Abstract

The power spectral density of surface-relief variations on polished optical surfaces across microscopic through to macroscopic spatial scales is calculated from measurements on substrates that are being produced for the Laser Interferometer Gravitational-Wave Observatory (LIGO). These spectra give a guide to the scattering properties of the surface, which in turn critically influence the performance of LIGO. Measurements obtained by use of a full-aperture interferometer and an interference microscope with two different objectives are combined to produce one-dimensional power spectral density representations of the surfaces across spatial frequencies ranging from 0.1 to 8000 cm-1. These measurements from different instruments are in good agreement with an analytic power spectrum that varies as ν -1.5, where ν is the spatial frequency. Some anomalies in the power spectral density spectra can be related to aspects of the polishing process.

© 1999 Optical Society of America

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  1. A. Abramovici, W. Althouse, R. Drever, Y. Gursel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
    [CrossRef] [PubMed]
  2. C. J. Walsh, A. J. Leistner, J. Seckold, B. F. Oreb, D. I. Farrant, “Fabrication and measurement of optics for LIGO,” Appl. Opt. 38, 2870–2879 (1999).
    [CrossRef]
  3. J. M. Bennett, L. Mattson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).
  4. J. E. Harvey, K. L. Lewotsky, A. Kotha, “Performance predictions of a Schwarzschild imaging microscope for soft x-ray applications,” Opt. Eng. 35, 2423–2436 (1996).
    [CrossRef]
  5. E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526 (1988).
    [CrossRef] [PubMed]
  6. H. Toebben, G. Ringel, F. Kratz, D.-R. Schmitt, “Use of power spectral density (PSD) to specify optical surfaces,” in Specification, Production, and Testing of Optical Components and Systems, A. E. Gee, J. Houee, eds., Proc. SPIE2775, 240–250 (1996).
    [CrossRef]
  7. J. K. Lawson, D. M. Aikens, R. E. English, C. R. Wolfe, “Power spectral density specifications for high-power laser systems,” in Specifications, Production, and Testing of Optical Components and Systems, A. E. Gee, J. Houee, eds., Proc. SPIE2775, 345–356 (1996).
    [CrossRef]
  8. G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, Calif., 1969).
  9. E. Spiller, R. A. McCorkle, J. S. Wilczynski, L. Golub, G. Nystrom, P. Z. Takacs, C. Welch, “Normal-incidence soft x-ray telescopes,” Opt. Eng. 30, 1109–1115 (1991).
    [CrossRef]
  10. V. T. Vorburger, C. J. Evans, V. W. Tsai, J. Fu, E. C. Williams, R. G. Dixson, P. J. Sullivan, T. McWaid, “Finish and figure metrology for soft x-ray optics,” in Proceedings of Japan Society for Precision Engineering U.S.-Japan Workshop on Soft X-ray Optics: Technical Challenges, T. Namioka, H. Kinoshita, K. Ito, eds. (Japan Society for Precision Engineering, Tokyo, 1997), p. 298.
  11. W. Wong, D. Wang, R. T. Benoit, P. Berthol, “A comparison of low scatter PSD derived from multiple wavelength BRDF’s and WYKO profilometer data,” in Optical Scatter: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. SPIE1530, 86–103 (1991).
  12. E. Marx, I. J. Malik, Y. E. Strauser, T. C. Bristow, N. S. Poduje, J. C. Stover, “Round robin determination of power spectral densities of different Si wafer surfaces,” in Flatness, Roughness, and Discrete Defects Characterization for Computer Disks, Wafers, and Flat Panel Displays II, J. C. Stover, ed., Proc. SPIE3275, 26–36 (1998).
    [CrossRef]
  13. R. Weiss, “Spectral analysis of coated optics phase maps,” (Laser Interferometer Gravitational-wave Observatory, California Institute of Technology, Pasadena, Calif., 1997); this report is available from the LIGO on-line document index at http://www.ligo.caltech.edu .
  14. P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “A 300-mm aperture phase shifting fizeau interferometer,” Opt. Eng. (to be published).
  15. vision software, Version 1.8 [Veeco Corporation (formerly WYKO Corporation), East Elvira Road, Tucson, Ariz., 1996). Mention of this product does not constitute endorsement by CSIRO.
  16. J. C. Wyant, C. L. Koliopolous, B. Bhushan, D. Basila, “Development of a three-dimensional noncontact optical profiler,” J. Tribol. 108, 1–8 (1986). The TOPO instrument is manufactured by WYKO Corporation, East Elvira Road, Tucson, Arizona. Mention of this product does not constitute endorsement by CSIRO.
    [CrossRef]
  17. “Standard practice for estimating the power spectral density function and related finish parameters from surface profile data,” (American Society for Testing and Materials, Pittsburgh, Pa., 1998).
  18. J. M. Elson, J. M. Bennett, “Calculation of the power spectral density from surface-profile data,” Appl. Opt. 34, 201–208 (1995).
    [CrossRef] [PubMed]
  19. W. H. Press, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).
  20. E. L. Church, P. Z. Takacs, “The optimal estimation of finish parameters,” in Optical Scatter: Applications, Measurements, and Theory, J. C. Stover, ed., Proc. SPIE1530, 71–86 (1991).
  21. W. D. Dixon, F. J. Massey, Introduction to Statistical Analysis (McGraw-Hill, New York, 1957).
  22. J. C. Wyant, K. Creath, “Basic wave-front theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1992), Vol. 11, pp. 1–53.
  23. E. L. Church, P. Z. Takacs, “Use of an optical-profiling instrument for the measurement of figure and finish of optical quality surfaces,” Wear 109, 241–257 (1986).
    [CrossRef]
  24. Talysurf (Rank Taylor Hobson Pty. Ltd., Leicester, UK). Mention of this product does not constitute endorsement by CSIRO.
  25. K. Creath, “Error sources in phase measuring interferometry,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation, J. Tsujiuchi, eds., Proc. SPIE1720, 428–435 (1992).
    [CrossRef]
  26. A. J. Leistner, “Teflon polishers: their manufacture and use,” Appl. Opt. 15, 293–298 (1976).
  27. S. Whitcomb, G. Billingsley, J. Carri, A. Golovitser, D. Jungwirth, W. Kells, H. Yamamoto, B. Bochner, Y. Hefetz, P. Saha, R. Weiss, “Optics development for LIGO,” paper presented at the TAMA Workshop on Gravitational Wave Detection, Saitama, Japan, 11–12 November 1996; also published as . (Laser Interferometry Gravitational-Wave Observatory, California Institute of Technology, Pasadena, Calif., 1996).

