Abstract

Modal analysis of waveguides and resonators by integral-equation formulations can be hindered by the existence of spurious solutions. In this paper, spurious solutions are shown to be eliminated by introduction of a Rayleigh-quotient based matrix singularity measure. Once the spurious solutions are eliminated, the true modes may be determined efficiently and reliably, even in the presence of degeneracy, by an adaptive search algorithm. Analysis examples that demonstrate the efficacy of the method include an elliptical dielectric waveguide, two unequal touching dielectric cylinders, a plasmonic waveguide, and a realistic micro-structured optical fiber. A freely downloadable version of an optical waveguide mode solver based on this article is available.

© 2007 Optical Society of America

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References

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  1. K. Hayata, M. Koshiba, M. Eguchi, and M. Suzuki, "Vectorial Finite-Element Method without any spurious solutions for DielectricWaveguiding problems using transverse magnetic-field component," IEEE Trans. Microwave Theory Tech. 34, 1120-1124 (1986).
    [CrossRef]
  2. B. Rahman, F. Fernandez, and J. Davies, "Review of finite element methods for microwave and optical waveguides," Proc. IEEE 79, 1442-1448 (1991).
    [CrossRef]
  3. J. T. Chen, T. W. Lin, K. H. Chen, and S. W. Chyuan, "True and spurious eigen solutions for the problems with the mixed-type boundary conditions using BEMs," Finite Elem. Anal. Des. 40, 1521-1549 (2004).
    [CrossRef]
  4. C. Vassallo, "1993-1995 optical mode solvers," Opt. Quantum Electron. 29, 95-114 (1997).
    [CrossRef]
  5. Y. Leviatan, A. Boag, and A. Boag, "Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies-theory and numerical solution," IEEE Trans. Antennas Propag. 36, 1722-1734 (1988).
    [CrossRef]
  6. A. Hochman and Y. Leviatan, "Analysis of strictly bound modes in photonic crystal fibers by use of a sourcemodel technique," J. Opt. Soc. Am. A 21, 1073-1081 (2004).
    [CrossRef]
  7. W. Schroeder and I. Wolff, "The origin of spurious modes in numerical solutions of electromagnetic field eigenvalue problems," IEEE Trans. Microwave Theory Tech. 42, 644-653 (1994).
    [CrossRef]
  8. W. Schroeder and I. Wolff, "Full-wave analysis of the influence of conductor shape and structure details on losses in coplanar waveguide," IEEE MTT S Int.Microwave Symp. Dig. 3, 1273-1276 (1995).
  9. O. Bucci, G. D’Elia, and M. Santojanni, "Non-redundant implementation of method of auxiliary sources for smooth 2D geometries," Electron. Lett. 41, 22 (2005).
    [CrossRef]
  10. B. E. Spielman and R. F. Harrington, "Waveguides of Arbitrary Cross Section by Solution of a Nonlinear Integral Eigenvalue Equation," IEEE Trans. Microwave Theory Tech. 20, 578-585 (1972).
    [CrossRef]
  11. V. Labay and J. Bornemann, "Matrix singular value decomposition for pole-free solutions of homogeneous matrix equations as applied to numerical modeling methods," IEEE Microwave Guid. Wave Lett. 2, 49-51 (1992).
    [CrossRef]
  12. W. Schroeder and I. Wolff, "A reliable and efficient numerical method for indirect eigenvalue problems arising in waveguide and resonator analysis," Int. J. Numer. Model. 12, 197-208 (1999).
    [CrossRef]
  13. N. Don, S. Germani,M. Bozzi, A. Kirilenko, and L. Perregrini, "Determination of the mode spectrum of arbitrarily shaped waveguides using the eigenvalue-tracking method," Microwave Opt. Technol. Lett. 48, 553 (2006).
    [CrossRef]
  14. The Source-Model Technique Package (SMTP) may be downloaded at: http://www.ee.technion.ac.il/leviatan/smtp/index.htm.
  15. Z. Altman, H. Cory, and Y. Leviatan, "Cutoff frequencies of dielectric waveguides using the multifilament current model," Microwave Opt. Technol. Lett. 3, 294-295 (1990).
    [CrossRef]
  16. M. Golberg, Solution Methods for Integral Equations: Theory and Applications (Plenum Press, 1979).
