Abstract

A novel separable effective adiabatic basis (SEAB) for the solution of the transverse vector wave equation by the variational method is presented. The basis is constructed by a suitably modified adiabatic approximation. The method of SEAB construction is applicable to the waveguides of a general cross section. By calculating scalar modes in rectangular and rib waveguides, we show that the use of SEAB entails computational effort several orders of magnitude less than the use of the more conventional Fourier basis. As an illustrative example, the polarized modes of a rib waveguide are calculated.

© 2004 Optical Society of America

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References

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  1. K. S. Chiang, C. H. Kwan, and K. M. Lo, “Effective-index method with built-in perturbation correction for the vector modes of rectangular-core optical waveguides,” J. Lightwave Technol. 17, 716–722 (1999).
    [CrossRef]
  2. C. H. Kwan and K. S. Chiang, “Study of polarization-dependent coupling in optical waveguide directional couplers by the effective-index method with built-in perturbation correction,” J. Lightwave Technol. 20, 1018–1026 (2002).
    [CrossRef]
  3. W. Huang, H. A. Haus, and H. N. Yoon, “Analysis of buried-channel waveguides and couplers: scalar solution and po- larization correction,” J. Lightwave Technol. 8, 642–648 (1990).
    [CrossRef]
  4. W. Huang and H. A. Haus, “A simple variational approach to optical rib waveguides,” J. Lightwave Technol. 9, 56–61 (1991).
    [CrossRef]
  5. C. H. Henry and 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]
  6. M. A. Forastiere and G. C. Righini, “Scalar analysis of general dielectric waveguides by Fourier decomposition method,” J. Lightwave Technol. 17, 362–368 (1999).
    [CrossRef]
  7. K. M. Lo and E. H. Li, “Solutions of the quasi-vector wave equation for optical waveguides in a mapped infinite domains by the Galerkin’s method,” J. Lightwave Technol. 16, 937–944 (1998).
    [CrossRef]
  8. R. L. Gallawa, I. C. Goyal, Y. Tu, and A. 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]
  9. T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and K. Rottwit, “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]
  10. M. V. Berry, “Quantum adiabatic anholonomy” in Anomalies, Phases, Defects, U. M. Bregola, G. Marmo, and G. Morandi, eds. (Bibliopolis, Naples, Italy, 1990), pp. 125–181.
  11. A. Yariv, Optical Electronics, 3rd ed. (CBS College, New York, 1985).
  12. N. Moiseyev, “Quantum theory of resonances,” Phys. Rep. 302, 212–293 (1998).
    [CrossRef]
  13. R. März, Integrated Optics (Artech House, Norwood, Mass., 1995).
  14. A. S. Davydov, Quantum Mechanics (Pergamon, New York, 1965).
  15. A. Sharma, “Analysis of integrated optical waveguides: variational method and effective-index method with built-in perturbation correction,” J. Opt. Soc. Am. A 18, 1383–1387 (2001).
    [CrossRef]
  16. J. R. Kuttler and V. G. Sigillito, “Eigenvalues of the Laplacian in two dimensions,” SIAM Rev. 26, 163–193 (1984).
    [CrossRef]

2002 (1)

2001 (1)

1999 (2)

1998 (2)

1993 (1)

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and K. Rottwit, “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 (2)

W. Huang and H. A. Haus, “A simple variational approach to optical rib waveguides,” J. Lightwave Technol. 9, 56–61 (1991).
[CrossRef]

R. L. Gallawa, I. C. Goyal, Y. Tu, and A. 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]

1990 (1)

W. Huang, H. A. Haus, and H. N. Yoon, “Analysis of buried-channel waveguides and couplers: scalar solution and po- larization correction,” J. Lightwave Technol. 8, 642–648 (1990).
[CrossRef]

1989 (1)

C. H. Henry and 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]

1984 (1)

J. R. Kuttler and V. G. Sigillito, “Eigenvalues of the Laplacian in two dimensions,” SIAM Rev. 26, 163–193 (1984).
[CrossRef]

Bjarklev, A.

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and K. Rottwit, “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]

Chiang, K. S.

Forastiere, M. A.

Gallawa, R. L.

R. L. Gallawa, I. C. Goyal, Y. Tu, and A. 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]

Ghatak, A. K.

R. L. Gallawa, I. C. Goyal, Y. Tu, and A. 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.

R. L. Gallawa, I. C. Goyal, Y. Tu, and A. 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]

Haus, H. A.

W. Huang and H. A. Haus, “A simple variational approach to optical rib waveguides,” J. Lightwave Technol. 9, 56–61 (1991).
[CrossRef]

W. Huang, H. A. Haus, and H. N. Yoon, “Analysis of buried-channel waveguides and couplers: scalar solution and po- larization correction,” J. Lightwave Technol. 8, 642–648 (1990).
[CrossRef]

Henry, C. H.

