Abstract

Full-wave simulations of optical waveguides are often intractable due to their large electrical size. Naively focussing on a smaller part of the waveguide, e.g. to study coupling, offers no solution given the non-negligible interaction with the remaining parts of the structure. Thereto, in this paper, the coordinate stretching formulation of a perfectly matched layer is integrated into a method of moments based boundary integral equation solver in order to damp the interaction between multiple parts, allowing to focus on the part of interest. The new technique is validated using the classical example of scattering by a wedge. By truncation of the simulation domain to merely ten wavelengths from the tip, the advocated method is found to be both efficient and accurate compared to a traditional, analytical solution technique. Next, the method is applied to model a silicon polarization beam splitter excited by a Gaussian beam.

© 2016 Optical Society of America

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References

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  1. J. Fostier and F. Olyslager, “A GRID computer implementation of the multilevel fast multipole algorithm for full-wave analysis of optical devices,” IEICE Trans. Commun. E90B, 2430–2438 (2007).
    [Crossref]
  2. M. Stallein, “Coupling efficiency of Gaussian beams into step-index waveguides–an improved ray-optical approach,” J. Lightwave Technol. 26, 2937–2945 (2008).
    [Crossref]
  3. M. Stallein, “Coupling of a Gaussian beam into a planar slab waveguide using the mode matching method,” in “Progress in Electromagnetic Research Symposium (PIERS),” (2004), pp. 309–312.
  4. G. F. Herrmann, “Numerical computation of diffraction coefficients,” IEEE Trans. Antennas Propag. 35, 53–61 (1987).
    [Crossref]
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    [Crossref]
  7. W. Bogaerts, P. Bienstman, D. Taillaert, R. Baets, and D. De Zutter, “Out-of-plane scattering in photonic crystal slabs,” IEEE Photon. Technol. Lett. 13, 565–567 (2001).
    [Crossref]
  8. W. Bogaerts, R. Baets, P. Dumon, V. Wiaux, S. Beckx, D. Taillaert, B. Luyssaert, J. Van Campenhout, P. Bienstman, and D. Van Thourhout, “Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology,” J. Lightwave Technol. 23, 401–412 (2005).
    [Crossref]
  9. W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15, 363–369 (1997).
    [Crossref]
  10. P. Y. Ufimtsev, “Fast convergent integrals for nonuniform currents on wedge faces,” Electromagn. 18, 289–313 (1998).
    [Crossref]
  11. D. Dai, “Silicon polarization beam splitter based on an asymmetrical evanescent coupling system with three optical waveguides,” J. Lightwave Technol. 30, 3281–3287 (2012).
    [Crossref]
  12. F. Olyslager, D. De Zutter, and K. Blomme, “Rigorous analysis of the propagation characteristics of general lossless and lossy multiconductor transmission lines in multilayered media,” IEEE Trans. Microw. Theory Techn. 41, 79–88 (1993).
    [Crossref]
  13. L. Knockaert and D. De Zutter, “On the stretching of Maxwell’s equations in general orthogonal coordinate systems and the perfectly matched layer,” Microw. Opt. Technol. Lett. 24, 31–34 (2000).
    [Crossref]
  14. R. Torretti, Philosophy of Geometry from Riemann to Poincaré (Springer, 1978).
    [Crossref]
  15. D. Pissoort, D. Vande Ginste, and F. Olyslager, “Including PML-based absorbing boundary conditions in the MLFMA,” IEEE Antennas Wireless Propagat. Lett. 4, 312–315 (2005).
    [Crossref]
  16. S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
    [Crossref]
  17. F. Collino and P. B. Monk, “Optimizing the perfectly matched layer,” Comput. Methods in Appl. Mech. Eng. 164, 157–171 (1998).
    [Crossref]

2015 (1)

G. Apaydin and L. Sevgi, “A novel wedge diffraction modeling using method of moments (MoM),” ACES J. 30, 1053–1058 (2015).

