Abstract

Optical design and tolerancing of aspheric or free-form surfaces require attention to surface form, structured surface errors, and nonstructured errors. We describe structured surface error profiles and effects on the image point-spread function using harmonic (Fourier) decomposition. Surface errors over the beam footprint map onto the pupil, where multiple structured surface frequencies mix to create sum and difference diffraction orders in the image plane at each field point. Difference frequencies widen the central lobe of the point-spread function and summation frequencies create ghost images.

© 2010 Optical Society of America

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  1. Y. E. Tohme, “Grinding aspheric and freeform micro-optical molds,” Proc. SPIE 6462, 64620K (2007).
    [CrossRef]
  2. D. Aikens, J. E. DeGroote, and R. N. Youngworth, “Specification and control of mid-spatial frequency wavefront errors in optical systems,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA1.
  3. A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE 7018, 701835 (2008).
    [CrossRef]
  4. H. Choi, W.-C. Kim, S.-H. Lee, N.-C. Park, and Y.-P. Park, “Effects of fabrication errors in the diffractive optical element on the modulation transfer function of a hybrid lens,” J. Opt. Soc. Am. A 25, 2764-2766 (2008).
    [CrossRef]
  5. M. Pfaff, “High-speed fabrication of aspheres and optical free-form surfaces,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OThD6.
  6. P. Murphy, “Methods and challenges in quantifying mid-spatial frequencies,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA3.
  7. S. Wang, S. To, C. F. Cheung, and W. B. Lee, “A study of the scallop generation mechanism in ultra-precision raster milling,” Key Eng. Mater. 364-366, 1262-1267 (2008).
    [CrossRef]
  8. Y. Tohme, “Trends in ultra-precision machining of freeform optical surfaces,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OThC6.
  9. R. N. Youngworth and B. D. Stone, “Simple estimates for the effects of mid-spatial-frequency surface errors on image quality,” Appl. Opt. 39, 2198-2209 (2000).
    [CrossRef]
  10. M. Schulz, I. Weingaertner, C. Elster, and J. Gerhardt, “Low- and mid-spatial-frequency component measurement for aspheres,” Proc. SPIE 5188, 287-295 (2003),
    [CrossRef]
  11. M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32, 19-23 (1980).
    [CrossRef]
  12. J. M. Tamkin, “A study of image artifacts caused by structured mid-spatial frequency fabrication errors on optical surfaces,” Ph.D. dissertation (University of Arizona, 2010).
  13. J. W. Goodman, Introduction to Fourier optics, 3rd ed., (Roberts & Co., 2005).
  14. Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399-416 (1907).
    [CrossRef]
  15. Harmonic decomposition is also called Fourier decomposition. The former usage will be maintained to prevent confusion with Fourier transform.
  16. E. Marx, T. A. Germer, T. V. Vorburger, and B. C. Park, “Angular distribution of light scattered from a sinusoidal grating,” Appl. Opt. 39, 4473-4485 (2000).
    [CrossRef]
  17. J. E. Harvey, A. Krywonos, and D. Bogunovic, “Nonparaxial scalar treatment of sinusoidal phase gratings,” J. Opt. Soc. Am. A 23, 858-865 (2006).
    [CrossRef]
  18. V. Greco, G. Molesini, and F. Quercioli, “Telescopes of Galileo,” Appl. Opt. 32, 6219-6226 (1993).
    [CrossRef] [PubMed]
  19. R. Smith, A Compleat System of Opticks in Four Books, viz A Popular, a Mathematical, a Mechanical and Philosophical Treatise (Cambridge University Press, 1738).
  20. F. Twyman, Prism and Lens Making (Hilger & Watts, 1952).
  21. T. T. Saito, “Machining of optics: an introduction,” Appl. Opt. 14, 1773-1776 (1975).
    [CrossRef] [PubMed]
  22. R. J. Noll, “Effect of mid and high spatial frequencies on optical performance,” Opt. Eng. 18, 137-142 (1979).
  23. U. Birnbaum, H. Bernitzki, O. Falkenstörfer, H. Lauth, R. Schreiner, and T. Waak, “Manufacturing of high-precision aspheres,” Proc. SPIE 6149, 61490H (2006).
    [CrossRef]
  24. S. To, H. Wang, B. Li, and C. F. Cheung, “An empirical approach to the identification of sources of machining errors in ultra-precision raster milling,” Key Eng. Mater. 364-366, 986-991 (2008).
    [CrossRef]
  25. Z. Q. Yin, S. To, and W. B. Lee, “Wear characteristics of diamond tool in ultraprecision raster milling,” Int. J. Adv. Manuf. Technol. 44, 638-647 (2009).
    [CrossRef]
  26. S. Rakuff and P. Beaudet, “Thermal and structural deformations during diamond turning of rotationally symmetric structured surfaces,” J. Manuf. Sci. Eng. 130, 041004 (2008).
    [CrossRef]
  27. M. L. Barkman, B. S. Dutterer, M. A. Davies, and T. J. Suleski, “Free-form machining for micro-imaging systems,” Proc. SPIE , 6883, 68830G (2008).
    [CrossRef]
  28. M. N. Cheng, C. F. Cheung, and W. B. Lee, “A study of factors affecting surface quality in ultra-precision raster milling,” Key Eng. Mater. 339, 400-406 (2007).
    [CrossRef]
  29. G. W. Forbes and C. P. Brophy, “Asphere, O asphere, how shall we describe thee?,” Proc. SPIE 7100, 710002 (2008).
    [CrossRef]
  30. J. R. Rogers, “Slope error tolerances for optical surfaces,” presented at the OptiFab Conference (SPIE, 2008), paper TD04-4, pp. 15-17.
  31. E. L. Church and J. M. Zavada, “Residual surface roughness of diamond-turned optics,” Appl. Opt. 14, 1788-1795 (1975).
    [CrossRef] [PubMed]
  32. J. C. Stover and J. E. Harvey, “Limitations of Rayleigh Rice perturbation theory for describing surface scatter,” Proc. SPIE 6672, 66720B (2007).
    [CrossRef]
  33. S. Shikama, “Effects of corrugation of aspherical mirrors on the optical performance of imaging optics with the mirror near an image plane,” Opt. Eng. 43, 3068-3076 (2004).
    [CrossRef]
  34. M. Shibuya, N. Watanabe, M. Yamamoto, T. Fukui, H. Ezaki, T. Kiire, and S. Nakadate, “Classification of undulated wavefront aberration in projection optics by considering its physical effects,” Opt. Eng. 46, 053001 (2007).
    [CrossRef]
  35. R. C. Juergens, R. H. Shepard III, and J. P. Schaefer, “Simulation of single-point diamond turning fabrication process errors,” Proc. SPIE 5174, 93-104 (2003).
    [CrossRef]
  36. T. B. A. Senior, “Scattering of electromagnetic waves by a corrugated sheet,” Can. J. Phys. 37, 787-797 (1959).
    [CrossRef]
  37. R. C. McPhedran and D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” J. Mod. Opt. 21, 413-421 (1974).
    [CrossRef]
  38. A. Wirgin, “Scattering from sinusoidal gratings: an evaluation of the Kirchhoff approximation,” J. Opt. Soc. Am. 73, 1028-1041 (1983).
    [CrossRef]
  39. M. J. Lighthill, Introduction to Fourier Analysis and Generalised Functions, Cambridge Monographs on Mechanics and Applied Mathematics (Cambridge University Press, 1958).
  40. H. H. Barrett, Foundations of Image Science, H.H.Barrett and K.J.Myers, eds. (Wiley-Interscience, 2004).
  41. J. Mathews and R. L. Walker, Mathematical Methods of Physics, 2nd ed. (W. A. Benjamin, 1970).
  42. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th expanded ed. (Cambridge University Press, 1999).
    [PubMed]
  43. C. An, Q. Xu, F. Zhang, and J. Zhang, “Calculation and structural analysis for the rigidity of air spindle in the single point diamond turning lathe,” Proc. SPIE 6722, 67222U (2007).
    [CrossRef]

