appendix A5, Physics of Speckle Pattern Formation in the Images of Rough Objects

Excerpt

Let us describe the formation of the coherent image of a rough surface illuminated by a plane monochromatic wave with the wavelength λ. Consider a somewhat idealized case where the rough surface is flat and the roughness height standard deviation is large compared to the wavelength, σ≫λ. The surface will be considered as a strongly deformed mirror (see Figs. A5.1–A5.3). In this case, it can be viewed as a set of separate elements, which can be approximated by convex or concave parabolic mirrors smoothly joined together. Their diameters are random and, on average, equal to the correlation radius of the surface roughness height ℓ. Their focal lengths are also random and, on the average, are equal to ℓ²/2σ. In this representation, the rough surface is a source of secondary spherical waves coming from separate elements and focusing at points placed on different sides of the flat mean surface. In Figs. A5.1–A5.3, these are random points A, B, C, D, F, K, which we will call points of random focusing. Their optical conjugates are points placed on different sides of the image plane. The image plane itself is optically conjugate to the flat mean surface. Each point of the coherent image of the rough surface is formed due to the superposition of waves focused by the imaging system to this point of the image plane.

Further, let us analyze how the coherent image of the rough surface is formed for different sizes d_ρ of the imaging system aperture. Here, one can distinguish between three typical cases.

Case 1: 0<d_ρ<(λr_c)/ℓ. Here, r_c is the distance from the imaging system aperture to the flat mean surface (Fig. A5.1). From these inequalities, it follows that the size (λr_c)/d_ρ of the minimum resolvable domain of the rough surface, according to the Rayleigh criterion, exceeds the roughness height correlation radius ℓ. In the figure, this is the domain between points A and K. Under this condition, the field at each point of the image plane is a superposition of a large number of waves scattered by that domain. All the waves are focused into the area shown in Fig. A5.1 by an oval. According to the central limit theorem, the resulting field distribution is close to Gaussian, and its intensity contrast C or, in other words, the contrast of the speckle pattern in the coherent image, is close to unity (C ≈ 1).