Abstract

A boundary variation method for the analysis of both infinite periodic and finite aperiodic waveguide grating couplers in two dimensions is introduced. Based on a previously introduced boundary variation method for the analysis of metallic and transmission gratings [J. Opt. Soc. Am. A 10, 2307, 2551 (1993)], a numerical algorithm suitable for waveguide grating couplers is derived. Examples of the analysis of purely periodic grating couplers are given that illustrate the convergence of the scheme. An analysis of the use of the proposed method for focusing waveguide grating couplers is given, and a comparison with a highly accurate spectral collocation method yields excellent agreement and illustrates the attractiveness of the proposed boundary variation method in terms of speed and achievable accuracy.

© 2000 Optical Society of America

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References

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  1. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  2. B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 1592–1598 (1994).
    [CrossRef]
  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]
  4. 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, 1131–1141 (1999).
    [CrossRef]
  5. K. H. Dridi, A. Bjarklev, “Optical electromagnetic vector-field modeling for the accurate analysis of finite diffractive structures of high complexity,” Appl. Opt. 38, 1668–1676 (1999).
    [CrossRef]
  6. 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]
  7. 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]
  8. 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]
  9. 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]
  10. S. Ramo, J. R. Whinnery, T. van Duzer, Fields and Waves in Communications Electronics, 3rd ed. (Wiley, New York, 1993).
  11. O. Bruno, F. Reitich, “Solution of a boundary value problem for Helmholtz equation via variation of the boundary into the complex domain,” Proc. R. Soc. Edinburgh, Sect. A: Math. 122, 317–340 (1992).
    [CrossRef]
  12. S. A. Schelknuoff, “Some equivalence theorems of electromagnetics and their application to radiation problems,” Bell Syst. Tech. J. 15, 92–112 (1936).
    [CrossRef]
  13. P. G. Dinesen, J. S. Hesthaven, J. P. Lynov, L. Lading, “Pseudo-spectral method for the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 26, 1124–1130 (1999).
    [CrossRef]

1999 (4)

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, J. P. Lynov, L. Lading, “Pseudo-spectral method for the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 26, 1124–1130 (1999).
[CrossRef]

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, 1131–1141 (1999).
[CrossRef]

K. H. Dridi, A. Bjarklev, “Optical electromagnetic vector-field modeling for the accurate analysis of finite diffractive structures of high complexity,” Appl. Opt. 38, 1668–1676 (1999).
[CrossRef]

1996 (1)

1994 (1)

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 1592–1598 (1994).
[CrossRef]

1993 (3)

1992 (1)

O. Bruno, F. Reitich, “Solution of a boundary value problem for Helmholtz equation via variation of the boundary into the complex domain,” Proc. R. Soc. Edinburgh, Sect. A: Math. 122, 317–340 (1992).
[CrossRef]

1985 (1)

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

Bjarklev, A.

Bruno, O.

O. Bruno, F. Reitich, “Solution of a boundary value problem for Helmholtz equation via variation of the boundary into the complex domain,” Proc. R. Soc. Edinburgh, Sect. A: Math. 122, 317–340 (1992).
[CrossRef]

Bruno, O. P.

Dinesen, P. G.

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, J. P. Lynov, L. Lading, “Pseudo-spectral method for the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 26, 1124–1130 (1999).
[CrossRef]

Dridi, K. H.

Gallagher, N. C.

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 1592–1598 (1994).
[CrossRef]

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.

Hesthaven, J. S.

P. G. Dinesen, J. S. Hesthaven, J. P. Lynov, L. Lading, “Pseudo-spectral method for the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 26, 1124–1130 (1999).
[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]

Hirayama, K.

Lading, L.

P. G. Dinesen, J. S. Hesthaven, J. P. Lynov, L. Lading, “Pseudo-spectral method for the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 26, 1124–1130 (1999).
[CrossRef]

Lichtenberg, B.

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 1592–1598 (1994).
[CrossRef]

Lynov, J. P.

