Abstract

We present a robust iterative technique for solving complex transcendental dispersion equations routinely encountered in integrated optics. Our method especially befits the multilayer dielectric and plasmonic waveguides forming the basis structures for a host of contemporary nanophotonic devices. The solution algorithm ports seamlessly from the real to the complex domain—i.e., no extra complexity results when dealing with leaky structures or those with material/metal loss. Unlike several existing numerical approaches, our algorithm exhibits markedly-reduced sensitivity to the initial guess and allows for straightforward implementation on a pocket calculator.

© 2009 Optical Society of America

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References

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  1. M. L. Brongersma and P. G. Kik, eds., Surface Plasmon Nanophotonics, Springer series in Optical Sciences (Springer, 2007) Vol. 131.
  2. R. Zia, J. A. Schuller, and M. L. Brongersma, "Plasmonics: The next chip-scale technology," Maters. Today 9, 20-27 (2006).
    [CrossRef]
  3. R. A. Pala, J. S. White, E. S. Barnard, J. Liu, and M. L. Brongersma, "Design of plasmonic thin-film solar cells with broadband absorption enhancements," Adv. Mater. 21, 1-6 (2009).
    [CrossRef]
  4. W. H. Press, S. A. Teukolsky,W. J. Vetterling, and B. P. Flannery, Numerical recipes in C++, The art of scientific computing (Cambridge University Press, 2002), 2nd ed.
  5. A. W. Snyder and J. Love, Optical Waveguide Theory (Science Paperbacks, 1983).
  6. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, "Guiding of a one-dimensional guiding of a one-dimensional optical beam with nanometer diameter," Opt. Lett. 22, 475-477 (1997).
    [CrossRef] [PubMed]
  7. J.-C. Weeber, Y. Lacroute, and A. Dereux, "Optical near-field distributions of surface plasmon waveguide modes," Phys. Rev. B 68, 115401 (2003).
    [CrossRef]
  8. R. Zia, A. Chandran, and M. L. Brongersma, "Dielectric waveguide model for guided surface polaritons," Opt. Lett. 30, 1473-1475 (2005).
    [CrossRef] [PubMed]
  9. R. Zia, J. A. Schuller, and M. L. Brongersma, "Near-field characterization of guided polariton propagation and cutoff in surface plasmon waveguides," Phys. Rev. B 74, 165415 (2006).
    [CrossRef]
  10. R. Zia, M. D. Selker, and M. L. Brongersma, "Leaky and bound modes of surface plasmon waveguides," Phys. Rev. B 71, 165431 (2005).
    [CrossRef]
  11. G. Veronis and S. Fan, "Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides," Appl. Phys. Lett. 87, 131102 (2005).
    [CrossRef]
  12. G. Veronis and S. Fan, "Guided subwavelength plasmonic mode supported by a slot in a thin metal film," Opt. Lett. 30, 3359-3361 (2005).
    [CrossRef]
  13. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006).
    [CrossRef] [PubMed]
  14. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, "Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method," J. Lightwave Technol. 17, 929-941 (1999).
    [CrossRef]
  15. R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, "Geometries and materials for subwavelength surface plasmon modes," J. Opt. Soc. Am. A 21, 2442-2446 (2004).
    [CrossRef]
  16. S. E. Kocabas¸, G. Veronis, D. A. B. Miller, and S. Fan, "Modal analysis and coupling in metal-insulator-metal waveguides," Phys. Rev. B 79, 035120 (2009).
    [CrossRef]
  17. J. P. McKelvey, "Simple iterative procedures for solving transcendental equations with the electronic slide rule," Am. J. Phys. 43, 331-334 (1975).
    [CrossRef]
  18. J. Dugundji and A. Granas, Fixed Point Theory (Springer-Verlag, 2003).
  19. C. R. Pollock, Fundamentals of Optoelectronics (McGraw-Hill Professional Publishing, 2003).
  20. J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986).
    [CrossRef]
  21. Q1Q2. H. Raether, "Surface plasmons on smooth and rough surfaces and on gratings," Springer Tracts Mod. Phys. 111, 1-133 (1988).

