Abstract

We investigate TE-wave propagation in a hollow waveguide with a graded dielectric layer, described using a hyperbolic tangent function. General formulae for the electric field components of the TE-waves, applicable to hollow waveguides with arbitrary cross sectional shapes, are presented. We illustrate the exact analytical results for the electric field components in the special case of a rectangular waveguide. Furthermore, we derive exact analytical results for the reflection and transmission coefficients valid for waveguides of arbitrary cross sectional shapes. Finally, we show that the obtained reflection and transmission coefficients are in exact asymptotic agreement with those obtained for a very thin homogeneous dielectric layer using mode-matching and cascading. The proposed method gives analytical results that are directly applicable without the need of mode-matching, and it has the ability to model realistic, smooth transitions.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. S. A. Maier, “Plasmonics: Fundamentals and Applications,” (Springer-Verlag, 2007).
  2. S. Link and M. A. El-Sayed, “Shape and size dependence of radiative non-radiative and photothermal properties of gold nanocrystals,” Int. Rev. Phys. Chem. 19(3), 409–453 (2000).
    [Crossref]
  3. N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. 85(21), 5040–5042 (2004).
    [Crossref]
  4. O. D. Miller, A. G. Polimeridis, M. T. H. Reid, C. W. Hsu, B. G. DeLacy, J. D. Joannopoulos, M. Soljacic, and S. G. Johnson, “Fundamental limits to optical response in absorptive systems,” Opt. Express 24(4), 3329–3364 (2016).
    [Crossref]
  5. T. Lund, M. F. Callaghan, P. Williams, M. Turmaine, C. Bachmann, T. Rademacher, I. M. Roitt, and R. Bayford, “The influence of ligand organization on the rate of uptake of gold nanoparticles by colorectal cancer cells,” Biomaterials 32(36), 9776–9784 (2011).
    [Crossref]
  6. G. W. Hanson, R. C. Monreal, and S. P. Apell, “Electromagnetic absorption mechanisms in metal nanospheres: Bulk and surface effects in radiofrequency-terahertz heating of nanoparticles,” J. Appl. Phys. 109(12), 124306 (2011).
    [Crossref]
  7. S. J. Corr, M. Raoof, Y. Mackeyev, S. Phounsavath, M. A. Cheney, B. T. Cisneros, M. Shur, M. Gozin, P. J. McNally, L. J. Wilson, and S. A. Curley, “Citrate-capped gold nanoparticle electrophoretic heat production in response to a time-varying radiofrequency electric-field,” J. Phys. Chem. C 116(45), 24380–24389 (2012).
    [Crossref]
  8. E. Sassaroli, K. C. P. Li, and B. E. O’Neil, “Radio frequency absorption in gold nanoparticle suspensions: a phenomenological study,” J. Phys. D: Appl. Phys. 45(7), 075303 (2012).
    [Crossref]
  9. C. B. Collins, R. S. McCoy, B. J. Ackerson, G. J. Collins, and C. J. Ackerson, “Radiofrequency heating pathways for gold nanoparticles,” Nanoscale 6(15), 8459–8472 (2014).
    [Crossref]
  10. S. Nordebo, M. Dalarsson, Y. Ivanenko, D. Sjöberg, and R. Bayford, “On the physical limitations for radio frequency absorption in gold nanoparticle suspensions,” J. Phys. D: Appl. Phys. 50(15), 155401 (2017).
    [Crossref]
  11. S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz,” Phys. Med. Biol. 41(11), 2251–2269 (1996).
    [Crossref]
  12. M. Dalarsson, S. Nordebo, D. Sjöberg, and R. Bayford, “Absorption and optimal plasmonic resonances for small ellipsoidal particles in lossy media,” J. Phys. D: Appl. Phys. 50(34), 345401 (2017).
    [Crossref]
  13. Y. Ivanenko, M. Gustafsson, B. L. G. Jonsson, A. Luger, B. Nilsson, S. Nordebo, and J. Toft, “Passive approximation and optimization using B-splines,” arXiv 1711.07937 (2017).
  14. Y. Ivanenko, M. Dalarsson, S. Nordebo, and R. Bayford, “On the Plasmonic Resonances in a Layered Waveguide Structure,” in Proceedings of The 12th International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials’ 2018 (2018).
  15. M. Dalarsson and P. Tassin, “Analytical solution for wave propagation through a graded index interface between a right-handed and a left-handed material,” Opt. Express 17(8), 6747 (2009).
    [Crossref]
  16. M. Dalarsson, M. Norgren, and Z. Jakšić, “Lossy gradient index metamaterial with sinusoidal periodicity of refractive index: case of constant impedance throughout the structure,” J. Nanophotonics 5(1), 051804 (2011).
    [Crossref]
  17. M. Dalarsson, M. Norgren, N. Dončov, and Z. Jakšić, “Lossy gradient index transmission optics with arbitrary periodic permittivity and permeability and constant impedance throughout the structure,” J. Opt. 14(6), 065102 (2012).
    [Crossref]
  18. M. Dalarsson, M. Norgren, T. Asenov, N. Dončov, and Z. Jakšić, “Exact analytical solution for fields in gradient index metamaterials with different loss factors in negative and positive refractive index segments,” J. Nanophotonics 7(1), 073086 (2013).
    [Crossref]
  19. M. Dalarsson, M. Norgren, and Z. Jakšić, “Exact analytical solution for fields in a lossy cylindrical structure with linear gradient index metamaterials,” Prog. Electromagn. Res. 151, 109–117 (2015).
    [Crossref]
  20. M. Dalarsson and Z. Jakšić, “Exact analytical solution for fields in a lossy cylindrical structure with hyperbolic tangent gradient index metamaterials,” Opt. Quantum Electron. 48(3), 209 (2016).
    [Crossref]
  21. M. Dalarsson, “General theory of wave propagation through graded interfaces between positive- and negative-refractive-index media,” Phys. Rev. A 96(4), 043848 (2017).
    [Crossref]
  22. M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables,” (Dover Books, New York, 1965).

