Abstract

Non-Gaussian speckle contrast from a phase-perturbed random object field in a spatially partially coherent system is simulated. A quasi-monochromatic extended incoherent source is modeled as a collection of independent point sources distributed on a regular grid. The source illuminates a phase screen object in a Kohler configuration. Speckle is calculated from the incoherent sum of irradiances in the image plane generated from the point sources. Simulated speckle contrasts are verified by an experiment with a fractallike rough surface distribution that is fabricated using a grayscale maskless lithography tool. Characteristics of the simulation method and physical quantities affecting speckle contrast are discussed.

© 2009 Optical Society of America

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  1. J. C. Dainty, A. E. Ennos, M. Francon, J. W. Goodman, T. S. McKechnie, and G. Parry, Laser Speckle and Related Phenomena (Springer-Verlag, 1984).
  2. J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2007).
  3. H. Fujii and T. Asakura, “Effect of the point spread function on the average contrast of image speckle patterns,” Opt. Commun. 21, 80-84 (1977).
    [Crossref]
  4. H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32-38 (1974).
    [Crossref]
  5. T. S. McKechnie, “Image-plane speckle in partially coherent illumination,” Opt. Quantum Electron. 8, 61-67 (1976).
    [Crossref]
  6. H. M. Pedersen, “Theory of speckle dependence on surface roughness,” J. Opt. Soc. Am. 66, 1204-1210 (1976).
    [Crossref]
  7. D. G. Voelz, K. A. Bush, and P. S. Idell, “Illumination coherence effects in laser-speckle imaging: modeling and experimental demonstration,” Appl. Opt. 36, 1781-1788 (1997).
    [Crossref] [PubMed]
  8. H. Ohtsubo and T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315-319 (1978).
    [Crossref]
  9. P. J. Chandley and H. M. Escamilla, “Speckle from a rough surface when the illuminated region contains few correlation areas: the effect of changing the surface height variance,” Opt. Commun. 29, 151-154 (1979).
    [Crossref]
  10. D. L. Jordan, R. C. Hollins, and E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B: Photophys. Laser Chem. 31, 179-186 (1983).
    [Crossref]
  11. E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. (Bellingham) 23, 453-461 (1984).
  12. J. Uozumi and T. Asakura, “The first-order statistics of partially developed non-Gaussian speckle patterns,” J. Opt. 12, 177-186 (1981).
    [Crossref]
  13. B. M. Levine, “Non-Gaussian speckle caused by thin phase screens of large root-mean-square phase variations and long single-scale autocorrelations,” J. Opt. Soc. Am. A 3, 1283-1292 (1986).
    [Crossref]
  14. E. Jakeman and P. N. Pussy, “Significance of K distribution in scattering experiments,” Phys. Rev. Lett. 40, 546-550 (1978).
    [Crossref]
  15. R. Barakat, “Weak-scatterer generalization of the K-density function with application to laser scattering in atmospheric turbulence,” J. Opt. Soc. Am. A 3, 401-409 (1986).
    [Crossref]
  16. L. Weng, J. M. Reid, P. Mohana Shankar, and K. Soetanto, “Ultrasound speckle analysis based on the K distribution,” J. Acoust. Soc. Am. 89, 2992-2995 (1991).
    [Crossref] [PubMed]
  17. I. A. Popov, N. V. Sidorovsky, and L. M. Veselov, “Experimental study of intensity probability density function in the speckle pattern formed by a small number of scatterers,” Opt. Commun. 97, 304-306 (1993).
    [Crossref]
  18. P. K. Murphy, J. P. Allebach, and N. C. Gallagher, “Effect of optical aberrations on laser speckle,” J. Opt. Soc. Am. A 3, 215-222 (1986).
    [Crossref]
  19. D. Kang, E. Clarkson, and T. D. Milster, “Effect of optical aberrations on Gaussian laser speckle,” Opt. Express 17, 3084-3100 (2009).
    [Crossref] [PubMed]
  20. I. A. Popov, N. V. Sidorovsky, and L. M. Veselov, “Statistical properties of non-Gaussian intensity fluctuations in the image plane of an optical system,” Opt. Commun. 134, 289-300 (1997).
    [Crossref]
  21. K. A. Stetson, “The vulnerability of speckle photography to lens aberrations,” J. Opt. Soc. Am. 67, 1587-1590 (1977).
    [Crossref]
  22. R. D. Bahuguna, K. K. Gupta, and K. Singh, “Study of laser speckles in the presence of spherical aberration,” J. Opt. Soc. Am. 69, 877-882 (1979).
    [Crossref]
  23. R. D. Bahuguna, K. K. Gupta, and K. Singh, “Speckle patterns of weak diffusers: effect of spherical aberration,” Appl. Opt. 19, 1874-1878 (1980).
    [Crossref] [PubMed]
  24. A. Majumdar and C. L. Tien, “Fractal characterization and simulation of rough surfaces,” Wear 136, 313-327 (1990).
    [Crossref]
  25. J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces,” Phys. Rev. Lett. 70, 57-60 (1993).
    [Crossref] [PubMed]
  26. M. V. Berry, “Diffractals,” J. Phys. A 12, 781-797 (1979).
    [Crossref]
  27. E. Jakeman, “Fresnel scattering by a corrugated random surface with fractal slope,” J. Opt. Soc. Am. 72, 1034-1041 (1982).
    [Crossref]
  28. E. Marx, J. J. Malik, Y. E. Strausser, T. Bristow, N. Poduje, and J. C. Stover, “Power spectral densities: A multiple technique study of different Si wafer surface,” J. Vac. Sci. Technol. B 20, 31-41 (2002).
    [Crossref]
  29. E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc. SPIE 1530, 71-85 (1991).
    [Crossref]
  30. G. Palasantzas, “Roughness spectrum and surface width of self-affine fractal surfaces via the K-correlation model,” Phys. Rev. B 48, 14472-14478 (1993).
    [Crossref]
  31. H. N. Yang, Y. P. Zhao, A. Chan, T. M. Lu, and G. C. Wang, “Sampling-induced hidden cycles in correlated random rough surfaces,” Phys. Rev. B 56, 4224-4232 (1997).
    [Crossref]
  32. E. Jakeman and J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B: Photophys. Laser Chem. 26, 125-131 (1981).
    [Crossref]
  33. B. M. Levine and J. C. Dainty, “Non-Gaussian image plane speckle: Measurements from diffusers of known statistics,” Opt. Commun. 45, 252-257 (1983).
    [Crossref]
  34. J. M. Tamkin, B. Bagwell, B. Kimbrough, G. Jabbour, and M. Descour, “High speed gray scale laser direct write technology for micro-optic fabrication,” Proc. SPIE 4984, 210-218 (2003).
    [Crossref]
  35. J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30-34 (1975).
    [Crossref]
  36. D. Kang and T. D. Milster, “Effect of optical aberration on Gaussian speckle in a partially coherent imaging system,” submitted to J. Opt. Soc. Am. A.
  37. D. S. Goodman and A. E. Rosenbluth, “Condenser aberrations in Kohler illumination,” Proc. SPIE 922, 108-134 (1988).

