Abstract

The interference effect between dual diffractive cylindrical microlenses is investigated based on the boundary-element method. The interference patterns and intensity distributions of the near field and the middle-distance field are presented. The influence of various factors such as the wavelength of illuminating light, the size of the individual microlens, the beam aperture of the incident light, the preset focal length, and the refractive index of microlens material, on the interference results is studied in detail. The results demonstrate that the interference effect is dependent on the spacing between dual microlenses and surface-relief structures. We also indicate how to diminish the interference effect. It is believed that this work will provide useful information for designing diffractive microlens arrays with submicrometer-scale dimensions.

© 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. V. P. Koronkevich, I. G. Pal’chikova, “Modern zone plates,” Avtometriya 1, 85–100 (1992).
  6. V. Moreno, J. F. Roman, J. R. Salgueiro, “High efficiency diffractive lenses: deduction of kinoform profile,” Am. J. Phys. 65, 556–562 (1997).
    [CrossRef]
  7. A. Wang, A. Prata, “Lenslet analysis by rigorous vector diffraction theory,” J. Opt. Soc. Am. A 12, 1161–1169 (1995).
    [CrossRef]
  8. P. Blattner, H. P. Herzig, “Rigorous diffraction theory applied to microlenses,” J. Mod. Opt. 45, 1395–1403 (1998).
    [CrossRef]
  9. 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]
  10. 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]
  11. 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]
  12. 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]
  13. B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 3518–3526 (1994).
    [CrossRef]
  14. E. Noponen, J. Turunen, A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434–443 (1993).
    [CrossRef]
  15. 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]
  16. J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A 12, 2145–2158 (1995).
    [CrossRef]
  17. 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]
  18. S. Kagami, I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 455–461 (1984).
    [CrossRef]
  19. K. Yashiro, S. Ohkawa, “Boundary-element method for electromagnetic field problems,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
    [CrossRef]
  20. M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43–47.
  21. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chaps. 3, 4, and 6.
  22. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980), Chap. 6.

1999 (3)

1998 (3)

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 electromagnetic 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 electromagnetic 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.

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 electromagnetic 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]

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 electromagnetic 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)

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

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.

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Figures (9)

Fig. 1
Fig. 1

Schematic of a two-dimensional scattering problem in a system of two diffractive cylindrical microlenses with surface-relief structures.

Fig. 2
Fig. 2

Diffracted field intensity in region S2 for a single diffractive microlens system determined by the boundary-element method. (a) Regional plot of the normalized intensity distribution plotted in a 256 gray-level representation. Bright regions indicate areas of high field intensity, and dark regions correspond to areas of low intensity. (b) Line-scan plot of the electric-field intensity in the focal plane.

Fig. 3
Fig. 3

Diffracted field intensity in region S2 for a dual-diffractive-microlens system determined by the boundary-element method. (a) Regional plot of the normalized intensity distribution plotted in a 256 gray-level representation for the spacing between two microlenses of l=0. Bright regions indicate areas of high field intensity, and dark regions correspond to areas of low intensity. (b) Line-scan plot of the electric-field intensity in the focal plane.

Fig. 4
Fig. 4

(a), (b) Same as Figs. 3(a) and 3(b) except for l=4λ0; (c), (d) same as Figs. 3(a) and 3(b) except for l=8λ0. (e) Diffraction efficiency versus the spacing between dual microlenses.

Fig. 5
Fig. 5

(a), (b) Same as Figs. 4(a) and 4(b) except for D=27.5 μm; (c), (d) same as Figs. 4(c) and 4(d) except for D=27.5 μm. (e) Diffraction efficiency versus the microlens size for l=4λ0.

Fig. 6
Fig. 6

(a), (b) Same as Figs. 4(a) and 4(b) except for λ0=1.17 μm; (c), (d) same as Figs. 4(c) and 4(d) except for λ0=1.17 μm. (e) Diffraction efficiency versus the incident light wavelength for l=4λ0, D=16.8 μm.

Fig. 7
Fig. 7

(a), (b) Same as Figs. 4(a) and 4(b) except for the window function of the incident light, as defined by Eq. (10c); (c), (d) same as Figs. 4(c) and 4(d) except for the window function of the incident light, as defined by Eq. (10c).  

Fig. 8
Fig. 8

(a), (b) Same as Figs. 4(a) and 4(b) except for f=36 μm; (c), (d) same as Figs. 4(c) and 4(d) except for f=36 μm.

Fig. 9
Fig. 9

(a), (b) Same as Figs. 4(a) and 4(b) except for n1/n2=1.8; (c), (d) same as Figs. 4(c) and 4(d) except for n1/n2=1.8.

Equations (24)

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

-ϕ1t(r1)+Γ[ϕΓ(rΓ)nˆ  G1(r1, rΓ)
-p1G1(r1, rΓ)ψΓ(rΓ)]dl=-ϕinc(r1),
r1S1,
ϕ2t(r2)+Γ[ϕΓ(rΓ)nˆ  G2(r2, rΓ)
-p2G2(r2, rΓ)ψΓ(rΓ)]dl=0,
r2S2,
Gi(ri, rΓ)=(-j/4)H0(2)(ki|ri-rΓ|)(i=1, 2),
ϕ1t=ϕ2t=ϕΓ,
(1/p1)nˆ  ϕ1t=(1/p2)nˆ  ϕ2t=ψΓ.
θΓ2π-1ϕΓ(rΓ)+ΓϕΓ(rΓ) G1(rΓ, rΓ)n
-p1G1(rΓ, rΓ)ψΓ(rΓ)dl=-ϕinc(rΓ),
θΓ2πϕΓ(rΓ)+ΓϕΓ(rΓ) G2(rΓ, rΓ)n
-p2G2(rΓ, rΓ)ψΓ(rΓ)dl=0,
h(x)=n2n1-n2 (f 2+x2-f-mλ2),
xm|x|min(xm+1, D/2)
hx±D+l2=n2n1-n2 (f 2+x2-f-mλ2),
xm|x|min(xm+1, D/2)
xm=[2mf λ2+(mλ2)2]1/2.
w(x)=10|x|D2exp-10|x|-D22D2|x|D2+s0D2+s<|x|<,
w(x)
=10|x|D+l2exp-10|x|-D22D+l2|x|D+l2+s0D+l2+s<|x|<
w(x)
=1l2|x|D+l2exp-10|x|-D22D+l2|x|D+l2+sl2|x|l2+s0otherwise.
η=Σk-d/2d/2I2k(x2, y2)dx2-RRI1(x1, y1)dx1,

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