Abstract

The Seidel aberrations of a rotationally-symmetric diffractive lens with an arbitrary phase profile are presented. It is shown that by a proper choice of phase function and aperture stop position, third-order coma and astigmatism can be eliminated for any chosen conjugate ratio. Since a diffractive lens has an inherent zero value for the Petzval sum, the image plane is flat in both tangential and sagittal meridians. The substrate curvature of the lens may be chosen to introduce a prescribed amount of distortion to allow for use as a Fourier transform lens or a laser scan lens. Examples are given of lens performance in finite conjugate imaging and laser scanning, where the fθ condition is satisfied.

© 1991 Optical Society of America

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References

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  1. Lord Rayleigh, laboratory notebook, 11 Apr. 1871, quoted in R. W. Wood, Physical Optics (Macmillan, New York, 1934), pp. 37–38.
  2. J. Kirz, “Phase Zone Plates for X-Rays and the Extreme UV,” J. Opt. Soc. Am. 64, 301–309 (1974).
    [CrossRef]
  3. L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The Kinoform: a New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150–155 (1969); J. A. Jordan, P. M. Hirsch, L. B. Lesem, D. L. Van Rooy, “Kinoform Lenses,” Appl. Opt. 9, 1883–1887 (1970).
    [CrossRef] [PubMed]
  4. W. B. Veldkamp, G. J. Swanson, D. C. Shaver, “High Efficiency binary Lenses,” Opt. Commun. 53, 353–358 (1985); G. J. Swanson, W. B. Veldkamp, “Binary Lenses for Use at 10.6 Micrometers,” Opt. Eng. 24, 791–795 (1985); G. J. Swanson, W. B. Veldkamp, “Diffractive Optical Elements for Use in Infrared Systems,” Opt. Eng. 28, 605–608 (1989).
    [CrossRef]
  5. K. Miyamoto, “The Phase Fresnel Lens,” J. Opt. Soc. Am. 51, 17–20 (1961).
    [CrossRef]
  6. 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,” 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, 41–42 (1989).
    [CrossRef]
  7. L. d’Auria, J. P. Huignard, A. M. Roy, E. Spitz, “Photolithographic Fabrication of Thin Film Lenses,” Opt. Commun. 5, 232–235 (1972).
    [CrossRef]
  8. V. P. Koronkevich, “Computer Synthesis of Diffraction Optical Elements,” in Optical Processing and Computing, H. H. Arsenault, T. Szoplik, B. Macukow, Eds (Academic, Boston, 1989), pp. 277–313.
  9. D. A. Buralli, J. R. Rogers, “Some Fundamental Limitations of Achromatic Holographic Systems,” J. Opt. Soc. Am. A 6, 1863–1868 (1989).
    [CrossRef]
  10. A. I. Tudorovskii, “An Objective with a Phase Plate,” Opt. Spectrosc. 6, 126–133 (1959).
  11. H. Madjidi-Zolbanine, C. Froehly, “Holographic Correction of Both Chromatic and Spherical Aberrations of Single Glass Lenses,” Appl. Opt. 18, 2385–2393 (1979).
    [CrossRef] [PubMed]
  12. G. M. Morris, “Diffraction Theory for an Achromatic Fourier Transformation,” Appl. Opt. 20, 2017–2025 (1981).
    [CrossRef] [PubMed]
  13. T. Stone, N. George, “Hybrid Diffractive–Refractive Lenses and Achromats,” Appl. Opt. 27, 2960–2971 (1988).
    [CrossRef] [PubMed]
  14. T. A. Fritz, J. A. Cox, “Diffractive Optics for Broadband Infrared Imagers: Design Examples,” Proc. Soc. Photo-Opt. Instrum. Eng. 1052, 25–31 (1989).
  15. D. Falkis, G. M. Morris, “Broadband Imaging with Holographic Lenses,” Opt. Eng. 28, 592–598 (1989).
  16. W. C. Sweatt, “Describing Holographic Optical Elements as Lenses,” J. Opt. Soc. Am. 67, 803–808 (1977).
    [CrossRef]
  17. W. A. Kleinhans, “Aberrations of Curved Zone Plates and Fresnel Lenses,” Appl. Opt. 16, 1701–1704 (1977).
    [CrossRef] [PubMed]
  18. W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, 1986), pp. 130–140.
  19. Ref. 18, pp. 226–234.
  20. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 77–83.
  21. Ref. 17, pp. 1702.
  22. Ref. 18, pp. 148–152.
  23. Ref. 18, pp. 241–246.
  24. Ref. 18, p. 146.
  25. D. C. Sinclair, “Designing Diffractive Optics using the Sweatt Model,” Sinclair Optics Design Notes, Vol. 1, No. 1 (Winter1990).
  26. Ref. 18, pp. 152–153.
  27. super-oslo is a trademark of Sinclair Optics, 6780 Palmyra Rd., Fairport, NY 14450.
  28. K. von Bieren, “Lens Design for Optical Fourier Transform Systems,” Appl. Opt. 10, 2739–2742 (1971).
    [CrossRef] [PubMed]
  29. Ref. 18, pp. 93–98.
  30. D. A. Buralli, G. M. Morris, “Design of a Wide Field Diffractive Landscape Lens,” Appl. Opt. 28, 3950–3959 (1989).
    [CrossRef] [PubMed]
  31. R. E. Hopkins, M. J. Buzawa, “Optics for Laser Scanning,” Opt. Eng. 15, 90–94 (1976).
  32. See Ref. 31, pp. 93–94.
  33. D. A. Buralli, G. M. Morris, J. R. Rogers, “Optical Performance of Holographic Kinoforms,” Appl. Opt. 28, 976–983 (1989).
    [CrossRef] [PubMed]
  34. H. Sonnenberg, “Laser-Scanning Parameters and Latitudes in Laser Xerography,” Appl. Opt. 21, 1745–1751 (1982).
    [CrossRef] [PubMed]

