Abstract

If a rough surface is illuminated by a coherent lightwave of wavelength λ1, it is not possible to determine the surface profile from the phases of the speckle field formed by the scattered light. If the rough surface is illuminated, however, by an additional coherent wave of wavelength λ1, the phase differences between the two speckle fields do contain information about the macroscopic surface profile even if subject to a statistical error. It is shown that (1) the macroscopic surface profile may be determined from the phase differences if the effective wavelength Λ = λ1λ2/|λ1−λ2| is sufficiently larger than the standard deviation of the microscopic profile of the illuminated surface, and (2) the statistical error is reasonably small if the phase measurements are obtained from speckles of sufficient intensity. Using a heterodyne interferometer we demonstrate the feasibility of this technique. In the first experiment we determine the radius of curvature of a rough spherical surface. In the second experiment the macroscopic surface contour on two ophthalmic lenses of the same power variation, one with a grounded surface and the other with a polished surface, was determined.

© 1985 Optical Society of America

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References

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  1. H. Fujii, T. Asakura, “Effect of Surface Roughness on the Statistical Distribution of Image Speckle Intensity,” Opt. Commun. 11, 35 (1974);J. W. Goodman, “Dependence of Image Speckle Contrast on Surface Roughness,” Opt. Commun. 14, 324 (1975).
    [CrossRef]
  2. J. M. Burch, “Dependence of Image Speckle Contrast on Surface Roughness,” in Optical Instruments and Techniques, J. H. Dickson, Ed. (Oriel Press, Newcastle-upon-Tyne, 1970), p. 213.
  3. C. R. Munnerlyn, M. Latta, “Rough Surface Interferometry Using a CO2 Laser Source,” Appl. Opt. 7, 1858 (1968).
    [CrossRef] [PubMed]
  4. O. Kwon, J. C. Wyant, C. R. Hayslett, “Rough Surface Interferometry at 10.6 μm,” Appl. Opt. 19, 1862 (1980).
    [CrossRef] [PubMed]
  5. M. V. R. K. Murty, R. P. Shukla, “An Oblique Incidence Interferometer,” Opt. Eng. 15, 461 (1976).
    [CrossRef]
  6. K. G. Birch, “Oblique Incidence Interferometry Applied to Non-Optical Surfaces,” J. Phys. E 6, 1045 (1973).
    [CrossRef]
  7. G. Schulz, J. Schwider, “Interferometric Testing of Smooth Surfaces,” Prog. Opt. 13, 93 (1976).
    [CrossRef]
  8. C. Polhemus, “Two-Wavelength Interferometry,” Appl. Opt. 12, 2071 (1973).
    [CrossRef] [PubMed]
  9. J. C. Wyant, “Testing Aspherics Using Two-Wavelength Holography,” Appl. Opt. 10, 2113 (1971).
    [CrossRef] [PubMed]
  10. W. Schmidt, A. F. Fercher, “Holographic Generation of Depth Contours Using a Flash-Lamp-Pumped Dye Laser,” Opt. Commun. 3, 363 (1971).
    [CrossRef]
  11. F. M. Kuchel, H. J. Tiziani, “Real-Time Contour Holography Using BSO Crystals,” Opt. Commun. 38, 17 (1981).
    [CrossRef]
  12. J. S. Zelenka, J. R. Varner, “A New Method for Generating Depth Contours Holographically,” Appl. Opt. 7, 2107 (1968).
    [CrossRef] [PubMed]
  13. A. F. Fercher, H. Z. Hu, “Two-Wavelength Heterodyne Interferometry,” in Optoelectronics in Engineering, W. Waidelich, Ed. (Springer, Berlin, 1984), p. 142.
  14. R. E. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U.P., London, 1983).
  15. H. Z. Hu, “Polarization Heterodyne Interferometry Using a Simple Rotating Analyzer. 1: Theory and Error Analysis,” Appl. Opt. 22, 2052 (1983).
    [CrossRef] [PubMed]
  16. E. W. Marchand, E. Wolf, “Radiometry with Sources of Any State of Coherence,” J. Opt. Soc. Am. 64, 1219 (1974).
    [CrossRef]
  17. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 328.
  18. S. Donati, G. Martini, “Speckle-Pattern Intensity and Phase: Second-order Conditional Statistics,” J. Opt. Soc. Am. 69, 1690 (1979).
    [CrossRef]
  19. J. W. Goodman, “Statiscal Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975), p. 9.
    [CrossRef]
  20. J. W. Goodman, Stanford Electronic Laboratories TR 2303-1, SEL-63-140 (1963).
  21. N. George, A. Jain, “Space and Wavelength Dependence of Speckle Intensity,” Appl. Phys. 4, 201 (1974).
    [CrossRef]
  22. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 59.
  23. I. S. Reed, “On a Moment Theorem for Complex Gaussian Processes,”\ IRE Trans. Inf. Theory IT-8, 194 (1962).
    [CrossRef]
  24. M. Giglio, S. Musazzi, U. Perini, “Surface Roughness Measurements by Means of Speckle Wavelength Decorrelation,” Opt. Commun. 28, 166 (1979).
    [CrossRef]