2009

R. A. Pala, J. S. White, E. S. Barnard, J. Liu, and M. L. Brongersma, "Design of plasmonic thin-film solar cells with broadband absorption enhancements," Adv. Mater. 21, 1-6 (2009).
[CrossRef]

S. E. Kocabas¸, G. Veronis, D. A. B. Miller, and S. Fan, "Modal analysis and coupling in metal-insulator-metal waveguides," Phys. Rev. B 79, 035120 (2009).
[CrossRef]

2006

R. Zia, J. A. Schuller, and M. L. Brongersma, "Plasmonics: The next chip-scale technology," Maters. Today 9, 20-27 (2006).
[CrossRef]

R. Zia, J. A. Schuller, and M. L. Brongersma, "Near-field characterization of guided polariton propagation and cutoff in surface plasmon waveguides," Phys. Rev. B 74, 165415 (2006).
[CrossRef]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

2005

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

G. Veronis and S. Fan, "Guided subwavelength plasmonic mode supported by a slot in a thin metal film," Opt. Lett. 30, 3359-3361 (2005).
[CrossRef]

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

G. Veronis and S. Fan, "Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides," Appl. Phys. Lett. 87, 131102 (2005).
[CrossRef]

2004

2003

J.-C. Weeber, Y. Lacroute, and A. Dereux, "Optical near-field distributions of surface plasmon waveguide modes," Phys. Rev. B 68, 115401 (2003).
[CrossRef]

1999

1997

1988

Q1Q2. H. Raether, "Surface plasmons on smooth and rough surfaces and on gratings," Springer Tracts Mod. Phys. 111, 1-133 (1988).

1986

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

1975

J. P. McKelvey, "Simple iterative procedures for solving transcendental equations with the electronic slide rule," Am. J. Phys. 43, 331-334 (1975).
[CrossRef]

Anemogiannis, E.

Barnard, E. S.

R. A. Pala, J. S. White, E. S. Barnard, J. Liu, and M. L. Brongersma, "Design of plasmonic thin-film solar cells with broadband absorption enhancements," Adv. Mater. 21, 1-6 (2009).
[CrossRef]

Bozhevolnyi, S. I.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Brongersma, M. L.

R. A. Pala, J. S. White, E. S. Barnard, J. Liu, and M. L. Brongersma, "Design of plasmonic thin-film solar cells with broadband absorption enhancements," Adv. Mater. 21, 1-6 (2009).
[CrossRef]

R. Zia, J. A. Schuller, and M. L. Brongersma, "Near-field characterization of guided polariton propagation and cutoff in surface plasmon waveguides," Phys. Rev. B 74, 165415 (2006).
[CrossRef]

R. Zia, J. A. Schuller, and M. L. Brongersma, "Plasmonics: The next chip-scale technology," Maters. Today 9, 20-27 (2006).
[CrossRef]

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

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

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

Burke, J. J.

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

Catrysse, P. B.

Chandran, A.

Dereux, A.

J.-C. Weeber, Y. Lacroute, and A. Dereux, "Optical near-field distributions of surface plasmon waveguide modes," Phys. Rev. B 68, 115401 (2003).
[CrossRef]

Devaux, E.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Ebbesen, T. W.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Fan, S.

S. E. Kocabas¸, G. Veronis, D. A. B. Miller, and S. Fan, "Modal analysis and coupling in metal-insulator-metal waveguides," Phys. Rev. B 79, 035120 (2009).
[CrossRef]

G. Veronis and S. Fan, "Guided subwavelength plasmonic mode supported by a slot in a thin metal film," Opt. Lett. 30, 3359-3361 (2005).
[CrossRef]

G. Veronis and S. Fan, "Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides," Appl. Phys. Lett. 87, 131102 (2005).
[CrossRef]

Gaylord, T. K.

Glytsis, E. N.

Kobayashi, T.

Kocabas¸, S. E.

S. E. Kocabas¸, G. Veronis, D. A. B. Miller, and S. Fan, "Modal analysis and coupling in metal-insulator-metal waveguides," Phys. Rev. B 79, 035120 (2009).
[CrossRef]

Lacroute, Y.

J.-C. Weeber, Y. Lacroute, and A. Dereux, "Optical near-field distributions of surface plasmon waveguide modes," Phys. Rev. B 68, 115401 (2003).
[CrossRef]

Laluet, J.-Y.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Liu, J.

R. A. Pala, J. S. White, E. S. Barnard, J. Liu, and M. L. Brongersma, "Design of plasmonic thin-film solar cells with broadband absorption enhancements," Adv. Mater. 21, 1-6 (2009).
[CrossRef]

McKelvey, J. P.

