Abstract

Small angle scattering in and near the specular direction has recently been measured with a monostatic laser interferometric reflectometer for a laser beam incident normal to the surface. Employing the Ricean statistics of the amplitude distribution for coherent glint with an additive scattered component of speckles, the scattered portion has been partitioned from the total reflected signal at 1.06 and 10.6 μm. Such measurements are basic to surface metrology and useful in assessing the performance of mirrors used in lasers and astronomical telescopes.

© 1990 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. A. Temple, D. P. Arndt, J. M. Bennett, J. M. Elson, P. C. Archibald, S. J. Hixson, “Near-Angle Scatter and Total Integrated Scatter from Single-Point Diamond-Turned Copper Surfaces Measured at 2.7 μm Wavelength,” in Technical Digest, Topical Meeting on High Power Laser Optical Components (Optical Society of America, Washington, DC, 1984).
  2. J. B. Breckinridge, T. G. Kuper, R. V. Shack, “Space Telescope Low Scattered Camera—A Model,” Proc. Soc. Photo-Opt. Instrum. Eng. 331, 395–403 (1982).
  3. S. J. Wein, W. L. Wolfe, “Gaussian-Apodized Apertures and Small-Angle Scatter Measurement,” Opt. Eng. 28, 273–280 (1989).
    [Crossref]
  4. W. W. Lee, L. M. Sherr, M. K. Barsh, “Stray Light Analysis and Suppression in Small Angle BRDF/BTDF Measurement,” Proc. Soc. Photo-Opt. Instrum. Eng. 675, 32–00 (1988).
  5. C. L. Vernold, “Application and Verification of Wavelength Scaling for Near-Specular Scatter Predictions,” Proc. Soc. Photo-Opt. Instrum. Eng. 1165, 18–00 (1989).
  6. Z. H. Gu, R. S. Dummer, A. A. Maradudin, A. R. McGurn, “Experimental Investigation of the Opposition Effect in the Scattering of Light from a Random Rough Metal Surface,” Appl. Opt. 28, 537–543 (1989).
    [Crossref] [PubMed]
  7. Z. H. Gu, R. S. Dummer, A. A. Maradudin, A. R. McGurn, “Enhanced Backscattering of Light from a Randomly Rough Metal Surface,” submitted to Radio Sci. (Sept.1989).
  8. D. L. Fried, “Statistics of the Laser Radar Cross-Section of a Randomly Rough Target,” in Technical Digest, Topical Meeting on Image Processing (Optical Society of America, Washington, DC, 1976).
  9. R. O. Rice, “Mathematical Analysis of Random Noise,” in Noise and Stochastic Processes, N. Wax, Ed. (Publisher, New York, 1954), pp. 236–240.
  10. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960), pp. 414–416.
  11. J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” Laser Speckle, J. C. Dainty, Ed. (Springer-Verlag, New York, 1984), pp. 68–74.
  12. E. R. Mendez, authod to add affiliation; verbal communication.
  13. E. T. Copson, Functions of a Complex Variable (Oxford U.P., London, 1935), p. 260.

1989 (3)

S. J. Wein, W. L. Wolfe, “Gaussian-Apodized Apertures and Small-Angle Scatter Measurement,” Opt. Eng. 28, 273–280 (1989).
[Crossref]

C. L. Vernold, “Application and Verification of Wavelength Scaling for Near-Specular Scatter Predictions,” Proc. Soc. Photo-Opt. Instrum. Eng. 1165, 18–00 (1989).

Z. H. Gu, R. S. Dummer, A. A. Maradudin, A. R. McGurn, “Experimental Investigation of the Opposition Effect in the Scattering of Light from a Random Rough Metal Surface,” Appl. Opt. 28, 537–543 (1989).
[Crossref] [PubMed]

1988 (1)

W. W. Lee, L. M. Sherr, M. K. Barsh, “Stray Light Analysis and Suppression in Small Angle BRDF/BTDF Measurement,” Proc. Soc. Photo-Opt. Instrum. Eng. 675, 32–00 (1988).

1982 (1)

J. B. Breckinridge, T. G. Kuper, R. V. Shack, “Space Telescope Low Scattered Camera—A Model,” Proc. Soc. Photo-Opt. Instrum. Eng. 331, 395–403 (1982).

Archibald, P. C.

P. A. Temple, D. P. Arndt, J. M. Bennett, J. M. Elson, P. C. Archibald, S. J. Hixson, “Near-Angle Scatter and Total Integrated Scatter from Single-Point Diamond-Turned Copper Surfaces Measured at 2.7 μm Wavelength,” in Technical Digest, Topical Meeting on High Power Laser Optical Components (Optical Society of America, Washington, DC, 1984).

