Abstract

Diffractive lenses differ from conventional optical elements in that they can produce more than one image because of the presence of more than one diffraction order. These spurious, defocused images serve to lower the contrast of the desired image. We show that a quantity that we define as the integrated efficiency serves as a useful figure of merit to describe diffractive lenses. The integrated efficiency is shown to be the limiting value for the optical transfer function; in most cases it serves as an overall scale factor for the transfer function. We discuss both monochromatic and polychromatic applications of the integrated efficiency and provide examples to demonstrate its utility.

© 1992 Optical Society of America

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References

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  1. P. P. Clark, C. Londono, “Production of kinoforms by single point diamond machining,” Opt. News, 15, 39–40 (1989); J. A. Futhey, “Diffractive bifocal intraocular lens,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 142–149 (1989); G. M. Morris, D. A. Buralli, “Wide field diffractive lenses for imaging, scanning, and Fourier transformation,” Opt. News 15(12), 41–42 (1989).
    [CrossRef]
  2. L. d’Auria, J. P. Huignard, A. M. Roy, E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972); G. J. Swanson, W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng 28, 605–608 (1989).
    [CrossRef]
  3. V. P. Koronkevich, “Computer synthesis of diffraction optical elements,” Optical Processing and Computing, H. H. Arsenault, T. Szoplik, B. Macukow, eds. (Academic, Orlando, Fla., 1989), pp. 277–313.
  4. For example, see the proceedings of several recent conferences on holographic and diffractive optics: Proc. Soc. Photo-Opt. Instrum. Eng. 883, 1052, 1136, 1211, and selected papers in Proc. Soc. Photo-Opt. Instrum. Eng. 1354.
  5. D. A. Buralli, G. M. Morris, J. R. Rogers, “Optical performance of holographic kinoforms,” Appl. Opt. 28, 976–983 (1989).
    [CrossRef] [PubMed]
  6. There are many papers that describe results from the rigorous electromagnetic grating theory. See, for example, E. G. Loewen, M. Nevière, D. Maystre, “Grating efficiency theory as it applies to blazed and holographic gratings,” Appl. Opt. 16, 2711–2721 (1977); R. Petit, ed., Electromagnetic Theory of Gratings, (Springer-Verlag, Berlin, 1980).
    [CrossRef] [PubMed]
  7. K. Rosenhauer, K. Rosenbruch, “Flare and optical transfer function,” Appl. Opt. 7, 283–287 (1968).
    [CrossRef] [PubMed]
  8. O. A. Barteneva, “Effect of scattered light on the photographic image quality,” Sov. J. Opt. Technol. 44, 197–199 (1977).
  9. J. J. Jakubowski, “Methodology for quantifying flare in a microdensitometer,” Opt. Eng. 19, 122–131 (1980).
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), pp. 90–96.
  11. W. B. Wetherell, “The calculation of image quality,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, Orlando, Fla., 1980), Vol. VIII, pp. 202–215.
  12. G. J. Swanson, “Binary optics technology: the theory and design of multi-level diffractive optical elements,” Tech. Rep. 854 (Lincoln Laboratory, MIT, Lexington, Mass., 1989); W. -H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics XVI, E. Wolf., ed. (North-Holland, Amsterdam, 1978).
    [CrossRef]
  13. Similar transfer functions for zone plates have been published in M. J. Simpson, A. G. Michette, “Considerations of zone plate optics for soft x-ray microscopy,” Opt. Acta 31, 1417–1426 (1984) and in A. G. Michette, Optical Systems for Soft X-Rays (Plenum, New York, 1986), pp. 186–188.
    [CrossRef]
  14. J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 116–124 (1990).
  15. D. A. Buralli, G. M. Morris, “Design of two- and three-element diffractive Keplerian telescopes,” Appl. Opt. 31, 38–43 (1992).
    [CrossRef] [PubMed]
  16. Ref. 10, pp. 70–74.
  17. Ref. 11, pp. 225–226.
  18. See Eq. (11) of Ref. 5.
  19. J. P. Jennings, F. J. Busselle, S. G. Shaw, “Observed differences in MTF results between line-spread and interferometric measurements for IR lenses,” in Infrared Technology and Applications, L. R. Baker, A. Masson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.590, 138–143 (1985).