P. G. Dinesen, J. S. Hesthaven, J. P. Lynov, L. Lading, “Pseudo-spectral method for the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 26, 1124–1130 (1999).
[CrossRef]

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]

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.

Ramo, S.

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

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]

Shi, 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).

Wilson, D. W.

Appl. Opt. (1)

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]

J. Comput. Phys. (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]

J. Opt. Soc. Am. A (6)

Opt. Eng. (1)

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. 33, 1592–1598 (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]

Proc. R. Soc. Edinburgh, Sect. A: Math. (1)

O. Bruno, F. Reitich, “Solution of a boundary value problem for Helmholtz equation via variation of the boundary into the complex domain,” Proc. R. Soc. Edinburgh, Sect. A: Math. 122, 317–340 (1992).
[CrossRef]

Other (1)

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

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Figures (7)

Fig. 1
Fig. 1

Thin-film optical waveguide comprising a surface-relief grating for coupling from guided-wave to free-space radiation.

Fig. 2
Fig. 2

Power in the -1st diffraction order as a function of grating depth for a grating period of 0.7036 corresponding to perpendicular diffraction outcoupling.

Fig. 3
Fig. 3

Power output in (top) -1st, (center) -2nd, and (bottom) -3rd diffraction orders as a function of grating period Λ of the surface relief.  

Fig. 4
Fig. 4

Far-field radiation pattern for four lengths, L, of the computational domain.

Fig. 5
Fig. 5

Far-field radiation pattern for various resolutions used in the Fourier series [Eq. (12)] to represent the surface relief. F, number of Fourier modes (24–48 here).

Fig. 6
Fig. 6

Far-field radiation patterns for a FGC computed by the BV method (dashed curve) and by the SC method (solid curve).

Fig. 7
Fig. 7

Near-field radiation patterns for surface-relief amplitudes (a) 0.1, (b) 0.2, and (c) 0.3 by the BV method (dashed curves) and the SC (solid curves) methods.

Tables (3)

Tables Icon

Table 1 Power in the -1st Diffraction Order for Different Numbers of Terms [M,M] in the Padé Approximation to the Power-Series Expansion

Tables Icon

Table 2 Convergence for Periodic Grating Coupler Analysis a

Tables Icon

Table 3 Key Figures for SC and Proposed BV Computations a

Equations (38)