2012 (1)

2008 (1)

2007 (1)

J. Fostier and F. Olyslager, “A GRID computer implementation of the multilevel fast multipole algorithm for full-wave analysis of optical devices,” IEICE Trans. Commun. E90B, 2430–2438 (2007).
[Crossref]

2005 (2)

2001 (1)

W. Bogaerts, P. Bienstman, D. Taillaert, R. Baets, and D. De Zutter, “Out-of-plane scattering in photonic crystal slabs,” IEEE Photon. Technol. Lett. 13, 565–567 (2001).
[Crossref]

2000 (2)

H. Derudder, F. Olyslager, and D. De Zutter, “Efficient calculation of far-field patterns of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 36, 798–799 (2000).
[Crossref]

L. Knockaert and D. De Zutter, “On the stretching of Maxwell’s equations in general orthogonal coordinate systems and the perfectly matched layer,” Microw. Opt. Technol. Lett. 24, 31–34 (2000).
[Crossref]

1998 (2)

P. Y. Ufimtsev, “Fast convergent integrals for nonuniform currents on wedge faces,” Electromagn. 18, 289–313 (1998).
[Crossref]

F. Collino and P. B. Monk, “Optimizing the perfectly matched layer,” Comput. Methods in Appl. Mech. Eng. 164, 157–171 (1998).
[Crossref]

1997 (1)

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15, 363–369 (1997).
[Crossref]

1996 (1)

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[Crossref]

1993 (1)

F. Olyslager, D. De Zutter, and K. Blomme, “Rigorous analysis of the propagation characteristics of general lossless and lossy multiconductor transmission lines in multilayered media,” IEEE Trans. Microw. Theory Techn. 41, 79–88 (1993).
[Crossref]

1987 (1)

G. F. Herrmann, “Numerical computation of diffraction coefficients,” IEEE Trans. Antennas Propag. 35, 53–61 (1987).
[Crossref]

Apaydin, G.

G. Apaydin and L. Sevgi, “A novel wedge diffraction modeling using method of moments (MoM),” ACES J. 30, 1053–1058 (2015).

Baets, R.

Beckx, S.

Bienstman, P.

Blomme, K.

F. Olyslager, D. De Zutter, and K. Blomme, “Rigorous analysis of the propagation characteristics of general lossless and lossy multiconductor transmission lines in multilayered media,” IEEE Trans. Microw. Theory Techn. 41, 79–88 (1993).
[Crossref]

Bogaerts, W.

Chew, W. C.

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15, 363–369 (1997).
[Crossref]

Collino, F.

F. Collino and P. B. Monk, “Optimizing the perfectly matched layer,” Comput. Methods in Appl. Mech. Eng. 164, 157–171 (1998).
[Crossref]

Dai, D.

De Zutter, D.

W. Bogaerts, P. Bienstman, D. Taillaert, R. Baets, and D. De Zutter, “Out-of-plane scattering in photonic crystal slabs,” IEEE Photon. Technol. Lett. 13, 565–567 (2001).
[Crossref]

L. Knockaert and D. De Zutter, “On the stretching of Maxwell’s equations in general orthogonal coordinate systems and the perfectly matched layer,” Microw. Opt. Technol. Lett. 24, 31–34 (2000).
[Crossref]

H. Derudder, F. Olyslager, and D. De Zutter, “Efficient calculation of far-field patterns of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 36, 798–799 (2000).
[Crossref]

F. Olyslager, D. De Zutter, and K. Blomme, “Rigorous analysis of the propagation characteristics of general lossless and lossy multiconductor transmission lines in multilayered media,” IEEE Trans. Microw. Theory Techn. 41, 79–88 (1993).
[Crossref]

Derudder, H.

H. Derudder, F. Olyslager, and D. De Zutter, “Efficient calculation of far-field patterns of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 36, 798–799 (2000).
[Crossref]

Dumon, P.

Fostier, J.

J. Fostier and F. Olyslager, “A GRID computer implementation of the multilevel fast multipole algorithm for full-wave analysis of optical devices,” IEICE Trans. Commun. E90B, 2430–2438 (2007).
[Crossref]

Gedney, S. D.

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[Crossref]

Herrmann, G. F.

G. F. Herrmann, “Numerical computation of diffraction coefficients,” IEEE Trans. Antennas Propag. 35, 53–61 (1987).
[Crossref]

Jin, J. M.

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15, 363–369 (1997).
[Crossref]

Knockaert, L.

L. Knockaert and D. De Zutter, “On the stretching of Maxwell’s equations in general orthogonal coordinate systems and the perfectly matched layer,” Microw. Opt. Technol. Lett. 24, 31–34 (2000).
[Crossref]

Luyssaert, B.

Michielssen, E.

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15, 363–369 (1997).
[Crossref]

Monk, P. B.

