Abstract

The third-order aberrations of a diffractive optical element with paraxial zone spacings are derived as a function of aperture stop position. It is shown that by placing the stop in the front focal plane, coma and astigmatism are identically zero, assuming an infinitely distant object. In addition, since the element is diffractive, the Petzval sum is also zero. Modulation transfer function comparisons with other lenses are given. The correction of spherical aberration using an aspheric plate located in the aperture stop and nonmonochromatic imaging performance are discussed. The distortion of the resulting system is shown to be the proper amount for use as a Fourier transform lens. An estimate for the space–bandwidth product of this Fourier transform system is given.

© 1989 Optical Society of America

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References

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  1. K. Kamiya, “Theory of Fresnel Zone Plate,” Sci. Light 12, 35–49 (1963).
  2. R. W. Meier, “Magnification and Third-Order Aberrations in Holography,” J. Opt. Soc. Am. 55, 987–992 (1965).
  3. E. B. Champagne, “Nonparaxial Imaging, Magnification, and Aberration Properties in Holography” J. Opt. Soc. Am. 57, 51–55 (1967).
    [CrossRef]
  4. M. Young, “Zone Plates and Their Aberrations,” J. Opt. Soc. Am. 62, 972–976 (1972).
    [CrossRef]
  5. W. C. Sweatt, “Describing Holographic and Optical Elements as Lenses,” J. Opt. Soc. Am. 67, 803–808 (1977).
    [CrossRef]
  6. W. A. Kleinhans, “Aberrations of Curved Zone Plates and Fresnel Lenses,” Appl. Opt. 16, 1701–1704 (1977).
    [CrossRef] [PubMed]
  7. S. T. Bobrov, Yu. G. Turkevich, “Method of Calculating the Wave Aberrations of Complex Holographic Systems,” Opt. Spectrosc. USSR 46, 555–557 (1979).
  8. M. A. Gan, “Third-Order Aberrations and the Fundamental Parameters of Axisymmetrical Holographic Elements,” Opt. Spectrosc. USSR 47, 419–422 (1979).
  9. L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The Kinoform: a New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150–155 (1969).
    [CrossRef]
  10. J. A. Jordan, P. M. Hirsch, L. B. Lesem, D. L. Van Rooy, “Kinoform Lenses,” Appl Opt. 9, 1883–1887 (1970).
    [PubMed]
  11. D. A. Buralli, G. M. Morris, J. R. Rogers, “Optical Performance of Holographic Kinoforms,” Appl. Opt. 28, 976–983 (1989).
    [CrossRef] [PubMed]
  12. W. T. Welford, Abberations of Optical Systems (Hilger, Bristol, 1986), pp. 226–234.
  13. Reference 12, pp. 130–140.
  14. R. Kingslake, Lens Design Fundamentals (Academic, Orlando, FL, 1978), p. 113.
  15. Reference 12, pp. 148–152.
  16. Reference 14, pp. 211–215.
  17. Reference 14, pp. 293.
  18. Super-Oslo is a trademark of Sinclair Optics, 6780 Palmyra Rd, Fairport, NY 14450.
  19. F. D. Cruickshank, G. A. Hills, “Use of Optical Aberration Coefficients in Optical Design,” J. Opt. Soc. Am. 50, 379–387 (1960).
    [CrossRef]
  20. W. T. Welford, “Aplanatic Hologram Lenses on Spherical Surfaces” Opt. Commun. 9, 268–269 (1973);see also W. T. Welford, “Isoplanatism and Holography” Opt. Commun. 8, 239–243 (1973).
    [CrossRef]
  21. R.W. Smith, “A Flat Field Holographic Lens with no First Order Astigmatism,” Opt. Commun. 19, 245–247 (1976);see also R. W. Smith, “Astigmatism Free Holographic Lens Elements” Opt. Commun. 21, 102–105 (1977);R. W. Smith, “The s and t Formulae for Holographic Lens Elements,” Opt. Commun. 21, 106–109 (1977).
    [CrossRef]
  22. Reference 12, pp. 152–153.
  23. E. H. Linfoot, “On the Optics of the Schmidt Camera,” Mon. Not. R. Astron. Soc. 109, 279–297 (1949).
  24. Phosphor data taken from “Optical Characteristics of Cathode Ray Tube Screens,” TEPAC Publication 116, published by the EIA Tube Engineering Panel Advisory Council (Dec.1980).
  25. B. A. F. Blandford, “A New Lens System for Use in Optical Data-Processing,” in Optical Instruments and Techniques 1969, J. H. Dickson, Ed. (Oriel, Newcastle upon Tyne, 1970), pp. 435–443.
  26. K. von Bieren, “Lens Design for Optical Fourier Transform Systems,” Appl. Opt. 10, 2739–2742 (1971).
    [CrossRef] [PubMed]
  27. Reference 12, pp. 93–98.
  28. M. Reiss, “The cos4 Law of Illumination,” J. Opt. Soc. Am. 35, 283–288 (1945).
    [CrossRef]
  29. I. C. Gardner, “Validity of the Cosine-Fourth-Power Law of Illumination,” J. Res. Natl. Bur. Stand. 39, 213–219 (1947).
    [CrossRef]
  30. M. Reiss, “Notes on the cos4 Law of Illumination,” J. Opt. Soc. Am. 38, 980–986 (1948).
    [CrossRef] [PubMed]
  31. Reference 12, pp. 241–246.
  32. Y. Matsui, S. Minami, S. Yamaguchi, “Fourier Transform Lens System,” U.S. Patent4,189,214 (19Feb.1980).
  33. Reference 12, p. 244.
  34. Reference 12, p. 116.
  35. J. Kedmi, A. A. Friesem, “Optimal Holographic Fourier-Transform Lens,” Appl. Opt. 23, 4015–4019 (1984).
    [CrossRef] [PubMed]
  36. A. W. Lohmann, “Parallel Interfacing of Integrated Optics with Free-Space Optics” Optik 76, 53–56 (1987).