J. P. McKelvey, "Simple iterative procedures for solving transcendental equations with the electronic slide rule," Am. J. Phys. 43, 331-334 (1975).
[CrossRef]

Miller, D. A. B.

S. E. Kocabas¸, G. Veronis, D. A. B. Miller, and S. Fan, "Modal analysis and coupling in metal-insulator-metal waveguides," Phys. Rev. B 79, 035120 (2009).
[CrossRef]

Morimoto, A.

Pala, R. A.

R. A. Pala, J. S. White, E. S. Barnard, J. Liu, and M. L. Brongersma, "Design of plasmonic thin-film solar cells with broadband absorption enhancements," Adv. Mater. 21, 1-6 (2009).
[CrossRef]

Raether, H.

Q1Q2. H. Raether, "Surface plasmons on smooth and rough surfaces and on gratings," Springer Tracts Mod. Phys. 111, 1-133 (1988).

Schuller, J. A.

R. Zia, J. A. Schuller, and M. L. Brongersma, "Near-field characterization of guided polariton propagation and cutoff in surface plasmon waveguides," Phys. Rev. B 74, 165415 (2006).
[CrossRef]

R. Zia, J. A. Schuller, and M. L. Brongersma, "Plasmonics: The next chip-scale technology," Maters. Today 9, 20-27 (2006).
[CrossRef]

Selker, M. D.

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

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

Stegeman, G. I.

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

Takahara, J.

Taki, H.

Tamir, T.

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

Veronis, G.

S. E. Kocabas¸, G. Veronis, D. A. B. Miller, and S. Fan, "Modal analysis and coupling in metal-insulator-metal waveguides," Phys. Rev. B 79, 035120 (2009).
[CrossRef]

G. Veronis and S. Fan, "Guided subwavelength plasmonic mode supported by a slot in a thin metal film," Opt. Lett. 30, 3359-3361 (2005).
[CrossRef]

G. Veronis and S. Fan, "Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides," Appl. Phys. Lett. 87, 131102 (2005).
[CrossRef]

Volkov, V. S.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Weeber, J.-C.

J.-C. Weeber, Y. Lacroute, and A. Dereux, "Optical near-field distributions of surface plasmon waveguide modes," Phys. Rev. B 68, 115401 (2003).
[CrossRef]

White, J. S.

R. A. Pala, J. S. White, E. S. Barnard, J. Liu, and M. L. Brongersma, "Design of plasmonic thin-film solar cells with broadband absorption enhancements," Adv. Mater. 21, 1-6 (2009).
[CrossRef]

Yamagishi, S.

Zia, R.

R. Zia, J. A. Schuller, and M. L. Brongersma, "Plasmonics: The next chip-scale technology," Maters. Today 9, 20-27 (2006).
[CrossRef]

R. Zia, J. A. Schuller, and M. L. Brongersma, "Near-field characterization of guided polariton propagation and cutoff in surface plasmon waveguides," Phys. Rev. B 74, 165415 (2006).
[CrossRef]

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

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

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

Adv. Mater.

R. A. Pala, J. S. White, E. S. Barnard, J. Liu, and M. L. Brongersma, "Design of plasmonic thin-film solar cells with broadband absorption enhancements," Adv. Mater. 21, 1-6 (2009).
[CrossRef]

Am. J. Phys.

J. P. McKelvey, "Simple iterative procedures for solving transcendental equations with the electronic slide rule," Am. J. Phys. 43, 331-334 (1975).
[CrossRef]

Appl. Phys. Lett.

G. Veronis and S. Fan, "Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides," Appl. Phys. Lett. 87, 131102 (2005).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. A

Maters. Today

R. Zia, J. A. Schuller, and M. L. Brongersma, "Plasmonics: The next chip-scale technology," Maters. Today 9, 20-27 (2006).
[CrossRef]

Nature

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, "Channel plasmon subwavelength waveguide components including interferometers and ring resonators," Nature 440, 508-511 (2006).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev. B

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

S. E. Kocabas¸, G. Veronis, D. A. B. Miller, and S. Fan, "Modal analysis and coupling in metal-insulator-metal waveguides," Phys. Rev. B 79, 035120 (2009).
[CrossRef]

J.-C. Weeber, Y. Lacroute, and A. Dereux, "Optical near-field distributions of surface plasmon waveguide modes," Phys. Rev. B 68, 115401 (2003).
[CrossRef]

R. Zia, J. A. Schuller, and M. L. Brongersma, "Near-field characterization of guided polariton propagation and cutoff in surface plasmon waveguides," Phys. Rev. B 74, 165415 (2006).
[CrossRef]

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

Springer Tracts Mod. Phys.