2009 (1)

2003 (1)

J. M. Tamkin, B. Bagwell, B. Kimbrough, G. Jabbour, and M. Descour, “High speed gray scale laser direct write technology for micro-optic fabrication,” Proc. SPIE 4984, 210-218 (2003).
[Crossref]

2002 (1)

E. Marx, J. J. Malik, Y. E. Strausser, T. Bristow, N. Poduje, and J. C. Stover, “Power spectral densities: A multiple technique study of different Si wafer surface,” J. Vac. Sci. Technol. B 20, 31-41 (2002).
[Crossref]

1997 (3)

H. N. Yang, Y. P. Zhao, A. Chan, T. M. Lu, and G. C. Wang, “Sampling-induced hidden cycles in correlated random rough surfaces,” Phys. Rev. B 56, 4224-4232 (1997).
[Crossref]

I. A. Popov, N. V. Sidorovsky, and L. M. Veselov, “Statistical properties of non-Gaussian intensity fluctuations in the image plane of an optical system,” Opt. Commun. 134, 289-300 (1997).
[Crossref]

D. G. Voelz, K. A. Bush, and P. S. Idell, “Illumination coherence effects in laser-speckle imaging: modeling and experimental demonstration,” Appl. Opt. 36, 1781-1788 (1997).
[Crossref] [PubMed]

1993 (3)

