Abstract

We present a method to design binary diffractive microlenses with subwavelength structures, based on the finite-difference time-domain method and the genetic algorithm, also accounting for limitations on feature size and aspect ratio imposed by fabrication. The focusing efficiency of the microlens designed by this method is close to that of the convex lens and much higher than that of the binary Fresnel lens designed by a previous method. Although the optimized structure appears to be a binary Fresnel lens qualitatively, it is hard to quantitatively derive directly from the convex Fresnel lens. The design of a microlens with reduced chromatic aberration is also presented.

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References

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  1. V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements (Wiley Series in Lasers and Applications) (John Wiley & Sons, 2002).
  2. D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2003).
  3. H. Kikuta, H. Toyota, and W. Yu, “Optical Elements with Subwavelength Structured Surfaces,” Opt. Rev. 10(2), 63–73 (2003).
    [CrossRef]
  4. G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. 28(12), 971–973 (2003).
    [CrossRef] [PubMed]
  5. Y. Okuno, M. Fujimoto, and T. Matsuda, “Numerical evaluation of diffractive optical elements with binary subwavelength structures,” IEIC Tech. Rep. 100, 157–162 (2001) (Japanese).
  6. J. N. Mait, D. W. Prather, and M. S. Mirotznik, “Binary subwavelength diffractive-lens design,” Opt. Lett. 23(17), 1343–1345 (1998).
    [CrossRef]
  7. D. W. Prather, J. N. Mait, M. S. Mirotznik, and J. P. Collins, “Vector-Based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A 15(6), 1599–1607 (1998).
    [CrossRef]
  8. D. A. Pommet, M. G. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11(6), 1827–1834 (1994).
    [CrossRef]
  9. A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,” IEEE Trans 22, 191–202 (1980).
  10. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12(5), 1068–1076 (1995).
    [CrossRef]
  11. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
    [CrossRef] [PubMed]
  12. M. Mitchell, An Introduction to Genetic Algorithms (MIT Press, 1998).
  13. E. G. Johnson and M. A. G. Abushagur, “Microgenetic-algorithm optimization methods applied to dielectric gratings,” J. Opt. Soc. Am. A 12(5), 1152–1160 (1995).
    [CrossRef]
  14. C. F. Huang and H. M. Li, ““Design optimization of chip antennas using the GA-FDTD approach”, Int. J. RF Microw,” Computer-Aided Engineering 15, 116–127 (2005).
  15. H. Jimbow, C. Yatabe, K. L. Ishikawa, Y. Yamada, and K. Masuda, “Design of subwavelength diffractive optical elements using genetic algorithm and FDTD method,” IEEJ Trans. EIS 127(9), 1298–1303 (2007) (Japanese).
    [CrossRef]
  16. A. Taflove, and S. C. Hagness, Computional Electrodynamics: The Finite-Difference Time-Domain Method, Chap. 12 (Artech House, 2005).
  17. J. P. Berenger, “A perfectly matched layer for the absorption of electro-magnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
    [CrossRef]
  18. K. Ono and K. Eriguchi, “Modeling of plasma-surface interactions and profile evolution during dry etching,” J. Plasma Fusion Res. 85, 165–176 (2009) (Japanese).

2009 (1)

K. Ono and K. Eriguchi, “Modeling of plasma-surface interactions and profile evolution during dry etching,” J. Plasma Fusion Res. 85, 165–176 (2009) (Japanese).

2007 (1)

H. Jimbow, C. Yatabe, K. L. Ishikawa, Y. Yamada, and K. Masuda, “Design of subwavelength diffractive optical elements using genetic algorithm and FDTD method,” IEEJ Trans. EIS 127(9), 1298–1303 (2007) (Japanese).
[CrossRef]

2005 (1)

C. F. Huang and H. M. Li, ““Design optimization of chip antennas using the GA-FDTD approach”, Int. J. RF Microw,” Computer-Aided Engineering 15, 116–127 (2005).

2003 (2)

H. Kikuta, H. Toyota, and W. Yu, “Optical Elements with Subwavelength Structured Surfaces,” Opt. Rev. 10(2), 63–73 (2003).
[CrossRef]

G. Saavedra, W. D. Furlan, and J. A. Monsoriu, “Fractal zone plates,” Opt. Lett. 28(12), 971–973 (2003).
[CrossRef] [PubMed]

2001 (1)

Y. Okuno, M. Fujimoto, and T. Matsuda, “Numerical evaluation of diffractive optical elements with binary subwavelength structures,” IEIC Tech. Rep. 100, 157–162 (2001) (Japanese).