Q1Q2. H. Raether, "Surface plasmons on smooth and rough surfaces and on gratings," Springer Tracts Mod. Phys. 111, 1-133 (1988).

Other

M. L. Brongersma and P. G. Kik, eds., Surface Plasmon Nanophotonics, Springer series in Optical Sciences (Springer, 2007) Vol. 131.

W. H. Press, S. A. Teukolsky,W. J. Vetterling, and B. P. Flannery, Numerical recipes in C++, The art of scientific computing (Cambridge University Press, 2002), 2nd ed.

A. W. Snyder and J. Love, Optical Waveguide Theory (Science Paperbacks, 1983).

J. Dugundji and A. Granas, Fixed Point Theory (Springer-Verlag, 2003).

C. R. Pollock, Fundamentals of Optoelectronics (McGraw-Hill Professional Publishing, 2003).

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Figures (11)

Fig. 1.
Fig. 1.

Graphs of the left- (red) and the right- (blue) hand sides of Eq. (4). The red dot indicates the approximate location of the solution near x⋍0.5.

Fig. 2.
Fig. 2.

Solution of the example transcendental equation via the iterative method. The graphs (a), (d), and (g) show the left- (magenta) and the right- (gray) hand sides of Eq. (5), (6), and (7). The red dot indicates the position of the initial guess. Convergence/divergence behavior of the real [(b), (e), (h)] and imaginary [(c), (f), (i)] parts of the iterates.

Fig. 3.
Fig. 3.

Criterion for convergence of the iterative solution. The magenta line in both figures is the curve y=x and the blue lines are two different cases of y=f (x). The solution (a) diverges for |f′(x)|≥1 and (d) converges for |f′(x)|<1. (b), (c), (e), and (f) show how the convergence/divergence is reflected in the behavior of the real and the imaginary parts of the successive iterates.

Fig. 4.
Fig. 4.

Geometries and modes of three-layer infinite slab waveguide structures. Parts (e–i) plot the typical magnetic field profiles.

Fig. 5.
Fig. 5.

Categories of three-layer slab waveguides for the purposes of iterative solution.

Fig. 6.
Fig. 6.

Iterative solution of a strong-confinement waveguide using Eq. (12). (a) The convergence plot depicting the normalized LHS (magenta) and RHS (gray) of Eq. (12); f (k) on the y-axis of of (a) refers to the function on the right hand side. The convergence of the real and imaginary parts of the effective index are shown in (b) and (c), respectively.

Fig. 7.
Fig. 7.

(a–c) Iterative solution of a weak-confinement waveguide using Eq. (12). (a) The LHS and RHS of Eq. (12); f (k) on the y-axis denotes the RHS of Eq. (12). Notice the apparent convergence of the real part (b) and the oscillatory divergence in the imaginary part (c) of the effective index. (d–f) Solution of the same weak-confinement waveguide problem using Eq. (15). (d) The LHS and RHS of Eq. (15); f (k) on the y-axis denotes the RHS of Eq. (15). (e) and (f) show the convergence of the real and imaginary parts of the effective index.

Fig. 8.
Fig. 8.

(a) Typical geometry of a SOI photonic wire waveguide. (b) The two steps used in determining the mode index of the PW waveguide using the effective index method. The procedure is illustrated here for a Ey -polarized mode; steps are similar for a Hy -polarized mode.

Fig. 9.
Fig. 9.

(a) Photonic wire waveguide structure used for evaluation of the effective index method (EIM). (b) The relative error in the calculation of the mode index using EIM as a function of the waveguide width. Relative error is defined as |niter-nFEM|/nFEM. The electric field of the fundamental mode of a SOI photonic wire waveguide calculated using (c) the finite element method and (d) the EIM. The operating wavelength is 1550 nm for all calculations.

Fig. 10.
Fig. 10.