2017 (3)

S. Nordebo, M. Dalarsson, Y. Ivanenko, D. Sjöberg, and R. Bayford, “On the physical limitations for radio frequency absorption in gold nanoparticle suspensions,” J. Phys. D: Appl. Phys. 50(15), 155401 (2017).
[Crossref]

M. Dalarsson, S. Nordebo, D. Sjöberg, and R. Bayford, “Absorption and optimal plasmonic resonances for small ellipsoidal particles in lossy media,” J. Phys. D: Appl. Phys. 50(34), 345401 (2017).
[Crossref]

M. Dalarsson, “General theory of wave propagation through graded interfaces between positive- and negative-refractive-index media,” Phys. Rev. A 96(4), 043848 (2017).
[Crossref]

2016 (2)

M. Dalarsson and Z. Jakšić, “Exact analytical solution for fields in a lossy cylindrical structure with hyperbolic tangent gradient index metamaterials,” Opt. Quantum Electron. 48(3), 209 (2016).
[Crossref]

O. D. Miller, A. G. Polimeridis, M. T. H. Reid, C. W. Hsu, B. G. DeLacy, J. D. Joannopoulos, M. Soljacic, and S. G. Johnson, “Fundamental limits to optical response in absorptive systems,” Opt. Express 24(4), 3329–3364 (2016).
[Crossref]

2015 (1)

M. Dalarsson, M. Norgren, and Z. Jakšić, “Exact analytical solution for fields in a lossy cylindrical structure with linear gradient index metamaterials,” Prog. Electromagn. Res. 151, 109–117 (2015).
[Crossref]

2014 (1)

C. B. Collins, R. S. McCoy, B. J. Ackerson, G. J. Collins, and C. J. Ackerson, “Radiofrequency heating pathways for gold nanoparticles,” Nanoscale 6(15), 8459–8472 (2014).
[Crossref]

2013 (1)