1989 (6)

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,” 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, 41–42 (1989).
[CrossRef]

T. A. Fritz, J. A. Cox, “Diffractive Optics for Broadband Infrared Imagers: Design Examples,” Proc. Soc. Photo-Opt. Instrum. Eng. 1052, 25–31 (1989).

D. Falkis, G. M. Morris, “Broadband Imaging with Holographic Lenses,” Opt. Eng. 28, 592–598 (1989).

D. A. Buralli, J. R. Rogers, “Some Fundamental Limitations of Achromatic Holographic Systems,” J. Opt. Soc. Am. A 6, 1863–1868 (1989).
[CrossRef]

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

D. A. Buralli, G. M. Morris, “Design of a Wide Field Diffractive Landscape Lens,” Appl. Opt. 28, 3950–3959 (1989).
[CrossRef] [PubMed]

1988 (1)

1985 (1)

W. B. Veldkamp, G. J. Swanson, D. C. Shaver, “High Efficiency binary Lenses,” Opt. Commun. 53, 353–358 (1985); G. J. Swanson, W. B. Veldkamp, “Binary Lenses for Use at 10.6 Micrometers,” Opt. Eng. 24, 791–795 (1985); G. J. Swanson, W. B. Veldkamp, “Diffractive Optical Elements for Use in Infrared Systems,” Opt. Eng. 28, 605–608 (1989).
[CrossRef]

1982 (1)

1981 (1)

1979 (1)

1977 (2)

1976 (1)

R. E. Hopkins, M. J. Buzawa, “Optics for Laser Scanning,” Opt. Eng. 15, 90–94 (1976).

1974 (1)

1972 (1)

L. d’Auria, J. P. Huignard, A. M. Roy, E. Spitz, “Photolithographic Fabrication of Thin Film Lenses,” Opt. Commun. 5, 232–235 (1972).
[CrossRef]

1971 (1)

1969 (1)

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The Kinoform: a New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150–155 (1969); J. A. Jordan, P. M. Hirsch, L. B. Lesem, D. L. Van Rooy, “Kinoform Lenses,” Appl. Opt. 9, 1883–1887 (1970).
[CrossRef] [PubMed]

1961 (1)

1959 (1)

A. I. Tudorovskii, “An Objective with a Phase Plate,” Opt. Spectrosc. 6, 126–133 (1959).

Buralli, D. A.

Buzawa, M. J.

R. E. Hopkins, M. J. Buzawa, “Optics for Laser Scanning,” Opt. Eng. 15, 90–94 (1976).

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,” 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, 41–42 (1989).
[CrossRef]

Cox, J. A.