2010

J. M. Tamkin, “A study of image artifacts caused by structured mid-spatial frequency fabrication errors on optical surfaces,” Ph.D. dissertation (University of Arizona, 2010).

2009

Z. Q. Yin, S. To, and W. B. Lee, “Wear characteristics of diamond tool in ultraprecision raster milling,” Int. J. Adv. Manuf. Technol. 44, 638-647 (2009).
[CrossRef]

2008

S. Rakuff and P. Beaudet, “Thermal and structural deformations during diamond turning of rotationally symmetric structured surfaces,” J. Manuf. Sci. Eng. 130, 041004 (2008).
[CrossRef]

M. L. Barkman, B. S. Dutterer, M. A. Davies, and T. J. Suleski, “Free-form machining for micro-imaging systems,” Proc. SPIE , 6883, 68830G (2008).
[CrossRef]

D. Aikens, J. E. DeGroote, and R. N. Youngworth, “Specification and control of mid-spatial frequency wavefront errors in optical systems,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA1.

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE 7018, 701835 (2008).
[CrossRef]

M. Pfaff, “High-speed fabrication of aspheres and optical free-form surfaces,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OThD6.

P. Murphy, “Methods and challenges in quantifying mid-spatial frequencies,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA3.

S. Wang, S. To, C. F. Cheung, and W. B. Lee, “A study of the scallop generation mechanism in ultra-precision raster milling,” Key Eng. Mater. 364-366, 1262-1267 (2008).
[CrossRef]

Y. Tohme, “Trends in ultra-precision machining of freeform optical surfaces,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OThC6.

G. W. Forbes and C. P. Brophy, “Asphere, O asphere, how shall we describe thee?,” Proc. SPIE 7100, 710002 (2008).
[CrossRef]

J. R. Rogers, “Slope error tolerances for optical surfaces,” presented at the OptiFab Conference (SPIE, 2008), paper TD04-4, pp. 15-17.

S. To, H. Wang, B. Li, and C. F. Cheung, “An empirical approach to the identification of sources of machining errors in ultra-precision raster milling,” Key Eng. Mater. 364-366, 986-991 (2008).
[CrossRef]

H. Choi, W.-C. Kim, S.-H. Lee, N.-C. Park, and Y.-P. Park, “Effects of fabrication errors in the diffractive optical element on the modulation transfer function of a hybrid lens,” J. Opt. Soc. Am. A 25, 2764-2766 (2008).
[CrossRef]

2007

C. An, Q. Xu, F. Zhang, and J. Zhang, “Calculation and structural analysis for the rigidity of air spindle in the single point diamond turning lathe,” Proc. SPIE 6722, 67222U (2007).
[CrossRef]

J. C. Stover and J. E. Harvey, “Limitations of Rayleigh Rice perturbation theory for describing surface scatter,” Proc. SPIE 6672, 66720B (2007).
[CrossRef]

M. Shibuya, N. Watanabe, M. Yamamoto, T. Fukui, H. Ezaki, T. Kiire, and S. Nakadate, “Classification of undulated wavefront aberration in projection optics by considering its physical effects,” Opt. Eng. 46, 053001 (2007).
[CrossRef]

Y. E. Tohme, “Grinding aspheric and freeform micro-optical molds,” Proc. SPIE 6462, 64620K (2007).
[CrossRef]

M. N. Cheng, C. F. Cheung, and W. B. Lee, “A study of factors affecting surface quality in ultra-precision raster milling,” Key Eng. Mater. 339, 400-406 (2007).
[CrossRef]

2006

U. Birnbaum, H. Bernitzki, O. Falkenstörfer, H. Lauth, R. Schreiner, and T. Waak, “Manufacturing of high-precision aspheres,” Proc. SPIE 6149, 61490H (2006).
[CrossRef]

