Abstract

The statistical properties of speckles in paraxial optical systems depend on the system parameters. In particular, the speckle orientation and the lateral dependence (x and y) of the longitudinal speckle size can vary significantly. For example, the off-axis longitudinal correlation length remains equal to the on-axis size for speckles in a Fourier transform system, while it decreases dramatically as the observation position moves off axis in a Fresnel system. In this paper, we review the speckle correlation function in general linear canonical transform (LCT) systems, clearly demonstrating that speckle properties can be controlled by introducing different optical components, i.e., lenses and sections of free space. Using a series of numerical simulations, we examine how the correlation function changes for some typical LCT systems. The integrating effect of the camera pixel and the impact this has on the measured first- and second-order statistics of the speckle intensities is also examined theoretically. A series of experimental results are then presented to confirm several of these predictions. First, the effect the pixel size has on the measured first-order speckle statistics is demonstrated, and second, the orientation of speckles in a Fourier transform system is measured, showing that the speckles lie parallel to the optical axis.

© 2012 Optical Society of America

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References

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2011 (4)

2010 (1)

2009 (2)

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, 1855–1864 (2009).
[CrossRef]

X. Zhao and Z. Gao, “Surface roughness measurement using spatial-average analysis of objective speckle pattern in specular direction,” Opt. Lasers Eng. 47, 1307–1316(2009).
[CrossRef]

2007 (1)

2006 (3)

1999 (2)

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

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, 1402–1412 (1999).
[CrossRef]

1996 (1)

M. Lehmann, “Phase-shifting speckle interferometry with unresolved speckles: a theoretical investigation,” Opt. Commun. 128, 325–340 (1996).
[CrossRef]

1994 (1)

1993 (2)

1992 (1)

1990 (1)

1987 (1)

Abramowitz, M.

M. Abramowitz and A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1972).

Benckert, L. R.

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Cerbino, R.

Chiang, F. P.

Ezawa, T.

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Gao, Z.

X. Zhao and Z. Gao, “Surface roughness measurement using spatial-average analysis of objective speckle pattern in specular direction,” Opt. Lasers Eng. 47, 1307–1316(2009).
[CrossRef]

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 1st ed. (Roberts, 2007).

Gopinathan, U.

Grum, T. P.

Hansen, R. S.

Hanson, S. G.

Healy, J. J.

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid Pub. 6, 11034 (2011).
[CrossRef]

Hennelly, B. M.

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid Pub. 6, 11034 (2011).
[CrossRef]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O’Neill, and J. T. Sheridan, “Generalized Yamaguchi correlation factor for coherent quadratic phase speckle metrology systems with an aperture,” Opt. Lett. 31, 3444–3446 (2006).
[CrossRef]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O’Neill, and J. T. Sheridan, “Paraxial speckle-based metrology system with an aperture,” J. Opt. Soc. Am. A 23, 2861–2870 (2006).
[CrossRef]

Iwamoto, S.

Jueptner, W.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques, 1st ed. (Springer, 2004).

Kelly, D. P.

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid Pub. 6, 11034 (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, 1904–1908 (2011).
[CrossRef]

D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields. Part I. Theory and a numerical investigation,” J. Opt. Soc. Am. A 28, 1896–1903 (2011).
[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, 1855–1864 (2009).
[CrossRef]

D. P. Kelly, J. E. Ward, U. Gopinathan, and J. T. Sheridan, “Controlling speckle using lenses and free space,” Opt. Lett. 32, 3394–3396 (2007).
[CrossRef]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O’Neill, and J. T. Sheridan, “Paraxial speckle-based metrology system with an aperture,” J. Opt. Soc. Am. A 23, 2861–2870 (2006).
[CrossRef]

D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O’Neill, and J. T. Sheridan, “Generalized Yamaguchi correlation factor for coherent quadratic phase speckle metrology systems with an aperture,” Opt. Lett. 31, 3444–3446 (2006).
[CrossRef]

Kirchner, M.

Lehmann, M.

M. Lehmann, “Phase-shifting speckle interferometry with unresolved speckles: a theoretical investigation,” Opt. Commun. 128, 325–340 (1996).
[CrossRef]

Leushacke, L.

Li, D.

Li, Q. B.

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Mas, D.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Miyamoto, Y.

Naik, D. N.

O’Neill, F. T.

Peuser, J.

Rastogi, P. K.

P. K. Rastogi, Digital Speckle Pattern Interferometry and Related Techniques, 1st ed. (Wiley, 2001).

Rhodes, W. T.

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Rose, B.

Scheffold, F.

Schnars, U.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques, 1st ed. (Springer, 2004).

Sheridan, J. T.

