Abstract

We present a numerical scheme for the analysis of periodic dielectric waveguides using Floquet–Bloch theory. The problem of finding the fundamental propagation modes is reduced to a nonlinear eigenvalue problem involving Dirichlet-to-Neumann maps. This approach leads to much smaller matrix problems than the ones that have appeared previously. By an increase of the discretization fineness, any desired precision of the method can be achieved. We discuss an eigensolver and extend the conventional rule to choose the branches of the transverse wave numbers. This ensures analytic dependence on the Floquet multiplier and convergence of the nonlinear solver. We demonstrate that even for a complicated multilayer waveguide structure the propagation factors can be calculated within seconds to several digits of accuracy.

© 2002 Optical Society of America

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References

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  1. R. E. Collin, F. J. Zucker, Antenna Theory, Part 2, Vol. 7 of Inter-University Electronics Series (McGraw-Hill, New York, 1969).
  2. J. Jacobsen, “Analytical, numerical, and experimental investigation of guided waves on a periodically strip-loaded dielectric slab,” IEEE Trans. Antennas Propag. AP-18, 379–388 (1970).
    [CrossRef]
  3. K. C. Chang, V. Shah, T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. A 70, 804–813 (1980).
    [CrossRef]
  4. W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
    [CrossRef]
  5. B. E. Little, W. P. Huang, S. K. Chaudhuri, “A multiple-scale analysis of grating-assisted couplers,” J. Lightwave Technol. 9, 1254–1263 (1991).
    [CrossRef]
  6. B. E. Little, H. A. Haus, “A variational coupled-mode theory for periodic waveguides,” IEEE J. Quantum Electron. 31, 2258–2264 (1995).
    [CrossRef]
  7. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  8. M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  9. N. P. K. Cotter, T. W. Priest, J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
    [CrossRef]
  10. S. Peng, G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
    [CrossRef]
  11. R. E. Jorgenson, R. Mittra, “Efficient calculation of the free-space periodic Green’s function,” IEEE Trans. Antennas Propag. 38, 633–642 (1990).
    [CrossRef]
  12. A. F. Peterson, “An outward-looking differential equation formulation for scattering from one-dimensional periodic diffraction gratings,” Electromagnetics 14, 227–238 (1994).
    [CrossRef]
  13. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [CrossRef]
  14. G. Hadjicostas, J. K. Butler, G. A. Evans, N. W. Carlson, R. Amantea, “A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 26, 893–902 (1990).
    [CrossRef]
  15. J. K. Butler, W. E. Ferguson, G. A. Evans, P. Stabile, A. Rosen, “A boundary element technique applied to the analysis of waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 28, 1701–1707 (1992).
    [CrossRef]
  16. J. Tausch, J. Butler, “Floquet multipliers of periodic waveguides via Dirichlet-to-Neumann maps,” J. Comput. Phys. 159, 90–102 (2000).
    [CrossRef]
  17. J. L. Volakis, A. Chatterjee, L. C. Kempel, Finite Element Method for Electromagnetics (IEEE Press, Piscataway, N.J., 1998).
  18. E. Allgower, K. Georg, “Continuation and path following,” in Acta Numerica 1993, A. Iserles, ed. (Cambridge U. Press, Cambridge, UK, 1993), pp. 1–64.
  19. N. H. Sun, J. K. Butler, G. A. Evans, L. Pang, “Analysis of grating-assisted directional couplers using the Floquet–Bloch theory,” J. Lightwave Technol. 15, 2301–2315 (1997).
    [CrossRef]

2000 (1)

J. Tausch, J. Butler, “Floquet multipliers of periodic waveguides via Dirichlet-to-Neumann maps,” J. Comput. Phys. 159, 90–102 (2000).
[CrossRef]

1997 (1)

N. H. Sun, J. K. Butler, G. A. Evans, L. Pang, “Analysis of grating-assisted directional couplers using the Floquet–Bloch theory,” J. Lightwave Technol. 15, 2301–2315 (1997).
[CrossRef]