1999 (1)

1996 (1)

J. E. Harvey, K. L. Lewotsky, A. Kotha, “Performance predictions of a Schwarzschild imaging microscope for soft x-ray applications,” Opt. Eng. 35, 2423–2436 (1996).
[CrossRef]

1995 (1)

1992 (1)

A. Abramovici, W. Althouse, R. Drever, Y. Gursel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

1991 (1)

E. Spiller, R. A. McCorkle, J. S. Wilczynski, L. Golub, G. Nystrom, P. Z. Takacs, C. Welch, “Normal-incidence soft x-ray telescopes,” Opt. Eng. 30, 1109–1115 (1991).
[CrossRef]

1988 (1)

1986 (2)

J. C. Wyant, C. L. Koliopolous, B. Bhushan, D. Basila, “Development of a three-dimensional noncontact optical profiler,” J. Tribol. 108, 1–8 (1986). The TOPO instrument is manufactured by WYKO Corporation, East Elvira Road, Tucson, Arizona. Mention of this product does not constitute endorsement by CSIRO.
[CrossRef]

E. L. Church, P. Z. Takacs, “Use of an optical-profiling instrument for the measurement of figure and finish of optical quality surfaces,” Wear 109, 241–257 (1986).
[CrossRef]

1976 (1)

Abramovici, A.

A. Abramovici, W. Althouse, R. Drever, Y. Gursel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Aikens, D. M.

J. K. Lawson, D. M. Aikens, R. E. English, C. R. Wolfe, “Power spectral density specifications for high-power laser systems,” in Specifications, Production, and Testing of Optical Components and Systems, A. E. Gee, J. Houee, eds., Proc. SPIE2775, 345–356 (1996).
[CrossRef]

Althouse, W.

A. Abramovici, W. Althouse, R. Drever, Y. Gursel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Basila, D.

J. C. Wyant, C. L. Koliopolous, B. Bhushan, D. Basila, “Development of a three-dimensional noncontact optical profiler,” J. Tribol. 108, 1–8 (1986). The TOPO instrument is manufactured by WYKO Corporation, East Elvira Road, Tucson, Arizona. Mention of this product does not constitute endorsement by CSIRO.
[CrossRef]

Bennett, J. M.

J. M. Elson, J. M. Bennett, “Calculation of the power spectral density from surface-profile data,” Appl. Opt. 34, 201–208 (1995).
[CrossRef] [PubMed]

J. M. Bennett, L. Mattson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

Benoit, R. T.

W. Wong, D. Wang, R. T. Benoit, P. Berthol, “A comparison of low scatter PSD derived from multiple wavelength BRDF’s and WYKO profilometer data,” in Optical Scatter: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. SPIE1530, 86–103 (1991).

Berthol, P.

W. Wong, D. Wang, R. T. Benoit, P. Berthol, “A comparison of low scatter PSD derived from multiple wavelength BRDF’s and WYKO profilometer data,” in Optical Scatter: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. SPIE1530, 86–103 (1991).

Bhushan, B.

J. C. Wyant, C. L. Koliopolous, B. Bhushan, D. Basila, “Development of a three-dimensional noncontact optical profiler,” J. Tribol. 108, 1–8 (1986). The TOPO instrument is manufactured by WYKO Corporation, East Elvira Road, Tucson, Arizona. Mention of this product does not constitute endorsement by CSIRO.
[CrossRef]

Billingsley, G.

S. Whitcomb, G. Billingsley, J. Carri, A. Golovitser, D. Jungwirth, W. Kells, H. Yamamoto, B. Bochner, Y. Hefetz, P. Saha, R. Weiss, “Optics development for LIGO,” paper presented at the TAMA Workshop on Gravitational Wave Detection, Saitama, Japan, 11–12 November 1996; also published as . (Laser Interferometry Gravitational-Wave Observatory, California Institute of Technology, Pasadena, Calif., 1996).

Bochner, B.

S. Whitcomb, G. Billingsley, J. Carri, A. Golovitser, D. Jungwirth, W. Kells, H. Yamamoto, B. Bochner, Y. Hefetz, P. Saha, R. Weiss, “Optics development for LIGO,” paper presented at the TAMA Workshop on Gravitational Wave Detection, Saitama, Japan, 11–12 November 1996; also published as . (Laser Interferometry Gravitational-Wave Observatory, California Institute of Technology, Pasadena, Calif., 1996).

