Abstract

The first-order probability density functions of displayed speckle patterns in electronic speckle pattern interferometry are calculated. We show that in specular-reference-beam setups the brightness in the displayed pattern is always (χ)2 distributed. For speckle-reference-beam setups the probability density depends on the effective resolution of the recorded speckle pattern. This function is calculated in the case of fully resolved patterns and shown to approach the (χ)2 density function as the size of the aperture increases. The statistics obtained with specular-and speckle-reference-beam setups are compared. General expressions are derived for the average value and the standard deviation of the monitor brightness. In specular-reference-beam setups the speckle contrast is found always to be equal to 2, and in speckle-reference-beam setups it is found to be in the region from 2 to 5. In setups in which speckle-reduction techniques have been used the speckle contrast decreases below these numbers. We also calculate this reduced speckle contrast.

© 1981 Optical Society of America

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References

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  1. J. M. Burch, in Optical Instruments and Techniques, J. H. Dickenson, ed. (Oriel, Newcastle upon Tyne, England, 1970), p. 213.
  2. H. J. Gerritsen, W. J. Hannan, and E. G. Ramberg, “Elimination of speckle noise in holograms with redundancy,” Appl. Opt. 7, 2301–2311 (1968).
    [Crossref] [PubMed]
  3. J. C. Dainty and W. T. Welford, “Reduction of speckle in image plane hologram reconstruction by moving pupils,” Opt. Commun. 3, 289–294 (1971).
    [Crossref]
  4. A. A. Kostanyan, V. A. Medvedev, and V. N. Filinow, “Calculating the statistical characteristics of diffuse noise in holographic images,” Opt. Spectrosc. 46, 558–562 (1979).
  5. G. Å. Slettemoen, “Optimal signal processing in electronic speckle pattern interferometry,” Opt. Commun. 23, 213–216 (1977).
    [Crossref]
  6. G. Å. Slettemoen, “General analysis of fringe contrast in electronic speckle pattern interferometry,” Opt. Acta 26, 313–327 (1979).
    [Crossref]
  7. G. Å. Slettemoen, “Electronic speckle pattern interferometric system based on a speckle reference beam,” Appl. Opt. 19, 616–623 (1980).
    [Crossref] [PubMed]
  8. K. A. Stetson, “A rigorous treatment of the fringes of hologram interferometry,” Optik 29, 386–400 (1969).
  9. E. N. Leith and J. U. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. 52, 1123–1130 (1962).
    [Crossref]
  10. J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–76.
    [Crossref]
  11. W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).
  12. W. Martienssen and S. Spiller, “Holographic reconstruction without granulation,” Phys. Lett. 24A, 126–127 (1967).
  13. I. M. Ryshik and I. S. Gradstein, Tables of Series, Products, and Integrals (VEB Deutscher Verlag der Wissenschaften, Berlin, 1963).
  14. H. M. Pedersen, University of Trondheim, Norwegian Institute of Technology, N-7034 Trondheim NTH, Norway, personal communication.
  15. R. L. Stratonovich, Topics in the Theory of Random Noise, Vol. I (Gordon and Breach, New York, 1963).
  16. N. George and et al., “Speckle noise in displays,” J. Opt. Soc. Am. 66, 1282–1290 (1976).
    [Crossref]
  17. S. Lowenthal and H. Arsenault, “Image formation for coherent diffuse objects: statistical properties,” J. Opt. Soc. Am. 60, 1478–1483 (1970).
    [Crossref]
  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  19. H. M. Pedersen, “Theory of speckle dependence on surface roughness,” J. Opt. Soc. Am. 66, 1204–1210 (1976).
    [Crossref]

1980 (1)

1979 (2)

A. A. Kostanyan, V. A. Medvedev, and V. N. Filinow, “Calculating the statistical characteristics of diffuse noise in holographic images,” Opt. Spectrosc. 46, 558–562 (1979).

G. Å. Slettemoen, “General analysis of fringe contrast in electronic speckle pattern interferometry,” Opt. Acta 26, 313–327 (1979).
[Crossref]

1977 (1)

G. Å. Slettemoen, “Optimal signal processing in electronic speckle pattern interferometry,” Opt. Commun. 23, 213–216 (1977).
[Crossref]

1976 (2)

1971 (1)

J. C. Dainty and W. T. Welford, “Reduction of speckle in image plane hologram reconstruction by moving pupils,” Opt. Commun. 3, 289–294 (1971).
[Crossref]

1970 (1)

1969 (1)

K. A. Stetson, “A rigorous treatment of the fringes of hologram interferometry,” Optik 29, 386–400 (1969).

1968 (1)

1967 (1)

W. Martienssen and S. Spiller, “Holographic reconstruction without granulation,” Phys. Lett. 24A, 126–127 (1967).

1962 (1)

Arsenault, H.