1989 (1)

1987 (1)

A. W. Lohmann, “Parallel Interfacing of Integrated Optics with Free-Space Optics” Optik 76, 53–56 (1987).

1984 (1)

1980 (1)

Phosphor data taken from “Optical Characteristics of Cathode Ray Tube Screens,” TEPAC Publication 116, published by the EIA Tube Engineering Panel Advisory Council (Dec.1980).

1979 (2)

S. T. Bobrov, Yu. G. Turkevich, “Method of Calculating the Wave Aberrations of Complex Holographic Systems,” Opt. Spectrosc. USSR 46, 555–557 (1979).

M. A. Gan, “Third-Order Aberrations and the Fundamental Parameters of Axisymmetrical Holographic Elements,” Opt. Spectrosc. USSR 47, 419–422 (1979).

1977 (2)

1976 (1)

R.W. Smith, “A Flat Field Holographic Lens with no First Order Astigmatism,” Opt. Commun. 19, 245–247 (1976);see also R. W. Smith, “Astigmatism Free Holographic Lens Elements” Opt. Commun. 21, 102–105 (1977);R. W. Smith, “The s and t Formulae for Holographic Lens Elements,” Opt. Commun. 21, 106–109 (1977).
[CrossRef]

1973 (1)

W. T. Welford, “Aplanatic Hologram Lenses on Spherical Surfaces” Opt. Commun. 9, 268–269 (1973);see also W. T. Welford, “Isoplanatism and Holography” Opt. Commun. 8, 239–243 (1973).
[CrossRef]

1972 (1)

1971 (1)

1970 (1)

J. A. Jordan, P. M. Hirsch, L. B. Lesem, D. L. Van Rooy, “Kinoform Lenses,” Appl Opt. 9, 1883–1887 (1970).
[PubMed]

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).
[CrossRef]