Bristow, T. C.

E. Marx, I. J. Malik, Y. E. Strauser, T. C. Bristow, N. S. Poduje, J. C. Stover, “Round robin determination of power spectral densities of different Si wafer surfaces,” in Flatness, Roughness, and Discrete Defects Characterization for Computer Disks, Wafers, and Flat Panel Displays II, J. C. Stover, ed., Proc. SPIE3275, 26–36 (1998).
[CrossRef]

Carri, J.

S. Whitcomb, G. Billingsley, J. Carri, A. Golovitser, D. Jungwirth, W. Kells, H. Yamamoto, B. Bochner, Y. Hefetz, P. Saha, R. Weiss, “Optics development for LIGO,” paper presented at the TAMA Workshop on Gravitational Wave Detection, Saitama, Japan, 11–12 November 1996; also published as . (Laser Interferometry Gravitational-Wave Observatory, California Institute of Technology, Pasadena, Calif., 1996).

Church, E. L.

E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526 (1988).
[CrossRef] [PubMed]

E. L. Church, P. Z. Takacs, “Use of an optical-profiling instrument for the measurement of figure and finish of optical quality surfaces,” Wear 109, 241–257 (1986).
[CrossRef]

E. L. Church, P. Z. Takacs, “The optimal estimation of finish parameters,” in Optical Scatter: Applications, Measurements, and Theory, J. C. Stover, ed., Proc. SPIE1530, 71–86 (1991).

Creath, K.

K. Creath, “Error sources in phase measuring interferometry,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation, J. Tsujiuchi, eds., Proc. SPIE1720, 428–435 (1992).
[CrossRef]

J. C. Wyant, K. Creath, “Basic wave-front theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1992), Vol. 11, pp. 1–53.

Dixon, W. D.

W. D. Dixon, F. J. Massey, Introduction to Statistical Analysis (McGraw-Hill, New York, 1957).

Dixson, R. G.

V. T. Vorburger, C. J. Evans, V. W. Tsai, J. Fu, E. C. Williams, R. G. Dixson, P. J. Sullivan, T. McWaid, “Finish and figure metrology for soft x-ray optics,” in Proceedings of Japan Society for Precision Engineering U.S.-Japan Workshop on Soft X-ray Optics: Technical Challenges, T. Namioka, H. Kinoshita, K. Ito, eds. (Japan Society for Precision Engineering, Tokyo, 1997), p. 298.

Drever, R.

A. Abramovici, W. Althouse, R. Drever, Y. Gursel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Elson, J. M.

English, R. E.

J. K. Lawson, D. M. Aikens, R. E. English, C. R. Wolfe, “Power spectral density specifications for high-power laser systems,” in Specifications, Production, and Testing of Optical Components and Systems, A. E. Gee, J. Houee, eds., Proc. SPIE2775, 345–356 (1996).
[CrossRef]

Evans, C. J.

V. T. Vorburger, C. J. Evans, V. W. Tsai, J. Fu, E. C. Williams, R. G. Dixson, P. J. Sullivan, T. McWaid, “Finish and figure metrology for soft x-ray optics,” in Proceedings of Japan Society for Precision Engineering U.S.-Japan Workshop on Soft X-ray Optics: Technical Challenges, T. Namioka, H. Kinoshita, K. Ito, eds. (Japan Society for Precision Engineering, Tokyo, 1997), p. 298.

Fairman, P. S.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “A 300-mm aperture phase shifting fizeau interferometer,” Opt. Eng. (to be published).

Farrant, D. I.

C. J. Walsh, A. J. Leistner, J. Seckold, B. F. Oreb, D. I. Farrant, “Fabrication and measurement of optics for LIGO,” Appl. Opt. 38, 2870–2879 (1999).
[CrossRef]

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “A 300-mm aperture phase shifting fizeau interferometer,” Opt. Eng. (to be published).

Freund, C. H.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “A 300-mm aperture phase shifting fizeau interferometer,” Opt. Eng. (to be published).

Fu, J.

V. T. Vorburger, C. J. Evans, V. W. Tsai, J. Fu, E. C. Williams, R. G. Dixson, P. J. Sullivan, T. McWaid, “Finish and figure metrology for soft x-ray optics,” in Proceedings of Japan Society for Precision Engineering U.S.-Japan Workshop on Soft X-ray Optics: Technical Challenges, T. Namioka, H. Kinoshita, K. Ito, eds. (Japan Society for Precision Engineering, Tokyo, 1997), p. 298.

Gilliand, Y.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “A 300-mm aperture phase shifting fizeau interferometer,” Opt. Eng. (to be published).

Golovitser, A.

S. Whitcomb, G. Billingsley, J. Carri, A. Golovitser, D. Jungwirth, W. Kells, H. Yamamoto, B. Bochner, Y. Hefetz, P. Saha, R. Weiss, “Optics development for LIGO,” paper presented at the TAMA Workshop on Gravitational Wave Detection, Saitama, Japan, 11–12 November 1996; also published as . (Laser Interferometry Gravitational-Wave Observatory, California Institute of Technology, Pasadena, Calif., 1996).

Golub, L.

E. Spiller, R. A. McCorkle, J. S. Wilczynski, L. Golub, G. Nystrom, P. Z. Takacs, C. Welch, “Normal-incidence soft x-ray telescopes,” Opt. Eng. 30, 1109–1115 (1991).
[CrossRef]

Gursel, Y.

