Abstract

Possibilities of optical noncontact diagnostics of random phase objects are studied, based on measurements of transverse coherence function, scintillation index, and amplitude and phase dispersion of the field resulting from interaction with the object. The advantages of these methods are increased speed and accuracy compared with commonly used methods. Interference measurements of second- and higher-order correlation parameters of the field phase is demonstrated which can be used to find the corresponding probability density distribution function for objects with phase statistics differing from Gaussian. The sensitivity threshold of the methods is estimated to be ~0.005 μm when measuring surfaces with slight roughness.

© 1990 Optical Society of America

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References

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  1. S. M. Rytov, Yu. A. Kravtsov, B. I. Tatarsky, Introduction to Statistical Radiophysics. Part II. Random Fields (Izd. Nauka, Moscow, 1978), p. 464, in Russian.
  2. H. P. Baltes, Inverse Source Problems in Optics (Springer-Verlag, Berlin, 1978).
    [CrossRef]
  3. V. N. Moiseyev, V. I. Mandrosov, “Informativeness of Speckled Coherent Images,” Zarubezhnaya Radioelektronika 2, 3–21 (1982).
  4. H. Fujii, F. Asakura, Y. Shindo, “Measurement of Surface Roughness Properties by Means of Laser Speckle Techniques,” Opt. Commun. 16, 68–72 (1976).
    [CrossRef]
  5. H. Fujii, J. Uozumi, T. Asakura, “Computer Simulation Study of Image Speckle Patterns with Relation to Object Surface Profile,” J. Opt. Soc. Am. 66, 1222–1236 (1976).
    [CrossRef]
  6. J. W. Goodman, “Dependence of Image Speckle Contrast of Surface Roughness,” Opt. Commun. 14, 324–327 (1975).
    [CrossRef]
  7. O. V. Angelsky, V. G. Zhitaryuk, P. P. Maksimyak, “On the Possibility of Optical Correlation Measurement of Inhomogeneous Phase Statistical Surfaces,” Opt. Spektrosk. 60, 1013–1017 (1986).
  8. S. A. Akhmanov, Yu. Ye. Dyakov, A. S. Chirkin, Introduction to Statistical Radiophysics and Optics (Izd. Nauka, Moscow, 1981), p. 640, in Russian.
  9. O. V. Angelsky, P. P. Maksimyak, T. O. Perun, “Optical Correlation Systems for Studying Weak Surface Roughness,” in Fringe ’89 Automatic Processing of Fringe Patterns (Akademie-Verlag, Berlin, 1989), p. 53.
  10. O. V. Angelsky, I. I. Magun, P. P. Maksimyak, “Optical Correlation Methods in the Studies of Inhomogeneous Phase Samples,” in Proceedings, Third International Symposium on Modern Optics, Budapest (1988), p. 337.
  11. M. G. Kendall, The Advanced Theory of Statistics. Vol. 1. Distribution Theory, A.Stuart Stuart, (Griffin Co. Ltd., London1960).

1986 (1)

O. V. Angelsky, V. G. Zhitaryuk, P. P. Maksimyak, “On the Possibility of Optical Correlation Measurement of Inhomogeneous Phase Statistical Surfaces,” Opt. Spektrosk. 60, 1013–1017 (1986).

1982 (1)

V. N. Moiseyev, V. I. Mandrosov, “Informativeness of Speckled Coherent Images,” Zarubezhnaya Radioelektronika 2, 3–21 (1982).

1976 (2)

H. Fujii, F. Asakura, Y. Shindo, “Measurement of Surface Roughness Properties by Means of Laser Speckle Techniques,” Opt. Commun. 16, 68–72 (1976).
[CrossRef]

H. Fujii, J. Uozumi, T. Asakura, “Computer Simulation Study of Image Speckle Patterns with Relation to Object Surface Profile,” J. Opt. Soc. Am. 66, 1222–1236 (1976).
[CrossRef]

1975 (1)

J. W. Goodman, “Dependence of Image Speckle Contrast of Surface Roughness,” Opt. Commun. 14, 324–327 (1975).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, Yu. Ye. Dyakov, A. S. Chirkin, Introduction to Statistical Radiophysics and Optics (Izd. Nauka, Moscow, 1981), p. 640, in Russian.

