Abstract

Mid-spatial-frequency surface errors can be introduced by various manufacturing processes. These errors bridge the gap between traditional figure and finish errors. Although the effects of mid-spatial-frequency errors on the imagery of an optical system can be modeled with a ray-based approach, simply tracing rays provides little insight. We present an alternative method that treats surface errors as perturbations to the nominal surface profile. This approach, combined with standard statistical methods, allows one to make simple back-of-the-envelope predictions of the effects of mid-spatial-frequency errors for various measures of optical performance. Two examples illustrating the effectiveness of this approach are presented.

© 2000 Optical Society of America

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References

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  1. A. Tabenkin, “Surface finish: a machinist’s tool. A design necessity,” Modern Machine Shop, 71 (March1999), pp. 80–88.
  2. Such a categorization of errors appears, for example, in W. B. Wetherell, “Effects of mirror surface ripple on image quality,” in International Conference on Advanced Technology Optical Telescopes, G. R. Burbidge, L. D. Barr, eds., Proc. SPIE332, 335–351 (1982).
    [CrossRef]
  3. For example, this terminology is used in Ref. 1 and in R. S. Hahn, R. P. Lindsay, “Principles of grinding … part V: grinding chatter,” Machinery Magazine (November1971), pp. 48–53.
  4. J. K. Lawson, R. C. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aiken, R. E. English, “Specification of optical components using the power spectral density function,” in Optical Manufacturing and Testing, V. J. Doherty, H. P. Stahl, eds., Proc. SPIE2536, 38–50 (1995).
    [CrossRef]
  5. D. M. Aikens, C. R. Wolfe, J. K. Lawson, “The use of power spectral density (PSD) functions in specifying optics for the National Ignition Facility,” in International Conference on Optical Fabrication and Testing, T. Kasai, ed., Proc. SPIE2576, 281–292 (1995).
    [CrossRef]
  6. J. E. Harvey, A. Kotha, “Scattering effects from residual optical fabrication errors,” in International Conference on Optical Fabrication and Testing, T. Kasai, ed., Proc. SPIE2576, 155–174 (1995).
    [CrossRef]
  7. J. E. Harvey, C. L. Vernold, “Transfer function characterization of scattering surfaces: revisited,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 113–127 (1997).
    [CrossRef]
  8. C. L. Vernold, J. E. Harvey, “Comparison of Harvey–Shack scatter theory with experimental measurements,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 128–138 (1997).
    [CrossRef]
  9. J. E. Harvey, C. L. Vernold, “Modifying the Harvey–Shack surface scatter theory,” in Scattering and Surface Roughness II, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 326–332 (1998).
    [CrossRef]
  10. A description of Fermat’s principle can be found, for example, in E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Sec. 4.2.4.
  11. B. D. Stone, “Perturbations of optical systems,” J. Opt. Soc. Am. A 14, 2837–2849 (1997).
    [CrossRef]
  12. M. P. Rimmer, “Analysis of perturbed lens systems,” Appl. Opt. 9, 533–537 (1970).
    [CrossRef] [PubMed]
  13. H. H. Hopkins, H. J. Tiziani, “A theoretical and experimental study of lens centring errors and their influence on optical image quality,” Brit. J. Appl. Phys. 17, 33–54 (1966).
    [CrossRef]
  14. A discussion of the wave aberration function can be found in W. T. Welford, Aberrations of Optical Systems (Adam Hilger, New York, 1991), Chap. 7.
  15. R. J. Noll, “Effect of mid- and high-spatial frequencies on optical performance,” Opt. Eng. 18, 137–142 (1979).
    [CrossRef]
  16. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Sec. 8.3.
  17. See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6.
  18. The expression for the PSF is treated in more detail in Refs. 15 and 16.
  19. The starting lens (with only spherical surfaces) is the superachromat documented in Fig. 14 of R. D. Sigler, “Glass selection for airspaced apochromats using the Buchdahl dispersion equation,” Appl. Opt. 25, 4311–4320 (1986). This lens is also described in W. J. Smith, Modern Lens Design: A Resource Manual (McGraw-Hill, New York, 1992), Fig. 10.5. For this example, aspheres were added to two of the surfaces and the lens reoptimized. During optimization, we only varied the aspheric coefficients. The figure of merit used for optimization was the mean-square wave-front error at the helium d-line (λ = 587.6 nm).
    [CrossRef]
  20. See, for example, M. Laikin, Lens Design (Marcel Dekker, New York, 1995), Sec. 1.3.
  21. For a summary of this procedure, see G. H. Spencer, M. V. R. K. Murty, “General ray-tracing procedure,” J. Opt. Soc. Am. 52, 672–678 (1962).
    [CrossRef]
  22. The lens is taken from T. P. Fjeldsted, “Four element infrared objective lens,” U.S. patent4,380,363 (19April1983). This lens is also described in W. J. Smith, Modern Lens Design: A Resource Manual (McGraw-Hill, New York, 1992), Fig. 21.9.
  23. The phase maps used here are obtained with a commercial interferometer. They were spatial filtered to eliminate low-spatial-frequency errors and then scaled to the same rms value.
  24. OSLO is a registered trademark of Sinclair Optics, Inc., 6780 Palmyra Road, Fairport, N.Y. 14450.

