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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 4 — Feb. 13, 2012
  • pp: 4726–4737
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Speckle size of light scattered from 3D rough objects

Geng Zhang, Zhensen Wu, and Yanhui Li  »View Author Affiliations


Optics Express, Vol. 20, Issue 4, pp. 4726-4737 (2012)
http://dx.doi.org/10.1364/OE.20.004726


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Abstract

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 OSA

1. 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

1. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena (Springer, 1975), vol. 9, pp. 9–75.

]. 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

2. Q. B. Li and F. P. Chiang, “A new formula for fringe localization in holographic interferometry,” Opt. Lasers Eng. 8, 1–21 (1988).

], and laser speckle interferometry [3

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). [CrossRef]

]. 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

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). [CrossRef]

]. 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

5. T. Yoshimura, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am. A 3(7), 1032–1054 (1986). [CrossRef]

]. 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 speckle-interferometric measurement technique [6

6. L. Leushacke and M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. A 7(5), 827–832 (1990). [CrossRef]

]. 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

7. Q. B. Li and F. P. Chiang, “Three-dimensional dimension of laser speckle,” Appl. Opt. 31(29), 6287–6291 (1992). [CrossRef] [PubMed]

]. Yoshimura and Iwamoto researched the 3D space-time cross-correlation function for free-space geometry under Gaussian beam illumination [8

8. T. Yoshimura and S. Iwamoto, “Dynamic properties of three-dimensional speckles,” J. Opt. Soc. Am. A 10(2), 324–328 (1993). [CrossRef]

]. 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

9. M. Ibrahim, J. Uozumi, and T. Asakura, “Longitudinal correlation properties of speckles produced by Ring-Slit illumination,” Opt. Rev. 5(3), 129–137 (1998). [CrossRef]

]. 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

10. H. T. Yura, B. Rose, and S. G. Hanson, “Dynamic laser speckle in complex ABCD optical systems,” J. Opt. Soc. Am. A 15(5), 1160–1166 (1998). [CrossRef]

], later they investigated the three-dimensional 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

11. H. T. Yura, S. G. Hanson, R. S. Hansen, and B. Rose, “Three-dimensional speckle dynamics in paraxial optical systems,” J. Opt. Soc. Am. A 16(6), 1402–1412 (1999). [CrossRef]

]. 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

12. G. J. Guo, S. K. Li, and Q. L. Tan, “Statistical properties of laser speckles generated from far rough surfaces,” Int. J. Infrared Millim. Waves 22(8), 1177–1191 (2001). [CrossRef]

]. 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

13. K. Chu and N. George, “Correlation function for speckle size in the right-half-space,” Opt. Commun. 276(1), 1–7 (2007). [CrossRef]

]. The modified space-time correlation function of the light-intensity fluctuations was introduced to estimate the correlation parameters of a dynamic speckle pattern [14

14. D. V. Semenov, S. V. Miridonov, E. Nippolainen, and A. A. Kamshilin, “Statistical properties of dynamic speckles formed by a deflecting laser beam,” Opt. Express 16(2), 1238–1249 (2008). [CrossRef] [PubMed]

].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

15. T. Xu and G. R. Bashford, “Further progress on lateral flow estimation using speckle size variation with scan direction,” in IEEE International Ultrosonics Symposium Proceedings, pp. 1383–1386 (2009).

]. 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

16. J. E. Ward, D. P. Kelly, and J. T. Sheridan, “Three-dimensional speckle size in generalized optical systems with limiting apertures,” J. Opt. Soc. Am. A 26(8), 1855–1864 (2009). [CrossRef] [PubMed]

]. 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

17. D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields. Part I. theory and numerical investigation,” J. Opt. Soc. Am. A 28(9), 1896–1903 (2011). [CrossRef]

, 18

18. D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields. Part II. experimental investigation,” J. Opt. Soc. Am. A 28(9), 1904–1908 (2011). [CrossRef]

].

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

19. D. R. Dunmeyer, Laser Speckle Modeling for Three-Dimensional Metrology and LADAR, M. Eng. dissertation (Massachusetts Institute of Technology, 2001).