Angelsky, O. V.

O. V. Angelsky, V. G. Zhitaryuk, P. P. Maksimyak, “On the Possibility of Optical Correlation Measurement of Inhomogeneous Phase Statistical Surfaces,” Opt. Spektrosk. 60, 1013–1017 (1986).

O. V. Angelsky, P. P. Maksimyak, T. O. Perun, “Optical Correlation Systems for Studying Weak Surface Roughness,” in Fringe ’89 Automatic Processing of Fringe Patterns (Akademie-Verlag, Berlin, 1989), p. 53.

O. V. Angelsky, I. I. Magun, P. P. Maksimyak, “Optical Correlation Methods in the Studies of Inhomogeneous Phase Samples,” in Proceedings, Third International Symposium on Modern Optics, Budapest (1988), p. 337.

Asakura, F.

H. Fujii, F. Asakura, Y. Shindo, “Measurement of Surface Roughness Properties by Means of Laser Speckle Techniques,” Opt. Commun. 16, 68–72 (1976).
[CrossRef]

Asakura, T.

Baltes, H. P.

H. P. Baltes, Inverse Source Problems in Optics (Springer-Verlag, Berlin, 1978).
[CrossRef]

Chirkin, A. S.

S. A. Akhmanov, Yu. Ye. Dyakov, A. S. Chirkin, Introduction to Statistical Radiophysics and Optics (Izd. Nauka, Moscow, 1981), p. 640, in Russian.

Dyakov, Yu. Ye.

S. A. Akhmanov, Yu. Ye. Dyakov, A. S. Chirkin, Introduction to Statistical Radiophysics and Optics (Izd. Nauka, Moscow, 1981), p. 640, in Russian.

Fujii, H.

H. Fujii, J. Uozumi, T. Asakura, “Computer Simulation Study of Image Speckle Patterns with Relation to Object Surface Profile,” J. Opt. Soc. Am. 66, 1222–1236 (1976).
[CrossRef]

H. Fujii, F. Asakura, Y. Shindo, “Measurement of Surface Roughness Properties by Means of Laser Speckle Techniques,” Opt. Commun. 16, 68–72 (1976).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Dependence of Image Speckle Contrast of Surface Roughness,” Opt. Commun. 14, 324–327 (1975).
[CrossRef]

Kendall, M. G.

M. G. Kendall, The Advanced Theory of Statistics. Vol. 1. Distribution Theory, A.Stuart Stuart, (Griffin Co. Ltd., London1960).

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov, B. I. Tatarsky, Introduction to Statistical Radiophysics. Part II. Random Fields (Izd. Nauka, Moscow, 1978), p. 464, in Russian.

Magun, I. I.

O. V. Angelsky, I. I. Magun, P. P. Maksimyak, “Optical Correlation Methods in the Studies of Inhomogeneous Phase Samples,” in Proceedings, Third International Symposium on Modern Optics, Budapest (1988), p. 337.

Maksimyak, P. P.

O. V. Angelsky, V. G. Zhitaryuk, P. P. Maksimyak, “On the Possibility of Optical Correlation Measurement of Inhomogeneous Phase Statistical Surfaces,” Opt. Spektrosk. 60, 1013–1017 (1986).

O. V. Angelsky, I. I. Magun, P. P. Maksimyak, “Optical Correlation Methods in the Studies of Inhomogeneous Phase Samples,” in Proceedings, Third International Symposium on Modern Optics, Budapest (1988), p. 337.

O. V. Angelsky, P. P. Maksimyak, T. O. Perun, “Optical Correlation Systems for Studying Weak Surface Roughness,” in Fringe ’89 Automatic Processing of Fringe Patterns (Akademie-Verlag, Berlin, 1989), p. 53.