I. A. Popov, N. V. Sidorovsky, and L. M. Veselov, “Experimental study of intensity probability density function in the speckle pattern formed by a small number of scatterers,” Opt. Commun. 97, 304-306 (1993).
[Crossref]

J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces,” Phys. Rev. Lett. 70, 57-60 (1993).
[Crossref] [PubMed]

G. Palasantzas, “Roughness spectrum and surface width of self-affine fractal surfaces via the K-correlation model,” Phys. Rev. B 48, 14472-14478 (1993).
[Crossref]

1991 (2)

E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc. SPIE 1530, 71-85 (1991).
[Crossref]

L. Weng, J. M. Reid, P. Mohana Shankar, and K. Soetanto, “Ultrasound speckle analysis based on the K distribution,” J. Acoust. Soc. Am. 89, 2992-2995 (1991).
[Crossref] [PubMed]

1990 (1)

A. Majumdar and C. L. Tien, “Fractal characterization and simulation of rough surfaces,” Wear 136, 313-327 (1990).
[Crossref]

1988 (1)

D. S. Goodman and A. E. Rosenbluth, “Condenser aberrations in Kohler illumination,” Proc. SPIE 922, 108-134 (1988).

1986 (3)

1984 (1)

E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. (Bellingham) 23, 453-461 (1984).

1983 (2)

D. L. Jordan, R. C. Hollins, and E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B: Photophys. Laser Chem. 31, 179-186 (1983).
[Crossref]

B. M. Levine and J. C. Dainty, “Non-Gaussian image plane speckle: Measurements from diffusers of known statistics,” Opt. Commun. 45, 252-257 (1983).
[Crossref]

1982 (1)

1981 (2)

E. Jakeman and J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B: Photophys. Laser Chem. 26, 125-131 (1981).
[Crossref]

J. Uozumi and T. Asakura, “The first-order statistics of partially developed non-Gaussian speckle patterns,” J. Opt. 12, 177-186 (1981).
[Crossref]

1980 (1)

1979 (3)

R. D. Bahuguna, K. K. Gupta, and K. Singh, “Study of laser speckles in the presence of spherical aberration,” J. Opt. Soc. Am. 69, 877-882 (1979).
[Crossref]

M. V. Berry, “Diffractals,” J. Phys. A 12, 781-797 (1979).
[Crossref]

P. J. Chandley and H. M. Escamilla, “Speckle from a rough surface when the illuminated region contains few correlation areas: the effect of changing the surface height variance,” Opt. Commun. 29, 151-154 (1979).
[Crossref]

1978 (2)

H. Ohtsubo and T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315-319 (1978).
[Crossref]

E. Jakeman and P. N. Pussy, “Significance of K distribution in scattering experiments,” Phys. Rev. Lett. 40, 546-550 (1978).
[Crossref]

1977 (2)

H. Fujii and T. Asakura, “Effect of the point spread function on the average contrast of image speckle patterns,” Opt. Commun. 21, 80-84 (1977).
[Crossref]

K. A. Stetson, “The vulnerability of speckle photography to lens aberrations,” J. Opt. Soc. Am. 67, 1587-1590 (1977).
[Crossref]

1976 (2)

T. S. McKechnie, “Image-plane speckle in partially coherent illumination,” Opt. Quantum Electron. 8, 61-67 (1976).
[Crossref]

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

1975 (1)

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30-34 (1975).
[Crossref]

1974 (1)

H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32-38 (1974).
[Crossref]

Allebach, J. P.

Asakura, T.

J. Uozumi and T. Asakura, “The first-order statistics of partially developed non-Gaussian speckle patterns,” J. Opt. 12, 177-186 (1981).
[Crossref]

H. Ohtsubo and T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315-319 (1978).
[Crossref]

H. Fujii and T. Asakura, “Effect of the point spread function on the average contrast of image speckle patterns,” Opt. Commun. 21, 80-84 (1977).
[Crossref]

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30-34 (1975).
[Crossref]

H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32-38 (1974).
[Crossref]

Bagwell, B.

J. M. Tamkin, B. Bagwell, B. Kimbrough, G. Jabbour, and M. Descour, “High speed gray scale laser direct write technology for micro-optic fabrication,” Proc. SPIE 4984, 210-218 (2003).
[Crossref]

Bahuguna, R. D.

Barakat, R.

Berry, M. V.

