Abstract

A novel spectral method with variable transformation, the adaptive Hermite–Gauss decomposition method (A-HGDM), has been developed and applied to the analysis of three-dimensional (3D) dielectric structures. The proposed method includes an optimization strategy to automatically find the quasi-optimum numerical parameters of the variable transformation with low computational effort. The technique has been tested by analyzing two typical 3D dielectric structures: the rectangular step-index waveguide and the rib-waveguide directional coupler. In both cases, the A-HGDM increases the accuracy of the Hermite–Gauss decomposition method (HGDM), especially when the mode is near cutoff, and improves the computational efficiency of previously published optimization strategies (optimized HGDM).

© 2003 Optical Society of America

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References

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  1. C. H. Henry, B. H. Verbeek, “Solution of the scalar wave equation for arbitrarily shaped dielectric waveguides by two-dimensional Fourier analysis,” J. Lightwave Technol. 7, 308–313 (1989).
    [CrossRef]
  2. S. J. Hewlett, F. Ladouceur, “Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff,” J. Lightwave Technol. 13, 375–383 (1995).
    [CrossRef]
  3. J. G. Wangüemert, I. Molina, “Analysis of dielectric waveguides by a modified Fourier decomposition method with adaptive mapping parameters,” J. Lightwave Technol. 19, 1614–1627 (2001).
    [CrossRef]
  4. I. Molina, J. G. Wangüemert, “Variable transformed series expansion approach for the analysis of nonlinear guided waves in planar dielectric waveguides,” J. Lightwave Technol. 16, 1354–1363 (1998).
    [CrossRef]
  5. R. Gallawa, C. Goyal, Y. Tu, K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin’s method with Hermite–Gauss basis functions,” IEEE J. Quantum Electron. 27, 518–522 (1991).
    [CrossRef]
  6. A. Weisshar, J. Li, R. L. Gallawa, I. C. Goyal, Y. Tu, K. Ghatak, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite–Gauss basis functions,” J. Lightwave Technol. 13, 1795–1800 (1995).
    [CrossRef]
  7. T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, 429–433 (1993).
    [CrossRef]
  8. J. P. Boyd, Chebyshev and Fourier Spectral Methods (Springer-Verlag, Berlin, 1989).
  9. J. G. Wangüemert, “Desarrollo y validación de métodos espectrales para el análisis y diseño de dispositivos ópticos lineales y no lineales,” Ph.D. thesis (Universidad de Málaga, Málaga, Spain, 1999).

2001

1998

1995

S. J. Hewlett, F. Ladouceur, “Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff,” J. Lightwave Technol. 13, 375–383 (1995).
[CrossRef]

A. Weisshar, J. Li, R. L. Gallawa, I. C. Goyal, Y. Tu, K. Ghatak, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite–Gauss basis functions,” J. Lightwave Technol. 13, 1795–1800 (1995).
[CrossRef]

1993

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, 429–433 (1993).
[CrossRef]

1991

R. Gallawa, C. Goyal, Y. Tu, K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin’s method with Hermite–Gauss basis functions,” IEEE J. Quantum Electron. 27, 518–522 (1991).
[CrossRef]

1989

C. H. Henry, B. H. Verbeek, “Solution of the scalar wave equation for arbitrarily shaped dielectric waveguides by two-dimensional Fourier analysis,” J. Lightwave Technol. 7, 308–313 (1989).
[CrossRef]

Bjarklev, A.

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, 429–433 (1993).
[CrossRef]

Boyd, J. P.

J. P. Boyd, Chebyshev and Fourier Spectral Methods (Springer-Verlag, Berlin, 1989).

Gallawa, R.

R. Gallawa, C. Goyal, Y. Tu, K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin’s method with Hermite–Gauss basis functions,” IEEE J. Quantum Electron. 27, 518–522 (1991).
[CrossRef]

Gallawa, R. L.

