Speckle size of light scattered from 3D rough objects

From scalar Helmholtz integral relation and by coordinate system transformation, this paper begins with a derivation of the far-zone speckle field in the observation plane perpendicular to the scattering direction from an arbitrarily shaped conducting rough object illuminated by a plane wave illumination, followed by the spatial correlation function of the speckle intensity to obtain the speckle size from the objects. Especially, the specific expressions for the speckle sizes of light backscattered from spheres, cylinders and cones are obtained in detail showing that the speckle size along one direction in the observation plane is proportional to the incident wavelength and the distance between the object and the observation plane, and is inverse proportional to the maximal illuminated dimension of the object parallel to the direction. In addition, the shapes of the speckle of the rough objects with different shapes are different. The investigation on the speckle size in this paper will be useful for the statistical properties of speckle from complicated rough objects and the speckle imaging to target detection and identification. ©2012 Optical Society of America OCIS codes: (290.5880) Scattering, rough surfaces; (030.6140) Speckle; (030.6600) Statistical optics References and links 1. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena (Springer, 1975), vol. 9, pp. 9–75. 2. Q. B. Li and F. P. Chiang, “A new formula for fringe localization in holographic interferometry,” Opt. Lasers Eng. 8, 1–21 (1988). 3. D. W. Li, F. P. Chiang, and J. B. Chen, “Statistical analysis of one-beam subjective laser speckle interferometry,” J. Opt. Soc. Am. A 2(5), 657–666 (1985). 4. T. Asakura and N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. (Berl.) 25(3), 179–194 (1981). 5. T. 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Introduction
Illumination of an optically rough object by a coherent light produces a grainy structure in space which is known as a speckle pattern [1]. Statistical properties of the resultant speckle fields are usually examined by studying the space-time cross-correlation function in the observation plane that is perpendicular to the optical axis. One of the statistical properties is the average size of speckle which defines the extent of the spatial correlation of the pattern, and any two points apart beyond that are regarded as practically uncorrelated. The average speckle sizes are important to several areas of our interest, e.g., laser holographic interferometry [2], and laser speckle interferometry [3]. Asakura and Takai systematically discussed the space-time correlation function of the dynamic speckles produced by a moving diffuse object under Gaussian beam illumination [4]. A useful study of the statistical properties of dynamic speckles was presented by Yoshimura, in which other illumination conditions and optical configuration geometries were also examined [5]. Leushacke and Kirchner investigated the 3D structure of static speckles under plane wave illumination for rectangular and circular apertures which could lead to a greater accuracy of different speckleinterferometric measurement technique [6]. Li and Chiang also studied 3D speckle and measured the lateral and on-axis longitudinal speckle size, respectively, in a free space from a circular scattering area with some supporting results from an experimental investigation [7]. Yoshimura and Iwamoto researched the 3D space-time cross-correlation function for freespace geometry under Gaussian beam illumination [8]. The longitudinal statistical properties of speckle patterns produced under illumination with a circular aperture and ring-slit apertures having different ratios of the inner to the outer radius was investigated, and also the experimental results were given to prove the theory [9]. By ABCD-matrix theory and the paraxial approximation of the Huygens-Fresnel formulation of wave optics, Yura et al. derived the general analytic expressions for the mean spot size and both the mean speckle size and the temporal coherence length and gave a general description of both speckle boiling and speckle translation in an arbitrary observation plane [10], later they investigated the threedimensional speckle dynamics results from an in-plane translation, an out-of-plane rotation, and an in-plane rotation of a diffuse scattering object illuminated by a Gaussian-shaped laser beam [11]. The speckle size and the degree of correlation of the speckle intensities from far diffuse objects illuminated by Gaussian beam were studied showing to be independent of the surface roughness, and only determined by the laser beam waist [12]. The spatial correlation coefficient at any point in the hemisphere were calculated using the Rayleigh-Sommerfeld-Smythe integral formula and also the non-paraxial formulas for the speckle sizes were obtained [13]. The modified space-time correlation function of the light-intensity fluctuations was introduced to estimate the correlation parameters of a dynamic speckle pattern [14].The relationship between the speckle size and blood flow velocity was investigated when the blood flow velocity is close to and greater than scan velocity [15]. For determining axial and lateral speckle sizes, the analytical formulas was derived using the linear canonical transform and ABCD ray matrix techniques to describe these general optical systems and then extended to non-axial speckles [16]. Latterly, the space cross-correlation function of speckle was studied and the prediction of the model in this paper were verified against experimental results for both lateral and longitudinal speckle decorrelations and on-and off-axis cases [17,18].
Essentially, these papers above most concentrated on the investigation of the time and spatial statistical properties of speckles from rough plane surface in the observation plane. Actually, the shape and the size of the speckle also depends greatly on that of the scattering objects [19]. Berlasso et al. derived in detail the autocorrelation function of the intensity scattered from cylindrical slightly rough surfaces based on the Kirchhoff scalar diffraction theory and discussed the relationship of the speckle size with the size of the cylinder [20,21]. In this paper, we will provide the specific expressions for the speckle size from 3D optically rough objects, such as spheres, cylinders, cones, to show its dependence on the shape and the size of the scattering objects.
In this paper, the average size of the speckle in a free space produced by 3D conducting rough objects is investigated to illustrate the relationship of the speckle size with the shape and the size of the scattering objects. From the scalar Helmholtz integral relation, the speckle field in the far field from an arbitrarily shaped object in the observation plane perpendicular to the scattering direction is derived detailed first, and then the spatial correlation function of the speckle intensity. At last, taking rough spheres and cylinders and cones for examples, the explicit formulas for the speckle size in the observation are obtained.

