Abstract

A modified Galerkin method is used to study the modal behavior of generic integrated optical waveguides down to first-order mode cutoff. The scalar Helmholtz equation is solved through nonlinear mapping of the transverse plane and subsequent Fourier decomposition. The differential equation is thus transformed into the eigenproblem for a specific finite-dimension linear operator. The largest eigenvalues, corresponding to the lowest-order guided modes, are in turn determined by an iterative Arnoldi procedure. Therefore actual diagonalization of a huge coefficient matrix is avoided, and a very large number of field frequency components can be considered.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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. D. Marcuse, “Solution of the vector wave equation for general dielectric waveguides by the Galerkin method,” IEEE J. Quantum Electron. 28, 459–465 (1992).
    [CrossRef]
  3. R. G. Grimes, J. G. Lewis, H. D. Simon, “A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems,” SIAM J. Matrix Anal. Appl. 15, 228–272 (1994).
    [CrossRef]
  4. M. A. Forastiere, G. C. Righini, “Scalar analysis of general dielectric waveguides by Fourier decomposition method and Lanczos reduction,” J. Lightwave Technol. 17, 362–368 (1999).
    [CrossRef]
  5. 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]
  6. R. W. Freund, M. H. Gutknecht, N. M. Nachtigal, “An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 137–158 (1993).
    [CrossRef]
  7. W. Arnoldi, “The principle of minimized iterations in the solution of the matrix eigenvalue problem,” Q. Appl. Math. 9, 17–29 (1951).
  8. S. F. Ashby, T. A. Manteuffel, P. Saylor, “A taxonomy for conjugate gradient methods,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 27, 1542–1568 (1990).
    [CrossRef]
  9. See, for instance, MATLAB 5.0 by The Mathworks Inc., Natick, Massachusetts, 1993.
  10. R. B. Lehoucq, D. C. Sorensen, “Deflation techniques within an implicitly restarted iteration,” SIAM J. Matrix Anal. Appl. 17, 789–821 (1996).
    [CrossRef]
  11. G. R. Hadley, R. E. Smith, “Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions,” J. Lightwave Technol. 13, 465–499 (1999).
    [CrossRef]
  12. BPM_CAD 3.0 by Optiwave Corporation, Nepean, Ontario, Canada, 1997 ( www.optiwave.com ).
  13. S. Pelli, M. Brenci, M. Fossi, G. C. Righini, C. Duverger, M. Montagna, R. Rolli, M. Ferrari, “Optical and spectroscopic characterization of Er/Yb-activated planar waveguides,” in Rare-Earth-Doped Materials and Devices IV, S. Jiang, ed., Proc. SPIE3942, 139–145 (2000).
    [CrossRef]

1999

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

G. R. Hadley, R. E. Smith, “Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions,” J. Lightwave Technol. 13, 465–499 (1999).
[CrossRef]

1996

R. B. Lehoucq, D. C. Sorensen, “Deflation techniques within an implicitly restarted iteration,” SIAM J. Matrix Anal. Appl. 17, 789–821 (1996).
[CrossRef]

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]

1994

R. G. Grimes, J. G. Lewis, H. D. Simon, “A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems,” SIAM J. Matrix Anal. Appl. 15, 228–272 (1994).
[CrossRef]

1993

R. W. Freund, M. H. Gutknecht, N. M. Nachtigal, “An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 137–158 (1993).
[CrossRef]

1992

D. Marcuse, “Solution of the vector wave equation for general dielectric waveguides by the Galerkin method,” IEEE J. Quantum Electron. 28, 459–465 (1992).
[CrossRef]

1990

S. F. Ashby, T. A. Manteuffel, P. Saylor, “A taxonomy for conjugate gradient methods,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 27, 1542–1568 (1990).
[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]

1951

W. Arnoldi, “The principle of minimized iterations in the solution of the matrix eigenvalue problem,” Q. Appl. Math. 9, 17–29 (1951).

Arnoldi, W.

