Zernike theory. Illustrative calculations are presented for near-critical xenon using the Redlich-Kwong equation of state and parameters obtained from independent measurements. These results indicate ${\mathit{n}}_{2}$ values of ${10}^{\mathrm{\ensuremath{-}}15}$--${10}^{\mathrm{\ensuremath{-}}14}$ ${\mathrm{m}}^{2}$/W and response times in the 1--10-\ensuremath{\mu}sec range. Calculations are presented for the phase-conjugate reflectivity in degenerate four-wave mixing that include both nonsaturable background loss and light-scattering noise. Reflectivities in excess of unity are predicted for 1.06-\ensuremath{\mu}m wavelength and cw pump intensities in the 10-kW/${\mathrm{cm}}^{2}$ range.

DOI
10.1103/physreva.42.2235

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The connection between light scattering and nonlinear optical response is examined quantitatively for Kerr media and is generalized to include isotropic materials with nonlocal interactions near a critical point. A number of important quantities, including the nonlinear refractive index (n2{\mathit{n}}_{2}n2​) and the four-wave-mixing coefficient, are shown to acquire increasing grating wave-vector (q) dependence as the critical point is approached. This dependence is identical to the q dependence obtained for light scattering in the Ornstein-Zernike theory. Illustrative calculations are presented for near-critical xenon using the Redlich-Kwong equation of state and parameters obtained from independent measurements. These results indicate n2{\mathit{n}}_{2}n2​ values of 10\ensuremath−15{10}^{\mathrm{\ensuremath{-}}15}10\ensuremath−15--10\ensuremath−14{10}^{\mathrm{\ensuremath{-}}14}10\ensuremath−14 m2{\mathrm{m}}^{2}m2/W and response times in the 1--10-\ensuremath{\mu}sec range. Calculations are presented for the phase-conjugate reflectivity in degenerate four-wave mixing that include both nonsaturable background loss and light-scattering noise. Reflectivities in excess of unity are predicted for 1.06-\ensuremath{\mu}m wavelength and cw pump intensities in the 10-kW/cm2{\mathrm{cm}}^{2}cm2 range.
Thermal light-scattering fluctuations are examined as a fundamental noise source inherent in nonlinear optical processes involving Kerr and artificial Kerr media. Stochastic models are introduced to simulate noise in optical phase conjugation via degenerate four-wave mixing, and in signal amplification via nondegenerate two-wave-mixing processes. Significant noise fluctuations in the amplitude and phase components of the processed signals are shown to occur even with incident signal powers in the milliwatt range. For time-dependent processes, including nondegenerate two-wave mixing, the fluctuation-dissipation theorem is shown to generalize a previously derived static susceptibility relation between light-scattering and nonlinear optical response to media exhibiting both power dissipation and nonlocal interaction.
This paper examines thermal (light-scattering) fluctuations as a dominant noise source in nonlinear optical processes. Fundamental limits to conjugate wave fidelity and signal power requirement are obtained through the statistical thermodynamic treatment of light-scattering noise in four-wave mixing based on the fluctuation-dissipation theorem. Several types of nonlinear media are examined including artificial Kerr suspensions, isotropic Kerr media, and fluids near a critical point. Where measurements are available, excellent quantitative agreement is obtained between theory and experiment.
The exact theory of degenerate four-wave mixing in a Kerr-like medium in the Bragg scattering regime is used to study the reflectivity of phase-conjugate mirrors pumped by intense laser beams. The analysis takes into account pump depletion, nonlinear phase modulation and the relaxation time of the nonlinearity. The steady-state response exhibits multivalued reflectivity, hysteresis and potential bistability; however, the transient response shows that at most one value of the phase-conjugate reflectivity is stable. Periodic or chaotic pulsations are predicted beyond threshold values of pump intensity. The onset of a natural oscillation in the Kerr-like medium is also predicted in the absence of a continuous probe beam; the resulting field amplitude is stable and weakly sensitive upon the presence of external noise.
We consider degenerate four-wave mixing in a Kerr medium with a standard noncollinear geometry and the polarization directions of all the beams parallel to one another. The two pump beams, the probe beam, and the phase-conjugate beam are denoted by the subscripts 1,2,4 and 3, respectively. The nonlinear medium is placed between z = 0 and z = 1; the values of the intensities of the beams at z = 0 and 1 are denoted by Ii0 andIi1, where i = 1,2,3,4. Beams 2 and 4 are incident at z = 1; and beam 1 is incident at z = 0. We have determined the dependence of the phase-conjugate reflectivity R = I31/I41 and the pump transmission T = I11/I10 on the following parameters: the probe ratio, q=I41/(I10 + I21); the pump ratio, r = I21/I 1 0; the absorption coefficient–path length product, al; and the coupling constant of the medium, q. In the absence of nonlinear refraction, pump depletion, and linear absorption in the medium, R = tan2q. When all of these effects are taken into account, the maximum value of R is given by Rmax = exp (-2α1)/(q(1 + r)). The minimum value that T can reach is proportional to q for q < 1. Even for small values of q, multistable solutions for R and T are obtained when q is greater than a threshold value, which we denote by qth. We have determined the dependence of qth on the parameters q and r.
A theory of degenerate four-wave mixing in absorbing media is developed in terms of a general nonlinear susceptibility. The predictions of the theory are examined in both high and low reflectivity regimes and the specific nonlinearity-inducing mechanisms of thermally induced refractive index changes, the Kerr effect, and saturable absorption are considered. Such a theory is of particular relevance to degenerate four-wave mixing experiments in the ultraviolet and the predictions of the theory are compared with results obtained from such experiments.