Abstract

Commercial software in modern interferometers used in optical testing frequently fit the wave-front or surface-figure error to Zernike polynomials; typically 37 coefficients are provided. We provide visual representations of these data in a form that may help optical fabricators decide how to improve their process or how to optimize system assembly.

© 1995 Optical Society of America

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References

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  1. P. Z. Takacs, E. L. Church, “Figure and finish characterization of high performance metal mirrors,” in Proceedings of the ASPE Spring Topical Meeting on Metal Plating for Precision Finishing Operations (American Society of Precision Engineering, Raleigh, N.C., 1991) pp. 110–117.
  2. E. L. Church, P. Z. Takacs, “Specification of surface figure and finish in terms of system performance,” Appl. Opt. 32, 3344–3353 (1993).
    [CrossRef] [PubMed]
  3. E. Spiller, R. A. McCorkle, L. Golub, G. Nystrom, P. Z. Takacs, C. Welch, “Normal incidence soft x-ray telescopes,” Opt. Eng. 30, 1109–1115 (1991).
    [CrossRef]
  4. P. R. Reid, A. L. Nonnemacher, “Alternative set of surface descriptors for grazing incidence optics,” Opt. Eng. 29, 637–640 (1990).
    [CrossRef]
  5. K. J. Stout, P. J. Sullivan, W. P. Dong, E. Mainsah, N. Luo, T. Mathia, H. Zahouani, “The development of methods for the characterization of roughness in three dimensions,” EC Rep. EUR 15178 EN (Commission of the European Communities, Dissemination of Scientific and Technical Knowledge Unit, Directorate-General Information Technologies and Industries and Telecommunications, Luxembourg, 1993).
  6. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1993).
  7. V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” in Eng. Lab. News, Suppl. to Opt. Photon. News 5, (Nov.1995).
  8. W. Swantner, W. W. Chow, “Gram–Schmidt orthonormalization of Zernike polynomials for general aperture shapes,” Appl. Opt. 33, 1832–1837 (1994).
    [CrossRef] [PubMed]
  9. J. Wyant, K. Creath, “Basic wavefront aberration theory for optical metrology,” in Vol. 11 of Applied Optics and Optical Engineering Series, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1992), pp. 1–53.
  10. C-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Vol. 10 of Applied Optics and Optical Engineering Series, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987), pp. 193–221.
  11. J. Loomis, Fringe Users Manual (Optical Sciences Center, University of Arizona, Tucson, Ariz.); D. Anderson, Fringe Manual Version 3 (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1982).
  12. J. Y. Wang, D. E. Silva, “Wavefront interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [CrossRef] [PubMed]
  13. F. Zernike, Lawrence Livermore National Laboratory, Livermore, Calif. 94550 (personal communication, 1994).
  14. One anonymous reviewer of the manuscript of this paper promoted an alternative terminology using, for example, C31 and S31 for the sine and cosine terms of coma and Rn for rotationally symmetric terms of maximum order n. The authors have some sympathy with the reviewer’s suggestion that this terminology offers unambiguous communication of the characteristics of the polynomial term under discussion.
  15. C. J. Evans, “Cryogenic diamond turning of stainless steel,” CIRP Ann. Int. Inst. Prod. Eng. Res. 40, 511–575 (1991).

1995 (1)

V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” in Eng. Lab. News, Suppl. to Opt. Photon. News 5, (Nov.1995).

1994 (1)

1993 (1)

1991 (2)

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

C. J. Evans, “Cryogenic diamond turning of stainless steel,” CIRP Ann. Int. Inst. Prod. Eng. Res. 40, 511–575 (1991).

1990 (1)

P. R. Reid, A. L. Nonnemacher, “Alternative set of surface descriptors for grazing incidence optics,” Opt. Eng. 29, 637–640 (1990).
[CrossRef]

1980 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1993).

Chow, W. W.

Church, E. L.

E. L. Church, P. Z. Takacs, “Specification of surface figure and finish in terms of system performance,” Appl. Opt. 32, 3344–3353 (1993).
[CrossRef] [PubMed]

P. Z. Takacs, E. L. Church, “Figure and finish characterization of high performance metal mirrors,” in Proceedings of the ASPE Spring Topical Meeting on Metal Plating for Precision Finishing Operations (American Society of Precision Engineering, Raleigh, N.C., 1991) pp. 110–117.

Creath, K.