1997 (1)

1986 (1)

1979 (1)

R. J. Noll, “Effect of mid- and high-spatial frequencies on optical performance,” Opt. Eng. 18, 137–142 (1979).
[CrossRef]

1970 (1)

1966 (1)

H. H. Hopkins, H. J. Tiziani, “A theoretical and experimental study of lens centring errors and their influence on optical image quality,” Brit. J. Appl. Phys. 17, 33–54 (1966).
[CrossRef]

1962 (1)

Aiken, D. M.

J. K. Lawson, R. C. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aiken, R. E. English, “Specification of optical components using the power spectral density function,” in Optical Manufacturing and Testing, V. J. Doherty, H. P. Stahl, eds., Proc. SPIE2536, 38–50 (1995).
[CrossRef]

Aikens, D. M.

D. M. Aikens, C. R. Wolfe, J. K. Lawson, “The use of power spectral density (PSD) functions in specifying optics for the National Ignition Facility,” in International Conference on Optical Fabrication and Testing, T. Kasai, ed., Proc. SPIE2576, 281–292 (1995).
[CrossRef]

English, R. E.

J. K. Lawson, R. C. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aiken, R. E. English, “Specification of optical components using the power spectral density function,” in Optical Manufacturing and Testing, V. J. Doherty, H. P. Stahl, eds., Proc. SPIE2536, 38–50 (1995).
[CrossRef]

Fjeldsted, T. P.

The lens is taken from T. P. Fjeldsted, “Four element infrared objective lens,” U.S. patent4,380,363 (19April1983). This lens is also described in W. J. Smith, Modern Lens Design: A Resource Manual (McGraw-Hill, New York, 1992), Fig. 21.9.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Sec. 8.3.

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6.

Hahn, R. S.

For example, this terminology is used in Ref. 1 and in R. S. Hahn, R. P. Lindsay, “Principles of grinding … part V: grinding chatter,” Machinery Magazine (November1971), pp. 48–53.

Harvey, J. E.

J. E. Harvey, A. Kotha, “Scattering effects from residual optical fabrication errors,” in International Conference on Optical Fabrication and Testing, T. Kasai, ed., Proc. SPIE2576, 155–174 (1995).
[CrossRef]

J. E. Harvey, C. L. Vernold, “Transfer function characterization of scattering surfaces: revisited,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 113–127 (1997).
[CrossRef]

C. L. Vernold, J. E. Harvey, “Comparison of Harvey–Shack scatter theory with experimental measurements,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 128–138 (1997).
[CrossRef]

J. E. Harvey, C. L. Vernold, “Modifying the Harvey–Shack surface scatter theory,” in Scattering and Surface Roughness II, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 326–332 (1998).
[CrossRef]

Hecht, E.

A description of Fermat’s principle can be found, for example, in E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Sec. 4.2.4.

Hopkins, H. H.

H. H. Hopkins, H. J. Tiziani, “A theoretical and experimental study of lens centring errors and their influence on optical image quality,” Brit. J. Appl. Phys. 17, 33–54 (1966).
[CrossRef]

Kotha, A.

