Abstract

A fibrous radial structure of speckle patterns produced at the far-field diffraction plane by various diffuse objects under illumination of polychromatic light is studied theoretically and experimentally by means of the spatial correlation function of the speckle intensity. It is shown that the fibrous radial structure of speckle patterns strongly depends on both the surface roughness of diffuse objects under illumination and the radial distance from the center of patterns at the far-field diffraction plane.

© 1980 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Hariharan, Opt. Acta 19, 791 (1972).
    [CrossRef]
  2. G. N. Ramachandran, Proc. Indian Acad. Sci. 18, 190 (1943).
  3. W. Martienssen, E. Spiller, Naturwissenschaften 52, 53 (1965).
    [CrossRef]
  4. G. Schiffner, Proc. IEEE 53, 1245 (1965).
    [CrossRef]
  5. H. Fujii, T. Asakura, Optik 39, 99 (1973).
  6. H. Fujii, T. Asakura, Optik 39, 284 (1974).
  7. G. Parry, Opt. Commun. 12, 75 (1974).
    [CrossRef]
  8. H. M. Pedersen, Opt. Acta 22, 523 (1975).
    [CrossRef]
  9. C. T. Stansberg, Appl. Opt. 18, 4051 (1979).
    [CrossRef] [PubMed]
  10. E. Jakeman, P. N. Pusey, J. Phys. A: Gen. Phys. 8, 369 (1975).
    [CrossRef]
  11. E. Mensel, B. Stoffregen, Optik 46, 203 (1976).
  12. J. W. Goodman, Stanfrod Electronics Laboratory Technical Report SEL-03-140 (TR.2303-1) (1963).
  13. K. Nakagawa, T. Asakura, Opt. Commun. 27, 207 (1978).
    [CrossRef]
  14. K. Nakagawa, T. Asakura, Opt. Acta 26, 951 (1979).
    [CrossRef]
  15. K. Nakagawa, T. Asakura, Appl. Opt. 18, 3725 (1979).

1979 (3)

1978 (1)

K. Nakagawa, T. Asakura, Opt. Commun. 27, 207 (1978).
[CrossRef]

1976 (1)

E. Mensel, B. Stoffregen, Optik 46, 203 (1976).

1975 (2)

E. Jakeman, P. N. Pusey, J. Phys. A: Gen. Phys. 8, 369 (1975).
[CrossRef]

H. M. Pedersen, Opt. Acta 22, 523 (1975).
[CrossRef]

1974 (2)

H. Fujii, T. Asakura, Optik 39, 284 (1974).

G. Parry, Opt. Commun. 12, 75 (1974).
[CrossRef]

1973 (1)

H. Fujii, T. Asakura, Optik 39, 99 (1973).

1972 (1)

P. Hariharan, Opt. Acta 19, 791 (1972).
[CrossRef]

1965 (2)

W. Martienssen, E. Spiller, Naturwissenschaften 52, 53 (1965).
[CrossRef]

G. Schiffner, Proc. IEEE 53, 1245 (1965).
[CrossRef]

1943 (1)

G. N. Ramachandran, Proc. Indian Acad. Sci. 18, 190 (1943).

Asakura, T.

K. Nakagawa, T. Asakura, Opt. Acta 26, 951 (1979).
[CrossRef]

K. Nakagawa, T. Asakura, Appl. Opt. 18, 3725 (1979).

K. Nakagawa, T. Asakura, Opt. Commun. 27, 207 (1978).
[CrossRef]

H. Fujii, T. Asakura, Optik 39, 284 (1974).

H. Fujii, T. Asakura, Optik 39, 99 (1973).

Fujii, H.

H. Fujii, T. Asakura, Optik 39, 284 (1974).

H. Fujii, T. Asakura, Optik 39, 99 (1973).

Goodman, J. W.

J. W. Goodman, Stanfrod Electronics Laboratory Technical Report SEL-03-140 (TR.2303-1) (1963).

Hariharan, P.

P. Hariharan, Opt. Acta 19, 791 (1972).
[CrossRef]

Jakeman, E.

E. Jakeman, P. N. Pusey, J. Phys. A: Gen. Phys. 8, 369 (1975).
[CrossRef]

Martienssen, W.

W. Martienssen, E. Spiller, Naturwissenschaften 52, 53 (1965).
[CrossRef]

Mensel, E.

E. Mensel, B. Stoffregen, Optik 46, 203 (1976).

Nakagawa, K.

