Abstract

We propose a rigorous electromagnetic design of two-dimensional and finite-aperture diffractive optical elements (DOEs) that employs an effective iterative optimization algorithm in conjunction with a rigorous electromagnetic computational model: the finite-difference time-domain method. The iterative optimization process, the finite-difference time-domain method, and the angular spectrum propagation method are discussed in detail. Without any approximation based on the scalar theory, the algorithm can produce rigorous design results, both numerical and graphical, with fast convergence, reasonable computational cost, and good design quality. Using our iterative algorithm, we designed a diffractive cylindrical lens and a 1-to-2-beam fanner for normal-incidence TE-mode illumination, thus showing that the optimization algorithm is valid and competent for rigorously designing diffractive optical elements. Concerning the problem of fabrication, we also evaluated the performance of the DOE when the DOE profile is discrete.

© 2003 Optical Society of America

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References

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  1. Feature issue, “Diffractive optics applications,” Appl. Opt. 34, 2399–2559 (1995).
    [CrossRef]
  2. D. A. Pommet, M. G. Moharam, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
    [CrossRef]
  3. N. Sergienko, J. Turunen, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).
    [CrossRef]
  4. B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
    [CrossRef]
  5. K. Hirayama, E. N. Glytsis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
    [CrossRef]
  6. J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Scalar integral diffraction methods: unification, accuracy, and comparison with a rigorous boundary element method with application to diffractive cylindrical lenses,” J. Opt. Soc. Am. A 15, 1822–1837 (1998).
    [CrossRef]
  7. D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modeling-diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
    [CrossRef]
  8. D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
    [CrossRef]
  9. E. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
    [CrossRef]
  10. D. W. Prather, S. Shi, “Formulation and application ofthe finite-difference time-domain method for the analysis of axially symmetric DOEs,” J. Opt. Soc. Am. A 16, 1131–1142 (1999).
    [CrossRef]
  11. L. Gur, D. Mendlovic, “Diffraction limited domain flat-top generator,” Opt. Commun. 145, 237–248 (1998).
    [CrossRef]
  12. D. W. Prather, J. N. Mait, “Vector-based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A 15, 1599–1607 (1998).
    [CrossRef]
  13. J. Jiang, G. P. Nordin, “A rigorous unidirectional method for designing finite aperture diffractive optical elements,” Opt. Express 7, 237–242 (2000).
    [CrossRef] [PubMed]
  14. M. E. Testorf, M. A. Fiddy, “Efficient optimization of diffractive optical elements based on rigorous diffraction models,” J. Opt. Soc. Am. A 18, 2908–2914 (2001).
    [CrossRef]
  15. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  16. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holography,” Opt. Eng. 19, 297–306 (1980).
    [CrossRef]
  17. S. Buhling, F. Wyrowski, “Improved transmission design algorithms by utilizing variable-strength projections,” J. Mod. Opt. 49, 1871–1892 (2002).
    [CrossRef]
  18. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).
  19. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 1995).
  20. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  21. G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge U. Press, Cambridge, UK, 1997).
  22. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  23. S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
    [CrossRef] [PubMed]
  24. E. G. Johnson, M. A. G. Abushagur, “Microgenetic-algorithm optimization methods applied to dielectric gratings,” J. Opt. Soc. Am. A 12, 1152–1160 (1995).
    [CrossRef]
  25. J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A 12, 2145–2158 (1995).
    [CrossRef]
  26. D. W. Prather, D. Pustai, Shouyuan Shi, “Performance of multilevel diffractive lenses as a function of f-number,” Appl. Opt. 40, 207–210 (2001).
    [CrossRef]

2002 (1)

S. Buhling, F. Wyrowski, “Improved transmission design algorithms by utilizing variable-strength projections,” J. Mod. Opt. 49, 1871–1892 (2002).
[CrossRef]

2001 (2)

2000 (1)

1999 (3)

N. Sergienko, J. Turunen, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).
[CrossRef]

E. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
[CrossRef]

