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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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)

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]

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]

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, 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]

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 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]

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, 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]

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]

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 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]

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, 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]

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]

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)

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]

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]

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]

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]

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]

T. Yoshimura and S. Iwamoto, “Dynamic properties of three-dimensional speckles,” J. Opt. Soc. Am. A 10, 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, 1855–1864 (2009).
[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]

Opt. Commun. (2)

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

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]

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).

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

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

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

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

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

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