Abstract

In this paper, we present a full-vector finite difference method to solve for optical modes in one and two dimensional subwavelength plasmonic waveguides. We have used the Implicitly Restarted Arnoldi method to directly calculate the propagation constants of the dominant modes. The method has low computational complexity and can be applied to accurately model complex geometries and structures with fast-varying field profiles. When applied to solve for purely bounded modes, our method automatically separates evanescent and low-loss guided modes.

© 2006 Optical Society of America

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  1. M. Hochberg, T. Baehr-Jones, C. Walker, and A. Scherer, "Integrated plasmon and dielectric waveguides," Opt. Express 12, 5481-5486 (2004).
    [CrossRef] [PubMed]
  2. K. Tanaka, M. Tanaka, and T. Sugiyama, "Simulation of practical nanometric circuits based on surface plasmon polariton gap waveguide," Opt. Express 13, 256-266 (2005).
    [CrossRef] [PubMed]
  3. G. I. Stegeman, R. F. Wallias, and A. Maradudin, "Excitation of surface polaritons by end-fire coupling," Opt. Lett. 8, 386 (1983).
    [CrossRef] [PubMed]
  4. R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, "Experimental observation of plasmon-polariton waves supported by a thin metal film of finite width," Opt. Lett. 25, 844-846 (2000).
    [CrossRef]
  5. R. Zia,M. D. Selker, P. B. Catrysse, andM. L. Brongersma, "Geometries and materials for subwavelength surface plasmon modes," J. Opt. Soc. Am. A 21, 2442-2446 (2004).
    [CrossRef]
  6. S. J. Al-Bader, "Optical transmission on metallic wires-fundamental modes," IEEE J. Quantum Electron 40, 325-329 (2004).
    [CrossRef]
  7. P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric strucutres," J. Phys. Rev. B 61, 10484-10503 (2000).
    [CrossRef]
  8. P. Berini, A. Stohr, K. Wu, D. Jager, "Normal mode analysis and characterization of an InGaAs/GaAs MQW field-induced optical waveguide including electrode effects," J. Lightwave Technol. 14, 2422-2435 (1996).
    [CrossRef]
  9. R. Zia, M. D. Selker, and M. L. Brongersma, "Leaky and bound modes of surface plasmon waveguide," Phys. Rev. B 71, 165431 (2005).
    [CrossRef]
  10. C. Chen, P. Berini, D. Feng, S. Tanev, and V. Tzolov, "Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media," Opt. Express 7, 260-272 (2000).
    [CrossRef] [PubMed]
  11. J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," J. Phys. Rev. B 72, 075405 (2005).
    [CrossRef]
  12. I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, "Metallic photonic crystals at optical wavelength," J. Phys. Rev. B 62, 15299-15302 (2000).
    [CrossRef]
  13. P. Lusse, P. Stuwe, J. Schule, and H.-G. Unger, "Analysis of vectorial mode fields in optical waveguides by new finite difference method," J. Lightwave Technol. 12, 487-494 (1994).
    [CrossRef]
  14. K. Ramm, P. Lusse, and H.-G. Unger, "Multigrid eigenvalue solver for mode calculation of planar optical waveguides," IEEE Photonics Technol. Lett. 9, 967-969 (1997).
    [CrossRef]
  15. V. Hernandez, J. E. Roman, A. Tomas, and V. Vidal, "A Survey of Software for Sparse Eigenvalue Problems," Technical report, Universidad Politecnica de Valencia, (2005).
  16. W. J. Stewart and A. Jennings, "Algorithm 570: LOPSI: A Simultaneous Iteration Method for Real Matrices [F2]," ACM Trans. Math. Softw. 7, 230-232 (1981).
    [CrossRef]
  17. R. B. Lehoucq and D. C. Sorensen, "Deflation techniques within an implicitly restarted iteration," SIAM J. Matrix Anal. Appl. 17, 789-821 (1996).
    [CrossRef]
  18. D. C. Sorensen, "Implicit application of polynomial filters in a K-step Arnoldi method," SIAM J. Matrix Anal. Appl. 13, 357-385, (1992).
    [CrossRef]
  19. R. Radke, "A MATLAB implementation of the implicitly restarted Arnoldi method for solving large scale eigenvalue problems," Technical report, Dept. of Applied and Computational Mathematics, Rice University, Houston, TX, (1996).
  20. R. Zia, A. Chandran, and M. L. Brongersma, "Dielectric waveguide model for guided surface polaritons," Opt. Lett. 30, 1473-1475 (2005).
    [CrossRef] [PubMed]

2005

R. Zia, M. D. Selker, and M. L. Brongersma, "Leaky and bound modes of surface plasmon waveguide," Phys. Rev. B 71, 165431 (2005).
[CrossRef]