1995 (4)

1994 (2)

A. F. Peterson, “An outward-looking differential equation formulation for scattering from one-dimensional periodic diffraction gratings,” Electromagnetics 14, 227–238 (1994).
[CrossRef]

W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
[CrossRef]

1992 (1)

J. K. Butler, W. E. Ferguson, G. A. Evans, P. Stabile, A. Rosen, “A boundary element technique applied to the analysis of waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 28, 1701–1707 (1992).
[CrossRef]

1991 (1)

B. E. Little, W. P. Huang, S. K. Chaudhuri, “A multiple-scale analysis of grating-assisted couplers,” J. Lightwave Technol. 9, 1254–1263 (1991).
[CrossRef]

1990 (2)

G. Hadjicostas, J. K. Butler, G. A. Evans, N. W. Carlson, R. Amantea, “A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 26, 893–902 (1990).
[CrossRef]

R. E. Jorgenson, R. Mittra, “Efficient calculation of the free-space periodic Green’s function,” IEEE Trans. Antennas Propag. 38, 633–642 (1990).
[CrossRef]

1982 (1)

1980 (1)

K. C. Chang, V. Shah, T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. A 70, 804–813 (1980).
[CrossRef]

1975 (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

1970 (1)

J. Jacobsen, “Analytical, numerical, and experimental investigation of guided waves on a periodically strip-loaded dielectric slab,” IEEE Trans. Antennas Propag. AP-18, 379–388 (1970).
[CrossRef]

Allgower, E.

E. Allgower, K. Georg, “Continuation and path following,” in Acta Numerica 1993, A. Iserles, ed. (Cambridge U. Press, Cambridge, UK, 1993), pp. 1–64.

Amantea, R.

G. Hadjicostas, J. K. Butler, G. A. Evans, N. W. Carlson, R. Amantea, “A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 26, 893–902 (1990).
[CrossRef]

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Butler, J.

J. Tausch, J. Butler, “Floquet multipliers of periodic waveguides via Dirichlet-to-Neumann maps,” J. Comput. Phys. 159, 90–102 (2000).
[CrossRef]

Butler, J. K.

N. H. Sun, J. K. Butler, G. A. Evans, L. Pang, “Analysis of grating-assisted directional couplers using the Floquet–Bloch theory,” J. Lightwave Technol. 15, 2301–2315 (1997).
[CrossRef]

J. K. Butler, W. E. Ferguson, G. A. Evans, P. Stabile, A. Rosen, “A boundary element technique applied to the analysis of waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 28, 1701–1707 (1992).
[CrossRef]

G. Hadjicostas, J. K. Butler, G. A. Evans, N. W. Carlson, R. Amantea, “A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 26, 893–902 (1990).
[CrossRef]

Carlson, N. W.

G. Hadjicostas, J. K. Butler, G. A. Evans, N. W. Carlson, R. Amantea, “A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 26, 893–902 (1990).
[CrossRef]

Chang, K. C.

K. C. Chang, V. Shah, T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. A 70, 804–813 (1980).
[CrossRef]

Chatterjee, A.

J. L. Volakis, A. Chatterjee, L. C. Kempel, Finite Element Method for Electromagnetics (IEEE Press, Piscataway, N.J., 1998).

Chaudhuri, S. K.

B. E. Little, W. P. Huang, S. K. Chaudhuri, “A multiple-scale analysis of grating-assisted couplers,” J. Lightwave Technol. 9, 1254–1263 (1991).
[CrossRef]

Collin, R. E.

R. E. Collin, F. J. Zucker, Antenna Theory, Part 2, Vol. 7 of Inter-University Electronics Series (McGraw-Hill, New York, 1969).

Cotter, N. P. K.

Evans, G. A.