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

Dong, W. P.

K. J. Stout, P. J. Sullivan, W. P. Dong, E. Mainsah, N. Luo, T. Mathia, H. Zahouani, “The development of methods for the characterization of roughness in three dimensions,” EC Rep. EUR 15178 EN (Commission of the European Communities, Dissemination of Scientific and Technical Knowledge Unit, Directorate-General Information Technologies and Industries and Telecommunications, Luxembourg, 1993).

Evans, C. J.

C. J. Evans, “Cryogenic diamond turning of stainless steel,” CIRP Ann. Int. Inst. Prod. Eng. Res. 40, 511–575 (1991).

Golub, L.

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

Kim, C-J.

C-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Vol. 10 of Applied Optics and Optical Engineering Series, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987), pp. 193–221.

Loomis, J.

J. Loomis, Fringe Users Manual (Optical Sciences Center, University of Arizona, Tucson, Ariz.); D. Anderson, Fringe Manual Version 3 (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1982).

Luo, N.

K. J. Stout, P. J. Sullivan, W. P. Dong, E. Mainsah, N. Luo, T. Mathia, H. Zahouani, “The development of methods for the characterization of roughness in three dimensions,” EC Rep. EUR 15178 EN (Commission of the European Communities, Dissemination of Scientific and Technical Knowledge Unit, Directorate-General Information Technologies and Industries and Telecommunications, Luxembourg, 1993).

Mahajan, V. N.

V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” in Eng. Lab. News, Suppl. to Opt. Photon. News 5, (Nov.1995).

Mainsah, E.

K. J. Stout, P. J. Sullivan, W. P. Dong, E. Mainsah, N. Luo, T. Mathia, H. Zahouani, “The development of methods for the characterization of roughness in three dimensions,” EC Rep. EUR 15178 EN (Commission of the European Communities, Dissemination of Scientific and Technical Knowledge Unit, Directorate-General Information Technologies and Industries and Telecommunications, Luxembourg, 1993).

Mathia, T.

K. J. Stout, P. J. Sullivan, W. P. Dong, E. Mainsah, N. Luo, T. Mathia, H. Zahouani, “The development of methods for the characterization of roughness in three dimensions,” EC Rep. EUR 15178 EN (Commission of the European Communities, Dissemination of Scientific and Technical Knowledge Unit, Directorate-General Information Technologies and Industries and Telecommunications, Luxembourg, 1993).

McCorkle, R. A.

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

Nonnemacher, A. L.

P. R. Reid, A. L. Nonnemacher, “Alternative set of surface descriptors for grazing incidence optics,” Opt. Eng. 29, 637–640 (1990).
[CrossRef]

Nystrom, G.

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

Reid, P. R.

P. R. Reid, A. L. Nonnemacher, “Alternative set of surface descriptors for grazing incidence optics,” Opt. Eng. 29, 637–640 (1990).
[CrossRef]

Shannon, R. R.

C-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Vol. 10 of Applied Optics and Optical Engineering Series, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987), pp. 193–221.

Silva, D. E.

Spiller, E.

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

Stout, K. J.

K. J. Stout, P. J. Sullivan, W. P. Dong, E. Mainsah, N. Luo, T. Mathia, H. Zahouani, “The development of methods for the characterization of roughness in three dimensions,” EC Rep. EUR 15178 EN (Commission of the European Communities, Dissemination of Scientific and Technical Knowledge Unit, Directorate-General Information Technologies and Industries and Telecommunications, Luxembourg, 1993).

Sullivan, P. J.

K. J. Stout, P. J. Sullivan, W. P. Dong, E. Mainsah, N. Luo, T. Mathia, H. Zahouani, “The development of methods for the characterization of roughness in three dimensions,” EC Rep. EUR 15178 EN (Commission of the European Communities, Dissemination of Scientific and Technical Knowledge Unit, Directorate-General Information Technologies and Industries and Telecommunications, Luxembourg, 1993).

Swantner, W.

Takacs, P. Z.

E. L. Church, P. Z. Takacs, “Specification of surface figure and finish in terms of system performance,” Appl. Opt. 32, 3344–3353 (1993).
[CrossRef] [PubMed]

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

P. Z. Takacs, E. L. Church, “Figure and finish characterization of high performance metal mirrors,” in Proceedings of the ASPE Spring Topical Meeting on Metal Plating for Precision Finishing Operations (American Society of Precision Engineering, Raleigh, N.C., 1991) pp. 110–117.