A. Abramovici, W. Althouse, R. Drever, Y. Gursel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Harvey, J. E.

J. E. Harvey, K. L. Lewotsky, A. Kotha, “Performance predictions of a Schwarzschild imaging microscope for soft x-ray applications,” Opt. Eng. 35, 2423–2436 (1996).
[CrossRef]

Hefetz, Y.

S. Whitcomb, G. Billingsley, J. Carri, A. Golovitser, D. Jungwirth, W. Kells, H. Yamamoto, B. Bochner, Y. Hefetz, P. Saha, R. Weiss, “Optics development for LIGO,” paper presented at the TAMA Workshop on Gravitational Wave Detection, Saitama, Japan, 11–12 November 1996; also published as . (Laser Interferometry Gravitational-Wave Observatory, California Institute of Technology, Pasadena, Calif., 1996).

Jenkins, G. M.

G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, Calif., 1969).

Jungwirth, D.

S. Whitcomb, G. Billingsley, J. Carri, A. Golovitser, D. Jungwirth, W. Kells, H. Yamamoto, B. Bochner, Y. Hefetz, P. Saha, R. Weiss, “Optics development for LIGO,” paper presented at the TAMA Workshop on Gravitational Wave Detection, Saitama, Japan, 11–12 November 1996; also published as . (Laser Interferometry Gravitational-Wave Observatory, California Institute of Technology, Pasadena, Calif., 1996).

Kawamura, S.

A. Abramovici, W. Althouse, R. Drever, Y. Gursel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Kells, W.

S. Whitcomb, G. Billingsley, J. Carri, A. Golovitser, D. Jungwirth, W. Kells, H. Yamamoto, B. Bochner, Y. Hefetz, P. Saha, R. Weiss, “Optics development for LIGO,” paper presented at the TAMA Workshop on Gravitational Wave Detection, Saitama, Japan, 11–12 November 1996; also published as . (Laser Interferometry Gravitational-Wave Observatory, California Institute of Technology, Pasadena, Calif., 1996).

Koliopolous, C. L.

J. C. Wyant, C. L. Koliopolous, B. Bhushan, D. Basila, “Development of a three-dimensional noncontact optical profiler,” J. Tribol. 108, 1–8 (1986). The TOPO instrument is manufactured by WYKO Corporation, East Elvira Road, Tucson, Arizona. Mention of this product does not constitute endorsement by CSIRO.
[CrossRef]

Kotha, A.

J. E. Harvey, K. L. Lewotsky, A. Kotha, “Performance predictions of a Schwarzschild imaging microscope for soft x-ray applications,” Opt. Eng. 35, 2423–2436 (1996).
[CrossRef]

Kratz, F.

H. Toebben, G. Ringel, F. Kratz, D.-R. Schmitt, “Use of power spectral density (PSD) to specify optical surfaces,” in Specification, Production, and Testing of Optical Components and Systems, A. E. Gee, J. Houee, eds., Proc. SPIE2775, 240–250 (1996).
[CrossRef]

Lawson, J. K.

J. K. Lawson, D. M. Aikens, R. E. English, C. R. Wolfe, “Power spectral density specifications for high-power laser systems,” in Specifications, Production, and Testing of Optical Components and Systems, A. E. Gee, J. Houee, eds., Proc. SPIE2775, 345–356 (1996).
[CrossRef]

Leistner, A. J.

C. J. Walsh, A. J. Leistner, J. Seckold, B. F. Oreb, D. I. Farrant, “Fabrication and measurement of optics for LIGO,” Appl. Opt. 38, 2870–2879 (1999).
[CrossRef]

A. J. Leistner, “Teflon polishers: their manufacture and use,” Appl. Opt. 15, 293–298 (1976).

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “A 300-mm aperture phase shifting fizeau interferometer,” Opt. Eng. (to be published).

Lewotsky, K. L.

J. E. Harvey, K. L. Lewotsky, A. Kotha, “Performance predictions of a Schwarzschild imaging microscope for soft x-ray applications,” Opt. Eng. 35, 2423–2436 (1996).
[CrossRef]

Malik, I. J.

E. Marx, I. J. Malik, Y. E. Strauser, T. C. Bristow, N. S. Poduje, J. C. Stover, “Round robin determination of power spectral densities of different Si wafer surfaces,” in Flatness, Roughness, and Discrete Defects Characterization for Computer Disks, Wafers, and Flat Panel Displays II, J. C. Stover, ed., Proc. SPIE3275, 26–36 (1998).
[CrossRef]

Marx, E.

E. Marx, I. J. Malik, Y. E. Strauser, T. C. Bristow, N. S. Poduje, J. C. Stover, “Round robin determination of power spectral densities of different Si wafer surfaces,” in Flatness, Roughness, and Discrete Defects Characterization for Computer Disks, Wafers, and Flat Panel Displays II, J. C. Stover, ed., Proc. SPIE3275, 26–36 (1998).
[CrossRef]

Massey, F. J.

W. D. Dixon, F. J. Massey, Introduction to Statistical Analysis (McGraw-Hill, New York, 1957).

Mattson, L.

J. M. Bennett, L. Mattson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

McCorkle, R. A.

E. Spiller, R. A. McCorkle, J. S. Wilczynski, L. Golub, G. Nystrom, P. Z. Takacs, C. Welch, “Normal-incidence soft x-ray telescopes,” Opt. Eng. 30, 1109–1115 (1991).
[CrossRef]

McWaid, T.