M. V. Berry, “Diffractals,” J. Phys. A 12, 781-797 (1979).
[Crossref]

Bristow, T.

E. Marx, J. J. Malik, Y. E. Strausser, T. Bristow, N. Poduje, and J. C. Stover, “Power spectral densities: A multiple technique study of different Si wafer surface,” J. Vac. Sci. Technol. B 20, 31-41 (2002).
[Crossref]

Bruynseraede, Y.

J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces,” Phys. Rev. Lett. 70, 57-60 (1993).
[Crossref] [PubMed]

Bush, K. A.

Chan, A.

H. N. Yang, Y. P. Zhao, A. Chan, T. M. Lu, and G. C. Wang, “Sampling-induced hidden cycles in correlated random rough surfaces,” Phys. Rev. B 56, 4224-4232 (1997).
[Crossref]

Chandley, P. J.

P. J. Chandley and H. M. Escamilla, “Speckle from a rough surface when the illuminated region contains few correlation areas: the effect of changing the surface height variance,” Opt. Commun. 29, 151-154 (1979).
[Crossref]

Church, E. L.

E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc. SPIE 1530, 71-85 (1991).
[Crossref]

Clarkson, E.

Dainty, J. C.

B. M. Levine and J. C. Dainty, “Non-Gaussian image plane speckle: Measurements from diffusers of known statistics,” Opt. Commun. 45, 252-257 (1983).
[Crossref]

J. C. Dainty, A. E. Ennos, M. Francon, J. W. Goodman, T. S. McKechnie, and G. Parry, Laser Speckle and Related Phenomena (Springer-Verlag, 1984).

Descour, M.

J. M. Tamkin, B. Bagwell, B. Kimbrough, G. Jabbour, and M. Descour, “High speed gray scale laser direct write technology for micro-optic fabrication,” Proc. SPIE 4984, 210-218 (2003).
[Crossref]

Ennos, A. E.

J. C. Dainty, A. E. Ennos, M. Francon, J. W. Goodman, T. S. McKechnie, and G. Parry, Laser Speckle and Related Phenomena (Springer-Verlag, 1984).

Escamilla, H. M.

P. J. Chandley and H. M. Escamilla, “Speckle from a rough surface when the illuminated region contains few correlation areas: the effect of changing the surface height variance,” Opt. Commun. 29, 151-154 (1979).
[Crossref]

Francon, M.

J. C. Dainty, A. E. Ennos, M. Francon, J. W. Goodman, T. S. McKechnie, and G. Parry, Laser Speckle and Related Phenomena (Springer-Verlag, 1984).

Fujii, H.

H. Fujii and T. Asakura, “Effect of the point spread function on the average contrast of image speckle patterns,” Opt. Commun. 21, 80-84 (1977).
[Crossref]

H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32-38 (1974).
[Crossref]

Gallagher, N. C.

Goodman, D. S.

D. S. Goodman and A. E. Rosenbluth, “Condenser aberrations in Kohler illumination,” Proc. SPIE 922, 108-134 (1988).

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2007).

J. C. Dainty, A. E. Ennos, M. Francon, J. W. Goodman, T. S. McKechnie, and G. Parry, Laser Speckle and Related Phenomena (Springer-Verlag, 1984).

Gupta, K. K.

Heyvaert, I.

J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces,” Phys. Rev. Lett. 70, 57-60 (1993).
[Crossref] [PubMed]

Hollins, R. C.

D. L. Jordan, R. C. Hollins, and E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B: Photophys. Laser Chem. 31, 179-186 (1983).
[Crossref]

Idell, P. S.

Jabbour, G.

J. M. Tamkin, B. Bagwell, B. Kimbrough, G. Jabbour, and M. Descour, “High speed gray scale laser direct write technology for micro-optic fabrication,” Proc. SPIE 4984, 210-218 (2003).
[Crossref]

Jakeman, E.

E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. (Bellingham) 23, 453-461 (1984).

D. L. Jordan, R. C. Hollins, and E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B: Photophys. Laser Chem. 31, 179-186 (1983).
[Crossref]

E. Jakeman, “Fresnel scattering by a corrugated random surface with fractal slope,” J. Opt. Soc. Am. 72, 1034-1041 (1982).
[Crossref]

E. Jakeman and J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B: Photophys. Laser Chem. 26, 125-131 (1981).
[Crossref]

E. Jakeman and P. N. Pussy, “Significance of K distribution in scattering experiments,” Phys. Rev. Lett. 40, 546-550 (1978).
[Crossref]

Jordan, D. L.

