Abstract

We describe a method to compute S-matrix interface terms using a selection of eigenmodes. When solving the modal equation, the computation of left and right eigenvectors leads to rectangular eigenmodes matrices. Expressions of S-matrix interface terms are then expressed so as to allow for a significant reduction of the computation cost. The reduction is even further decreased in the case of the B-spline modal method, which deals with sparse matrices. Its convergence is illustrated on a high-density plasmonic structure and compared to a full modal method.

© 2010 Optical Society of America

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  1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
    [CrossRef] [PubMed]
  2. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, Nat. Mater. 9, 193 (2010).
    [CrossRef] [PubMed]
  3. R. Haïdar, G. Vincent, N. Guérineau, S. Collin, S. Velghe, and J. Primot, Opt. Express 13, 9941 (2005).
    [CrossRef] [PubMed]
  4. L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Appl. Phys. Lett. 95, 071112 (2009).
    [CrossRef]
  5. L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, Nano Lett. 9, 235 (2009).
  6. P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, Proc. SPIE 7717, 771703 (2010).
    [CrossRef]
  7. P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, J. Opt. Soc. Am. A 27, 696 (2010).
    [CrossRef]
  8. D. Y. K. Ko and J. R. Sambles, J. Opt. Soc. Am. A 5, 1863(1988).
    [CrossRef]
  9. F. Pardo, P. Bouchon, R. Haïdar, and J. L. Pelouard, “Huge reduction of computation cost of S-matrix interface terms,” presented at Polaritons Workshop 2009, Aussoix, France, 4–7 May 2009.
  10. P. Lalanne and G. M. Morris, J. Opt. Soc. Am. A 13, 779(1996).
    [CrossRef]
  11. L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Phys. Rev. Lett. 103, 033902 (2009).
    [CrossRef] [PubMed]
  12. E. D. Palik and G. Ghosh, Handbook of Optical Constants of Solids (Academic, 1985).
  13. J. P. Hugonin and P. Lalanne, Reticolo Software for Grating Analysis (Institut d’Optique, 2005).

2010 (3)

P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, Proc. SPIE 7717, 771703 (2010).
[CrossRef]

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, Nat. Mater. 9, 193 (2010).
[CrossRef] [PubMed]

P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, J. Opt. Soc. Am. A 27, 696 (2010).
[CrossRef]

2009 (3)

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Appl. Phys. Lett. 95, 071112 (2009).
[CrossRef]

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, Nano Lett. 9, 235 (2009).

2005 (1)

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef] [PubMed]

1996 (1)

1988 (1)

Barnard, E. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, Nat. Mater. 9, 193 (2010).
[CrossRef] [PubMed]

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, Nano Lett. 9, 235 (2009).

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef] [PubMed]

Bouchon, P.

P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, J. Opt. Soc. Am. A 27, 696 (2010).
[CrossRef]

P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, Proc. SPIE 7717, 771703 (2010).
[CrossRef]

F. Pardo, P. Bouchon, R. Haïdar, and J. L. Pelouard, “Huge reduction of computation cost of S-matrix interface terms,” presented at Polaritons Workshop 2009, Aussoix, France, 4–7 May 2009.

Brongersma, M. L.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, Nat. Mater. 9, 193 (2010).
[CrossRef] [PubMed]

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, Nano Lett. 9, 235 (2009).

Cai, W.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, Nat. Mater. 9, 193 (2010).
[CrossRef] [PubMed]

Catrysse, P. B.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Appl. Phys. Lett. 95, 071112 (2009).
[CrossRef]

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, Nano Lett. 9, 235 (2009).

Collin, S.

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef] [PubMed]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef] [PubMed]

Fan, S.

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, Nano Lett. 9, 235 (2009).

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Appl. Phys. Lett. 95, 071112 (2009).
[CrossRef]

Ghosh, G.

E. D. Palik and G. Ghosh, Handbook of Optical Constants of Solids (Academic, 1985).

Guérineau, N.

Haïdar, R.

P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, J. Opt. Soc. Am. A 27, 696 (2010).
[CrossRef]

P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, Proc. SPIE 7717, 771703 (2010).
[CrossRef]

R. Haïdar, G. Vincent, N. Guérineau, S. Collin, S. Velghe, and J. Primot, Opt. Express 13, 9941 (2005).
[CrossRef] [PubMed]

F. Pardo, P. Bouchon, R. Haïdar, and J. L. Pelouard, “Huge reduction of computation cost of S-matrix interface terms,” presented at Polaritons Workshop 2009, Aussoix, France, 4–7 May 2009.

Hugonin, J. P.

J. P. Hugonin and P. Lalanne, Reticolo Software for Grating Analysis (Institut d’Optique, 2005).

Jun, Y. C.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, Nat. Mater. 9, 193 (2010).
[CrossRef] [PubMed]

Ko, D. Y. K.

Lalanne, P.

P. Lalanne and G. M. Morris, J. Opt. Soc. Am. A 13, 779(1996).
[CrossRef]

J. P. Hugonin and P. Lalanne, Reticolo Software for Grating Analysis (Institut d’Optique, 2005).

Morris, G. M.