V. T. Vorburger, C. J. Evans, V. W. Tsai, J. Fu, E. C. Williams, R. G. Dixson, P. J. Sullivan, T. McWaid, “Finish and figure metrology for soft x-ray optics,” in Proceedings of Japan Society for Precision Engineering U.S.-Japan Workshop on Soft X-ray Optics: Technical Challenges, T. Namioka, H. Kinoshita, K. Ito, eds. (Japan Society for Precision Engineering, Tokyo, 1997), p. 298.

Nystrom, G.

E. Spiller, R. A. McCorkle, J. S. Wilczynski, L. Golub, G. Nystrom, P. Z. Takacs, C. Welch, “Normal-incidence soft x-ray telescopes,” Opt. Eng. 30, 1109–1115 (1991).
[CrossRef]

Oreb, B. F.

C. J. Walsh, A. J. Leistner, J. Seckold, B. F. Oreb, D. I. Farrant, “Fabrication and measurement of optics for LIGO,” Appl. Opt. 38, 2870–2879 (1999).
[CrossRef]

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “A 300-mm aperture phase shifting fizeau interferometer,” Opt. Eng. (to be published).

Poduje, N. S.

E. Marx, I. J. Malik, Y. E. Strauser, T. C. Bristow, N. S. Poduje, J. C. Stover, “Round robin determination of power spectral densities of different Si wafer surfaces,” in Flatness, Roughness, and Discrete Defects Characterization for Computer Disks, Wafers, and Flat Panel Displays II, J. C. Stover, ed., Proc. SPIE3275, 26–36 (1998).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Raab, F.

A. Abramovici, W. Althouse, R. Drever, Y. Gursel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Ringel, G.

H. Toebben, G. Ringel, F. Kratz, D.-R. Schmitt, “Use of power spectral density (PSD) to specify optical surfaces,” in Specification, Production, and Testing of Optical Components and Systems, A. E. Gee, J. Houee, eds., Proc. SPIE2775, 240–250 (1996).
[CrossRef]

Saha, P.

S. Whitcomb, G. Billingsley, J. Carri, A. Golovitser, D. Jungwirth, W. Kells, H. Yamamoto, B. Bochner, Y. Hefetz, P. Saha, R. Weiss, “Optics development for LIGO,” paper presented at the TAMA Workshop on Gravitational Wave Detection, Saitama, Japan, 11–12 November 1996; also published as . (Laser Interferometry Gravitational-Wave Observatory, California Institute of Technology, Pasadena, Calif., 1996).

Schmitt, D.-R.

H. Toebben, G. Ringel, F. Kratz, D.-R. Schmitt, “Use of power spectral density (PSD) to specify optical surfaces,” in Specification, Production, and Testing of Optical Components and Systems, A. E. Gee, J. Houee, eds., Proc. SPIE2775, 240–250 (1996).
[CrossRef]

Seckold, J.

Seckold, J. A.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “A 300-mm aperture phase shifting fizeau interferometer,” Opt. Eng. (to be published).

Shoemaker, D.

A. Abramovici, W. Althouse, R. Drever, Y. Gursel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Sievers, L.

A. Abramovici, W. Althouse, R. Drever, Y. Gursel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Spero, R.

A. Abramovici, W. Althouse, R. Drever, Y. Gursel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Spiller, E.

E. Spiller, R. A. McCorkle, J. S. Wilczynski, L. Golub, G. Nystrom, P. Z. Takacs, C. Welch, “Normal-incidence soft x-ray telescopes,” Opt. Eng. 30, 1109–1115 (1991).
[CrossRef]

Stover, J. C.

E. Marx, I. J. Malik, Y. E. Strauser, T. C. Bristow, N. S. Poduje, J. C. Stover, “Round robin determination of power spectral densities of different Si wafer surfaces,” in Flatness, Roughness, and Discrete Defects Characterization for Computer Disks, Wafers, and Flat Panel Displays II, J. C. Stover, ed., Proc. SPIE3275, 26–36 (1998).
[CrossRef]

Strauser, Y. E.

E. Marx, I. J. Malik, Y. E. Strauser, T. C. Bristow, N. S. Poduje, J. C. Stover, “Round robin determination of power spectral densities of different Si wafer surfaces,” in Flatness, Roughness, and Discrete Defects Characterization for Computer Disks, Wafers, and Flat Panel Displays II, J. C. Stover, ed., Proc. SPIE3275, 26–36 (1998).
[CrossRef]

Sullivan, P. J.

V. T. Vorburger, C. J. Evans, V. W. Tsai, J. Fu, E. C. Williams, R. G. Dixson, P. J. Sullivan, T. McWaid, “Finish and figure metrology for soft x-ray optics,” in Proceedings of Japan Society for Precision Engineering U.S.-Japan Workshop on Soft X-ray Optics: Technical Challenges, T. Namioka, H. Kinoshita, K. Ito, eds. (Japan Society for Precision Engineering, Tokyo, 1997), p. 298.

Takacs, P. Z.

E. Spiller, R. A. McCorkle, J. S. Wilczynski, L. Golub, G. Nystrom, P. Z. Takacs, C. Welch, “Normal-incidence soft x-ray telescopes,” Opt. Eng. 30, 1109–1115 (1991).
[CrossRef]

E. L. Church, P. Z. Takacs, “Use of an optical-profiling instrument for the measurement of figure and finish of optical quality surfaces,” Wear 109, 241–257 (1986).
[CrossRef]

E. L. Church, P. Z. Takacs, “The optimal estimation of finish parameters,” in Optical Scatter: Applications, Measurements, and Theory, J. C. Stover, ed., Proc. SPIE1530, 71–86 (1991).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Thorne, K.