W. Arnoldi, “The principle of minimized iterations in the solution of the matrix eigenvalue problem,” Q. Appl. Math. 9, 17–29 (1951).

Ashby, S. F.

S. F. Ashby, T. A. Manteuffel, P. Saylor, “A taxonomy for conjugate gradient methods,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 27, 1542–1568 (1990).
[CrossRef]

Brenci, M.

S. Pelli, M. Brenci, M. Fossi, G. C. Righini, C. Duverger, M. Montagna, R. Rolli, M. Ferrari, “Optical and spectroscopic characterization of Er/Yb-activated planar waveguides,” in Rare-Earth-Doped Materials and Devices IV, S. Jiang, ed., Proc. SPIE3942, 139–145 (2000).
[CrossRef]

Duverger, C.

S. Pelli, M. Brenci, M. Fossi, G. C. Righini, C. Duverger, M. Montagna, R. Rolli, M. Ferrari, “Optical and spectroscopic characterization of Er/Yb-activated planar waveguides,” in Rare-Earth-Doped Materials and Devices IV, S. Jiang, ed., Proc. SPIE3942, 139–145 (2000).
[CrossRef]

Ferrari, M.

S. Pelli, M. Brenci, M. Fossi, G. C. Righini, C. Duverger, M. Montagna, R. Rolli, M. Ferrari, “Optical and spectroscopic characterization of Er/Yb-activated planar waveguides,” in Rare-Earth-Doped Materials and Devices IV, S. Jiang, ed., Proc. SPIE3942, 139–145 (2000).
[CrossRef]

Forastiere, M. A.

Fossi, M.

S. Pelli, M. Brenci, M. Fossi, G. C. Righini, C. Duverger, M. Montagna, R. Rolli, M. Ferrari, “Optical and spectroscopic characterization of Er/Yb-activated planar waveguides,” in Rare-Earth-Doped Materials and Devices IV, S. Jiang, ed., Proc. SPIE3942, 139–145 (2000).
[CrossRef]

Freund, R. W.

R. W. Freund, M. H. Gutknecht, N. M. Nachtigal, “An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 137–158 (1993).
[CrossRef]

Grimes, R. G.

R. G. Grimes, J. G. Lewis, H. D. Simon, “A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems,” SIAM J. Matrix Anal. Appl. 15, 228–272 (1994).
[CrossRef]

Gutknecht, M. H.

R. W. Freund, M. H. Gutknecht, N. M. Nachtigal, “An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 137–158 (1993).
[CrossRef]

Hadley, G. R.

G. R. Hadley, R. E. Smith, “Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions,” J. Lightwave Technol. 13, 465–499 (1999).
[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]

Lehoucq, R. B.

R. B. Lehoucq, D. C. Sorensen, “Deflation techniques within an implicitly restarted iteration,” SIAM J. Matrix Anal. Appl. 17, 789–821 (1996).
[CrossRef]

Lewis, J. G.

R. G. Grimes, J. G. Lewis, H. D. Simon, “A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems,” SIAM J. Matrix Anal. Appl. 15, 228–272 (1994).
[CrossRef]

Manteuffel, T. A.

S. F. Ashby, T. A. Manteuffel, P. Saylor, “A taxonomy for conjugate gradient methods,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 27, 1542–1568 (1990).
[CrossRef]

Marcuse, D.

D. Marcuse, “Solution of the vector wave equation for general dielectric waveguides by the Galerkin method,” IEEE J. Quantum Electron. 28, 459–465 (1992).
[CrossRef]

Montagna, M.

S. Pelli, M. Brenci, M. Fossi, G. C. Righini, C. Duverger, M. Montagna, R. Rolli, M. Ferrari, “Optical and spectroscopic characterization of Er/Yb-activated planar waveguides,” in Rare-Earth-Doped Materials and Devices IV, S. Jiang, ed., Proc. SPIE3942, 139–145 (2000).
[CrossRef]

Nachtigal, N. M.