Burch, J. M.

J. M. Burch, in Optical Instruments and Techniques, J. H. Dickenson, ed. (Oriel, Newcastle upon Tyne, England, 1970), p. 213.

Dainty, J. C.

J. C. Dainty and W. T. Welford, “Reduction of speckle in image plane hologram reconstruction by moving pupils,” Opt. Commun. 3, 289–294 (1971).
[Crossref]

Davenport, W. B.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

Filinow, V. N.

A. A. Kostanyan, V. A. Medvedev, and V. N. Filinow, “Calculating the statistical characteristics of diffuse noise in holographic images,” Opt. Spectrosc. 46, 558–562 (1979).

George, N.

Gerritsen, H. J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–76.
[Crossref]

Gradstein, I. S.

I. M. Ryshik and I. S. Gradstein, Tables of Series, Products, and Integrals (VEB Deutscher Verlag der Wissenschaften, Berlin, 1963).

Hannan, W. J.

Kostanyan, A. A.

A. A. Kostanyan, V. A. Medvedev, and V. N. Filinow, “Calculating the statistical characteristics of diffuse noise in holographic images,” Opt. Spectrosc. 46, 558–562 (1979).

Leith, E. N.

Lowenthal, S.

Martienssen, W.

W. Martienssen and S. Spiller, “Holographic reconstruction without granulation,” Phys. Lett. 24A, 126–127 (1967).

Medvedev, V. A.

A. A. Kostanyan, V. A. Medvedev, and V. N. Filinow, “Calculating the statistical characteristics of diffuse noise in holographic images,” Opt. Spectrosc. 46, 558–562 (1979).

Pedersen, H. M.

H. M. Pedersen, “Theory of speckle dependence on surface roughness,” J. Opt. Soc. Am. 66, 1204–1210 (1976).
[Crossref]

H. M. Pedersen, University of Trondheim, Norwegian Institute of Technology, N-7034 Trondheim NTH, Norway, personal communication.

Ramberg, E. G.

Root, W. L.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

Ryshik, I. M.

I. M. Ryshik and I. S. Gradstein, Tables of Series, Products, and Integrals (VEB Deutscher Verlag der Wissenschaften, Berlin, 1963).

Slettemoen, G. Å.

G. Å. Slettemoen, “Electronic speckle pattern interferometric system based on a speckle reference beam,” Appl. Opt. 19, 616–623 (1980).
[Crossref] [PubMed]

G. Å. Slettemoen, “General analysis of fringe contrast in electronic speckle pattern interferometry,” Opt. Acta 26, 313–327 (1979).
[Crossref]

G. Å. Slettemoen, “Optimal signal processing in electronic speckle pattern interferometry,” Opt. Commun. 23, 213–216 (1977).
[Crossref]

Spiller, S.

W. Martienssen and S. Spiller, “Holographic reconstruction without granulation,” Phys. Lett. 24A, 126–127 (1967).

Stetson, K. A.

K. A. Stetson, “A rigorous treatment of the fringes of hologram interferometry,” Optik 29, 386–400 (1969).

Stratonovich, R. L.

R. L. Stratonovich, Topics in the Theory of Random Noise, Vol. I (Gordon and Breach, New York, 1963).

Upatnieks, J. U.

Welford, W. T.

J. C. Dainty and W. T. Welford, “Reduction of speckle in image plane hologram reconstruction by moving pupils,” Opt. Commun. 3, 289–294 (1971).
[Crossref]

Appl. Opt. (2)

J. Opt. Soc. Am. (4)

Opt. Acta (1)

G. Å. Slettemoen, “General analysis of fringe contrast in electronic speckle pattern interferometry,” Opt. Acta 26, 313–327 (1979).
[Crossref]

Opt. Commun. (2)

J. C. Dainty and W. T. Welford, “Reduction of speckle in image plane hologram reconstruction by moving pupils,” Opt. Commun. 3, 289–294 (1971).
[Crossref]

G. Å. Slettemoen, “Optimal signal processing in electronic speckle pattern interferometry,” Opt. Commun. 23, 213–216 (1977).
[Crossref]

Opt. Spectrosc. (1)

A. A. Kostanyan, V. A. Medvedev, and V. N. Filinow, “Calculating the statistical characteristics of diffuse noise in holographic images,” Opt. Spectrosc. 46, 558–562 (1979).

