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.

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References

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  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).

2011 (3)

2009 (1)

2008 (1)

2007 (1)

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

2002 (1)

2001 (1)

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]

2000 (1)

1999 (1)

1998 (2)

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]

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]

1993 (1)

1992 (2)

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

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).

1990 (1)

1989 (1)

1988 (1)

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

1986 (1)

1985 (1)

1981 (1)

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]

Ariel, E. D.

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).

Asakura, T.

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]

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]

Berlasso, R.

Berlasso, R. G.

Bernabeu, M. E.

Chen, J. B.

Chiang, F. P.

Chu, K.

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

Gaggioli, N. G.

George, N.

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

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]

Guo, G. J.

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]

Hallerman, G. R.

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).

Hansen, R. S.

Hanson, S. G.

Ibrahim, M.

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]

Iwamoto, S.

Kamshilin, A. A.

Kelly, D. P.

Kirchner, M.

Leushacke, L.

Li, D.

Li, D. W.

Li, Q. B.

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

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

Li, S. K.

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]

Miridonov, S. V.

Nippolainen, E.

Payson, H. C.

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).

Perez Quintián, F.

Quintián, F. P.

Raffo, C. A.

Rebollo, M. A.

Rose, B.

Sánchez, B. L.

Semenov, D. V.

Sheridan, J. T.

Shirley, L. G.

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).

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]

Takai, N.

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]

Tan, Q. L.

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]

Uozumi, J.

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]

Vivilecchia, J. R.

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).

Ward, J. E.

Wu, Z.

Yoshimura, T.

Yura, H. T.

Zhang, G.

Appl. Opt. (3)

Appl. Phys. (Berl.) (1)

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]

Int. J. Infrared Millim. Waves (1)

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]

J. Opt. Soc. Am. A (10)

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]

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]

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]

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

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]

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]

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

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]

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]

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]

Linc. Lab. J. (1)

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).

Opt. Commun. (1)

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

Opt. Express (2)

Opt. Lasers Eng. (1)

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

Opt. Rev. (1)

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]

Other (6)

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

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).

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

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

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

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

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Figures (11)

Fig. 1
Fig. 1

scattering geometry for a rough object.

Fig. 2
Fig. 2

Geometrical arrangement.

Fig. 3
Fig. 3

Geometric representation of the speckle statistics d ( ξ d , η d ) and d || ( ζ d ) .

Fig. 4
Fig. 4

Simulated speckle patterns from rough objects (left: rough plane; middle: rough sphere; right: rough cylinder).

Fig. 5
Fig. 5

Coordinate system for a sphere.

Fig. 6
Fig. 6

Correlation function | Γ( ρ ) | from a sphere versusρ.

Fig. 7
Fig. 7

Coordinate system for a cylinder.

Fig. 8
Fig. 8

Correlation function | Γ( ρ ) | from a cylinder versusρ.

Fig. 9
Fig. 9

Coordinate system for a cone.

Fig. 10
Fig. 10

Correlation function | Γ( ρ ) | from a cone versusρ.

Fig. 11
Fig. 11

Normalized correlation function of the speckle intensity γ 12 ( ρ ) from cone versusρ.

Equations (40)

