Abstract

We adapt an efficient finite-difference procedure for determining complex propagation constants to the analysis of modes in planar waveguides. The method requires solving a single rather than multiple eigenvalue equations and does not require prior knowledge of either the nature of the solutions or the position of the modal eigenvalues in the complex plane.

© 2001 Optical Society of America

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References

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  1. S. M. Saad, “Review of numerical methods for the analysis of arbitrarily-shaped microwave and optical dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 894–899 (1985).
    [CrossRef]
  2. K. S. Chiang, “Review of numerical and approximate methods for the analysis of general optical dielectric waveguides,” Opt. Quantum Electron. 26, S113–S134 (1994).
    [CrossRef]
  3. E. Anemogiannis, E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
    [CrossRef]
  4. J. Chilwell, I. Hodgkinson, “Thin-films field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguide,” J. Opt. Soc. Am. A 1, 742–753 (1984).
    [CrossRef]
  5. L. M. Walpita, “Solutions for planar optical waveguide equations by selecting zero elements in a characteristic matrix,” J. Opt. Soc. Am. A 2, 595–602 (1985).
    [CrossRef]
  6. K.-H. Schlereth, M. Tacke, “The complex propagation constant of multilayer waveguide: an algorithm for a personal computer,” IEEE J. Quantum Electron. 26, 627–630 (1990).
    [CrossRef]
  7. C. A. Hulse, A. Knoesen, “Iterative calculation of complex propagation constants of modes in multilayer planar waveguides,” IEEE J. Quantum Electron. 28, 2682–2684 (1992).
    [CrossRef]
  8. R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” IEEE J. Quantum Electron. 28, 1520–1526 (1992).
    [CrossRef]
  9. R. E. Smith, G. W. Forbes, S. N. Houde-Walter, “Unfolding the multivalued planar waveguide dispersion relation,” IEEE J. Quantum Electron. 29, 1031–1034 (1993).
    [CrossRef]
  10. E. Anemogiannis, E. N. Glytsis, T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures,” J. Lightwave Technol. 12, 2080–2084 (1994).
    [CrossRef]
  11. E. Anemogiannis, E. N. Glytsis, T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999).
    [CrossRef]
  12. M. S. Stern, “Semivectorial polarised finite difference method for optical waveguide with arbitrary index profiles,” IEE Proc. J. 135, 56–63 (1988).
  13. A. Delâge, “Modelling of semiconductor rib wave guides by a finite difference method,” Can. J. Phys. 69, 512–519 (1991).
    [CrossRef]
  14. G. R. Hadley, R. E. Smith, “Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions,” J. Lightwave Technol. 13, 465–469 (1995).
    [CrossRef]
  15. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 11.

1999

1995

G. R. Hadley, R. E. Smith, “Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions,” J. Lightwave Technol. 13, 465–469 (1995).
[CrossRef]

1994

K. S. Chiang, “Review of numerical and approximate methods for the analysis of general optical dielectric waveguides,” Opt. Quantum Electron. 26, S113–S134 (1994).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures,” J. Lightwave Technol. 12, 2080–2084 (1994).
[CrossRef]

1993

R. E. Smith, G. W. Forbes, S. N. Houde-Walter, “Unfolding the multivalued planar waveguide dispersion relation,” IEEE J. Quantum Electron. 29, 1031–1034 (1993).
[CrossRef]

1992

E. Anemogiannis, E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
[CrossRef]

C. A. Hulse, A. Knoesen, “Iterative calculation of complex propagation constants of modes in multilayer planar waveguides,” IEEE J. Quantum Electron. 28, 2682–2684 (1992).
[CrossRef]

R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” IEEE J. Quantum Electron. 28, 1520–1526 (1992).
[CrossRef]

1991

A. Delâge, “Modelling of semiconductor rib wave guides by a finite difference method,” Can. J. Phys. 69, 512–519 (1991).
[CrossRef]

1990

K.-H. Schlereth, M. Tacke, “The complex propagation constant of multilayer waveguide: an algorithm for a personal computer,” IEEE J. Quantum Electron. 26, 627–630 (1990).
[CrossRef]

1988

M. S. Stern, “Semivectorial polarised finite difference method for optical waveguide with arbitrary index profiles,” IEE Proc. J. 135, 56–63 (1988).