Optik (1)

K. A. Stetson, “A rigorous treatment of the fringes of hologram interferometry,” Optik 29, 386–400 (1969).

Phys. Lett. (1)

W. Martienssen and S. Spiller, “Holographic reconstruction without granulation,” Phys. Lett. 24A, 126–127 (1967).

Other (7)

I. M. Ryshik and I. S. Gradstein, Tables of Series, Products, and Integrals (VEB Deutscher Verlag der Wissenschaften, Berlin, 1963).

H. M. Pedersen, University of Trondheim, Norwegian Institute of Technology, N-7034 Trondheim NTH, Norway, personal communication.

R. L. Stratonovich, Topics in the Theory of Random Noise, Vol. I (Gordon and Breach, New York, 1963).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. W. Goodman, in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–76.
[Crossref]

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

J. M. Burch, in Optical Instruments and Techniques, J. H. Dickenson, ed. (Oriel, Newcastle upon Tyne, England, 1970), p. 213.

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Figures (5)

Fig. 1
Fig. 1

Schematic drawing of two ESPI setups. (a) Specular-reference ESPI, (b) speckle-reference ESPI.

Fig. 2
Fig. 2

p0(Im) is the probability density function of the monitor brightness for a specular-reference ESPI. The broken line shows the probability density for a fully developed speckle pattern.

Fig. 3
Fig. 3

p1(Im) is the probability density function of the monitor brightness for a speckle-reference ESPI (the cross-interference speckle pattern fully resolved). This function is compared with the probability density p0(Im) for a specular-reference ESPI.

Fig. 4
Fig. 4

(a) Multislit aperture that is used in speckle-reference ESPI. The effective aperture used in the object 1 imaging system is denoted by 1, and the effective aperture used in the object 2 imaging system is denoted by 2. (b) Modification of the aperture in (a). (c) Schematic of Young’s experiment. The maximum distance d that gives resolvable interference fringes at the TV camera is denote by L in (a) and (b).

Fig. 5
Fig. 5

Photographs of the monitor display. The setup is a speckle-reference ESPI, and N is the number of elementary pairs of small-circular apertures. The distance between any elementary pairs is larger than the TV-camera resolution limit L.

Equations (52)