M. Dalarsson, M. Norgren, T. Asenov, N. Dončov, and Z. Jakšić, “Exact analytical solution for fields in gradient index metamaterials with different loss factors in negative and positive refractive index segments,” J. Nanophotonics 7(1), 073086 (2013).
[Crossref]

2012 (3)

M. Dalarsson, M. Norgren, N. Dončov, and Z. Jakšić, “Lossy gradient index transmission optics with arbitrary periodic permittivity and permeability and constant impedance throughout the structure,” J. Opt. 14(6), 065102 (2012).
[Crossref]

S. J. Corr, M. Raoof, Y. Mackeyev, S. Phounsavath, M. A. Cheney, B. T. Cisneros, M. Shur, M. Gozin, P. J. McNally, L. J. Wilson, and S. A. Curley, “Citrate-capped gold nanoparticle electrophoretic heat production in response to a time-varying radiofrequency electric-field,” J. Phys. Chem. C 116(45), 24380–24389 (2012).
[Crossref]

E. Sassaroli, K. C. P. Li, and B. E. O’Neil, “Radio frequency absorption in gold nanoparticle suspensions: a phenomenological study,” J. Phys. D: Appl. Phys. 45(7), 075303 (2012).
[Crossref]

2011 (3)

T. Lund, M. F. Callaghan, P. Williams, M. Turmaine, C. Bachmann, T. Rademacher, I. M. Roitt, and R. Bayford, “The influence of ligand organization on the rate of uptake of gold nanoparticles by colorectal cancer cells,” Biomaterials 32(36), 9776–9784 (2011).
[Crossref]

G. W. Hanson, R. C. Monreal, and S. P. Apell, “Electromagnetic absorption mechanisms in metal nanospheres: Bulk and surface effects in radiofrequency-terahertz heating of nanoparticles,” J. Appl. Phys. 109(12), 124306 (2011).
[Crossref]

M. Dalarsson, M. Norgren, and Z. Jakšić, “Lossy gradient index metamaterial with sinusoidal periodicity of refractive index: case of constant impedance throughout the structure,” J. Nanophotonics 5(1), 051804 (2011).
[Crossref]

2009 (1)

2004 (1)

N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. 85(21), 5040–5042 (2004).
[Crossref]

2000 (1)

S. Link and M. A. El-Sayed, “Shape and size dependence of radiative non-radiative and photothermal properties of gold nanocrystals,” Int. Rev. Phys. Chem. 19(3), 409–453 (2000).
[Crossref]

1996 (1)

S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz,” Phys. Med. Biol. 41(11), 2251–2269 (1996).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables,” (Dover Books, New York, 1965).

Ackerson, B. J.

C. B. Collins, R. S. McCoy, B. J. Ackerson, G. J. Collins, and C. J. Ackerson, “Radiofrequency heating pathways for gold nanoparticles,” Nanoscale 6(15), 8459–8472 (2014).
[Crossref]

Ackerson, C. J.

C. B. Collins, R. S. McCoy, B. J. Ackerson, G. J. Collins, and C. J. Ackerson, “Radiofrequency heating pathways for gold nanoparticles,” Nanoscale 6(15), 8459–8472 (2014).
[Crossref]

Apell, S. P.

G. W. Hanson, R. C. Monreal, and S. P. Apell, “Electromagnetic absorption mechanisms in metal nanospheres: Bulk and surface effects in radiofrequency-terahertz heating of nanoparticles,” J. Appl. Phys. 109(12), 124306 (2011).
[Crossref]

Asenov, T.

M. Dalarsson, M. Norgren, T. Asenov, N. Dončov, and Z. Jakšić, “Exact analytical solution for fields in gradient index metamaterials with different loss factors in negative and positive refractive index segments,” J. Nanophotonics 7(1), 073086 (2013).
[Crossref]

Bachmann, C.

T. Lund, M. F. Callaghan, P. Williams, M. Turmaine, C. Bachmann, T. Rademacher, I. M. Roitt, and R. Bayford, “The influence of ligand organization on the rate of uptake of gold nanoparticles by colorectal cancer cells,” Biomaterials 32(36), 9776–9784 (2011).
[Crossref]

Bayford, R.