A. Abramovici, W. Althouse, R. Drever, Y. Gursel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Toebben, H.

H. Toebben, G. Ringel, F. Kratz, D.-R. Schmitt, “Use of power spectral density (PSD) to specify optical surfaces,” in Specification, Production, and Testing of Optical Components and Systems, A. E. Gee, J. Houee, eds., Proc. SPIE2775, 240–250 (1996).
[CrossRef]

Tsai, V. W.

V. T. Vorburger, C. J. Evans, V. W. Tsai, J. Fu, E. C. Williams, R. G. Dixson, P. J. Sullivan, T. McWaid, “Finish and figure metrology for soft x-ray optics,” in Proceedings of Japan Society for Precision Engineering U.S.-Japan Workshop on Soft X-ray Optics: Technical Challenges, T. Namioka, H. Kinoshita, K. Ito, eds. (Japan Society for Precision Engineering, Tokyo, 1997), p. 298.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Vogt, R.

A. Abramovici, W. Althouse, R. Drever, Y. Gursel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Vorburger, V. T.

V. T. Vorburger, C. J. Evans, V. W. Tsai, J. Fu, E. C. Williams, R. G. Dixson, P. J. Sullivan, T. McWaid, “Finish and figure metrology for soft x-ray optics,” in Proceedings of Japan Society for Precision Engineering U.S.-Japan Workshop on Soft X-ray Optics: Technical Challenges, T. Namioka, H. Kinoshita, K. Ito, eds. (Japan Society for Precision Engineering, Tokyo, 1997), p. 298.

Walsh, C. J.

C. J. Walsh, A. J. Leistner, J. Seckold, B. F. Oreb, D. I. Farrant, “Fabrication and measurement of optics for LIGO,” Appl. Opt. 38, 2870–2879 (1999).
[CrossRef]

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “A 300-mm aperture phase shifting fizeau interferometer,” Opt. Eng. (to be published).

Wang, D.

W. Wong, D. Wang, R. T. Benoit, P. Berthol, “A comparison of low scatter PSD derived from multiple wavelength BRDF’s and WYKO profilometer data,” in Optical Scatter: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. SPIE1530, 86–103 (1991).

Ward, B. K.

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “A 300-mm aperture phase shifting fizeau interferometer,” Opt. Eng. (to be published).

Watts, D. G.

G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, Calif., 1969).

Weiss, R.

A. Abramovici, W. Althouse, R. Drever, Y. Gursel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

R. Weiss, “Spectral analysis of coated optics phase maps,” (Laser Interferometer Gravitational-wave Observatory, California Institute of Technology, Pasadena, Calif., 1997); this report is available from the LIGO on-line document index at http://www.ligo.caltech.edu .

S. Whitcomb, G. Billingsley, J. Carri, A. Golovitser, D. Jungwirth, W. Kells, H. Yamamoto, B. Bochner, Y. Hefetz, P. Saha, R. Weiss, “Optics development for LIGO,” paper presented at the TAMA Workshop on Gravitational Wave Detection, Saitama, Japan, 11–12 November 1996; also published as . (Laser Interferometry Gravitational-Wave Observatory, California Institute of Technology, Pasadena, Calif., 1996).

Welch, C.

E. Spiller, R. A. McCorkle, J. S. Wilczynski, L. Golub, G. Nystrom, P. Z. Takacs, C. Welch, “Normal-incidence soft x-ray telescopes,” Opt. Eng. 30, 1109–1115 (1991).
[CrossRef]

Whitcomb, S.

A. Abramovici, W. Althouse, R. Drever, Y. Gursel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

S. Whitcomb, G. Billingsley, J. Carri, A. Golovitser, D. Jungwirth, W. Kells, H. Yamamoto, B. Bochner, Y. Hefetz, P. Saha, R. Weiss, “Optics development for LIGO,” paper presented at the TAMA Workshop on Gravitational Wave Detection, Saitama, Japan, 11–12 November 1996; also published as . (Laser Interferometry Gravitational-Wave Observatory, California Institute of Technology, Pasadena, Calif., 1996).

Wilczynski, J. S.

E. Spiller, R. A. McCorkle, J. S. Wilczynski, L. Golub, G. Nystrom, P. Z. Takacs, C. Welch, “Normal-incidence soft x-ray telescopes,” Opt. Eng. 30, 1109–1115 (1991).
[CrossRef]

Williams, E. C.

V. T. Vorburger, C. J. Evans, V. W. Tsai, J. Fu, E. C. Williams, R. G. Dixson, P. J. Sullivan, T. McWaid, “Finish and figure metrology for soft x-ray optics,” in Proceedings of Japan Society for Precision Engineering U.S.-Japan Workshop on Soft X-ray Optics: Technical Challenges, T. Namioka, H. Kinoshita, K. Ito, eds. (Japan Society for Precision Engineering, Tokyo, 1997), p. 298.

Wolfe, C. R.

J. K. Lawson, D. M. Aikens, R. E. English, C. R. Wolfe, “Power spectral density specifications for high-power laser systems,” in Specifications, Production, and Testing of Optical Components and Systems, A. E. Gee, J. Houee, eds., Proc. SPIE2775, 345–356 (1996).
[CrossRef]

Wong, W.