N. H. Sun, J. K. Butler, G. A. Evans, L. Pang, “Analysis of grating-assisted directional couplers using the Floquet–Bloch theory,” J. Lightwave Technol. 15, 2301–2315 (1997).
[CrossRef]

J. K. Butler, W. E. Ferguson, G. A. Evans, P. Stabile, A. Rosen, “A boundary element technique applied to the analysis of waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 28, 1701–1707 (1992).
[CrossRef]

G. Hadjicostas, J. K. Butler, G. A. Evans, N. W. Carlson, R. Amantea, “A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 26, 893–902 (1990).
[CrossRef]

Ferguson, W. E.

J. K. Butler, W. E. Ferguson, G. A. Evans, P. Stabile, A. Rosen, “A boundary element technique applied to the analysis of waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 28, 1701–1707 (1992).
[CrossRef]

Gaylord, T. K.

Georg, K.

E. Allgower, K. Georg, “Continuation and path following,” in Acta Numerica 1993, A. Iserles, ed. (Cambridge U. Press, Cambridge, UK, 1993), pp. 1–64.

Grann, E. B.

Hadjicostas, G.

G. Hadjicostas, J. K. Butler, G. A. Evans, N. W. Carlson, R. Amantea, “A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 26, 893–902 (1990).
[CrossRef]

Haus, H. A.

B. E. Little, H. A. Haus, “A variational coupled-mode theory for periodic waveguides,” IEEE J. Quantum Electron. 31, 2258–2264 (1995).
[CrossRef]

Huang, W. P.

W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
[CrossRef]

B. E. Little, W. P. Huang, S. K. Chaudhuri, “A multiple-scale analysis of grating-assisted couplers,” J. Lightwave Technol. 9, 1254–1263 (1991).
[CrossRef]

Jacobsen, J.

J. Jacobsen, “Analytical, numerical, and experimental investigation of guided waves on a periodically strip-loaded dielectric slab,” IEEE Trans. Antennas Propag. AP-18, 379–388 (1970).
[CrossRef]

Jorgenson, R. E.

R. E. Jorgenson, R. Mittra, “Efficient calculation of the free-space periodic Green’s function,” IEEE Trans. Antennas Propag. 38, 633–642 (1990).
[CrossRef]

Kempel, L. C.

J. L. Volakis, A. Chatterjee, L. C. Kempel, Finite Element Method for Electromagnetics (IEEE Press, Piscataway, N.J., 1998).

Little, B. E.

B. E. Little, H. A. Haus, “A variational coupled-mode theory for periodic waveguides,” IEEE J. Quantum Electron. 31, 2258–2264 (1995).
[CrossRef]

B. E. Little, W. P. Huang, S. K. Chaudhuri, “A multiple-scale analysis of grating-assisted couplers,” J. Lightwave Technol. 9, 1254–1263 (1991).
[CrossRef]

Mittra, R.

R. E. Jorgenson, R. Mittra, “Efficient calculation of the free-space periodic Green’s function,” IEEE Trans. Antennas Propag. 38, 633–642 (1990).
[CrossRef]

Moharam, M. G.

Morris, G. M.

Pang, L.

N. H. Sun, J. K. Butler, G. A. Evans, L. Pang, “Analysis of grating-assisted directional couplers using the Floquet–Bloch theory,” J. Lightwave Technol. 15, 2301–2315 (1997).
[CrossRef]

Peng, S.

Peng, S. T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Peterson, A. F.

A. F. Peterson, “An outward-looking differential equation formulation for scattering from one-dimensional periodic diffraction gratings,” Electromagnetics 14, 227–238 (1994).
[CrossRef]

Pommet, D. A.

Priest, T. W.

Rosen, A.

J. K. Butler, W. E. Ferguson, G. A. Evans, P. Stabile, A. Rosen, “A boundary element technique applied to the analysis of waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 28, 1701–1707 (1992).
[CrossRef]

Sambles, J. R.

Shah, V.

K. C. Chang, V. Shah, T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. A 70, 804–813 (1980).
[CrossRef]

Stabile, P.