K. Nakagawa, T. Asakura, Opt. Acta 26, 951 (1979).
[CrossRef]

K. Nakagawa, T. Asakura, Appl. Opt. 18, 3725 (1979).

K. Nakagawa, T. Asakura, Opt. Commun. 27, 207 (1978).
[CrossRef]

Parry, G.

G. Parry, Opt. Commun. 12, 75 (1974).
[CrossRef]

Pedersen, H. M.

H. M. Pedersen, Opt. Acta 22, 523 (1975).
[CrossRef]

Pusey, P. N.

E. Jakeman, P. N. Pusey, J. Phys. A: Gen. Phys. 8, 369 (1975).
[CrossRef]

Ramachandran, G. N.

G. N. Ramachandran, Proc. Indian Acad. Sci. 18, 190 (1943).

Schiffner, G.

G. Schiffner, Proc. IEEE 53, 1245 (1965).
[CrossRef]

Spiller, E.

W. Martienssen, E. Spiller, Naturwissenschaften 52, 53 (1965).
[CrossRef]

Stansberg, C. T.

Stoffregen, B.

E. Mensel, B. Stoffregen, Optik 46, 203 (1976).

Appl. Opt. (2)

J. Phys. A: Gen. Phys. (1)

E. Jakeman, P. N. Pusey, J. Phys. A: Gen. Phys. 8, 369 (1975).
[CrossRef]

Naturwissenschaften (1)

W. Martienssen, E. Spiller, Naturwissenschaften 52, 53 (1965).
[CrossRef]

Opt. Acta (3)

P. Hariharan, Opt. Acta 19, 791 (1972).
[CrossRef]

H. M. Pedersen, Opt. Acta 22, 523 (1975).
[CrossRef]

K. Nakagawa, T. Asakura, Opt. Acta 26, 951 (1979).
[CrossRef]

Opt. Commun. (2)

K. Nakagawa, T. Asakura, Opt. Commun. 27, 207 (1978).
[CrossRef]

G. Parry, Opt. Commun. 12, 75 (1974).
[CrossRef]

Optik (3)

H. Fujii, T. Asakura, Optik 39, 99 (1973).

H. Fujii, T. Asakura, Optik 39, 284 (1974).

E. Mensel, B. Stoffregen, Optik 46, 203 (1976).

Proc. IEEE (1)

G. Schiffner, Proc. IEEE 53, 1245 (1965).
[CrossRef]

Proc. Indian Acad. Sci. (1)

G. N. Ramachandran, Proc. Indian Acad. Sci. 18, 190 (1943).

Other (1)

J. W. Goodman, Stanfrod Electronics Laboratory Technical Report SEL-03-140 (TR.2303-1) (1963).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Light scattering geometry for producing the fibrous radial structure of speckle patterns at the far-field diffraction plane.

Fig. 2
Fig. 2

Geometry for calculating the spatial correlation of speckle intensity at the far-field diffraction plane.

Fig. 3
Fig. 3

Two-dimensional correlation areas αs/αd of polychromatic speckle patterns around the radial positions of (a) |r|/αd = 20 and (b) |r|/αd = 40 at the far-field diffraction plane for two diffuse objects with surface roughness σL = 0.60 μm (solid curve) and 1.67 μm (dotted curve).

Fig. 4
Fig. 4

Schematic of the experimental arrangement for measuring the 2-D spatial correlation function of polychromatic speckle patterns at the far-field diffraction plane.

Fig. 5
Fig. 5

Geometry for measuring the 2-D spatial correlation at the far-field diffraction plane. Points A and B indicate the positions of photomultiplier Ph1 and Ph2 in Fig. 4. If time delay τ is set for the photocurrent from photomultiplier Ph2, the correlation between the photocurrents from Ph1 at A and Ph2 at B corresponds to that between the photocurrents from Ph1 at A and Ph2 at B′.

Fig. 6
Fig. 6

Correlation functions of speckle intensity fluctuations for a diffuse object with surface roughness σh = 1.21 μm at radial position |r|/αd = 40. The correlations are taken for three separations of |δr(0)|/αd = (a) 0, (b) 1.27, and (c) 2.54 from position |r|/αd = 40.

Fig. 7
Fig. 7

Correlation functions of speckle intensity fluctuations obtained theoretically from Eq. (11). The conditions of the diffuse object, the radial position, and the two-point separations are the same as in Fig. 6.