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid Pub. 6, 11034 (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, 1904–1908 (2011).
[CrossRef]

D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields. Part I. Theory and a numerical investigation,” J. Opt. Soc. Am. A 28, 1896–1903 (2011).
[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, 1855–1864 (2009).
[CrossRef]

D. P. Kelly, J. E. Ward, U. Gopinathan, and J. T. Sheridan, “Controlling speckle using lenses and free space,” Opt. Lett. 32, 3394–3396 (2007).
[CrossRef]

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O’Neill, and J. T. Sheridan, “Paraxial speckle-based metrology system with an aperture,” J. Opt. Soc. Am. A 23, 2861–2870 (2006).
[CrossRef]

D. P. Kelly, J. E. Ward, U. Gopinathan, B. M. Hennelly, F. T. O’Neill, and J. T. Sheridan, “Generalized Yamaguchi correlation factor for coherent quadratic phase speckle metrology systems with an aperture,” Opt. Lett. 31, 3444–3446 (2006).
[CrossRef]

Singh, R. K.

Sjödahl, M.

Skipetrov, S. E.

Stegun, A.

M. Abramowitz and A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1972).

Takeda, M.

Ward, J. E.

Weber, B.

Yoshimura, T.

Yura, H. T.

Zakharov, P.

Zhao, X.

X. Zhao and Z. Gao, “Surface roughness measurement using spatial-average analysis of objective speckle pattern in specular direction,” Opt. Lasers Eng. 47, 1307–1316(2009).
[CrossRef]

Appl. Opt. (2)

J. Eur. Opt. Soc. Rapid Pub. (1)

D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5D imaging performance of digital holographic systems,” J. Eur. Opt. Soc. Rapid Pub. 6, 11034 (2011).
[CrossRef]

J. Opt. Soc. Am. A (9)

H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
[CrossRef]

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

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, 1402–1412 (1999).
[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, 1855–1864 (2009).
[CrossRef]

D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O’Neill, and J. T. Sheridan, “Paraxial speckle-based metrology system with an aperture,” J. Opt. Soc. Am. A 23, 2861–2870 (2006).
[CrossRef]

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

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

H. T. Yura, S. G. Hanson, and T. P. Grum, “Speckle: statistics and interferometric decorrelation effects in complex ABCD optical systems,” J. Opt. Soc. Am. A 10, 316–323 (1993).
[CrossRef]

Opt. Commun. (2)

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

M. Lehmann, “Phase-shifting speckle interferometry with unresolved speckles: a theoretical investigation,” Opt. Commun. 128, 325–340 (1996).
[CrossRef]

Opt. Eng. (1)

D. P. Kelly, B. M. Hennelly, W. T. Rhodes, and J. T. Sheridan, “Analytical and numerical analysis of linear optical systems,” Opt. Eng. 45, 088201 (2006).
[CrossRef]

Opt. Express (2)

Opt. Lasers Eng. (1)

X. Zhao and Z. Gao, “Surface roughness measurement using spatial-average analysis of objective speckle pattern in specular direction,” Opt. Lasers Eng. 47, 1307–1316(2009).
[CrossRef]

Opt. Lett. (2)

Other (6)

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 1st ed. (Roberts, 2007).

Wolfram Research, http://www.wolfram.com/ .

M. Abramowitz and A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1972).

P. K. Rastogi, Digital Speckle Pattern Interferometry and Related Techniques, 1st ed. (Wiley, 2001).

Mathworks, http://www.mathworks.com/help/toolbox/images/ref/normxcorr2.html .

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques, 1st ed. (Springer, 2004).

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Figures (11)

Fig. 1.
Fig. 1.

Static speckle formation that propagates through real-valued ABCD optical system. The illuminating spot in the object plane is Gaussian soft aperture with diameter 2 w .

Fig. 2.
Fig. 2.

Schematic demonstration of the orientation of speckle grains (ellipses) in FST configuration and in FT configuration ( x z plane). The dashed lines inside the ellipses denote the correlation lengths of the speckle grain in the corresponding directions. In the right half of this LCT system, the speckle grains all lie parallel to the optical axis and have the same longitudinal correlation lengths.

Fig. 3.
Fig. 3.

Correlation coefficient of integrated speckle intensity as the ratio of pixel size ( A D ) to lateral speckle size ( A C ) is varied. Circular and square dots, correlation coefficients calculated when A D / A C = 0.202 and A D / A C = π / 8 , respectively; dashed curves: Gaussian fitting of the corresponding dots. Solid curve, correlation coefficient calculated when A D / A C = 0 , which corresponds to the ideal pointlike detection. (Inset) Autocorrelation coefficient of integrated speckle intensity as a function of A D / A C .

Fig. 4.
Fig. 4.

Schematic representation of (a) an FST configuration, (b) an FT configuration when z 1 = z 2 = f , and an SLS, when z 1 f .

Fig. 5.
Fig. 5.