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E s ( r s )= S k ^ n ^ exp[ ik V n ^ ς( r c ) ] exp[ ik( | r s r c |+ k ^ r c ) ] | r s r c | dS
| r s r c |= ( x r x 1 ) 2 + ( y r y 1 ) 2 + [ z r z( x 1 , y 1 ) ] 2
| r s r c |= r sc R+ x 1 2 + y 1 2 + z 1 2 2R x 1 x r + y 1 y r + z 1 z r R
R>> x 1 2 + y 1 2 + z 1 2 2R x 1 x r + y 1 y r + z 1 z r R = r c 2 2R r c r s R
| r s r c |R= x r 2 + y r 2 + z r 2
E s ( r s )= exp( ikR ) R S k ^ n ^ exp( ik V n ^ ς )exp[ ik( r c 2 2R r c r s R + k ^ r c ) ]dS
E s ( r s )= exp( ikR ) R S k ^ n ^ exp( ik V n ^ ς )exp[ ik( r c r s R + k ^ r c ) ]dS
( x r y r z r )=( cos θ s cos φ s sin φ s sin θ s cos φ s cos θ s sin φ s cos φ s sin θ s sin φ s sin θ s 0 cos θ s )( ξ η R )
C 12 = I 1 I 2 I 1 I 2 = E s1 E s1 * E s2 E s2 * | E s1 | 2 | E s2 | 2
E s1 E s1 * E s2 E s2 * = | E s1 | 2 | E s2 | 2 + | E s1 E s2 * | 2
γ 12 = C 12 / ( C 11 C 22 ) 1/2 = | E s1 E s2 * | 2 / ( I 1 I 2 )
E s ( ξ,η )= S dS n x exp( ik V n ^ ς )exp( ik x 1 )exp[ ik( x 1 cos φ s + y 1 sin φ s ) ] ×exp[ ikη( x 1 sin φ s + y 1 cos φ s ) /R ]exp( ikξ z 1 /R )
E s1 E s2 * = S d S 1 d S 2 n x1 n x2 exp[ ik( V n ^ 1 ς 1 V n ^ 2 ς 2 ) ] exp[ ik( x 1 x 2 ) ] ×exp{ ik[ ( x 1 x 2 )cos φ s +( y 1 y 2 )sin φ s ] }exp[ ik( ξ 1 z 1 ξ 2 z 2 ) /R ] ×exp{ ik[ ( η 1 x 1 + η 2 x 2 )sin φ s +( η 1 y 1 η 2 y 2 )cos φ s ] /R }
exp[ ik( V n ^ 1 ς 1 V n ^ 2 ς 2 ) ] =δ( x 1 x 2 , y 1 y 2 )
E s1 E s2 * = S dS n x 2 exp{ ik[ ξ d z+ η d ( xsin φ s +ycos φ s ) ] /R }
E s1 E s2 * = a 2 0 π dθ sin 3 θ π/2 + φ s π/2 dφ cos 2 φexp{ ika[ ξ d cosθ+ η d sinθsin( φ φ s ) ] /R }
E s1 E s2 * = a 2 0 π dθ sin 3 θexp( i ρ ξ cosθ ) π/2 π/2 dφ cos 2 φ = 2π a 2 [ ρ ξ cos( ρ ξ )+sin( ρ ξ ) ] / ρ ξ 3
E s1 E s2 * = a 2 0 π dθ sin 3 θ π/2 π/2 dφ cos 2 φexp( i ρ η sinθsinφ ) = 2π a 2 [ ρ η cos( ρ η )+sin( ρ η ) ] / ρ η 3
I 1 = E s1 E s1 * = a 2 0 π dθ sin 3 θ π/2 π/2 dφ cos 2 φ = 2π a 2 /3
Γ( ρ )= E s1 E s2 * / E s1 E s1 *
Γ( ρ )= 3[ ρcos( ρ )+sin( ρ ) ] / ρ 3
ξ d = η d = 4.5R / ka =0.716 λR /a
E s1 E s2 * = S dS n x 2 exp{ ik[ ξ d z+ η d asin( φ φ s ) ] /R }
E s1 E s2 * =a b/2 b/2 dz π/2 π/2 dφ cos 2 φexp{ ik( ξ d z+ η d asinφ ) /R }
E s1 E s2 * =a π/2 π/2 dφ cos 2 φ b/2 b/2 exp( ik ξ d z /R )dz = πab 2 sinc( ρ ξ )
E s1 E s2 * =a b/2 b/2 dz π/2 π/2 dφ cos 2 φexp( ika η d sinφ /R ) = πab J 1 ( ρ η ) / ρ η
π/2 π/2 dφ cos 2 φexp( iasinφ ) = π J 1 ( a ) /a ( a>0 )
E s1 E s1 * =a π/2 π/2 dφ cos 2 φ b/2 b/2 dz = πab /2
Γ( ρ ξ )=sinc( ρ ξ ) ρ ξ =b ξ d /λR
Γ( ρ η )= J 1 ( ρ η ) / 2 ρ η ρ η = 2πa η d / λR
ξ d =λR/b η d = 0.61λR /a
f( x,y,z )= x 2 + y 2 z 2 t g 2 α
n ^ = f / | f | = ( cosφ,sinφ,tgα ) / 1+t g 2 α
E s1 E s2 * =sinα 0 h zdz π/2 π/2 dφ cos 2 φexp[ ikz( ξ d +tgαsinφ η d ) /R ]
I 1 = E s1 E s1 * =sinα 0 h zdz π/2 π/2 dφ cos 2 φ = π h 2 sinα /4
0 A exp( iBx )xdx = [ 1+exp( iAB )( 1+iAB ) ] / B 2
Γ( ρ ξ )= 2[ 1+exp( i ρ ξ )( 1+i ρ ξ ) ] / ρ 2
0 b J 1 ( ax )dx = [ 1 J 0 ( ab ) ] /a
Γ( ρ η )= 4[ 1 J 0 ( ρ η ) ] / ρ η 2
ξ d = 0.65λR /h η d =0.44 λR / htgα =0.44 λR /a

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