J. E. Harvey, A. Kotha, “Scattering effects from residual optical fabrication errors,” in International Conference on Optical Fabrication and Testing, T. Kasai, ed., Proc. SPIE2576, 155–174 (1995).
[CrossRef]

Laikin, M.

See, for example, M. Laikin, Lens Design (Marcel Dekker, New York, 1995), Sec. 1.3.

Lawson, J. K.

D. M. Aikens, C. R. Wolfe, J. K. Lawson, “The use of power spectral density (PSD) functions in specifying optics for the National Ignition Facility,” in International Conference on Optical Fabrication and Testing, T. Kasai, ed., Proc. SPIE2576, 281–292 (1995).
[CrossRef]

J. K. Lawson, R. C. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aiken, R. E. English, “Specification of optical components using the power spectral density function,” in Optical Manufacturing and Testing, V. J. Doherty, H. P. Stahl, eds., Proc. SPIE2536, 38–50 (1995).
[CrossRef]

Lindsay, R. P.

For example, this terminology is used in Ref. 1 and in R. S. Hahn, R. P. Lindsay, “Principles of grinding … part V: grinding chatter,” Machinery Magazine (November1971), pp. 48–53.

Manes, K. R.

J. K. Lawson, R. C. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aiken, R. E. English, “Specification of optical components using the power spectral density function,” in Optical Manufacturing and Testing, V. J. Doherty, H. P. Stahl, eds., Proc. SPIE2536, 38–50 (1995).
[CrossRef]

Murty, M. V. R. K.

Noll, R. J.

R. J. Noll, “Effect of mid- and high-spatial frequencies on optical performance,” Opt. Eng. 18, 137–142 (1979).
[CrossRef]

Rimmer, M. P.

Sigler, R. D.

Spencer, G. H.

Stone, B. D.

Tabenkin, A.

A. Tabenkin, “Surface finish: a machinist’s tool. A design necessity,” Modern Machine Shop, 71 (March1999), pp. 80–88.

Tiziani, H. J.

H. H. Hopkins, H. J. Tiziani, “A theoretical and experimental study of lens centring errors and their influence on optical image quality,” Brit. J. Appl. Phys. 17, 33–54 (1966).
[CrossRef]

Trenholme, J. B.

J. K. Lawson, R. C. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aiken, R. E. English, “Specification of optical components using the power spectral density function,” in Optical Manufacturing and Testing, V. J. Doherty, H. P. Stahl, eds., Proc. SPIE2536, 38–50 (1995).
[CrossRef]

Vernold, C. L.

J. E. Harvey, C. L. Vernold, “Transfer function characterization of scattering surfaces: revisited,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 113–127 (1997).
[CrossRef]

C. L. Vernold, J. E. Harvey, “Comparison of Harvey–Shack scatter theory with experimental measurements,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 128–138 (1997).
[CrossRef]

J. E. Harvey, C. L. Vernold, “Modifying the Harvey–Shack surface scatter theory,” in Scattering and Surface Roughness II, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 326–332 (1998).
[CrossRef]

Welford, W. T.

A discussion of the wave aberration function can be found in W. T. Welford, Aberrations of Optical Systems (Adam Hilger, New York, 1991), Chap. 7.

Wetherell, W. B.

Such a categorization of errors appears, for example, in W. B. Wetherell, “Effects of mirror surface ripple on image quality,” in International Conference on Advanced Technology Optical Telescopes, G. R. Burbidge, L. D. Barr, eds., Proc. SPIE332, 335–351 (1982).
[CrossRef]

Wolfe, C. R.

D. M. Aikens, C. R. Wolfe, J. K. Lawson, “The use of power spectral density (PSD) functions in specifying optics for the National Ignition Facility,” in International Conference on Optical Fabrication and Testing, T. Kasai, ed., Proc. SPIE2576, 281–292 (1995).
[CrossRef]

Wolfe, R. C.

