Abstract

A boundary variation method for the fast and accurate modeling of three-dimensional waveguide grating couplers is presented. The algorithm is verified by detailed comparisons with the results of a rigorous spectral collocation method, showing excellent agreement. Examples of the modeling of large waveguide grating couplers are given to illustrate the applicability and versatility of the method.

© 2001 Optical Society of America

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References

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  1. J. S. Hesthaven, P. G. Dinesen, J.-P. Lynov, “Spectral collocation time-domain modeling of diffractive optical elements,” J. Comput. Phys. 155, 287–306 (1999).
    [CrossRef]
  2. P. G. Dinesen, J. S. Hesthaven, “Rigorous 3-d analysis of focusing grating couplers using a spectral collocation method,” in Diffractive Optics and Micro-Optics, Vol. 41 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 81–83.
  3. P. G. Dinesen, J. S. Hesthaven, “Fast and accurate modeling of waveguide grating couplers,” J. Opt. Soc. Am. A 17, 1565–1572 (2000).
    [CrossRef]
  4. O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168–1175 (1993).
    [CrossRef]
  5. O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
    [CrossRef]
  6. R. Syms, J. Cozens, Optical Guided Waves and Devices (McGraw-Hill, New York, 1993), p. 114.
  7. O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. II. Finitely conducting gratings, Padé approximants, and singularities,” J. Opt. Soc. Am. A 10, 2307–2316 (1993).
    [CrossRef]
  8. S. A. Schelknuoff, “Some equivalence theorems of electromagnetics and their application to radiation problems,” Bell Syst. Tech. J. 15, 92–112 (1936).
    [CrossRef]
  9. 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]

2000

1999

J. S. Hesthaven, P. G. Dinesen, J.-P. Lynov, “Spectral collocation time-domain modeling of diffractive optical elements,” J. Comput. Phys. 155, 287–306 (1999).
[CrossRef]

1997

1993

1936

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

Bruno, O. P.

Cozens, J.

R. Syms, J. Cozens, Optical Guided Waves and Devices (McGraw-Hill, New York, 1993), p. 114.

Dinesen, P. G.

P. G. Dinesen, J. S. Hesthaven, “Fast and accurate modeling of waveguide grating couplers,” J. Opt. Soc. Am. A 17, 1565–1572 (2000).
[CrossRef]

J. S. Hesthaven, P. G. Dinesen, J.-P. Lynov, “Spectral collocation time-domain modeling of diffractive optical elements,” J. Comput. Phys. 155, 287–306 (1999).
[CrossRef]

P. G. Dinesen, J. S. Hesthaven, “Rigorous 3-d analysis of focusing grating couplers using a spectral collocation method,” in Diffractive Optics and Micro-Optics, Vol. 41 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 81–83.

Hesthaven, J. S.

P. G. Dinesen, J. S. Hesthaven, “Fast and accurate modeling of waveguide grating couplers,” J. Opt. Soc. Am. A 17, 1565–1572 (2000).
[CrossRef]

J. S. Hesthaven, P. G. Dinesen, J.-P. Lynov, “Spectral collocation time-domain modeling of diffractive optical elements,” J. Comput. Phys. 155, 287–306 (1999).
[CrossRef]

P. G. Dinesen, J. S. Hesthaven, “Rigorous 3-d analysis of focusing grating couplers using a spectral collocation method,” in Diffractive Optics and Micro-Optics, Vol. 41 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 81–83.

Lynov, J.-P.

J. S. Hesthaven, P. G. Dinesen, J.-P. Lynov, “Spectral collocation time-domain modeling of diffractive optical elements,” J. Comput. Phys. 155, 287–306 (1999).
[CrossRef]

Mait, J. N.

Mirotznik, M. S.

Prather, D. W.

Reitich, F.

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]

Syms, R.

R. Syms, J. Cozens, Optical Guided Waves and Devices (McGraw-Hill, New York, 1993), p. 114.

Bell Syst. Tech. J.

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

J. Comput. Phys.

J. S. Hesthaven, P. G. Dinesen, J.-P. Lynov, “Spectral collocation time-domain modeling of diffractive optical elements,” J. Comput. Phys. 155, 287–306 (1999).
[CrossRef]

J. Opt. Soc. Am. A

Other

R. Syms, J. Cozens, Optical Guided Waves and Devices (McGraw-Hill, New York, 1993), p. 114.

P. G. Dinesen, J. S. Hesthaven, “Rigorous 3-d analysis of focusing grating couplers using a spectral collocation method,” in Diffractive Optics and Micro-Optics, Vol. 41 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 81–83.

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Figures (8)

Fig. 1
Fig. 1

Waveguide grating coupler with two-dimensional surface relief for controlled coupling between a thin-film guided wave and free-space radiation.

