Abstract

In Part I [J. Opt. Soc. Am. A 28, 1896 (2011) of this paper, the physical model for fully developed speckle is examined based on two critical assumptions. (i) It is assumed that in the object plane, the speckle field is delta correlated, and (ii) it is assumed that the speckle field in the observation plane can be described as a Gaussian random process. A satisfactory simulation technique, based on the assumption that spatial averaging can be used to replace ensemble averaging, is also presented. In this part a detailed experimental investigation of the three-dimensional speckle properties is performed using spatial averaging. The results provide solid verification for the predictions presented in Part I. The results are not only of theoretical interest but have practical implications. Techniques for locating and aligning the optical system axis with the camera center, and for measuring out-of- plane displacement, are demonstrated.

© 2011 Optical Society of America

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References

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  1. 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).
  2. L. Leushacke and M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. 7, 827–832 (1990).
    [CrossRef]
  3. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 1st ed. (Roberts, 2007).
  4. T. Yoshimura and S. Iwamoto, “Dynamic properties of three-dimensional speckles,” J. Opt. Soc. Am. 10, 324–328 (1993).
    [CrossRef]
  5. Q. B. Li and F. P. Chiang, “Three-dimensional dimension of laser speckle,” Appl. Opt. 31, 6287–6291 (1992).
    [CrossRef] [PubMed]
  6. A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. 78, 063806 (2008).
    [CrossRef]
  7. G. P. Weigelt and B. Stoffregen, “The longitudinal correlation of three-dimensional speckle intensity distribution,” Optik 48, 399–408 (1977).
  8. C. E. Halford, W. L. Gamble, and N. George, “Experimental investigation of the longitudinal characteristics of laser speckle,” Opt. Eng. 26, 1263–1264 (1987).
  9. J. E. Ward, D. P. Kelly, and J. T. Sheridan, “An alignment technique based on the speckle correlation properties of Fresnel transforming optical systems,” Proc. SPIE 7068, 70680L (2008).
    [CrossRef]
  10. J. E. Ward, “Optical metrology, speckle control, and the reduction of image degradation due to atmospheric turbulence,” Ph.D. dissertation (University College Dublin, 2009).
  11. T. Fricke-Begemann, “Three-dimensional deformation field measurement with digital speckle correlation,” Appl. Opt. 42, 6783–6796 (2003).
    [CrossRef] [PubMed]
  12. D. P. Kelly, B. M. Hennelly, and J. T. Sheridan, “Magnitude and direction of motion with speckle correlation and the optical fractional Fourier transform,” Appl. Opt. 44, 2720–2727 (2005).
    [CrossRef] [PubMed]
  13. 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. 23, 2861–2870 (2006).
    [CrossRef]
  14. B. Bhaduri, C. Quan, C. J. Tay, and M. Sjödahl, “Simultaneous measurement of translation and tilt using digital speckle photography,” Appl. Opt. 49, 3573–3579 (2010).
    [CrossRef] [PubMed]
  15. Wolfram Research, http://reference.wolfram.com/mathematica/ref/Interpolation.html.

2010

2008

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. 78, 063806 (2008).
[CrossRef]

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “An alignment technique based on the speckle correlation properties of Fresnel transforming optical systems,” Proc. SPIE 7068, 70680L (2008).
[CrossRef]

2006

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. 23, 2861–2870 (2006).
[CrossRef]

2005

2003

1993

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

1992

1990

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

1987

C. E. Halford, W. L. Gamble, and N. George, “Experimental investigation of the longitudinal characteristics of laser speckle,” Opt. Eng. 26, 1263–1264 (1987).

1977

G. P. Weigelt and B. Stoffregen, “The longitudinal correlation of three-dimensional speckle intensity distribution,” Optik 48, 399–408 (1977).

Bhaduri, B.

Chiang, F. P.

Ferri, F.

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. 78, 063806 (2008).
[CrossRef]

Fricke-Begemann, T.

Gamble, W. L.

C. E. Halford, W. L. Gamble, and N. George, “Experimental investigation of the longitudinal characteristics of laser speckle,” Opt. Eng. 26, 1263–1264 (1987).

Gatti, A.

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. 78, 063806 (2008).
[CrossRef]

George, N.

C. E. Halford, W. L. Gamble, and N. George, “Experimental investigation of the longitudinal characteristics of laser speckle,” Opt. Eng. 26, 1263–1264 (1987).

Goodman, J. W.

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

Gopinathan, U.

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. 23, 2861–2870 (2006).
[CrossRef]

Halford, C. E.

C. E. Halford, W. L. Gamble, and N. George, “Experimental investigation of the longitudinal characteristics of laser speckle,” Opt. Eng. 26, 1263–1264 (1987).

Hennelly, B. M.

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. 23, 2861–2870 (2006).
[CrossRef]

D. P. Kelly, B. M. Hennelly, and J. T. Sheridan, “Magnitude and direction of motion with speckle correlation and the optical fractional Fourier transform,” Appl. Opt. 44, 2720–2727 (2005).
[CrossRef] [PubMed]

Iwamoto, S.