D. W. Prather, S. Shi, “Formulation and application ofthe finite-difference time-domain method for the analysis of axially symmetric DOEs,” J. Opt. Soc. Am. A 16, 1131–1142 (1999).
[CrossRef]

1998 (3)

L. Gur, D. Mendlovic, “Diffraction limited domain flat-top generator,” Opt. Commun. 145, 237–248 (1998).
[CrossRef]

D. W. Prather, J. N. Mait, “Vector-based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A 15, 1599–1607 (1998).
[CrossRef]

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Scalar integral diffraction methods: unification, accuracy, and comparison with a rigorous boundary element method with application to diffractive cylindrical lenses,” J. Opt. Soc. Am. A 15, 1822–1837 (1998).
[CrossRef]

1997 (1)

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

1996 (1)

K. Hirayama, E. N. Glytsis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
[CrossRef]

1995 (3)

Feature issue, “Diffractive optics applications,” Appl. Opt. 34, 2399–2559 (1995).
[CrossRef]

E. G. Johnson, M. A. G. Abushagur, “Microgenetic-algorithm optimization methods applied to dielectric gratings,” J. Opt. Soc. Am. A 12, 1152–1160 (1995).
[CrossRef]

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

1994 (3)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

D. A. Pommet, M. G. Moharam, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
[CrossRef]

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
[CrossRef]

1983 (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

1980 (1)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holography,” Opt. Eng. 19, 297–306 (1980).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Abushagur, M. A. G.

E. G. Johnson, M. A. G. Abushagur, “Microgenetic-algorithm optimization methods applied to dielectric gratings,” J. Opt. Soc. Am. A 12, 1152–1160 (1995).
[CrossRef]

Bendickson, J. M.

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Scalar integral diffraction methods: unification, accuracy, and comparison with a rigorous boundary element method with application to diffractive cylindrical lenses,” J. Opt. Soc. Am. A 15, 1822–1837 (1998).
[CrossRef]

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Buhling, S.

S. Buhling, F. Wyrowski, “Improved transmission design algorithms by utilizing variable-strength projections,” J. Mod. Opt. 49, 1871–1892 (2002).
[CrossRef]

Elston, S. J.

E. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
[CrossRef]

Fiddy, M. A.

Fienup, J. R.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holography,” Opt. Eng. 19, 297–306 (1980).
[CrossRef]

Friberg, A. T.

N. Sergienko, J. Turunen, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).
[CrossRef]

Gallagher, N. C.

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
[CrossRef]

Gaylord, T. K.

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Scalar integral diffraction methods: unification, accuracy, and comparison with a rigorous boundary element method with application to diffractive cylindrical lenses,” J. Opt. Soc. Am. A 15, 1822–1837 (1998).
[CrossRef]

K. Hirayama, E. N. Glytsis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
[CrossRef]

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Glytsis, E. N.

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Scalar integral diffraction methods: unification, accuracy, and comparison with a rigorous boundary element method with application to diffractive cylindrical lenses,” J. Opt. Soc. Am. A 15, 1822–1837 (1998).
[CrossRef]

K. Hirayama, E. N. Glytsis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
[CrossRef]

Goodman, J. W.

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

Grann, E. B.

D. A. Pommet, M. G. Moharam, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
[CrossRef]

Gur, L.

L. Gur, D. Mendlovic, “Diffraction limited domain flat-top generator,” Opt. Commun. 145, 237–248 (1998).
[CrossRef]

Hirayama, K.

K. Hirayama, E. N. Glytsis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
[CrossRef]

Jiang, J.

Johnson, E. G.

E. G. Johnson, M. A. G. Abushagur, “Microgenetic-algorithm optimization methods applied to dielectric gratings,” J. Opt. Soc. Am. A 12, 1152–1160 (1995).
[CrossRef]

Kettunen, V.

N. Sergienko, J. Turunen, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).
[CrossRef]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Kriezis, E. E.

E. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
[CrossRef]

Kuittinen, M.