1967 (1)

1965 (1)

1963 (1)

K. Kamiya, “Theory of Fresnel Zone Plate,” Sci. Light 12, 35–49 (1963).

1960 (1)

1949 (1)

E. H. Linfoot, “On the Optics of the Schmidt Camera,” Mon. Not. R. Astron. Soc. 109, 279–297 (1949).

1948 (1)

1947 (1)

I. C. Gardner, “Validity of the Cosine-Fourth-Power Law of Illumination,” J. Res. Natl. Bur. Stand. 39, 213–219 (1947).
[CrossRef]

1945 (1)

Blandford, B. A. F.

B. A. F. Blandford, “A New Lens System for Use in Optical Data-Processing,” in Optical Instruments and Techniques 1969, J. H. Dickson, Ed. (Oriel, Newcastle upon Tyne, 1970), pp. 435–443.

Bobrov, S. T.

S. T. Bobrov, Yu. G. Turkevich, “Method of Calculating the Wave Aberrations of Complex Holographic Systems,” Opt. Spectrosc. USSR 46, 555–557 (1979).

Buralli, D. A.

Champagne, E. B.

Cruickshank, F. D.

Friesem, A. A.

Gan, M. A.

M. A. Gan, “Third-Order Aberrations and the Fundamental Parameters of Axisymmetrical Holographic Elements,” Opt. Spectrosc. USSR 47, 419–422 (1979).

Gardner, I. C.

I. C. Gardner, “Validity of the Cosine-Fourth-Power Law of Illumination,” J. Res. Natl. Bur. Stand. 39, 213–219 (1947).
[CrossRef]

Hills, G. A.

Hirsch, P. M.

J. A. Jordan, P. M. Hirsch, L. B. Lesem, D. L. Van Rooy, “Kinoform Lenses,” Appl Opt. 9, 1883–1887 (1970).
[PubMed]

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The Kinoform: a New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

Jordan, J. A.

J. A. Jordan, P. M. Hirsch, L. B. Lesem, D. L. Van Rooy, “Kinoform Lenses,” Appl Opt. 9, 1883–1887 (1970).
[PubMed]

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The Kinoform: a New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

Kamiya, K.

K. Kamiya, “Theory of Fresnel Zone Plate,” Sci. Light 12, 35–49 (1963).

Kedmi, J.

Kingslake, R.

R. Kingslake, Lens Design Fundamentals (Academic, Orlando, FL, 1978), p. 113.

Kleinhans, W. A.

Lesem, L. B.

J. A. Jordan, P. M. Hirsch, L. B. Lesem, D. L. Van Rooy, “Kinoform Lenses,” Appl Opt. 9, 1883–1887 (1970).
[PubMed]

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The Kinoform: a New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

Linfoot, E. H.

E. H. Linfoot, “On the Optics of the Schmidt Camera,” Mon. Not. R. Astron. Soc. 109, 279–297 (1949).

Lohmann, A. W.

A. W. Lohmann, “Parallel Interfacing of Integrated Optics with Free-Space Optics” Optik 76, 53–56 (1987).

Matsui, Y.

Y. Matsui, S. Minami, S. Yamaguchi, “Fourier Transform Lens System,” U.S. Patent4,189,214 (19Feb.1980).

Meier, R. W.

Minami, S.

Y. Matsui, S. Minami, S. Yamaguchi, “Fourier Transform Lens System,” U.S. Patent4,189,214 (19Feb.1980).

Morris, G. M.

Reiss, M.

Rogers, J. R.

Smith, R.W.

R.W. Smith, “A Flat Field Holographic Lens with no First Order Astigmatism,” Opt. Commun. 19, 245–247 (1976);see also R. W. Smith, “Astigmatism Free Holographic Lens Elements” Opt. Commun. 21, 102–105 (1977);R. W. Smith, “The s and t Formulae for Holographic Lens Elements,” Opt. Commun. 21, 106–109 (1977).
[CrossRef]

Sweatt, W. C.