Mandrosov, V. I.

V. N. Moiseyev, V. I. Mandrosov, “Informativeness of Speckled Coherent Images,” Zarubezhnaya Radioelektronika 2, 3–21 (1982).

Moiseyev, V. N.

V. N. Moiseyev, V. I. Mandrosov, “Informativeness of Speckled Coherent Images,” Zarubezhnaya Radioelektronika 2, 3–21 (1982).

Perun, T. O.

O. V. Angelsky, P. P. Maksimyak, T. O. Perun, “Optical Correlation Systems for Studying Weak Surface Roughness,” in Fringe ’89 Automatic Processing of Fringe Patterns (Akademie-Verlag, Berlin, 1989), p. 53.

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, B. I. Tatarsky, Introduction to Statistical Radiophysics. Part II. Random Fields (Izd. Nauka, Moscow, 1978), p. 464, in Russian.

Shindo, Y.

H. Fujii, F. Asakura, Y. Shindo, “Measurement of Surface Roughness Properties by Means of Laser Speckle Techniques,” Opt. Commun. 16, 68–72 (1976).
[CrossRef]

Tatarsky, B. I.

S. M. Rytov, Yu. A. Kravtsov, B. I. Tatarsky, Introduction to Statistical Radiophysics. Part II. Random Fields (Izd. Nauka, Moscow, 1978), p. 464, in Russian.

Uozumi, J.

Zhitaryuk, V. G.

O. V. Angelsky, V. G. Zhitaryuk, P. P. Maksimyak, “On the Possibility of Optical Correlation Measurement of Inhomogeneous Phase Statistical Surfaces,” Opt. Spektrosk. 60, 1013–1017 (1986).

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

H. Fujii, F. Asakura, Y. Shindo, “Measurement of Surface Roughness Properties by Means of Laser Speckle Techniques,” Opt. Commun. 16, 68–72 (1976).
[CrossRef]

J. W. Goodman, “Dependence of Image Speckle Contrast of Surface Roughness,” Opt. Commun. 14, 324–327 (1975).
[CrossRef]

Opt. Spektrosk. (1)

O. V. Angelsky, V. G. Zhitaryuk, P. P. Maksimyak, “On the Possibility of Optical Correlation Measurement of Inhomogeneous Phase Statistical Surfaces,” Opt. Spektrosk. 60, 1013–1017 (1986).

Zarubezhnaya Radioelektronika (1)

V. N. Moiseyev, V. I. Mandrosov, “Informativeness of Speckled Coherent Images,” Zarubezhnaya Radioelektronika 2, 3–21 (1982).

Other (6)

S. M. Rytov, Yu. A. Kravtsov, B. I. Tatarsky, Introduction to Statistical Radiophysics. Part II. Random Fields (Izd. Nauka, Moscow, 1978), p. 464, in Russian.

H. P. Baltes, Inverse Source Problems in Optics (Springer-Verlag, Berlin, 1978).
[CrossRef]

S. A. Akhmanov, Yu. Ye. Dyakov, A. S. Chirkin, Introduction to Statistical Radiophysics and Optics (Izd. Nauka, Moscow, 1981), p. 640, in Russian.

O. V. Angelsky, P. P. Maksimyak, T. O. Perun, “Optical Correlation Systems for Studying Weak Surface Roughness,” in Fringe ’89 Automatic Processing of Fringe Patterns (Akademie-Verlag, Berlin, 1989), p. 53.

O. V. Angelsky, I. I. Magun, P. P. Maksimyak, “Optical Correlation Methods in the Studies of Inhomogeneous Phase Samples,” in Proceedings, Third International Symposium on Modern Optics, Budapest (1988), p. 337.

M. G. Kendall, The Advanced Theory of Statistics. Vol. 1. Distribution Theory, A.Stuart Stuart, (Griffin Co. Ltd., London1960).

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Figures (5)

Fig. 1
Fig. 1

Schematic view of the setup for measuring the scintillation index: 1, sample; 2, beam splitter cube; 3, micron-size diaphragm; 4, photomultiplier.