A. Weisshar, J. Li, R. L. Gallawa, I. C. Goyal, Y. Tu, K. Ghatak, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite–Gauss basis functions,” J. Lightwave Technol. 13, 1795–1800 (1995).
[CrossRef]

Ghatak, K.

A. Weisshar, J. Li, R. L. Gallawa, I. C. Goyal, Y. Tu, K. Ghatak, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite–Gauss basis functions,” J. Lightwave Technol. 13, 1795–1800 (1995).
[CrossRef]

R. Gallawa, C. Goyal, Y. Tu, K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin’s method with Hermite–Gauss basis functions,” IEEE J. Quantum Electron. 27, 518–522 (1991).
[CrossRef]

Goyal, C.

R. Gallawa, C. Goyal, Y. Tu, K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin’s method with Hermite–Gauss basis functions,” IEEE J. Quantum Electron. 27, 518–522 (1991).
[CrossRef]

Goyal, I. C.

A. Weisshar, J. Li, R. L. Gallawa, I. C. Goyal, Y. Tu, K. Ghatak, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite–Gauss basis functions,” J. Lightwave Technol. 13, 1795–1800 (1995).
[CrossRef]

Henry, C. H.

C. H. Henry, B. H. Verbeek, “Solution of the scalar wave equation for arbitrarily shaped dielectric waveguides by two-dimensional Fourier analysis,” J. Lightwave Technol. 7, 308–313 (1989).
[CrossRef]

Hewlett, S. J.

S. J. Hewlett, F. Ladouceur, “Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff,” J. Lightwave Technol. 13, 375–383 (1995).
[CrossRef]

Ladouceur, F.

S. J. Hewlett, F. Ladouceur, “Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff,” J. Lightwave Technol. 13, 375–383 (1995).
[CrossRef]

Li, J.

A. Weisshar, J. Li, R. L. Gallawa, I. C. Goyal, Y. Tu, K. Ghatak, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite–Gauss basis functions,” J. Lightwave Technol. 13, 1795–1800 (1995).
[CrossRef]

Lumholt, O.

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, 429–433 (1993).
[CrossRef]

Molina, I.

Pedersen, B.

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, 429–433 (1993).
[CrossRef]

Povlsen, J. H.

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, 429–433 (1993).
[CrossRef]

Rasmussen, T.

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, 429–433 (1993).
[CrossRef]

Rottwitt, K.

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, 429–433 (1993).
[CrossRef]

Tu, Y.

A. Weisshar, J. Li, R. L. Gallawa, I. C. Goyal, Y. Tu, K. Ghatak, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite–Gauss basis functions,” J. Lightwave Technol. 13, 1795–1800 (1995).
[CrossRef]

R. Gallawa, C. Goyal, Y. Tu, K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin’s method with Hermite–Gauss basis functions,” IEEE J. Quantum Electron. 27, 518–522 (1991).
[CrossRef]

Verbeek, B. H.

C. H. Henry, B. H. Verbeek, “Solution of the scalar wave equation for arbitrarily shaped dielectric waveguides by two-dimensional Fourier analysis,” J. Lightwave Technol. 7, 308–313 (1989).
[CrossRef]

Wangüemert, J. G.

Weisshar, A.

A. Weisshar, J. Li, R. L. Gallawa, I. C. Goyal, Y. Tu, K. Ghatak, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite–Gauss basis functions,” J. Lightwave Technol. 13, 1795–1800 (1995).
[CrossRef]

IEEE J. Quantum Electron.