S. Nordebo, M. Dalarsson, Y. Ivanenko, D. Sjöberg, and R. Bayford, “On the physical limitations for radio frequency absorption in gold nanoparticle suspensions,” J. Phys. D: Appl. Phys. 50(15), 155401 (2017).
[Crossref]

M. Dalarsson, S. Nordebo, D. Sjöberg, and R. Bayford, “Absorption and optimal plasmonic resonances for small ellipsoidal particles in lossy media,” J. Phys. D: Appl. Phys. 50(34), 345401 (2017).
[Crossref]

T. Lund, M. F. Callaghan, P. Williams, M. Turmaine, C. Bachmann, T. Rademacher, I. M. Roitt, and R. Bayford, “The influence of ligand organization on the rate of uptake of gold nanoparticles by colorectal cancer cells,” Biomaterials 32(36), 9776–9784 (2011).
[Crossref]

Y. Ivanenko, M. Dalarsson, S. Nordebo, and R. Bayford, “On the Plasmonic Resonances in a Layered Waveguide Structure,” in Proceedings of The 12th International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials’ 2018 (2018).

Callaghan, M. F.

T. Lund, M. F. Callaghan, P. Williams, M. Turmaine, C. Bachmann, T. Rademacher, I. M. Roitt, and R. Bayford, “The influence of ligand organization on the rate of uptake of gold nanoparticles by colorectal cancer cells,” Biomaterials 32(36), 9776–9784 (2011).
[Crossref]

Cheney, M. A.

S. J. Corr, M. Raoof, Y. Mackeyev, S. Phounsavath, M. A. Cheney, B. T. Cisneros, M. Shur, M. Gozin, P. J. McNally, L. J. Wilson, and S. A. Curley, “Citrate-capped gold nanoparticle electrophoretic heat production in response to a time-varying radiofrequency electric-field,” J. Phys. Chem. C 116(45), 24380–24389 (2012).
[Crossref]

Cisneros, B. T.

S. J. Corr, M. Raoof, Y. Mackeyev, S. Phounsavath, M. A. Cheney, B. T. Cisneros, M. Shur, M. Gozin, P. J. McNally, L. J. Wilson, and S. A. Curley, “Citrate-capped gold nanoparticle electrophoretic heat production in response to a time-varying radiofrequency electric-field,” J. Phys. Chem. C 116(45), 24380–24389 (2012).
[Crossref]

Collins, C. B.

C. B. Collins, R. S. McCoy, B. J. Ackerson, G. J. Collins, and C. J. Ackerson, “Radiofrequency heating pathways for gold nanoparticles,” Nanoscale 6(15), 8459–8472 (2014).
[Crossref]

Collins, G. J.

C. B. Collins, R. S. McCoy, B. J. Ackerson, G. J. Collins, and C. J. Ackerson, “Radiofrequency heating pathways for gold nanoparticles,” Nanoscale 6(15), 8459–8472 (2014).
[Crossref]

Corr, S. J.

S. J. Corr, M. Raoof, Y. Mackeyev, S. Phounsavath, M. A. Cheney, B. T. Cisneros, M. Shur, M. Gozin, P. J. McNally, L. J. Wilson, and S. A. Curley, “Citrate-capped gold nanoparticle electrophoretic heat production in response to a time-varying radiofrequency electric-field,” J. Phys. Chem. C 116(45), 24380–24389 (2012).
[Crossref]

Curley, S. A.

S. J. Corr, M. Raoof, Y. Mackeyev, S. Phounsavath, M. A. Cheney, B. T. Cisneros, M. Shur, M. Gozin, P. J. McNally, L. J. Wilson, and S. A. Curley, “Citrate-capped gold nanoparticle electrophoretic heat production in response to a time-varying radiofrequency electric-field,” J. Phys. Chem. C 116(45), 24380–24389 (2012).
[Crossref]

Dalarsson, M.