Turkevich, Yu. G.

S. T. Bobrov, Yu. G. Turkevich, “Method of Calculating the Wave Aberrations of Complex Holographic Systems,” Opt. Spectrosc. USSR 46, 555–557 (1979).

Van Rooy, D. L.

J. A. Jordan, P. M. Hirsch, L. B. Lesem, D. L. Van Rooy, “Kinoform Lenses,” Appl Opt. 9, 1883–1887 (1970).
[PubMed]

von Bieren, K.

Welford, W. T.

W. T. Welford, “Aplanatic Hologram Lenses on Spherical Surfaces” Opt. Commun. 9, 268–269 (1973);see also W. T. Welford, “Isoplanatism and Holography” Opt. Commun. 8, 239–243 (1973).
[CrossRef]

W. T. Welford, Abberations of Optical Systems (Hilger, Bristol, 1986), pp. 226–234.

Yamaguchi, S.

Y. Matsui, S. Minami, S. Yamaguchi, “Fourier Transform Lens System,” U.S. Patent4,189,214 (19Feb.1980).

Young, M.

Appl Opt. (1)

J. A. Jordan, P. M. Hirsch, L. B. Lesem, D. L. Van Rooy, “Kinoform Lenses,” Appl Opt. 9, 1883–1887 (1970).
[PubMed]

Appl. Opt. (4)

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).
[CrossRef]

J. Opt. Soc. Am. (7)

J. Res. Natl. Bur. Stand. (1)

I. C. Gardner, “Validity of the Cosine-Fourth-Power Law of Illumination,” J. Res. Natl. Bur. Stand. 39, 213–219 (1947).
[CrossRef]

Mon. Not. R. Astron. Soc. (1)

E. H. Linfoot, “On the Optics of the Schmidt Camera,” Mon. Not. R. Astron. Soc. 109, 279–297 (1949).

Opt. Commun. (2)

W. T. Welford, “Aplanatic Hologram Lenses on Spherical Surfaces” Opt. Commun. 9, 268–269 (1973);see also W. T. Welford, “Isoplanatism and Holography” Opt. Commun. 8, 239–243 (1973).
[CrossRef]

R.W. Smith, “A Flat Field Holographic Lens with no First Order Astigmatism,” Opt. Commun. 19, 245–247 (1976);see also R. W. Smith, “Astigmatism Free Holographic Lens Elements” Opt. Commun. 21, 102–105 (1977);R. W. Smith, “The s and t Formulae for Holographic Lens Elements,” Opt. Commun. 21, 106–109 (1977).
[CrossRef]

Opt. Spectrosc. USSR (2)

S. T. Bobrov, Yu. G. Turkevich, “Method of Calculating the Wave Aberrations of Complex Holographic Systems,” Opt. Spectrosc. USSR 46, 555–557 (1979).

M. A. Gan, “Third-Order Aberrations and the Fundamental Parameters of Axisymmetrical Holographic Elements,” Opt. Spectrosc. USSR 47, 419–422 (1979).

Optik (1)

A. W. Lohmann, “Parallel Interfacing of Integrated Optics with Free-Space Optics” Optik 76, 53–56 (1987).

Sci. Light (1)

K. Kamiya, “Theory of Fresnel Zone Plate,” Sci. Light 12, 35–49 (1963).

TEPAC Publication 116 (1)

Phosphor data taken from “Optical Characteristics of Cathode Ray Tube Screens,” TEPAC Publication 116, published by the EIA Tube Engineering Panel Advisory Council (Dec.1980).

Other (14)

B. A. F. Blandford, “A New Lens System for Use in Optical Data-Processing,” in Optical Instruments and Techniques 1969, J. H. Dickson, Ed. (Oriel, Newcastle upon Tyne, 1970), pp. 435–443.