R. Gallawa, C. Goyal, Y. Tu, K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin’s method with Hermite–Gauss basis functions,” IEEE J. Quantum Electron. 27, 518–522 (1991).
[CrossRef]

J. Lightwave Technol.

A. Weisshar, J. Li, R. L. Gallawa, I. C. Goyal, Y. Tu, K. Ghatak, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite–Gauss basis functions,” J. Lightwave Technol. 13, 1795–1800 (1995).
[CrossRef]

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, 429–433 (1993).
[CrossRef]

C. H. Henry, B. H. Verbeek, “Solution of the scalar wave equation for arbitrarily shaped dielectric waveguides by two-dimensional Fourier analysis,” J. Lightwave Technol. 7, 308–313 (1989).
[CrossRef]

S. J. Hewlett, F. Ladouceur, “Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff,” J. Lightwave Technol. 13, 375–383 (1995).
[CrossRef]

I. Molina, J. G. Wangüemert, “Variable transformed series expansion approach for the analysis of nonlinear guided waves in planar dielectric waveguides,” J. Lightwave Technol. 16, 1354–1363 (1998).
[CrossRef]

J. G. Wangüemert, I. Molina, “Analysis of dielectric waveguides by a modified Fourier decomposition method with adaptive mapping parameters,” J. Lightwave Technol. 19, 1614–1627 (2001).
[CrossRef]

Other

J. P. Boyd, Chebyshev and Fourier Spectral Methods (Springer-Verlag, Berlin, 1989).

J. G. Wangüemert, “Desarrollo y validación de métodos espectrales para el análisis y diseño de dispositivos ópticos lineales y no lineales,” Ph.D. thesis (Universidad de Málaga, Málaga, Spain, 1999).

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Figures (10)

Fig. 1
Fig. 1

Basic idea behind the optimization criterion: (a) electric field distribution in the original domain; (b), (c) spectral coefficients in the transformed domain obtained with two different sets of transformation parameters pi=(αxi, oxi) and pj=(αxj, oxj), respectively.

Fig. 2
Fig. 2

Optimization algorithm flow chart.

Fig. 3
Fig. 3

Geometry of the rectangular step-index waveguide: (a) refractive index in the original domain xy, (b) normalized refractive index in the normalized original domain xy.

Fig. 4
Fig. 4

Rectangular step-index waveguide: normalized propagation constant versus normalized frequency for the fundamental mode (Nx=Ny=7).

Fig. 5
Fig. 5

Rectangular step-index waveguide (Vx=Vy=1): convergence rates of the normalized propagation constant versus the number of coefficients for the fundamental mode.

Fig. 6
Fig. 6

Rectangular step-index waveguide (Vx=Vy=1, Nx=Ny=7): contour map of the quotient b/bmax (%). The path followed by the optimization algorithm is superimposed onto the map.

Fig. 7
Fig. 7

Geometry of a rib-waveguide asymmetric directional coupler (λ=1.55 µm).

Fig. 8
Fig. 8

Symmetric rib-waveguide directional coupler (2W2=3 µm): (a) electric field contour maps, (b) A-HGDM with Nx=Ny=10; (c), (d) HGDM with Nx=Ny=10; (e), (f) HGDM with Nx=Ny=25.

Fig. 9
Fig. 9

Directional coupler: convergence of the effective refractive index (Neff=β/k0) as a function of the number of coefficients.

Fig. 10
Fig. 10

Rectangular step-index waveguide: computation time required to formulate and solve the eigenvalue problem once versus the number of coefficients. The computer used was a Pentium IV 1.7 GHz.

Tables (1)

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Table 1 Summary of Spectral Methods with Variable Transformation

Equations (51)