Fig. 2
Fig. 2

Schematic view of the setup for measuring the transverse coherence function: 1, laser; 2, telescopic system; 3,6, beam splitters; 4,5, mirrors; 7, sample; 8, Mach-Zehnder interferometer; 9, objective lens; 10,11, aperture and field diaphragms; 12, photodetector.

Fig. 3
Fig. 3

Schematic view of the setup for measuring the correlation moments: 1, laser; 2, telescopic system; 3, interferometer; 4,5,6, polarizers controlling the intensity of the beams; 7, sample; 8, objective lens; 9,10, aperture and field diaphragms; 11, photodetector.

Fig. 4
Fig. 4

Dependence of 〈 J s (x,y)〉/ J 0 (curve 1), σ φ 2 (curve 2), φ A 2 (curve 3), and A ˜ ( x , y ) A ¯ ( x , y ) φ ˜ 2 ( x , y ) (curve 4) on recording zone D = (2Z)/(klφ0) for a germanium single crystal sample.

Fig. 5
Fig. 5

Distribution functions F of surface irregularities from (2) profilometric and (3) optical measurements. Gaussian distribution (1) is also shown.

Tables (1)

Tables Icon

Table I Surface Inhomogeneity Phase Dispersion for Some Fused Quartz Samples Obtained from Profilometric ( σ φ 0 h 2), Transverse Coherence Function ( σ φ 0 k 2), Scintillation Index ( σ φ 0 β 2), and Normalized Amplitude Variance( σ φ 0 A 2) Measurements

Equations (23)

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

β 2 = 2 σ φ 0 2 ,
β 2 = σ J 2 J ¯ 2 ,
Γ ( ρ ) = exp { σ φ 0 2 [ k φ 0 ( ρ ) - 1 ] } ,
A ( x , y ) 2 ¯ = A ( x , y ) 2 = A o 2 = J 0 ,
σ A 2 = A 2 ( x , y ) - A ( x , y ) 2 A ( x , y ) 2 .
Z 1 = k a l φ 0 ,
Z m = k l φ 0 2 / σ φ 0 .
β 2 = 2 σ φ 0 2 ,             σ A 2 = ½ σ φ 0 2
Z > Z m = k l φ 0 2 / σ φ 0 .
O Z Z 1
Z Z < Z 1
J s ( x , y ) = A o 2 + A 2 ( x , y ) + 2 A o A ( x , y ) cos φ ( x , y ) ,
J s ( x , y ) J 0 = A 2 ( x , y ) - A ( x , y ) 2 A ( x , y ) 2 + A ˜ ( x , y ) A ¯ ( x , y ) φ ˜ 2 ( x , y ) + φ ˜ 2 ( x , y ) ,
σ A 2 = A 2 ( x , y ) - A ( x , y ) 2 A ( x , y ) 2
A ˜ ( x , y ) A ¯ ( x , y ) φ ˜ 2 ( x , y )
σ A 2 = A ˜ ( x , y ) A ¯ ( x , y ) φ ˜ 2 ( x , y ) = 0.
J s ( x , y ) J 0 = σ φ 0 2 .
A ˜ ( x , y ) A ¯ ( x , y ) φ ˜ 2 ( x , y )
k a = [ φ ( x , y ) - φ ¯ ( x , y ) ] 3 σ φ 3 = [ J s ( x , y ) ] 3 / 2 J 0 3 / 2 ; k k = [ φ ( x , y ) - φ ( x , y ) ] 4 σ φ 4 = J s 2 ( x , y ) J 0 2 - 3.
F ( h ) = 1 2 π - + θ ( u ) exp ( - i u h ) d u .
A ˜ ( x , y ) A ¯ ( x , y ) φ ˜ 2 ( x , y )
A ˜ ( x , y ) A ¯ ( x , y ) φ ˜ 2 ( x , y )
A ˜ ( x , y ) A ¯ ( x , y ) φ ˜ 2 ( x , y )

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