]. 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

20. R. Berlasso, F. Perez Quintián, M. A. Rebollo, C. A. Raffo, and N. G. Gaggioli, “Study of speckle size of light scattered from cylindrical rough surfaces,” Appl. Opt. 39(31), 5811–5819 (2000). [CrossRef] [PubMed]

, 21

21. R. G. Berlasso, F. P. Quintián, M. A. Rebollo, N. G. Gaggioli, B. L. Sánchez, and M. E. Bernabeu, “Speckle size of light scattered from slightly rough cylindrical surfaces,” Appl. Opt. 41(10), 2020–2027 (2002). [CrossRef] [PubMed]

]. 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.

2. Speckle field from rough objects in the observation plane perpendicular to the scattered direction

A plane waveEi(r)=exp(ikk^r)illuminates a rough convex conducting object; the scattering geometry is illustrated by Fig. 1
Fig. 1 scattering geometry for a rough object.
. The surface S is the unperturbed surface, n^ is the corresponding external normal, rcis its vector distance and θiis the local incident angle at rc while Sis the roughened surface which is the surface S plus a random fluctuationς(rc), N^is its corresponding normal, andris the vector distance, θi is the incident angle at r.k^ andk^sare the incident unit vector and the scattering unit vector, respectively. k=2π/λis the wave number,λis 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

22. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press,1978).

, 23

23. G. Zhang and Z. Wu, “Two-frequency mutual coherence function of scattering from arbitrarily shaped rough objects,” Opt. Express 19(8), 7007–7019 (2011). [CrossRef] [PubMed]

]
Es(rs)=Sk^n^exp[ikVn^ς(rc)]exp[ik(|rsrc|+k^rc)]|rsrc|dS
(1)
omitting the nonessential factor before the integral.

The incident wave vector isk^=(sinθicosφi,sinθisinφi,cosθi), and the scattering wave vector isk^s=(sinθscosφs,sinθssinφs,cosθs), (θi,φi) is the incident direction, while (θs,φs)is the scattering direction, andV=k^k^s.

In Eq. (1), rsis the vector between the receiver point P(xr,yr,zr)and the origin of the object while |rsrc|is the distance between the point P and the point on the object surface
|rsrc|=(xrx1)2+(yry1)2+[zrz(x1,y1)]2
(2)
wherez(x1,y1)is the curve function of the object. Equation (2)can be series expanded as
|rsrc|=rscR+x12+y12+z122Rx1xr+y1yr+z1zrR
(3)
neglected the higher-order terms, andR=xr2+yr2+zr2is the distance of the receiver point from the origin. Provided that the size of the object is much smaller than the distance Rthat the following inequality is available

R>>x12+y12+z122Rx1xr+y1yr+z1zrR=rc22RrcrsR
(4)

Therefore,|rsrc|in the denominators in Eq. (1) can be simplified as

|rsrc|R=xr2+yr2+zr2
(5)

Substituting Eqs. (3)(5) into Eq. (1), we get the approximate expression for the speckle field from a diffuse convex surface

Es(rs)=exp(ikR)RSk^n^exp(ikVn^ς)exp[ik(rc22RrcrsR+k^rc)]dS
(6)

Sincerc<<R, Eq. (6) can be further approximated as

Es(rs)=exp(ikR)RSk^n^exp(ikVn^ς)exp[ik(rcrsR+k^rc)]dS
(7)

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 to obtain first. The observation coordinate systemoξηζis illustrated in Fig. 2
Fig. 2 Geometrical arrangement.
, choosing the direction of the scattered light as the ζ-axis, and the ξoη plane as the observation plane, obeing the origin of the coordinate systemoξηζ, the observation point P close to the ζ-axis.

Then the transformation relation between the object coordinate systemoxyzand the observation coordinate system oξηζis

(xryrzr)=(cosθscosφssinφssinθscosφscosθssinφscosφssinθssinφssinθs0cosθs)(ξηR)
(8)

Inserting Eq. (8) into Eq. (7), the speckle field in the observation plane perpendicular to the scattering direction can be finally obtained.

3. 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

24. R. K. Erf, Speckle Metrology (Speckle Metrology, 1978).

], 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

1. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena (Springer, 1975), vol. 9, pp. 9–75.