Fig. 2
Fig. 2

Line scans of far-field radiation from a surface-relief grating along (a) ϕ=0 and (b) θ=π/2, comparing the proposed boundary variation method (solid curves) and a rigorous spectral collocation method (dashed curves).

Fig. 3
Fig. 3

Contour plot of the field variable, Ey, in an aperture plane one wavelength above the surface relief, computed by the spectral collocation method (solid gray-scale contours) and the boundary variation method (line contours).

Fig. 4
Fig. 4

Contour plots of the near-field radiation from a three-dimensional grating coupler, illustrating focusing at 20λ away from the surface relief.

Fig. 5
Fig. 5

Contour plots of Ey of near-field radiation from a three-dimensional grating coupler for (a) y=0 and (b) z=0. Gaussian truncation widths are wz=wy=16λ.

Fig. 6
Fig. 6

Contour plot of the Ey field component in the focal plane at x=47.

Fig. 7
Fig. 7

Far-field radiation from a focusing grating coupler with multiplexed surface relief.

Fig. 8
Fig. 8

Contour plots of Ey at several distances from the surface relief for a multiplexed grating.

Tables (1)

Tables Icon

Table 1 Comparison of Memory Usage and Computational Time for the Boundary Variation Method and the Spectral Collocation Method for Three-Dimensional Surface-Relief Scattering Problems

Equations (43)

