Abstract

There are a variety of common situations in which specification of a one-dimensional modulation transfer function (MTF) or two orthogonal profiles of the 2D MTF are not adequate descriptions of the image quality performance of an optical system. These include systems with an asymmetric on-axis impulse response, systems with off-axis aberrations, systems with surfaces that include mid-spatial frequency errors, and freeform systems. In this paper, we develop the concept of the Minimum Modulation Curve (MMC). Starting with the two-dimensional MTF in polar form, the minimum MTF for any azimuth angle is plotted as a function of the radial spatial frequency. This can be presented in a familiar form similar to an MTF curve and is useful in the context of guaranteeing that a given MTF specification is met for any possible orientation of spatial frequencies in the image. In this way, an MMC may be of value in specifying the required performance of an optical system. We illustrate application of the MMC using profile data for surfaces with mid-spatial frequency errors.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. J. W. Goodman, Introduction to Fourier Optics, (Roberts and Company, 2005).
  2. H. Aryan and T. J. Suleski, “Non-directional modulation transfer function for optical surfaces with asymmetric mid-spatial frequency errors,” in Optical Design and Fabrication 2019 (Freeform, OFT), OSA Technical Digest (online) (Optical Society of America, 2019), paper OT1A.2.
  3. O. Hadar, A. Dogariu, and G. D. Boreman, “Angular dependence of sampling modulation transfer function,” Appl. Opt. 36(28), 7210–7216 (1997).
    [Crossref] [PubMed]
  4. R. Rhorer and C. Evans, “Fabrication of optics by diamond turning,” in Handbook of Optics, Third Edition Volume II: Design, Fabrication and Testing, Sources and Detectors, Radiometry and Photometry, M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, and E. Van Stryland, eds. (McGraw-Hill, 2009), Chap. 10.
  5. W. B. Lee, C. F. Cheung, and S. To, “Materials-induced vibration in single point diamond turning,” in Machining Dynamics, K. Cheng, ed. (Springer, 2009), pp. 263–282.
  6. C. R. Dunn and D. D. Walker, “Pseudo-random tool paths for CNC sub-aperture polishing and other applications,” Opt. Express 16(23), 18942–18949 (2008).
    [Crossref] [PubMed]
  7. A. Sohn, L. Lamonds, and K. Garrard, “Modeling of Vibration in Single-Point Diamond Turning,” in Proceedings of the American Society of Precision Engineering (ASPE, 2006), pp. 15–20.
  8. H. Aryan, K. Liang, M. A. Alonso, and T. J. Suleski, “Predictive models for the Strehl ratio of diamond-machined optics,” Appl. Opt. 58(12), 3272–3276 (2019).
    [Crossref] [PubMed]
  9. G. W. Forbes, “Never-ending struggles with mid-spatial frequencies,” Proc. SPIE 9525, 95251B (2015).
  10. Z. Hosseinimakarem, H. Aryan, A. Davies, and C. Evans, “Considering a Zernike polynomial representation for spatial frequency content of optical surfaces,” in Imaging and Applied Optics 2015, OSA Technical Digest (online) (Optical Society of America, 2015), paper FT2B.2.
  11. J. A. Shultz, H. Aryan, J. D. Owen, M. A. Davies, and T. J. Suleski, “Impacts of sub-aperture manufacturing techniques on the performance of freeform optics,” in Proceedings ASPE/ASPEN Spring Topical Meeting: Manufacture and Metrology of Structured and Freeform Surfaces for Functional Applications (2017), paper 0061.
  12. E. L. Church and J. M. Zavada, “Residual surface roughness of diamond-turned optics,” Appl. Opt. 14(8), 1788–1795 (1975).
    [Crossref] [PubMed]
  13. J. C. Stover, “Roughness characterization of smooth machined surfaces by light scattering,” Appl. Opt. 14(8), 1796–1802 (1975).
    [Crossref] [PubMed]
  14. J. P. Marioge and S. Slansky, “Effect of figure and waviness on image quality,” J. Opt. 14(4), 189–198 (1983).
    [Crossref]
  15. J. M. Tamkin, T. D. Milster, and W. Dallas, “Theory of modulation transfer function artifacts due to mid-spatial-frequency errors and its application to optical tolerancing,” Appl. Opt. 49(25), 4825–4835 (2010).
    [Crossref] [PubMed]
  16. J. M. Tamkin, “A Study of Image Artifacts Caused by Structured Mid-spatial Frequency Fabrication Errors on Optical Surfaces,” dissertation, The University of Arizona, (2010).
  17. J. Filhaber, “Mid-spatial-frequency errors: the hidden culprit of poor optical performance,” Laser Focus World 49(8), 32 (2013).
  18. M. A. Alonso and G. W. Forbes, “Strehl ratio as the Fourier transform of a probability density of error differences,” Opt. Lett. 41(16), 3735–3738 (2016).
    [Crossref] [PubMed]
  19. F. Tinker and K. Xin, “Correlation of mid-spatial features to image performance in aspheric mirrors,” Proc. SPIE 8837, 88370N (2013).
    [Crossref]
  20. G. Boreman, Modulation Transfer Function in Optical and Electro-Optical Systems (SPIE Press, 2001).
  21. T. S. Ross, “Limitations and applicability of the Maréchal approximation,” Appl. Opt. 48(10), 1812–1818 (2009).
    [Crossref] [PubMed]
  22. H. Aryan, C. J. Evans, and T. J. Suleski, “On the Use of ISO 10110-8 for Specification of Optical Surfaces with Mid-Spatial Frequency Errors,” in Optical Design and Fabrication 2017 (Freeform, IODC, OFT), OSA Technical Digest (online) (Optical Society of America, 2017), paper OW4B.2.
  23. W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000), pp. 383–385.
  24. B. Dube, R. Cicala, A. Closz, and J. P. Rolland, “How good is your lens? Assessing performance with MTF full-field displays,” Appl. Opt. 56(20), 5661–5667 (2017).
    [Crossref] [PubMed]