S. Nordebo, M. Dalarsson, Y. Ivanenko, D. Sjöberg, and R. Bayford, “On the physical limitations for radio frequency absorption in gold nanoparticle suspensions,” J. Phys. D: Appl. Phys. 50(15), 155401 (2017).
[Crossref]

M. Dalarsson, S. Nordebo, D. Sjöberg, and R. Bayford, “Absorption and optimal plasmonic resonances for small ellipsoidal particles in lossy media,” J. Phys. D: Appl. Phys. 50(34), 345401 (2017).
[Crossref]

M. Dalarsson, “General theory of wave propagation through graded interfaces between positive- and negative-refractive-index media,” Phys. Rev. A 96(4), 043848 (2017).
[Crossref]

M. Dalarsson and Z. Jakšić, “Exact analytical solution for fields in a lossy cylindrical structure with hyperbolic tangent gradient index metamaterials,” Opt. Quantum Electron. 48(3), 209 (2016).
[Crossref]

M. Dalarsson, M. Norgren, and Z. Jakšić, “Exact analytical solution for fields in a lossy cylindrical structure with linear gradient index metamaterials,” Prog. Electromagn. Res. 151, 109–117 (2015).
[Crossref]

M. Dalarsson, M. Norgren, T. Asenov, N. Dončov, and Z. Jakšić, “Exact analytical solution for fields in gradient index metamaterials with different loss factors in negative and positive refractive index segments,” J. Nanophotonics 7(1), 073086 (2013).
[Crossref]

M. Dalarsson, M. Norgren, N. Dončov, and Z. Jakšić, “Lossy gradient index transmission optics with arbitrary periodic permittivity and permeability and constant impedance throughout the structure,” J. Opt. 14(6), 065102 (2012).
[Crossref]

M. Dalarsson, M. Norgren, and Z. Jakšić, “Lossy gradient index metamaterial with sinusoidal periodicity of refractive index: case of constant impedance throughout the structure,” J. Nanophotonics 5(1), 051804 (2011).
[Crossref]

M. Dalarsson and P. Tassin, “Analytical solution for wave propagation through a graded index interface between a right-handed and a left-handed material,” Opt. Express 17(8), 6747 (2009).
[Crossref]

Y. Ivanenko, M. Dalarsson, S. Nordebo, and R. Bayford, “On the Plasmonic Resonances in a Layered Waveguide Structure,” in Proceedings of The 12th International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials’ 2018 (2018).

DeLacy, B. G.

Doncov, N.

M. Dalarsson, M. Norgren, T. Asenov, N. Dončov, and Z. Jakšić, “Exact analytical solution for fields in gradient index metamaterials with different loss factors in negative and positive refractive index segments,” J. Nanophotonics 7(1), 073086 (2013).
[Crossref]

M. Dalarsson, M. Norgren, N. Dončov, and Z. Jakšić, “Lossy gradient index transmission optics with arbitrary periodic permittivity and permeability and constant impedance throughout the structure,” J. Opt. 14(6), 065102 (2012).
[Crossref]

El-Sayed, M. A.

S. Link and M. A. El-Sayed, “Shape and size dependence of radiative non-radiative and photothermal properties of gold nanocrystals,” Int. Rev. Phys. Chem. 19(3), 409–453 (2000).
[Crossref]

Gabriel, C.

S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz,” Phys. Med. Biol. 41(11), 2251–2269 (1996).
[Crossref]

Gabriel, S.

S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz,” Phys. Med. Biol. 41(11), 2251–2269 (1996).
[Crossref]

Gozin, M.

S. J. Corr, M. Raoof, Y. Mackeyev, S. Phounsavath, M. A. Cheney, B. T. Cisneros, M. Shur, M. Gozin, P. J. McNally, L. J. Wilson, and S. A. Curley, “Citrate-capped gold nanoparticle electrophoretic heat production in response to a time-varying radiofrequency electric-field,” J. Phys. Chem. C 116(45), 24380–24389 (2012).
[Crossref]

Gustafsson, M.

Y. Ivanenko, M. Gustafsson, B. L. G. Jonsson, A. Luger, B. Nilsson, S. Nordebo, and J. Toft, “Passive approximation and optimization using B-splines,” arXiv 1711.07937 (2017).

