Abstract

A pseudospectral method for the analysis of diffractive optical elements is presented. This method is a full-vectorial direct solution of the time-domain Maxwell equations based on a spectral approximation of the spatial derivatives employed within a multidomain framework. The method exhibits little numerical dispersion, and only a few points per wavelength are needed to accurately resolve the propagation of the optical field over long distances. A comparison with the analytic solution for a thin-film waveguide is performed, and examples of analyses of grating couplers are given to demonstrate the feasibility of the method.

© 1999 Optical Society of America

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References

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    [CrossRef] [PubMed]
  2. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  3. K. Hirayama, E. N. Glytsis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A 14, 907–917 (1997).
    [CrossRef]
  4. 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]
  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. M. Schmitz, O. Bryngdahl, “Rigorous concept for the design of diffractive microlenses with high numerical apertures,” J. Opt. Soc. Am. A 14, 901–906 (1997).
    [CrossRef]
  7. M. Mirotznik, J. N. Mait, D. W. Prather, W. A. Beck, “Three-dimensional vector-based analysis of subwavelength diffractive optical elements using the finite-difference time-domain (FDTD) method,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 91–93.
  8. S. Ura, T. Suhara, H. Nishihara, “Aberration characterizations of a focusing grating coupler in an integrated-optic disk pickup device,” Appl. Opt. 26, 4777–4782 (1987).
    [CrossRef] [PubMed]
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    [CrossRef]
  11. R. W. Ziolkowski, “The incorporation of microscopic material models into the FDTD approach for ultrafast optical pulse simulations,” IEEE Trans. Antennas Propag. 45, 375–391 (1997).
    [CrossRef]
  12. B. Yang, D. Gottlieb, J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
    [CrossRef]
  13. A. Taflove, Computational Electrodynamics—The Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 1995).
  14. J. S. Hesthaven, P. G. Dinesen, J.-P. Lynov, “Parallel pseudospectral time-domain modeling of diffractive optical elements,” submitted to J. Comput. Vision.
  15. M. H. Carpenter, C. A. Kennedy, “Fourth order 2N-storage Runge–Kutta scheme,” (NASA, Washington, D.C., 1994).
  16. D. Funaro, Polynomial Approximation of Differential Equations, Vol. 8 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1992).
  17. W. J. Gordon, C. A. Hall, “Transfinite element methods: blending-function interpolation over arbitrary curved element domains,” Numer. Math. 21, 109–129 (1973).
    [CrossRef]
  18. J. S. Hesthaven, “A stable penalty method for the compressible Navier–Stokes equations. III. Multi-dimensional domain decomposition schemes,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Comput. 20, 62–93 (1999).
    [CrossRef]
  19. B. Yang, D. Gottlieb, J. S. Hesthaven, “On the use of PML ABC’s in spectral time-domain simulations of electromagnetic scattering,” in Proceedings of the 13th Annual Review of Progress in Applied Computational Electromagnetics (ACES ’97), E. C. Michielssen, ed. (Applied Computational Electromagnetics Society, Monterey, Calif., 1997), Vol. 2, pp. 926–933.
  20. S. A. Schelknuoff, “Some equivalence theorems of electromagnetics and their application to radiation problems,” Bell Syst. Tech. J. 15, 92–112 (1936).
    [CrossRef]
  21. S. Ramo, J. R. Whinnery, T. van Duzer, Fields and Waves in Communications Electronics, 3rd ed. (Wiley, New York, 1993).

1999 (1)

J. S. Hesthaven, “A stable penalty method for the compressible Navier–Stokes equations. III. Multi-dimensional domain decomposition schemes,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Comput. 20, 62–93 (1999).
[CrossRef]

1997 (5)

1996 (1)

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 (1)

1994 (1)

J. J. M. Braat, M. O. E. Laurijs, “Geometrical optics design and aberration analysis of a focusing grating coupler,” Opt. Eng. 33, 1037–1043 (1994).
[CrossRef]

1992 (1)

1987 (1)

1985 (1)

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

1973 (1)

W. J. Gordon, C. A. Hall, “Transfinite element methods: blending-function interpolation over arbitrary curved element domains,” Numer. Math. 21, 109–129 (1973).
[CrossRef]

1936 (1)

S. A. Schelknuoff, “Some equivalence theorems of electromagnetics and their application to radiation problems,” Bell Syst. Tech. J. 15, 92–112 (1936).
[CrossRef]

Beck, W. A.