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

Kelly, D. P.

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “An alignment technique based on the speckle correlation properties of Fresnel transforming optical systems,” Proc. SPIE 7068, 70680L (2008).
[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. 23, 2861–2870 (2006).
[CrossRef]

D. P. Kelly, B. M. Hennelly, and J. T. Sheridan, “Magnitude and direction of motion with speckle correlation and the optical fractional Fourier transform,” Appl. Opt. 44, 2720–2727 (2005).
[CrossRef] [PubMed]

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).

Kirchner, M.

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

Leushacke, L.

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

Li, D.

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).

Li, Q. B.

Magatti, D.

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. 78, 063806 (2008).
[CrossRef]

O’Neill, F. T.

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. 23, 2861–2870 (2006).
[CrossRef]

Quan, C.

Sheridan, J. T.

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “An alignment technique based on the speckle correlation properties of Fresnel transforming optical systems,” Proc. SPIE 7068, 70680L (2008).
[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. 23, 2861–2870 (2006).
[CrossRef]

D. P. Kelly, B. M. Hennelly, and J. T. Sheridan, “Magnitude and direction of motion with speckle correlation and the optical fractional Fourier transform,” Appl. Opt. 44, 2720–2727 (2005).
[CrossRef] [PubMed]

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).

Sjödahl, M.

Stoffregen, B.

G. P. Weigelt and B. Stoffregen, “The longitudinal correlation of three-dimensional speckle intensity distribution,” Optik 48, 399–408 (1977).

Tay, C. J.

Ward, J. E.

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “An alignment technique based on the speckle correlation properties of Fresnel transforming optical systems,” Proc. SPIE 7068, 70680L (2008).
[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. 23, 2861–2870 (2006).
[CrossRef]

J. E. Ward, “Optical metrology, speckle control, and the reduction of image degradation due to atmospheric turbulence,” Ph.D. dissertation (University College Dublin, 2009).

Weigelt, G. P.

G. P. Weigelt and B. Stoffregen, “The longitudinal correlation of three-dimensional speckle intensity distribution,” Optik 48, 399–408 (1977).

Yoshimura, T.

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

Appl. Opt.

J. Opt. Soc. Am.

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. 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. 7, 827–832 (1990).
[CrossRef]

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

Opt. Eng.

C. E. Halford, W. L. Gamble, and N. George, “Experimental investigation of the longitudinal characteristics of laser speckle,” Opt. Eng. 26, 1263–1264 (1987).

Optik

G. P. Weigelt and B. Stoffregen, “The longitudinal correlation of three-dimensional speckle intensity distribution,” Optik 48, 399–408 (1977).

Phys. Rev.

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. 78, 063806 (2008).
[CrossRef]

Proc. SPIE

J. E. Ward, D. P. Kelly, and J. T. Sheridan, “An alignment technique based on the speckle correlation properties of Fresnel transforming optical systems,” Proc. SPIE 7068, 70680L (2008).
[CrossRef]

Other

J. E. Ward, “Optical metrology, speckle control, and the reduction of image degradation due to atmospheric turbulence,” Ph.D. dissertation (University College Dublin, 2009).

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).

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

Wolfram Research, http://reference.wolfram.com/mathematica/ref/Interpolation.html.

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Figures (5)

Fig. 1
Fig. 1

Optical arrangement for the static speckle formation in the Fresnel regime under illumination of a Gaussian beam.

Fig. 2
Fig. 2

Illustration of two longitudinally displaced speckle pattern images: (a) at z = z 0 and (b)  z = z 0 + ε . Each has been divided into 25 subimages in order to observe the longitudinal correlation coefficients variations along the radial directions.

Fig. 3
Fig. 3

Experimental results of the longitudinal correlation coefficients for the subimage pairs. (a) Results for Step 1, the central active area of the camera is divided into 25 subimages, each subimage has 409 × 409 pixels. (b) Final results for Step 2, the central image region of size 605 × 605 is divided into 25 subimages, each subimage has 121 × 121 pixels.

Fig. 4
Fig. 4

Lateral speckle correlation function for an off-axis field at Q 1 = ( x 1 , 0 ) versus lateral shifting γ when ε = 0 mm , ε = 0.5 mm , ε = 1 mm . Dots: discrete correlation coefficients from the experiment. Dashed curve: cubic spline interpolation based on the experimental data. Solid curve: theoretical continuous speckle correlation.

Fig. 5
Fig. 5

Longitudinal speckle correlation function for an off-axis field at Q = ( x , 0 ) versus longitudinal displacement ε when x = 0 mm (on-axis), x = 2.22 mm , x = 4.44 mm , and x = 6.66 mm . Dots: discrete correlation coefficients from the experiment. Solid curve: theoretical continuous speckle correlation.

Tables (1)

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Table 1 Out-of-Plane Displacement Measurement Results

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