T. A. Fritz, J. A. Cox, “Diffractive Optics for Broadband Infrared Imagers: Design Examples,” Proc. Soc. Photo-Opt. Instrum. Eng. 1052, 25–31 (1989).

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).
[CrossRef]

Falkis, D.

D. Falkis, G. M. Morris, “Broadband Imaging with Holographic Lenses,” Opt. Eng. 28, 592–598 (1989).

Fritz, T. A.

T. A. Fritz, J. A. Cox, “Diffractive Optics for Broadband Infrared Imagers: Design Examples,” Proc. Soc. Photo-Opt. Instrum. Eng. 1052, 25–31 (1989).

Froehly, C.

George, N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 77–83.

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The Kinoform: a New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150–155 (1969); J. A. Jordan, P. M. Hirsch, L. B. Lesem, D. L. Van Rooy, “Kinoform Lenses,” Appl. Opt. 9, 1883–1887 (1970).
[CrossRef] [PubMed]

Hopkins, R. E.

R. E. Hopkins, M. J. Buzawa, “Optics for Laser Scanning,” Opt. Eng. 15, 90–94 (1976).

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).
[CrossRef]

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The Kinoform: a New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150–155 (1969); J. A. Jordan, P. M. Hirsch, L. B. Lesem, D. L. Van Rooy, “Kinoform Lenses,” Appl. Opt. 9, 1883–1887 (1970).
[CrossRef] [PubMed]

Kirz, J.

Kleinhans, W. A.

Koronkevich, V. P.

V. P. Koronkevich, “Computer Synthesis of Diffraction Optical Elements,” in Optical Processing and Computing, H. H. Arsenault, T. Szoplik, B. Macukow, Eds (Academic, Boston, 1989), pp. 277–313.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The Kinoform: a New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150–155 (1969); J. A. Jordan, P. M. Hirsch, L. B. Lesem, D. L. Van Rooy, “Kinoform Lenses,” Appl. Opt. 9, 1883–1887 (1970).
[CrossRef] [PubMed]

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,” 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, 41–42 (1989).
[CrossRef]

Madjidi-Zolbanine, H.

Miyamoto, K.

Morris, G. M.

Rogers, J. R.

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).
[CrossRef]

Shaver, D. C.

W. B. Veldkamp, G. J. Swanson, D. C. Shaver, “High Efficiency binary Lenses,” Opt. Commun. 53, 353–358 (1985); G. J. Swanson, W. B. Veldkamp, “Binary Lenses for Use at 10.6 Micrometers,” Opt. Eng. 24, 791–795 (1985); G. J. Swanson, W. B. Veldkamp, “Diffractive Optical Elements for Use in Infrared Systems,” Opt. Eng. 28, 605–608 (1989).
[CrossRef]

Sinclair, D. C.

D. C. Sinclair, “Designing Diffractive Optics using the Sweatt Model,” Sinclair Optics Design Notes, Vol. 1, No. 1 (Winter1990).

Sonnenberg, H.

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).
[CrossRef]

Stone, T.

Swanson, G. J.

W. B. Veldkamp, G. J. Swanson, D. C. Shaver, “High Efficiency binary Lenses,” Opt. Commun. 53, 353–358 (1985); G. J. Swanson, W. B. Veldkamp, “Binary Lenses for Use at 10.6 Micrometers,” Opt. Eng. 24, 791–795 (1985); G. J. Swanson, W. B. Veldkamp, “Diffractive Optical Elements for Use in Infrared Systems,” Opt. Eng. 28, 605–608 (1989).
[CrossRef]

Sweatt, W. C.

Tudorovskii, A. I.

A. I. Tudorovskii, “An Objective with a Phase Plate,” Opt. Spectrosc. 6, 126–133 (1959).

Veldkamp, W. B.

W. B. Veldkamp, G. J. Swanson, D. C. Shaver, “High Efficiency binary Lenses,” Opt. Commun. 53, 353–358 (1985); G. J. Swanson, W. B. Veldkamp, “Binary Lenses for Use at 10.6 Micrometers,” Opt. Eng. 24, 791–795 (1985); G. J. Swanson, W. B. Veldkamp, “Diffractive Optical Elements for Use in Infrared Systems,” Opt. Eng. 28, 605–608 (1989).
[CrossRef]

von Bieren, K.

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, 1986), pp. 130–140.