M. Mirotznik, J. N. Mait, D. W. Prather, W. A. Beck, “Three-dimensional vector-based analysis of subwavelength diffractive optical elements using the finite-difference time-domain (FDTD) method,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 91–93.

Borsboom, P.-P.

Braat, J. J. M.

J. J. M. Braat, M. O. E. Laurijs, “Geometrical optics design and aberration analysis of a focusing grating coupler,” Opt. Eng. 33, 1037–1043 (1994).
[CrossRef]

Bryngdahl, O.

Carpenter, M. H.

M. H. Carpenter, C. A. Kennedy, “Fourth order 2N-storage Runge–Kutta scheme,” (NASA, Washington, D.C., 1994).

Dinesen, P. G.

J. S. Hesthaven, P. G. Dinesen, J.-P. Lynov, “Parallel pseudospectral time-domain modeling of diffractive optical elements,” submitted to J. Comput. Vision.

Farn, M. W.

Frankena, H. J.

Funaro, D.

D. Funaro, Polynomial Approximation of Differential Equations, Vol. 8 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1992).

Gaylord, T. K.

K. Hirayama, E. N. Glytsis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A 14, 907–917 (1997).
[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.

Gordon, W. J.

W. J. Gordon, C. A. Hall, “Transfinite element methods: blending-function interpolation over arbitrary curved element domains,” Numer. Math. 21, 109–129 (1973).
[CrossRef]

Gottlieb, D.

B. Yang, D. Gottlieb, J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[CrossRef]

B. Yang, D. Gottlieb, J. S. Hesthaven, “On the use of PML ABC’s in spectral time-domain simulations of electromagnetic scattering,” in Proceedings of the 13th Annual Review of Progress in Applied Computational Electromagnetics (ACES ’97), E. C. Michielssen, ed. (Applied Computational Electromagnetics Society, Monterey, Calif., 1997), Vol. 2, pp. 926–933.

Hall, C. A.

W. J. Gordon, C. A. Hall, “Transfinite element methods: blending-function interpolation over arbitrary curved element domains,” Numer. Math. 21, 109–129 (1973).
[CrossRef]

Hesthaven, J. S.

J. S. Hesthaven, “A stable penalty method for the compressible Navier–Stokes equations. III. Multi-dimensional domain decomposition schemes,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Comput. 20, 62–93 (1999).
[CrossRef]

B. Yang, D. Gottlieb, J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[CrossRef]

B. Yang, D. Gottlieb, J. S. Hesthaven, “On the use of PML ABC’s in spectral time-domain simulations of electromagnetic scattering,” in Proceedings of the 13th Annual Review of Progress in Applied Computational Electromagnetics (ACES ’97), E. C. Michielssen, ed. (Applied Computational Electromagnetics Society, Monterey, Calif., 1997), Vol. 2, pp. 926–933.

J. S. Hesthaven, P. G. Dinesen, J.-P. Lynov, “Parallel pseudospectral time-domain modeling of diffractive optical elements,” submitted to J. Comput. Vision.

Hirayama, K.

Kennedy, C. A.

M. H. Carpenter, C. A. Kennedy, “Fourth order 2N-storage Runge–Kutta scheme,” (NASA, Washington, D.C., 1994).

Laurijs, M. O. E.

J. J. M. Braat, M. O. E. Laurijs, “Geometrical optics design and aberration analysis of a focusing grating coupler,” Opt. Eng. 33, 1037–1043 (1994).
[CrossRef]

Lynov, J.-P.

J. S. Hesthaven, P. G. Dinesen, J.-P. Lynov, “Parallel pseudospectral time-domain modeling of diffractive optical elements,” submitted to J. Comput. Vision.

Mait, J. N.

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]

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]

M. Mirotznik, J. N. Mait, D. W. Prather, W. A. Beck, “Three-dimensional vector-based analysis of subwavelength diffractive optical elements using the finite-difference time-domain (FDTD) method,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 91–93.