F. Collino and P. B. Monk, “Optimizing the perfectly matched layer,” Comput. Methods in Appl. Mech. Eng. 164, 157–171 (1998).
[Crossref]

Olyslager, F.

J. Fostier and F. Olyslager, “A GRID computer implementation of the multilevel fast multipole algorithm for full-wave analysis of optical devices,” IEICE Trans. Commun. E90B, 2430–2438 (2007).
[Crossref]

D. Pissoort, D. Vande Ginste, and F. Olyslager, “Including PML-based absorbing boundary conditions in the MLFMA,” IEEE Antennas Wireless Propagat. Lett. 4, 312–315 (2005).
[Crossref]

H. Derudder, F. Olyslager, and D. De Zutter, “Efficient calculation of far-field patterns of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 36, 798–799 (2000).
[Crossref]

F. Olyslager, D. De Zutter, and K. Blomme, “Rigorous analysis of the propagation characteristics of general lossless and lossy multiconductor transmission lines in multilayered media,” IEEE Trans. Microw. Theory Techn. 41, 79–88 (1993).
[Crossref]

Pissoort, D.

D. Pissoort, D. Vande Ginste, and F. Olyslager, “Including PML-based absorbing boundary conditions in the MLFMA,” IEEE Antennas Wireless Propagat. Lett. 4, 312–315 (2005).
[Crossref]

Sevgi, L.

G. Apaydin and L. Sevgi, “A novel wedge diffraction modeling using method of moments (MoM),” ACES J. 30, 1053–1058 (2015).

Stallein, M.

M. Stallein, “Coupling efficiency of Gaussian beams into step-index waveguides–an improved ray-optical approach,” J. Lightwave Technol. 26, 2937–2945 (2008).
[Crossref]

M. Stallein, “Coupling of a Gaussian beam into a planar slab waveguide using the mode matching method,” in “Progress in Electromagnetic Research Symposium (PIERS),” (2004), pp. 309–312.

Taillaert, D.

Torretti, R.

R. Torretti, Philosophy of Geometry from Riemann to Poincaré (Springer, 1978).
[Crossref]

Ufimtsev, P. Y.

P. Y. Ufimtsev, “Fast convergent integrals for nonuniform currents on wedge faces,” Electromagn. 18, 289–313 (1998).
[Crossref]

Van Campenhout, J.

Van Thourhout, D.

Vande Ginste, D.

D. Pissoort, D. Vande Ginste, and F. Olyslager, “Including PML-based absorbing boundary conditions in the MLFMA,” IEEE Antennas Wireless Propagat. Lett. 4, 312–315 (2005).
[Crossref]

Wiaux, V.

ACES J. (1)

G. Apaydin and L. Sevgi, “A novel wedge diffraction modeling using method of moments (MoM),” ACES J. 30, 1053–1058 (2015).

Comput. Methods in Appl. Mech. Eng. (1)

F. Collino and P. B. Monk, “Optimizing the perfectly matched layer,” Comput. Methods in Appl. Mech. Eng. 164, 157–171 (1998).
[Crossref]

Electromagn. (1)

P. Y. Ufimtsev, “Fast convergent integrals for nonuniform currents on wedge faces,” Electromagn. 18, 289–313 (1998).
[Crossref]

Electron. Lett. (1)

H. Derudder, F. Olyslager, and D. De Zutter, “Efficient calculation of far-field patterns of waveguide discontinuities using perfectly matched layers,” Electron. Lett. 36, 798–799 (2000).
[Crossref]

IEEE Antennas Wireless Propagat. Lett. (1)

D. Pissoort, D. Vande Ginste, and F. Olyslager, “Including PML-based absorbing boundary conditions in the MLFMA,” IEEE Antennas Wireless Propagat. Lett. 4, 312–315 (2005).
[Crossref]

IEEE Photon. Technol. Lett. (1)

W. Bogaerts, P. Bienstman, D. Taillaert, R. Baets, and D. De Zutter, “Out-of-plane scattering in photonic crystal slabs,” IEEE Photon. Technol. Lett. 13, 565–567 (2001).
[Crossref]

IEEE Trans. Antennas Propag. (2)

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[Crossref]

G. F. Herrmann, “Numerical computation of diffraction coefficients,” IEEE Trans. Antennas Propag. 35, 53–61 (1987).
[Crossref]

IEEE Trans. Microw. Theory Techn. (1)