Hanson, G. W.

G. W. Hanson, R. C. Monreal, and S. P. Apell, “Electromagnetic absorption mechanisms in metal nanospheres: Bulk and surface effects in radiofrequency-terahertz heating of nanoparticles,” J. Appl. Phys. 109(12), 124306 (2011).
[Crossref]

Hsu, C. W.

Ivanenko, Y.

S. Nordebo, M. Dalarsson, Y. Ivanenko, D. Sjöberg, and R. Bayford, “On the physical limitations for radio frequency absorption in gold nanoparticle suspensions,” J. Phys. D: Appl. Phys. 50(15), 155401 (2017).
[Crossref]

Y. Ivanenko, M. Gustafsson, B. L. G. Jonsson, A. Luger, B. Nilsson, S. Nordebo, and J. Toft, “Passive approximation and optimization using B-splines,” arXiv 1711.07937 (2017).

Y. Ivanenko, M. Dalarsson, S. Nordebo, and R. Bayford, “On the Plasmonic Resonances in a Layered Waveguide Structure,” in Proceedings of The 12th International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials’ 2018 (2018).

Jakšic, Z.

M. Dalarsson and Z. Jakšić, “Exact analytical solution for fields in a lossy cylindrical structure with hyperbolic tangent gradient index metamaterials,” Opt. Quantum Electron. 48(3), 209 (2016).
[Crossref]

M. Dalarsson, M. Norgren, and Z. Jakšić, “Exact analytical solution for fields in a lossy cylindrical structure with linear gradient index metamaterials,” Prog. Electromagn. Res. 151, 109–117 (2015).
[Crossref]

M. Dalarsson, M. Norgren, T. Asenov, N. Dončov, and Z. Jakšić, “Exact analytical solution for fields in gradient index metamaterials with different loss factors in negative and positive refractive index segments,” J. Nanophotonics 7(1), 073086 (2013).
[Crossref]

M. Dalarsson, M. Norgren, N. Dončov, and Z. Jakšić, “Lossy gradient index transmission optics with arbitrary periodic permittivity and permeability and constant impedance throughout the structure,” J. Opt. 14(6), 065102 (2012).
[Crossref]

M. Dalarsson, M. Norgren, and Z. Jakšić, “Lossy gradient index metamaterial with sinusoidal periodicity of refractive index: case of constant impedance throughout the structure,” J. Nanophotonics 5(1), 051804 (2011).
[Crossref]

Joannopoulos, J. D.

Johnson, S. G.

Jonsson, B. L. G.

Y. Ivanenko, M. Gustafsson, B. L. G. Jonsson, A. Luger, B. Nilsson, S. Nordebo, and J. Toft, “Passive approximation and optimization using B-splines,” arXiv 1711.07937 (2017).

Lau, R. W.

S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz,” Phys. Med. Biol. 41(11), 2251–2269 (1996).
[Crossref]

Lawandy, N. M.

N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. 85(21), 5040–5042 (2004).
[Crossref]

Li, K. C. P.

E. Sassaroli, K. C. P. Li, and B. E. O’Neil, “Radio frequency absorption in gold nanoparticle suspensions: a phenomenological study,” J. Phys. D: Appl. Phys. 45(7), 075303 (2012).
[Crossref]

Link, S.

S. Link and M. A. El-Sayed, “Shape and size dependence of radiative non-radiative and photothermal properties of gold nanocrystals,” Int. Rev. Phys. Chem. 19(3), 409–453 (2000).
[Crossref]

Luger, A.

Y. Ivanenko, M. Gustafsson, B. L. G. Jonsson, A. Luger, B. Nilsson, S. Nordebo, and J. Toft, “Passive approximation and optimization using B-splines,” arXiv 1711.07937 (2017).

Lund, T.

