Abstract

We apply the rigorous electromagnetic grating theory and an electromagnetic wave-propagation model with Kirchhoff boundary conditions to analyze and optimize the performance of arrays of cylindrical dielectric multilevel diffractive lenses. We show that the diffraction efficiency of a high-numerical-aperture lens array can be improved if the surface-relief profile is constructed with the use of rigorously optimized grating groove structures, which differ significantly from a multilevel approximation of a triangular profile.

© 1993 Optical Society of America

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References

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  1. H. Nishihara, T. Suhara, “Micro Fresnel lenses,” in Progress in Optics XXIV, E. Wolf, ed. (North-Holland, Amsterdam, 1987), pp. 1–40.
    [CrossRef]
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  3. M. Haruna, M. Takahashi, K. Wakahayashi, H. Nishihara, “Laser beam lithographed micro-Fresnel lenses,” Appl. Opt. 29, 5120–5125 (1990).
    [CrossRef] [PubMed]
  4. L. D’Auria, J. P. Huignard, A. M. Roy, E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972).
    [CrossRef]
  5. J. Jahns, S. J. Walker, “Two-dimensional arrays of microlenses by thin film deposition,” Appl. Opt. 29, 931–936 (1990).
    [CrossRef] [PubMed]
  6. T. Shiono, M. Kitagawa, K. Setsune, T. Mitsuyu, “Reflection micro-Fresnel lenses and their use in an integrated focus sensor,” Appl. Opt. 28, 3434–3442 (1989).
    [CrossRef] [PubMed]
  7. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  8. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984), pp. 1–67.
    [CrossRef]
  9. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  10. N. C. Gallagher, S. S. Naqvi, “Diffractive optics: scalar and non-scalar design analysis,” in Holographic Optics: Optically and Computer-Generated, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum Eng.1052, 32–40 (1989).
    [CrossRef]
  11. 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. N. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 116–124 (1990).
    [CrossRef]
  12. A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction analysis of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
    [CrossRef]
  13. A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336, (1992).
    [CrossRef] [PubMed]
  14. E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
    [CrossRef]
  15. A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, J. Tuovinen, “Rigorous diffraction theory of binary optical interconnects,” in Holographic Optics III: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1507, 224–238 (1991).
    [CrossRef]
  16. E. G. Johnson, A. D. Kathman, “Rigorous electromagnetic modeling of diffractive optical elements,” in International Conference on the Application and Theory of Periodic Structures, M. Lerner, W. R. McKinney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1545, 209–216 (1991).
    [CrossRef]
  17. E. Noponen, J. Turunen, A. Vasara, “Parametric optimization of multilevel diffractive optical elements by electromagnetic theory,” Appl. Opt. 31, 5910–5912 (1992).
    [CrossRef] [PubMed]
  18. J. Turunen, J. Fagerholm, A. Vasara, M. R. Taghizadeh, “Detour-phase kino-form interconnects: the concept and fabrication considerations,” J. Opt. Soc. Am. A 7, 1202–1208 (1990).
    [CrossRef]
  19. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
    [CrossRef] [PubMed]
  20. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1507 (1966).
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  21. F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with complex dielectric constant,” J. Opt. Soc. Am. 63, 37–45 (1973).
    [CrossRef]
  22. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  23. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  24. W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62, 1195–1201 (1972).
    [CrossRef]
  25. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).
  26. A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
    [CrossRef]
  27. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Phys. Soc. London Section A 253, 358–379 (1959).
    [CrossRef]
  28. P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics VI, E. Wolf, ed. (North-Holland, Amsterdam, 1967), pp. 53–69.
    [CrossRef]
  29. P. Vincent, “A finite difference method for dielectric and conducting cross gratings,” Opt. Commun. 26, 293–296 (1978).
    [CrossRef]
  30. G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Cross gratings: a theory and its implementation,” Appl. Phys. 18, 39–52 (1979).
    [CrossRef]
  31. R. C. McPhedran, D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977).
    [CrossRef]
  32. M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  33. S. T. Han, Y.-L. Tsao, R. M. Walser, M. E Becker, “Electromagnetic scattering of two-dimensional surface-relief diffraction gratings,” Appl. Opt. 31, 2343–2352 (1992).
    [CrossRef] [PubMed]