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," J. Phys. Rev. B 72, 075405 (2005).
[CrossRef]

K. Tanaka, M. Tanaka, and T. Sugiyama, "Simulation of practical nanometric circuits based on surface plasmon polariton gap waveguide," Opt. Express 13, 256-266 (2005).
[CrossRef] [PubMed]

R. Zia, A. Chandran, and M. L. Brongersma, "Dielectric waveguide model for guided surface polaritons," Opt. Lett. 30, 1473-1475 (2005).
[CrossRef] [PubMed]

2004

2000

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric strucutres," J. Phys. Rev. B 61, 10484-10503 (2000).
[CrossRef]

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, "Metallic photonic crystals at optical wavelength," J. Phys. Rev. B 62, 15299-15302 (2000).
[CrossRef]

R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, "Experimental observation of plasmon-polariton waves supported by a thin metal film of finite width," Opt. Lett. 25, 844-846 (2000).
[CrossRef]

C. Chen, P. Berini, D. Feng, S. Tanev, and V. Tzolov, "Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media," Opt. Express 7, 260-272 (2000).
[CrossRef] [PubMed]

1997

K. Ramm, P. Lusse, and H.-G. Unger, "Multigrid eigenvalue solver for mode calculation of planar optical waveguides," IEEE Photonics Technol. Lett. 9, 967-969 (1997).
[CrossRef]

1996

P. Berini, A. Stohr, K. Wu, D. Jager, "Normal mode analysis and characterization of an InGaAs/GaAs MQW field-induced optical waveguide including electrode effects," J. Lightwave Technol. 14, 2422-2435 (1996).
[CrossRef]

R. B. Lehoucq and D. C. Sorensen, "Deflation techniques within an implicitly restarted iteration," SIAM J. Matrix Anal. Appl. 17, 789-821 (1996).
[CrossRef]

1994

P. Lusse, P. Stuwe, J. Schule, and H.-G. Unger, "Analysis of vectorial mode fields in optical waveguides by new finite difference method," J. Lightwave Technol. 12, 487-494 (1994).
[CrossRef]

1992

D. C. Sorensen, "Implicit application of polynomial filters in a K-step Arnoldi method," SIAM J. Matrix Anal. Appl. 13, 357-385, (1992).
[CrossRef]

1983

1981

W. J. Stewart and A. Jennings, "Algorithm 570: LOPSI: A Simultaneous Iteration Method for Real Matrices [F2]," ACM Trans. Math. Softw. 7, 230-232 (1981).
[CrossRef]

Al-Bader, S. J.

S. J. Al-Bader, "Optical transmission on metallic wires-fundamental modes," IEEE J. Quantum Electron 40, 325-329 (2004).
[CrossRef]

Atwater, H. A.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," J. Phys. Rev. B 72, 075405 (2005).
[CrossRef]

Baehr-Jones, T.

Berini, P.

C. Chen, P. Berini, D. Feng, S. Tanev, and V. Tzolov, "Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media," Opt. Express 7, 260-272 (2000).
[CrossRef] [PubMed]

R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, "Experimental observation of plasmon-polariton waves supported by a thin metal film of finite width," Opt. Lett. 25, 844-846 (2000).
[CrossRef]

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric strucutres," J. Phys. Rev. B 61, 10484-10503 (2000).
[CrossRef]

P. Berini, A. Stohr, K. Wu, D. Jager, "Normal mode analysis and characterization of an InGaAs/GaAs MQW field-induced optical waveguide including electrode effects," J. Lightwave Technol. 14, 2422-2435 (1996).
[CrossRef]

Berolo, E.

Biswas, R.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, "Metallic photonic crystals at optical wavelength," J. Phys. Rev. B 62, 15299-15302 (2000).
[CrossRef]

Brongersma, M. L.

R. Zia, M. D. Selker, and M. L. Brongersma, "Leaky and bound modes of surface plasmon waveguide," Phys. Rev. B 71, 165431 (2005).
[CrossRef]

R. Zia, A. Chandran, and M. L. Brongersma, "Dielectric waveguide model for guided surface polaritons," Opt. Lett. 30, 1473-1475 (2005).
[CrossRef] [PubMed]

Catrysse, P. B.

Chandran, A.

Charbonneau, R.

Chen, C.

Dionne, J. A.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," J. Phys. Rev. B 72, 075405 (2005).
[CrossRef]

El-Kady, I.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, "Metallic photonic crystals at optical wavelength," J. Phys. Rev. B 62, 15299-15302 (2000).
[CrossRef]

Feng, D.