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0=neff-(λ/Λ)
ΔE+k2E=0,
H=1iωμ0(×E).
E±=r=-s=-Br,s± exp(±iαr,s±x+iγsy+iβrz),
βr=β0+rKz,
γs=γ0+sKy,
(αr,s±)2+βr2+γs2=(k±)2.
x(z, y)=fδ(z, y)=δf(z, y).
Br,s±=p=0 dp,(r,s)±δp,
Ey,inc=Ai exp(ikix-iβz)+Bi exp(-ikix-iβz).
ki2+β2=ni2k02,
Ey,inc,0=B0 exp(ik0x-iβz),
Ey,inc,1=A1 exp(-ik1x-iβz)+B1 exp(ik1x-iβz).
nδ×(E+-E-)=nδ×yˆ(Ey,inc,1-Ey,inc,0)=-(nδ×yˆ)[A1 exp(-ik1x-iβz)+B1×exp(ik1x-iβz)-B0 exp(ik0x-iβz)]nδ×(×E+-×E-)=nδ×(×yˆEy,inc,1-×yˆEy,inc,0)nδ×βωμ0xˆ[A1exp(-ik1x-iβz)+B1×exp(ik1x-iβz)-B0exp(ik0x-iβz)]+zˆ 1iωμ0[-ik1A1 exp(-ik1x-iβz)+ik1B1×exp(ik1x-iβz)-ik0B0 exp(ik0x-iβz)]
nδ=1l(1,-δfy,-δfz),
E3+-E3-+δfz(E1+-E1-)=0,
E2+-E2-+δfy(E1+-E1-)=A1 exp(-ik1x-iβz)+B1 exp(ik1x-iβz)-B0 exp(ik0x-iβz),
Ex2+-Ex2--(Ey1+-Ey1-)+δfz(Ey3+-Ey3--Ez2-+Ez2-)=-ik1A1 exp(-ik1x-iβz)+ik1B1×exp(ik1x-iβz)-ik0B0 exp(ik0x-iβz)+δfziβ[A1 exp(-ik1x-iβz)+B1×exp(ik1x-iβz)-B0 exp(ik0x-iβz)],
Ex3+-Ex3--(Ez1+-Ez1-)+δfy[Ey3+-Ey3--(Ez2+-Ez2-)]=iβδfy[A1 exp(-ik1x-iβz)+B1 exp(ik1x-iβz)-B0 exp(ik0x-iβz)].
·E±=0,
Ex1++Ey2++Ez3+=0,
Ex1-+Ey2-+Ez3-=0.
1n! n(E2+-E2-)δn=-p=0n-1 fn-p(n-p)! n-pxn-p 1p! p(E2+-E2-)δp-fyp=0n-1 fn-p-1(n-1-p)! n-1-pxn-1-p×1p! p(E1+-E1-)δp+[A1(-ik1)n+B1(ik1)n-B0(ik0)n] fnn!exp(-iβz),
1n! nδn[Ex2+-Ex2--(Ey1+-Ey1-)]=-p=0n-1 fn-p(n-p)! n-pxn-p×1p! p[Ex2+-Ex2--(Ey1+-Ey1-)]δp-fzp=0n-1 fn-p-1(n-1-p)! n-1-pxn-1-p×1p! p(Ey3+-Ey3--Ez2++Ez2-)δp+[-A1(-ik1)n+1+B1(ik1)n+1+B0(ik0)n+1] fnn! exp(-iβz)+βfy[A1(-ik1)n-1+B1(ik1)n-1-B0(ik0)n-1] fn-1(n-1)! exp(-iβz).
f(y, z)ll!=t=-lFylFyq=-lFzlFz Cl,(t,q) exp[i(Kyty+Kzqz)],
fz fl-1(l-1)!=-lF<t,q<lF Cl,(t,q)(iKzqz)×exp[i(Kyty+Kzqz)],
fy fl-1(l-1)!=-lF<t,q<lF Cl,(t,q)(iKyty)×exp[i(Kyty+Kzqz)].
1p! pE±δp(x, y, z; δ=0)=r,s 1p! pBr,s±δp(0)exp(±iαr,s±x+iγsy-iβrz)=r,s dp,(r,s)± exp(±iαr,s±x+iγsy-iβrz),
r,s[dp,(r,s)2+-dp,(r,s)2-]exp[i(γsy+βrz)]=-p=0n-1-(n-p)F<t,q<(n-p)F Cn-p,(t,q) exp[i(Kyty+Kzqz)]l,m[(iαl,m+)n-pdp,(l,m)2+-(-iαl,m-)n-pdp,(l,m)2-]×exp[i(γsy+βrz)]-p=0n-1-(n-p)F<t,q<(n-p)F Cn-p,(t,q)(iKyt)exp[i(Kyty+Kzqz)]×l,m[(iαl,m+)n-p-1dp,(l,m)1+-(-iαl,m-)n-p-1dp,(l,m)1-]exp[i(γsy+βrz)]+p=0n-1-(n-p)F<t,q<(n-p)F Cn-p,(t,q) exp[i(Kyty+Kzqz)][A1(-ik1)n+B1(ik1)n-B0(ik0)n]exp(-iβz),
r,s{iαr,s+dp,(r,s)2++iαr,s-dp,(r,s)2--[iγsdp,(r,s)1+-iγsdp,(r,s)1-]}exp[i(γsy+βrz)]=-p=0n-1-(n-p)F<t,q<(n-p)F Cn-p,(t,q)exp[i(Kyty+Kzqz)]×l,m[(iαl,m+)n-p+1dp,(l,m)2+-(-iαl,m-)n-p+1dp,(l,m)2--(iαl,m+)n-p(iγs)dp,(l,m)1++(iαl,m-)n-p(iγs)dp,(l,m)1-]×exp[i(γsy+βrz)]-p=0n-1-(n-p)F<t,q<(n-p)F Cn-p,(t,q)(iKzq)exp[i(Kyty+Kzqz)]×l,m((iαl,m+)n-p-1[iγsdp,(l,m)3+-(-iαl,m-)n-p-1iγsdp,(l,m)3--(iαl,m+)n-p-1(iβr)dp,(l,m)2++(iαl,m-)n-p-1(iβr)dp,(l,m)2-]exp[i(γsy+βrz)]+p=0n-1-(n-p)F<t,q<(n-p)F Cn-p,(t,q) exp[i(Kyty+Kzqz)]×[-A1(-ik1)n+1+B1(ik1)n+1+B0(ik0)n+1]exp(-iβz)+βp=0n-1-(n-p)F<t,q<(n-p)F Cn-p,(t,q)×(iKyt)exp[i(Kyty+Kzqz)][A1(-ik1)n-1+B1(ik1)n-1-B0(ik0)n-1]exp(-iβz).