J. E. Harvey, A. Krywonos, and D. Bogunovic, “Nonparaxial scalar treatment of sinusoidal phase gratings,” J. Opt. Soc. Am. A 23, 858-865 (2006).
[CrossRef]

2005

J. W. Goodman, Introduction to Fourier optics, 3rd ed., (Roberts & Co., 2005).

2004

S. Shikama, “Effects of corrugation of aspherical mirrors on the optical performance of imaging optics with the mirror near an image plane,” Opt. Eng. 43, 3068-3076 (2004).
[CrossRef]

H. H. Barrett, Foundations of Image Science, H.H.Barrett and K.J.Myers, eds. (Wiley-Interscience, 2004).

2003

R. C. Juergens, R. H. Shepard III, and J. P. Schaefer, “Simulation of single-point diamond turning fabrication process errors,” Proc. SPIE 5174, 93-104 (2003).
[CrossRef]

M. Schulz, I. Weingaertner, C. Elster, and J. Gerhardt, “Low- and mid-spatial-frequency component measurement for aspheres,” Proc. SPIE 5188, 287-295 (2003),
[CrossRef]

2000

1999

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th expanded ed. (Cambridge University Press, 1999).
[PubMed]

1993

1983

1980

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32, 19-23 (1980).
[CrossRef]

1979

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

1975

1974

R. C. McPhedran and D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” J. Mod. Opt. 21, 413-421 (1974).
[CrossRef]

1970

J. Mathews and R. L. Walker, Mathematical Methods of Physics, 2nd ed. (W. A. Benjamin, 1970).

1959

T. B. A. Senior, “Scattering of electromagnetic waves by a corrugated sheet,” Can. J. Phys. 37, 787-797 (1959).
[CrossRef]

1958

M. J. Lighthill, Introduction to Fourier Analysis and Generalised Functions, Cambridge Monographs on Mechanics and Applied Mathematics (Cambridge University Press, 1958).

1952

F. Twyman, Prism and Lens Making (Hilger & Watts, 1952).

1907

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399-416 (1907).
[CrossRef]

1738

R. Smith, A Compleat System of Opticks in Four Books, viz A Popular, a Mathematical, a Mechanical and Philosophical Treatise (Cambridge University Press, 1738).

Aikens, D.

D. Aikens, J. E. DeGroote, and R. N. Youngworth, “Specification and control of mid-spatial frequency wavefront errors in optical systems,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA1.

An, C.

C. An, Q. Xu, F. Zhang, and J. Zhang, “Calculation and structural analysis for the rigidity of air spindle in the single point diamond turning lathe,” Proc. SPIE 6722, 67222U (2007).
[CrossRef]

Barkman, M. L.

M. L. Barkman, B. S. Dutterer, M. A. Davies, and T. J. Suleski, “Free-form machining for micro-imaging systems,” Proc. SPIE , 6883, 68830G (2008).
[CrossRef]

Barrett, H. H.

H. H. Barrett, Foundations of Image Science, H.H.Barrett and K.J.Myers, eds. (Wiley-Interscience, 2004).

Beaucamp, A.

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE 7018, 701835 (2008).
[CrossRef]

Beaudet, P.

S. Rakuff and P. Beaudet, “Thermal and structural deformations during diamond turning of rotationally symmetric structured surfaces,” J. Manuf. Sci. Eng. 130, 041004 (2008).
[CrossRef]

Bernitzki, H.

U. Birnbaum, H. Bernitzki, O. Falkenstörfer, H. Lauth, R. Schreiner, and T. Waak, “Manufacturing of high-precision aspheres,” Proc. SPIE 6149, 61490H (2006).
[CrossRef]

Birnbaum, U.

U. Birnbaum, H. Bernitzki, O. Falkenstörfer, H. Lauth, R. Schreiner, and T. Waak, “Manufacturing of high-precision aspheres,” Proc. SPIE 6149, 61490H (2006).
[CrossRef]

Bogunovic, D.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th expanded ed. (Cambridge University Press, 1999).
[PubMed]

Brophy, C. P.

G. W. Forbes and C. P. Brophy, “Asphere, O asphere, how shall we describe thee?,” Proc. SPIE 7100, 710002 (2008).
[CrossRef]

Cheng, M. N.

M. N. Cheng, C. F. Cheung, and W. B. Lee, “A study of factors affecting surface quality in ultra-precision raster milling,” Key Eng. Mater. 339, 400-406 (2007).
[CrossRef]

Cheung, C. F.

S. Wang, S. To, C. F. Cheung, and W. B. Lee, “A study of the scallop generation mechanism in ultra-precision raster milling,” Key Eng. Mater. 364-366, 1262-1267 (2008).
[CrossRef]

S. To, H. Wang, B. Li, and C. F. Cheung, “An empirical approach to the identification of sources of machining errors in ultra-precision raster milling,” Key Eng. Mater. 364-366, 986-991 (2008).
[CrossRef]

M. N. Cheng, C. F. Cheung, and W. B. Lee, “A study of factors affecting surface quality in ultra-precision raster milling,” Key Eng. Mater. 339, 400-406 (2007).
[CrossRef]

Choi, H.

Church, E. L.

Davies, M. A.

M. L. Barkman, B. S. Dutterer, M. A. Davies, and T. J. Suleski, “Free-form machining for micro-imaging systems,” Proc. SPIE , 6883, 68830G (2008).
[CrossRef]

DeGroote, J. E.

D. Aikens, J. E. DeGroote, and R. N. Youngworth, “Specification and control of mid-spatial frequency wavefront errors in optical systems,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA1.

Dutterer, B. S.

M. L. Barkman, B. S. Dutterer, M. A. Davies, and T. J. Suleski, “Free-form machining for micro-imaging systems,” Proc. SPIE , 6883, 68830G (2008).
[CrossRef]

Elster, C.

M. Schulz, I. Weingaertner, C. Elster, and J. Gerhardt, “Low- and mid-spatial-frequency component measurement for aspheres,” Proc. SPIE 5188, 287-295 (2003),
[CrossRef]

Ezaki, H.