F. Olyslager, D. De Zutter, and K. Blomme, “Rigorous analysis of the propagation characteristics of general lossless and lossy multiconductor transmission lines in multilayered media,” IEEE Trans. Microw. Theory Techn. 41, 79–88 (1993).
[Crossref]

IEICE Trans. Commun. (1)

J. Fostier and F. Olyslager, “A GRID computer implementation of the multilevel fast multipole algorithm for full-wave analysis of optical devices,” IEICE Trans. Commun. E90B, 2430–2438 (2007).
[Crossref]

J. Lightwave Technol. (3)

Microw. Opt. Technol. Lett. (2)

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15, 363–369 (1997).
[Crossref]

L. Knockaert and D. De Zutter, “On the stretching of Maxwell’s equations in general orthogonal coordinate systems and the perfectly matched layer,” Microw. Opt. Technol. Lett. 24, 31–34 (2000).
[Crossref]

Other (2)

R. Torretti, Philosophy of Geometry from Riemann to Poincaré (Springer, 1978).
[Crossref]

M. Stallein, “Coupling of a Gaussian beam into a planar slab waveguide using the mode matching method,” in “Progress in Electromagnetic Research Symposium (PIERS),” (2004), pp. 309–312.

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Figures (9)

Fig. 1
Fig. 1 Object illuminated by an electromagnetic wave.
Fig. 2
Fig. 2 Illustration of the PML’s coordinate stretching formalism.
Fig. 3
Fig. 3 Geometry used for the optimization tests of the PML layer.
Fig. 4
Fig. 4 Results of the optimization tests. The points A–E correspond to the pertinent points in Fig. 3. For type III stretching, D = 4.
Fig. 5
Fig. 5 Pareto fronts for demands on accuracy and damping when adding a PML layer.
Fig. 6
Fig. 6 RE on the z-oriented electric field compared to the analytical solution.
Fig. 7
Fig. 7 Polarization beam splitter with three silicon slabs.
Fig. 8
Fig. 8 Effective indices of guided modes in a silicon slab waveguide.
Fig. 9
Fig. 9 Field density in a polarization beam splitter with three optical waveguides (the colors are an indication for the field strength).

Tables (2)

Tables Icon

Table 1 Three different types of coordinate stretching for x > x0. Here, σ ( x ) = σ max ( x x 0 D ) m.

Tables Icon

Table 2 Computational resources for the simulation of the silicon PBS of Fig. 7.

Equations (15)

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

e z i lim r C + C d c [ e z G 0 n j k 0 2 ω 0 G 0 h t ] = lim r C C d c [ e z G n j k 1 2 ω 0 n 1 2 G h t ] ,
h t i lim r C + C d c [ j ω 0 k 0 2 e z 2 G 0 n n G 0 n h t ] = lim r C C d c [ j ω 0 n 1 2 k 1 2 e z 2 G n n G n h t ] ,
G ( r ; r ) = j 4 H 0 ( 2 ) ( k 1 d ( r , r ) ) ,
x ˜ = x r + j x i = 0 x d x χ ˜ ( x ) ,
x x ˜ = 1 χ ˜ ( x ) x .
( 2 x ˜ 2 + k x 2 ) ϕ ˜ ( x ) = 0 ,
ϕ ˜ ( x ) = A exp ( j k x x ˜ ) + B exp ( j k x x ˜ ) ,
x ˜ = { x , x x 0 x r + j x i ( x i 0 ) , x > x 0
A exp ( j k x x r ) exp ( k x x i ) ,
d ( r ˜ , r ˜ ) = ( x ˜ x ˜ ) 2 + ( y y ) 2 = [ ( x r x r ) 2 ( x i x i ) 2 + ( y y ) 2 + 2 j ( x r x r ) ( x i x i ) ] 1 / 2 .
p ˜ = ( 1 s ) p ˜ 0 + s p ˜ 1 p ˜ 0 + s l ˜ t ˜ ,
n = n n ˜ ( 1 χ ˜ ( x ) x x ^ + y y ^ ) .
j z ( r ) = 4 j 0 c 0 2 ν r ω l = 1 l ν exp ( j l π 2 ν ) J l / ν ( k r ) sin ( l ν ϕ 0 ) ,
j z ( r ) = 4 j 0 c 0 2 ν r ω l = 1 ( 1 ) l + 1 l ν exp ( j l π 2 ν ) J l / ν ( k r ) sin ( l ν ϕ 0 ) .
RE = | solution reference reference | .

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