Abstract

We present a three-dimensional (3D) analysis of subwavelength diffractive optical elements (DOE’s), using the finite-difference time-domain (FDTD) method. To this end we develop and apply efficient 3D FDTD methods that exploit DOE properties, such as symmetry. An axisymmetric method is validated experimentally and is used to validate the more general 3D method. Analyses of subwavelength gratings and lenses, both with and without rotational symmetry, are presented in addition to a 2 × 2 subwavelength focusing array generator.

© 2000 Optical Society of America

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References

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  1. M. W. Farn, “Binary gratings with increased efficiency,” Appl. Opt. 31, 4453–4458 (1992).
    [CrossRef] [PubMed]
  2. P. Lalanne, S. Astilean, P. Chavel, E. Cambril, H. Launois, “Blazed-binary subwavelength gratings with efficiencies larger than those of conventional echelette gratings,” Opt. Lett. 23, 1081–1083 (1998).
    [CrossRef]
  3. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  4. M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite-boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1322 (1996).
    [CrossRef]
  5. 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]
  6. 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]
  7. 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).
  8. A. Taflove, Computational Electromagnetics: The Finite-Difference Time Domain Method (Artech House, Boston, Mass., 1995).
  9. R. Luebbers, R. F. Hunsberger, K. Kunz, R. Standler, M. Schneider, “A frequency dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–229 (1990).
    [CrossRef]
  10. K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).
  11. D. Davidson, R. Ziolkowski, “Body-of-revolution finite-difference time-domain modeling of space-time focusing by a three-dimensional lens,” J. Opt. Soc. Am. A 11, 1471–1490 (1994).
    [CrossRef]
  12. Y. Chen, R. Mittra, P. Harms, “Finite-difference time-domain algorithm for solving maxwell’s equations in rotationally symmetric geometries,” IEEE Trans. Microwave Theory Tech. MTT-44, 832–839 (1996).
    [CrossRef]
  13. D. W. Prather, S. Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,” J. Opt. Soc. Am. A 16, 1131–1142 (1999).
    [CrossRef]
  14. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  15. D. S. Katz, E. T. Thiele, A. Taflove, “Validation and extension to three dimensions of the berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guid. Wave Lett. 4, 268–270 (1994).
    [CrossRef]
  16. J. N. Mait, D. W. Prather, M. S. Mirotznik, “Binary subwavelength diffractive-lens design,” Opt. Lett. 31, 1343–1345 (1998).
    [CrossRef]
  17. S. Tibuleac, R. Magnusson, “Reflection and transmission guided-mode resonance filters,” J. Opt. Soc. Am. A 14, 1617–1626 (1997).
    [CrossRef]

1999 (1)

1998 (2)

1997 (2)

1996 (3)

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-boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1322 (1996).
[CrossRef]

Y. Chen, R. Mittra, P. Harms, “Finite-difference time-domain algorithm for solving maxwell’s equations in rotationally symmetric geometries,” IEEE Trans. Microwave Theory Tech. MTT-44, 832–839 (1996).
[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. S. Katz, E. T. Thiele, A. Taflove, “Validation and extension to three dimensions of the berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guid. Wave Lett. 4, 268–270 (1994).
[CrossRef]

D. Davidson, R. Ziolkowski, “Body-of-revolution finite-difference time-domain modeling of space-time focusing by a three-dimensional lens,” J. Opt. Soc. Am. A 11, 1471–1490 (1994).
[CrossRef]

1992 (1)

1990 (1)

R. Luebbers, R. F. Hunsberger, K. Kunz, R. Standler, M. Schneider, “A frequency dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–229 (1990).
[CrossRef]

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

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).

Astilean, S.

Berenger, J. P.

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

Cambril, E.

Chavel, P.

Chen, Y.

Y. Chen, R. Mittra, P. Harms, “Finite-difference time-domain algorithm for solving maxwell’s equations in rotationally symmetric geometries,” IEEE Trans. Microwave Theory Tech. MTT-44, 832–839 (1996).
[CrossRef]

Davidson, D.

Farn, M. W.

Gaylord, T. K.