M. Shibuya, N. Watanabe, M. Yamamoto, T. Fukui, H. Ezaki, T. Kiire, and S. Nakadate, “Classification of undulated wavefront aberration in projection optics by considering its physical effects,” Opt. Eng. 46, 053001 (2007).
[CrossRef]

Falkenstörfer, O.

U. Birnbaum, H. Bernitzki, O. Falkenstörfer, H. Lauth, R. Schreiner, and T. Waak, “Manufacturing of high-precision aspheres,” Proc. SPIE 6149, 61490H (2006).
[CrossRef]

Forbes, G. W.

G. W. Forbes and C. P. Brophy, “Asphere, O asphere, how shall we describe thee?,” Proc. SPIE 7100, 710002 (2008).
[CrossRef]

Freeman, R.

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE 7018, 701835 (2008).
[CrossRef]

Fukui, T.

M. Shibuya, N. Watanabe, M. Yamamoto, T. Fukui, H. Ezaki, T. Kiire, and S. Nakadate, “Classification of undulated wavefront aberration in projection optics by considering its physical effects,” Opt. Eng. 46, 053001 (2007).
[CrossRef]

Gaylord, T. K.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32, 19-23 (1980).
[CrossRef]

Gerhardt, J.

M. Schulz, I. Weingaertner, C. Elster, and J. Gerhardt, “Low- and mid-spatial-frequency component measurement for aspheres,” Proc. SPIE 5188, 287-295 (2003),
[CrossRef]

Germer, T. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier optics, 3rd ed., (Roberts & Co., 2005).

Greco, V.

Harvey, J. E.

J. C. Stover and J. E. Harvey, “Limitations of Rayleigh Rice perturbation theory for describing surface scatter,” Proc. SPIE 6672, 66720B (2007).
[CrossRef]

J. E. Harvey, A. Krywonos, and D. Bogunovic, “Nonparaxial scalar treatment of sinusoidal phase gratings,” J. Opt. Soc. Am. A 23, 858-865 (2006).
[CrossRef]

Juergens, R. C.

R. C. Juergens, R. H. Shepard III, and J. P. Schaefer, “Simulation of single-point diamond turning fabrication process errors,” Proc. SPIE 5174, 93-104 (2003).
[CrossRef]

Kiire, T.

M. Shibuya, N. Watanabe, M. Yamamoto, T. Fukui, H. Ezaki, T. Kiire, and S. Nakadate, “Classification of undulated wavefront aberration in projection optics by considering its physical effects,” Opt. Eng. 46, 053001 (2007).
[CrossRef]

Kim, W.-C.

Krywonos, A.

Lauth, H.

U. Birnbaum, H. Bernitzki, O. Falkenstörfer, H. Lauth, R. Schreiner, and T. Waak, “Manufacturing of high-precision aspheres,” Proc. SPIE 6149, 61490H (2006).
[CrossRef]

Lee, S.-H.

Lee, W. B.

Z. Q. Yin, S. To, and W. B. Lee, “Wear characteristics of diamond tool in ultraprecision raster milling,” Int. J. Adv. Manuf. Technol. 44, 638-647 (2009).
[CrossRef]

S. Wang, S. To, C. F. Cheung, and W. B. Lee, “A study of the scallop generation mechanism in ultra-precision raster milling,” Key Eng. Mater. 364-366, 1262-1267 (2008).
[CrossRef]

M. N. Cheng, C. F. Cheung, and W. B. Lee, “A study of factors affecting surface quality in ultra-precision raster milling,” Key Eng. Mater. 339, 400-406 (2007).
[CrossRef]

Li, B.

S. To, H. Wang, B. Li, and C. F. Cheung, “An empirical approach to the identification of sources of machining errors in ultra-precision raster milling,” Key Eng. Mater. 364-366, 986-991 (2008).
[CrossRef]

Lighthill, M. J.

M. J. Lighthill, Introduction to Fourier Analysis and Generalised Functions, Cambridge Monographs on Mechanics and Applied Mathematics (Cambridge University Press, 1958).

Magnusson, R.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32, 19-23 (1980).
[CrossRef]

Marx, E.

Mathews, J.

J. Mathews and R. L. Walker, Mathematical Methods of Physics, 2nd ed. (W. A. Benjamin, 1970).

Maystre, D.

R. C. McPhedran and D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” J. Mod. Opt. 21, 413-421 (1974).
[CrossRef]

McPhedran, R. C.

R. C. McPhedran and D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” J. Mod. Opt. 21, 413-421 (1974).
[CrossRef]

Moharam, M. G.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32, 19-23 (1980).
[CrossRef]

Molesini, G.

Morton, R.

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE 7018, 701835 (2008).
[CrossRef]

Murphy, P.

P. Murphy, “Methods and challenges in quantifying mid-spatial frequencies,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA3.

Nakadate, S.

M. Shibuya, N. Watanabe, M. Yamamoto, T. Fukui, H. Ezaki, T. Kiire, and S. Nakadate, “Classification of undulated wavefront aberration in projection optics by considering its physical effects,” Opt. Eng. 46, 053001 (2007).
[CrossRef]

Noll, R. J.

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

Park, B. C.

Park, N.-C.

Park, Y.-P.

Pfaff, M.

M. Pfaff, “High-speed fabrication of aspheres and optical free-form surfaces,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OThD6.

Ponudurai, K.

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE 7018, 701835 (2008).
[CrossRef]

Quercioli, F.

Rakuff, S.

S. Rakuff and P. Beaudet, “Thermal and structural deformations during diamond turning of rotationally symmetric structured surfaces,” J. Manuf. Sci. Eng. 130, 041004 (2008).
[CrossRef]

Rayleigh, Lord

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399-416 (1907).
[CrossRef]

Rogers, J. R.

J. R. Rogers, “Slope error tolerances for optical surfaces,” presented at the OptiFab Conference (SPIE, 2008), paper TD04-4, pp. 15-17.

Saito, T. T.

Schaefer, J. P.