T. Lund, M. F. Callaghan, P. Williams, M. Turmaine, C. Bachmann, T. Rademacher, I. M. Roitt, and R. Bayford, “The influence of ligand organization on the rate of uptake of gold nanoparticles by colorectal cancer cells,” Biomaterials 32(36), 9776–9784 (2011).
[Crossref]

Mackeyev, Y.

S. J. Corr, M. Raoof, Y. Mackeyev, S. Phounsavath, M. A. Cheney, B. T. Cisneros, M. Shur, M. Gozin, P. J. McNally, L. J. Wilson, and S. A. Curley, “Citrate-capped gold nanoparticle electrophoretic heat production in response to a time-varying radiofrequency electric-field,” J. Phys. Chem. C 116(45), 24380–24389 (2012).
[Crossref]

Maier, S. A.

S. A. Maier, “Plasmonics: Fundamentals and Applications,” (Springer-Verlag, 2007).

McCoy, R. S.

C. B. Collins, R. S. McCoy, B. J. Ackerson, G. J. Collins, and C. J. Ackerson, “Radiofrequency heating pathways for gold nanoparticles,” Nanoscale 6(15), 8459–8472 (2014).
[Crossref]

McNally, P. J.

S. J. Corr, M. Raoof, Y. Mackeyev, S. Phounsavath, M. A. Cheney, B. T. Cisneros, M. Shur, M. Gozin, P. J. McNally, L. J. Wilson, and S. A. Curley, “Citrate-capped gold nanoparticle electrophoretic heat production in response to a time-varying radiofrequency electric-field,” J. Phys. Chem. C 116(45), 24380–24389 (2012).
[Crossref]

Miller, O. D.

Monreal, R. C.

G. W. Hanson, R. C. Monreal, and S. P. Apell, “Electromagnetic absorption mechanisms in metal nanospheres: Bulk and surface effects in radiofrequency-terahertz heating of nanoparticles,” J. Appl. Phys. 109(12), 124306 (2011).
[Crossref]

Nilsson, B.

Y. Ivanenko, M. Gustafsson, B. L. G. Jonsson, A. Luger, B. Nilsson, S. Nordebo, and J. Toft, “Passive approximation and optimization using B-splines,” arXiv 1711.07937 (2017).

Nordebo, S.

M. Dalarsson, S. Nordebo, D. Sjöberg, and R. Bayford, “Absorption and optimal plasmonic resonances for small ellipsoidal particles in lossy media,” J. Phys. D: Appl. Phys. 50(34), 345401 (2017).
[Crossref]

S. Nordebo, M. Dalarsson, Y. Ivanenko, D. Sjöberg, and R. Bayford, “On the physical limitations for radio frequency absorption in gold nanoparticle suspensions,” J. Phys. D: Appl. Phys. 50(15), 155401 (2017).
[Crossref]

Y. Ivanenko, M. Gustafsson, B. L. G. Jonsson, A. Luger, B. Nilsson, S. Nordebo, and J. Toft, “Passive approximation and optimization using B-splines,” arXiv 1711.07937 (2017).

Y. Ivanenko, M. Dalarsson, S. Nordebo, and R. Bayford, “On the Plasmonic Resonances in a Layered Waveguide Structure,” in Proceedings of The 12th International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials’ 2018 (2018).

Norgren, M.

M. Dalarsson, M. Norgren, and Z. Jakšić, “Exact analytical solution for fields in a lossy cylindrical structure with linear gradient index metamaterials,” Prog. Electromagn. Res. 151, 109–117 (2015).
[Crossref]

M. Dalarsson, M. Norgren, T. Asenov, N. Dončov, and Z. Jakšić, “Exact analytical solution for fields in gradient index metamaterials with different loss factors in negative and positive refractive index segments,” J. Nanophotonics 7(1), 073086 (2013).
[Crossref]

M. Dalarsson, M. Norgren, N. Dončov, and Z. Jakšić, “Lossy gradient index transmission optics with arbitrary periodic permittivity and permeability and constant impedance throughout the structure,” J. Opt. 14(6), 065102 (2012).
[Crossref]

M. Dalarsson, M. Norgren, and Z. Jakšić, “Lossy gradient index metamaterial with sinusoidal periodicity of refractive index: case of constant impedance throughout the structure,” J. Nanophotonics 5(1), 051804 (2011).
[Crossref]

O’Neil, B. E.