], and the exact same result could be found using geometric arguments discussed by Lyle Shirley et al [25

25. L. G. Shirley, E. D. Ariel, G. R. Hallerman, H. C. Payson, and J. R. Vivilecchia, “Advanced techniques for target discirmination using laser speckle,” Linc. Lab. J. 5(3), 367–440 (1992).

]. The geometry of the speckle size is shown by Fig. 3
Fig. 3 Geometric representation of the speckle statistics d(ξd,ηd)andd||(ζd).
below, whered(ξd,ηd)is the size vector of the speckle in the observation plane, i.e., the lateral speckle size, andd||(ζd)is the longitudinal speckle size. Thus the ability to predict and control the speckle size can be used to improve system performance.

However, when the scattering objects are three-dimensional curve surface, the speckle pattern will be different from that of the rough plane surface [19

19. D. R. Dunmeyer, Laser Speckle Modeling for Three-Dimensional Metrology and LADAR, M. Eng. dissertation (Massachusetts Institute of Technology, 2001).

], whose simulated results has been shown in Fig. 4
Fig. 4 Simulated speckle patterns from rough objects (left: rough plane; middle: rough sphere; right: rough cylinder).
. 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

1. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena (Springer, 1975), vol. 9, pp. 9–75.

], 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 isI(P)=Es(P)Es*(P), hence the spatial correlation functionC12of the speckle intensity fluctuation can be expressed as
C12=I1I2I1I2=Es1Es1*Es2Es2*|Es1|2|Es2|2
(9)
whereEs1andEs2are the speckle fields at different pointsP1andP2, 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

26. L. G. Shirley and N. George, “Speckle from a cascade of two thin diffusers,” J. Opt. Soc. Am. A 6(6), 765–781 (1989). [CrossRef]

], the fourth-order moment in Eq. (9) can be obtained as
Es1Es1*Es2Es2*=|Es1|2|Es2|2+|Es1Es2*|2
(10)
then the normalized spatial correlation function of the speckle intensity can be written in terms of the correlation function of the speckle field

γ12=C12/(C11C22)1/2=|Es1Es2*|2/(I1I2)
(11)

By Eq. (7) and Eq. (8), the functionγ12can then be attained. For simplicity, it is only considered the case that the incident direction is(π/2,π),k^=(1,0,0)and the scattering direction is(π/2,φs), k^s=(cosφs,sinφs,0). The speckle field in the observation plane is
Es(ξ,η)=SdSnxexp(ikVn^ς)exp(ikx1)exp[ik(x1cosφs+y1sinφs)]×exp[ikη(x1sinφs+y1cosφs)/R]exp(ikξz1/R)
(12)
here the factor before the integral is also neglected.

Then we can get

Es1Es2*=SdS1dS2nx1nx2exp[ik(Vn^1ς1Vn^2ς2)]exp[ik(x1x2)]×exp{ik[(x1x2)cosφs+(y1y2)sinφs]}exp[ik(ξ1z1ξ2z2)/R]×exp{ik[(η1x1+η2x2)sinφs+(η1y1η2y2)cosφs]/R}
(13)

Since the scattered fields on the diffuser surface are delta correlated

exp[ik(Vn^1ς1Vn^2ς2)]=δ(x1x2,y1y2)
(14)

Let ξd=ξ1ξ2,ηd=η1η2, Eq. (13) can reduce to

Es1Es2*=SdSnx2exp{ik[ξdz+ηd(xsinφs+ycosφs)]/R}
(15)

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.

3.1 Rough spheres

We are now to calculate the correlation coefficient for the speckle from rough spheres of radius a, as in Fig. 5
Fig. 5 Coordinate system for a sphere.
below. The coordinate of the point is(a,θ,φ), then making an integral transformationx=asinθcosφ,y=asinθsinφ,z=acosθand the normal vector for the sphere isn^=(sinθcosφ,sinθsinφ,cosθ), and then Eq. (15) is transformed into

Es1Es2*=a20πdθsin3θπ/2+φsπ/2dφcos2φexp{ika[ξdcosθ+ηdsinθsin(φφs)]/R}
(16)

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, φs=0. Afterwards, letηd=0andξd=0, respectively, the corresponding speckle sizes ξdandηd can be calculated.
Es1Es2*=a20πdθsin3θexp(iρξcosθ)π/2π/2dφcos2φ=2πa2[ρξcos(ρξ)+sin(ρξ)]/ρξ3
(17)
whereρξ=kaξd/R, and ρη=kaηd/R