  17. C. Müller, Foundations of the Mathematical Theory of Electromagnetic (Springer-Verlag, 1969).
  18. S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, "Highly efficient full-vectorial integral equation solution for the bound, leaky, and complex modes of dielectric waveguides," IEEE J. Sel. Top. Quantum Electron. 8, 1225-1232 (1990).
  19. H. Cheng, W. Y. Crutchfield, M. Doery, and L. Greengard, "Fast, accurate integral equation methods for the analysis of photonic crystal fibers," Opt. Express. 12, 3791-3805 (2004).
    [CrossRef] [PubMed]
  20. D. L. Young, S. P. Hu, C. W. Chen, C. M. Fan, and K. Murugesan, "Analysis of elliptical waveguides by the method of fundamental solutions," Microwave Opt. Technol. Lett. 44, 552-558 (2005).
    [CrossRef]
  21. R. Lehoucq, D. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM Publications, Philadelphia, 1998).
  22. C. F. Van Loan, "Generalizing the singular value decomposition," SIAM J. Numer. Anal. 13, 76-83 (1976).
    [CrossRef]
  23. R. Brent, Algorithms for minimization without derivatives (Dover Publications, 2002).
  24. B. N. Datta, Numerical Linear Algebra and Applications (Brooks/Cole, Pacific Grove, CA, 1994).
  25. C. G. Broyden, "The convergence of a class of double-rank minimization algorithms," J. Inst. Math. Appl. 6, 76-90 (1970).
    [CrossRef]
  26. G. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins Univ. Pr, 1996).
  27. K. Lo, R. McPhedran, I. Bassett, and G. Milton, "An electromagnetic theory of dielectric waveguides with multiple embedded cylinders," J. Lightwave Technol. 12, 396-410 (1994).
    [CrossRef]
  28. M. J. Steel and R. OsgoodJr, "Polarization and dispersive properties of elliptical-hole photoniccrystal fibers," J. Lightwave Technol. 19, 495-503 (2001).
    [CrossRef]
  29. P. R. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides-I: Summary of results," IEEE Trans. Microwave Theory and Tech. 23, 421-429 (1975).
    [CrossRef]
  30. D. Marcuse, Light Transmition Optics, 2nd ed. (Rober E. Krieger Publishing Company, Malabar, Florida, 1989).
  31. J. Kottmann, O. Martin, D. Smith, and S. Schultz, "Plasmon resonances of silver nanowires with a nonregular cross section," Phys. Rev. B 64, 235,402 (2001).
    [CrossRef]
  32. J. Krenn and J. Weeber, "Surface plasmon polaritons in metal stripes and wires," Philos. Trans. R. Soc. London Ser. A:Mathematical, Physical and Engineering Sciences 362, 739-756 (2004).
    [CrossRef]
  33. H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. Aussenegg, and J. Krenn, "Silver Nanowires as Surface Plasmon Resonators," Phys. Rev. Lett. 95, 257,403 (2005).
    [CrossRef]
  34. L. Novotny and C. Hafner, "Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function," Phys. Rev. E 50, 4094-4106 (1994).
    [CrossRef]
  35. P. Johnson and R. Christy, "Optical Constants of the Noble Metals," Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]
  36. J. Burke, G. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986).
    [CrossRef]
  37. P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B 61, 10,484-10,503 (2000).
    [CrossRef]
  38. M. Szpulak, W. Urbanczyk, E. Serebryannikov, A. Zheltikov, A. Hochman, Y. Leviatan, R. Kotynski, and K. Panajotov, "Comparison of different methods for rigorous modeling of photonic crystal fibers," Opt. Express 14, 5699-5714 (2006).
    [CrossRef] [PubMed]
  39. A. Webb, F. Poletti, D. Richardson, J. Sahu, "Suspended-core holey fiber for evanescent-field sensing," Opt. Eng. 46, 010,503 (2007).
    [CrossRef]
  40. A. Hochman and Y. Leviatan, "Calculation of confinement losses in photonic crystal fibers by use of a sourcemodel technique," J. Opt. Soc. Am. B 22, 474-480 (2005).
    [CrossRef]

2007 (1)

A. Webb, F. Poletti, D. Richardson, J. Sahu, "Suspended-core holey fiber for evanescent-field sensing," Opt. Eng. 46, 010,503 (2007).
[CrossRef]