1985

S. M. Saad, “Review of numerical methods for the analysis of arbitrarily-shaped microwave and optical dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 894–899 (1985).
[CrossRef]

L. M. Walpita, “Solutions for planar optical waveguide equations by selecting zero elements in a characteristic matrix,” J. Opt. Soc. Am. A 2, 595–602 (1985).
[CrossRef]

1984

Anemogiannis, E.

E. Anemogiannis, E. N. Glytsis, T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures,” J. Lightwave Technol. 12, 2080–2084 (1994).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
[CrossRef]

Chiang, K. S.

K. S. Chiang, “Review of numerical and approximate methods for the analysis of general optical dielectric waveguides,” Opt. Quantum Electron. 26, S113–S134 (1994).
[CrossRef]

Chilwell, J.

Delâge, A.

A. Delâge, “Modelling of semiconductor rib wave guides by a finite difference method,” Can. J. Phys. 69, 512–519 (1991).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 11.

Forbes, G. W.

R. E. Smith, G. W. Forbes, S. N. Houde-Walter, “Unfolding the multivalued planar waveguide dispersion relation,” IEEE J. Quantum Electron. 29, 1031–1034 (1993).
[CrossRef]

R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” IEEE J. Quantum Electron. 28, 1520–1526 (1992).
[CrossRef]

Gaylord, T. K.

Glytsis, E. N.

E. Anemogiannis, E. N. Glytsis, T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures,” J. Lightwave Technol. 12, 2080–2084 (1994).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
[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, 465–469 (1995).
[CrossRef]

Hodgkinson, I.

Houde-Walter, S. N.

R. E. Smith, G. W. Forbes, S. N. Houde-Walter, “Unfolding the multivalued planar waveguide dispersion relation,” IEEE J. Quantum Electron. 29, 1031–1034 (1993).
[CrossRef]

R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” IEEE J. Quantum Electron. 28, 1520–1526 (1992).
[CrossRef]

Hulse, C. A.

C. A. Hulse, A. Knoesen, “Iterative calculation of complex propagation constants of modes in multilayer planar waveguides,” IEEE J. Quantum Electron. 28, 2682–2684 (1992).
[CrossRef]

Knoesen, A.

C. A. Hulse, A. Knoesen, “Iterative calculation of complex propagation constants of modes in multilayer planar waveguides,” IEEE J. Quantum Electron. 28, 2682–2684 (1992).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 11.

Saad, S. M.

S. M. Saad, “Review of numerical methods for the analysis of arbitrarily-shaped microwave and optical dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 894–899 (1985).
[CrossRef]

Schlereth, K.-H.

K.-H. Schlereth, M. Tacke, “The complex propagation constant of multilayer waveguide: an algorithm for a personal computer,” IEEE J. Quantum Electron. 26, 627–630 (1990).
[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, 465–469 (1995).
[CrossRef]

R. E. Smith, G. W. Forbes, S. N. Houde-Walter, “Unfolding the multivalued planar waveguide dispersion relation,” IEEE J. Quantum Electron. 29, 1031–1034 (1993).
[CrossRef]

R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” IEEE J. Quantum Electron. 28, 1520–1526 (1992).
[CrossRef]

Stern, M. S.

M. S. Stern, “Semivectorial polarised finite difference method for optical waveguide with arbitrary index profiles,” IEE Proc. J. 135, 56–63 (1988).

Tacke, M.

K.-H. Schlereth, M. Tacke, “The complex propagation constant of multilayer waveguide: an algorithm for a personal computer,” IEEE J. Quantum Electron. 26, 627–630 (1990).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 11.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 11.

Walpita, L. M.

Can. J. Phys.

A. Delâge, “Modelling of semiconductor rib wave guides by a finite difference method,” Can. J. Phys. 69, 512–519 (1991).
[CrossRef]

IEE Proc. J.

M. S. Stern, “Semivectorial polarised finite difference method for optical waveguide with arbitrary index profiles,” IEE Proc. J. 135, 56–63 (1988).