1992

1989

P. P. Clark, C. Londono, “Production of kinoforms by single point diamond machining,” Opt. News, 15, 39–40 (1989); J. A. Futhey, “Diffractive bifocal intraocular lens,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 142–149 (1989); G. M. Morris, D. A. Buralli, “Wide field diffractive lenses for imaging, scanning, and Fourier transformation,” Opt. News 15(12), 41–42 (1989).
[CrossRef]

D. A. Buralli, G. M. Morris, J. R. Rogers, “Optical performance of holographic kinoforms,” Appl. Opt. 28, 976–983 (1989).
[CrossRef] [PubMed]

1984

Similar transfer functions for zone plates have been published in M. J. Simpson, A. G. Michette, “Considerations of zone plate optics for soft x-ray microscopy,” Opt. Acta 31, 1417–1426 (1984) and in A. G. Michette, Optical Systems for Soft X-Rays (Plenum, New York, 1986), pp. 186–188.
[CrossRef]

1980

J. J. Jakubowski, “Methodology for quantifying flare in a microdensitometer,” Opt. Eng. 19, 122–131 (1980).

1977

1972

L. d’Auria, J. P. Huignard, A. M. Roy, E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972); G. J. Swanson, W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng 28, 605–608 (1989).
[CrossRef]

1968

Barteneva, O. A.

O. A. Barteneva, “Effect of scattered light on the photographic image quality,” Sov. J. Opt. Technol. 44, 197–199 (1977).

Bergstrom, J.

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 116–124 (1990).

Buralli, D. A.

Busselle, F. J.

J. P. Jennings, F. J. Busselle, S. G. Shaw, “Observed differences in MTF results between line-spread and interferometric measurements for IR lenses,” in Infrared Technology and Applications, L. R. Baker, A. Masson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.590, 138–143 (1985).

Clark, P. P.

P. P. Clark, C. Londono, “Production of kinoforms by single point diamond machining,” Opt. News, 15, 39–40 (1989); J. A. Futhey, “Diffractive bifocal intraocular lens,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 142–149 (1989); G. M. Morris, D. A. Buralli, “Wide field diffractive lenses for imaging, scanning, and Fourier transformation,” Opt. News 15(12), 41–42 (1989).
[CrossRef]

Cox, J. A.

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 116–124 (1990).

d’Auria, L.

L. d’Auria, J. P. Huignard, A. M. Roy, E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972); G. J. Swanson, W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng 28, 605–608 (1989).
[CrossRef]

Fritz, B.

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 116–124 (1990).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), pp. 90–96.

Huignard, J. P.

L. d’Auria, J. P. Huignard, A. M. Roy, E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972); G. J. Swanson, W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng 28, 605–608 (1989).
[CrossRef]

Jakubowski, J. J.

J. J. Jakubowski, “Methodology for quantifying flare in a microdensitometer,” Opt. Eng. 19, 122–131 (1980).

Jennings, J. P.

J. P. Jennings, F. J. Busselle, S. G. Shaw, “Observed differences in MTF results between line-spread and interferometric measurements for IR lenses,” in Infrared Technology and Applications, L. R. Baker, A. Masson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.590, 138–143 (1985).

Koronkevich, V. P.

V. P. Koronkevich, “Computer synthesis of diffraction optical elements,” Optical Processing and Computing, H. H. Arsenault, T. Szoplik, B. Macukow, eds. (Academic, Orlando, Fla., 1989), pp. 277–313.

Lee, J.

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 116–124 (1990).

Loewen, E. G.

Londono, C.

P. P. Clark, C. Londono, “Production of kinoforms by single point diamond machining,” Opt. News, 15, 39–40 (1989); J. A. Futhey, “Diffractive bifocal intraocular lens,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 142–149 (1989); G. M. Morris, D. A. Buralli, “Wide field diffractive lenses for imaging, scanning, and Fourier transformation,” Opt. News 15(12), 41–42 (1989).
[CrossRef]

Maystre, D.

Michette, A. G.