R. W. Freund, M. H. Gutknecht, N. M. Nachtigal, “An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 137–158 (1993).
[CrossRef]

Pelli, S.

S. Pelli, M. Brenci, M. Fossi, G. C. Righini, C. Duverger, M. Montagna, R. Rolli, M. Ferrari, “Optical and spectroscopic characterization of Er/Yb-activated planar waveguides,” in Rare-Earth-Doped Materials and Devices IV, S. Jiang, ed., Proc. SPIE3942, 139–145 (2000).
[CrossRef]

Righini, G. C.

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

S. Pelli, M. Brenci, M. Fossi, G. C. Righini, C. Duverger, M. Montagna, R. Rolli, M. Ferrari, “Optical and spectroscopic characterization of Er/Yb-activated planar waveguides,” in Rare-Earth-Doped Materials and Devices IV, S. Jiang, ed., Proc. SPIE3942, 139–145 (2000).
[CrossRef]

Rolli, R.

S. Pelli, M. Brenci, M. Fossi, G. C. Righini, C. Duverger, M. Montagna, R. Rolli, M. Ferrari, “Optical and spectroscopic characterization of Er/Yb-activated planar waveguides,” in Rare-Earth-Doped Materials and Devices IV, S. Jiang, ed., Proc. SPIE3942, 139–145 (2000).
[CrossRef]

Saylor, P.

S. F. Ashby, T. A. Manteuffel, P. Saylor, “A taxonomy for conjugate gradient methods,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 27, 1542–1568 (1990).
[CrossRef]

Simon, H. D.

R. G. Grimes, J. G. Lewis, H. D. Simon, “A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems,” SIAM J. Matrix Anal. Appl. 15, 228–272 (1994).
[CrossRef]

Smith, R. E.

G. R. Hadley, R. E. Smith, “Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions,” J. Lightwave Technol. 13, 465–499 (1999).
[CrossRef]

Sorensen, D. C.

R. B. Lehoucq, D. C. Sorensen, “Deflation techniques within an implicitly restarted iteration,” SIAM J. Matrix Anal. Appl. 17, 789–821 (1996).
[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]

IEEE J. Quantum Electron.

D. Marcuse, “Solution of the vector wave equation for general dielectric waveguides by the Galerkin method,” IEEE J. Quantum Electron. 28, 459–465 (1992).
[CrossRef]

J. Lightwave Technol.

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

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]

G. R. Hadley, R. E. Smith, “Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions,” J. Lightwave Technol. 13, 465–499 (1999).
[CrossRef]

Q. Appl. Math.

W. Arnoldi, “The principle of minimized iterations in the solution of the matrix eigenvalue problem,” Q. Appl. Math. 9, 17–29 (1951).

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.

S. F. Ashby, T. A. Manteuffel, P. Saylor, “A taxonomy for conjugate gradient methods,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 27, 1542–1568 (1990).
[CrossRef]

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

R. W. Freund, M. H. Gutknecht, N. M. Nachtigal, “An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 14, 137–158 (1993).
[CrossRef]

SIAM J. Matrix Anal. Appl.

R. G. Grimes, J. G. Lewis, H. D. Simon, “A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems,” SIAM J. Matrix Anal. Appl. 15, 228–272 (1994).
[CrossRef]

R. B. Lehoucq, D. C. Sorensen, “Deflation techniques within an implicitly restarted iteration,” SIAM J. Matrix Anal. Appl. 17, 789–821 (1996).
[CrossRef]

Other

See, for instance, MATLAB 5.0 by The Mathworks Inc., Natick, Massachusetts, 1993.

BPM_CAD 3.0 by Optiwave Corporation, Nepean, Ontario, Canada, 1997 ( www.optiwave.com ).

S. Pelli, M. Brenci, M. Fossi, G. C. Righini, C. Duverger, M. Montagna, R. Rolli, M. Ferrari, “Optical and spectroscopic characterization of Er/Yb-activated planar waveguides,” in Rare-Earth-Doped Materials and Devices IV, S. Jiang, ed., Proc. SPIE3942, 139–145 (2000).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Difference between the analytical and AGM effective index (na and nAGM, respectively) values (dots) for a single-layer planar waveguide as a function of the logarithm of the basis size. The line is a linear best fit of the data.

