Abstract

We suggest an approach for numerically studying the performance of cylindrical microlenses without a beam-shaping aperture based on the boundary-element method (BEM). We divide the infinite microlens boundary into two components: The first part is an infinite expanded flat interface excluding the curved interface, and the second part is only the originally curved microlens interface. The resulting transmitted field can be regarded as the composition of two fields: One is generated by the first boundary, and the other is contributed from the second boundary. We carry out numerical simulations for two microlens systems, with or without aperture. We find that, for the nonapertured system, an ideal focusing feature is still observed; however, the axial distribution of the transmitted field exhibits an oscillation, different from the apertured system. It is expected that the current approach may provide a useful technique for the analysis of micro-optical elements.

© 2001 Optical Society of America

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References

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  1. 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]
  2. K. Hirayama, E. N. Glytsis, T. K. Gaylord, D. W. Wilson, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
    [CrossRef]
  3. J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors,” J. Opt. Soc. Am. A 16, 113–130 (1999).
    [CrossRef]
  4. K. Hirayama, K. Igarashi, Y. Hayashi, E. N. Glytsis, T. K. Gaylord, “Finite-substrate-thickness cylindrical diffractive lenses: exact and approximate boundary-element methods,” J. Opt. Soc. Am. A 16, 1294–1302 (1999).
    [CrossRef]
  5. E. N. Glytsis, M. E. Harrigan, T. K. Gaylord, K. Hirayama, “Effects of fabrication errors on the performance of cylindrical diffractive lenses: rigorous boundary-element method and scalar approximation,” Appl. Opt. 37, 6591–6602 (1998).
    [CrossRef]
  6. E. N. Glytsis, M. E. Harrigan, K. Hirayama, T. K. Gaylord, “Collimating cylindrical diffractive lenses: rigorous electromagnetic analysis and scalar approximation,” Appl. Opt. 37, 34–43 (1998).
    [CrossRef]
  7. V. P. Koronkevich, I. G. Pal’chikova, “Modern zone plates,” Avtometriya 1, 85–100 (1992).
  8. V. Moreno, J. F. Roman, J. R. Salgueiro, “High efficiency diffractive lenses: deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
    [CrossRef]
  9. A. Wang, A. Prata, “Lenslet analysis by rigorous vector diffraction theory,” J. Opt. Soc. Am. A 12, 1161–1169 (1995).
    [CrossRef]
  10. P. Blattner, H. P. Herzig, “Rigorous diffraction theory applied to microlenses,” J. Mod. Opt. 45, 1395–1403 (1998).
    [CrossRef]
  11. D. W. Prather, S. Shi, J. S. Bergey, “Field stitching algorithm for the analysis of electrically large diffractive optical elements,” Opt. Lett. 24, 273–275 (1999).
    [CrossRef]
  12. 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]
  13. D. W. Prather, J. N. Mait, M. S. Mirotznik, J. P. Collins, “Vector-based synthesis of finite aperiodic subwavelength diffractive optical elements,” J. Opt. Soc. Am. A 15, 1599–1607 (1998).
    [CrossRef]
  14. M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element-boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
    [CrossRef]
  15. B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
    [CrossRef]
  16. E. Noponen, J. Turunen, A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434–443 (1993).
    [CrossRef]
  17. Y. Nakata, M. Koshiba, “Boundary-element analysis of plane-wave diffraction from groove-type dielectric and metallic gratings,” J. Opt. Soc. Am. A 7, 1494–1502 (1990).
    [CrossRef]
  18. J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A 12, 2145–2158 (1995).
    [CrossRef]
  19. D. A. Pommet, M. G. Moharam, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1837 (1994).
    [CrossRef]
  20. S. Kagami, I. Fukai, “Application of boundary-element method to electro-magnetic field problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 455–461 (1984).
    [CrossRef]
  21. K. Yashiro, S. Ohkawa, “Boundary element method for electromagnetic field problems,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
    [CrossRef]
  22. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chaps. 3, 4, and 6.
  23. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980), Chap. 6.
    [CrossRef]
  24. M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43–47.