2019 (1)

2017 (1)

2016 (1)

2015 (1)

G. W. Forbes, “Never-ending struggles with mid-spatial frequencies,” Proc. SPIE 9525, 95251B (2015).

2013 (2)

F. Tinker and K. Xin, “Correlation of mid-spatial features to image performance in aspheric mirrors,” Proc. SPIE 8837, 88370N (2013).
[Crossref]

J. Filhaber, “Mid-spatial-frequency errors: the hidden culprit of poor optical performance,” Laser Focus World 49(8), 32 (2013).

2010 (1)

2009 (1)

2008 (1)

1997 (1)

1983 (1)

J. P. Marioge and S. Slansky, “Effect of figure and waviness on image quality,” J. Opt. 14(4), 189–198 (1983).
[Crossref]

1975 (2)

Alonso, M. A.

Aryan, H.

H. Aryan, K. Liang, M. A. Alonso, and T. J. Suleski, “Predictive models for the Strehl ratio of diamond-machined optics,” Appl. Opt. 58(12), 3272–3276 (2019).
[Crossref] [PubMed]

J. A. Shultz, H. Aryan, J. D. Owen, M. A. Davies, and T. J. Suleski, “Impacts of sub-aperture manufacturing techniques on the performance of freeform optics,” in Proceedings ASPE/ASPEN Spring Topical Meeting: Manufacture and Metrology of Structured and Freeform Surfaces for Functional Applications (2017), paper 0061.

Boreman, G. D.

Church, E. L.

Cicala, R.

Closz, A.

Dallas, W.

Davies, M. A.

J. A. Shultz, H. Aryan, J. D. Owen, M. A. Davies, and T. J. Suleski, “Impacts of sub-aperture manufacturing techniques on the performance of freeform optics,” in Proceedings ASPE/ASPEN Spring Topical Meeting: Manufacture and Metrology of Structured and Freeform Surfaces for Functional Applications (2017), paper 0061.

Dogariu, A.

Dube, B.

Dunn, C. R.

Filhaber, J.