Es1Es2*=a20πdθsin3θπ/2π/2dφcos2φexp(iρηsinθsinφ)=2πa2[ρηcos(ρη)+sin(ρη)]/ρη3
(18)

From Eq. (17) and Eq. (18), it is clear that in the observation plane perpendicular to the backscattering direction, the speckle size for spheres along ξ-axis is same as that along η-axis. And the average speckle intensity from rough sphere is

I1=Es1Es1*=a20πdθsin3θπ/2π/2dφcos2φ=2πa2/3
(19)

Taking a new function
Γ(ρ)=Es1Es2*/Es1Es1*
(20)
utilizing Eq. (17) and Eq. (19) or Eq. (18) and Eq. (19), Eq. (20) can then be obtained
Γ(ρ)=3[ρcos(ρ)+sin(ρ)]/ρ3
(21)
whereρisρξorρη. It is clear that the relationship between the normalized spatial correlation of the speckle intensity and the functionΓ(ρ)isγ12(ρ)=|Γ(ρ)|2, thus the speckle size can also be obtained using the absolution of the functionΓ(ρ).

As illustrated by Fig. 6
Fig. 6 Correlation function|Γ(ρ)|from a sphere versusρ.
, whenρ=4.5,Γ(ρ)gets its first zero-value, then the speckle size will be

ξd=ηd=4.5R/ka=0.716λR/a
(22)

The speckle size inξ-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.

3.2 Rough cylinders

For the case of rough cylinders with radius a and length b, as in Fig. 7
Fig. 7 Coordinate system for a cylinder.
, with the point (a,φ,z) in the surface, the spatial correlation function of Eq. (15) can be rewritten as

Es1Es2*=SdSnx2exp{ik[ξdz+ηdasin(φφs)]/R}
(23)

For backscatteringφs=0, the normal at the point isn^=(cosφ,sinφ,0), transforming the integral of Eq. (23) over the surface-element into cylindrical coordinate integral as

Es1Es2*=ab/2b/2dzπ/2π/2dφcos2φexp{ik(ξdz+ηdasinφ)/R}
(24)

To calculate the speckle size alongξ-direction,ηd=0, Eq. (24) can be integrated into
Es1Es2*=aπ/2π/2dφcos2φb/2b/2exp(ikξdz/R)dz=πab2sinc(ρξ)
(25)
wheresinc(ρξ)=sin(πρξ)/(πρξ), andρξ=bξd/λR. Similarly, the speckle size alongη-direction with ξd=0can then be obtained
Es1Es2*=ab/2b/2dzπ/2π/2dφcos2φexp(ikaηdsinφ/R)=πabJ1(ρη)/ρη
(26)
ρη=2πaηd/λR, and the integral relationship [27

27. J. S. Gradshteyn and J. M. Ryzhik, Table of Integrals, Series and Products (Academic,1965).

] has been used
π/2π/2dφcos2φexp(iasinφ)=πJ1(a)/a(a>0)
(27)
J1(...)is the Bessel function of first-order. Also the mean scattered intensity from the rough cylinder can easily be gained

Es1Es1*=aπ/2π/2dφcos2φb/2b/2dz=πab/2
(28)

Then by Eq. (25) and (28), and by Eq. (26) and (28), respectively, the functionΓ(ρ)forρξ andρηcan be written as

Γ(ρξ)=sinc(ρξ)ρξ=bξd/λR
(29)
Γ(ρη)=J1(ρη)/2ρηρη=2πaηd/λR
(30)

The scattering geometry and the profile of the absolution of the functionΓ(ρ)are illustrated by Fig. 8
Fig. 8 Correlation function|Γ(ρ)|from a cylinder versusρ.
below.

It can be seen that, when ρξ=1 andρη=3.8317, the first zero-values of the functions are obtained, so that the speckle sizes along ξ-axis and η-size, respectively, are

ξd=λR/bηd=0.61λR/a
(31)

From the transformation in Eq. (8), we know that whenθs=π/2andφs=0, ξoη is perpendicular to thex-axis while ξ-axis is along negativez-axis andηalongy-axis, i.e., xr=R,yr=η,zr=ξ. By Eq. (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.