C. H. Henry and 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]

Huang, W.

W. Huang and H. A. Haus, “A simple variational approach to optical rib waveguides,” J. Lightwave Technol. 9, 56–61 (1991).
[CrossRef]

W. Huang, H. A. Haus, and H. N. Yoon, “Analysis of buried-channel waveguides and couplers: scalar solution and po- larization correction,” J. Lightwave Technol. 8, 642–648 (1990).
[CrossRef]

Kuttler, J. R.

J. R. Kuttler and V. G. Sigillito, “Eigenvalues of the Laplacian in two dimensions,” SIAM Rev. 26, 163–193 (1984).
[CrossRef]

Kwan, C. H.

Li, E. H.

Lo, K. M.

Lumholt, O.

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and K. Rottwit, “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]

Moiseyev, N.

N. Moiseyev, “Quantum theory of resonances,” Phys. Rep. 302, 212–293 (1998).
[CrossRef]

Pedersen, B.

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and K. Rottwit, “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, and K. Rottwit, “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, and K. Rottwit, “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]

Righini, G. C.

Rottwit, K.

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and K. Rottwit, “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]

Sharma, A.

Sigillito, V. G.

J. R. Kuttler and V. G. Sigillito, “Eigenvalues of the Laplacian in two dimensions,” SIAM Rev. 26, 163–193 (1984).
[CrossRef]

Tu, Y.

R. L. Gallawa, I. C. Goyal, Y. Tu, and A. 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 and 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]

Yoon, H. N.

W. Huang, H. A. Haus, and H. N. Yoon, “Analysis of buried-channel waveguides and couplers: scalar solution and po- larization correction,” J. Lightwave Technol. 8, 642–648 (1990).
[CrossRef]

IEEE J. Quantum Electron. (1)

R. L. Gallawa, I. C. Goyal, Y. Tu, and A. 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. (8)

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and K. Rottwit, “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]

W. Huang, H. A. Haus, and H. N. Yoon, “Analysis of buried-channel waveguides and couplers: scalar solution and po- larization correction,” J. Lightwave Technol. 8, 642–648 (1990).
[CrossRef]

W. Huang and H. A. Haus, “A simple variational approach to optical rib waveguides,” J. Lightwave Technol. 9, 56–61 (1991).
[CrossRef]

C. H. Henry and 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]

M. A. Forastiere and G. C. Righini, “Scalar analysis of general dielectric waveguides by Fourier decomposition method,” J. Lightwave Technol. 17, 362–368 (1999).
[CrossRef]

K. S. Chiang, C. H. Kwan, and K. M. Lo, “Effective-index method with built-in perturbation correction for the vector modes of rectangular-core optical waveguides,” J. Lightwave Technol. 17, 716–722 (1999).
[CrossRef]

K. M. Lo and E. H. Li, “Solutions of the quasi-vector wave equation for optical waveguides in a mapped infinite domains by the Galerkin’s method,” J. Lightwave Technol. 16, 937–944 (1998).
[CrossRef]

C. H. Kwan and K. S. Chiang, “Study of polarization-dependent coupling in optical waveguide directional couplers by the effective-index method with built-in perturbation correction,” J. Lightwave Technol. 20, 1018–1026 (2002).
[CrossRef]

J. Opt. Soc. Am. A (1)

Phys. Rep. (1)

N. Moiseyev, “Quantum theory of resonances,” Phys. Rep. 302, 212–293 (1998).
[CrossRef]

SIAM Rev. (1)

J. R. Kuttler and V. G. Sigillito, “Eigenvalues of the Laplacian in two dimensions,” SIAM Rev. 26, 163–193 (1984).
[CrossRef]

Other (4)

M. V. Berry, “Quantum adiabatic anholonomy” in Anomalies, Phases, Defects, U. M. Bregola, G. Marmo, and G. Morandi, eds. (Bibliopolis, Naples, Italy, 1990), pp. 125–181.

A. Yariv, Optical Electronics, 3rd ed. (CBS College, New York, 1985).

R. März, Integrated Optics (Artech House, Norwood, Mass., 1995).

A. S. Davydov, Quantum Mechanics (Pergamon, New York, 1965).

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Figures (7)

Fig. 1
Fig. 1

Schematic drawing of the rectangular single-mode waveguide. Waveguide dimensions are 4 µm×1 µm, ncore=1.495, nclad=1.445, bottom cladding of 9 µm, top cladding of 6 µm, and λ=1.55 µm.

Fig. 2
Fig. 2

Schematic drawing of the rib waveguide. The first core layer is 12 µm×0.6 µm, the second core layer is 8 µm×1.5 µm, and the intermediate cladding layer is 0.2 µm. The rest of the parameters are in the caption to Fig. 1.