Reference 12, pp. 93–98.

Reference 12, pp. 152–153.

Reference 12, pp. 241–246.

Y. Matsui, S. Minami, S. Yamaguchi, “Fourier Transform Lens System,” U.S. Patent4,189,214 (19Feb.1980).

Reference 12, p. 244.

Reference 12, p. 116.

W. T. Welford, Abberations of Optical Systems (Hilger, Bristol, 1986), pp. 226–234.

Reference 12, pp. 130–140.

R. Kingslake, Lens Design Fundamentals (Academic, Orlando, FL, 1978), p. 113.

Reference 12, pp. 148–152.

Reference 14, pp. 211–215.

Reference 14, pp. 293.

Super-Oslo is a trademark of Sinclair Optics, 6780 Palmyra Rd, Fairport, NY 14450.

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Figures (11)

Fig. 1
Fig. 1

Pictorial illustration of various paraxial quantities. The numerical value of all symbols is positive with the exception of u′.

Fig. 2
Fig. 2

Layout of telecentric paraxial diffractive lens. The aperture stop for the system is placed in the front focal plane of the lens.

Fig. 3
Fig. 3

Modulation transfer functions for f/5.6 Cooke triplet; (a) full field (9.0° off-axis); (b) 0.7 field (6.33° off-axis); (c) on-axis. The focal length is 50 mm, and the design wavelength is 0.58756 μm. The triplet system specifications are given in Table I. In each plot, three curves are given—the diffraction limited MTF and the system MTF for tangential and sagittal orientations of the target grating lines.

Fig. 4
Fig. 4

Modulation transfer functions for f/5.6 telecentric paraxial diffractive lens; (a) full field (9.0° off-axis); (b) 0.7 field (6.33° off-axis); (c) on-axis. The focal length is 50 mm, and the design wavelength is 0.58756 μm.

Fig. 5
Fig. 5

Modulation transfer functions for f/5.6 optically recorded HOE: (a) full field (9.0° off-axis); (b) 0.7 field (6.33° off-axis); (c) on-axis. The focal length is 50 mm, and the design wavelength is 0.58756 μm.

Fig. 6
Fig. 6

Modulation transfer functions for f/3.0 germanium landscape lens; (a) full field (10.0° off-axis); (b) 0.7 field (7.04° off-axis); (c) on-axis. The focal length is 50 mm, and the design wavelength is 10 μm.

Fig. 7
Fig. 7

Modulation transfer functions for f/3.0 telecentric paraxial diffractive lens; (a) full field (10.0° off-axis); (b) 0.7 field (7.04° off-axis); (c) on-axis. The focal length is 50 mm, and the design wavelength is 10 μm.

Fig. 8
Fig. 8

Layout of the telecentric paraxial diffractive lens with spherical aberration corrected by a Schmidt aspheric plate located in the aperture stop.

Fig. 9
Fig. 9

On-axis polychromatic modulation transfer functions for Gaussian line shapes of various standard deviations σ. The focal length is 50 mm, λpeak = 0.544 F system # μm, (a) σ = 1 nm; (b) σ = 2 nm; (c) σ = 5 nm; (d) σ = 10 nm.

Fig. 10
Fig. 10

On-axis polychromatic modulation transfer function for the primary yellowish green emission of a P43 phosphor. The focal length is 50 mm, λpeak = 0.544 μm, F system # = 8 .

Fig. 11
Fig. 11

Optical layout and parameter definitions for an optical Fourier transform. Spatial frequency F is related to object periodicity d by F = 1/d. For normally incident illumination, the grating equation reduces to sin(θ) = λ/d = λF.