3.3 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 angleα and height h. The coordinate system of the cone is shown in Fig. 9
Fig. 9 Coordinate system for a cone.
.

From Eq. (15), one can see that it is required to know the normal to the surface of the object first for the integral. Provided f(x,y,z)is the surface profile of the scattering cone,

f(x,y,z)=x2+y2z2tg2α
(32)

Letx=ztgαcosφ,y=ztgαsinφ,z=z, then the normal to the conical surface can be got

n^=f/|f|=(cosφ,sinφ,tgα)/1+tg2α
(33)

Inserting Eq. (33) into Eq. (15), the spatial correlation of the speckle field back scattered from the rough cone can be obtained

Es1Es2*=sinα0hzdzπ/2π/2dφcos2φexp[ikz(ξd+tgαsinφηd)/R]
(34)

By Eq. (34), the mean speckle intensity can be got

I1=Es1Es1*=sinα0hzdzπ/2π/2dφcos2φ=πh2sinα/4
(35)

Then with the integral relationship [27

27. J. S. Gradshteyn and J. M. Ryzhik, Table of Integrals, Series and Products (Academic,1965).

]
0Aexp(iBx)xdx=[1+exp(iAB)(1+iAB)]/B2
(36)
withρξ=2πhξd/λR, the spatial correlation function Γ(ρξ), as defined by Eq. (20), for the cone along ξ-axis, ηd=0, can be attained from Eq. (34)

Γ(ρξ)=2[1+exp(iρξ)(1+iρξ)]/ρ2
(37)

Accordingly, with Eq. (27) and the integral relationship [27

27. J. S. Gradshteyn and J. M. Ryzhik, Table of Integrals, Series and Products (Academic,1965).

]
0bJ1(ax)dx=[1J0(ab)]/a
(38)
andρη=2πhtgαηd/λR, the function Γ(ρη) along η-axis of the cone will be
Γ(ρη)=4[1J0(ρη)]/ρη2
(39)
J0(...)is the Bessel function of zero-order.

The profiles for the absolutions of the functions Γ(ρ)either along ξ- or η-axis are illustrated in Fig. 10
Fig. 10 Correlation function|Γ(ρ)|from a cone versusρ.
above. One can see that there is no zero-point in the profiles, thus to get the speckle size of the cone, we need to apply the other definition of the speckle size [20

20. R. Berlasso, F. Perez Quintián, M. A. Rebollo, C. A. Raffo, and N. G. Gaggioli, “Study of speckle size of light scattered from cylindrical rough surfaces,” Appl. Opt. 39(31), 5811–5819 (2000). [CrossRef] [PubMed]

], that the value of the spatial location difference at which the normalized spatial correlation function of the speckle intensityγ12(ρ) equals to exp(1), then the profiles of which is shown by Fig. 11
Fig. 11 Normalized correlation function of the speckle intensityγ12(ρ)from cone versusρ.
below.

From Fig. 11, when ρξ=4.11andρη=2.76, one can obtain the speckle sizes of the cone in the observation plane are
ξd=0.65λR/hηd=0.44λR/htgα=0.44λR/a
(40)
a=htgαis 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)).

4. 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.

Acknowledgments

The authors gratefully acknowledge support from the National Natural Science Foundation of China under Grant No. 61172031 and the Fundamental Research Funds for the Central Universities.

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). [CrossRef]

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). [CrossRef]

5.

T. Yoshimura, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am. A 3(7), 1032–1054 (1986). [CrossRef]

6.

L. Leushacke and M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. A 7(5), 827–832 (1990). [CrossRef]

7.

Q. B. Li and F. P. Chiang, “Three-dimensional dimension of laser speckle,” Appl. Opt. 31(29), 6287–6291 (1992). [CrossRef] [PubMed]

8.

T. Yoshimura and S. Iwamoto, “Dynamic properties of three-dimensional speckles,” J. Opt. Soc. Am. A 10(2), 324–328 (1993). [CrossRef]

9.

M. Ibrahim, J. Uozumi, and T. Asakura, “Longitudinal correlation properties of speckles produced by Ring-Slit illumination,” Opt. Rev. 5(3), 129–137 (1998). [CrossRef]

10.