Ho, K. M.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, "Metallic photonic crystals at optical wavelength," J. Phys. Rev. B 62, 15299-15302 (2000).
[CrossRef]

Hochberg, M.

Jager, D.

P. Berini, A. Stohr, K. Wu, D. Jager, "Normal mode analysis and characterization of an InGaAs/GaAs MQW field-induced optical waveguide including electrode effects," J. Lightwave Technol. 14, 2422-2435 (1996).
[CrossRef]

Jennings, A.

W. J. Stewart and A. Jennings, "Algorithm 570: LOPSI: A Simultaneous Iteration Method for Real Matrices [F2]," ACM Trans. Math. Softw. 7, 230-232 (1981).
[CrossRef]

Lehoucq, R. B.

R. B. Lehoucq and D. C. Sorensen, "Deflation techniques within an implicitly restarted iteration," SIAM J. Matrix Anal. Appl. 17, 789-821 (1996).
[CrossRef]

Lisicka-Shrzek, E.

Lusse, P.

K. Ramm, P. Lusse, and H.-G. Unger, "Multigrid eigenvalue solver for mode calculation of planar optical waveguides," IEEE Photonics Technol. Lett. 9, 967-969 (1997).
[CrossRef]

P. Lusse, P. Stuwe, J. Schule, and H.-G. Unger, "Analysis of vectorial mode fields in optical waveguides by new finite difference method," J. Lightwave Technol. 12, 487-494 (1994).
[CrossRef]

Maradudin, A.

Ramm, K.

K. Ramm, P. Lusse, and H.-G. Unger, "Multigrid eigenvalue solver for mode calculation of planar optical waveguides," IEEE Photonics Technol. Lett. 9, 967-969 (1997).
[CrossRef]

Scherer, A.

Schule, J.

P. Lusse, P. Stuwe, J. Schule, and H.-G. Unger, "Analysis of vectorial mode fields in optical waveguides by new finite difference method," J. Lightwave Technol. 12, 487-494 (1994).
[CrossRef]

Selker, M. D.

R. Zia, M. D. Selker, and M. L. Brongersma, "Leaky and bound modes of surface plasmon waveguide," Phys. Rev. B 71, 165431 (2005).
[CrossRef]

R. Zia,M. D. Selker, P. B. Catrysse, andM. L. Brongersma, "Geometries and materials for subwavelength surface plasmon modes," J. Opt. Soc. Am. A 21, 2442-2446 (2004).
[CrossRef]

Sigalas, M. M.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, "Metallic photonic crystals at optical wavelength," J. Phys. Rev. B 62, 15299-15302 (2000).
[CrossRef]

Sorensen, D. C.

R. B. Lehoucq and D. C. Sorensen, "Deflation techniques within an implicitly restarted iteration," SIAM J. Matrix Anal. Appl. 17, 789-821 (1996).
[CrossRef]

D. C. Sorensen, "Implicit application of polynomial filters in a K-step Arnoldi method," SIAM J. Matrix Anal. Appl. 13, 357-385, (1992).
[CrossRef]

Soukoulis, C. M.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, "Metallic photonic crystals at optical wavelength," J. Phys. Rev. B 62, 15299-15302 (2000).
[CrossRef]

Stegeman, G. I.

Stewart, W. J.

W. J. Stewart and A. Jennings, "Algorithm 570: LOPSI: A Simultaneous Iteration Method for Real Matrices [F2]," ACM Trans. Math. Softw. 7, 230-232 (1981).
[CrossRef]

Stohr, A.

P. Berini, A. Stohr, K. Wu, D. Jager, "Normal mode analysis and characterization of an InGaAs/GaAs MQW field-induced optical waveguide including electrode effects," J. Lightwave Technol. 14, 2422-2435 (1996).
[CrossRef]

Stuwe, P.

P. Lusse, P. Stuwe, J. Schule, and H.-G. Unger, "Analysis of vectorial mode fields in optical waveguides by new finite difference method," J. Lightwave Technol. 12, 487-494 (1994).
[CrossRef]

Sugiyama, T.

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," J. Phys. Rev. B 72, 075405 (2005).
[CrossRef]

Tanaka, K.

Tanaka, M.

Tanev, S.

Tzolov, V.

Unger, H.-G.

K. Ramm, P. Lusse, and H.-G. Unger, "Multigrid eigenvalue solver for mode calculation of planar optical waveguides," IEEE Photonics Technol. Lett. 9, 967-969 (1997).
[CrossRef]

P. Lusse, P. Stuwe, J. Schule, and H.-G. Unger, "Analysis of vectorial mode fields in optical waveguides by new finite difference method," J. Lightwave Technol. 12, 487-494 (1994).
[CrossRef]

Walker, C.

Wallias, R. F.