1992 (4)

1991 (1)

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction analysis of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

1990 (3)

1989 (1)

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1983 (1)

1979 (1)

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Cross gratings: a theory and its implementation,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

1978 (2)

K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
[CrossRef]

P. Vincent, “A finite difference method for dielectric and conducting cross gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

1977 (1)

R. C. McPhedran, D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

1973 (1)

1972 (2)

W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62, 1195–1201 (1972).
[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]

1967 (1)

1966 (1)

1965 (1)

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

1959 (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Phys. Soc. London Section A 253, 358–379 (1959).
[CrossRef]

Becker, M. E

Beckmann, P.

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics VI, E. Wolf, ed. (North-Holland, Amsterdam, 1967), pp. 53–69.
[CrossRef]

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. N. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 116–124 (1990).
[CrossRef]

Boivin, A.

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Burckhardt, C. B.

Carter, W. H.

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. N. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 116–124 (1990).
[CrossRef]

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]

Derrick, G. H.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Cross gratings: a theory and its implementation,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Fagerholm, J.

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. N. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 116–124 (1990).
[CrossRef]

Gallagher, N. C.

N. C. Gallagher, S. S. Naqvi, “Diffractive optics: scalar and non-scalar design analysis,” in Holographic Optics: Optically and Computer-Generated, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum Eng.1052, 32–40 (1989).
[CrossRef]

Gaylord, T. K.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
[CrossRef]

Goodman, J. W.

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

Han, S. T.

Haruna, M.

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]

Ichikawa, H.

Jaakkola, T.

Jahns, J.

Johnson, E. G.

E. G. Johnson, A. D. Kathman, “Rigorous electromagnetic modeling of diffractive optical elements,” in International Conference on the Application and Theory of Periodic Structures, M. Lerner, W. R. McKinney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1545, 209–216 (1991).
[CrossRef]

Kaspar, F. G.

Kathman, A. D.

E. G. Johnson, A. D. Kathman, “Rigorous electromagnetic modeling of diffractive optical elements,” in International Conference on the Application and Theory of Periodic Structures, M. Lerner, W. R. McKinney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1545, 209–216 (1991).
[CrossRef]

Kitagawa, M.

Knop, K.

Kuisma, S.

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. N. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 116–124 (1990).
[CrossRef]

Lohmann, A. W.

Maystre, D.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Cross gratings: a theory and its implementation,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

R. C. McPhedran, D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984), pp. 1–67.
[CrossRef]

McPhedran, R. C.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Cross gratings: a theory and its implementation,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

R. C. McPhedran, D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

Miller, J. M.

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336, (1992).
[CrossRef] [PubMed]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction analysis of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, J. Tuovinen, “Rigorous diffraction theory of binary optical interconnects,” in Holographic Optics III: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1507, 224–238 (1991).
[CrossRef]

Mitsuyu, T.

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
[CrossRef]

Naqvi, S. S.

N. C. Gallagher, S. S. Naqvi, “Diffractive optics: scalar and non-scalar design analysis,” in Holographic Optics: Optically and Computer-Generated, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum Eng.1052, 32–40 (1989).
[CrossRef]

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. N. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 116–124 (1990).
[CrossRef]

Nevière, M.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Cross gratings: a theory and its implementation,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Nishihara, H.

M. Haruna, M. Takahashi, K. Wakahayashi, H. Nishihara, “Laser beam lithographed micro-Fresnel lenses,” Appl. Opt. 29, 5120–5125 (1990).
[CrossRef] [PubMed]

H. Nishihara, T. Suhara, “Micro Fresnel lenses,” in Progress in Optics XXIV, E. Wolf, ed. (North-Holland, Amsterdam, 1987), pp. 1–40.
[CrossRef]

Noponen, E.

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336, (1992).
[CrossRef] [PubMed]

E. Noponen, J. Turunen, A. Vasara, “Parametric optimization of multilevel diffractive optical elements by electromagnetic theory,” Appl. Opt. 31, 5910–5912 (1992).
[CrossRef] [PubMed]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction analysis of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, J. Tuovinen, “Rigorous diffraction theory of binary optical interconnects,” in Holographic Optics III: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1507, 224–238 (1991).
[CrossRef]

Paris, D. P.