J. Filhaber, “Mid-spatial-frequency errors: the hidden culprit of poor optical performance,” Laser Focus World 49(8), 32 (2013).

Forbes, G. W.

Hadar, O.

Liang, K.

Marioge, J. P.

J. P. Marioge and S. Slansky, “Effect of figure and waviness on image quality,” J. Opt. 14(4), 189–198 (1983).
[Crossref]

Milster, T. D.

Owen, J. D.

J. A. Shultz, H. Aryan, J. D. Owen, M. A. Davies, and T. J. Suleski, “Impacts of sub-aperture manufacturing techniques on the performance of freeform optics,” in Proceedings ASPE/ASPEN Spring Topical Meeting: Manufacture and Metrology of Structured and Freeform Surfaces for Functional Applications (2017), paper 0061.

Rolland, J. P.

Ross, T. S.

Shultz, J. A.

J. A. Shultz, H. Aryan, J. D. Owen, M. A. Davies, and T. J. Suleski, “Impacts of sub-aperture manufacturing techniques on the performance of freeform optics,” in Proceedings ASPE/ASPEN Spring Topical Meeting: Manufacture and Metrology of Structured and Freeform Surfaces for Functional Applications (2017), paper 0061.

Slansky, S.

J. P. Marioge and S. Slansky, “Effect of figure and waviness on image quality,” J. Opt. 14(4), 189–198 (1983).
[Crossref]

Stover, J. C.

Suleski, T. J.

H. Aryan, K. Liang, M. A. Alonso, and T. J. Suleski, “Predictive models for the Strehl ratio of diamond-machined optics,” Appl. Opt. 58(12), 3272–3276 (2019).
[Crossref] [PubMed]

J. A. Shultz, H. Aryan, J. D. Owen, M. A. Davies, and T. J. Suleski, “Impacts of sub-aperture manufacturing techniques on the performance of freeform optics,” in Proceedings ASPE/ASPEN Spring Topical Meeting: Manufacture and Metrology of Structured and Freeform Surfaces for Functional Applications (2017), paper 0061.

Tamkin, J. M.

Tinker, F.

F. Tinker and K. Xin, “Correlation of mid-spatial features to image performance in aspheric mirrors,” Proc. SPIE 8837, 88370N (2013).
[Crossref]

Walker, D. D.

Xin, K.

F. Tinker and K. Xin, “Correlation of mid-spatial features to image performance in aspheric mirrors,” Proc. SPIE 8837, 88370N (2013).
[Crossref]

Zavada, J. M.

Appl. Opt. (7)

J. Opt. (1)

J. P. Marioge and S. Slansky, “Effect of figure and waviness on image quality,” J. Opt. 14(4), 189–198 (1983).
[Crossref]

Laser Focus World (1)

J. Filhaber, “Mid-spatial-frequency errors: the hidden culprit of poor optical performance,” Laser Focus World 49(8), 32 (2013).

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (2)

F. Tinker and K. Xin, “Correlation of mid-spatial features to image performance in aspheric mirrors,” Proc. SPIE 8837, 88370N (2013).
[Crossref]

G. W. Forbes, “Never-ending struggles with mid-spatial frequencies,” Proc. SPIE 9525, 95251B (2015).

Other (11)

Z. Hosseinimakarem, H. Aryan, A. Davies, and C. Evans, “Considering a Zernike polynomial representation for spatial frequency content of optical surfaces,” in Imaging and Applied Optics 2015, OSA Technical Digest (online) (Optical Society of America, 2015), paper FT2B.2.

J. A. Shultz, H. Aryan, J. D. Owen, M. A. Davies, and T. J. Suleski, “Impacts of sub-aperture manufacturing techniques on the performance of freeform optics,” in Proceedings ASPE/ASPEN Spring Topical Meeting: Manufacture and Metrology of Structured and Freeform Surfaces for Functional Applications (2017), paper 0061.

R. Rhorer and C. Evans, “Fabrication of optics by diamond turning,” in Handbook of Optics, Third Edition Volume II: Design, Fabrication and Testing, Sources and Detectors, Radiometry and Photometry, M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, and E. Van Stryland, eds. (McGraw-Hill, 2009), Chap. 10.