Fig. 8
Fig. 8

Two-dimensional correlation areas αs/αd of polychromatic speckle patterns around radial positions (a) |r|/αd = 20 and (b) |r|/αd = 40 at the far-field diffraction plane for two diffuse objects with surface roughness σh = 1.21 and 3.34 μm. The two symbols represent experimental data, and the curves indicate theoretical results.

Fig. 9
Fig. 9

Dependence of the correlation length along the radial direction on the radial distance from the speckle pattern center (optical axis) for three diffuse objects with surface roughness αh = 1.21, 2.08, and 3.34 μm. The three symbols represents experimental data, and the curves denote theoretical results.

Fig. 10
Fig. 10

Dependence of the correlation length along the radial direction on the surface roughness of diffuse objects at three radial distances: |r|/αd = 20, 30, and 40. The three symbols represent experimental data, and the curves denote theoretical results.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

I ( r ) = i = 1 N I ( r ; k i ) = i = 1 N | A ( r ; k i ) | 2 ,
A ( r ; k i ) = exp ( r 2 / w 0 2 ) exp [ i k i L ( r ) ] × exp ( i k i r · r / R ) d 2 r ,
L ( r ) = ( n 1 ) h ( r ) ,
g Δ I ( r 1 , r 2 ) = Δ I ( r 1 ) Δ I ( r 2 ) [ Δ I 2 ( r 1 ) Δ I 2 ( r 2 ) ] 1 / 2
Δ I ( r ) = I ( r ) I ( r )
g Δ I ( r 1 , r 2 ) = m = 1 N n = 1 M S m S n | g A ( r 1 , r 2 ; k m , k n ) | 2 [ m = 1 N n = 1 N S m S n | g A ( r 1 , r 1 ; k m , k n ) | 2 ] 1 / 2 [ m = 1 N n = 1 M S m S n | g A ( r 2 , r 2 ; k m , k n ) | 2 ] 1 / 2 ,
g A ( r 1 , r 2 ; k m , k n ) = A ( r 1 ; k m ) A * ( r 2 ; k n ) [ | A ( r 1 ; k m ) | 2 | A ( r 2 ; k n ) | 2 ] 1 / 2 .
ρ ( r 0 ) = L ( r ) L ( r + r 0 ) / σ L 2 = exp ( | r 0 | 2 / ξ 2 ) ,
exp [ σ L 2 k m k n ρ ( r 0 ) ] 1 + [ exp ( k m k n σ L 2 ) 1 ] exp ( k m k n σ L 2 | r 0 | 2 / ξ 2 ) ,
| g A ( r 1 , r 2 ; k m , k n ) | = exp [ 1 2 σ L 2 ( k m k n ) 2 ] × exp ( w 0 2 8 R 2 | k m r 1 k n r 2 | 2 ) .
g Δ I ( r 1 , r 2 ) = m = 1 N n = 1 N S m S n exp [ σ L 2 ( k m k n ) 2 ] exp [ ( w 0 2 / 4 R 2 ) | k m r 1 k n r 2 | 2 ] { m = 1 N n = 1 N S m S n exp [ ( k m k n ) 2 ( σ L 2 + w 0 2 | r 1 | 2 / 4 R 2 ) ] } 1 / 2 { m = 1 N n = 1 N S m S n exp [ ( k m k n ) 2 ( σ L 2 + w 0 2 | r 1 | 2 / 4 R 2 ) ] } 1 / 2 .
g Δ I ( r 1 , r 2 ) = m = 1 N n = 1 N S m S n exp [ 4 π 2 σ L 2 λ m 2 λ n 2 ( λ m λ n ) 2 ] exp [ λ N 2 α d 2 | r 1 λ m r 2 λ n | 2 ] { m = 1 N n = 1 N S m S n exp [ ( λ m λ n ) 2 λ m 2 λ n 2 ( 4 π 2 σ L 2 + | r 1 | 2 α d 2 λ N 2 ) ] } 1 / 2 { m = 1 N n = 1 N S m S n exp [ ( λ m λ n ) 2 λ m 2 λ n 2 ( 4 π 2 σ L 2 + | r 2 | 2 α d 2 λ N 2 ) ] } 1 / 2 .
| δ r ( τ 0 ) | = [ | δ r ( 0 ) | 2 + | r + δ r ( 0 ) | 2 ω 2 τ 0 2 ] 1 / 2 θ ( τ 0 ) = cos 1 [ | δ r ( 0 ) | / | δ r ( τ 0 ) | ] .

Metrics