R. C. Juergens, R. H. Shepard III, and J. P. Schaefer, “Simulation of single-point diamond turning fabrication process errors,” Proc. SPIE 5174, 93-104 (2003).
[CrossRef]

Schreiner, R.

U. Birnbaum, H. Bernitzki, O. Falkenstörfer, H. Lauth, R. Schreiner, and T. Waak, “Manufacturing of high-precision aspheres,” Proc. SPIE 6149, 61490H (2006).
[CrossRef]

Schulz, M.

M. Schulz, I. Weingaertner, C. Elster, and J. Gerhardt, “Low- and mid-spatial-frequency component measurement for aspheres,” Proc. SPIE 5188, 287-295 (2003),
[CrossRef]

Senior, T. B. A.

T. B. A. Senior, “Scattering of electromagnetic waves by a corrugated sheet,” Can. J. Phys. 37, 787-797 (1959).
[CrossRef]

Shepard, R. H.

R. C. Juergens, R. H. Shepard III, and J. P. Schaefer, “Simulation of single-point diamond turning fabrication process errors,” Proc. SPIE 5174, 93-104 (2003).
[CrossRef]

Shibuya, M.

M. Shibuya, N. Watanabe, M. Yamamoto, T. Fukui, H. Ezaki, T. Kiire, and S. Nakadate, “Classification of undulated wavefront aberration in projection optics by considering its physical effects,” Opt. Eng. 46, 053001 (2007).
[CrossRef]

Shikama, S.

S. Shikama, “Effects of corrugation of aspherical mirrors on the optical performance of imaging optics with the mirror near an image plane,” Opt. Eng. 43, 3068-3076 (2004).
[CrossRef]

Smith, R.

R. Smith, A Compleat System of Opticks in Four Books, viz A Popular, a Mathematical, a Mechanical and Philosophical Treatise (Cambridge University Press, 1738).

Stone, B. D.

Stover, J. C.

J. C. Stover and J. E. Harvey, “Limitations of Rayleigh Rice perturbation theory for describing surface scatter,” Proc. SPIE 6672, 66720B (2007).
[CrossRef]

Suleski, T. J.

M. L. Barkman, B. S. Dutterer, M. A. Davies, and T. J. Suleski, “Free-form machining for micro-imaging systems,” Proc. SPIE , 6883, 68830G (2008).
[CrossRef]

Tamkin, J. M.

J. M. Tamkin, “A study of image artifacts caused by structured mid-spatial frequency fabrication errors on optical surfaces,” Ph.D. dissertation (University of Arizona, 2010).

To, S.

Z. Q. Yin, S. To, and W. B. Lee, “Wear characteristics of diamond tool in ultraprecision raster milling,” Int. J. Adv. Manuf. Technol. 44, 638-647 (2009).
[CrossRef]

S. To, H. Wang, B. Li, and C. F. Cheung, “An empirical approach to the identification of sources of machining errors in ultra-precision raster milling,” Key Eng. Mater. 364-366, 986-991 (2008).
[CrossRef]

S. Wang, S. To, C. F. Cheung, and W. B. Lee, “A study of the scallop generation mechanism in ultra-precision raster milling,” Key Eng. Mater. 364-366, 1262-1267 (2008).
[CrossRef]

Tohme, Y.

Y. Tohme, “Trends in ultra-precision machining of freeform optical surfaces,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OThC6.

Tohme, Y. E.

Y. E. Tohme, “Grinding aspheric and freeform micro-optical molds,” Proc. SPIE 6462, 64620K (2007).
[CrossRef]

Twyman, F.

F. Twyman, Prism and Lens Making (Hilger & Watts, 1952).

Vorburger, T. V.

Waak, T.

U. Birnbaum, H. Bernitzki, O. Falkenstörfer, H. Lauth, R. Schreiner, and T. Waak, “Manufacturing of high-precision aspheres,” Proc. SPIE 6149, 61490H (2006).
[CrossRef]

Walker, D. D.

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE 7018, 701835 (2008).
[CrossRef]

Walker, R. L.

J. Mathews and R. L. Walker, Mathematical Methods of Physics, 2nd ed. (W. A. Benjamin, 1970).

Wang, H.

S. To, H. Wang, B. Li, and C. F. Cheung, “An empirical approach to the identification of sources of machining errors in ultra-precision raster milling,” Key Eng. Mater. 364-366, 986-991 (2008).
[CrossRef]

Wang, S.

S. Wang, S. To, C. F. Cheung, and W. B. Lee, “A study of the scallop generation mechanism in ultra-precision raster milling,” Key Eng. Mater. 364-366, 1262-1267 (2008).
[CrossRef]

Watanabe, N.

M. Shibuya, N. Watanabe, M. Yamamoto, T. Fukui, H. Ezaki, T. Kiire, and S. Nakadate, “Classification of undulated wavefront aberration in projection optics by considering its physical effects,” Opt. Eng. 46, 053001 (2007).
[CrossRef]

Weingaertner, I.

M. Schulz, I. Weingaertner, C. Elster, and J. Gerhardt, “Low- and mid-spatial-frequency component measurement for aspheres,” Proc. SPIE 5188, 287-295 (2003),
[CrossRef]

Wirgin, A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th expanded ed. (Cambridge University Press, 1999).
[PubMed]

Xu, Q.

C. An, Q. Xu, F. Zhang, and J. Zhang, “Calculation and structural analysis for the rigidity of air spindle in the single point diamond turning lathe,” Proc. SPIE 6722, 67222U (2007).
[CrossRef]

Yamamoto, M.

M. Shibuya, N. Watanabe, M. Yamamoto, T. Fukui, H. Ezaki, T. Kiire, and S. Nakadate, “Classification of undulated wavefront aberration in projection optics by considering its physical effects,” Opt. Eng. 46, 053001 (2007).
[CrossRef]

Yin, Z. Q.