Mirotznik, M.

M. Mirotznik, J. N. Mait, D. W. Prather, W. A. Beck, “Three-dimensional vector-based analysis of subwavelength diffractive optical elements using the finite-difference time-domain (FDTD) method,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 91–93.

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]

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]

Moharam, M. G.

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

Nishihara, H.

Prather, D. W.

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]

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]

M. Mirotznik, J. N. Mait, D. W. Prather, W. A. Beck, “Three-dimensional vector-based analysis of subwavelength diffractive optical elements using the finite-difference time-domain (FDTD) method,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 91–93.

Ramo, S.

S. Ramo, J. R. Whinnery, T. van Duzer, Fields and Waves in Communications Electronics, 3rd ed. (Wiley, New York, 1993).

Schelknuoff, S. A.

S. A. Schelknuoff, “Some equivalence theorems of electromagnetics and their application to radiation problems,” Bell Syst. Tech. J. 15, 92–112 (1936).
[CrossRef]

Schmitz, M.

Suhara, T.

Taflove, A.

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

Ura, S.

van Duzer, T.

S. Ramo, J. R. Whinnery, T. van Duzer, Fields and Waves in Communications Electronics, 3rd ed. (Wiley, New York, 1993).

Whinnery, J. R.

S. Ramo, J. R. Whinnery, T. van Duzer, Fields and Waves in Communications Electronics, 3rd ed. (Wiley, New York, 1993).

Yang, B.

B. Yang, D. Gottlieb, J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[CrossRef]

B. Yang, D. Gottlieb, J. S. Hesthaven, “On the use of PML ABC’s in spectral time-domain simulations of electromagnetic scattering,” in Proceedings of the 13th Annual Review of Progress in Applied Computational Electromagnetics (ACES ’97), E. C. Michielssen, ed. (Applied Computational Electromagnetics Society, Monterey, Calif., 1997), Vol. 2, pp. 926–933.

Ziolkowski, R. W.

R. W. Ziolkowski, “The incorporation of microscopic material models into the FDTD approach for ultrafast optical pulse simulations,” IEEE Trans. Antennas Propag. 45, 375–391 (1997).
[CrossRef]

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

S. A. Schelknuoff, “Some equivalence theorems of electromagnetics and their application to radiation problems,” Bell Syst. Tech. J. 15, 92–112 (1936).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

R. W. Ziolkowski, “The incorporation of microscopic material models into the FDTD approach for ultrafast optical pulse simulations,” IEEE Trans. Antennas Propag. 45, 375–391 (1997).
[CrossRef]

J. Comput. Phys. (1)

B. Yang, D. Gottlieb, J. S. Hesthaven, “Spectral simulation of electromagnetic wave scattering,” J. Comput. Phys. 134, 216–230 (1997).
[CrossRef]

J. Mod. Opt. (1)

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 (4)

Numer. Math. (1)

W. J. Gordon, C. A. Hall, “Transfinite element methods: blending-function interpolation over arbitrary curved element domains,” Numer. Math. 21, 109–129 (1973).
[CrossRef]

Opt. Eng. (1)

J. J. M. Braat, M. O. E. Laurijs, “Geometrical optics design and aberration analysis of a focusing grating coupler,” Opt. Eng. 33, 1037–1043 (1994).
[CrossRef]

Proc. IEEE (1)

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

SIAM (Soc. Ind. Appl. Math.) J. Sci. Comput. (1)

J. S. Hesthaven, “A stable penalty method for the compressible Navier–Stokes equations. III. Multi-dimensional domain decomposition schemes,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Comput. 20, 62–93 (1999).
[CrossRef]

Other (7)

B. Yang, D. Gottlieb, J. S. Hesthaven, “On the use of PML ABC’s in spectral time-domain simulations of electromagnetic scattering,” in Proceedings of the 13th Annual Review of Progress in Applied Computational Electromagnetics (ACES ’97), E. C. Michielssen, ed. (Applied Computational Electromagnetics Society, Monterey, Calif., 1997), Vol. 2, pp. 926–933.