1983

1981

F. M. Kuchel, H. J. Tiziani, “Real-Time Contour Holography Using BSO Crystals,” Opt. Commun. 38, 17 (1981).
[CrossRef]

1980

1979

S. Donati, G. Martini, “Speckle-Pattern Intensity and Phase: Second-order Conditional Statistics,” J. Opt. Soc. Am. 69, 1690 (1979).
[CrossRef]

M. Giglio, S. Musazzi, U. Perini, “Surface Roughness Measurements by Means of Speckle Wavelength Decorrelation,” Opt. Commun. 28, 166 (1979).
[CrossRef]

1976

M. V. R. K. Murty, R. P. Shukla, “An Oblique Incidence Interferometer,” Opt. Eng. 15, 461 (1976).
[CrossRef]

G. Schulz, J. Schwider, “Interferometric Testing of Smooth Surfaces,” Prog. Opt. 13, 93 (1976).
[CrossRef]

1974

H. Fujii, T. Asakura, “Effect of Surface Roughness on the Statistical Distribution of Image Speckle Intensity,” Opt. Commun. 11, 35 (1974);J. W. Goodman, “Dependence of Image Speckle Contrast on Surface Roughness,” Opt. Commun. 14, 324 (1975).
[CrossRef]

N. George, A. Jain, “Space and Wavelength Dependence of Speckle Intensity,” Appl. Phys. 4, 201 (1974).
[CrossRef]

E. W. Marchand, E. Wolf, “Radiometry with Sources of Any State of Coherence,” J. Opt. Soc. Am. 64, 1219 (1974).
[CrossRef]

1973

K. G. Birch, “Oblique Incidence Interferometry Applied to Non-Optical Surfaces,” J. Phys. E 6, 1045 (1973).
[CrossRef]

C. Polhemus, “Two-Wavelength Interferometry,” Appl. Opt. 12, 2071 (1973).
[CrossRef] [PubMed]

1971

J. C. Wyant, “Testing Aspherics Using Two-Wavelength Holography,” Appl. Opt. 10, 2113 (1971).
[CrossRef] [PubMed]

W. Schmidt, A. F. Fercher, “Holographic Generation of Depth Contours Using a Flash-Lamp-Pumped Dye Laser,” Opt. Commun. 3, 363 (1971).
[CrossRef]

1968

1962

I. S. Reed, “On a Moment Theorem for Complex Gaussian Processes,”\ IRE Trans. Inf. Theory IT-8, 194 (1962).
[CrossRef]

Asakura, T.

H. Fujii, T. Asakura, “Effect of Surface Roughness on the Statistical Distribution of Image Speckle Intensity,” Opt. Commun. 11, 35 (1974);J. W. Goodman, “Dependence of Image Speckle Contrast on Surface Roughness,” Opt. Commun. 14, 324 (1975).
[CrossRef]

Birch, K. G.

K. G. Birch, “Oblique Incidence Interferometry Applied to Non-Optical Surfaces,” J. Phys. E 6, 1045 (1973).
[CrossRef]

Burch, J. M.

J. M. Burch, “Dependence of Image Speckle Contrast on Surface Roughness,” in Optical Instruments and Techniques, J. H. Dickson, Ed. (Oriel Press, Newcastle-upon-Tyne, 1970), p. 213.

