Abstract

We have designed high-efficiency finite-aperture diffractive optical elements (DOE’s) with features on the order of or smaller than the wavelength of the incident illumination. The use of scalar diffraction theory is generally not considered valid for the design of DOE’s with such features. However, we have found several cases in which the use of a scalar-based design is, in fact, quite accurate. We also present a modified scalar-based iterative design method that incorporates the angular spectrum approach to design diffractive optical elements that operate in the near-field and have sub-wavelength features. We call this design method the iterative angular spectrum approach (IASA). Upon comparison with a rigorous electromagnetic analysis technique, specifically, the finite difference time-domain method (FDTD), we find that our scalar-based design method is surprisingly valid for DOE’s having sub-wavelength features.

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References

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  1. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon Press, New York, 1965).
  2. J. Goodman, Introduction to Fourier Optics (McGraw-Hill., New York, 1968).
  3. G.S. Smith. An Introduction to Classical Electromagnetic Radiation (Cambridge University Press, Cambridge, 1997).
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    [CrossRef] [PubMed]
  5. D.A. Pommet, M.G. Moharam, and E.B. Grann, "Limits of scalar diffraction theory for diffractive phase elements", Opt. Lett. 11, 1827-1834 (1995).
  6. M. G. Moharam and T. K. Gaylord, "Diffraction analysis of surface-relief gratings," J. Opt. Soc. Am. A 72, 1385-1392 (1982).
    [CrossRef]
  7. R.W. Gerchberg, W.O. Saxton. "A practical algorithm for the determination of phase from image and diffraction plane pictures" Optik 35, 237-246 (1971).
  8. Pierre St. Hilaire, "Phase profiles for holographic stereograms," Opt. Eng. 34, 83-89 (1995).
    [CrossRef]
  9. N.C. Gallagher and B. Liu, "Method for computing kinoforms that reduces image reconstruction error," Appl. Opt. 12, 2328-2335 (1973).
    [CrossRef] [PubMed]
  10. J.R. Fienup, "Iterative method applied to image reconstruction and to computer-generated holograms," Opt. Eng. 19, 297-306 (1980).
  11. F. Wyrowski, "Iterative Fourier-transform algorithm applied to computer holography," J. Opt. Soc. Am. A 5, 1058-1065 (1988).
    [CrossRef]
  12. F. Wyrowski and O. Bryngdahl, "Digital holography as part of diffractive optics," Rep. Prog. Phys. 54, 1481-1571 (1991).
    [CrossRef]
  13. J. N. Mait, "Understanding diffractive optic design in the scalar domain," J. Opt. Soc. Am. A 12, 2145- 2158 (1995).
    [CrossRef]
  14. I.O. Bohachevsky, M.E. Johnson, M.L. Stein, "Generalized simulated annealing for function optimization," Technometrics 28, 209-217 (1986).
    [CrossRef]
  15. Y. Lin, T.J. Kessler, G.N. Lawrence, "Design of continuous surface-relief phase plates by surface-based simulated annealing to achieve control of focal-plane irradiance," Opt. Lett. 21, 1703-1705 (1996).
    [CrossRef] [PubMed]
  16. B. K. Jennison, J. P. Allebach, and D. W. Sweeney, "Iterative approaches to computer-generated holog-raphy," Opt. Eng. 28, 629-637 (1989).
  17. J. Turunen, A. Vasara, and J. Westerholm, "Kinoform phase relief synthesis: a stochastic method," Opt. Eng. 28, 1162-1167 (1989).
  18. M.R. Feldman and C.C. Guest, "High-efficiency hologram encoding for generation of spot arrays," Opt. Lett. 14, 479-481 (1989).
    [CrossRef] [PubMed]
  19. J. Jiang and G. Nordin, "A rigorous unidirectional method for designing finite aperture diffractive optical elements," Opt. Express 7, 237-242 (2000), http://www.opticsexpress.org/oearchive/source/23164.htm.
    [CrossRef]
  20. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 1995).

Other (20)

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon Press, New York, 1965).

J. Goodman, Introduction to Fourier Optics (McGraw-Hill., New York, 1968).

G.S. Smith. An Introduction to Classical Electromagnetic Radiation (Cambridge University Press, Cambridge, 1997).