1998 (2)

1995 (2)

1994 (2)

J. P. Berenger, “A perfectly matched layer for the absorption of electro-magnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[CrossRef]

D. A. Pommet, M. G. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11(6), 1827–1834 (1994).
[CrossRef]

1983 (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[CrossRef] [PubMed]

1980 (1)

A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,” IEEE Trans 22, 191–202 (1980).

Abushagur, M. A. G.

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electro-magnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[CrossRef]

Collins, J. P.

Eriguchi, K.

K. Ono and K. Eriguchi, “Modeling of plasma-surface interactions and profile evolution during dry etching,” J. Plasma Fusion Res. 85, 165–176 (2009) (Japanese).

Fujimoto, M.

Y. Okuno, M. Fujimoto, and T. Matsuda, “Numerical evaluation of diffractive optical elements with binary subwavelength structures,” IEIC Tech. Rep. 100, 157–162 (2001) (Japanese).

Furlan, W. D.

Gaylord, T. K.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[CrossRef] [PubMed]

Grann, E. B.

Huang, C. F.

C. F. Huang and H. M. Li, ““Design optimization of chip antennas using the GA-FDTD approach”, Int. J. RF Microw,” Computer-Aided Engineering 15, 116–127 (2005).

Ishikawa, K. L.

H. Jimbow, C. Yatabe, K. L. Ishikawa, Y. Yamada, and K. Masuda, “Design of subwavelength diffractive optical elements using genetic algorithm and FDTD method,” IEEJ Trans. EIS 127(9), 1298–1303 (2007) (Japanese).
[CrossRef]

Jimbow, H.

H. Jimbow, C. Yatabe, K. L. Ishikawa, Y. Yamada, and K. Masuda, “Design of subwavelength diffractive optical elements using genetic algorithm and FDTD method,” IEEJ Trans. EIS 127(9), 1298–1303 (2007) (Japanese).
[CrossRef]

Johnson, E. G.

Kikuta, H.

H. Kikuta, H. Toyota, and W. Yu, “Optical Elements with Subwavelength Structured Surfaces,” Opt. Rev. 10(2), 63–73 (2003).
[CrossRef]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[CrossRef] [PubMed]

Li, H. M.

C. F. Huang and H. M. Li, ““Design optimization of chip antennas using the GA-FDTD approach”, Int. J. RF Microw,” Computer-Aided Engineering 15, 116–127 (2005).

Mait, J. N.

Masuda, K.

H. Jimbow, C. Yatabe, K. L. Ishikawa, Y. Yamada, and K. Masuda, “Design of subwavelength diffractive optical elements using genetic algorithm and FDTD method,” IEEJ Trans. EIS 127(9), 1298–1303 (2007) (Japanese).
[CrossRef]

Matsuda, T.

Y. Okuno, M. Fujimoto, and T. Matsuda, “Numerical evaluation of diffractive optical elements with binary subwavelength structures,” IEIC Tech. Rep. 100, 157–162 (2001) (Japanese).

Mirotznik, M. S.

Moharam, M. G.

Monsoriu, J. A.

Okuno, Y.

Y. Okuno, M. Fujimoto, and T. Matsuda, “Numerical evaluation of diffractive optical elements with binary subwavelength structures,” IEIC Tech. Rep. 100, 157–162 (2001) (Japanese).

Ono, K.

K. Ono and K. Eriguchi, “Modeling of plasma-surface interactions and profile evolution during dry etching,” J. Plasma Fusion Res. 85, 165–176 (2009) (Japanese).

Pommet, D. A.

Prather, D. W.

Saavedra, G.

Taflove, A.

A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,” IEEE Trans 22, 191–202 (1980).

Toyota, H.

H. Kikuta, H. Toyota, and W. Yu, “Optical Elements with Subwavelength Structured Surfaces,” Opt. Rev. 10(2), 63–73 (2003).
[CrossRef]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[CrossRef] [PubMed]

Yamada, Y.

H. Jimbow, C. Yatabe, K. L. Ishikawa, Y. Yamada, and K. Masuda, “Design of subwavelength diffractive optical elements using genetic algorithm and FDTD method,” IEEJ Trans. EIS 127(9), 1298–1303 (2007) (Japanese).
[CrossRef]

Yatabe, C.