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]

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Glytsis, E. N.

Harms, P.

Y. Chen, R. Mittra, P. Harms, “Finite-difference time-domain algorithm for solving maxwell’s equations in rotationally symmetric geometries,” IEEE Trans. Microwave Theory Tech. MTT-44, 832–839 (1996).
[CrossRef]

Hirayama, K.

Hunsberger, R. F.

R. Luebbers, R. F. Hunsberger, K. Kunz, R. Standler, M. Schneider, “A frequency dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–229 (1990).
[CrossRef]

Katz, D. S.

D. S. Katz, E. T. Thiele, A. Taflove, “Validation and extension to three dimensions of the berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guid. Wave Lett. 4, 268–270 (1994).
[CrossRef]

Kunz, K.

R. Luebbers, R. F. Hunsberger, K. Kunz, R. Standler, M. Schneider, “A frequency dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–229 (1990).
[CrossRef]

Kunz, K. S.

K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).

Lalanne, P.

Launois, H.

Luebbers, R.

R. Luebbers, R. F. Hunsberger, K. Kunz, R. Standler, M. Schneider, “A frequency dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–229 (1990).
[CrossRef]

Luebbers, R. J.

K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).

Magnusson, R.

Mait, J. N.

Mirotznik, M. S.

Mittra, R.

Y. Chen, R. Mittra, P. Harms, “Finite-difference time-domain algorithm for solving maxwell’s equations in rotationally symmetric geometries,” IEEE Trans. Microwave Theory Tech. MTT-44, 832–839 (1996).
[CrossRef]

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Prather, D. W.

Schneider, M.

R. Luebbers, R. F. Hunsberger, K. Kunz, R. Standler, M. Schneider, “A frequency dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–229 (1990).
[CrossRef]

Shi, S.

Standler, R.

R. Luebbers, R. F. Hunsberger, K. Kunz, R. Standler, M. Schneider, “A frequency dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–229 (1990).
[CrossRef]

Taflove, A.

D. S. Katz, E. T. Thiele, A. Taflove, “Validation and extension to three dimensions of the berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guid. Wave Lett. 4, 268–270 (1994).
[CrossRef]

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

Thiele, E. T.

D. S. Katz, E. T. Thiele, A. Taflove, “Validation and extension to three dimensions of the berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guid. Wave Lett. 4, 268–270 (1994).
[CrossRef]

Tibuleac, S.

Wilson, D. W.

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).

Ziolkowski, R.

Appl. Opt. (1)

IEEE Microwave Guid. Wave Lett. (1)

D. S. Katz, E. T. Thiele, A. Taflove, “Validation and extension to three dimensions of the berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave Guid. Wave Lett. 4, 268–270 (1994).
[CrossRef]

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).

IEEE Trans. Electromagn. Compat. (1)

R. Luebbers, R. F. Hunsberger, K. Kunz, R. Standler, M. Schneider, “A frequency dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–229 (1990).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

Y. Chen, R. Mittra, P. Harms, “Finite-difference time-domain algorithm for solving maxwell’s equations in rotationally symmetric geometries,” IEEE Trans. Microwave Theory Tech. MTT-44, 832–839 (1996).
[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)

M. S. Mirotznik, D. W. Prather, J. N. Mait, “A hybrid finite-boundary element method for the analysis of diffractive elements,” J. Mod. Opt. 43, 1309–1322 (1996).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Lett. (2)

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Other (2)

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

K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).

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Figures (9)

Fig. 1
Fig. 1

Position of electric and magnetic field components about unit Yee cell in rectangular and polar coordinates.

Fig. 2
Fig. 2

Illustration of DOE located within FDTD solution space and absorbing PML region.

Fig. 3
Fig. 3

Plane-wave scattering from dielectric sphere with a dielectric constant ∊ r = 2.25 and a radius of two free-space wavelengths, 2λ0, in size. Comparison of general 3D and axisymmetric FDTD codes with analytical solution. Plane-wave scattering from dielectric sphere with a dielectric constant ∊ r = 2.25 and a radius of two free-space wavelengths, 2λ0, in size. Comparison of general 3D and axisymmetric FDTD codes with analytical solution. BOR, boundary of revolution.