IEEE J. Quantum Electron.

K.-H. Schlereth, M. Tacke, “The complex propagation constant of multilayer waveguide: an algorithm for a personal computer,” IEEE J. Quantum Electron. 26, 627–630 (1990).
[CrossRef]

C. A. Hulse, A. Knoesen, “Iterative calculation of complex propagation constants of modes in multilayer planar waveguides,” IEEE J. Quantum Electron. 28, 2682–2684 (1992).
[CrossRef]

R. E. Smith, S. N. Houde-Walter, G. W. Forbes, “Mode determination for planar waveguides using the four-sheeted dispersion relation,” IEEE J. Quantum Electron. 28, 1520–1526 (1992).
[CrossRef]

R. E. Smith, G. W. Forbes, S. N. Houde-Walter, “Unfolding the multivalued planar waveguide dispersion relation,” IEEE J. Quantum Electron. 29, 1031–1034 (1993).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

S. M. Saad, “Review of numerical methods for the analysis of arbitrarily-shaped microwave and optical dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-33, 894–899 (1985).
[CrossRef]

J. Lightwave Technol.

E. Anemogiannis, E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10, 1344–1351 (1992).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, T. K. Gaylord, “Efficient solution of eigenvalue equations of optical waveguiding structures,” J. Lightwave Technol. 12, 2080–2084 (1994).
[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, 465–469 (1995).
[CrossRef]

E. Anemogiannis, E. N. Glytsis, T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Quantum Electron.

K. S. Chiang, “Review of numerical and approximate methods for the analysis of general optical dielectric waveguides,” Opt. Quantum Electron. 26, S113–S134 (1994).
[CrossRef]

Other

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 11.

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Figures (1)

Fig. 1
Fig. 1

Multilayer planar waveguide structure. The computational region is [y0, yN+1].

Tables (3)

Tables Icon

Table 1 TE Complex Effective Refractive Indices for a Multilayer Waveguide

Tables Icon

Table 2 TE0 Complex Effective Refractive Index for a Slab Waveguide with [y0, yN+1]=[-1 μm, 3 μm]

Tables Icon

Table 3 TE0 Effective Refractive Index for a Slab Waveguide, Discretization Step Fixed with Δy=0.03 μm

Equations (23)

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

E(y)=xˆE(y)exp[i(βz-ωt)],
d2dy2E(y)+[k2(y)-β2]E(y)=0,
Ej+1-2Ej+Ej-1(Δy)2+kj2Ej=β2Ej.
E0=exp(-α0Δy)E1,α0=(β2-k02)1/2,
EN+1=exp(-αN+1Δy)EN,αN+1=(β2-kN+12)1/2.
ME=β2E.
u=(β2-k02)1/2+(β2-kN+12)1/2
α0=u2+R22u,
αN+1=u2-R22u,
β2=kN+12+αN+12=u4+2(2kN+12-R2)u2+R44u2.
Re{α0(u)}>0,
Re{αN+1(u)}>0.
exp(-α0Δy)1-α0Δy+12α02(Δy)2,
exp(-αN+1Δy)1-αN+1Δy+12αN+12(Δy)2
(M4u4+M3u3+M2u2+M1u+M0)E=0,
M˜ϕ=uϕ,
ϕ=ϕ0ϕ1ϕ2ϕ3,ϕ0E,
M˜=010000100001-M4-1M0-M4-1M1-M4-1M2-M4-1M3.
[M0]ij=R4(Δy)2i=j=1ori=j=N2R4(Δy)2i=j=2 ,, N-10 otherwise,
[M1]ij=4R2Δy i=j=1-4R2Δyi=j=N0otherwise,
[M2]ij=(8kN+12-8ki2-6R2)(Δy)2+8i=j=1(8kN+12-8ki2-2R2)(Δy)2+8i=j=N(8kN+12-8ki2-4R2)(Δy)2+16i=j=2,, N-1-8i=j+1=2,, Nori+1=j=2 ,, N0otherwise,
[M3]ij=4Δyi=j=1ori=j=N0otherwise,
[M4]ij=(Δy)2i=j=1ori=j=N2(Δy)2i=j=2,, N-10otherwise.

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