W. Wong, D. Wang, R. T. Benoit, P. Berthol, “A comparison of low scatter PSD derived from multiple wavelength BRDF’s and WYKO profilometer data,” in Optical Scatter: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. SPIE1530, 86–103 (1991).

Wyant, J. C.

J. C. Wyant, C. L. Koliopolous, B. Bhushan, D. Basila, “Development of a three-dimensional noncontact optical profiler,” J. Tribol. 108, 1–8 (1986). The TOPO instrument is manufactured by WYKO Corporation, East Elvira Road, Tucson, Arizona. Mention of this product does not constitute endorsement by CSIRO.
[CrossRef]

J. C. Wyant, K. Creath, “Basic wave-front theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1992), Vol. 11, pp. 1–53.

Yamamoto, H.

S. Whitcomb, G. Billingsley, J. Carri, A. Golovitser, D. Jungwirth, W. Kells, H. Yamamoto, B. Bochner, Y. Hefetz, P. Saha, R. Weiss, “Optics development for LIGO,” paper presented at the TAMA Workshop on Gravitational Wave Detection, Saitama, Japan, 11–12 November 1996; also published as . (Laser Interferometry Gravitational-Wave Observatory, California Institute of Technology, Pasadena, Calif., 1996).

Zucker, M.

A. Abramovici, W. Althouse, R. Drever, Y. Gursel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Appl. Opt. (4)

J. Tribol. (1)

J. C. Wyant, C. L. Koliopolous, B. Bhushan, D. Basila, “Development of a three-dimensional noncontact optical profiler,” J. Tribol. 108, 1–8 (1986). The TOPO instrument is manufactured by WYKO Corporation, East Elvira Road, Tucson, Arizona. Mention of this product does not constitute endorsement by CSIRO.
[CrossRef]

Opt. Eng. (2)

E. Spiller, R. A. McCorkle, J. S. Wilczynski, L. Golub, G. Nystrom, P. Z. Takacs, C. Welch, “Normal-incidence soft x-ray telescopes,” Opt. Eng. 30, 1109–1115 (1991).
[CrossRef]

J. E. Harvey, K. L. Lewotsky, A. Kotha, “Performance predictions of a Schwarzschild imaging microscope for soft x-ray applications,” Opt. Eng. 35, 2423–2436 (1996).
[CrossRef]

Science (1)

A. Abramovici, W. Althouse, R. Drever, Y. Gursel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. Spero, K. Thorne, R. Vogt, R. Weiss, S. Whitcomb, M. Zucker, “LIGO: the Laser Interferometer Gravitational-Wave Observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Wear (1)

E. L. Church, P. Z. Takacs, “Use of an optical-profiling instrument for the measurement of figure and finish of optical quality surfaces,” Wear 109, 241–257 (1986).
[CrossRef]

Other (18)

Talysurf (Rank Taylor Hobson Pty. Ltd., Leicester, UK). Mention of this product does not constitute endorsement by CSIRO.

K. Creath, “Error sources in phase measuring interferometry,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation, J. Tsujiuchi, eds., Proc. SPIE1720, 428–435 (1992).
[CrossRef]

“Standard practice for estimating the power spectral density function and related finish parameters from surface profile data,” (American Society for Testing and Materials, Pittsburgh, Pa., 1998).

S. Whitcomb, G. Billingsley, J. Carri, A. Golovitser, D. Jungwirth, W. Kells, H. Yamamoto, B. Bochner, Y. Hefetz, P. Saha, R. Weiss, “Optics development for LIGO,” paper presented at the TAMA Workshop on Gravitational Wave Detection, Saitama, Japan, 11–12 November 1996; also published as . (Laser Interferometry Gravitational-Wave Observatory, California Institute of Technology, Pasadena, Calif., 1996).

J. M. Bennett, L. Mattson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

H. Toebben, G. Ringel, F. Kratz, D.-R. Schmitt, “Use of power spectral density (PSD) to specify optical surfaces,” in Specification, Production, and Testing of Optical Components and Systems, A. E. Gee, J. Houee, eds., Proc. SPIE2775, 240–250 (1996).
[CrossRef]

J. K. Lawson, D. M. Aikens, R. E. English, C. R. Wolfe, “Power spectral density specifications for high-power laser systems,” in Specifications, Production, and Testing of Optical Components and Systems, A. E. Gee, J. Houee, eds., Proc. SPIE2775, 345–356 (1996).
[CrossRef]

G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, Calif., 1969).

V. T. Vorburger, C. J. Evans, V. W. Tsai, J. Fu, E. C. Williams, R. G. Dixson, P. J. Sullivan, T. McWaid, “Finish and figure metrology for soft x-ray optics,” in Proceedings of Japan Society for Precision Engineering U.S.-Japan Workshop on Soft X-ray Optics: Technical Challenges, T. Namioka, H. Kinoshita, K. Ito, eds. (Japan Society for Precision Engineering, Tokyo, 1997), p. 298.

W. Wong, D. Wang, R. T. Benoit, P. Berthol, “A comparison of low scatter PSD derived from multiple wavelength BRDF’s and WYKO profilometer data,” in Optical Scatter: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. SPIE1530, 86–103 (1991).