Fig. 3
Fig. 3

Error estimate [see Eq. (19)] in the computation of the first bound mode of the rib waveguide (see Fig. 2) by the SEAB.

Fig. 4
Fig. 4

Error estimate [see Eq. (19)] in the computation of the first bound mode of the rib waveguide (see Fig. 2) by the sine basis.

Fig. 5
Fig. 5

Real part of the first bound TM mode of the rib waveguide. a) Ex, subdominant component; b) Ey, dominant component; c) waveguide’s sketch.

Fig. 6
Fig. 6

Real part of the first TE mode of the rib waveguide. a) Ex, dominant component; b) Ey, subdominant component; c) waveguide’s sketch.

Fig. 7
Fig. 7

Representative waveguide with a step-index profile used to demonstrate the calculation of matrix elements of Θˆ. Coordinates xi and yj mark the position of intervals at which the change in the refractive index takes place.

Tables (6)

Tables Icon

Table 1 Effective Refractive Index, neff, and the Computation Error of the Scalar Wave Equation Solution in the Single-Mode Rectangular Waveguidea

Tables Icon

Table 2 Effective Refractive Index; neff, and the Computation Error of the First Bound Mode in the Rib Waveguidea

Tables Icon

Table 3 Effective Refractive Index, neff, and the Computation Error of the Last Bound Mode in the Rib Waveguidea

Tables Icon

Table 4 Results of the Full Vectorial Calculations for the Single-Mode Rectangular Waveguidea

Tables Icon

Table 5 Results of the Full Vectorial Calculations of the First Bound Mode in the Rib Waveguidea

Tables Icon

Table 6 Results of the Full Vectorial Calculations of the Last Bound Mode in the Rib Waveguidea

Equations (27)

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××E(r)-ω2c2n2(r)E(r)=0,
·E(r)=-E(r)·[ln n2(r)],
2+k2n2(x, y)+fx x+fxx-β2fy x+fxy0fx y+fxy2+k2n2(x, y)+fy y+fyy-β20iβfxiβfy2+k2n2(x, y)-β2E(x, y)=0,
ΘˆE(x, y)=β2E(x, y).
Θˆ=2+k2n2(x, y)002+k2n2(x, y)+fx x+fxxfy x+fxyfx y+fxyfy y+fyy.
E1|E2=1i (E1*××E2-E2××E1*)zdxdy.
E1|E2=E,1*E,2dxdy.
(EL|ER)=dxdyEL·ER,
|ER=j=1NCjRχj,
EL|=j=1NCjLχj*.
j=1NCjR(Θij-ES¯ij)=0,
2x2+2y2+k2n2(x, y)-β2ϕ(x, y)=0
2y2+k2n2(x, y)Ym(y; x)=βm2(x)Ym(y; x),
2x2+βm2(x)Xnm(x)=(βnmad)2Xnm(x).
2x2+β12(x)Xi1(x)=(βi1ad)2Xi1(x).
2y2+β2(y)-βj2Yj(y)=0,
β2(y)=(β11ad)2+dxX1(x)*[k2n2(x, y)-β12(x)]X1(x).
Θkl=dxdyEk(x, y)ΘˆEl(x, y),
ϕlsin(x, y)=2(LxLy)1/2 sin2πiLxxsin2πjLyy,
Er(n)=1-in|ai|2,
El(x)(x, y)=(XiYj, 0),
El(y)(x, y)=(0, XiYj),
Vˆ=fx x+fxxfy x+fxyfx y+fxyfy y+fyy,
Vlm(xx)=[El(x)|Vˆ|Em(x)]=4p=12yin(xp)yfin(xp)dyYi*(y)Yj(y) n(xp+, y)-n(xp-, y)n(xp+, y)+n(xp-, y) dXk*(x)dxXl(x)x=xp.
Vlmyy=[El(y)|Vˆ|Em(y)]=4p=13xin(yp)xfin(yp)dxXi*(x)Xj(x) n(x, yp+)-n(x, yp-)n(x, yp+)+n(x, yp-) dYk*(y)dyYl(y)y=yp,
Vlm(xy)=[El(x)|Vˆ|Em(y)]=4p=13xin(yp)xfin(yp)dxXi*(x) dXjdx(x) n(x, yp+)-n(x, yp-)n(x, yp+)+n(x, yp-)[Yk*(y)Yl(y)]|y=yp.
Vlm(yx)=[El(y)|Vˆ|Em(x)]=4p=12yin(xp)yin(xp)dyYi*(y) dYjdy(y) n(xp+, y)-n(xp-, y)n(xp+, y)+n(xp-, y)[Xk*(x)Xl(x)]|x=xp.

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