Wang, J. Y.

Welch, C.

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

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1993).

Wyant, J.

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

Zahouani, H.

K. J. Stout, P. J. Sullivan, W. P. Dong, E. Mainsah, N. Luo, T. Mathia, H. Zahouani, “The development of methods for the characterization of roughness in three dimensions,” EC Rep. EUR 15178 EN (Commission of the European Communities, Dissemination of Scientific and Technical Knowledge Unit, Directorate-General Information Technologies and Industries and Telecommunications, Luxembourg, 1993).

Zernike, F.

F. Zernike, Lawrence Livermore National Laboratory, Livermore, Calif. 94550 (personal communication, 1994).

Appl. Opt. (3)

CIRP Ann. Int. Inst. Prod. Eng. Res. (1)

C. J. Evans, “Cryogenic diamond turning of stainless steel,” CIRP Ann. Int. Inst. Prod. Eng. Res. 40, 511–575 (1991).

Eng. Lab. News, Suppl. to Opt. Photon. News (1)

V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” in Eng. Lab. News, Suppl. to Opt. Photon. News 5, (Nov.1995).

Opt. Eng. (2)

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

P. R. Reid, A. L. Nonnemacher, “Alternative set of surface descriptors for grazing incidence optics,” Opt. Eng. 29, 637–640 (1990).
[CrossRef]

Other (8)

K. J. Stout, P. J. Sullivan, W. P. Dong, E. Mainsah, N. Luo, T. Mathia, H. Zahouani, “The development of methods for the characterization of roughness in three dimensions,” EC Rep. EUR 15178 EN (Commission of the European Communities, Dissemination of Scientific and Technical Knowledge Unit, Directorate-General Information Technologies and Industries and Telecommunications, Luxembourg, 1993).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1993).

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

C-J. Kim, R. R. Shannon, “Catalog of Zernike polynomials,” in Vol. 10 of Applied Optics and Optical Engineering Series, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1987), pp. 193–221.

J. Loomis, Fringe Users Manual (Optical Sciences Center, University of Arizona, Tucson, Ariz.); D. Anderson, Fringe Manual Version 3 (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1982).

P. Z. Takacs, E. L. Church, “Figure and finish characterization of high performance metal mirrors,” in Proceedings of the ASPE Spring Topical Meeting on Metal Plating for Precision Finishing Operations (American Society of Precision Engineering, Raleigh, N.C., 1991) pp. 110–117.

F. Zernike, Lawrence Livermore National Laboratory, Livermore, Calif. 94550 (personal communication, 1994).

One anonymous reviewer of the manuscript of this paper promoted an alternative terminology using, for example, C31 and S31 for the sine and cosine terms of coma and Rn for rotationally symmetric terms of maximum order n. The authors have some sympathy with the reviewer’s suggestion that this terminology offers unambiguous communication of the characteristics of the polynomial term under discussion.

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Figures (8)

Fig. 1
Fig. 1

Surface figure of part A reconstructed from fitted Zernike polynomials.

Fig. 2
Fig. 2

Surface figure of part B.

Fig. 3
Fig. 3

Coefficients for part A (a) positioned according to their radial and azimuthal orders, (b) projected onto the XZ and the YZ planes.

Fig. 4
Fig. 4

Part A, rms surface contributions.

Fig. 5
Fig. 5

Part A, angular position of first peak.

Fig. 6
Fig. 6

Part B, rms surface contributions.

Fig. 7
Fig. 7

Part B, angular position of first peak.

Fig. 8
Fig. 8

Part B, residual after 36 Zernikes are fitted.

Equations (9)

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

z = a n m ( R n m cos m θ ) , z = a n - m ( R n - m sin m θ )
z = a n 0 ( R n 0 )
k = [ 2 ( n + 1 ) ] 1 / 2 ,
k = ( n + 1 ) 1 / 2 .
z = a n m f ( r ) cos m θ + a n - m f ( r ) sin m θ ,
d z / d θ = - m a n m f ( r ) sin m θ + m a n - m f ( r ) cos m θ = 0 ,
sin m θ / cos m θ = a n - m / a n m ,
θ = ( 1 / m ) arctan ( a n - m / a n m ) .
( 26 ) 1 / 2 r 12 cos 12 θ , ( 26 ) 1 / 2 r 12 sin 12 θ .

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