Abstract

For the sake of numerical performance, we hybridize two common approaches often used in electromagnetic computations, namely the finite-element method and the aperiodic Fourier modal method. To that end, we propose an extension of the classical S-matrix formalism to numerical situations, which requires handling different mathematical representations of the electromagnetic fields. As shown with a three-dimensional example, the proposed G-matrix formalism is stable and allows for an enhanced performance in terms of numerical accuracy and efficiency.

© 2008 Optical Society of America

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  1. J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).
  2. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, J. Opt. Soc. Am. A 12, 1068 (1995).
    [CrossRef]
  3. J. P. Hugonin and P. Lalanne, J. Opt. Soc. Am. A 22, 1844 (2005).
    [CrossRef]
  4. G. Lecamp, J. P. Hugonin, and P. Lalanne, Opt. Express 15, 11042 (2007).
    [CrossRef] [PubMed]
  5. E. Popov, M. Nevière, B. Gralak, and G. Tayeb, J. Opt. Soc. Am. A 19, 33 (2002).
    [CrossRef]
  6. H. Kim, I. M. Lee, and B. Lee, J. Opt. Soc. Am. A 24, 2313 (2007).
    [CrossRef]
  7. K. Dossou, M. A. Byrne, and L. C. Botten, J. Comput. Phys. 219, 120 (2006).
    [CrossRef]
  8. T. Delort and D. Maystre, J. Opt. Soc. Am. A 10, 2592 (1993).
    [CrossRef]
  9. P. Lalanne and J. P. Hugonin, J. Opt. Soc. Am. A 17, 1033 (2000).
    [CrossRef]
  10. With the MATLAB software, we have implemented several methods using LU and QR factorizations for the Gauss-Jordan elimination; see W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1989), Chap. 3.
  11. Q. Cao, P. Lalanne, and J. P. Hugonin, J. Opt. Soc. Am. A 19, 335 (2002).
    [CrossRef]
  12. M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
    [CrossRef]
  13. We note that accurately calculating the W1 modal field on the hole discontinuities is crucial for estimating the inevitable extrinsic loss in the slow-light regime, see L. C. Andreani and D. Gerace, Phys. Status Solidi B 244, 3528 (2007).
    [CrossRef]
  14. C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, IEEE Photonics Technol. Lett. 15, 1243 (2003).
    [CrossRef]

2007 (4)

G. Lecamp, J. P. Hugonin, and P. Lalanne, Opt. Express 15, 11042 (2007).
[CrossRef] [PubMed]

H. Kim, I. M. Lee, and B. Lee, J. Opt. Soc. Am. A 24, 2313 (2007).
[CrossRef]

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

We note that accurately calculating the W1 modal field on the hole discontinuities is crucial for estimating the inevitable extrinsic loss in the slow-light regime, see L. C. Andreani and D. Gerace, Phys. Status Solidi B 244, 3528 (2007).
[CrossRef]

2006 (1)

K. Dossou, M. A. Byrne, and L. C. Botten, J. Comput. Phys. 219, 120 (2006).
[CrossRef]

2005 (1)

2003 (1)

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, IEEE Photonics Technol. Lett. 15, 1243 (2003).
[CrossRef]

2002 (2)

2000 (1)

1995 (1)

1993 (1)

Andreani, L. C.

We note that accurately calculating the W1 modal field on the hole discontinuities is crucial for estimating the inevitable extrinsic loss in the slow-light regime, see L. C. Andreani and D. Gerace, Phys. Status Solidi B 244, 3528 (2007).
[CrossRef]

Baida, F. I.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Besbes, M.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Bienstman, P.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Botten, L. C.

K. Dossou, M. A. Byrne, and L. C. Botten, J. Comput. Phys. 219, 120 (2006).
[CrossRef]

Byrne, M. A.

K. Dossou, M. A. Byrne, and L. C. Botten, J. Comput. Phys. 219, 120 (2006).
[CrossRef]

Cao, Q.

Delort, T.

Dossou, K.

K. Dossou, M. A. Byrne, and L. C. Botten, J. Comput. Phys. 219, 120 (2006).
[CrossRef]

Flannery, B. P.

With the MATLAB software, we have implemented several methods using LU and QR factorizations for the Gauss-Jordan elimination; see W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1989), Chap. 3.

Gaylord, T. K.

Gerace, D.

We note that accurately calculating the W1 modal field on the hole discontinuities is crucial for estimating the inevitable extrinsic loss in the slow-light regime, see L. C. Andreani and D. Gerace, Phys. Status Solidi B 244, 3528 (2007).
[CrossRef]

Gralak, B.

Granet, G.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Grann, E. B.

Guizal, B.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Helfert, S.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Hugonin, J. P.

G. Lecamp, J. P. Hugonin, and P. Lalanne, Opt. Express 15, 11042 (2007).
[CrossRef] [PubMed]

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

J. P. Hugonin and P. Lalanne, J. Opt. Soc. Am. A 22, 1844 (2005).
[CrossRef]

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, IEEE Photonics Technol. Lett. 15, 1243 (2003).
[CrossRef]

Q. Cao, P. Lalanne, and J. P. Hugonin, J. Opt. Soc. Am. A 19, 335 (2002).
[CrossRef]

P. Lalanne and J. P. Hugonin, J. Opt. Soc. Am. A 17, 1033 (2000).
[CrossRef]

Janssen, O. T. A.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).