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I ( x , y ) = A 1 ( x , y ) A 1 * ( x , y ) + A 2 ( x , y ) A 2 * ( x , y ) + 2 Re [ A 1 * ( x , y ) A 2 ( x , y ) M ( x , y ) ] = I 1 ( x , y ) + I 2 ( x , y ) + I c ( x , y ) ,
I c ( x , y ) = h ( x - x 1 , y - y 1 ) I c ( x 1 , y 1 ) d x 1 d y 1 .
I m ( r ) = [ I c ( r ) ] 2 = h ( r - r ) h ( r - r ) I c ( r ) I c ( r ) d r d r .
I m = 4 [ Re ( A 1 * A 2 M ) ] 2 = 4 M 2 [ A 1 r A 2 r + A 1 i A 2 i ] 2 .
p r , i ( A r , A i ) = 1 π I exp [ - ( A r ) 2 + ( A i ) 2 I ] = p ( A r ) p ( A i ) ,
I m = 4 I 2 ( A 1 r ) 2 .
p ( A 1 r ) = 1 ( π I 1 ) 1 / 2 exp [ - ( A 1 r ) 2 / I 1 ] .
p y ( y ) = 1 y p x ( y ) for y 0 , p y ( y ) = 0 for y < 0.
p 0 ( I m ) = 1 2 ( π I 1 I 2 I m ) 1 / 2 exp ( - I m / 4 I 1 I 2 ) for I m 0 , p 0 ( I m ) = 0 for I m < 0.
P ( I r < I a r ) = - 0 ( I a r / A 2 r ) - p ( A 1 r ) p ( A 2 r ) d A 1 r d A 2 r + 0 + - ( I a r / A 2 r ) p ( A 1 r ) p ( A 2 r ) d A 1 r d A 2 r = 2 0 + - ( I a r / A 2 r ) p ( A 1 r ) p ( A 2 r ) d A 1 r d A 2 r .
p r ( I r ) = 2 0 p ( I r / A 2 r ) p ( A 2 r ) 1 A 2 r d A 2 r .
p r ( I r ) = 2 π ( I 1 I 2 ) 1 / 2 × 0 1 A 2 r exp [ - A 2 r 2 I 2 - ( I r ) 2 I 1 ( A 2 r ) 2 ] d A 2 r .
p ( I r ) = 2 π ( I 1 I 2 ) 1 / 2 K 0 [ 2 I r ( I 1 I 2 ) 1 / 2 ] .
F c ( f ) = F r ( f ) F i ( f ) 2 π = F r 2 ( f ) 2 π .
F ( f ) = 1 2 π - + e - i f t p ( t ) d t .
p s ( t ) = ( t 2 q ) s Γ ( ½ ) Γ ( s + ½ ) K s ( q t ) , F s ( f ) = 1 2 ( f 2 + q 2 ) s + 1 / 2 .
F c ( f ) = 1 2 π I 1 I 2 ( f 2 + 1 I 1 I 2 ) .
p c ( I c ) = 1 Γ ( ½ ) 2 ( I 1 I 2 ) 1 / 4 | I c I 1 I 2 | 1 / 2 × K 1 / 2 ( I c I 1 I 2 ) .
p 1 ( I m ) = 1 Γ ( ½ ) 2 I 1 I 2 ( I m I 1 I 2 ) - 1 / 2 × K 1 / 2 ( I m I 1 I 2 ) for I m 0 , p 1 ( I m ) = 0 for I m < 0.
p 1 ( I m ) = 1 2 ( I 1 I 2 I m ) 1 / 2 exp [ - ( I m / I 1 I 2 ) 1 / 2 ] for I m 0 , p 1 ( I m ) = 0 for I m < 0.
I m ( r ) = 4 I 2 [ h ( r - r ) A 1 r ( r ) d r ] 2 .
I c = 2 n = 1 N n = 1 N ( A 2 n r A 2 n - 1 r + A 2 n i A 2 n - 1 i ) .
I c = 2 n = 1 N - 1 [ A 2 n r ( A 2 n - 1 r + A 2 n + 1 r ) + A 2 n i ( A 2 n - 1 i + A 2 n + 1 i ) ] + A 2 N r A 2 N - 1 r + A 2 N i A 2 N - 1 i = 2 n = 1 N - 1 ( A 2 n r A 2 n - 1 r + A 2 n i A 2 n - 1 i ) + 2 n = 1 N - 1 ( A 2 n r A 2 n + 1 r + A 2 n i A 2 n + 1 i ) + 2 ( A 2 N r A 2 N - 1 r + A 2 N i A 2 N - 1 i ) .
p 0 ( I m ) = 1 ( 2 π I m I m ) 1 / 2 exp ( - I m / 2 I m ) for I m 0 , p 0 ( I m ) = 0 for I m < 0.
I m = 2 I 1 I 2 γ 12 2 ,
γ 12 2 = P 1 ( ν ) 2 H ( ν ) 2 d ν P 1 ( ν ) 2 d ν .
σ m = 2 I m = 2 2 I 1 I 2 γ 12 2 .
C = σ m / I m = 2 .
I m = h ( r - r ) h ( r - r ) I c ( r ) I c ( r ) d r d r .