D. L. Jordan, R. C. Hollins, and E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B: Photophys. Laser Chem. 31, 179-186 (1983).
[Crossref]

Kang, D.

D. Kang, E. Clarkson, and T. D. Milster, “Effect of optical aberrations on Gaussian laser speckle,” Opt. Express 17, 3084-3100 (2009).
[Crossref] [PubMed]

D. Kang and T. D. Milster, “Effect of optical aberration on Gaussian speckle in a partially coherent imaging system,” submitted to J. Opt. Soc. Am. A.

Kimbrough, B.

J. M. Tamkin, B. Bagwell, B. Kimbrough, G. Jabbour, and M. Descour, “High speed gray scale laser direct write technology for micro-optic fabrication,” Proc. SPIE 4984, 210-218 (2003).
[Crossref]

Krim, J.

J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces,” Phys. Rev. Lett. 70, 57-60 (1993).
[Crossref] [PubMed]

Levine, B. M.

B. M. Levine, “Non-Gaussian speckle caused by thin phase screens of large root-mean-square phase variations and long single-scale autocorrelations,” J. Opt. Soc. Am. A 3, 1283-1292 (1986).
[Crossref]

B. M. Levine and J. C. Dainty, “Non-Gaussian image plane speckle: Measurements from diffusers of known statistics,” Opt. Commun. 45, 252-257 (1983).
[Crossref]

Lu, T. M.

H. N. Yang, Y. P. Zhao, A. Chan, T. M. Lu, and G. C. Wang, “Sampling-induced hidden cycles in correlated random rough surfaces,” Phys. Rev. B 56, 4224-4232 (1997).
[Crossref]

Majumdar, A.

A. Majumdar and C. L. Tien, “Fractal characterization and simulation of rough surfaces,” Wear 136, 313-327 (1990).
[Crossref]

Malik, J. J.

E. Marx, J. J. Malik, Y. E. Strausser, T. Bristow, N. Poduje, and J. C. Stover, “Power spectral densities: A multiple technique study of different Si wafer surface,” J. Vac. Sci. Technol. B 20, 31-41 (2002).
[Crossref]

Marx, E.

E. Marx, J. J. Malik, Y. E. Strausser, T. Bristow, N. Poduje, and J. C. Stover, “Power spectral densities: A multiple technique study of different Si wafer surface,” J. Vac. Sci. Technol. B 20, 31-41 (2002).
[Crossref]

McKechnie, T. S.

T. S. McKechnie, “Image-plane speckle in partially coherent illumination,” Opt. Quantum Electron. 8, 61-67 (1976).
[Crossref]

J. C. Dainty, A. E. Ennos, M. Francon, J. W. Goodman, T. S. McKechnie, and G. Parry, Laser Speckle and Related Phenomena (Springer-Verlag, 1984).

McWhirter, J. G.

E. Jakeman and J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B: Photophys. Laser Chem. 26, 125-131 (1981).
[Crossref]

Milster, T. D.

D. Kang, E. Clarkson, and T. D. Milster, “Effect of optical aberrations on Gaussian laser speckle,” Opt. Express 17, 3084-3100 (2009).
[Crossref] [PubMed]

D. Kang and T. D. Milster, “Effect of optical aberration on Gaussian speckle in a partially coherent imaging system,” submitted to J. Opt. Soc. Am. A.

Murphy, P. K.

Ohtsubo, H.

H. Ohtsubo and T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315-319 (1978).
[Crossref]

Ohtsubo, J.

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30-34 (1975).
[Crossref]

Palasantzas, G.

G. Palasantzas, “Roughness spectrum and surface width of self-affine fractal surfaces via the K-correlation model,” Phys. Rev. B 48, 14472-14478 (1993).
[Crossref]

Parry, G.

J. C. Dainty, A. E. Ennos, M. Francon, J. W. Goodman, T. S. McKechnie, and G. Parry, Laser Speckle and Related Phenomena (Springer-Verlag, 1984).

Pedersen, H. M.

Poduje, N.