Speckle field from rough objects in the observation plane perpendicular to the scattered direction
⋅ illuminates a rough convex conducting object; the scattering geometry is illustrated by Fig. 1. The surface S is the unperturbed surface, n is the corresponding external normal, c r is its vector distance and i θ is the local incident angle at c r while S′ is the roughened surface which is the surface S plus a random fluctuation ( ) c r ς , N is its corresponding normal, and r′ is the vector distance, i θ ′ is the incident angle at r′ .k andˆs k are the incident unit vector and the scattering unit vector, respectively. 2 / k π λ = is the wave number, λ is the wavelength used. The time harmonic factor is omitted for convenience. According to the scalar Helmholtz integral relation, the speckle field from a rough object at a receiver point P in the far field can be expressed as [22,23] omitting the nonessential factor before the integral. The incident wave vector is ( ) sin cos ,sin sin , cos where ( ) 1 1 , z x y is the curve function of the object. Equation (2)can be series expanded as neglected the higher-order terms, and 2 2 2 r r r R x y z = + + is the distance of the receiver point from the origin. Provided that the size of the object is much smaller than the distance R that the following inequality is available Therefore, s c r r − in the denominators in Eq. (1) can be simplified as (1), we get the approximate expression for the speckle field from a diffuse convex surface Since c r R << , Eq. (6) can be further approximated as The chief purpose of the study in this paper is the spatial correlation function of the speckle in the observation plane; therefore, the field distribution on the observation is needed Then the transformation relation between the object coordinate system oxyz and the Inserting Eq. (8) into Eq. (7), the speckle field in the observation plane perpendicular to the scattering direction can be finally obtained.