-p=0n-1l,m-(n-p)F<t,q<(n-p)F[(iαl,m+)n-pdp,(l,m)2+-(-iαl,m-)n-pdp,(l,m)2-]Cn-p,(t,q) exp[i(Kyty+Kzqz)]exp[i(γsy+βrz)]-p=0n-1-(n-p)F<t,q<(n-p)Fl,m[(iαl,m+)n-p-1dp,(l,m)1+-(-iαl,m-)n-p-1dp,(l,m)1-](iKyt)Cn-p,(t,q) exp[i(Kyty+Kzqz)]exp[i(γsy+βrz)]+-(n-p)F<t,q<(n-p)F[A1(-ik1)n+B1(ik1)n-B0(ik0)n]Cp,(t,q) exp[i(γsy+βrz)],
-p=0n-1l,m-(n-p)F<t,q<(n-p)F[(iαl,m+)n-p+1dp,(l,m)2+-(-iαl,m-)n-p+1dp,(l,m)2--(iαl,m+)n-p(iγs)dp,(l,m)1++(iαl,m-)n-p(iγs)dp,(l,m)1-]Cn-p,(t,q) exp[i(Kyty+Kzqz)]exp[i(γsy+βrz)]-p=0n-1l,m-(n-p)F<t,q<(n-p)F×[(iαl,m+)n-p-1(iγs)dp,(l,m)3+-(-iαl,m-)n-p-1iγsdp,(l,m)3--(iαl,m+)n-p-1(iβr)dp,(l,m)2++(iαl,m-)n-p-1(iβr)dp,(l,m)2-]×(iKzq)Cn-p,(t,q) exp[i(Kyty+Kzqz)]exp[i(γsy+βrz)]+p=0n-1-(n-p)F<t,q<(n-p)F×{-A1(-ik1)n+1+B1(ik1)n+1+B0(ik0)n+1+β[A1(-ik1)n-1+B1(ik1)n-1-B0(ik0)n-1]}×Cn,(t,q) exp[i(γsy+βrz)].
exp[i(Kyty+Kzqz)]exp[i(γsy+βrz)]=exp[i(γs+ty+βr+qz)],
dn,(r,s)2+-dn,(r,s)2-=[-B0(ik0)n+A1(-ik1)n+B1(ik1)n]Cn,(r,s)-p=0n-1l=max[-pF,r-(n-p)F]min[pF,r+(n-p)F]m=max[-pF,s-(n-p)F]min[pF,s+(n-p)F]{(iαl,m+)n-pdp,(l,m)2+-(-iαl,m-)n-pdp,(l,m)2-+[(iαl,m+)n-1-pdp,(l,m)1+-(-iαl,m-)n-1-pdp,(l,m)1-][iKy(s-m)]}Cn-p,(r-t,s-m),
αr,s+dn,(r,s)2++αr,s-dn,(r,s)2--γsdn,(r,s)1++γsdn,(r,s)1-={-A1(-ik1)n+1+B1(ik1)n+1+B0(ik0)n+1+iβ[A1(-ik1)n-1+B1(ik1)n-1-B0(ik0)n-1](iKzr)}Cn,(r,s)-p=0n-1l=max[-pF,r-(n-p)F]min[pF,r+(n-p)F]m=max[-pF,s-(n-p)F]min[pF,s+(n-p)F]{αl,m+(iαl,m+)n-pdp,(r,s)2++αl,m-(-iαl,m-)n-pdp,(r,s)2--γs(iαl,m+)n-pdp,(r,s)1++γs(-iαl,m-)n-pdp,(r,s)1-+[γs(iαl,m+)n-p-1dp,(r,s)3+-γs(-iαl,m-)n-p-1dp,(r,s)3-βr(iαl,m+)n-p-1dp,(r,s)2++βr(-iαl,m-)n-p-1dp,(r,s)2-][iKz(r-l)]}Cn-p,(r-l,s-m),
dp,(l,m)i,±=0,|l|>pF,|m|>pF.
dn,(r,s)3+-dn,(r,s)3-=-p=0n-1l=max[-pF,r-(n-p)F]min[pF,r+(n-p)F]m=max[-pF,s-(n-p)F]min[pF,s+(n-p)F]{(iαl,m+)n-pdp,(r,s)3+-(-iαl,m-)n-pdp,(r,s)3-+[(iαl,m+)n-1-pdp,(l,m)1+-(-iαl,m-)n-1-pdp,(l,m)1-][iKz(r-l)]}Cn-p,(r-l,s-m),
αr,s+dn,(r,s)3++αr,s-dn,(r,s)3--βrdn,(r,s)1++βrdn,(r,s)1-=[A1(-ik1)n-1+B1(ik1)n-1-B0(ik0)n-1]iβ(iK1r)Cn,(r,s)-p=0n-1l=max[-pF,r-(n-p)F]min[pF,r+(n-p)F]m=max[-pF,s-(n-p)F]min[pF,s+(n-p)F]{βr(iαl,m+)n-pdp,(r,s)1+-βr(-iαl,m-)n-pdp,(r,s)1--αl,m+(iαl,m+)n-pdp,(r,s)3+-αl,m-(-iαl,m-)n-pdp,(r,s)3-+[γs(iαl,m+)n-1-pdp,(l,m)3+-γs(-iαl,m-)n-1-pdp,(l,m)3--βr(iαl,m+)n-1-pdp,(l,m)2++βr(-iαl,m-)n-1-pdp,(l,m)2-][iKy(s-m)]}Cn-p,(r-l,s-m).
αr,s+dn,(r,s)1++γsdn,(r,s)2+-βrdn,(r,s)3+=0,
-αr,s-dn,(r,s)1-+γsdn,(r,s)2--βrdn,(r,s)3-=0,
fδ(z, y)=A exp-z-z0wz2-y-y0wy2×cos(2π{a0+a1[r(z, y)-r0]}×[r(z, y)-r0]).
r(z, y)=[(z-zc)2+(y-yc)2]1/2.
fδ(z, y)=A exp-zwz2-ywy2×cos[Kz(r-r0)+2πa1(r-r0)2]×cos[-1/2(r-r0)+2πa1(r-r0)2].

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