1999 (3)

1998 (5)

1997 (2)

V. Moreno, J. F. Roman, J. R. Salgueiro, “High efficiency diffractive lenses: deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[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]

1996 (2)

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

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element-boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

1995 (2)

1994 (2)

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (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, 1827–1837 (1994).
[CrossRef]

1993 (1)

1992 (1)

V. P. Koronkevich, I. G. Pal’chikova, “Modern zone plates,” Avtometriya 1, 85–100 (1992).

1990 (1)

1985 (1)

K. Yashiro, S. Ohkawa, “Boundary element method for electromagnetic field problems,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
[CrossRef]

1984 (1)

S. Kagami, I. Fukai, “Application of boundary-element method to electro-magnetic field problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 455–461 (1984).
[CrossRef]

Bendickson, J. M.

Bergey, J. S.

Blattner, P.

P. Blattner, H. P. Herzig, “Rigorous diffraction theory applied to microlenses,” J. Mod. Opt. 45, 1395–1403 (1998).
[CrossRef]

Collins, J. P.

Fukai, I.

S. Kagami, I. Fukai, “Application of boundary-element method to electro-magnetic field problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 455–461 (1984).
[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.

Glytsis, E. N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chaps. 3, 4, and 6.

Grann, E. B.

Harrigan, M. E.

Hayashi, Y.

Herzig, H. P.

P. Blattner, H. P. Herzig, “Rigorous diffraction theory applied to microlenses,” J. Mod. Opt. 45, 1395–1403 (1998).
[CrossRef]

Hirayama, K.

Igarashi, K.

Kagami, S.

S. Kagami, I. Fukai, “Application of boundary-element method to electro-magnetic field problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 455–461 (1984).
[CrossRef]

Koronkevich, V. P.

V. P. Koronkevich, I. G. Pal’chikova, “Modern zone plates,” Avtometriya 1, 85–100 (1992).

Koshiba, M.

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.

Mirotznik, M. S.

Moharam, M. G.

Moreno, V.

V. Moreno, J. F. Roman, J. R. Salgueiro, “High efficiency diffractive lenses: deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[CrossRef]

Nakata, Y.

Noponen, E.

Ohkawa, S.

K. Yashiro, S. Ohkawa, “Boundary element method for electromagnetic field problems,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
[CrossRef]

Pal’chikova, I. G.

V. P. Koronkevich, I. G. Pal’chikova, “Modern zone plates,” Avtometriya 1, 85–100 (1992).

Pommet, D. A.

Prata, A.

Prather, D. W.

Roman, J. F.

V. Moreno, J. F. Roman, J. R. Salgueiro, “High efficiency diffractive lenses: deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[CrossRef]

Salgueiro, J. R.

V. Moreno, J. F. Roman, J. R. Salgueiro, “High efficiency diffractive lenses: deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[CrossRef]

Shi, S.

Turunen, J.

Vasara, A.

Wang, A.

Wilson, D. W.

Yashiro, K.

K. Yashiro, S. Ohkawa, “Boundary element method for electromagnetic field problems,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
[CrossRef]

Am. J. Phys. (1)

V. Moreno, J. F. Roman, J. R. Salgueiro, “High efficiency diffractive lenses: deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
[CrossRef]

Appl. Opt. (2)

Avtometriya (1)

V. P. Koronkevich, I. G. Pal’chikova, “Modern zone plates,” Avtometriya 1, 85–100 (1992).

IEEE Trans. Antennas Propag. (1)

K. Yashiro, S. Ohkawa, “Boundary element method for electromagnetic field problems,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. Kagami, I. Fukai, “Application of boundary-element method to electro-magnetic field problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 455–461 (1984).
[CrossRef]

J. Mod. Opt. (2)

P. Blattner, H. P. Herzig, “Rigorous diffraction theory applied to microlenses,” J. Mod. Opt. 45, 1395–1403 (1998).
[CrossRef]

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite element-boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1321 (1996).
[CrossRef]

J. Opt. Soc. Am. A (11)

D. A. Pommet, M. G. Moharam, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1837 (1994).
[CrossRef]

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors,” J. Opt. Soc. Am. A 16, 113–130 (1999).
[CrossRef]