Kim, H.

Lalanne, P.

G. Lecamp, J. P. Hugonin, and P. Lalanne, Opt. Express 15, 11042 (2007).
[CrossRef] [PubMed]

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

J. P. Hugonin and P. Lalanne, J. Opt. Soc. Am. A 22, 1844 (2005).
[CrossRef]

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, IEEE Photonics Technol. Lett. 15, 1243 (2003).
[CrossRef]

Q. Cao, P. Lalanne, and J. P. Hugonin, J. Opt. Soc. Am. A 19, 335 (2002).
[CrossRef]

P. Lalanne and J. P. Hugonin, J. Opt. Soc. Am. A 17, 1033 (2000).
[CrossRef]

Lecamp, G.

Lee, B.

Lee, I. M.

Maystre, D.

Moharam, M. G.

Moreau, A.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Nevière, M.

Nugrowati, A. M.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Pereira, S. F.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Pommet, D. A.

Popov, E.

Press, W. H.

With the MATLAB software, we have implemented several methods using LU and QR factorizations for the Gauss-Jordan elimination; see W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1989), Chap. 3.

Rodier, J. C.

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, IEEE Photonics Technol. Lett. 15, 1243 (2003).
[CrossRef]

Sauvan, C.

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, IEEE Photonics Technol. Lett. 15, 1243 (2003).
[CrossRef]

Seideman, T.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Sukharev, M.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Talneau, A.

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, IEEE Photonics Technol. Lett. 15, 1243 (2003).
[CrossRef]

Tayeb, G.

Teukolsky, S. A.

With the MATLAB software, we have implemented several methods using LU and QR factorizations for the Gauss-Jordan elimination; see W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1989), Chap. 3.

Urbach, H. P.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

van de Nes, A. S.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

van Haver, S.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Van Labeke, D.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

Vetterling, W. T.

With the MATLAB software, we have implemented several methods using LU and QR factorizations for the Gauss-Jordan elimination; see W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1989), Chap. 3.

Xu, M.

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

C. Sauvan, P. Lalanne, J. C. Rodier, J. P. Hugonin, and A. Talneau, IEEE Photonics Technol. Lett. 15, 1243 (2003).
[CrossRef]

J. Comput. Phys. (1)

K. Dossou, M. A. Byrne, and L. C. Botten, J. Comput. Phys. 219, 120 (2006).
[CrossRef]

J. Eur. Opt. Soc. Rapid Publ. (1)

M. Besbes, J. P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Janssen, A. M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, A. Moreau, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, and D. Van Labeke, J. Eur. Opt. Soc. Rapid Publ. 2, 07022 (2007).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Express (1)

Phys. Status Solidi B (1)

We note that accurately calculating the W1 modal field on the hole discontinuities is crucial for estimating the inevitable extrinsic loss in the slow-light regime, see L. C. Andreani and D. Gerace, Phys. Status Solidi B 244, 3528 (2007).
[CrossRef]

Other (2)

With the MATLAB software, we have implemented several methods using LU and QR factorizations for the Gauss-Jordan elimination; see W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1989), Chap. 3.

J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).

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Figures (2)

Fig. 1
Fig. 1

Sketch of a typical geometry considered for illustrating the relevancy of combining the FEM and the a-FMM. For z i < z < z i + 1 , the geometry encompasses metallic inclusions shown as vertical rices. The cross-section planes labeled z = z i i = 1 N represent cross-section planes related to G-matrix operations. At z = z 1 and z = z N , modal field representations are used to satisfy the outgoing wave conditions in z-periodic and z-invariant waveguides, respectively. For computational purpose, complex coordinate transforms, in the transverse x and y directions, map infinite space to finite space.

Fig. 2
Fig. 2

3D field calculation of the gap-guided mode of a W1 waveguide in a semiconductor ( n = 3.5 ) membrane in air with the (a) a-FMM and (b) HYB methods. The vertical axis represents the ratio E A E B of the z-electric-field modulus at points A and B (right inset) located in the median plane of the membrane. E A E B is plotted as a function of the total number N of Fourier harmonics in (a) for different slicing numbers S of the two inner rows (9 slices being used for the other hole rows). In (b), E A E B is plotted as a function of the total number of elements used to discretize the inner first rows for two values of N ( N = 220 and 2050 ) . The results hold for a period a = 420 nm , a membrane thickness t = 220 nm , and a hole radius r = 0.3 a . The wavelength is λ = 1.55 μ m .

Equations (6)

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

d d z [ Ψ ] = D [ Ψ ] ,
L [ Ψ ] = R [ Ψ ] ,
L [ Ψ ] R [ Ψ ] = [ 0 ] ,
L [ Ψ ] R [ Ψ ] = [ 0 ] .
L [ Ψ ] = R [ Ψ ] .
L = [ S 11 0 S 21 I ] , R = [ I S 12 0 S 22 ] ,

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