Similar transfer functions for zone plates have been published in M. J. Simpson, A. G. Michette, “Considerations of zone plate optics for soft x-ray microscopy,” Opt. Acta 31, 1417–1426 (1984) and in A. G. Michette, Optical Systems for Soft X-Rays (Plenum, New York, 1986), pp. 186–188.
[CrossRef]

Morris, G. M.

Nelson, S.

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 116–124 (1990).

Nevière, M.

Rogers, J. R.

Rosenbruch, K.

Rosenhauer, K.

Roy, A. M.

L. d’Auria, J. P. Huignard, A. M. Roy, E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972); G. J. Swanson, W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng 28, 605–608 (1989).
[CrossRef]

Shaw, S. G.

J. P. Jennings, F. J. Busselle, S. G. Shaw, “Observed differences in MTF results between line-spread and interferometric measurements for IR lenses,” in Infrared Technology and Applications, L. R. Baker, A. Masson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.590, 138–143 (1985).

Simpson, M. J.

Similar transfer functions for zone plates have been published in M. J. Simpson, A. G. Michette, “Considerations of zone plate optics for soft x-ray microscopy,” Opt. Acta 31, 1417–1426 (1984) and in A. G. Michette, Optical Systems for Soft X-Rays (Plenum, New York, 1986), pp. 186–188.
[CrossRef]

Spitz, E.

L. d’Auria, J. P. Huignard, A. M. Roy, E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972); G. J. Swanson, W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng 28, 605–608 (1989).
[CrossRef]

Swanson, G. J.

G. J. Swanson, “Binary optics technology: the theory and design of multi-level diffractive optical elements,” Tech. Rep. 854 (Lincoln Laboratory, MIT, Lexington, Mass., 1989); W. -H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics XVI, E. Wolf., ed. (North-Holland, Amsterdam, 1978).
[CrossRef]

Werner, T.

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 116–124 (1990).

Wetherell, W. B.

W. B. Wetherell, “The calculation of image quality,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, Orlando, Fla., 1980), Vol. VIII, pp. 202–215.

Appl. Opt.

Opt. Acta

Similar transfer functions for zone plates have been published in M. J. Simpson, A. G. Michette, “Considerations of zone plate optics for soft x-ray microscopy,” Opt. Acta 31, 1417–1426 (1984) and in A. G. Michette, Optical Systems for Soft X-Rays (Plenum, New York, 1986), pp. 186–188.
[CrossRef]

Opt. Commun.

L. d’Auria, J. P. Huignard, A. M. Roy, E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972); G. J. Swanson, W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng 28, 605–608 (1989).
[CrossRef]

Opt. Eng.

J. J. Jakubowski, “Methodology for quantifying flare in a microdensitometer,” Opt. Eng. 19, 122–131 (1980).

Opt. News

P. P. Clark, C. Londono, “Production of kinoforms by single point diamond machining,” Opt. News, 15, 39–40 (1989); J. A. Futhey, “Diffractive bifocal intraocular lens,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 142–149 (1989); G. M. Morris, D. A. Buralli, “Wide field diffractive lenses for imaging, scanning, and Fourier transformation,” Opt. News 15(12), 41–42 (1989).
[CrossRef]

Sov. J. Opt. Technol.

O. A. Barteneva, “Effect of scattered light on the photographic image quality,” Sov. J. Opt. Technol. 44, 197–199 (1977).

Other

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 116–124 (1990).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), pp. 90–96.

W. B. Wetherell, “The calculation of image quality,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, Orlando, Fla., 1980), Vol. VIII, pp. 202–215.

G. J. Swanson, “Binary optics technology: the theory and design of multi-level diffractive optical elements,” Tech. Rep. 854 (Lincoln Laboratory, MIT, Lexington, Mass., 1989); W. -H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics XVI, E. Wolf., ed. (North-Holland, Amsterdam, 1978).
[CrossRef]

V. P. Koronkevich, “Computer synthesis of diffraction optical elements,” Optical Processing and Computing, H. H. Arsenault, T. Szoplik, B. Macukow, eds. (Academic, Orlando, Fla., 1989), pp. 277–313.

For example, see the proceedings of several recent conferences on holographic and diffractive optics: Proc. Soc. Photo-Opt. Instrum. Eng. 883, 1052, 1136, 1211, and selected papers in Proc. Soc. Photo-Opt. Instrum. Eng. 1354.