D. A. Gremaux and N. C. Gallagher, "Limits of scalar diffraction theory for conducting gratings," Appl. Opt. 32, 1948-1953 (1993).
[CrossRef] [PubMed]

D.A. Pommet, M.G. Moharam, and E.B. Grann, "Limits of scalar diffraction theory for diffractive phase elements", Opt. Lett. 11, 1827-1834 (1995).

M. G. Moharam and T. K. Gaylord, "Diffraction analysis of surface-relief gratings," J. Opt. Soc. Am. A 72, 1385-1392 (1982).
[CrossRef]

R.W. Gerchberg, W.O. Saxton. "A practical algorithm for the determination of phase from image and diffraction plane pictures" Optik 35, 237-246 (1971).

Pierre St. Hilaire, "Phase profiles for holographic stereograms," Opt. Eng. 34, 83-89 (1995).
[CrossRef]

N.C. Gallagher and B. Liu, "Method for computing kinoforms that reduces image reconstruction error," Appl. Opt. 12, 2328-2335 (1973).
[CrossRef] [PubMed]

J.R. Fienup, "Iterative method applied to image reconstruction and to computer-generated holograms," Opt. Eng. 19, 297-306 (1980).

F. Wyrowski, "Iterative Fourier-transform algorithm applied to computer holography," J. Opt. Soc. Am. A 5, 1058-1065 (1988).
[CrossRef]

F. Wyrowski and O. Bryngdahl, "Digital holography as part of diffractive optics," Rep. Prog. Phys. 54, 1481-1571 (1991).
[CrossRef]

J. N. Mait, "Understanding diffractive optic design in the scalar domain," J. Opt. Soc. Am. A 12, 2145- 2158 (1995).
[CrossRef]

I.O. Bohachevsky, M.E. Johnson, M.L. Stein, "Generalized simulated annealing for function optimization," Technometrics 28, 209-217 (1986).
[CrossRef]

Y. Lin, T.J. Kessler, G.N. Lawrence, "Design of continuous surface-relief phase plates by surface-based simulated annealing to achieve control of focal-plane irradiance," Opt. Lett. 21, 1703-1705 (1996).
[CrossRef] [PubMed]

B. K. Jennison, J. P. Allebach, and D. W. Sweeney, "Iterative approaches to computer-generated holog-raphy," Opt. Eng. 28, 629-637 (1989).

J. Turunen, A. Vasara, and J. Westerholm, "Kinoform phase relief synthesis: a stochastic method," Opt. Eng. 28, 1162-1167 (1989).

M.R. Feldman and C.C. Guest, "High-efficiency hologram encoding for generation of spot arrays," Opt. Lett. 14, 479-481 (1989).
[CrossRef] [PubMed]

J. Jiang and G. Nordin, "A rigorous unidirectional method for designing finite aperture diffractive optical elements," Opt. Express 7, 237-242 (2000), http://www.opticsexpress.org/oearchive/source/23164.htm.
[CrossRef]

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 1995).

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Figures (11)

Fig. 1.
Fig. 1.

Diffractive optic geometry

Fig. 2.
Fig. 2.

Heuristically designed DOE profiles

Fig. 3.
Fig. 3.

Irradiance profiles for heuristically designed DOE profiles

Fig. 4.
Fig. 4.

Total field amplitudes for heuristic gratings

Fig. 5.
Fig. 5.

Total field phases for heuristic gratings

Fig. 6.
Fig. 6.

Angular spectrum magnitudes for heuristic gratings just past DOE

Fig. 7.
Fig. 7.

Propagating field amplitudes for heuristic gratings just past DOE

Fig. 8.
Fig. 8.

Propagating field phases for heuristic gratings just past DOE

Fig. 9.
Fig. 9.

IASA DOE profiles

Fig. 10.
Fig. 10.

Irradiances for IASA profiles

Fig. 11.
Fig. 11.

IASA: multiple peak beamfanners

Tables (3)

Tables Icon

Table 1. Diffraction efficiencies and errors for heuristically designed 1–2 beamfanners

Tables Icon

Table 2. Diffraction efficiencies and errors for 1-2 beamfanners designed via IASA

Tables Icon

Table 3. Diffraction efficiencies and errors for 1–3 and 1–4 beamfanners

Equations (3)

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η = apertures Idx all space Idx
Error s = η TE or TM η scalar η scalar
TE / TM difference = η TM η TE η TE

Metrics