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Δu+k2u=0,
Ei=Aiexp(ikix-iβz)+Biexp(-ikix-iβz),
ki2+β2=ni2k02,
x=fδ(z)=δf(z)
u+-u-=A0exp[ik0δf(z)-iβz]+B0exp[-ik0δf(z)-iβz]-A1exp[ik1δf(z)-iβz]-B1exp[-ik1δf(z)-iβz],
u+n-u-n=n {A0exp[ik0δf(z)-iβz]+B0exp[-ik0δf(z)-iβz]}×n {-A1exp[ik1δf(z)-iβz]-B1exp[-ik1δf(z)-iβz]},
u±(z, x, δ)=n=0un±(z, x)δn.
u±(z, x, δ)=r=- Br±(δ)exp(±iαr±x-iβrz),
un±(z, x)=r=-dn,r±exp(±iαr±x-iβrz).
dn,r±=1n!dnBr±dδn (0),
Br±(δ)=n=0 dn,r±δn
fl(z)l!=r=-lFlFCl,rexp(iKrx)
βr=β+rK,
(αr±)2+βr2=(ki)2,
k=0nf(z)n-k(n-k)!n-kxn-k1k!ku+δk(z, 0, 0)
-n-kxn-k1k!ku-δk(z, 0, 0)
=1n! [-B0(ik0)n+A1(-ik1)n+B1(ik1)n]f(z)nexp(-iβz).
nδ=1(1+δ2f(z)2)1/2 (-δf(z), 1),
k=0nf(z)n-k(n-k)!n-k+1xn-k+11k!ku+δk(z, 0, 0)
-n-k+1xn-k+11k!ku-δk(z, 0, 0)-k=0n-1f(z)f(z)n-k-1(n-k-1)!n-kxn-k-1z1k!ku+δk(z, 0, 0)
-n-kxn-k-1z1k!ku-δk(z, 0, 0)
=1n! {-B0[iβn(ik0)n-1f(z)f(z)n-1+(ik0)n+1f(z)n]+A1[iβn(-ik1)n-1f(z)f(z)n-1-(-ik1)n+1f(z)n]+B1[iβn(ik1)n-1f(z)f(z)n-1+(ik1)n+1f(z)n]}×exp(-iβz).
uk±(z, x)=1k!ku±δk (z, x, 0)
un+-un-
=1n! [-B0(ik0)n+A1(-ik1)n+B1(ik1)n]f(z)n×exp(-iβz)-k=0n-1f(z)n-k(n-k)!n-kuk+xn-k-n-kuk-xn-k,
un+x-un-x
=1n! {-B0[iβn(ik0)n-1f(z)f(z)n-1+(ik0)n+1f(z)n]+A1[iβn(-ik1)n-1f(z)f(z)n-1-(-ik1)n+1f(z)n]+B1[iβn(ik1)n-1f(z)f(z)n-1+(ik1)n+1f(z)n]}exp(-iβz)+k=0n-1f(z)f(z)n-k-1(n-k-1)!n-kuk+xn-k-1z-n-kuk-xn-k-1z-k=0n-1f(z)n-k(n-k)!n-k+1uk+xn-k+1-n-k+1uk-xn-k+1.
r=-(dn,r+-dn,r-)exp(-iβrz)
=[-B0(ik0)n+A1(-ik1)n+B1(ik1)n]×r=-nFnF Cn,rexp(-iβrz)-k=0n-1r=-(n-k)F(n-k)FCn-k,rexp(iKrz)×r=-[(iαr+)n-kdk,r+-(-iαr-)n-kdk,r-]exp(-iβrz).
 
r=-(iαr+dn,r++iαr-dn,r-)exp(-iβrz)
=r=-nFnF Cn,r{-B0[iβ(ik0)n-1(iKr)+(ik0)n+1]+A1[iβ(-ik1)n-1(iKr)-(-ik1)n+1]+B1[iβ(ik1)n-1(iKr)+(+ik1)n+1]}exp(-iβrz)+k=0n-1r=-(n-k)F(n-k)F Cn-k,riKr exp(iKrz)×r=-[(iαr+)n-k-1(-iβr)dk,r+-(-iαr-)n-k-1(-iβr)dk,r-]exp(-iβrz)-k=0n-1r=-(n-k)F(n-k)F Cn-k,rexp(iKrx)×r=-[(iαr+)n-k+1dk,r+-(-iαr-)n-k+1dk,r-]exp(-iβrz).
dn,r+-dn,r-
=[-B0(ik0)n+A1(-ik1)n+B1(ik1)n]Cn,r-k=0n-1q=max[-kF,r-(n-k)F]min[kF,r+(n-k)F] Cn-k,r-q[(iαq+)n-kdk,q+-(-iαq)n-kdk,q-],
iαr+dn,r++iαr-dn,r-
={-B0(ik0)n-1[βKr+(ik0)2]+A1(-ik1)n-1[βKr+(-ik1)2]+B1(ik1)n-1[βKr+(ik1)2]}Cn,r+k=0n-1q=max[-kF,r-(n-k)F]min[kF,r+(n-k)F] Cn-k,r-q{[iK(r-q)]×(-iβq)[(iαq+)n-k-1dk,q+-(-iαq-)n-k-1dk,q-]-[(iαq+)n-k+1dk,q+-(-iαq-)n-k+1dk,q-]}.
fδ(z)=A cos2πΛ z.
fδ(z)=A exp-z-z0w2×cos{2π[a0+a1(z-z0)](z-z0)},

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