J. K. Lawson, R. C. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aiken, R. E. English, “Specification of optical components using the power spectral density function,” in Optical Manufacturing and Testing, V. J. Doherty, H. P. Stahl, eds., Proc. SPIE2536, 38–50 (1995).
[CrossRef]

Appl. Opt. (2)

Brit. J. Appl. Phys. (1)

H. H. Hopkins, H. J. Tiziani, “A theoretical and experimental study of lens centring errors and their influence on optical image quality,” Brit. J. Appl. Phys. 17, 33–54 (1966).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

R. J. Noll, “Effect of mid- and high-spatial frequencies on optical performance,” Opt. Eng. 18, 137–142 (1979).
[CrossRef]

Other (18)

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Sec. 8.3.

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6.

The expression for the PSF is treated in more detail in Refs. 15 and 16.

See, for example, M. Laikin, Lens Design (Marcel Dekker, New York, 1995), Sec. 1.3.

The lens is taken from T. P. Fjeldsted, “Four element infrared objective lens,” U.S. patent4,380,363 (19April1983). This lens is also described in W. J. Smith, Modern Lens Design: A Resource Manual (McGraw-Hill, New York, 1992), Fig. 21.9.

The phase maps used here are obtained with a commercial interferometer. They were spatial filtered to eliminate low-spatial-frequency errors and then scaled to the same rms value.

OSLO is a registered trademark of Sinclair Optics, Inc., 6780 Palmyra Road, Fairport, N.Y. 14450.

A. Tabenkin, “Surface finish: a machinist’s tool. A design necessity,” Modern Machine Shop, 71 (March1999), pp. 80–88.

Such a categorization of errors appears, for example, in W. B. Wetherell, “Effects of mirror surface ripple on image quality,” in International Conference on Advanced Technology Optical Telescopes, G. R. Burbidge, L. D. Barr, eds., Proc. SPIE332, 335–351 (1982).
[CrossRef]

For example, this terminology is used in Ref. 1 and in R. S. Hahn, R. P. Lindsay, “Principles of grinding … part V: grinding chatter,” Machinery Magazine (November1971), pp. 48–53.

J. K. Lawson, R. C. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aiken, R. E. English, “Specification of optical components using the power spectral density function,” in Optical Manufacturing and Testing, V. J. Doherty, H. P. Stahl, eds., Proc. SPIE2536, 38–50 (1995).
[CrossRef]

D. M. Aikens, C. R. Wolfe, J. K. Lawson, “The use of power spectral density (PSD) functions in specifying optics for the National Ignition Facility,” in International Conference on Optical Fabrication and Testing, T. Kasai, ed., Proc. SPIE2576, 281–292 (1995).
[CrossRef]

J. E. Harvey, A. Kotha, “Scattering effects from residual optical fabrication errors,” in International Conference on Optical Fabrication and Testing, T. Kasai, ed., Proc. SPIE2576, 155–174 (1995).
[CrossRef]

J. E. Harvey, C. L. Vernold, “Transfer function characterization of scattering surfaces: revisited,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 113–127 (1997).
[CrossRef]

C. L. Vernold, J. E. Harvey, “Comparison of Harvey–Shack scatter theory with experimental measurements,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 128–138 (1997).
[CrossRef]

J. E. Harvey, C. L. Vernold, “Modifying the Harvey–Shack surface scatter theory,” in Scattering and Surface Roughness II, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 326–332 (1998).
[CrossRef]

A description of Fermat’s principle can be found, for example, in E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), Sec. 4.2.4.

A discussion of the wave aberration function can be found in W. T. Welford, Aberrations of Optical Systems (Adam Hilger, New York, 1991), Chap. 7.

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Figures (11)

Fig. 1
Fig. 1

Nominal and perturbed surfaces in a system. Rays from a given object point that pass through a given point on the pupil are also illustrated for both the nominal and the perturbed surface.

Fig. 2
Fig. 2

Telephoto lens used in the first example.

Fig. 3
Fig. 3

Through-frequency MTF’s plotted for the nominal telephoto system of the first example. The Strehl ratios for the axial (S 1), 0.7-field (S 2), and full-field (S 3) points are indicated.

Fig. 4
Fig. 4

Height maps for the algebraic surface errors: (a) spokes, rms of 0.17 λ; (b) rings, rms of 0.18 λ; (c) both rings and spokes, rms = 0.12 λ. The rms error associated with the various error patterns are also indicated for the case in which the peak-to-valley amplitude of the error is λ/2 (λ = 587.6 nm).