Fig. 4
Fig. 4

Experimental setup for validating the axisymmetric FDTD formulation.

Fig. 5
Fig. 5

Validation of the FDTD method by comparison of the measured and the computed values for a precision pinhole of 71 µm at an axial location of 1.2 mm.

Fig. 6
Fig. 6

Analysis of a subwavelength binary Fresnel zone plate: (a) lens profile, (b) line scans of the electric field intensities in the focal plane, (c) image plot of the electric field intensity in the x, z plane, and (d) image plot of the electric field intensity in the x, y focal plane. BOR, boundary of revolution.

Fig. 7
Fig. 7

Analysis of a 2 × 2 spot diffractive spot generator: (a) lens profile, (b) image plot of the electric field intensity in the x, y focal plane, and (c) line scans of the electric field magnitudes in the focal plane.

Fig. 8
Fig. 8

Analysis of a nonaxisymmetric diffractive lens: (a) lens profile, (b) line scans of the electric field intensities in the focal plane, (c) image plot of the electric field intensity in the x, z plane, and (d) image plot of the electric field intensity in the x, y focal plane.

Fig. 9
Fig. 9

Analysis of frequency-selective surface containing subwavelength infinite dielectric grating: (a) structure geometry, (b) electric field magnitude of incident field as a function of time, (c) electric field magnitude of reflected field as a function of time, and (d) frequency spectrum of reflection coefficient.

Tables (1)

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Table 1 Memory Requirements of Four of The Most Common Numerical Techniques in EMa

Equations (15)

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Ht=-1μ ×E,  Et=-1 ×H-σE,
Hxt=1μEzy-Eyz,  Ext=1Hzy-Hyz-σEx,
Hyt=1μExz-Ezx,  Eyt=1Hxz-Hzx-σEy,
Hzt=1μEyx-Exy,  Ezt=1Hyx-Hxy-σEz.
Ex|i,j,kn+1=1-σi,j,kΔt/2i,j,k1+σi,j,kΔt/2i,j,k Ex|i,j,kn+Δt/i,j,k1+σi,j,kΔt/2i,j,k1ΔyHz|i,j+1/2,kn+1/2-Hz|i,j-1/2,kn+1/2-Δt/i,j,k1+σi,j,kΔt/2i,j,k1ΔzHy|i,j,k+1/2n+1/2-Hy|i,j,k-1/2n+1/2,
Eρ, ϕ, z, t=k=1 E1kρ, z, tcoskϕ+E2kρ, z, tsinkϕ,
Hρ, ϕ, z, t=k=1 H1kρ, z, tcoskϕ+H2kρ, z, tsinkϕ,
μ Hρ,kt=kρ Ez,k+Eϕ,kz,  μ Hϕ,kt=-Eρ,kz+Ez,kρ,  μ Hz,kt=-1ρρEϕ,kρ-kρ Eρ,k,
 Eρ,kt=-kρ Hz,k-Hϕ,kz+σEρ,k,   Eϕ,kt=Hρ,kz-Hz,kρ+σEϕ,k,   Ez,kt=1ρρHϕ,kρ-kρ Hρ,k+σEz,k.
Hρ,kni, j=Hρ,kn-1i, j+kΔtμρ0i Ez,kn-1/2i, j+ΔtμΔzEϕ,kn-1/2i, j+1-Eϕ,kn-1/2i, j,
Hϕ,kni, j=Hϕ,kn-1i, j-kΔtμΔzEρ,kn-1/2i, j+1-Eρ,kn-1/2i, j+ΔtμΔρEz,kn-1/2i+1, j-Ez,kn-1/2i, j,
Hz,kni, j=Hz,kn-1i, j-kΔtμρi Eρ,kn-1/2i, j-ΔtμΔρρiρ0i+1Eϕ,kn-1/2i+1, j-ρ0iEϕ,kn-1/2i, j,
Er=Eir+SEscds,
Esc=-jωμGnˆ×H-nˆ×E×G-nˆ·EG,
Gr-r=14πexp-jk|r-r||r-r|,

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