Z. Q. Yin, S. To, and W. B. Lee, “Wear characteristics of diamond tool in ultraprecision raster milling,” Int. J. Adv. Manuf. Technol. 44, 638-647 (2009).
[CrossRef]

Youngworth, R. N.

D. Aikens, J. E. DeGroote, and R. N. Youngworth, “Specification and control of mid-spatial frequency wavefront errors in optical systems,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA1.

R. N. Youngworth and B. D. Stone, “Simple estimates for the effects of mid-spatial-frequency surface errors on image quality,” Appl. Opt. 39, 2198-2209 (2000).
[CrossRef]

Zavada, J. M.

Zhang, F.

C. An, Q. Xu, F. Zhang, and J. Zhang, “Calculation and structural analysis for the rigidity of air spindle in the single point diamond turning lathe,” Proc. SPIE 6722, 67222U (2007).
[CrossRef]

Zhang, J.

C. An, Q. Xu, F. Zhang, and J. Zhang, “Calculation and structural analysis for the rigidity of air spindle in the single point diamond turning lathe,” Proc. SPIE 6722, 67222U (2007).
[CrossRef]

Appl. Opt.

Can. J. Phys.

T. B. A. Senior, “Scattering of electromagnetic waves by a corrugated sheet,” Can. J. Phys. 37, 787-797 (1959).
[CrossRef]

Int. J. Adv. Manuf. Technol.

Z. Q. Yin, S. To, and W. B. Lee, “Wear characteristics of diamond tool in ultraprecision raster milling,” Int. J. Adv. Manuf. Technol. 44, 638-647 (2009).
[CrossRef]

J. Manuf. Sci. Eng.

S. Rakuff and P. Beaudet, “Thermal and structural deformations during diamond turning of rotationally symmetric structured surfaces,” J. Manuf. Sci. Eng. 130, 041004 (2008).
[CrossRef]

J. Mod. Opt.

R. C. McPhedran and D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” J. Mod. Opt. 21, 413-421 (1974).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Key Eng. Mater.

M. N. Cheng, C. F. Cheung, and W. B. Lee, “A study of factors affecting surface quality in ultra-precision raster milling,” Key Eng. Mater. 339, 400-406 (2007).
[CrossRef]

S. To, H. Wang, B. Li, and C. F. Cheung, “An empirical approach to the identification of sources of machining errors in ultra-precision raster milling,” Key Eng. Mater. 364-366, 986-991 (2008).
[CrossRef]

S. Wang, S. To, C. F. Cheung, and W. B. Lee, “A study of the scallop generation mechanism in ultra-precision raster milling,” Key Eng. Mater. 364-366, 1262-1267 (2008).
[CrossRef]

Opt. Commun.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32, 19-23 (1980).
[CrossRef]

Opt. Eng.

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

S. Shikama, “Effects of corrugation of aspherical mirrors on the optical performance of imaging optics with the mirror near an image plane,” Opt. Eng. 43, 3068-3076 (2004).
[CrossRef]

M. Shibuya, N. Watanabe, M. Yamamoto, T. Fukui, H. Ezaki, T. Kiire, and S. Nakadate, “Classification of undulated wavefront aberration in projection optics by considering its physical effects,” Opt. Eng. 46, 053001 (2007).
[CrossRef]

Proc. R. Soc. London Ser. A

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399-416 (1907).
[CrossRef]

Proc. SPIE

U. Birnbaum, H. Bernitzki, O. Falkenstörfer, H. Lauth, R. Schreiner, and T. Waak, “Manufacturing of high-precision aspheres,” Proc. SPIE 6149, 61490H (2006).
[CrossRef]

M. Schulz, I. Weingaertner, C. Elster, and J. Gerhardt, “Low- and mid-spatial-frequency component measurement for aspheres,” Proc. SPIE 5188, 287-295 (2003),
[CrossRef]

M. L. Barkman, B. S. Dutterer, M. A. Davies, and T. J. Suleski, “Free-form machining for micro-imaging systems,” Proc. SPIE , 6883, 68830G (2008).
[CrossRef]

Y. E. Tohme, “Grinding aspheric and freeform micro-optical molds,” Proc. SPIE 6462, 64620K (2007).
[CrossRef]

A. Beaucamp, R. Freeman, R. Morton, K. Ponudurai, and D. D. Walker, “Removal of diamond-turning signatures on x-ray mandrels and metal optics by fluid-jet polishing,” Proc. SPIE 7018, 701835 (2008).
[CrossRef]

R. C. Juergens, R. H. Shepard III, and J. P. Schaefer, “Simulation of single-point diamond turning fabrication process errors,” Proc. SPIE 5174, 93-104 (2003).
[CrossRef]

G. W. Forbes and C. P. Brophy, “Asphere, O asphere, how shall we describe thee?,” Proc. SPIE 7100, 710002 (2008).
[CrossRef]

C. An, Q. Xu, F. Zhang, and J. Zhang, “Calculation and structural analysis for the rigidity of air spindle in the single point diamond turning lathe,” Proc. SPIE 6722, 67222U (2007).
[CrossRef]

J. C. Stover and J. E. Harvey, “Limitations of Rayleigh Rice perturbation theory for describing surface scatter,” Proc. SPIE 6672, 66720B (2007).
[CrossRef]

Other

R. Smith, A Compleat System of Opticks in Four Books, viz A Popular, a Mathematical, a Mechanical and Philosophical Treatise (Cambridge University Press, 1738).

F. Twyman, Prism and Lens Making (Hilger & Watts, 1952).

J. R. Rogers, “Slope error tolerances for optical surfaces,” presented at the OptiFab Conference (SPIE, 2008), paper TD04-4, pp. 15-17.

M. J. Lighthill, Introduction to Fourier Analysis and Generalised Functions, Cambridge Monographs on Mechanics and Applied Mathematics (Cambridge University Press, 1958).