Fig. 5
Fig. 5

MTF plots for various errors applied to the first and the last surfaces of the telephoto lens. The actual Strehl ratios for the axial (S 1), 0.7-field (S 2), and full-field (S 3) points are also given with the corresponding predicted value in parentheses.

Fig. 6
Fig. 6

IR objective lens.

Fig. 7
Fig. 7

Nominal IR objective lens performance: (a) Through-frequency MTF plots and Strehl ratios for the axial (S 1), 0.7-field (S 2), and full-field (S 3) points. (b) EE plots.

Fig. 8
Fig. 8

Surface error height maps utilized in the second example. The peak-to-valley error is indicated for the case in which the rms values are all scaled to λ/200 (at the design wavelength of 3.7 µm).

Fig. 9
Fig. 9

Through-frequency MTF plots and Strehl ratios (predicted values in parentheses) of the perturbed IR objective. The results for three field points are shown on separate plots: (a) axial-field point, (b) 0.7-field point, (c) full-field point.

Fig. 10
Fig. 10

EE plots for the perturbed IR objective. (a) Axial-field point, (b) 0.7-field point, (c) full-field point.

Fig. 11
Fig. 11

Plots of the axial (S 1) and the full-field (S 3) Strehl ratios versus a change in the amplitude of the patterns on the surfaces of the IR objective.

Tables (1)

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Table 1 Aspheric Coefficients for the Telephoto Lens

Equations (33)

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N=-f/x, -f/y, 1,
Wih, ρ=ni cosθih, ρ-ni cosθih, ρεiyih, ρ+Oεi2.
Wh, ρ=W0h, ρ+Wmidh, ρ,
Wmidh, ρ=i=1N Wih, ρ,
σW2h := W-W¯2¯=W2¯-W¯2,
q¯ := pupil qdρpupildρ.
σW2h=W02¯+ i=1NWi2¯+2i=1NW0Wi¯+2i=1Nj=i+1NWiWj¯-W0¯+i=1NWi¯2.
σW2h=W02¯+i=1NWi2¯+2i=1Nj=i+1NWiWj¯-W0¯2.
σW2h=W02¯+i=1NWi2¯-W0¯2=σW, 02h+i=1NWi2¯.
σφ2h=2πλ2i=1NWi2¯.
Wh, ρ=W0h, ρexpiφh, ρ.
Hv= WρW*ρ-λfvdρ|Wρ|2dρ,
Hv= W0ρW0*ρ-λfvexpiφρ-φρ-λfvdρ|W0ρ|2dρ.
Hv= W0ρW0*ρ-λfvdρ|W0ρ|2dρ×expiφρ-φρ-λfv
=H0vHrv,
H0v= W0ρW0*ρ-λfvdρ |W0ρ|2dρ,
Hrv=expiφρ-φρ-λfv.
Hv=H0vexp-σφ21-γφλfv,
γφΔx, Δy :=  φx, yφx-Δx, y-Δydxdy φ2x, ydxdy.
σφ2h=2πλ2×i=1Nni cosθih, ρ-ni cosθih, ρ2εi2yih, ρ¯.
ni cosθih, ρ-ni cosθih, ρ2
σφ2h=2πλ2i=1Nni cosθ¯ih-ni cosθ¯ih2×εi2yih, ρ¯,
σφ2h2πλ2i=1Nni-ni2σi2,
σi2 := clear aperture εi2yidxidyiclear aperturedxidyi.
HvH0vexp-2πλ2i=1Nni-ni2σi2.
hx, yh0x, yexp-2πλ2i=1Nni-ni2σi2,
Lx, y; R=φ=02πr=0R rhx+r cos φ, y+r sin φdφdr.
Lx, y; RL0x, y; R×exp-2πλ2i=1Nni-ni2σi2,
z=cr21+1-c2r21/2+Ar4+Br6+Cr8+Dr10,
εspoke=12 ε1 cosη1tan-1yx+x2+y21/2ymax θ1+φ1×1-exp-x2+y2/w2,
εring=12 ε2 cosη22π sin-1c2x2+y21/2sin-1|cymax|+φ2,
12 ε1 cosη1tan-1yx+x2+y21/2ymax θ1+φ1.
1-exp-x2+y2/w2,

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