E. Marx, I. J. Malik, Y. E. Strauser, T. C. Bristow, N. S. Poduje, J. C. Stover, “Round robin determination of power spectral densities of different Si wafer surfaces,” in Flatness, Roughness, and Discrete Defects Characterization for Computer Disks, Wafers, and Flat Panel Displays II, J. C. Stover, ed., Proc. SPIE3275, 26–36 (1998).
[CrossRef]

R. Weiss, “Spectral analysis of coated optics phase maps,” (Laser Interferometer Gravitational-wave Observatory, California Institute of Technology, Pasadena, Calif., 1997); this report is available from the LIGO on-line document index at http://www.ligo.caltech.edu .

P. S. Fairman, B. K. Ward, B. F. Oreb, D. I. Farrant, Y. Gilliand, C. H. Freund, A. J. Leistner, J. A. Seckold, C. J. Walsh, “A 300-mm aperture phase shifting fizeau interferometer,” Opt. Eng. (to be published).

vision software, Version 1.8 [Veeco Corporation (formerly WYKO Corporation), East Elvira Road, Tucson, Ariz., 1996). Mention of this product does not constitute endorsement by CSIRO.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

E. L. Church, P. Z. Takacs, “The optimal estimation of finish parameters,” in Optical Scatter: Applications, Measurements, and Theory, J. C. Stover, ed., Proc. SPIE1530, 71–86 (1991).

W. D. Dixon, F. J. Massey, Introduction to Statistical Analysis (McGraw-Hill, New York, 1957).

J. C. Wyant, K. Creath, “Basic wave-front theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1992), Vol. 11, pp. 1–53.

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Figures (10)

Fig. 1
Fig. 1

Gray-scale phase maps of the front surface of an ITM. The aperture is 240 mm. On the left-hand side is the surface as measured, whereas on the right-hand side is the phase map for the surface with power, astigmatism, third-order coma, and spherical aberrations removed. The scale on the right-hand side of each image shows the surface height.

Fig. 2
Fig. 2

Data sets collected with the TOPO 40× (top) and 2.5× (bottom) objectives. Tilt and curvature have been removed from these data sets as per the discussion in Subsection 3.C.

Fig. 3
Fig. 3

(a) PSD calculated from the average of 12 PSD’s that were measured at different locations across the surface of an ETM. (b) PSD calculated by the windowing of the ACF of a single data set on the same surface with N/ K = 4, yielding 12 degrees of freedom according to Eq. (7). (c) Data from (a) and (b) overlaid on the same graph (solid curves). The uncertainty associated with an 80% confidence limit for the PSD of (a) is represented by the dotted curves. The units of the PSD are in meters cubed.

Fig. 4
Fig. 4

PSD’s calculated from TOPO 40× data with 33 degrees of freedom (dotted curve) and 500 degrees of freedom (solid curve) and the associated spectral resolutions of 3 and 48 cm-1, respectively. The standard deviations associated with an 80% confidence level are shown as vertical lines at the top right.

Fig. 5
Fig. 5

PSD spectra for an ITM polished substrate with and without astigmatism; coma and spherical aberrations were removed. The power was removed from the data set prior to analysis. The rms of the surface variations was 0.84 nm (terms included) compared with 0.53 nm (terms removed).

Fig. 6
Fig. 6

Ratio of the response of the WYKO TOPO 2.5× objective compared with that of a Talysurf scanning-stylus instrument. The noise is both instrumental (principally in the Talysurf device) and the result of the response functions’ for each instrument being derived from measurements at differing locations on the substrate. An analytic response function [expression (10)] is shown for comparison, with ν 0 = 350 cm-1. The rise in the response curve above ∼800 cm-1 is not well fitted by the analytic curve and possibly is due to the TOPO response’s becoming noise limited.

Fig. 7
Fig. 7

Average of (a) 10 x-chord scans and (b) 10 radial scans extracted from the phase map measured by the LADI.

Fig. 8
Fig. 8

PSD functions calculated from the LADI data for (a) four ETM surfaces, (b) four FM surfaces, (c) three RM surfaces. The 80% confidence limits are shown in the top right-hand side of each graph.

Fig. 9
Fig. 9

PSD’s calculated for two ETM surfaces, two RM surfaces, and two FM surfaces overlaid on the same graph: (a) results for the 2.5× objective, (b) results for the 40× objective.

Fig. 10
Fig. 10

PSD functions for (a) a FM, (b) the RM, (c) the ETM, (d) the ITM for which the results from the LADI and the TOPO measurements (2.5× and 40×) were combined, with corrections as discussed in Subsection 4.A. The uncertainty associated with the 80% confidence limit is shown for each plot.

Tables (1)

Tables Icon

Table 1 Values of the Exponent n in Eq. (12) Obtained by Fitting of an Analytic Curve to the PSD Curves from the LIGO and the TOPO 2.5× and 40× Dataa

Equations (13)

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

acfiest=1Nk=0N-1-|i| zkzk+|i|,
PSDνest=δl i=-N-1i=N-1acfiestwiexpj2πνiδl,
wi=1-|i|KiK0otherwise.
FFTacfw; q=12N1/2i=-N-1N-1×expj2πiq/Nacfi wi.
PSDνqest=FFTacfw; q δl 2N1/2,
νq=qNδl.
σ12=1Nδlq=q1q=q2PSDνqest1/2.
1Ni=0N-1zi2=q=1N/2PSDνqΔν,
n=3 NK.
b=32K δl.
PSDcν=PSDmν1-νd0sinπνdNπνdN-1,
Rν1-νν02,
PSDν=Aνn.

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