Ref. 10, pp. 70–74.

Ref. 11, pp. 225–226.

See Eq. (11) of Ref. 5.

J. P. Jennings, F. J. Busselle, S. G. Shaw, “Observed differences in MTF results between line-spread and interferometric measurements for IR lenses,” in Infrared Technology and Applications, L. R. Baker, A. Masson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.590, 138–143 (1985).

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Figures (7)

Fig. 1
Fig. 1

Schematic illustration of the point-spread function for an optical system that contains diffractive elements. The point-spread function comprises two components: a focused component caused by the diffraction order of interest and a background component, not necessarily uniform, of large spatial extent caused by the other diffraction orders.

Fig. 2
Fig. 2

Optical layout of single diffractive lens that images an infinitely distant object. The annotation m refers to the order of diffraction.

Fig. 3
Fig. 3

Local diffraction efficiency as a function of pupil coordinate u for the imaging example discussed in the text. The pupil width is D = 20 mm, the exit pupil–image plane distance is R = 100 mm, the wavelength is λ = 0.5 μm, and the local diffraction efficiency is ηlocal(u) = cos2(au).

Fig. 4
Fig. 4

Modulation transfer functions (MTF’s) for the imaging example. The wavelength is 0.5 μm and the system f-number is F/5.

Fig. 5
Fig. 5

Point-spread functions (PSF’s) for the imaging example. IS is the value of the Strehl intensity for each value of a.

Fig. 6
Fig. 6

Scalar value of the diffraction efficiency ηscalar as a function of the wavelength-detuning parameter λ0/λ for the diffraction orders m = 0, 1, and 2.

Fig. 7
Fig. 7

Polychromatic modulation transfer functions (MTF’s) for two spectral bands. a)F/5 system (circular exit pupil) with a center wavelength of λ0 = 0.55 μm and a spectral bandwidth of 0.40–0.70 μm. The value of ηint,poly is 0.914. b) F/2 system (circular exit pupil) with a center wavelength of λ0 = 10.0 μm and a spectral bandwidth of 8.0–12.0 μm. The value of ηint,poly is 0.955.

Tables (2)

Tables Icon

Table 1 Integrated Efficiency, Full-Width-at-Half-Power Points, and Energy Enclosed Within the First Zeros of the Diffraction-Limited Point-Spread Function for the Example Pupil Functions Described in the Text

Tables Icon

Table 2 Approximatea and Exactb Polychromatic Integrated Efficiencies for Several Spectral Regions, Based on the Scalar Prediction of Diffraction Efficiency

Equations (37)