Arndt, D. P.

P. A. Temple, D. P. Arndt, J. M. Bennett, J. M. Elson, P. C. Archibald, S. J. Hixson, “Near-Angle Scatter and Total Integrated Scatter from Single-Point Diamond-Turned Copper Surfaces Measured at 2.7 μm Wavelength,” in Technical Digest, Topical Meeting on High Power Laser Optical Components (Optical Society of America, Washington, DC, 1984).

Barsh, M. K.

W. W. Lee, L. M. Sherr, M. K. Barsh, “Stray Light Analysis and Suppression in Small Angle BRDF/BTDF Measurement,” Proc. Soc. Photo-Opt. Instrum. Eng. 675, 32–00 (1988).

Bennett, J. M.

P. A. Temple, D. P. Arndt, J. M. Bennett, J. M. Elson, P. C. Archibald, S. J. Hixson, “Near-Angle Scatter and Total Integrated Scatter from Single-Point Diamond-Turned Copper Surfaces Measured at 2.7 μm Wavelength,” in Technical Digest, Topical Meeting on High Power Laser Optical Components (Optical Society of America, Washington, DC, 1984).

Breckinridge, J. B.

J. B. Breckinridge, T. G. Kuper, R. V. Shack, “Space Telescope Low Scattered Camera—A Model,” Proc. Soc. Photo-Opt. Instrum. Eng. 331, 395–403 (1982).

Copson, E. T.

E. T. Copson, Functions of a Complex Variable (Oxford U.P., London, 1935), p. 260.

Dummer, R. S.

Z. H. Gu, R. S. Dummer, A. A. Maradudin, A. R. McGurn, “Experimental Investigation of the Opposition Effect in the Scattering of Light from a Random Rough Metal Surface,” Appl. Opt. 28, 537–543 (1989).
[Crossref] [PubMed]

Z. H. Gu, R. S. Dummer, A. A. Maradudin, A. R. McGurn, “Enhanced Backscattering of Light from a Randomly Rough Metal Surface,” submitted to Radio Sci. (Sept.1989).

Elson, J. M.

P. A. Temple, D. P. Arndt, J. M. Bennett, J. M. Elson, P. C. Archibald, S. J. Hixson, “Near-Angle Scatter and Total Integrated Scatter from Single-Point Diamond-Turned Copper Surfaces Measured at 2.7 μm Wavelength,” in Technical Digest, Topical Meeting on High Power Laser Optical Components (Optical Society of America, Washington, DC, 1984).

Fried, D. L.

D. L. Fried, “Statistics of the Laser Radar Cross-Section of a Randomly Rough Target,” in Technical Digest, Topical Meeting on Image Processing (Optical Society of America, Washington, DC, 1976).

Goodman, J. W.

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” Laser Speckle, J. C. Dainty, Ed. (Springer-Verlag, New York, 1984), pp. 68–74.

Gu, Z. H.

Z. H. Gu, R. S. Dummer, A. A. Maradudin, A. R. McGurn, “Experimental Investigation of the Opposition Effect in the Scattering of Light from a Random Rough Metal Surface,” Appl. Opt. 28, 537–543 (1989).
[Crossref] [PubMed]

Z. H. Gu, R. S. Dummer, A. A. Maradudin, A. R. McGurn, “Enhanced Backscattering of Light from a Randomly Rough Metal Surface,” submitted to Radio Sci. (Sept.1989).

Hixson, S. J.

P. A. Temple, D. P. Arndt, J. M. Bennett, J. M. Elson, P. C. Archibald, S. J. Hixson, “Near-Angle Scatter and Total Integrated Scatter from Single-Point Diamond-Turned Copper Surfaces Measured at 2.7 μm Wavelength,” in Technical Digest, Topical Meeting on High Power Laser Optical Components (Optical Society of America, Washington, DC, 1984).

Kuper, T. G.

J. B. Breckinridge, T. G. Kuper, R. V. Shack, “Space Telescope Low Scattered Camera—A Model,” Proc. Soc. Photo-Opt. Instrum. Eng. 331, 395–403 (1982).

Lee, W. W.

W. W. Lee, L. M. Sherr, M. K. Barsh, “Stray Light Analysis and Suppression in Small Angle BRDF/BTDF Measurement,” Proc. Soc. Photo-Opt. Instrum. Eng. 675, 32–00 (1988).

Maradudin, A. A.