Wood, R. W.

Lord Rayleigh, laboratory notebook, 11 Apr. 1871, quoted in R. W. Wood, Physical Optics (Macmillan, New York, 1934), pp. 37–38.

Appl. Opt. (8)

IBM J. Res. Dev. (1)

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The Kinoform: a New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150–155 (1969); J. A. Jordan, P. M. Hirsch, L. B. Lesem, D. L. Van Rooy, “Kinoform Lenses,” Appl. Opt. 9, 1883–1887 (1970).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (3)

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

Opt. Commun. (2)

W. B. Veldkamp, G. J. Swanson, D. C. Shaver, “High Efficiency binary Lenses,” Opt. Commun. 53, 353–358 (1985); G. J. Swanson, W. B. Veldkamp, “Binary Lenses for Use at 10.6 Micrometers,” Opt. Eng. 24, 791–795 (1985); G. J. Swanson, W. B. Veldkamp, “Diffractive Optical Elements for Use in Infrared Systems,” Opt. Eng. 28, 605–608 (1989).
[CrossRef]

L. d’Auria, J. P. Huignard, A. M. Roy, E. Spitz, “Photolithographic Fabrication of Thin Film Lenses,” Opt. Commun. 5, 232–235 (1972).
[CrossRef]

Opt. Eng. (2)

D. Falkis, G. M. Morris, “Broadband Imaging with Holographic Lenses,” Opt. Eng. 28, 592–598 (1989).

R. E. Hopkins, M. J. Buzawa, “Optics for Laser Scanning,” Opt. Eng. 15, 90–94 (1976).

Opt. News (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,” 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, 41–42 (1989).
[CrossRef]

Opt. Spectrosc. (1)

A. I. Tudorovskii, “An Objective with a Phase Plate,” Opt. Spectrosc. 6, 126–133 (1959).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

T. A. Fritz, J. A. Cox, “Diffractive Optics for Broadband Infrared Imagers: Design Examples,” Proc. Soc. Photo-Opt. Instrum. Eng. 1052, 25–31 (1989).

Other (14)

See Ref. 31, pp. 93–94.

Ref. 18, pp. 93–98.

V. P. Koronkevich, “Computer Synthesis of Diffraction Optical Elements,” in Optical Processing and Computing, H. H. Arsenault, T. Szoplik, B. Macukow, Eds (Academic, Boston, 1989), pp. 277–313.

W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, 1986), pp. 130–140.

Ref. 18, pp. 226–234.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 77–83.

Ref. 17, pp. 1702.

Ref. 18, pp. 148–152.

Ref. 18, pp. 241–246.

Ref. 18, p. 146.

D. C. Sinclair, “Designing Diffractive Optics using the Sweatt Model,” Sinclair Optics Design Notes, Vol. 1, No. 1 (Winter1990).

Ref. 18, pp. 152–153.

super-oslo is a trademark of Sinclair Optics, 6780 Palmyra Rd., Fairport, NY 14450.

Lord Rayleigh, laboratory notebook, 11 Apr. 1871, quoted in R. W. Wood, Physical Optics (Macmillan, New York, 1934), pp. 37–38.

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Figures (7)

Fig. 1
Fig. 1

Illustration of various paraxial quantities. In the figure, l and u′ are negative as shown, while the other quantities are positive as shown.

Fig. 2
Fig. 2

Construction used to find the position of the entrance pupil.

Fig. 3
Fig. 3

Finite conjugate imaging example discussed in the text.

Fig. 4
Fig. 4

Modulation transfer functions for a 3:1 imaging lens. The plots are given for three field positions: (a) on-axis, (b) 0.7 field, and (c) full field. In each part of the figure, three curves are drawn: the diffraction limit and lens MTF for tangential (T) and sagittal (S) orientations of the target grating. The S and T curves are formally identical for the on-axis case and nearly indistinguishable in (b) and (c).

Fig. 5
Fig. 5

Layout of the diffractive laser scan lens.

Fig. 6
Fig. 6

Evaluation of the diffractive scan lens discussed in the text: (a) geometric rms spot size as a function of scan angle; (b) scan error as a function of scan angle. Scan error is defined as the difference between the centroid of the spot diagram and the product f · θ for a given angle.