H. H. Barrett, Foundations of Image Science, H.H.Barrett and K.J.Myers, eds. (Wiley-Interscience, 2004).

J. Mathews and R. L. Walker, Mathematical Methods of Physics, 2nd ed. (W. A. Benjamin, 1970).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th expanded ed. (Cambridge University Press, 1999).
[PubMed]

M. Pfaff, “High-speed fabrication of aspheres and optical free-form surfaces,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OThD6.

P. Murphy, “Methods and challenges in quantifying mid-spatial frequencies,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA3.

D. Aikens, J. E. DeGroote, and R. N. Youngworth, “Specification and control of mid-spatial frequency wavefront errors in optical systems,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA1.

J. M. Tamkin, “A study of image artifacts caused by structured mid-spatial frequency fabrication errors on optical surfaces,” Ph.D. dissertation (University of Arizona, 2010).

J. W. Goodman, Introduction to Fourier optics, 3rd ed., (Roberts & Co., 2005).

Y. Tohme, “Trends in ultra-precision machining of freeform optical surfaces,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OThC6.

Harmonic decomposition is also called Fourier decomposition. The former usage will be maintained to prevent confusion with Fourier transform.

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Figures (9)

Fig. 1
Fig. 1

Test lens with a beam footprint on surfaces A, B, C, and D (top) along with a beam footprint on each surface (bottom) for three field positions. The effective focal length is 50 mm, f / 10 , and field of view is 50 ° .

Fig. 2
Fig. 2

Simulated field-dependent PSFs due to rotationally symmetric structured MSF errors on surface D of Fig. 1: period, 0.5 mm,; footprint diameter, 4.8 mm; PV error, 200 nm. Left, on axis; right, 25 deg off axis.

Fig. 3
Fig. 3

(a) Relationship between the MSF grating with period p and the diffracted beam location at image plane s. For infinite conjugate, D is the diameter of pupil and f is the focal length of the lens. θ is the convergence angle of the beam, and θ is the diffraction angle. (b) The grating can be located on any surface within the optical system, where the diffraction angle becomes scaled by the local magnification as in Eq. (5).

Fig. 4
Fig. 4

Diffraction efficiency for the first four orders, according to Eq. (2), as a function of PV surface height β (in μm): n = 1.5 , λ = 500 nm .

Fig. 5
Fig. 5

PSF two sinusoid ripples, φ β = 0.2   waves (0.671 rad), zero to peak. ξ 1 = 20   cycles , ξ 2 = 25   cycles . The horizontal scale is the spatial frequency multiplied by the aperture size, which gives the number of cycles across the aperture. The cross terms have a frequency of 5 and 45 cycles for subtractive and additive terms, respectively.

Fig. 6
Fig. 6

Four sinusoids, equal amplitude (60, 54, 48, and 42 cycles across aperture). (a) 0.2 waves peak surface height for each component. Note that six cycle cross terms overlap. Phases add linearly, but peak amplitude is dependent on J 1 ( φ l ( l + 1 ) ) . (b) 0.3 waves peak for each component. Nonlinear amplification of peak cause first-order ( m = 1 ) cross terms to exceed first-order amplitudes. (c) Frequency separation of sinusoids reduced to two cycles: 60, 58, 56, and 54 cycles.

Fig. 7
Fig. 7

Effect of clipped surface phase heights. Surface seven parameters with φ β = 0.2   waves : (a) no truncation, 0.056 waves RMS; (b) surface height clipped by 50%, 0.0553 waves RMS; (c) surface height clipped by 75%, 0.054 waves RMS.

Fig. 8
Fig. 8

Multiple sinusoid components. Surface profile in waves for surface B. Six spatial components: 20, 18, 14.8, 12.2, 10.8, and 8 cycles across the 32 mm aperture. Each of the six components has 25 nm PV surface height, resulting in a 240 nm PV error over the surface.

Fig. 9
Fig. 9

(a), (c) PSF and (b), (d) pupil wavefront for the MSF error on surface B in Fig. 1.

Tables (1)

Tables Icon

Table 1 Prescription for the Lens in Fig. 1

Equations (42)