J. K. Butler, W. E. Ferguson, G. A. Evans, P. Stabile, A. Rosen, “A boundary element technique applied to the analysis of waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 28, 1701–1707 (1992).
[CrossRef]

Sun, N. H.

N. H. Sun, J. K. Butler, G. A. Evans, L. Pang, “Analysis of grating-assisted directional couplers using the Floquet–Bloch theory,” J. Lightwave Technol. 15, 2301–2315 (1997).
[CrossRef]

Tamir, T.

K. C. Chang, V. Shah, T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. A 70, 804–813 (1980).
[CrossRef]

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Tausch, J.

J. Tausch, J. Butler, “Floquet multipliers of periodic waveguides via Dirichlet-to-Neumann maps,” J. Comput. Phys. 159, 90–102 (2000).
[CrossRef]

Volakis, J. L.

J. L. Volakis, A. Chatterjee, L. C. Kempel, Finite Element Method for Electromagnetics (IEEE Press, Piscataway, N.J., 1998).

Zucker, F. J.

R. E. Collin, F. J. Zucker, Antenna Theory, Part 2, Vol. 7 of Inter-University Electronics Series (McGraw-Hill, New York, 1969).

Electromagnetics (1)

A. F. Peterson, “An outward-looking differential equation formulation for scattering from one-dimensional periodic diffraction gratings,” Electromagnetics 14, 227–238 (1994).
[CrossRef]

IEEE J. Quantum Electron. (3)

B. E. Little, H. A. Haus, “A variational coupled-mode theory for periodic waveguides,” IEEE J. Quantum Electron. 31, 2258–2264 (1995).
[CrossRef]

G. Hadjicostas, J. K. Butler, G. A. Evans, N. W. Carlson, R. Amantea, “A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 26, 893–902 (1990).
[CrossRef]

J. K. Butler, W. E. Ferguson, G. A. Evans, P. Stabile, A. Rosen, “A boundary element technique applied to the analysis of waveguides with periodic surface corrugations,” IEEE J. Quantum Electron. 28, 1701–1707 (1992).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

R. E. Jorgenson, R. Mittra, “Efficient calculation of the free-space periodic Green’s function,” IEEE Trans. Antennas Propag. 38, 633–642 (1990).
[CrossRef]

J. Jacobsen, “Analytical, numerical, and experimental investigation of guided waves on a periodically strip-loaded dielectric slab,” IEEE Trans. Antennas Propag. AP-18, 379–388 (1970).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

J. Comput. Phys. (1)

J. Tausch, J. Butler, “Floquet multipliers of periodic waveguides via Dirichlet-to-Neumann maps,” J. Comput. Phys. 159, 90–102 (2000).
[CrossRef]

J. Lightwave Technol. (2)

N. H. Sun, J. K. Butler, G. A. Evans, L. Pang, “Analysis of grating-assisted directional couplers using the Floquet–Bloch theory,” J. Lightwave Technol. 15, 2301–2315 (1997).
[CrossRef]

B. E. Little, W. P. Huang, S. K. Chaudhuri, “A multiple-scale analysis of grating-assisted couplers,” J. Lightwave Technol. 9, 1254–1263 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Other (3)

J. L. Volakis, A. Chatterjee, L. C. Kempel, Finite Element Method for Electromagnetics (IEEE Press, Piscataway, N.J., 1998).

E. Allgower, K. Georg, “Continuation and path following,” in Acta Numerica 1993, A. Iserles, ed. (Cambridge U. Press, Cambridge, UK, 1993), pp. 1–64.

R. E. Collin, F. J. Zucker, Antenna Theory, Part 2, Vol. 7 of Inter-University Electronics Series (McGraw-Hill, New York, 1969).

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Figures (10)

Fig. 1
Fig. 1

Typical waveguide geometry. Light is traveling in the z direction.

Fig. 2
Fig. 2

Decomposition of the infinite strip.