2006 (2)

M. Szpulak, W. Urbanczyk, E. Serebryannikov, A. Zheltikov, A. Hochman, Y. Leviatan, R. Kotynski, and K. Panajotov, "Comparison of different methods for rigorous modeling of photonic crystal fibers," Opt. Express 14, 5699-5714 (2006).
[CrossRef] [PubMed]

N. Don, S. Germani,M. Bozzi, A. Kirilenko, and L. Perregrini, "Determination of the mode spectrum of arbitrarily shaped waveguides using the eigenvalue-tracking method," Microwave Opt. Technol. Lett. 48, 553 (2006).
[CrossRef]

2005 (4)

O. Bucci, G. D’Elia, and M. Santojanni, "Non-redundant implementation of method of auxiliary sources for smooth 2D geometries," Electron. Lett. 41, 22 (2005).
[CrossRef]

D. L. Young, S. P. Hu, C. W. Chen, C. M. Fan, and K. Murugesan, "Analysis of elliptical waveguides by the method of fundamental solutions," Microwave Opt. Technol. Lett. 44, 552-558 (2005).
[CrossRef]

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. Aussenegg, and J. Krenn, "Silver Nanowires as Surface Plasmon Resonators," Phys. Rev. Lett. 95, 257,403 (2005).
[CrossRef]

A. Hochman and Y. Leviatan, "Calculation of confinement losses in photonic crystal fibers by use of a sourcemodel technique," J. Opt. Soc. Am. B 22, 474-480 (2005).
[CrossRef]

2004 (4)

A. Hochman and Y. Leviatan, "Analysis of strictly bound modes in photonic crystal fibers by use of a sourcemodel technique," J. Opt. Soc. Am. A 21, 1073-1081 (2004).
[CrossRef]

J. T. Chen, T. W. Lin, K. H. Chen, and S. W. Chyuan, "True and spurious eigen solutions for the problems with the mixed-type boundary conditions using BEMs," Finite Elem. Anal. Des. 40, 1521-1549 (2004).
[CrossRef]

J. Krenn and J. Weeber, "Surface plasmon polaritons in metal stripes and wires," Philos. Trans. R. Soc. London Ser. A:Mathematical, Physical and Engineering Sciences 362, 739-756 (2004).
[CrossRef]

H. Cheng, W. Y. Crutchfield, M. Doery, and L. Greengard, "Fast, accurate integral equation methods for the analysis of photonic crystal fibers," Opt. Express. 12, 3791-3805 (2004).
[CrossRef] [PubMed]

2001 (2)

J. Kottmann, O. Martin, D. Smith, and S. Schultz, "Plasmon resonances of silver nanowires with a nonregular cross section," Phys. Rev. B 64, 235,402 (2001).
[CrossRef]

M. J. Steel and R. OsgoodJr, "Polarization and dispersive properties of elliptical-hole photoniccrystal fibers," J. Lightwave Technol. 19, 495-503 (2001).
[CrossRef]

2000 (1)

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B 61, 10,484-10,503 (2000).
[CrossRef]

1999 (1)

W. Schroeder and I. Wolff, "A reliable and efficient numerical method for indirect eigenvalue problems arising in waveguide and resonator analysis," Int. J. Numer. Model. 12, 197-208 (1999).
[CrossRef]

1997 (1)

C. Vassallo, "1993-1995 optical mode solvers," Opt. Quantum Electron. 29, 95-114 (1997).
[CrossRef]

1995 (1)

W. Schroeder and I. Wolff, "Full-wave analysis of the influence of conductor shape and structure details on losses in coplanar waveguide," IEEE MTT S Int.Microwave Symp. Dig. 3, 1273-1276 (1995).