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P ( u , v ) = t m = 1 ( u , v ) exp [ i k n W m = 1 ( u , v ) ] + t BG ( u , v ) exp [ i k n W BG ( u , v ) ] ,
η local ( u , v ) = t m = 1 ( u , v ) 2 .
η int = 1 A pupil - - η local ( u , v ) d u d v ,
h m = 1 ( x , y ) = 1 λ R - - t m = 1 ( u , v ) exp [ i k W m = 1 ( u , v ) ] × exp [ - i 2 π λ R ( x u + y v ) ] d u d v ,
h BG ( x , y ) = 1 λ R - - t BG ( u , v ) exp [ i k W BG ( u , v ) ] × exp [ - i 2 π λ R ( x u + y v ) ] d u d v ,
I ( x , y ) = h m = 1 ( x , y ) 2 + h BG ( x , y ) 2 + h m = 1 * ( x , y ) h BG ( x , y ) + h m = 1 ( x , y ) h BG * ( x , y ) .
I ( x , y ) h m = 1 ( x , y ) 2 + h BG ( x , y ) 2 .
- - h m = 1 1 ( x , y ) 2 d x d y + - - h BG 1 ( x , y ) 2 d x d y = 1 ,
I 1 ( x , y ) h m = 1 1 ( x , y ) 2 + h BG 1 ( x , y ) 2 .
η int = - - h m = 1 1 ( x , y ) 2 d x d y .
OTF ( f x , f y ) = - - I 1 ( x , y ) × exp [ - i 2 π ( f x x + f y y ) ] d x d y .
OTF ( f x , f y ) = OTF m = 1 ( f x , f y ) + OTF BG ( f x , f y ) .
OTF ( 0 , 0 ) = - - h m = 1 1 ( x , y ) 2 d x d y + - - h BG 1 ( x , y ) 2 d x d y = 1.
OTF m = 1 ( 0 , 0 ) = η int ,
OTF BG ( 0 , 0 ) = 1 - η int .
OTF m = 1 ( f x , f y ) = OTF m = 1 ( 0 , 0 ) + [ OTF m = 1 f x ] f x = f y = 0 f x + [ OTF m = 1 f y ] f x = f y = 0 f y + .
OTF ( f x , f y ) 1 A pupil - - P m = 1 ( u , v ) P m = 1 ( u , v ) d u d v + ( 1 - η int ) f x , 0 f y , 0 ,
OTF ( f x , f y ) η int - - P m = 1 ( u , v ) P m = 1 ( u , v ) d u d v - - t m = 1 ( u , v ) 2 d u d v + ( 1 - η int ) δ f x 0 δ f y , 0 .
t m = 1 ( u ) = cos ( a u ) ,
t BG ( u ) = sin ( a u ) .
η local ( u ) = cos 2 ( a u ) .
η int = 1 2 [ 1 + sin ( a D ) a D ] ,
I ( x ) = 1 D 2 { sin 2 [ ( a - 2 π x λ R ) D 2 ] ( a - 2 π x λ R ) 2 + sin 2 [ ( a + 2 π x λ R ) D 2 ] ( a + 2 π x λ R ) 2 } + 1 D 2 { cos ( 2 π D x λ R ) - cos ( a D ) ( a - 2 π x λ R ) ( a + 2 π x λ R ) + λ R sin 2 ( a x ) } , x ( D / 2 ) ,
I ( x ) = 1 D 2 { sin 2 [ ( a - 2 π x λ R ) D 2 ] ( a - 2 π x λ R ) 2 + sin 2 [ ( a + 2 π x λ R ) D 2 ] ( a + 2 π x λ R ) 2 + cos ( 2 π D x λ R ) - cos ( a D ) ( a - 2 π x λ R ) ( a + 2 π x λ R ) } ,             x > ( D / 2 ) .
OTF ( f x ) = 1 2 D { ( D - λ R f x ) cos ( a λ R f x ) + sin [ a ( D - λ R f x ) ] a + sin ( π D f x ) π f x } - 1 2 D { sin [ D ( a - π f x ) ] 2 ( a - π f x ) + sin [ D ( a + π f x ) ] 2 ( a + π f x ) } , f x D / ( λ R ) .
η int , poly = λ min λ max η int ( λ ) d λ λ max - λ min .
OTF poly ( f x , f y ) = λ min λ max S ( λ ) OTF ( f x , f y ; λ ) d λ λ min λ max S ( λ ) d λ ,
OTF poly ( f x , f y ) η int , poly OTF m = 1 , poly ( f x , f y ) + ( 1 - η int , poly ) δ f x , 0 δ f y , 0 .
η scalar ( λ ) = sin 2 [ π ( λ 0 λ - m ) ] [ π ( λ 0 λ - m ) ] 2 ,
η int , poly 1 + π 2 3 λ 0 ( λ min + λ max - λ 0 ) - π 2 9 λ 0 2 ( λ min 2 + λ min λ max + λ max 2 ) .
f ( x ) = m = - c m exp ( i 2 π m f 0 x ) ,
1 L 0 L f ( x ) 2 d x = m = - c m 2 .
m = - c m 2 = 1             for x D / 2.
F ( f x ) = m = - c m D sinc [ D ( f x - m f 0 ) ] ,
F ( f x ) 2 = m = - c m 2 D 2 sinc 2 [ D ( f x - m f 0 ) ] + m , n = - m n ( c m c n * ) D 2 × sinc [ D ( f x - m f 0 ) ] sinc [ D ( f x - n f 0 ) ] .
- D 2 sinc [ D ( f x - m f 0 ) ] sinc [ D ( f x - n f 0 ) ] d f x = D sinc [ D f 0 ( m - n ) ] .
- - h m = 1 * ( x , y ) h BG ( x , y ) d x d y = - - h m = 1 ( x , y ) h BG * ( x , y ) d x d y = 0.

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