K. Hirayama, K. Igarashi, Y. Hayashi, E. N. Glytsis, T. K. Gaylord, “Finite-substrate-thickness cylindrical diffractive lenses: exact and approximate boundary-element methods,” J. Opt. Soc. Am. A 16, 1294–1302 (1999).
[CrossRef]

D. W. Prather, J. N. Mait, M. S. Mirotznik, J. P. Collins, “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]

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]

Y. Nakata, M. Koshiba, “Boundary-element analysis of plane-wave diffraction from groove-type dielectric and metallic gratings,” J. Opt. Soc. Am. A 7, 1494–1502 (1990).
[CrossRef]

E. Noponen, J. Turunen, A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434–443 (1993).
[CrossRef]

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

A. Wang, A. Prata, “Lenslet analysis by rigorous vector diffraction theory,” J. Opt. Soc. Am. A 12, 1161–1169 (1995).
[CrossRef]

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

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. Lett. (1)

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chaps. 3, 4, and 6.

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980), Chap. 6.
[CrossRef]

M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43–47.

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of a 2D scattering problem in a cylindrical system. (a) Arbitrary curved boundary Γ O between two homogeneous isotropic media corresponding to two semi-infinite regions, (b) infinitely expanded plane boundary ΓI without the curved interface, (c) curved boundary ΓII, (d) approximate boundary ΓI, (e) infinitely expanded planar interface ΓI1, (f) finite planar interface ΓI2. The dot symbol indicates zero field boundary.

Fig. 2
Fig. 2

Diffracted field intensity on region S 2 for diffractive microlens determined by the boundary element method. (a) Regional plot of the total intensity distribution plotted in a 256-gray-level representation, where the incident light wave is not blocked by the aperture, and the dark (bright) regions indicate the areas of high (low) field intensity. (b) Axial intensity distribution on region S 2. (c) Lateral electric field intensity on the focal plane. (d), (e), and (f) are the same as (a), (b), and (c) except for the incident light wave blocked by the aperture.

Fig. 3
Fig. 3

Diffracted field intensity on region S 2 for the refractive microlens. (a) Regional plot of the total intensity distribution plotted in a 256-gray-level representation, where the incident light wave is not blocked by the aperture, and the dark (bright) regions indicate the areas of high (low) field intensity. (b) Axial intensity distribution on region S 2. (c) Lateral electric field intensity on the real (preset) focal plane; the solid (dashed) curve represents the intensity distribution on the real (expected) focal plane. (d), (e), and (f) are the same as (a), (b), and (c) except for the incident wave blocked by the aperture.

Equations (13)

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

-E1zr1+ΓOE1zrΓnˆ·G1r1, rΓ-G1r1, rΓnˆ·E1zrΓdl=-Ezincr1,  r1S1,
E2zr2+ΓOE2zrΓnˆ·G2r2, rΓ-G2r2, rΓnˆ·E2zrΓdl=0,  r2S2,
θΓ2π-1E1zrΓ+ΓOE1zrΓnˆ·G1rΓ, rΓ-G1rΓ, rΓnˆ·E1zrΓdl=-EzincrΓ,
θΓ2πE2zrΓ+ΓOE2zrΓnˆ·G2rΓ, rΓ-G2rΓ, rΓnˆ·E2zrΓdl=0, rΓrΓΓO,
Giri, rΓ=-j/4H02ki|ri-rΓ|,  i=1, 2,
E1zrΓ=E2zrΓ=EzΓ,
nˆ·E1zrΓ=nˆ·E2zrΓ=nˆ·EzΓ.
ΓO=ΓI+ΓII=ΓI1-ΓI2+ΓII.
E2zr2=ΓOdl=ΓI1dl-ΓI2dl+ΓIIdl,
=G2r2, rΓnˆ·E2zrΓ-E2zrΓnˆ·G2r2, rΓ.
E2zI1=ΓI1dl, E2zI2=ΓI2dl, E2zII=ΓIIdl;
E2zr2=E2zI1-E2zI2+E2zII.
E2zI1=E02n1n1+n2exp-i2πyn2λ0,

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