1994 (3)

W. Schroeder and I. Wolff, "The origin of spurious modes in numerical solutions of electromagnetic field eigenvalue problems," IEEE Trans. Microwave Theory Tech. 42, 644-653 (1994).
[CrossRef]

L. Novotny and C. Hafner, "Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function," Phys. Rev. E 50, 4094-4106 (1994).
[CrossRef]

K. Lo, R. McPhedran, I. Bassett, and G. Milton, "An electromagnetic theory of dielectric waveguides with multiple embedded cylinders," J. Lightwave Technol. 12, 396-410 (1994).
[CrossRef]

1992 (1)

V. Labay and J. Bornemann, "Matrix singular value decomposition for pole-free solutions of homogeneous matrix equations as applied to numerical modeling methods," IEEE Microwave Guid. Wave Lett. 2, 49-51 (1992).
[CrossRef]

1991 (1)

B. Rahman, F. Fernandez, and J. Davies, "Review of finite element methods for microwave and optical waveguides," Proc. IEEE 79, 1442-1448 (1991).
[CrossRef]

1990 (2)

Z. Altman, H. Cory, and Y. Leviatan, "Cutoff frequencies of dielectric waveguides using the multifilament current model," Microwave Opt. Technol. Lett. 3, 294-295 (1990).
[CrossRef]

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, "Highly efficient full-vectorial integral equation solution for the bound, leaky, and complex modes of dielectric waveguides," IEEE J. Sel. Top. Quantum Electron. 8, 1225-1232 (1990).

1986 (2)

K. Hayata, M. Koshiba, M. Eguchi, and M. Suzuki, "Vectorial Finite-Element Method without any spurious solutions for DielectricWaveguiding problems using transverse magnetic-field component," IEEE Trans. Microwave Theory Tech. 34, 1120-1124 (1986).
[CrossRef]

J. Burke, G. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986).
[CrossRef]

1976 (1)

C. F. Van Loan, "Generalizing the singular value decomposition," SIAM J. Numer. Anal. 13, 76-83 (1976).
[CrossRef]

1975 (1)

P. R. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides-I: Summary of results," IEEE Trans. Microwave Theory and Tech. 23, 421-429 (1975).
[CrossRef]

1972 (2)

P. Johnson and R. Christy, "Optical Constants of the Noble Metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

B. E. Spielman and R. F. Harrington, "Waveguides of Arbitrary Cross Section by Solution of a Nonlinear Integral Eigenvalue Equation," IEEE Trans. Microwave Theory Tech. 20, 578-585 (1972).
[CrossRef]

1970 (1)

C. G. Broyden, "The convergence of a class of double-rank minimization algorithms," J. Inst. Math. Appl. 6, 76-90 (1970).
[CrossRef]

Electron. Lett. (1)

O. Bucci, G. D’Elia, and M. Santojanni, "Non-redundant implementation of method of auxiliary sources for smooth 2D geometries," Electron. Lett. 41, 22 (2005).
[CrossRef]

Finite Elem. Anal. Des. (1)

J. T. Chen, T. W. Lin, K. H. Chen, and S. W. Chyuan, "True and spurious eigen solutions for the problems with the mixed-type boundary conditions using BEMs," Finite Elem. Anal. Des. 40, 1521-1549 (2004).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, "Highly efficient full-vectorial integral equation solution for the bound, leaky, and complex modes of dielectric waveguides," IEEE J. Sel. Top. Quantum Electron. 8, 1225-1232 (1990).

IEEE Microwave Guid. Wave Lett. (1)

V. Labay and J. Bornemann, "Matrix singular value decomposition for pole-free solutions of homogeneous matrix equations as applied to numerical modeling methods," IEEE Microwave Guid. Wave Lett. 2, 49-51 (1992).
[CrossRef]

IEEE Trans. Microwave Theory and Tech. (1)

P. R. McIsaac, "Symmetry-induced modal characteristics of uniform waveguides-I: Summary of results," IEEE Trans. Microwave Theory and Tech. 23, 421-429 (1975).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

K. Hayata, M. Koshiba, M. Eguchi, and M. Suzuki, "Vectorial Finite-Element Method without any spurious solutions for DielectricWaveguiding problems using transverse magnetic-field component," IEEE Trans. Microwave Theory Tech. 34, 1120-1124 (1986).
[CrossRef]

B. E. Spielman and R. F. Harrington, "Waveguides of Arbitrary Cross Section by Solution of a Nonlinear Integral Eigenvalue Equation," IEEE Trans. Microwave Theory Tech. 20, 578-585 (1972).
[CrossRef]

W. Schroeder and I. Wolff, "The origin of spurious modes in numerical solutions of electromagnetic field eigenvalue problems," IEEE Trans. Microwave Theory Tech. 42, 644-653 (1994).
[CrossRef]