Donati, S.

Fercher, A. F.

W. Schmidt, A. F. Fercher, “Holographic Generation of Depth Contours Using a Flash-Lamp-Pumped Dye Laser,” Opt. Commun. 3, 363 (1971).
[CrossRef]

A. F. Fercher, H. Z. Hu, “Two-Wavelength Heterodyne Interferometry,” in Optoelectronics in Engineering, W. Waidelich, Ed. (Springer, Berlin, 1984), p. 142.

Fujii, H.

H. Fujii, T. Asakura, “Effect of Surface Roughness on the Statistical Distribution of Image Speckle Intensity,” Opt. Commun. 11, 35 (1974);J. W. Goodman, “Dependence of Image Speckle Contrast on Surface Roughness,” Opt. Commun. 14, 324 (1975).
[CrossRef]

George, N.

N. George, A. Jain, “Space and Wavelength Dependence of Speckle Intensity,” Appl. Phys. 4, 201 (1974).
[CrossRef]

Giglio, M.

M. Giglio, S. Musazzi, U. Perini, “Surface Roughness Measurements by Means of Speckle Wavelength Decorrelation,” Opt. Commun. 28, 166 (1979).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 59.

J. W. Goodman, “Statiscal Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975), p. 9.
[CrossRef]

J. W. Goodman, Stanford Electronic Laboratories TR 2303-1, SEL-63-140 (1963).

Hayslett, C. R.

Hu, H. Z.

H. Z. Hu, “Polarization Heterodyne Interferometry Using a Simple Rotating Analyzer. 1: Theory and Error Analysis,” Appl. Opt. 22, 2052 (1983).
[CrossRef] [PubMed]

A. F. Fercher, H. Z. Hu, “Two-Wavelength Heterodyne Interferometry,” in Optoelectronics in Engineering, W. Waidelich, Ed. (Springer, Berlin, 1984), p. 142.

Jain, A.

N. George, A. Jain, “Space and Wavelength Dependence of Speckle Intensity,” Appl. Phys. 4, 201 (1974).
[CrossRef]

Jones, R. E.

R. E. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U.P., London, 1983).

Kuchel, F. M.

F. M. Kuchel, H. J. Tiziani, “Real-Time Contour Holography Using BSO Crystals,” Opt. Commun. 38, 17 (1981).
[CrossRef]

Kwon, O.

Latta, M.

Marchand, E. W.

Martini, G.

Munnerlyn, C. R.

Murty, M. V. R. K.

M. V. R. K. Murty, R. P. Shukla, “An Oblique Incidence Interferometer,” Opt. Eng. 15, 461 (1976).
[CrossRef]

Musazzi, S.

M. Giglio, S. Musazzi, U. Perini, “Surface Roughness Measurements by Means of Speckle Wavelength Decorrelation,” Opt. Commun. 28, 166 (1979).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 328.

Perini, U.

M. Giglio, S. Musazzi, U. Perini, “Surface Roughness Measurements by Means of Speckle Wavelength Decorrelation,” Opt. Commun. 28, 166 (1979).
[CrossRef]

Polhemus, C.

Reed, I. S.

I. S. Reed, “On a Moment Theorem for Complex Gaussian Processes,”\ IRE Trans. Inf. Theory IT-8, 194 (1962).
[CrossRef]

Schmidt, W.

W. Schmidt, A. F. Fercher, “Holographic Generation of Depth Contours Using a Flash-Lamp-Pumped Dye Laser,” Opt. Commun. 3, 363 (1971).
[CrossRef]

Schulz, G.

G. Schulz, J. Schwider, “Interferometric Testing of Smooth Surfaces,” Prog. Opt. 13, 93 (1976).
[CrossRef]

Schwider, J.

G. Schulz, J. Schwider, “Interferometric Testing of Smooth Surfaces,” Prog. Opt. 13, 93 (1976).
[CrossRef]

Shukla, R. P.

M. V. R. K. Murty, R. P. Shukla, “An Oblique Incidence Interferometer,” Opt. Eng. 15, 461 (1976).
[CrossRef]

Tiziani, H. J.