H. T. Yura, B. Rose, and S. G. Hanson, “Dynamic laser speckle in complex ABCD optical systems,” J. Opt. Soc. Am. A 15(5), 1160–1166 (1998). [CrossRef]

11.

H. T. Yura, S. G. Hanson, R. S. Hansen, and B. Rose, “Three-dimensional speckle dynamics in paraxial optical systems,” J. Opt. Soc. Am. A 16(6), 1402–1412 (1999). [CrossRef]

12.

G. J. Guo, S. K. Li, and Q. L. Tan, “Statistical properties of laser speckles generated from far rough surfaces,” Int. J. Infrared Millim. Waves 22(8), 1177–1191 (2001). [CrossRef]

13.

K. Chu and N. George, “Correlation function for speckle size in the right-half-space,” Opt. Commun. 276(1), 1–7 (2007). [CrossRef]

14.

D. V. Semenov, S. V. Miridonov, E. Nippolainen, and A. A. Kamshilin, “Statistical properties of dynamic speckles formed by a deflecting laser beam,” Opt. Express 16(2), 1238–1249 (2008). [CrossRef] [PubMed]

15.

T. Xu and G. R. Bashford, “Further progress on lateral flow estimation using speckle size variation with scan direction,” in IEEE International Ultrosonics Symposium Proceedings, pp. 1383–1386 (2009).

16.

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “Three-dimensional speckle size in generalized optical systems with limiting apertures,” J. Opt. Soc. Am. A 26(8), 1855–1864 (2009). [CrossRef] [PubMed]

17.

D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields. Part I. theory and numerical investigation,” J. Opt. Soc. Am. A 28(9), 1896–1903 (2011). [CrossRef]

18.

D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields. Part II. experimental investigation,” J. Opt. Soc. Am. A 28(9), 1904–1908 (2011). [CrossRef]

19.

D. R. Dunmeyer, Laser Speckle Modeling for Three-Dimensional Metrology and LADAR, M. Eng. dissertation (Massachusetts Institute of Technology, 2001).

20.

R. Berlasso, F. Perez Quintián, M. A. Rebollo, C. A. Raffo, and N. G. Gaggioli, “Study of speckle size of light scattered from cylindrical rough surfaces,” Appl. Opt. 39(31), 5811–5819 (2000). [CrossRef] [PubMed]

21.

R. G. Berlasso, F. P. Quintián, M. A. Rebollo, N. G. Gaggioli, B. L. Sánchez, and M. E. Bernabeu, “Speckle size of light scattered from slightly rough cylindrical surfaces,” Appl. Opt. 41(10), 2020–2027 (2002). [CrossRef] [PubMed]

22.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press,1978).

23.

G. Zhang and Z. Wu, “Two-frequency mutual coherence function of scattering from arbitrarily shaped rough objects,” Opt. Express 19(8), 7007–7019 (2011). [CrossRef] [PubMed]

24.

R. K. Erf, Speckle Metrology (Speckle Metrology, 1978).

25.

L. G. Shirley, E. D. Ariel, G. R. Hallerman, H. C. Payson, and J. R. Vivilecchia, “Advanced techniques for target discirmination using laser speckle,” Linc. Lab. J. 5(3), 367–440 (1992).

26.

L. G. Shirley and N. George, “Speckle from a cascade of two thin diffusers,” J. Opt. Soc. Am. A 6(6), 765–781 (1989). [CrossRef]

27.

J. S. Gradshteyn and J. M. Ryzhik, Table of Integrals, Series and Products (Academic,1965).

OCIS Codes
(030.6140) Coherence and statistical optics : Speckle
(030.6600) Coherence and statistical optics : Statistical optics
(290.5880) Scattering : Scattering, rough surfaces

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: January 3, 2012
Revised Manuscript: February 2, 2012
Manuscript Accepted: February 7, 2012
Published: February 9, 2012

Citation
Geng Zhang, Zhensen Wu, and Yanhui Li, "Speckle size of light scattered from 3D rough objects," Opt. Express 20, 4726-4737 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-4726