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2ϕ(x, y)x2+2ϕ(x, y)y2+k02n2(x, y)ϕ(x, y)=β2ϕ(x, y),
1Vx22ϕ(x, y)x2+1Vy22ϕ(x, y)y2+n2(x, y)ϕ(x, y)=bϕ(x, y),
u=f(x),v=g(y).
1Vx2f1(u)2ϕu2+f2(u)ϕu+1Vy2g1(u)2ϕv2+g2(v)ϕv+n2(u, v)ϕ=bϕ,
f1(u)=ux2,f2(u)=2ux2,
g1(v)=vy2,g2(v)=2vy2.
ϕ(u, v)=k=0NϕkFk(u, v),
[M]{Φ}=b{Φ},
[M]=1Vx2{[P(f1(u))][DDu]+[P(f2(u))][Du]}+1Vy2{[P(g1(v))][DDv]+[P(g2(v))][Dv]}+[P(n2(u, v))],
DDu|i,k=Ω0Fi*(u, v)2Fk(u, v)u2dudv,
DDv|i,k=Ω0Fi*(u, v)2Fk(u, v)v2dudv,
Du|i,k=Ω0Fi*(u, v)Fk(u, v)ududv,
Dv|i,k=Ω0Fi*(u, v)Fk(u, v)vdudv,
P(h(u, v))|i,k=Ω0h(u,v)Fi*(u,v)Fk(u,v)dudv,
x=f-1(u),y=g-1(v).
Bm(x, σx)=Hm(x/σx)exp[-(x/σx)2/2](π1/22mm!)1/2,
Bn(y, σy)=Hn(y/σy)exp[-(y/σy)2/2](π1/22nn!)1/2.
u=αx(x-ox),v=αy(y-oy).
u=Vxx,v=Vyy
u=1σxx,v=1σyy
[M]=αx21Vx2[DDu]+αy21Vy2[DDv]+[P(n2(u, v))].
Var(p)=mnm2n2|ϕm,n(p)|2mn|ϕm,n(p)|2,
ϕ(u, v)k=0NϕkFk(u, v),
ϕk=ϕ(u, v), Fk*(u, v)Fk(u, v), Fk*(u, v)=Ω0ϕ(u, v)Fk*(u, v)dudvΩ0Fk(u, v)Fk*(u, v)dudv.
Fk(u, v)=Bm(u)Bn(v)=Hm(u)exp(-u2/2)(π1/22mm!)1/2Hn(v)exp(-v2/2)(π1/22nn!)1/2,
m(0,, Nx-1),n(0,, Ny-1),
ϕ(u, v)m=0Nx-1n=0Ny-1ϕm,nBm(u)Bn(v).
m=k mod Nx,n=k div Nx,
k=nNx+m.
[UV]={U}{V}t,
{U}={u0, u1,, umax}t,
{V}={v0, v1,, vmax}t,
[ϕ([UV])]=[Tinv({V})][Φ][Tinv({U})]t,
[Tinv({U})]=B0(u0)B1(u0)BNx-1(u0)B0(u1)B1(u1)BNx-1(u1)B0(umax)B1(umax)BNx-1(umax),
[Tinv({V})]=B0(v0)B1(v0)BNy-1(v0)B0(v1)B1(v1)BNy-1(v1)B0(vmax)B1(vmax)BNy-1(vmax).
ϕm,n=pqwpwqϕ(up, vq)Bm(up)Bn(vq)pqwpwq[Bm(up)]2[Bn(vq)]2,
[Φ]=[T({V})][ϕ([UV])][T({U})]t,
T({U})=inverse(Tinv({U})),
T({V})=inverse(Tinv({V})).
ψ(u, v)=2ϕ(u, v)u2,
{Ψ}=[DDu]{Φ},
DDu|i,k=Ω0Fi*(u, v)2Fk(u, v)u2dudv=Ω0Br(u)Bs(v)2Bm(u)Bn(v)u2dudv.
2Bm(u)u2=u2Bm(u)-(2m+1)Bm(u),
-u2Bm(u)Br(u)du=12[(m+1)(m+2)]1/2δm,r+2+12(2m+1)δm,r+12[m(m-1)]1/2δm,r-2,
DDu|i,k=12[(m+1)(m+2)]1/2δi,k+2-12(2m+1)δi,k+12[m(m-1)]1/2δi,k-2.
ψ(u, v)=2ϕ(u, v)v2,
{Ψ}=[DDv]{Φ},
DDv|i,k=12[(n+1)(n+2)]1/2δi,k+2Nx-12(2n+1)δi,k+12[n(n-1)]1/2δi,k-2Nx.
ψ(u, v)=f(u, v)ϕ(u, v);
{Ψ}=[P(f(u, v))]{Φ},
P(f(u, v))|i,k=Ω0f(u, v)Br(u)Bs(v)Bm(u)Bn(v)dudv.

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