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sin θ i sin θ m = m λ p ,
s θ f .
d 2.44 λ f D ,
s d = θ D 2.44 λ = D 2.44 p .
s 3 = z i 1 θ M 2 M 3 ,
M 2 = z i 2 z o 2 , M 3 = z i 3 z o 3 .
U s ( x s ) = A o exp [ i ϕ ( x s ) ] .
ϕ ( x s ) = 2 π h ( x s ) ( n 1 ) λ γ i ,
h ( x s ) = 1 2 β sin ( 2 π x s ξ o ) ,
φ β = 2 π ( n 1 ) λ β 2 = π ( n 1 ) β λ .
U s ( x s ) = A o rect ( x s D ) exp [ i φ β sin ( 2 π x s ξ o ) ] .
U s ( x s ) = A o rect ( x s D ) m = c m exp ( i m 2 π x s ξ o ) ,
c m = ξ o 1 / 2 ξ o 1 / 2 ξ o exp [ i φ β sin ( 2 π x s ξ o ) ] exp ( i m 2 π x s ξ o ) d x .
J m ( z ) = 1 2 π 0 2 π exp [ i ( z sin θ m θ ) ] d θ .
z = φ β , θ = 2 π x s ξ o
c m = J m ( φ β ) .
U s ( x s ) = A o rect ( x s D ) m = J m ( φ β ) exp ( i m 2 π x s ξ o ) .
U i ( x i ) = A o D m = J m ( φ β ) D / 2 D / 2 exp ( i m 2 π x s ξ o ) exp ( i 2 π ξ x s ) d x s | ξ = x i λ z i = A o D m = J m ( φ β ) δ ( ξ m ξ o ) * sinc ( ξ D ) | ξ = x i λ z i .
η m ( β ) = J m 2 ( π ( n 1 ) β λ ) = J m 2 ( φ β ) ,
h ( x s ) = 1 2 β o + 1 2 = 1 β A cos ( 2 π ξ o x s ) + 1 2 = 1 β B sin ( 2 π ξ o x s ) ,
φ ( x s ) = φ o + = 1 φ A cos ( 2 π ξ o x s ) + = 1 φ B sin ( 2 π ξ o x s ) ,
φ A = ξ o 1 / 2 ξ o 1 / 2 ξ o φ ( x s ) cos ( 2 π ξ o x s ) d x s , φ B = ξ o 1 / 2 ξ o 1 / 2 ξ o φ ( x s ) sin ( 2 π ξ o x s ) d x s .
U s ( x s ) = A o rect ( x s D ) exp { i [ = 1 φ A cos ( 2 π ξ o x s ) + = 1 φ B sin ( 2 π ξ o x s ) ] } ,
exp { i [ = 1 φ A cos ( 2 π ξ o x s ) ] } = = 1 exp { i [ φ A cos ( 2 π ξ o x s ) ] } = = 1 U s A ( x s ) , exp { i [ = 1 φ B sin ( 2 π ξ o x s ) ] } = = 1 exp { i [ φ B sin ( 2 π ξ o x s ) ] } = = 1 U s B ( x s ) .
U s ( x s ) = A o rect ( x s D ) = 1 U s A ( x s ) = 1 U s B ( x s ) .
U i ( x i ) = A o F { rect ( x s D ) U s 1 A ( x s ) U s 2 A ( x s ) U s 1 B ( x s ) U s 2 B ( x s ) 2 } = sinc ( ξ D ) * U i 1 A ( ξ ) * U i 2 A ( ξ ) * U i 1 B ( ξ ) * U i 2 B ( ξ ) * | ξ = x i λ z i .
U s B ( x s ) = exp [ i φ B sin ( 2 π ξ o x s ) ] = m = c m , exp ( i 2 π m ξ o x s )
c m , = ξ o 1 / 2 ξ o 1 / 2 ξ o exp [ i φ B sin ( 2 π ξ o x s ) ] exp ( i 2 π m ξ o x s ) d x s = J m [ φ B ] .
U i B ( x i ) = A o D F { rect ( x s D ) l = 1 [ m = J m ( φ B ) exp ( i 2 π m l ξ o x s ) ] } = A o D [ sinc ( ξ D ) * m = J m ( φ B ) δ ( ξ m ξ o ) * m = J m ( φ B ) δ ( ξ m ( + 1 ) ξ o ) * | ξ = x i λ z i ] .
J m ( z ) = i m 2 π 0 2 π exp [ i ( z cos θ m θ ) ] d θ .
φ ( x s ) = φ o + = 1 φ cos ( 2 π ξ o x s + γ ) ,
φ ( x s ) = φ o + = 1 [ φ cos ( 2 π ξ o x s ) cos ( γ ) φ sin ( 2 π ξ o x s ) sin ( γ ) ] .
φ A = φ cos ( γ ) , φ B = φ sin ( γ ) ,
γ = arctan ( φ B φ A ) , φ = ( φ A ) 2 + ( φ B ) 2 .
U s ( x s ) = A o rect ( x s D ) = 1 exp [ φ cos ( 2 π ξ o x s + γ ) ] = A o rect ( x s D ) U s 1 ( x s ) U s 2 ( x s )
U s ( x s ) = m = c m , exp [ i ( 2 π m ξ o x s + γ ) ] ,
c m , = ξ o 1 / 2 ξ o 1 / 2 ξ o exp [ φ cos ( 2 π ξ o x s + γ ) ] exp ( i [ 2 π m ξ o x s + γ ] ) d x s = exp ( i γ ) i m J m ( φ ) .
U i ( x i ) = A o D { m = i m J m ( φ ) sinc [ ( ξ m ξ o ) D ] * m = i m J m ( φ + 1 ) sinc [ ( ξ m ( + 1 ) ξ o ) D ] * } | ξ = x i λ z i .
U i B ( ξ ) = m = 1 1 J m ( φ β ) δ ( ξ m ξ o ) * m = 1 1 J m ( φ β ) δ ( ξ m μ ξ o ) * sinc ( ξ D ) = [ B 1 δ ( ξ + ( 1 ) ξ o ) + B 0 δ ( ξ ( 0 ) ξ o ) + B 1 δ ( ξ + ( 1 ) ξ o ) ] * [ B 1 δ ( ξ + μ ξ o ) + B 0 δ ( ξ ( 0 ) μ ξ o ) + B 1 δ ( ξ μ ξ o ) ] * sinc ( ξ D ) .
δ ξ o = δ ( ξ + ξ o ) , δ ξ o = δ ( ξ ξ o ) , δ μ ξ o = δ ( ξ + μ ξ o ) , δ μ ξ o = δ ( ξ μ ξ o ) ,
U i B ( ξ ) = ( B 1 δ ξ o + B 0 δ 0 + B 1 δ ξ o ) * ( B 1 δ μ ξ o + B 0 δ 0 + B 1 δ μ ξ o ) * sinc ( ξ D ) = [ ( B 1 δ f o * B 1 δ μ ξ o ) + ( B 1 δ ξ o * B 0 δ 0 ) + ( B 1 δ ξ o * B 1 δ μ ξ o ) + ( B 0 δ 0 * B 1 δ μ ξ o ) + ( B 0 δ 0 * B 0 δ 0 ) + ( B 0 δ 0 * B 1 δ μ ξ o ) + ( B 1 δ ξ o * B 1 δ μ ξ o ) + ( B 1 δ ξ o * B 0 δ 0 ) + ( B 1 δ ξ o * B 1 δ μ ξ o ) ] * sinc ( ξ D ) .
U i B ( ξ ) = { B 0 2 δ 0 + [ B 0 B 1 δ μ ξ o + B 0 B 1 δ ξ o + B 0 B 1 δ μ ξ o + B 0 B 1 δ ξ o ] + [ ( B 1 ) 2 δ ξ o + μ ξ o + ( B 1 ) 2 δ ξ o μ ξ o + ( B 1 ) 2 δ μ ξ o ξ o + ( B 1 ) 2 δ μ ξ o ξ o ] } * sinc ( ξ D ) .

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