E. Marx, J. J. Malik, Y. E. Strausser, T. Bristow, N. Poduje, and J. C. Stover, “Power spectral densities: A multiple technique study of different Si wafer surface,” J. Vac. Sci. Technol. B 20, 31-41 (2002).
[Crossref]

Popov, I. A.

I. A. Popov, N. V. Sidorovsky, and L. M. Veselov, “Statistical properties of non-Gaussian intensity fluctuations in the image plane of an optical system,” Opt. Commun. 134, 289-300 (1997).
[Crossref]

I. A. Popov, N. V. Sidorovsky, and L. M. Veselov, “Experimental study of intensity probability density function in the speckle pattern formed by a small number of scatterers,” Opt. Commun. 97, 304-306 (1993).
[Crossref]

Pussy, P. N.

E. Jakeman and P. N. Pussy, “Significance of K distribution in scattering experiments,” Phys. Rev. Lett. 40, 546-550 (1978).
[Crossref]

Reid, J. M.

L. Weng, J. M. Reid, P. Mohana Shankar, and K. Soetanto, “Ultrasound speckle analysis based on the K distribution,” J. Acoust. Soc. Am. 89, 2992-2995 (1991).
[Crossref] [PubMed]

Rosenbluth, A. E.

D. S. Goodman and A. E. Rosenbluth, “Condenser aberrations in Kohler illumination,” Proc. SPIE 922, 108-134 (1988).

Shankar, P. Mohana

L. Weng, J. M. Reid, P. Mohana Shankar, and K. Soetanto, “Ultrasound speckle analysis based on the K distribution,” J. Acoust. Soc. Am. 89, 2992-2995 (1991).
[Crossref] [PubMed]

Sidorovsky, N. V.

I. A. Popov, N. V. Sidorovsky, and L. M. Veselov, “Statistical properties of non-Gaussian intensity fluctuations in the image plane of an optical system,” Opt. Commun. 134, 289-300 (1997).
[Crossref]

I. A. Popov, N. V. Sidorovsky, and L. M. Veselov, “Experimental study of intensity probability density function in the speckle pattern formed by a small number of scatterers,” Opt. Commun. 97, 304-306 (1993).
[Crossref]

Singh, K.

Soetanto, K.

L. Weng, J. M. Reid, P. Mohana Shankar, and K. Soetanto, “Ultrasound speckle analysis based on the K distribution,” J. Acoust. Soc. Am. 89, 2992-2995 (1991).
[Crossref] [PubMed]

Stetson, K. A.

Stover, J. C.

E. Marx, J. J. Malik, Y. E. Strausser, T. Bristow, N. Poduje, and J. C. Stover, “Power spectral densities: A multiple technique study of different Si wafer surface,” J. Vac. Sci. Technol. B 20, 31-41 (2002).
[Crossref]

Strausser, Y. E.

E. Marx, J. J. Malik, Y. E. Strausser, T. Bristow, N. Poduje, and J. C. Stover, “Power spectral densities: A multiple technique study of different Si wafer surface,” J. Vac. Sci. Technol. B 20, 31-41 (2002).
[Crossref]

Takacs, P. Z.

E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc. SPIE 1530, 71-85 (1991).
[Crossref]

Tamkin, J. M.

J. M. Tamkin, B. Bagwell, B. Kimbrough, G. Jabbour, and M. Descour, “High speed gray scale laser direct write technology for micro-optic fabrication,” Proc. SPIE 4984, 210-218 (2003).
[Crossref]

Tien, C. L.

A. Majumdar and C. L. Tien, “Fractal characterization and simulation of rough surfaces,” Wear 136, 313-327 (1990).
[Crossref]

Uozumi, J.

J. Uozumi and T. Asakura, “The first-order statistics of partially developed non-Gaussian speckle patterns,” J. Opt. 12, 177-186 (1981).
[Crossref]

Van Haesendonck, C.

J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces,” Phys. Rev. Lett. 70, 57-60 (1993).
[Crossref] [PubMed]

Veselov, L. M.

I. A. Popov, N. V. Sidorovsky, and L. M. Veselov, “Statistical properties of non-Gaussian intensity fluctuations in the image plane of an optical system,” Opt. Commun. 134, 289-300 (1997).
[Crossref]

I. A. Popov, N. V. Sidorovsky, and L. M. Veselov, “Experimental study of intensity probability density function in the speckle pattern formed by a small number of scatterers,” Opt. Commun. 97, 304-306 (1993).
[Crossref]

Voelz, D. G.