F. M. Kuchel, H. J. Tiziani, “Real-Time Contour Holography Using BSO Crystals,” Opt. Commun. 38, 17 (1981).
[CrossRef]

Varner, J. R.

Wolf, E.

Wyant, J. C.

Wykes, C.

R. E. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U.P., London, 1983).

Zelenka, J. S.

Appl. Opt.

Appl. Phys.

N. George, A. Jain, “Space and Wavelength Dependence of Speckle Intensity,” Appl. Phys. 4, 201 (1974).
[CrossRef]

IRE Trans. Inf. Theory

I. S. Reed, “On a Moment Theorem for Complex Gaussian Processes,”\ IRE Trans. Inf. Theory IT-8, 194 (1962).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. E

K. G. Birch, “Oblique Incidence Interferometry Applied to Non-Optical Surfaces,” J. Phys. E 6, 1045 (1973).
[CrossRef]

Opt. Commun.

W. Schmidt, A. F. Fercher, “Holographic Generation of Depth Contours Using a Flash-Lamp-Pumped Dye Laser,” Opt. Commun. 3, 363 (1971).
[CrossRef]

F. M. Kuchel, H. J. Tiziani, “Real-Time Contour Holography Using BSO Crystals,” Opt. Commun. 38, 17 (1981).
[CrossRef]

H. Fujii, T. Asakura, “Effect of Surface Roughness on the Statistical Distribution of Image Speckle Intensity,” Opt. Commun. 11, 35 (1974);J. W. Goodman, “Dependence of Image Speckle Contrast on Surface Roughness,” Opt. Commun. 14, 324 (1975).
[CrossRef]

M. Giglio, S. Musazzi, U. Perini, “Surface Roughness Measurements by Means of Speckle Wavelength Decorrelation,” Opt. Commun. 28, 166 (1979).
[CrossRef]

Opt. Eng.

M. V. R. K. Murty, R. P. Shukla, “An Oblique Incidence Interferometer,” Opt. Eng. 15, 461 (1976).
[CrossRef]

Prog. Opt.

G. Schulz, J. Schwider, “Interferometric Testing of Smooth Surfaces,” Prog. Opt. 13, 93 (1976).
[CrossRef]

Other

A. F. Fercher, H. Z. Hu, “Two-Wavelength Heterodyne Interferometry,” in Optoelectronics in Engineering, W. Waidelich, Ed. (Springer, Berlin, 1984), p. 142.

R. E. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U.P., London, 1983).

J. W. Goodman, “Statiscal Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Berlin, 1975), p. 9.
[CrossRef]

J. W. Goodman, Stanford Electronic Laboratories TR 2303-1, SEL-63-140 (1963).

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965), p. 328.

J. M. Burch, “Dependence of Image Speckle Contrast on Surface Roughness,” in Optical Instruments and Techniques, J. H. Dickson, Ed. (Oriel Press, Newcastle-upon-Tyne, 1970), p. 213.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 59.

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Figures (7)

Fig. 1
Fig. 1

Two-wavelength heterodyne interferometer. R1,R2 are the reference detectors and S1,S2 are the scanning detectors for λ12, respectively.

Fig. 2
Fig. 2

Two-wavelength heterodyne interferometer adapted for rough surface interferometry.

Fig. 3
Fig. 3

Phase difference measurement results obtained from a grounded lens surface (σh = 5 μm, 〈I1〉 = 〈I2〉 = 〈I〉) and a smooth surface; Λ = 36.5 μm. Each point represents one measurement: (a) grounded surface, intensity threshold = 0.1 〈I〉; (b) grounded surface, intensity threshold = 〈I〉 (75% of the data points had to be rejected in this case); (c) smooth surface.

Fig. 4
Fig. 4

Random walk of the amplitudes A1 and A2 of the scattered light of wavelengths λ1 and λ2, respectively, in the complex plane.

Fig. 5
Fig. 5

Geometry of the speckle field. The z axis coincides with the optical axis of the interferometer. Points P of the speckle field are defined by position vector r. Points at the object surface are defined by vector ρ. z is the distance from the focus at the object surface to the image of the interferometer exit pinhole given by the focusing lens.