N. Sergienko, J. Turunen, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).
[CrossRef]

Lichtenberg, B.

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
[CrossRef]

Mait, J. N.

D. W. Prather, J. N. Mait, “Vector-based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A 15, 1599–1607 (1998).
[CrossRef]

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

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

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modeling-diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
[CrossRef]

Mendlovic, D.

L. Gur, D. Mendlovic, “Diffraction limited domain flat-top generator,” Opt. Commun. 145, 237–248 (1998).
[CrossRef]

Mirotznik, M. S.

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modeling-diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
[CrossRef]

Moharam, M. G.

D. A. Pommet, M. G. Moharam, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
[CrossRef]

Nordin, G. P.

Pommet, D. A.

D. A. Pommet, M. G. Moharam, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
[CrossRef]

Prather, D. W.

D. W. Prather, D. Pustai, Shouyuan Shi, “Performance of multilevel diffractive lenses as a function of f-number,” Appl. Opt. 40, 207–210 (2001).
[CrossRef]

D. W. Prather, S. Shi, “Formulation and application ofthe finite-difference time-domain method for the analysis of axially symmetric DOEs,” J. Opt. Soc. Am. A 16, 1131–1142 (1999).
[CrossRef]

D. W. Prather, J. N. Mait, “Vector-based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A 15, 1599–1607 (1998).
[CrossRef]

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modeling-diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
[CrossRef]

Pustai, D.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Sergienko, N.

N. Sergienko, J. Turunen, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).
[CrossRef]

Shi, S.

Shi, Shouyuan

Smith, G. S.

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

Taflove, A.

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

Testorf, M. E.

Turunen, J.

N. Sergienko, J. Turunen, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).
[CrossRef]

N. Sergienko, J. Turunen, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).
[CrossRef]

Vahimaa, P.

N. Sergienko, J. Turunen, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).
[CrossRef]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Wyrowski, F.

S. Buhling, F. Wyrowski, “Improved transmission design algorithms by utilizing variable-strength projections,” J. Mod. Opt. 49, 1871–1892 (2002).
[CrossRef]

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

J. Mod. Opt. (1)

N. Sergienko, J. Turunen, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, A. T. Friberg, “Comparison of electromagnetic and scalar methods for evaluation of diffractive lenses,” J. Mod. Opt. 46, 65–82 (1999).
[CrossRef]

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

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Scalar integral diffraction methods: unification, accuracy, and comparison with a rigorous boundary element method with application to diffractive cylindrical lenses,” J. Opt. Soc. Am. A 15, 1822–1837 (1998).
[CrossRef]

D. W. Prather, J. N. Mait, “Vector-based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A 15, 1599–1607 (1998).
[CrossRef]

J. Comput. Phys. (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Mod. Opt. (1)

S. Buhling, F. Wyrowski, “Improved transmission design algorithms by utilizing variable-strength projections,” J. Mod. Opt. 49, 1871–1892 (2002).
[CrossRef]

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

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

J. Opt. Soc. Am. A (3)

D. A. Pommet, M. G. Moharam, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
[CrossRef]

K. Hirayama, E. N. Glytsis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
[CrossRef]

E. G. Johnson, M. A. G. Abushagur, “Microgenetic-algorithm optimization methods applied to dielectric gratings,” J. Opt. Soc. Am. A 12, 1152–1160 (1995).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Eng. (1)

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
[CrossRef]

Opt. Commun. (2)

E. E. Kriezis, S. J. Elston, “Finite-difference time domain method for light wave propagation within liquid crystal devices,” Opt. Commun. 165, 99–105 (1999).
[CrossRef]

L. Gur, D. Mendlovic, “Diffraction limited domain flat-top generator,” Opt. Commun. 145, 237–248 (1998).
[CrossRef]

Opt. Eng. (1)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holography,” Opt. Eng. 19, 297–306 (1980).
[CrossRef]

Opt. Express (1)

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Science (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Other (4)

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

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

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

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary element method for vector modeling-diffractive optical elements,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Iterative optimization algorithm combined with the FDTD method for rigorous design of DOEs.