Int. J. Numer. Model. (1)

W. Schroeder and I. Wolff, "A reliable and efficient numerical method for indirect eigenvalue problems arising in waveguide and resonator analysis," Int. J. Numer. Model. 12, 197-208 (1999).
[CrossRef]

J. Inst. Math. Appl. (1)

C. G. Broyden, "The convergence of a class of double-rank minimization algorithms," J. Inst. Math. Appl. 6, 76-90 (1970).
[CrossRef]

J. Lightwave Technol. (2)

K. Lo, R. McPhedran, I. Bassett, and G. Milton, "An electromagnetic theory of dielectric waveguides with multiple embedded cylinders," J. Lightwave Technol. 12, 396-410 (1994).
[CrossRef]

M. J. Steel and R. OsgoodJr, "Polarization and dispersive properties of elliptical-hole photoniccrystal fibers," J. Lightwave Technol. 19, 495-503 (2001).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Mathematical, Physical and Engineering Sciences (1)

J. Krenn and J. Weeber, "Surface plasmon polaritons in metal stripes and wires," Philos. Trans. R. Soc. London Ser. A:Mathematical, Physical and Engineering Sciences 362, 739-756 (2004).
[CrossRef]

Microwave Opt. Technol. Lett. (3)

N. Don, S. Germani,M. Bozzi, A. Kirilenko, and L. Perregrini, "Determination of the mode spectrum of arbitrarily shaped waveguides using the eigenvalue-tracking method," Microwave Opt. Technol. Lett. 48, 553 (2006).
[CrossRef]

Z. Altman, H. Cory, and Y. Leviatan, "Cutoff frequencies of dielectric waveguides using the multifilament current model," Microwave Opt. Technol. Lett. 3, 294-295 (1990).
[CrossRef]

D. L. Young, S. P. Hu, C. W. Chen, C. M. Fan, and K. Murugesan, "Analysis of elliptical waveguides by the method of fundamental solutions," Microwave Opt. Technol. Lett. 44, 552-558 (2005).
[CrossRef]

Microwave Symp. Dig. (1)

W. Schroeder and I. Wolff, "Full-wave analysis of the influence of conductor shape and structure details on losses in coplanar waveguide," IEEE MTT S Int.Microwave Symp. Dig. 3, 1273-1276 (1995).

Opt. Eng. (1)

A. Webb, F. Poletti, D. Richardson, J. Sahu, "Suspended-core holey fiber for evanescent-field sensing," Opt. Eng. 46, 010,503 (2007).
[CrossRef]

Opt. Express (1)

Opt. Express. (1)

H. Cheng, W. Y. Crutchfield, M. Doery, and L. Greengard, "Fast, accurate integral equation methods for the analysis of photonic crystal fibers," Opt. Express. 12, 3791-3805 (2004).
[CrossRef] [PubMed]

Opt. Quantum Electron. (1)

C. Vassallo, "1993-1995 optical mode solvers," Opt. Quantum Electron. 29, 95-114 (1997).
[CrossRef]

Phys. Rev. B (4)

P. Johnson and R. Christy, "Optical Constants of the Noble Metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

J. Burke, G. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986).
[CrossRef]

P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B 61, 10,484-10,503 (2000).
[CrossRef]

J. Kottmann, O. Martin, D. Smith, and S. Schultz, "Plasmon resonances of silver nanowires with a nonregular cross section," Phys. Rev. B 64, 235,402 (2001).
[CrossRef]

Phys. Rev. E (1)

L. Novotny and C. Hafner, "Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function," Phys. Rev. E 50, 4094-4106 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. Aussenegg, and J. Krenn, "Silver Nanowires as Surface Plasmon Resonators," Phys. Rev. Lett. 95, 257,403 (2005).
[CrossRef]

Proc. IEEE (1)

B. Rahman, F. Fernandez, and J. Davies, "Review of finite element methods for microwave and optical waveguides," Proc. IEEE 79, 1442-1448 (1991).
[CrossRef]

SIAM J. Numer. Anal. (1)

C. F. Van Loan, "Generalizing the singular value decomposition," SIAM J. Numer. Anal. 13, 76-83 (1976).
[CrossRef]

Other (9)

R. Brent, Algorithms for minimization without derivatives (Dover Publications, 2002).

B. N. Datta, Numerical Linear Algebra and Applications (Brooks/Cole, Pacific Grove, CA, 1994).

R. Lehoucq, D. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (SIAM Publications, Philadelphia, 1998).

M. Golberg, Solution Methods for Integral Equations: Theory and Applications (Plenum Press, 1979).

C. Müller, Foundations of the Mathematical Theory of Electromagnetic (Springer-Verlag, 1969).

The Source-Model Technique Package (SMTP) may be downloaded at: http://www.ee.technion.ac.il/leviatan/smtp/index.htm.

