Speckle Pattern Formed by Laser Scattering from Particles in Gases*

SHERMAN G E E Arnold Engineering Development Center,

Arnold Air Force Station, Tennessee 37389 (Received 10 November 1966)

INDEX HEADINGS: Laser; Scattering.

THE spatial autocorrelation function and power spectral density of the speckle pattern produced by laser scattering

from a diffuse surface was established by Goldfìscher in a recent paper.1 He showed that at a given observation distance the spatial autocorrelation function (and hence the power spectral density) depends only on the intensity variation of the incident radiation over the illuminated area, and is independent of the random phase distribution over this area. It seems worthwhile to point out that these results also appear valid for laser scattering from particles in a gas.

Let us consider a scattering volume of gas V as illustrated by Fig. 1 in which are suspended particles having sizes comparable to the wavelength ( ~ l µ) of the incident radiation. Scattering from the suspended particles consequently predominates over scattering from the gas molecules. The particle size is assumed to be much smaller than the dimensions (Vlβ) of the scattering volume so that a sufficient number of particles can be contained in V to produce scattering similar to scattering from a diffuse surface. We assume a single-scattering process and that the di­mensions of the scattering volume are small compared with the observation distance R, i.e., V1/3≪R. Hence, the power returned from each random scatterer in the volume will be approximately the same, and only their phase contributions are important in determining the resultant speckle pattern. Further­more, we assume that the Brownian motion associated with each scatterer can be neglected so that scatterers in the gas are essen­tially immobile. This assumption is justifiable as long as the observation time is short compared to the mean time required for a particle to move a distance comparable with a wavelength.

Under the above conditions, then, scattering from particles in a gas resembles scattering from a diffuse surface. The only major difference is that in a gas the scatterers are distributed in depth as well as breadth, whereas on a diffuse surface the scatterers are distributed only in breadth. In both cases the scatterers are randomly positioned and radiate with random phase. Now let

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FIG. 1. Scattering geometry.

us consider all particles in V to be projected onto an equivalent surface S (Fig. 1) and account for their original positions by appropriate modification of the phase random variable of each scatterer. That is, if a scatterer was originally located a distance ξ from S, then the scattered radiation it produces is the same as would be produced if the particle was removed to ξ = 0 on S, except for the phase factor exp[ –j(k i – k s ) ∙ξ] , where ki and ks

are the incident and scattered wave numbers, respectively. Since the scatterers are randomly located in V, then ξ is a random vari­able. (More exactly, both ξ and the directional cosines are random variables since the observation angles for each scatterer in V, as seen by a single speckle at R, are slightly different.) The cor­responding phase factor then is random and can be absorbed in the phase random variable of each scatterer. Hence, with appropriate modification of the phase random variable, scattering from V may be considered to be equivalent to scattering from S where all particles in V have been removed to S. However, since the spatial autocorrelation function of the speckle pattern formed by scattering from a surface is independent of the random phase distribution over the illuminated area, then it follows that the autocorrelation function of the speckle pattern cannot distinguish between scattering from V and scattering from S when all particles in V have been removed to S. Consequently, the autocorrelation function of the speckle pattern must be identical in both cases.

We thus conclude that the autocorrelation function of the speckle pattern formed by laser scattering from particles in a gas and from a diffuse surface will be the same under fairly general conditions. Of course, the intensity of the speckle pattern formed in the two cases will most likely be quite different, de­pending on the comparative scattering cross sections of the diffuse surface and the particles in the gas.

* Research sponsored by Arnold Engineering Development Center, Air Force Systems Command, United States Air Force, under Contract AF 40(600)-1200. Further reproduction is authorized to satisfy needs of the U. S. Government. 1 L. I. Goldfischer, J. Opt. Soc. Am. 55, 247 (1966).

June 1967 L E T T E R S T O T H E E D I T O R 837