Palik, E. D.

E. D. Palik and G. Ghosh, Handbook of Optical Constants of Solids (Academic, 1985).

Pardo, F.

P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, Proc. SPIE 7717, 771703 (2010).
[CrossRef]

P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, J. Opt. Soc. Am. A 27, 696 (2010).
[CrossRef]

F. Pardo, P. Bouchon, R. Haïdar, and J. L. Pelouard, “Huge reduction of computation cost of S-matrix interface terms,” presented at Polaritons Workshop 2009, Aussoix, France, 4–7 May 2009.

Pelouard, J. L.

P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, Proc. SPIE 7717, 771703 (2010).
[CrossRef]

P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, J. Opt. Soc. Am. A 27, 696 (2010).
[CrossRef]

F. Pardo, P. Bouchon, R. Haïdar, and J. L. Pelouard, “Huge reduction of computation cost of S-matrix interface terms,” presented at Polaritons Workshop 2009, Aussoix, France, 4–7 May 2009.

Primot, J.

Sambles, J. R.

Schuller, J. A.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, Nat. Mater. 9, 193 (2010).
[CrossRef] [PubMed]

Velghe, S.

Verslegers, L.

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, Nano Lett. 9, 235 (2009).

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Appl. Phys. Lett. 95, 071112 (2009).
[CrossRef]

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

Vincent, G.

White, J. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, Nat. Mater. 9, 193 (2010).
[CrossRef] [PubMed]

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, Nano Lett. 9, 235 (2009).

Yu, Z.

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, Nano Lett. 9, 235 (2009).

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Appl. Phys. Lett. 95, 071112 (2009).
[CrossRef]

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Appl. Phys. Lett. 95, 071112 (2009).
[CrossRef]

J. Opt. Soc. Am. A (3)

Nano Lett. (1)

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, Nano Lett. 9, 235 (2009).

Nat. Mater. (1)

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, Nat. Mater. 9, 193 (2010).
[CrossRef] [PubMed]

Nature (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef] [PubMed]

Opt. Express (1)

Phys. Rev. Lett. (1)

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

Proc. SPIE (1)

P. Bouchon, F. Pardo, R. Haïdar, and J. L. Pelouard, Proc. SPIE 7717, 771703 (2010).
[CrossRef]

Other (3)

F. Pardo, P. Bouchon, R. Haïdar, and J. L. Pelouard, “Huge reduction of computation cost of S-matrix interface terms,” presented at Polaritons Workshop 2009, Aussoix, France, 4–7 May 2009.

E. D. Palik and G. Ghosh, Handbook of Optical Constants of Solids (Academic, 1985).

J. P. Hugonin and P. Lalanne, Reticolo Software for Grating Analysis (Institut d’Optique, 2005).

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Figures (4)

Fig. 1
Fig. 1

(a) Grating system: region 1 is a dielectric medium and region 2 is a compound metallic grating. (b) Schema of the S-matrix formalism for the interface between the two regions of the structure. The S-matrix links incoming waves from both regions (I) to the outgoing waves (O).

Fig. 2
Fig. 2

Scheme of the matrix equations that define the S submatrices. Size of the squares and rectangles are a representation of the dimensions of the different matrices. In the usual algorithm, all matrices are square ( N × N ) . In our algorithm, the number of modes is reduced so that the matrices involved in each equation are rectangular ( N × n ) and lead to smaller square S submatrices ( n × n ).

Fig. 3
Fig. 3

(a) Focusing with a 41 slits grating. The pattern of the grating is composed of 41 slits widths that vary linearly from 10 to 30 nm from the sides to the center. They are separated by 20 nm gold walls. The period is 1.63 μm . The incoming wave is normally incident, TM-polarized, at a wavelength of 2 μm ( ϵ m = 156 + 15.2 i ). (b) Electric field intensity in the waveguide array computed using BMM with a mesh of 664 knots and keeping 50 modes for the computation of S matrices. Focusing is achieved at a distance z = 12.2 μm .

Fig. 4
Fig. 4

Comparison of the convergence of RCWA and BMM with the reduced scattering algorithm for the computation of ϕ z / ϕ 0 as a function of the numbers of orders for RCWA and eigenmodes kept for BMM. ϕ z is defined as the vertical flux of the Poynting vector along the central slit ( 30 nm ) at z = 12.2 μm . ϕ 0 is the vertical flux of the Poynting vector over a period of the structure at z = 0 .

Equations (7)

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

e i = ϵ p 1 h i k i ,
H 1 V + H 1 S 11 V = H 2 S 21 V .
H 1 + H 1 S 11 = H 2 S 21 .
E 1 E 1 S 11 = E 2 S 21 ,
S 11 = 1 2 ( 1 + H 1 1 H 2 E 2 1 E 1 ) 1 ,
S 21 = E 2 1 E 1 ( 1 S 11 ) .
H p 1 M p = K p H p 1 .

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