Wu, K.

P. Berini, A. Stohr, K. Wu, D. Jager, "Normal mode analysis and characterization of an InGaAs/GaAs MQW field-induced optical waveguide including electrode effects," J. Lightwave Technol. 14, 2422-2435 (1996).
[CrossRef]

Zia, R.

ACM Trans. Math. Softw.

W. J. Stewart and A. Jennings, "Algorithm 570: LOPSI: A Simultaneous Iteration Method for Real Matrices [F2]," ACM Trans. Math. Softw. 7, 230-232 (1981).
[CrossRef]

IEEE J. Quantum Electron

S. J. Al-Bader, "Optical transmission on metallic wires-fundamental modes," IEEE J. Quantum Electron 40, 325-329 (2004).
[CrossRef]

IEEE Photonics Technol. Lett.

K. Ramm, P. Lusse, and H.-G. Unger, "Multigrid eigenvalue solver for mode calculation of planar optical waveguides," IEEE Photonics Technol. Lett. 9, 967-969 (1997).
[CrossRef]

J. Lightwave Technol.

P. Berini, A. Stohr, K. Wu, D. Jager, "Normal mode analysis and characterization of an InGaAs/GaAs MQW field-induced optical waveguide including electrode effects," J. Lightwave Technol. 14, 2422-2435 (1996).
[CrossRef]

P. Lusse, P. Stuwe, J. Schule, and H.-G. Unger, "Analysis of vectorial mode fields in optical waveguides by new finite difference method," J. Lightwave Technol. 12, 487-494 (1994).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. Rev. B

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric strucutres," J. Phys. Rev. B 61, 10484-10503 (2000).
[CrossRef]

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," J. Phys. Rev. B 72, 075405 (2005).
[CrossRef]

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, "Metallic photonic crystals at optical wavelength," J. Phys. Rev. B 62, 15299-15302 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B

R. Zia, M. D. Selker, and M. L. Brongersma, "Leaky and bound modes of surface plasmon waveguide," Phys. Rev. B 71, 165431 (2005).
[CrossRef]

SIAM J. Matrix Anal. Appl.

R. B. Lehoucq and D. C. Sorensen, "Deflation techniques within an implicitly restarted iteration," SIAM J. Matrix Anal. Appl. 17, 789-821 (1996).
[CrossRef]

D. C. Sorensen, "Implicit application of polynomial filters in a K-step Arnoldi method," SIAM J. Matrix Anal. Appl. 13, 357-385, (1992).
[CrossRef]

Other

R. Radke, "A MATLAB implementation of the implicitly restarted Arnoldi method for solving large scale eigenvalue problems," Technical report, Dept. of Applied and Computational Mathematics, Rice University, Houston, TX, (1996).

V. Hernandez, J. E. Roman, A. Tomas, and V. Vidal, "A Survey of Software for Sparse Eigenvalue Problems," Technical report, Universidad Politecnica de Valencia, (2005).

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Figures (3)

Fig. 1.
Fig. 1.

Comparison between the neff values computed using direct eigenvalue decomposition and the Implicitly Restarted Arnoldi method. In this example, we calculate the 13 eigenvalues with the largest real component.

Fig. 2.
Fig. 2.

Complex propagation constants of SPP between two metallic plates with varying gap width. The solid and the dashed lines are the real and 100x the imaginary parts of the effective index calculated using the method in [10]. Circles and squares shows the calculated real and 100x the imaginary parts of the effective index using the proposed method with periodic boundary conditions.

Fig. 3.
Fig. 3.

Variation of the Real(neff ) with t. The corner modes (MC ), slab symmetric mode (Sb ), slab antisymmetric mode (Ab ), slab symmetric mode (Sa ), fundamental upper branch strip mode (M 01), fundamental lower branch strip mode (M 00) and second lowest-order lower branch strip mode (M 10) are shown. The inset shows the geometry of the waveguide and defines the x and y directions.

Tables (1)

Tables Icon

Table 1. Simulation of a 1D MIM waveguide with t = 100nm

Equations (8)

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

2 H x x 2 + 2 H x y 2 + k 0 2 ( ε ( x , y ) n eff 2 ) H x = 0
2 H y x 2 + 2 H y y 2 + k 0 2 ( ε ( x , y ) n eff 2 ) H y = 0
H z = 1 i β ( H x x + H y y )
E z = i ω ε 0 ε ( H y x H y y )
L H ¯ = [ L xx L xy L yx L yy ] [ H ¯ x H ¯ y ] = β 2 [ H ¯ x H ¯ y ]
K n = [ b , Lb , L 2 b , , L n 1 b ] .
L V m = V m H m + f m e m T
H m = V m * L V m ,

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