Y. Leviatan, A. Boag, and A. Boag, "Generalized formulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies-theory and numerical solution," IEEE Trans. Antennas Propag. 36, 1722-1734 (1988).
[CrossRef]

G. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins Univ. Pr, 1996).

D. Marcuse, Light Transmition Optics, 2nd ed. (Rober E. Krieger Publishing Company, Malabar, Florida, 1989).

Supplementary Material (1)

» Media 1: AVI (3697 KB)     

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Figures (19)

Fig. 1.
Fig. 1.

Sources and testing points for a SMT analysis of a circular waveguide.

Fig. 2.
Fig. 2.

Singularities at the zeros of the Bessel functions. For this graph, N = 20 and α = 1.5. A very fine sampling grid of 1500 points was needed to reveal all the singularities.

Fig. 3.
Fig. 3.

Dependence of the shape of the singularity on (a) the number of sources and (b) the distance from the waveguide boundary. In (a), α= 1.5, and in (b), N = 30.

Fig. 4.
Fig. 4.

Singularities at the zeros of the Bessel functions, calculated with the proposed matrix singularity measure. For this graph, N = 20 and α= 1.5. The sampling grid is the same as that of Fig. 2, although a much coarser grid could have been used.

Fig. 5.
Fig. 5.

The proposed matrix singularity measure. Dependence of the shape of the singularity on the number of sources (a) and the distance from the waveguide boundary (b). In (a), α= 1.5, and in (b), N = 30. Note that the range of the abscissa is ten times that of Fig. 3.

Fig. 6.
Fig. 6.

Typical sawtooth-like dependence of the singularity measure, ΔE, on (a) the normalized propagation constant, or effective index, β/k 0, and on (b) the normalized wave vector, k0R. The structure analyzed is a round step-index fiber. The relative permittivities are: εrw = 1.82, and εrc = 1.52, for the waveguide and cladding regions, respectively. In (a) the radius R is equal to the free-space wavelength, λ 0. In (b) the normalized propagation constant, β/k 0, is 1.6.

Fig. 7.
Fig. 7.

Adaptive search algorithm for finding the extrema of ΔE(β).

Fig. 8.
Fig. 8.

Algorithm for attempting to determine whether ΔE(β) is monotonic by sampling it at most n max times.

Fig. 9.
Fig. 9.

Progress of the sampling scheme used to determine whether ΔE(β) is monotonic. A high resolution plot of ΔE is shown in (a) for the same step-index fiber of Fig. 6, but with λ 0 = 2R. In the first step of the sampling scheme (b), the endpoints and midpoint are sampled. Since the sample series is monotonic, two new samples are added at (β 0 + β 1)/2 and (β 1 + β 2)/2, as shown in (c). Since the sample series is still monotonic, the three less collinear consecutive points are found (β 3,β 4, and β 5) and two points are added in between them (d). This last refinement reveals a minimum and a maximum.

Fig. 10.
Fig. 10.

An animation summarizing the sampling algorithm (AVI, 3.7MB), for the same step-index fiber shown in Fig. 9. The current search interval is marked by the dotted lines. The rate of the animation is much slower than the actual computation. [Media 1]

Fig. 11.
Fig. 11.

The resolution of two close modes in the step-index fiber of Fig. 9. Assuming the sampling process missed the first minimum but found the second minimum, the neighborhood of the second minimum can be searched again, this time with the matrix [A] of Eq. (12). The first minimum can then be easily detected.

Fig. 12.
Fig. 12.

The normalized error, ΔE(β), near a minimum in the complex β plane. Contours of equal ΔE are plotted at the bottom. The contours are equally spaced in ΔE.