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Phys. Soc. London Section A 253, 358–379 (1959).
[CrossRef]

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]

Setsune, K.

Shiono, T.

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]

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

Suhara, T.

H. Nishihara, T. Suhara, “Micro Fresnel lenses,” in Progress in Optics XXIV, E. Wolf, ed. (North-Holland, Amsterdam, 1987), pp. 1–40.
[CrossRef]

Taghizadeh, M. R.

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336, (1992).
[CrossRef] [PubMed]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction analysis of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

J. Turunen, J. Fagerholm, A. Vasara, M. R. Taghizadeh, “Detour-phase kino-form interconnects: the concept and fabrication considerations,” J. Opt. Soc. Am. A 7, 1202–1208 (1990).
[CrossRef]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, J. Tuovinen, “Rigorous diffraction theory of binary optical interconnects,” in Holographic Optics III: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1507, 224–238 (1991).
[CrossRef]

Takahashi, M.

Tsao, Y.-L.

Tuovinen, J.

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, J. Tuovinen, “Rigorous diffraction theory of binary optical interconnects,” in Holographic Optics III: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1507, 224–238 (1991).
[CrossRef]

Turunen, J.

Vasara, A.

Vincent, P.

P. Vincent, “A finite difference method for dielectric and conducting cross gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Wakahayashi, K.

Walker, S. J.

Walser, R. M.

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. N. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 116–124 (1990).
[CrossRef]

Westerholm, J.

Wolf, E.

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Phys. Soc. London Section A 253, 358–379 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Appl. Opt. (7)

Appl. Phys. (2)

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Cross gratings: a theory and its implementation,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

R. C. McPhedran, D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (2)

Opt. Commun. (3)

L. D’Auria, J. P. Huignard, A. M. Roy, E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972).
[CrossRef]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction analysis of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

P. Vincent, “A finite difference method for dielectric and conducting cross gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Phys. Rev. B (1)

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Proc. Phys. Soc. London Section A (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Phys. Soc. London Section A 253, 358–379 (1959).
[CrossRef]

Other (11)

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics VI, E. Wolf, ed. (North-Holland, Amsterdam, 1967), pp. 53–69.
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

N. C. Gallagher, S. S. Naqvi, “Diffractive optics: scalar and non-scalar design analysis,” in Holographic Optics: Optically and Computer-Generated, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum Eng.1052, 32–40 (1989).
[CrossRef]

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. N. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 116–124 (1990).
[CrossRef]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, J. Tuovinen, “Rigorous diffraction theory of binary optical interconnects,” in Holographic Optics III: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1507, 224–238 (1991).
[CrossRef]

E. G. Johnson, A. D. Kathman, “Rigorous electromagnetic modeling of diffractive optical elements,” in International Conference on the Application and Theory of Periodic Structures, M. Lerner, W. R. McKinney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1545, 209–216 (1991).
[CrossRef]

H. Nishihara, T. Suhara, “Micro Fresnel lenses,” in Progress in Optics XXIV, E. Wolf, ed. (North-Holland, Amsterdam, 1987), pp. 1–40.
[CrossRef]

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

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984), pp. 1–67.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Geometry and notation: focusing of an electromagnetic plane wave by a diffractive-lens array.

Fig. 2
Fig. 2

Focal spot profiles in the geometrical-optics focal plane for an array of four-level F/1 lenses, with facet size d = 100 λ. Solid curves, rigorous boundary conditions; dashed curves, Kirchhoff boundary conditions. (a) Electric energy density, (b) magnetic energy density, (c) electromagnetic energy density, (d) axial component of the Poynting vector.

Fig. 3
Fig. 3

Axial-intensity profile of the four-level F/1 lens array, with d = 100 λ. Solid curves, rigorous boundary conditions; dashed curves, Kirchhoff boundary conditions. (a) Electric energy density, (b) magnetic energy density, (c) electromagnetic energy density, (d) axial component of the Poynting vector.

Fig. 4
Fig. 4

First-order diffraction efficiencies η1 of various multilevel gratings as a function of the normalized local period dL/λ.