Wang, G. C.

H. N. Yang, Y. P. Zhao, A. Chan, T. M. Lu, and G. C. Wang, “Sampling-induced hidden cycles in correlated random rough surfaces,” Phys. Rev. B 56, 4224-4232 (1997).
[Crossref]

Weng, L.

L. Weng, J. M. Reid, P. Mohana Shankar, and K. Soetanto, “Ultrasound speckle analysis based on the K distribution,” J. Acoust. Soc. Am. 89, 2992-2995 (1991).
[Crossref] [PubMed]

Yang, H. N.

H. N. Yang, Y. P. Zhao, A. Chan, T. M. Lu, and G. C. Wang, “Sampling-induced hidden cycles in correlated random rough surfaces,” Phys. Rev. B 56, 4224-4232 (1997).
[Crossref]

Zhao, Y. P.

H. N. Yang, Y. P. Zhao, A. Chan, T. M. Lu, and G. C. Wang, “Sampling-induced hidden cycles in correlated random rough surfaces,” Phys. Rev. B 56, 4224-4232 (1997).
[Crossref]

Appl. Opt. (2)

Appl. Phys. B: Photophys. Laser Chem. (2)

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[Crossref]

D. L. Jordan, R. C. Hollins, and E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B: Photophys. Laser Chem. 31, 179-186 (1983).
[Crossref]

J. Acoust. Soc. Am. (1)

L. Weng, J. M. Reid, P. Mohana Shankar, and K. Soetanto, “Ultrasound speckle analysis based on the K distribution,” J. Acoust. Soc. Am. 89, 2992-2995 (1991).
[Crossref] [PubMed]

J. Opt. (1)

J. Uozumi and T. Asakura, “The first-order statistics of partially developed non-Gaussian speckle patterns,” J. Opt. 12, 177-186 (1981).
[Crossref]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (3)

J. Phys. A (1)

M. V. Berry, “Diffractals,” J. Phys. A 12, 781-797 (1979).
[Crossref]

J. Vac. Sci. Technol. B (1)

E. Marx, J. J. Malik, Y. E. Strausser, T. Bristow, N. Poduje, and J. C. Stover, “Power spectral densities: A multiple technique study of different Si wafer surface,” J. Vac. Sci. Technol. B 20, 31-41 (2002).
[Crossref]

Opt. Commun. (8)

I. A. Popov, N. V. Sidorovsky, and L. M. Veselov, “Statistical properties of non-Gaussian intensity fluctuations in the image plane of an optical system,” Opt. Commun. 134, 289-300 (1997).
[Crossref]

J. Ohtsubo and T. Asakura, “Statistical properties of speckle intensity variations in the diffraction field under illumination of coherent light,” Opt. Commun. 14, 30-34 (1975).
[Crossref]

B. M. Levine and J. C. Dainty, “Non-Gaussian image plane speckle: Measurements from diffusers of known statistics,” Opt. Commun. 45, 252-257 (1983).
[Crossref]

I. A. Popov, N. V. Sidorovsky, and L. M. Veselov, “Experimental study of intensity probability density function in the speckle pattern formed by a small number of scatterers,” Opt. Commun. 97, 304-306 (1993).
[Crossref]

H. Ohtsubo and T. Asakura, “Measurement of surface roughness properties using speckle patterns with non-Gaussian statistics,” Opt. Commun. 25, 315-319 (1978).
[Crossref]

P. J. Chandley and H. M. Escamilla, “Speckle from a rough surface when the illuminated region contains few correlation areas: the effect of changing the surface height variance,” Opt. Commun. 29, 151-154 (1979).
[Crossref]

H. Fujii and T. Asakura, “Effect of the point spread function on the average contrast of image speckle patterns,” Opt. Commun. 21, 80-84 (1977).
[Crossref]

H. Fujii and T. Asakura, “A contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32-38 (1974).
[Crossref]

Opt. Eng. (Bellingham) (1)

E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. (Bellingham) 23, 453-461 (1984).