Fig. 7
Fig. 7

Evaluation of the planar diffractive scan lens discussed in the text: (a) geometric rms spot size as a function of scan angle; (b) scan error as a function of scan angle.

Equations (48)

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Φ ( r ) = 2 π ( A r 2 + G r 4 + ) ,
W ( h , ρ , cos ϕ p ) = S I ρ 4 + ½ S II h ρ 3 cos ϕ p + ½ S III h 2 ρ 2 cos 2 ϕ p + ¼ ( S III + S IV ) h 2 ρ 2 + ½ S V h 3 ρ cos ϕ p .
E = c 1 + c 2 c 1 - c 2 ,             T = u + u u - u .
S I = y 4 ϕ 3 4 [ ( n n - 1 ) 2 + n + 2 n ( n - 1 ) 2 E 2 + 4 ( n + 1 ) n ( n - 1 ) E T + 3 n + 2 n T 2 ] ;
S II = - y 2 ϕ 2 H 2 [ n + 1 n ( n - 1 ) E + 2 n + 1 n T ] ;
S III = H 2 ϕ ;
S IV = H 2 ϕ n ;
S V = 0.
B = c 1 + c 2 ( n - 1 ) ( c 1 - c 2 ) = c 1 + c 2 ϕ = E n - 1 .
B = 2 c s ϕ .
S I = y 4 ϕ 3 4 ( 1 + B 2 + 4 B T + 3 T 2 ) - 8 λ G p y 4 ;
S II = - y 2 ϕ 2 H 2 ( B + 2 T ) ;
S III = H 2 ϕ ;
S IV = 0 ;
S V = 0.
S I * = S I ,
S II * = S II + ɛ S I ,
S III * = S III + 2 ɛ S II + ɛ 2 S I ,
S IV * = S IV ,
S V * = S V + ɛ ( 3 S III + S IV ) + 3 ɛ 2 S II + ɛ 3 S I .
ɛ = δ y ¯ y ,
G = ϕ 0 3 ( 1 - T 2 ) 32 p 0 λ 0 ,
ɛ = 2 H y 2 ϕ 0 ( B + 2 T ) .
S I * = y 4 ϕ 0 3 4 ( B + 2 T ) 2 ,
S II * = S III * = S IV * = 0 ,
S V * = 2 H 3 y 2 ( B + 2 T ) .
y 30.4 λ 0 ϕ 0 3 ( B + 2 T ) 2 4 .
δ η η = S V * 2 H = H 2 y 2 ( B + 2 T ) ,
y ¯ - l = y ¯ object l - l ¯ ,
l ¯ = 2 ϕ 0 ( B + T - 1 ) .
l ¯ = 2 ϕ 0 ( B + T + 1 ) .
c 1 = ϕ 0 2 ( B + 1 n s - 1 ) ,
c 2 = ϕ 0 2 ( B - 1 n s - 1 ) .
z ( r ) = c r 2 1 + 1 - ( c r ) 2 + d r 4 + ,
d = ϕ 0 3 ( T 2 - 1 ) 32 Δ n ,
n s ( λ , p ) = p λ p 0 λ 0 [ n s ( λ 0 , p 0 ) - 1 ] + 1.
r m = 2 m λ 0 f ,
ɛ y = f [ sin ( θ ) - tan ( θ ) ] = f [ u ¯ h 1 + ( u ¯ h ) 2 ) - u ¯ h ] = - 1 2 f u ¯ 3 h 3 + ,
S V * = u ¯ 3 y .
ɛ y = 1 n u W ρ y
ɛ y , diffractive = - ½ f u ¯ 3 h 3 + ,
ɛ y = f θ - f tan ( θ ) = f [ tan - 1 ( u ¯ h ) - u ¯ h ] = - f u ¯ 3 h 3 + .
S V , scan * = y u ¯ 3 .
c s , scan = - ϕ 0 2 .
l ¯ = - 2 3 ϕ 0 .
G A = ϕ 0 3 ( 1 - 3 2 ) 8 λ 0 ,             l ¯ A = - 1 ϕ 0 ( 1 + 3 3 ) ;
G B = ϕ 0 3 ( 1 + 3 2 ) 8 λ 0 ,             l ¯ B = - 1 ϕ 0 ( 1 - 3 3 ) .
S II * = - ( 1 + 3 ) 2 u ¯ y 3 ϕ 0 2 .

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