Z. H. Gu, R. S. Dummer, A. A. Maradudin, A. R. McGurn, “Experimental Investigation of the Opposition Effect in the Scattering of Light from a Random Rough Metal Surface,” Appl. Opt. 28, 537–543 (1989).
[Crossref] [PubMed]

Z. H. Gu, R. S. Dummer, A. A. Maradudin, A. R. McGurn, “Enhanced Backscattering of Light from a Randomly Rough Metal Surface,” submitted to Radio Sci. (Sept.1989).

McGurn, A. R.

Z. H. Gu, R. S. Dummer, A. A. Maradudin, A. R. McGurn, “Experimental Investigation of the Opposition Effect in the Scattering of Light from a Random Rough Metal Surface,” Appl. Opt. 28, 537–543 (1989).
[Crossref] [PubMed]

Z. H. Gu, R. S. Dummer, A. A. Maradudin, A. R. McGurn, “Enhanced Backscattering of Light from a Randomly Rough Metal Surface,” submitted to Radio Sci. (Sept.1989).

Mendez, E. R.

E. R. Mendez, authod to add affiliation; verbal communication.

Middleton, D.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960), pp. 414–416.

Rice, R. O.

R. O. Rice, “Mathematical Analysis of Random Noise,” in Noise and Stochastic Processes, N. Wax, Ed. (Publisher, New York, 1954), pp. 236–240.

Shack, R. V.

J. B. Breckinridge, T. G. Kuper, R. V. Shack, “Space Telescope Low Scattered Camera—A Model,” Proc. Soc. Photo-Opt. Instrum. Eng. 331, 395–403 (1982).

Sherr, L. M.

W. W. Lee, L. M. Sherr, M. K. Barsh, “Stray Light Analysis and Suppression in Small Angle BRDF/BTDF Measurement,” Proc. Soc. Photo-Opt. Instrum. Eng. 675, 32–00 (1988).

Temple, P. A.

P. A. Temple, D. P. Arndt, J. M. Bennett, J. M. Elson, P. C. Archibald, S. J. Hixson, “Near-Angle Scatter and Total Integrated Scatter from Single-Point Diamond-Turned Copper Surfaces Measured at 2.7 μm Wavelength,” in Technical Digest, Topical Meeting on High Power Laser Optical Components (Optical Society of America, Washington, DC, 1984).

Vernold, C. L.

C. L. Vernold, “Application and Verification of Wavelength Scaling for Near-Specular Scatter Predictions,” Proc. Soc. Photo-Opt. Instrum. Eng. 1165, 18–00 (1989).

Wein, S. J.

S. J. Wein, W. L. Wolfe, “Gaussian-Apodized Apertures and Small-Angle Scatter Measurement,” Opt. Eng. 28, 273–280 (1989).
[Crossref]

Wolfe, W. L.

S. J. Wein, W. L. Wolfe, “Gaussian-Apodized Apertures and Small-Angle Scatter Measurement,” Opt. Eng. 28, 273–280 (1989).
[Crossref]

Appl. Opt. (1)

Opt. Eng. (1)

S. J. Wein, W. L. Wolfe, “Gaussian-Apodized Apertures and Small-Angle Scatter Measurement,” Opt. Eng. 28, 273–280 (1989).
[Crossref]

Proc. Soc. Photo-Opt. Instrum. Eng. (3)

W. W. Lee, L. M. Sherr, M. K. Barsh, “Stray Light Analysis and Suppression in Small Angle BRDF/BTDF Measurement,” Proc. Soc. Photo-Opt. Instrum. Eng. 675, 32–00 (1988).

C. L. Vernold, “Application and Verification of Wavelength Scaling for Near-Specular Scatter Predictions,” Proc. Soc. Photo-Opt. Instrum. Eng. 1165, 18–00 (1989).

J. B. Breckinridge, T. G. Kuper, R. V. Shack, “Space Telescope Low Scattered Camera—A Model,” Proc. Soc. Photo-Opt. Instrum. Eng. 331, 395–403 (1982).

Other (8)

P. A. Temple, D. P. Arndt, J. M. Bennett, J. M. Elson, P. C. Archibald, S. J. Hixson, “Near-Angle Scatter and Total Integrated Scatter from Single-Point Diamond-Turned Copper Surfaces Measured at 2.7 μm Wavelength,” in Technical Digest, Topical Meeting on High Power Laser Optical Components (Optical Society of America, Washington, DC, 1984).

Z. H. Gu, R. S. Dummer, A. A. Maradudin, A. R. McGurn, “Enhanced Backscattering of Light from a Randomly Rough Metal Surface,” submitted to Radio Sci. (Sept.1989).