Opt. Express (1)

Opt. Quantum Electron. (1)

T. S. McKechnie, “Image-plane speckle in partially coherent illumination,” Opt. Quantum Electron. 8, 61-67 (1976).
[Crossref]

Phys. Rev. B (2)

G. Palasantzas, “Roughness spectrum and surface width of self-affine fractal surfaces via the K-correlation model,” Phys. Rev. B 48, 14472-14478 (1993).
[Crossref]

H. N. Yang, Y. P. Zhao, A. Chan, T. M. Lu, and G. C. Wang, “Sampling-induced hidden cycles in correlated random rough surfaces,” Phys. Rev. B 56, 4224-4232 (1997).
[Crossref]

Phys. Rev. Lett. (2)

J. Krim, I. Heyvaert, C. Van Haesendonck, and Y. Bruynseraede, “Scanning tunneling microscopy observation of self-affine fractal roughness in ion-bombarded film surfaces,” Phys. Rev. Lett. 70, 57-60 (1993).
[Crossref] [PubMed]

E. Jakeman and P. N. Pussy, “Significance of K distribution in scattering experiments,” Phys. Rev. Lett. 40, 546-550 (1978).
[Crossref]

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E. L. Church and P. Z. Takacs, “The optimal estimation of finish parameters,” Proc. SPIE 1530, 71-85 (1991).
[Crossref]

J. M. Tamkin, B. Bagwell, B. Kimbrough, G. Jabbour, and M. Descour, “High speed gray scale laser direct write technology for micro-optic fabrication,” Proc. SPIE 4984, 210-218 (2003).
[Crossref]

D. S. Goodman and A. E. Rosenbluth, “Condenser aberrations in Kohler illumination,” Proc. SPIE 922, 108-134 (1988).

Wear (1)

A. Majumdar and C. L. Tien, “Fractal characterization and simulation of rough surfaces,” Wear 136, 313-327 (1990).
[Crossref]

Other (3)

D. Kang and T. D. Milster, “Effect of optical aberration on Gaussian speckle in a partially coherent imaging system,” submitted to J. Opt. Soc. Am. A.

J. C. Dainty, A. E. Ennos, M. Francon, J. W. Goodman, T. S. McKechnie, and G. Parry, Laser Speckle and Related Phenomena (Springer-Verlag, 1984).

J. W. Goodman, Speckle Phenomena in Optics (Roberts, 2007).

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Figures (5)

Fig. 1
Fig. 1

Simple scheme for a Kohler illumination system, which is a spatially partially coherent system.

Fig. 2
Fig. 2

Comparisons between simulations with developed theories for completely coherent illumination. (a) Non-Gaussian speckle contrast from a strong rough surface of σ l = 1.5 , H = 1 , and L cor = 0.5 μ m . (b) Gaussian speckle contrast from a moderately rough surface of σ l = 0.2 , H = 0.8 , and L cor = 0.025 μ m with defocus and spherical aberration. k m is the effective width of an object space coherent PSF. k m = 0.58 μ m for (b).

Fig. 3
Fig. 3

Simple scheme of a Kohler illumination experimental setup for measuring non-Gaussian speckle contrast.

Fig. 4
Fig. 4

Ideal simulated and fabricated roughness are shown in (a) and (b), respectively. Height distribution (b) is measured by a vertical scanning interferometer (Vecco NT9800). (c) and (d) show autocorrelation functions and PSDs from the simulated and fabricated roughness, respectively. Autocorrelation and PSD are 1D profiles of 2D measurements. Autocorrelation and PSD of the fabricated roughness are in relatively good agreement with the ideal values.

Fig. 5
Fig. 5

Simulated and experimental speckle contrasts are shown for (a) 25 mm and (b) 10 mm pupils. In (a), there is a large discrepancy between simulation and experiment. It is also shown that speckle contrasts from simulation are affected by defocus ( W 020 ) and spherical ( W 040 ) aberrations dramatically. Note that ratios of C s and σ c axes are the same for both (a) and (b).

Equations (8)

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

C s = σ I I ,
g s ( x i ) = 1 | m T | f s ( x o m T ) h ( x i x o ) d x o ,
f s ( x o ) = exp { i 2 π [ φ s ( x o ) + l ( x o ) q s ] } ,
I s ( x i ) = s = 1 N | g s ( x i ) | 2 ,
K u ( Δ x o ) = σ l 2 exp [ ( Δ x o L cor ) 2 H ]
k m = 2 λ π N A o ,
φ ( x o ) = 2 π ( n p 1 ) l ( x o ) ,
σ c = d s d p ,

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