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References

  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. A2(5), 657–666 (1985). [CrossRef]
  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). [CrossRef]
  5. T. Yoshimura, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am. A3(7), 1032–1054 (1986). [CrossRef]
  6. L. Leushacke and M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. A7(5), 827–832 (1990). [CrossRef]
  7. Q. B. Li and F. P. Chiang, “Three-dimensional dimension of laser speckle,” Appl. Opt.31(29), 6287–6291 (1992). [CrossRef] [PubMed]
  8. T. Yoshimura and S. Iwamoto, “Dynamic properties of three-dimensional speckles,” J. Opt. Soc. Am. A10(2), 324–328 (1993). [CrossRef]
  9. M. Ibrahim, J. Uozumi, and T. Asakura, “Longitudinal correlation properties of speckles produced by Ring-Slit illumination,” Opt. Rev.5(3), 129–137 (1998). [CrossRef]
  10. H. T. Yura, B. Rose, and S. G. Hanson, “Dynamic laser speckle in complex ABCD optical systems,” J. Opt. Soc. Am. A15(5), 1160–1166 (1998). [CrossRef]
  11. H. T. Yura, S. G. Hanson, R. S. Hansen, and B. Rose, “Three-dimensional speckle dynamics in paraxial optical systems,” J. Opt. Soc. Am. A16(6), 1402–1412 (1999). [CrossRef]
  12. G. J. Guo, S. K. Li, and Q. L. Tan, “Statistical properties of laser speckles generated from far rough surfaces,” Int. J. Infrared Millim. Waves22(8), 1177–1191 (2001). [CrossRef]
  13. K. Chu and N. George, “Correlation function for speckle size in the right-half-space,” Opt. Commun.276(1), 1–7 (2007). [CrossRef]
  14. D. V. Semenov, S. V. Miridonov, E. Nippolainen, and A. A. Kamshilin, “Statistical properties of dynamic speckles formed by a deflecting laser beam,” Opt. Express16(2), 1238–1249 (2008). [CrossRef] [PubMed]
  15. T. Xu and G. R. Bashford, “Further progress on lateral flow estimation using speckle size variation with scan direction,” in IEEE International Ultrosonics Symposium Proceedings, pp. 1383–1386 (2009).
  16. J. E. Ward, D. P. Kelly, and J. T. Sheridan, “Three-dimensional speckle size in generalized optical systems with limiting apertures,” J. Opt. Soc. Am. A26(8), 1855–1864 (2009). [CrossRef] [PubMed]
  17. D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields. Part I. theory and numerical investigation,” J. Opt. Soc. Am. A28(9), 1896–1903 (2011). [CrossRef]
  18. D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields. Part II. experimental investigation,” J. Opt. Soc. Am. A28(9), 1904–1908 (2011). [CrossRef]
  19. D. R. Dunmeyer, Laser Speckle Modeling for Three-Dimensional Metrology and LADAR, M. Eng. dissertation (Massachusetts Institute of Technology, 2001).
  20. R. Berlasso, F. Perez Quintián, M. A. Rebollo, C. A. Raffo, and N. G. Gaggioli, “Study of speckle size of light scattered from cylindrical rough surfaces,” Appl. Opt.39(31), 5811–5819 (2000). [CrossRef] [PubMed]
  21. R. G. Berlasso, F. P. Quintián, M. A. Rebollo, N. G. Gaggioli, B. L. Sánchez, and M. E. Bernabeu, “Speckle size of light scattered from slightly rough cylindrical surfaces,” Appl. Opt.41(10), 2020–2027 (2002). [CrossRef] [PubMed]
  22. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press,1978).
  23. G. Zhang and Z. Wu, “Two-frequency mutual coherence function of scattering from arbitrarily shaped rough objects,” Opt. Express19(8), 7007–7019 (2011). [CrossRef] [PubMed]
  24. R. K. Erf, Speckle Metrology (Speckle Metrology, 1978).
  25. L. G. Shirley, E. D. Ariel, G. R. Hallerman, H. C. Payson, and J. R. Vivilecchia, “Advanced techniques for target discirmination using laser speckle,” Linc. Lab. J.5(3), 367–440 (1992).
  26. L. G. Shirley and N. George, “Speckle from a cascade of two thin diffusers,” J. Opt. Soc. Am. A6(6), 765–781 (1989). [CrossRef]
  27. J. S. Gradshteyn and J. M. Ryzhik, Table of Integrals, Series and Products (Academic,1965).

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