Fig. 6
Fig. 6

Surface contours along the vertical meridian at the center of two ophthalmic lenses with a power variation of 1 diopter. D is the distance from the center of table rotation: (a) grounded surface (σh = 5 μm); each point represents one measurement; (b) polished surface.

Fig. 7
Fig. 7

Local radius of curvature of the surface of an ophthalmic lens with a power variation of 1 diopter measured along the central meridian and averaged over 1 cm: a, grounded surface (σh = 5 μm); b, polished surface. The results with the error bars were obtained from b with the help of an interference microscope.

Equations (20)

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Λ = λ 1 λ 2 / | λ 1 λ 2 | .
I ( x , y , t ) = a 2 + b 2 2 + a b sin [ 2 ω R t α ( x , y ) ] ,
I l ( x , y , t ) = a l 2 + b l 2 2 + a l b l sin [ 2 ω R t α l ( x , y ) ] ; l = 1 , 2 .
α 1 ( x , y ) α 2 ( x , y ) = 2 π Λ · Δ l ( x , y ) ,
z ( x , y ) = z 0 ( x , y ) + h ( x , y ) .
A l = x 1 x 2 exp { i 2 k l [ z 0 + h ( x ) ] } d x ; l = 1 , 2 ,
arg A 2 arg A 1 = 2 K z 0 + arg { x 1 x 2 exp [ i 2 k 2 h ( x ) ] d x } arg { x 1 x 2 exp [ i 2 k 1 h ( x ) ] d x } ,
1 x 2 x 1 x 1 x 2 exp [ i 2 k l h ( x ) ] d x exp [ i 2 k l h ( x ) ] = exp [ 2 k l 2 h 2 ] = exp [ 2 k l 2 σ h 2 ] ; l = 1 , 2 ,
A l ( r ) = j a j l · exp ( i ϕ j l ) ; l = 1 , 2
C I ( r P , r Q ) = I 1 ( r P ) I 2 ( r Q ) I 1 ( r P ) I 2 ( r Q ) ,
A l ( r P ) = exp ( i k l z ) i λ z · A l ( ρ ) exp [ i k l 2 z ( ρ t P ) 2 ] d 2 ρ ; l = 1 , 2 ,
C I ( P P , r Q ; λ 1 , λ 2 ) = I 1 ( r P ) I 2 ( r Q ) I 1 ( r P ) I 2 ( r Q ) = | A 1 ( r P ) A 2 * ( r Q ) | 2 ,
A 1 ( r P ) A 2 * ( r Q ) = exp [ i ( k 1 k 2 ) z ] λ 1 λ 2 z · A 1 ( ρ 1 ) A 2 * ( ρ 2 ) · exp { i π z · ( ρ 1 t P ) 2 λ 1 i π z · ( ρ 2 t Q ) 2 λ 2 ] · d 2 ρ 1 d 2 ρ 2 .
A 1 ( ρ 1 ) A 2 * ( ρ 2 ) = U 1 U 2 * exp { i 4 π [ h ( ρ 1 ) λ 1 h ( ρ 2 ) λ 2 ] } δ ( ρ 1 ρ 2 ) = U 1 U 2 * exp [ i 4 π h ( ρ 1 ) Λ ] δ ( ρ 1 ρ 2 ) .
A 1 ( ρ 1 ) A 2 * ( ρ 2 ) = U 1 U 2 * exp [ 8 π 2 σ h 2 λ 2 ] δ ( ρ 1 ρ 2 ) .
C A ( r P , r Q ; λ 1 , λ 2 ) = exp [ i ( k 1 k 2 ) z ] λ 1 λ 2 z · exp ( 8 π 2 σ h 2 Λ 2 ) · U 1 U 2 * exp [ i π 2 ( ρ t P ) 2 λ 1 i π 2 ( ρ t Q ) 2 λ 2 ] d 2 ρ .
C A ( λ 1 , λ 2 ) = exp [ i ( k 1 k 2 ) z ] λ 1 λ 2 z · exp ( 8 π 2 σ h 2 Λ 2 ) · U 1 U 2 * exp ( i π ρ 2 z Λ ) d 2 ρ .
c I = C I ( λ 1 , λ 2 ) I 1 I 2 1 ,
c I = exp ( 16 π 2 σ h 2 Λ 2 ) .
Λ 9 σ h .

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