I c ( r ) I c ( r ) = 4 Re [ A 1 * ( r ) A 2 ( r ) ] Re [ A 1 * ( r ) A 2 ( r ) ] = 2 Re [ A 1 ( r ) A 1 * ( r ) A 2 * ( r ) A 2 ( r ) ] .
A 1 ( r ) = h 1 ( r - r 1 ) Ā 1 ( r 1 ) d r 1 .
I c ( r ) I c ( r ) = 2 Re [ h 1 ( r - r 1 ) h 1 * ( r - r 2 ) Ā 1 ( r 1 ) Ā 1 * ( r 2 ) d r 1 d r 2 × h 2 * ( r - r 1 ) h 2 ( r - r 2 ) Ā 2 * ( r 1 ) Ā 2 ( r 2 ) d r 1 d r 2 ] .
Ā 1 * ( r 1 ) Ā 1 ( r 2 ) = Ī 1 δ ( r 1 - r 2 ) , Ā 2 * ( r 1 ) Ā 2 ( r 2 ) = Ī 2 δ ( r 1 - r 2 ) .
h ( r ) = P ( ν ) e 2 π i r ν d ν ,
I c ( r ) I c ( r ) = 2 Ī 1 Ī 2 Re [ P 1 ( ν ) 2 P 2 ( ν ) 2 × e 2 π i r ( ν - ν ) e 2 π i r ( ν - ν ) d ν d ν ] .
I m = 2 Ī 1 Ī 2 P 1 ( ν ) 2 P 2 ( ν ) 2 × H ( ν - ν ) 2 d ν d ν .
I m = 2 I 1 I 2 [ P 1 ( ν ) 2 * P 2 ( ν ) 2 ] H ( ν ) 2 d ν [ P 1 ( ν ) 2 * P 2 ( ν ) 2 ] d ν = 2 I 1 I 2 γ 12 2 .
σ m 2 = I m 2 - I m 2 .
I m 2 = h ( r - r ) h ( r - r ) h ( r - r ) h ( r - r i v ) × I c ( r ) I c ( r ) I c ( r ) I c ( r i v ) d r d r d r d r i v .
I c ( r ) I c ( r ) I c ( r ) I c ( r i v ) = [ A 1 * ( r ) A 2 ( r ) + A 1 ( r ) A 2 * ( r ) ] × [ A 1 * ( r ) A 2 ( r ) + A 1 ( r ) A 2 * ( r ) ] × [ A 1 * ( r ) A 2 ( r ) + A 1 ( r ) A 2 * ( r ) ] × [ A 1 * ( r i v ) A 2 ( r i v ) + A 1 ( r i v ) A 2 * ( r i v ) ] .
I c ( r ) I c ( r ) I c ( r ) I c ( r i v ) = 2 Re [ A 1 * ( r ) A 1 * ( r ) A 1 * ( r ) A 1 * ( r i v ) × A 2 ( r ) A 2 ( r ) A 2 ( r ) A 2 ( r i v ) ] + 8 Re [ A 1 * ( r ) A 1 * ( r ) A 1 * ( r ) A 1 ( r i v ) × A 2 ( r ) A 2 ( r ) A 2 ( r ) A 2 * ( r i v ) ] + 6 Re [ A 1 * ( r ) A 1 * ( r ) A 1 ( r ) A 1 ( r i v ) × A 2 ( r ) A 2 ( r ) A 2 * ( r ) A 2 * ( r i v ) ] .
x 1 x 2 x 3 x 4 = x 1 x 2 x 3 x 4 + x 1 x 3 x 2 x 4 + x 1 x 4 x 2 x 3 - 2 x 1 x 2 x 3 x 4 .
I c ( r ) I c ( r ) I c ( r ) I c ( r i v ) = 12 Re [ A 1 * ( r ) A 1 ( r ) A 1 * ( r ) A 1 ( r i v ) × A 2 ( r ) A 2 * ( r ) A 2 ( r ) A 2 * ( r i v ) ] + 12 Re [ A 1 * ( r ) A 1 ( r i v ) A 1 * ( r ) A 1 ( r ) × A 2 ( r ) A 2 * ( r ) A 2 ( r ) A 2 * ( r i v ) ] .
σ m 2 = I m 2 { 2 + 3 × [ 4 A 1 * ( r ) A 1 ( r i v ) A 1 * ( r ) A 1 ( r ) × A 2 ( r ) A 2 * ( r ) A 2 ( r ) A 2 * ( r i v ) × h ( r - r ) h ( r - r ) h ( r - r ) × h ( r - r i v ) d r d r d r d r i v ] / I m 2 } .
σ m 2 = I m 2 ( 2 + 3 τ ) ,
τ = P 1 ( ν ) 2 P 1 ( ν ) 2 P 2 ( ν ) 2 P 2 ( ν i v ) 2 H ( ν - ν ) H ( ν i v - ν ) H ( ν - ν ) H ( ν - ν i v ) d ν d ν d ν d ν i v { [ P 1 ( ν ) 2 * P 2 ( ν ) 2 ] H ( ν ) 2 d ν } 2 .
C = σ m / I m = ( 2 + 3 τ ) 1 / 2 .
P 1 ( ν ) = p = 0 N P 1 ( ν - p ν 0 ) , P 2 ( ν ) = k = 0 N P 2 ( ν - k ν 0 ) .
τ = 1 N τ ,
τ = 1 / N .
C red = [ ( 2 + 3 τ ) / T ] 1 / 2 .
C red = 2 / T .