Fig. 5
Fig. 5

Approximate model for diffraction by a dielectric triangular-profile grating. (a) Angle α is below the angle of total internal reflection, (b) α is above the angle of total internal reflection.

Fig. 6
Fig. 6

Results of the optimization of the local groove structure of a four-level grating. (a) First-order diffraction efficiency η1, (b) phase ψ of the first diffraction order in radians.

Fig. 7
Fig. 7

Surface-relief profile of an F/0.5 lens, with d = 100 λ. (a) Construction with Kirchhoff boundary conditions, (b) construction with rigorously optimized groove structures.

Fig. 8
Fig. 8

Spatial-power spectrum produced by the four-level F/1 lens array, with d = 100 λ. (a) Kirchhoff boundary conditions, (b) rigorous theory and construction with Kirchhoff boundary conditions, (c) rigorous theory and construction with local optimization results.

Fig. 9
Fig. 9

Peak value of the z component of the Poynting vector as a function of shifting the ideal profile with respect to the four quantization levels [ϕ0 in Eq. (12) ranges from −2π to 0]. The focal length is F = 100 λ. Dotted curves, Kirchhoff boundary conditions; solid curves, rigorous theory and construction with Kirchhoff boundary conditions; dashed curves, rigorous theory and construction with local grating-profile optimization results. (a) F/d = 0.25, (b) F/d = 0.5, (c) F/d = 1.

Fig. 10
Fig. 10

Off-axis aberrations of cylindrical F/1 lens arrays. (a) Angle of incidence θ = 0, (b) θ = 2°, (c) θ = 4°. Solid curves, results for the rigorously optimized profile in plane z = 100 λ; dashed curves, results for the nonoptimized profile in the plane z = 97.67 λ, where the peak efficiency is highest for the on-axis spot; dotted curves, prediction of the approximate theory for this profile in the same plane.

Equations (16)

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E I ( x , y , z ) = ŷ exp [ ikn ( x sin θ + z cos θ ) ] ,
H I ( x , y , z ) = n ( x ̂ cos θ + sin θ ) × exp [ ikn ( x sin θ + z cos θ ) ] ,
E D ( x , y , z ) = ŷ m = T m exp [ i ( γ m x + t m z ) ] ,
H D ( x , y , z ) = k 1 m = T m ( x ̂ t m + γ m ) exp [ i ( γ m x + t m z ) ] ,
γ m = k n sin θ + 2 π m / d ,
t m = { ( k 2 γ m 2 ) 1 / 2 if | γ m | k i ( γ m 2 k 2 ) 1 / 2 if | γ m | > k ,
T m = d 1 0 d E y ( x , y , 0 ) exp ( i γ m x ) d x ,
w e ( x , y , z ) = w e ( x , y , z ) D w e ( x , y , z ) I = 1 n 2 | m T m e m ( x , z ) | 2 ,
w h ( x , y , z ) = w h ( x , y , z ) D w h ( x , y , z ) I = 1 ( k n ) 2 [ | m T m t m e m ( x , z ) | 2 + | m T m γ m e m ( x , z ) | 2 ] ,
w ( x , y , z ) = w e ( x , y , z ) D + w h ( x , y , z ) D w e ( x , y , z ) I + w h ( x , y , z ) I = ½ [ w e ( x , y , z ) + w h ( x , y , z ) ] ,
S z ( x , y , z ) = S z ( x , y , z ) D S z ( x , y , z ) I = ( k n cos θ ) 1 [ m T m e m ( x , z ) n T n * t n * e n * ( x , z ) ] ,
ϕ ( x ) = 2 π λ { F [ F 2 + ( x d / 2 ) 2 ] 1 / 2 } + ϕ 0 ,
ϕ ( x ) = [ ϕ ( x q ) ] mod 2 π 2 π , when x ( x q , x q + 1 ) .
h ( x ) = λ 2 π ( n 1 ) ϕ ( x ) ,
E y ( x , y , 0 ) = 2 n n + 1 exp [ i ϕ ( x ) ]
d L ( x ) / λ = 2 π λ | d d x ϕ ( x ) | 1 = [ 1 + F 2 ( x d / 2 ) 2 ] 1 / 2 ,

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