Fig. 3
Fig. 3

Brillouin diagram (example 1) ω versus Re(γ) (left) and Im(γ) (right) (Λ=1).

Fig. 4
Fig. 4

First Bragg in Fig. 3 enlarged.

Fig. 5
Fig. 5

Second Bragg in Fig. 3 enlarged.

Fig. 6
Fig. 6

Re(γ) versus grating period Λ (in micrometers) for various discretization parameters. GADC, two modes near resonance.

Fig. 7
Fig. 7

Im(γ) versus grating period Λ (in micrometers) for various discretization parameters. GADC, two modes near resonance.

Fig. 8
Fig. 8

Distance of the modes versus grating period Λ (in micrometers) for various discretization parameters. GADC, near resonance.

Fig. 9
Fig. 9

Field distributions of modes A and B. GADC, near resonance.

Fig. 10
Fig. 10

Riemann surface of the function zz2+κ2. A neighborhood of the upper imaginary axis is shaded.

Tables (4)

Tables Icon

Table 1 Parameters for Example 1

Tables Icon

Table 2 Parameters for Example 2

Tables Icon

Table 3 Convergence of the Propagation Factor for Example 1 (ω=π, Λ=1 μm )

Tables Icon

Table 4 Total Solution Times (Rounded Up to the Next Whole Second) for Example 1 a

Equations (41)

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

Ψ(x, y, z)=exp(γz)Φ(x, y, z),
Δu+μω2u=0
u+=u-,1μ+u+n=1μ-u-nforTEmodes,
u+=u-,1+u+n=1-u-nforTMmodes
u(x, z+Λ)=exp(γΛ)u(x, z),
u|Γ2=exp(γΛ)u|Γ0,
unΓ2=-exp(γΛ) unΓ0.
u(x, z)=nZfn-exp[ikxnJ(x-x+)]exp(kznz),
xx+,
u(x, z)=nZfn+exp[ikxn0(x--x)]exp(kznx),
xx-,
kzn=2πinΛ+γ,
kxnj=±(kzn2+μjjω2)1/2.
fn±=1Λ0Λu(x±, z)exp(-kznz)dz.
unΓ3Text(γ)u|Γ3=nZikxnJfn+exp(kznz)onΓ3+nZikxn0fn-exp(kznz)onΓ3-.
u-nΓ3=Text(γ)u|Γ3.
Δu+μω2u=0inΩint,
u|Γ2 =exp(γΛ)u|Γ0,
unΓ2=-exp(γΛ)unΓ0,
unΓ3=Text(γ)u|Γ3,
Δu+μω2u=0inΩint,
u|Γ2=exp(γΛ)u|Γ0,
unΓ2=-exp(γΛ)unΓ0,
u|Γ3=f,
Tint(γ)funΓ3,
T(γ)Tint(γ)-Text(γ)
T(γ)f=0.
fp=|n|pfn+en++|n|pfn-en-,
en±(z)exp(kznz)onΓ3±0onΓ3±.
Tp(γp)fp=0,
f=en±
[Tint,p(γ)]n±,m±=0Λen±(z)u¯mnzΓ3±dz.
u(x, z)=nZexp(kznz)cos[kxnj(x-xj)]ϕnj+μjkxnjsin[xnj(x-xj)]ψnj.
ϕnj=un(xj),
ψnj=1μjunx (xj),
ϕnj+1ψnj+1=Tψnj(γ)ϕnjψnj,
Tnj(γ)=1001/μjcos(kxnjwj)sin(kxnjwj)/kxnj-sin(kxnjwj)kxnjcos(kxnjwj)×100μj,
ϕkj-ψkj-=Tnj-(γ)  Tn1(γ)1kxn0.
Text(γ)en±=μjψnj±ϕnj± en±.
F(x, γ)=Tp(γ)xx*x-1
Tp(γ¯)+δTp(γ¯)=δTp(γ¯)1δ I+Tp(γ¯)-1Tp(γ¯),

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