W. B. Lee, C. F. Cheung, and S. To, “Materials-induced vibration in single point diamond turning,” in Machining Dynamics, K. Cheng, ed. (Springer, 2009), pp. 263–282.

J. W. Goodman, Introduction to Fourier Optics, (Roberts and Company, 2005).

H. Aryan and T. J. Suleski, “Non-directional modulation transfer function for optical surfaces with asymmetric mid-spatial frequency errors,” in Optical Design and Fabrication 2019 (Freeform, OFT), OSA Technical Digest (online) (Optical Society of America, 2019), paper OT1A.2.

G. Boreman, Modulation Transfer Function in Optical and Electro-Optical Systems (SPIE Press, 2001).

A. Sohn, L. Lamonds, and K. Garrard, “Modeling of Vibration in Single-Point Diamond Turning,” in Proceedings of the American Society of Precision Engineering (ASPE, 2006), pp. 15–20.

H. Aryan, C. J. Evans, and T. J. Suleski, “On the Use of ISO 10110-8 for Specification of Optical Surfaces with Mid-Spatial Frequency Errors,” in Optical Design and Fabrication 2017 (Freeform, IODC, OFT), OSA Technical Digest (online) (Optical Society of America, 2017), paper OW4B.2.

W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2000), pp. 383–385.

J. M. Tamkin, “A Study of Image Artifacts Caused by Structured Mid-spatial Frequency Fabrication Errors on Optical Surfaces,” dissertation, The University of Arizona, (2010).

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Figures (8)

Fig. 1
Fig. 1 Illustrating the methodology of extracting data for each spatial frequency from a general 2D-MTF. The minimum value around each circle is extracted to generate the MMC for a given value of radial spatial frequency ρ.
Fig. 2
Fig. 2 With reference to the 2D MTF in Fig. 1(a), the minimum modulation curve (dash dot red) and horizontal MTF cross section (blue), (b) MTF standard deviation at each radial spatial frequency.
Fig. 3
Fig. 3 Two diamond machined surfaces with the same fabrication parameters; Diamond tool cusp errors: a tool-tip radius of 1 mm and Λ = 40 µm; Sinusoidal error of 1 cycle/mm with peak to valley (PV) of 150 nm to represent thermal errors. (a) Diamond turned. (b) Diamond milled.
Fig. 4
Fig. 4 Rayleigh-Sommerfeld simulations of PSF for the above examples for the (a) perfect lens, (b) diamond turned lens, (c) diamond milled lens.
Fig. 5
Fig. 5 2D-MTF simulations for the (a) perfect lens, (b) diamond turned lens, (c) diamond milled lens. Red color represents 1 and blue color represents 0 modulation in these figures.
Fig. 6
Fig. 6 Comparing (a) horizontal and (b) vertical cross section of the 2D-MTF for the diamond milled case from Fig. 3(b).
Fig. 7
Fig. 7 With reference to the 2D MTF in Fig. 5(c). (a) the minimum modulation curve (dash dot red) and horizontal MTF cross section (solid blue), (b) MTF standard deviation at each radial spatial frequency.
Fig. 8
Fig. 8 (a) Measured surface error with RMS of 44 nm over a 127 mm clear aperture. (b) Simulated on-axis 2D MTF for a lens with this surface error. (c) Comparing MMC (dash dot red) with horizontal cross section of MTF at ϕ = 0° (solid blue) and ϕ = 90° (dash green). (d) MTF standard deviation at each radial spatial frequency.

Equations (4)

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MTF( f x , f y )=| PSF(x,y) e j2π(x f x +y f y ) dxdy |,
MMC(ρ)= min ϕ[0,2π] { MTF(ρ,ϕ) },
σ MTF (ρ)= n=1 N [ MTF(ρ, ϕ n ) n=1 N MTF(ρ, ϕ n ) N ] 2 N ,
SR=exp[ (kΔnσ) 2 ],

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