Fig. 13.
Fig. 13.

Dispersion curves of the two touching cylinders analyzed in [27]. The material parameters are: εrc = 1.4572, and εrw = 1.4572/0.9, and the ratio of radii, R 1/R 2, is 1.3.

Fig. 14.
Fig. 14.

Real part of the z component of the complex Poynting vector, Re(Sz ), for the x-polarized mode of the first (i.e. fundamental) mode pair (a), and for the x-polarized mode of second mode pair. For the y-polarized modes, the plots are almost identical to the x-polarized plots shown. The parameters are as in Fig. 13, and the effective indices of the modes shown are nx = 1.5155 and ny = 1.5044, for V = 3.5. The plots have been normalized to unit total power in the cross-section shown.

Fig. 15.
Fig. 15.

Solver performance as function of the number of sources used. The number of modes found is shown in (a), the average error in continuity conditions per mode, ΔE¯, in (b), and the average computation time per mode on a 3.2 GHz Intel PC, T¯, in (c). For the elliptical waveguide analyzed, εrw = 2, and εrc = 1.

Fig. 16.
Fig. 16.

Real (a) and imaginary (b) parts of the effective index of the TM0 mode of circular and elliptical nano-wires made of silver. The free-space wavelength and permittivity are as in Table 2.

Fig. 17.
Fig. 17.

Real part of the z-component of the complex Poynting vector, Re(Sz ), for the x polarized mode of the first (i.e. fundamental) mode pair (β/k 0 = 1.1179, V = 0.38). The plots have been normalized to unit total power in the cross-section shown.

Fig. 18.
Fig. 18.

Fraction of the power carried in air as function of core diameter, at λ = 1.55μm. The range of core diameters follows [39].

Fig. 19.
Fig. 19.

Dispersion curves of the fundamental mode pair (plot on top of each other), and their difference, the birefringence. The range of V parallels the range of core diameters in Fig. 18.

Tables (5)

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Table 1. Convergence of the effective index. The normalized frequency, V, is 3.5. All the rest of the parameters are given in the caption of Fig. 13.

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Table 2. Convergence of the effective index and the estimate of the imaginary part, β 0i /k 0, of the TM0 mode of circular silver nano-wire. The free-space wavelength is 633nm, the radius of the wire is 50nm, and the relative permittivity of the wire (interpolated from [35]) is: – 18.3187697 – j0.5037517.

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Table 3. Effective indices at V = 1.

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Table 4. Effective indices at V = 2.

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Table 5. Effective indices at V = 3.

Equations (18)

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C J z ( r′ ) H 0 ( 2 ) ( k ρ r r′ ) d l = 0 .
k ρ = k 0 2 β 2 ,
L ( f ) = K ( r , r′ ) f ( r′ ) d l
E z = π ρ 0 k ρ 2 2 ω ε 0 J n ( k ρ ρ ) H n ( 2 ) ( k ρ ρ 0 ) e j n ϕ ; ρ ρ 0 ,
E z k ρ 2 ρ 0 2 j ω ε 0 n ( ρ ρ 0 ) n e j n ϕ ; ρ ρ 0 .
[ Z ] I = 0 ,
Δ E = rms ( [ Z ] I ) rms ( [ Z ˜ ] I ) ,
[ Z ] [ Z ] I = ξ [ Z ˜ ] [ Z ˜ ] I .
r 2 = 1 i = 1 3 [ Δ E ( β i ) Δ E f ( β i ) ] 2 i = 1 3 [ Δ E ( β i ) Δ E ( β i ) ] 2
[ A ] x = ξ [ B ] x .
x i [ B ] x j = 0 , i j .
[ A ] = [ Z ] [ Z ] + ξ 0 [ B ] I 1 I 1 [ B ] I 1 [ B ] I 1 ,
Δ E ( β ) a β β 0 .
[ Δ E ( β r ) ] 2 a 2 [ ( β r β 0 r ) 2 + β 0 i 2 ] ,
[ Δ E ( β r ) ] 2 p 2 β r 2 + p 1 β r + p 0 .
β 0 p 1 ( 2 p 2 ) j p 0 p 2 [ p 1 ( 2 p 2 ) ] 2 .
V = k 0 R 1 π ε r w ε r c ,
Δ β = n x n y ,

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