Lateral decorrelation between the off-axis speckles at P 1 = ( x 1 , 0 , z 0 ) and P 2 = ( x 1 + Δ x , 0 , z 0 + Δ z ) in FT configuration. Δ z in units of millimeters.

Fig. 6.
Fig. 6.

Longitudinal decorrelation of the speckles in the FT configuration. Note that the three plots, for various observation positions x = 0 (on-axis), x = 0.444 mm , and x = 0.888 mm (off-axis), overlap on top of one another.

Fig. 7.
Fig. 7.

Lateral decorrelation between the off-axis speckles at P 1 = ( x 1 , 0 , z 0 ) and P 2 = ( x 1 + Δ x , 0 , z 0 + Δ z ) in SLS configuration. Δ z in units of millimeters.

Fig. 8.
Fig. 8.

Longitudinal decorrelation of the speckles in the SLS. The three plots, for various observation positions x = 0 (on-axis), x = 2 mm , and x = 4 mm (off-axis), show that speckle longitudinal correlation length depends on the lateral observation position.

Fig. 9.
Fig. 9.

First-order statistics of the captured speckle intensities. The histograms are generated using 4096 intensity values. Dashed blue curves, PDFs of the intensity values predicted using Eq. (14). (a) For speckle pattern captured at z 0 = 80 mm ; in this case, A D / A C = 1.01 and M auto = 2.4975 . (b) For speckle pattern captured at z 0 = 250 mm ; in this case, A D / A C = 0.323 and M auto = 1.1420 . (c) For speckle pattern captured at z 0 = 400 mm ; in this case, A D / A C = 0.202 and M auto = 1.0548 .

Fig. 10.
Fig. 10.

Experimental measurements of the lateral speckle correlation coefficients in FT configuration as a function of lateral displacement Δ x of the two correlated speckle fields. Δ z in units of millimeters.

Fig. 11.
Fig. 11.

Experimental measurements of the longitudinal correlation coefficients in FT configuration, as a function of longitudinal displacement Δ z of the two correlated speckle fields. Note the correlation values for the various observation positions, x = 0 (on-axis), x = 3 mm , and x = 6 mm (off-axis), overlap on top of one another.

Equations (19)

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U ( ρ ) = U 0 ( r ) exp [ i k 2 B ( A r 2 2 r · ρ + D ρ 2 ) ] d 2 r ,
{ A B C D } FST = { 1 z 0 0 1 } ,
{ A B C D } FT = { 0 f 1 f 0 } ,
{ A B C D } CMT = { 1 0 1 f 1 } ,
{ A B C D } SLS = { 1 z 2 f z 2 + z 1 ( 1 z 2 f ) 1 f 1 z 1 f } .
I ( ρ 1 ) I ( ρ 2 ) I ( ρ 1 ) I ( ρ 2 ) = 1 + C I ( ρ , Δ ρ , Δ z ) ,
C I ( ρ , Δ ρ , Δ z ) = 1 1 + ( Δ z / l z ) 2 exp [ 1 ρ 0 2 | Δ ρ D Δ z ρ / B | 2 ] .
l z = 4 B ( B + D Δ z ) k w 2 ,
ρ 0 = 2 ( B + D Δ z ) k w [ 1 + ( Δ z / l z ) 2 ] 1 / 2 .
I a = 1 A D 2 + D a ( ρ ) I ( ρ ) d 2 ρ ,
I a 1 I a 2 = 1 A D 4 + + D a 1 ( ρ 1 ) D a 2 ( ρ 2 ) I ( ρ 1 ) I ( ρ 2 ) d 2 ρ 1 d 2 ρ 2 ,
M 1 = I a 1 I a 2 I a 1 I a 2 1 = 1 A D 4 + + D a 1 ( ρ 1 ) D a 2 ( ρ 2 ) C I ( ρ 1 , Δ ρ , Δ z ) d 2 ρ 1 d 2 ρ 2 ,
M auto 1 = 1 A D 4 + K D ( Δ ρ ) C I ( ρ 1 , Δ ρ , 0 ) d 2 Δ ρ ,
K D ( Δ ρ ) = + D a ( ρ 1 ) D a ( ρ 1 + Δ ρ ) d 2 ρ 1 .
C I ( ρ 1 , Δ ρ , 0 ) = exp [ | Δ ρ | 2 ( 2 B / k w ) 2 ] ,
D a ( x , y ) = { 1 A D 2 x , y A D 2 0 , otherwise ,
M auto 1 = A C 2 16 A D 4 exp ( 8 A D 2 A C 2 ) [ A C exp ( 4 A D 2 A C 2 ) [ A C 2 A D π erf ( 2 A D A C ) ] ] 2 ,
P ( I a ) = 1 Γ ( M auto ) ( M auto I a ) M auto I a ( M auto 1 ) exp ( M auto I a I a ) ,
Δ x Δ z = x 1 ( D B ) .

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