H. Jimbow, C. Yatabe, K. L. Ishikawa, Y. Yamada, and K. Masuda, “Design of subwavelength diffractive optical elements using genetic algorithm and FDTD method,” IEEJ Trans. EIS 127(9), 1298–1303 (2007) (Japanese).
[CrossRef]

Yu, W.

H. Kikuta, H. Toyota, and W. Yu, “Optical Elements with Subwavelength Structured Surfaces,” Opt. Rev. 10(2), 63–73 (2003).
[CrossRef]

Computer-Aided Engineering (1)

C. F. Huang and H. M. Li, ““Design optimization of chip antennas using the GA-FDTD approach”, Int. J. RF Microw,” Computer-Aided Engineering 15, 116–127 (2005).

IEEE Trans (1)

A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,” IEEE Trans 22, 191–202 (1980).

IEEJ Trans. EIS (1)

H. Jimbow, C. Yatabe, K. L. Ishikawa, Y. Yamada, and K. Masuda, “Design of subwavelength diffractive optical elements using genetic algorithm and FDTD method,” IEEJ Trans. EIS 127(9), 1298–1303 (2007) (Japanese).
[CrossRef]

IEIC Tech. Rep. (1)

Y. Okuno, M. Fujimoto, and T. Matsuda, “Numerical evaluation of diffractive optical elements with binary subwavelength structures,” IEIC Tech. Rep. 100, 157–162 (2001) (Japanese).

J. Comput. Phys. (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electro-magnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Plasma Fusion Res. (1)

K. Ono and K. Eriguchi, “Modeling of plasma-surface interactions and profile evolution during dry etching,” J. Plasma Fusion Res. 85, 165–176 (2009) (Japanese).

Opt. Lett. (2)

Opt. Rev. (1)

H. Kikuta, H. Toyota, and W. Yu, “Optical Elements with Subwavelength Structured Surfaces,” Opt. Rev. 10(2), 63–73 (2003).
[CrossRef]

Science (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 (1983).
[CrossRef] [PubMed]

Other (4)

M. Mitchell, An Introduction to Genetic Algorithms (MIT Press, 1998).

A. Taflove, and S. C. Hagness, Computional Electrodynamics: The Finite-Difference Time-Domain Method, Chap. 12 (Artech House, 2005).

V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements (Wiley Series in Lasers and Applications) (John Wiley & Sons, 2002).

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2003).

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Figures (10)

Fig. 1
Fig. 1

Schematic of the analytical domain rotationally symmetric about the z-axis. A plane wave is incident from the left to the microlens (blue region). The close-up on the right indicates the electromagnetic components on the BOD-FDTD calculation grid. (i, j) are indexes for the spatial coordinate (i and j for the radial and z directions, respectively).

Fig. 2
Fig. 2

Mapping between (a) Genotype and (b) Phenotype in Coding #1.

Fig. 3
Fig. 3

Mapping between (a) Genotype and (b) Phenotype in Coding #2.

Fig. 4
Fig. 4

(a) Structure obtained by the GA-FDTD method (b) Intensity distribution

Fig. 6
Fig. 6

(a) Focusing performance comparison of the GA-designed structure with the convex lens, its Fresnel lens, and binarization of the Fresnel lens in focal face (Z = 9.3 μm). The grid size used in FDTD calculations is 5 nm except for the black dashed curve. (b) Ellipticity of the spot shape.

Fig. 5
Fig. 5

(a) The convex lens and (b) its Fresnel lens and (c) binarization of the Fresnel lens [5]

Fig. 7
Fig. 7

(a) Trapezoidal gratings based on the GA-designed structure (b) intensity distribution

Fig. 8
Fig. 8

Evolution of the fitness as a function of generation for two differenct coding methods.

Fig. 9
Fig. 9

(a) Binary lens with reduced chromatic aberration at three wavelengths designed by the GA-FDTD method and intensity distributions for plane wave incidence with a wavelength of (b) 660 nm, (c) 532nm, (d) 445 nm

Fig. 10
Fig. 10

Focal distance (left axis) and efficiency (right axis) of two different microlenses (Figs. 4 and 9). The structure depicted in Fig. 4 does not work as a lens below 470 nm.

Equations (1)

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F = ( f d | f o f | ) f d × z = f f d f + f d r = 0 f r S z ( r , z ) ,

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