Tables (2)

Tables Icon

Table I System Specifications for 50-mm Focal Length Cooke Triplet

Tables Icon

Table II System Specifications for 50-mm Focal Length Germanium Landscape Lens

Equations (58)

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r m = 2 m λ 0 f + ( m λ 0 ) 2 .
r m , paraxial = 2 m λ 0 f .
W ( h , ρ , cos ϕ ) = S I ρ 4 + ½ S II h ρ 3 cos ϕ + ½ S III h 2 ρ 2 cos 2 ϕ + ¼ ( S III + S IV ) h 2 ρ 2 + ½ S V h 3 ρ cos ϕ .
S I = y 4 ϕ 3 4 [ ( n n 1 ) 2 + n + 2 n ( n 1 ) 2 B 2 + 4 ( n + 1 ) n ( n 1 ) B T + 3 n + 2 n T 2 ] + 8 G y 4 ( Δ n ) ;
S II = y 2 ϕ 2 H 2 [ n + 1 n ( n 1 ) B + 2 n + 1 n T ] ;
S III = H 2 ϕ ;
S IV = H 2 ϕ n ;
S V = 0 0.
B = c 1 + c 2 c 1 c 2 , T = u + u u u .
S I = y 4 f 3 ( λ λ 0 ) 3 ,
S II = y 3 u ¯ f 2 ( λ λ 0 ) 2 ,
S III = y 2 u ¯ 2 f ( λ λ 0 ) ,
S IV = 0 ,
S V = 0.
ϕ ( λ ) = λ λ 0 ϕ 0 = λ λ 0 1 f ,
H = u ¯ y ,
G = n 2 ( λ 0 ) 8 f 3 [ n ( λ 0 ) 1 ] 3 .
S I = y 4 f 3 ( λ 3 λ λ 0 2 λ 0 3 ) .
S I * = S I ,
S II * = S II + y ¯ y S I ,
S III * = S III + 2 y ¯ y S II + ( y ¯ y ) 2 S I ,
S IV * = S IV ,
S V * = S V + y ¯ y ( 3 S III + S IV ) + 3 ( y ¯ y ) 2 S II + ( y ¯ y ) 3 S I .
y ¯ = t u ¯ ,
S I * = y 4 f 3 ,
S II * = y 3 u ¯ ( t f ) f 3 ,
S III * = y 2 u ¯ 2 ( t f ) 2 f 3 ,
S IV * = 0 ,
S V * = y u ¯ 3 t ( 3 f 2 3 t f + t 2 ) f 3 .
S I * = y 4 f 3 ,
S II * = S III * = S IV * = 0 ,
S V * = y u ¯ 3 .
y + y ¯ a ,
y + f tan ( θ ) a .
f 2 y = F system # ,
f 2 a = F lens # ,
θ max = tan 1 [ 1 2 ( 1 F lens # 1 F system # ) ] .
δ S I = 8 G y 4 Δ ( n ) ,
y 4 f 3 + 8 G y 4 Δ ( n ) = 0 .
G = 1 8 f 3 Δ ( n ) .
| W 040 W 060 | = 8 ( F system # ) 2 ,
y = f sin ( θ ) f tan ( θ ) = f [ ( θ θ 3 6 + ) ( θ + θ 3 3 + ) ] = 1 2 f θ 3 .
y = σ 5 h 3 ,
σ 5 = ½ S V * n u ,
σ 5 = 1 2 f u ¯ 3 .
y 7.6 f 3 λ 4
λ F max = sin ( θ max ) .
F max = 1 λ sin { tan 1 [ 1 2 ( 1 F lens # 1 F system # ) ] } .
F max = 1 2 λ ( 1 F lens # 1 F system # ) .
SBP 1 D = 2 y F max = y λ ( 1 F lens # 1 F system # ) ,
W ( h , ρ , cos ϕ ) = W 020 ρ 2 + W 040 ρ 4 ,
X = 2 W 020 n u 2 .
X = f ( 4 F system # ) 2 .
SBP 1 D = y λ ( 1 F lens # 2 y f ) .
y = f 4 F lens # ,
y = a / 2 .
F system # = 2 F lens # .
t ( λ ) = λ 0 λ f .

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