Fig. 2
Fig. 2

Schematic diagram of our rigorous optimization algorithm.

Fig. 3
Fig. 3

Electric-field intensity at the focal plane for (a) the target intensity distribution and the real intensity distribution produced by our designed lens at the focal plane (b) the analytical lens’s normalized intensity distribution and that of the designed lens at the focal plane.

Fig. 4
Fig. 4

Profiles of the designed lens and the analytical lens.

Fig. 5
Fig. 5

Electric-field intensity distributions plotted in a gray-level representation (a) for our designed lens and (b) for the analytical, quadratic lens.

Fig. 6
Fig. 6

(a) Quantified lens profile (eight-level), (b) comparison of the intensity distribution produced by the continuous profile and that produced by the quantified (eight-level) profile.

Fig. 7
Fig. 7

(a) Normalized electric-field intensity at the focal plane for the target distribution and the real distribution made by our designed beam fanner, (b) Profile of the designed beam fanner.

Fig. 8
Fig. 8

Electric field intensity distributions of the designed beam fanner plotted in a gray-level representation (a) for the continuous profile and (b) for the quantified (eight-level) profile.

Fig. 9
Fig. 9

Normalized electric-field intensity at the focal plane for the target distribution and the distribution calculated by scalar theory.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

E z t = 1 H y x - H x y - σ E z ,
H x t = - 1 μ E z y + σ * H x ,
H y t = 1 μ E z x - σ * H y ,
μ 0 = μ , σ * = 0 .
E ¯ z n + 1 ( i ,   j ) = 2 ( i ,   j ) - σ ( i ,   j ) Δ t 2 ( i ,   j ) + σ ( i ,   j ) Δ t   E ¯ z n ( i ,   j ) + Δ t Δ s 0 μ 0 2 0 2 ( i ,   j ) + σ ( i ,   j ) Δ t × [ H y n + 1 / 2 ( i + 1 / 2 ,   j ) - H y n + 1 / 2 ( i - 1 / 2 ,   j ) + H x n + 1 / 2 ( i ,   j - 1 / 2 ) - H x n + 1 / 2 ( i ,   j + 1 / 2 ) ] ,
H x n + 1 / 2 ( i ,   j + 1 / 2 )
= H x n - 1 / 2 ( i ,   j + 1 / 2 ) + Δ t Δ s 0 μ 0 × [ E ¯ z n ( i ,   j ) - E ¯ z n ( i ,   j + 1 ) ] ,
H y n + 1 / 2 ( i + 1 / 2 ,   j )
= H y n - 1 / 2 ( i + 1 / 2 ,   j ) + Δ t Δ s 0 μ 0 × [ E ¯ z n ( i + 1 ,   j ) - E ¯ z n ( i ,   j ) ] ,
E zx t = 1 H y x - σ E zx ,
E zy t = 1 - H x y - σ E zy ,
H x t = - 1 μ ( E zx + E zy ) y + σ * H x ,
H y t = 1 μ ( E zx + E zy ) x + σ * H y ,
U ( f x ,   0 ) = Σ u ( x ,   0 ) exp ( j 2 π f x x ) d x ,
f x = cos ( α ) / λ ;
H ( f x ) = exp ( j 2 π y 0 1 - cos 2   α / λ ) ,
U ( f x ,   y 0 ) = U ( f x ,   0 ) H ( f x ) .
u ( f x ,   y 0 ) = DFT - 1 [ U ( f x ,   y 0 ) ] = DFT - 1 [ U ( f x ,   0 ) exp ( j 2 π y 0 1 - cos 2   α / λ ) ] ,
h ( x ) = λ φ ( x ) 2 π ( n - 1 ) ,
RMS = 1 total - 1 i = 1 total ( | u target ( i ) ( x i ) | 2 - | u ( i ) ( x i ) | 2 ) 2 1 / 2 ,
y ( x ) = n 2 n 1 - n 2   ( f 2 + x 2 - f ) .

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