S. Ramo, J. R. Whinnery, T. van Duzer, Fields and Waves in Communications Electronics, 3rd ed. (Wiley, New York, 1993).

M. Mirotznik, J. N. Mait, D. W. Prather, W. A. Beck, “Three-dimensional vector-based analysis of subwavelength diffractive optical elements using the finite-difference time-domain (FDTD) method,” in Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 91–93.

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

J. S. Hesthaven, P. G. Dinesen, J.-P. Lynov, “Parallel pseudospectral time-domain modeling of diffractive optical elements,” submitted to J. Comput. Vision.

M. H. Carpenter, C. A. Kennedy, “Fourth order 2N-storage Runge–Kutta scheme,” (NASA, Washington, D.C., 1994).

D. Funaro, Polynomial Approximation of Differential Equations, Vol. 8 of Lecture Notes in Physics (Springer-Verlag, Berlin, 1992).

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Figures (8)

Fig. 1
Fig. 1

Time-of-flight velocimeter comprising two FGC’s.

Fig. 2
Fig. 2

Mapping, Ψ1, between physical coordinates (z, x) and general curvilinear coordinates (ξ, η).

Fig. 3
Fig. 3

Pseudospectral grid for analysis of a thin-film optical waveguide. The axis coordinates are normalized as described in Section 2.

Fig. 4
Fig. 4

Grating coupler consisting of a thin-film waveguide with a corrugated top cladding layer. The domain decomposition of the computational domain is indicated.

Fig. 5
Fig. 5

Contour plot showing a snapshot of the Ey field component in a grating coupler. The dark shading corresponds to large negative values of the field, while the light shading corresponds to large positive values.

Fig. 6
Fig. 6

Comparison of the direct solution and the solution obtained by free-space integration for a grating coupler, for field components (a) Hz, (b) Hx, and (c) Ey. Solid curve, direct solution; dashed curve, solution obtained by free-space integration (solid and dashed curves are nearly indistinguishable).

Fig. 7
Fig. 7

Far-field radiation pattern from a grating coupler.

Fig. 8
Fig. 8

Line scan of the Poynting vector in the focal plane of the -1st diffraction order for a FGC. Distance from structure: solid curve, 200λ; dashed curve, 258λ; dashed–dotted curve, 300λ.

Tables (1)

Tables Icon

Table 1 Global L Error of All Three Field Components for a Thin-Film Waveguide Problem as a Function of the Number of Grid Points per Wavelength

Equations (25)

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

H˜zt˜=-cZ0E˜yx˜,
H˜xt˜=cZ0E˜yz˜,
E˜yt˜=cZ0 1n2H˜xz˜-H˜zx˜,
t=t˜ν,x=x˜/λ,z=z˜/λ.
Hx=H˜x,Hy=H˜y,Ez=Z0-1E˜z,
Hzt=-Eyx,
Hxt=Eyz,
Eyt=1n2Hxz-Hzx,
Ey1=Ey2,nˆ×H1=nˆ×H2,nˆ·H1=nˆ·H2,
Tk(z)=cos(k cos-1 z),
zi=-cosiπN,0iN.
(INf)(z)=i=0Nfigi(z).
gi(z)=(1-z2)TN(z)(-1)i+1ciN2(z-zi),
Dij=gj(zi)
dfdz(zi)d(INf)dz(zi)=j=0NDijf(zj),
(IN,Mf)(z, x)=i=0Nj=0Mf(zi, xj)gi(z)gj(x),
ξ=ξ(z, x),η=η(z, x),
qt+A(ξ) qξ+A(η) qη=0,
A(n)=00nx00-nznxn-2-nzn-20,
S(n)=-nˆxnˆz-nˆxnˆznˆxnˆzn-10-n-1,
S-1(n)=12-nˆxnˆzn2nˆz2nˆx0-nˆxnˆz-n,
R=S-1(n)q=R1R2R3=12-nˆxHz+nˆzHx+nEy2nˆzHz+2nˆxHx-nˆxHz+nˆzHx-nEy.
qt+A(ξ) qξ+A(η) qη-f(z, x)q=0,
h(z)=A exp-z-z0w2×cos{2π[a0+a1(z-z0)](z-z0)},
neff+mλΛ=sin θm,

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