D. L. Fried, “Statistics of the Laser Radar Cross-Section of a Randomly Rough Target,” in Technical Digest, Topical Meeting on Image Processing (Optical Society of America, Washington, DC, 1976).

R. O. Rice, “Mathematical Analysis of Random Noise,” in Noise and Stochastic Processes, N. Wax, Ed. (Publisher, New York, 1954), pp. 236–240.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960), pp. 414–416.

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” Laser Speckle, J. C. Dainty, Ed. (Springer-Verlag, New York, 1984), pp. 68–74.

E. R. Mendez, authod to add affiliation; verbal communication.

E. T. Copson, Functions of a Complex Variable (Oxford U.P., London, 1935), p. 260.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Simplified schematic of the basic experimental apparatus.

Fig. 2
Fig. 2

Component schematic of the monostatic bidirectional reflectometer.

Fig. 3
Fig. 3

Probability distribution of Ricean statistics.

Fig. 4
Fig. 4

Interpretation of the surface topology by Ricean parameters.

Fig. 5
Fig. 5

BRDF for small angles around the specular of a mirror at 10.6 μm.

Fig. 6
Fig. 6

Partition of the scattered portion of BRDF of a mirror at 10.6 μm.

Fig. 7
Fig. 7

Logarithmic display of the partition of the scattered component in the BRDF of an IR filter at 1.06 μm: ○ = coherent plus incoherent; □ = incoherent.

Equations (20)

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

f ( x ) = { x exp [ - ( x 2 + k 2 ) / 2 ] I o ( k x ) , x 0 , 0 otherwise ,
E { R σ 0 } = 0 x f ( x ) d x = exp ( - k 2 / 2 ) 0 x 2 exp ( - x 2 / 2 ) I 0 ( k x ) d x
E { R σ 0 } = ( π 2 ) 0.5 exp ( k 2 4 ) [ ( 1 + k 2 2 ) I 0 ( k 2 4 ) + k 2 2 I 1 ( k 2 4 ) ] ,
E { R 2 σ 0 2 } = 0 x 2 f ( x ) d x , E { R 2 σ 0 2 } = 2 + K 2 ,
σ R 2 = E { R 2 } - ( E { R } ) 2 .
σ R 2 σ 0 2 = 2 + k 2 - π 2 exp [ - ( k 2 2 ) ] · [ ( 1 + k 2 2 ) I 0 ( k 2 ) 2 + k 2 2 I 1 ( k 2 ) 2 ] 2 .
V ( t ) = N ( t ) + U cos ( ω t ) ,
N ( t ) = N r ( t ) cos ( ω t ) + N i ( t ) sin ( ω t ) , V ( t ) = R ( t ) cos ( ω t + θ ) ,
R ( t ) = [ N r ( t ) + U ] 2 + [ N i ( t ) ] 2 , θ = arctan [ - ( N r + U ) N i ] .
f ( x ) = { x exp [ - ( x 2 + k 2 ) / 2 ] I o ( k x ) , x 0 , 0 otherwise ,
E { R r } = 0 R r + 1 σ 0 2 exp [ - ( R 2 + U 2 ) / 2 σ 0 2 ] I 0 ( U R σ 0 2 ) d R .
R = x σ 0 ,
E { R r } = 0 x r + 1 σ 0 r exp [ - ( x 2 + k 2 ) / 2 ] I 0 ( k x ) d x
E { R r } = ( 2 σ 0 2 ) r / 2 Γ ( r 2 + 1 ) F 1 1 ( - r 2 ; 1 ; - k 2 2 ) ,
E { R } = 2 σ 0 2 Γ ( 3 2 ) F 1 1 ( - 1 2 ; 1 ; - k 2 2 ) .
Γ ( 3 2 ) = π 2 , F 1 1 ( - 1 2 ; 1 ; - k 2 2 ) = exp ( - k 2 4 ) [ ( 1 + k 2 2 ) I 0 ( k 2 4 ) + k 2 2 I 1 ( k 2 4 ) ] ,
E { R σ 0 } = ( π 2 ) 0.5 exp ( - k 2 4 ) [ ( 1 + k 2 2 ) I 0 ( k 2 4 ) + k 2 2 I 1 ( k 2 4 ) ] .
E { R 2 σ 0 2 } = 2 Γ ( 2 ) F 1 1 ( - 1 ; 1 ; - k 2 2 ) ,
F 1 1 ( - 1 ; 1 ; - k 2 2 ) = 1 + k 2 2 ,
E { R 2 σ 0 2 } = 2 + k 2 .

Metrics