Speckle size for 3D various shaped rough objects
In this section, the speckle size for various rough objects in the observation plane perpendicular to the scattering direction will be calculated from Eqs. (7) and (8). Being a critical parameter in many application [24], the speckle size from rough plane surface depends closely on the lenses, apertures, and sections of free space in an optical system and also the wavelength used [1], and the exact same result could be found using geometric arguments discussed by Lyle Shirley et al [25]. The geometry of the speckle size is shown by Fig. 3 below, where ( ) However, when the scattering objects are three-dimensional curve surface, the speckle pattern will be different from that of the rough plane surface [19], whose simulated results has been shown in Fig. 4. From the figure, one can see that different shapes of the scattering objects will result in different shapes of the speckle in the observation plane. Therefore, we will discuss the influence factors on the speckle sizes scattered from different 3D shaped rough objects. Following Goodman [1], the 'width' of the spatial correlation function of the speckle, i.e., the value of the spatial location difference when the correlation coefficient gets its first minimum, is defined as the speckle size. Consequently, the spatial correlation function of the speckle is utilized to study the speckle size.
The speckle intensity is ( ) ( ) ( ) * s s I P E P E P = , hence the spatial correlation function 12 C of the speckle intensity fluctuation can be expressed as where 1 s E and 2 s E are the speckle fields at different points 1 P and 2 P , respectively. In this paper, the critical assumptions have typically been made that the rough surface on the diffuser obeys Gaussian random distribution, and its roughness is larger than the incident wavelength. Accordingly, the speckle fields in the observation plane obey a complex Gaussian random process and the scattered fields from the diffuser surface are delta correlated.
By the assumptions above and with Gaussian moment theorem [26], the fourth-order moment in Eq. (9) can be obtained as here the factor before the integral is also neglected.
Since the scattered fields on the diffuser surface are delta correlated ( ) It can be seen clearly from Eq. (15) that we can get a closed-form solution for the average speckle size as long as the exact shape of the scattering object is known, hereby, in the following the speckle sizes of three typical objects(a sphere, a cylinder and a cone)are derived.

Rough spheres
We are now to calculate the correlation coefficient for the speckle from rough spheres of radius a, as in Fig. 5  For convenience to the integral, also in the other derivation of the speckle size below for both the cylinder and the cone, it is only considered the backscattering from the object, that is, 0 where ρ is ξ ρ or η ρ . It is clear that the relationship between the normalized spatial correlation of the speckle intensity and the function ( ) The speckle size in ξ -direction is same as that in η -direction, as show by the middle pattern in Fig. 4, it is positive proportional to the incident wavelength and the distance from the origin of the object to the observation plane, and is inverse proportional to the radius of the sphere. For the case of rough cylinders with radius a and length b, as in Fig. 7 The scattering geometry and the profile of the absolution of the function ( ) ρ Γ are illustrated by Fig. 8 below.  31), it is clear that the speckle size is closely dependent the dimension of the scattering object where it is along, that is, it has strong directional property, as illustrated by the right pattern of Fig. 4.

Rough cones
Additionally, conical objects are often used in practice, in the following we will consider the case that the speckle from rough cones with half-cone angle α and height h. The coordinate system of the cone is shown in Fig. 9.  a htgα = is the bottom radius of the cone. By Eq. (40), the speckle size of the cone in the observation plane closely depends on the incident wavelength, the distance from the object, and the maximal dimension of the object, which is consistent with that of the sphere (Eq. (22)) and the cylinder (Eq. (31)).

Conclusion
With the scalar Helmholtz integral relation, the speckle field in the far field from arbitrarily shaped 3D optically rough objects in the observation plane which is perpendicular to the scattering direction by coordinate system transformation is derived detailed, and then the speckle sizes are investigated by the spatial correlation function of the speckle intensity backscattered from 3D conducting rough objects based on Gaussian moment theorem. Taking three kinds of simple objects for example, sphere, cylinder and cone, the specific expressions for the speckle sizes are obtained to analyses their relationship with the shape and the size illuminated of the 3D objects, the incident condition and the distance from the object to the observation plane. The results show that the speckle size along one direction in the observation plane is proportional to the incident wavelength and the distance, and is inverse proportional to the maximal dimension of the object parallel to the direction. In addition, for different shape of 3D objects, the shapes of the speckle are different. The work in this paper will be further applied to investigate the statistical properties of speckle from complicated rough objects, the importance of these equations for the speckle size is immense, and will help us describe both the requirements for a full-size LADAR simulator and the computational simulation of speckle patterns to aid in three-dimensional imaging applications.