Fig. 2
Fig. 2

Absolute difference between the LP approximation (nLP) and the AGM (nAGM) effective index values for a step-index optical fiber as a function of the fiber core radius (curve a: the core was approximated by 200 rectangular domains, while 25×25 basis waves were used for the AGM calculations; curve b: 100 rectangular domains for the core approximation, 50×50 basis waves for the AGM calculations; curve c: 200 rectangular domains for the core approximation, 50×50 basis waves for the AGM calculations).

Fig. 3
Fig. 3

Sketch of the calculation window used in the channel waveguide problem. The transformed domain boundaries for the AGM calculation are indicated.

Fig. 4
Fig. 4

Comparison of AGM (solid curves) and ADI (dashed curves) LP11 mode field amplitudes for w=8 µm and d=2.2 µm (1.55-µm wavelength).

Fig. 5
Fig. 5

Field amplitude of LP01 (solid curves) and LP11 (dashed curves) guided modes at LP11 cutoff (1.55-µm wavelength, w=5.3528 µm, d=2.2 µm). Contour levels are at 10%–90% of maximum field amplitude.

Fig. 6
Fig. 6

Comparison of pump (solid curves) and signal (dashed curves) LP01 mode field amplitudes for w=2.29 µm and d=2.2 µm. Contour levels are at 10%–90% of maximum field amplitude.

Equations (29)

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

2x2+2y2+k2n(x, y)2-β2E(x, y)=0,
x=-αx cot(πu)τx(u)y=-αy cot(πv)τy(v)  u=τx-1(x)v=τy-1(y),
dudx22u2+d2udx2u+dvdy22v2+d2vdy2v+k2[n(u, v)]2-β2}E(u, v)=0.
φm(u)=2 sin(mπu),ϕn(v)=2 sin(nπv),
E(u, v)=m,namnφm(u)ϕn(v),
1αx2[σM][a]+1αy2[a][σN]T+k2kμh[ξk][a][ηk]=β2[a],
[n(u, v)]2=hμhPh(u)Qh(v)χR(h).
ξh;mmahbhPk(u)φm(u)φm(u)du,
ηh;nnrkskQh(v)φn(v)φn(v)dv.
I([a])=β2[a],
Km(I, v)span{v, I(v), I2(v), I3(v),, Im-1(v)}.
I(Vm)=VmHm+Em,
norm[I(x)-xθ]=norm{[I(Vm)-VmHm]y}=norm(Emy).
v=ibiui.
An(v)=λmaxnibiλiλmaxnui,
limnAn(v)=λmaxnbmaxumax.
yAn-1(v)  λapprox=yT·A(y)yT·y.
MI=1αx2[σM]+1αy2[σN]+k2hμh[ξh][ηh].
[n(x, y)]2=ns2+(ng2-ns2)g(x)f(y),y>01,y<0.
g(x)=12erfcxd-w2d-erfcxd+w2d,
f(y)=exp-yd2.
εji14δi,2-j+14δi, j-2-14δi, j+2-18δi,4-j-18δi, j-4+18δi, j+4,
γji38δij-14δi, j-2-14δi, j+2+14δi,2-j+116δi, j-4+116δi, j+4-116δi,4-j,
2ahbhup sin(iπu)sin(jπu)du,
2ahbhup sin(iπu)sin(jπu)du
={π(i-j)}1-p[Vp(i-j; bh)-Vp(i-j; ah)]-[π(i+j)]1-p[Vp(i+j; bh)-Vp(i+j; ah)]
=Up(i; bh)-Up(i; ah)ifij,ifi=j,
Vp(n; L)r=0pp!(p-r)!(πn)p-rLp-r sinnπL+rπ2,
Up(n; L)Lp-1p+1(2πn)1-pVp(2n; L).

Metrics