(a) Normalized left- and right-hand sides of Eq. (18); f (κ) refers to the RHS. Convergence of the real (b) and the imaginary (c) parts of the effective index for the fundamental gap-plasmon mode.

Fig. 11.
Fig. 11.

Convergence of the real and imaginary parts of κ, A, and B in Eq. (21) for the case of a 50 nm thick silver-silica-silver waveguide operating at 1550 nm.

Tables (5)

Tables Icon

Table 1. Definitions of various quantities and their expressions in terms of the material parameters and the perpendicular core wavevector k or κ.

Tables Icon

Table 2. Comparison of mode indices for a silicon-on-insulator (SOI) slab waveguide operating at 1550 nm computed using the iterative method and the Newton’s method as implemented by the FindRoot function in Mathematica™

Tables Icon

Table 3. Comparison of mode indices for an Al0.1Ga0.9As/GaAs/air slab waveguide operating at 1550 nm computed using the iterative method and the Newton’s method as implemented by the FindRoot function in Mathematica

Tables Icon

Table 4. Effective indices of various modes of MDM waveguides operating at 1550 nm obtained using the iterative method, with a comparison to the solutions calculated using the Newton’s method as implemented by the FindRoot function in Mathematica.

Tables Icon

Table 5. Effective indices of various modes of an DMD waveguide operating at 1550 nm obtained using the iterative method and their comparison with solutions obtained using the Newton’s method as implemented by the FindRoot function in Mathematica.

Equations (30)

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

F ( x ) = 0 .
x = f ( x ) .
x n + 1 = f ( x n ) .
sin x = 1 x .
x n + 1 = 1 sin x n .
x n + 1 = sin 1 ( 1 x n ) .
sin 2 x = 1 cos 2 x = ( 1 x ) 2 = 1 2 x + x 2 .
x n + 1 = 1 2 ( x n 2 + cos 2 x n ) .
tan ( kh ) = k ( p γ c + q γ s ) k 2 pq γ c γ s for dielectric waveguides .
tanh ( κh ) = κ ( p α c + q α s ) κ 2 pq α c α s for plasmonic waveguides .
tan ( kh / 2 ) = ( pq γ c γ s k 2 ) ± G c G s k ( p γ c + q γ s ) .
k n + 1 = 2 h { M π + tan 1 [ ( pq γ c n γ s n k n 2 ) ± G c n G s n k n ( p γ c n + q γ s n ) ] }
k n + 1 = 2 h [ M π + tan 1 ( p K c 2 / k n 2 1 ) ] for even modes .
k n + 1 = 2 h [ M π cot 1 ( p K c 2 / k n 2 1 ) ] for odd modes .
k 2 ± G c G s cos ( kh ) = pq γ c γ s .
k n + 1 = ( pq K c K s ) 2 ( G c n G s n ) 2 cos 2 ( k n h ) + ( p 2 q 2 1 ) k n 4 p 2 q 2 ( K c 2 + K s 2 ) 2 G c n G s n cos ( k n h )
k n + 1 = K c 1 + p 2 tan 2 ( k n h / 2 ) for even modes .
k n + 1 = K c 1 + p 2 cot 2 ( k n h / 2 ) for odd modes .
κ 2 + 2 S κ coth ( κh ) + pq α c α s = 0 .
κ n + 1 = S n coth ( κ n h ) ± S n 2 coth 2 ( κ n h ) pq α c n α s n
κ n + 1 = p κ n 2 + K c 2 tanh ( κ n h / 2 ) for even gap plasmon .
κ n + 1 = p κ n 2 + K c 2 coth ( κ n h / 2 ) for odd gap plasmon .
tanh κh = 2 A κ κ 2 + A 2 B 2 ,
a n = κ n coth κ n h ± B n 2 + κ n 2 csch 2 ( κ n h ) ,
b n = a n 2 + κ n 2 + 2 a n κ n coth ( κ n h ) ,
κ n + 1 = ( a n + b n ) 2 / p 2 + Q c 2 ,
A n + 1 = ( p ζ c , n + 1 + q ξ s , n + 1 ) / 2 ,
B n + 1 = ( p ζ c , n + 1 q ζ s , n + 1 ) / 2 .
κ n + 1 = Q c 1 p 2 tanh 2 ( κ n h / 2 ) for even plasmon mode .
κ n + 1 = Q c 1 p 2 coth 2 ( κ n h / 2 ) for odd plasmon mode .

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