E. Sassaroli, K. C. P. Li, and B. E. O’Neil, “Radio frequency absorption in gold nanoparticle suspensions: a phenomenological study,” J. Phys. D: Appl. Phys. 45(7), 075303 (2012).
[Crossref]

Phounsavath, S.

S. J. Corr, M. Raoof, Y. Mackeyev, S. Phounsavath, M. A. Cheney, B. T. Cisneros, M. Shur, M. Gozin, P. J. McNally, L. J. Wilson, and S. A. Curley, “Citrate-capped gold nanoparticle electrophoretic heat production in response to a time-varying radiofrequency electric-field,” J. Phys. Chem. C 116(45), 24380–24389 (2012).
[Crossref]

Polimeridis, A. G.

Rademacher, T.

T. Lund, M. F. Callaghan, P. Williams, M. Turmaine, C. Bachmann, T. Rademacher, I. M. Roitt, and R. Bayford, “The influence of ligand organization on the rate of uptake of gold nanoparticles by colorectal cancer cells,” Biomaterials 32(36), 9776–9784 (2011).
[Crossref]

Raoof, M.

S. J. Corr, M. Raoof, Y. Mackeyev, S. Phounsavath, M. A. Cheney, B. T. Cisneros, M. Shur, M. Gozin, P. J. McNally, L. J. Wilson, and S. A. Curley, “Citrate-capped gold nanoparticle electrophoretic heat production in response to a time-varying radiofrequency electric-field,” J. Phys. Chem. C 116(45), 24380–24389 (2012).
[Crossref]

Reid, M. T. H.

Roitt, I. M.

T. Lund, M. F. Callaghan, P. Williams, M. Turmaine, C. Bachmann, T. Rademacher, I. M. Roitt, and R. Bayford, “The influence of ligand organization on the rate of uptake of gold nanoparticles by colorectal cancer cells,” Biomaterials 32(36), 9776–9784 (2011).
[Crossref]

Sassaroli, E.

E. Sassaroli, K. C. P. Li, and B. E. O’Neil, “Radio frequency absorption in gold nanoparticle suspensions: a phenomenological study,” J. Phys. D: Appl. Phys. 45(7), 075303 (2012).
[Crossref]

Shur, M.

S. J. Corr, M. Raoof, Y. Mackeyev, S. Phounsavath, M. A. Cheney, B. T. Cisneros, M. Shur, M. Gozin, P. J. McNally, L. J. Wilson, and S. A. Curley, “Citrate-capped gold nanoparticle electrophoretic heat production in response to a time-varying radiofrequency electric-field,” J. Phys. Chem. C 116(45), 24380–24389 (2012).
[Crossref]

Sjöberg, D.

M. Dalarsson, S. Nordebo, D. Sjöberg, and R. Bayford, “Absorption and optimal plasmonic resonances for small ellipsoidal particles in lossy media,” J. Phys. D: Appl. Phys. 50(34), 345401 (2017).
[Crossref]

S. Nordebo, M. Dalarsson, Y. Ivanenko, D. Sjöberg, and R. Bayford, “On the physical limitations for radio frequency absorption in gold nanoparticle suspensions,” J. Phys. D: Appl. Phys. 50(15), 155401 (2017).
[Crossref]

Soljacic, M.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables,” (Dover Books, New York, 1965).

Tassin, P.

Toft, J.

Y. Ivanenko, M. Gustafsson, B. L. G. Jonsson, A. Luger, B. Nilsson, S. Nordebo, and J. Toft, “Passive approximation and optimization using B-splines,” arXiv 1711.07937 (2017).

Turmaine, M.

T. Lund, M. F. Callaghan, P. Williams, M. Turmaine, C. Bachmann, T. Rademacher, I. M. Roitt, and R. Bayford, “The influence of ligand organization on the rate of uptake of gold nanoparticles by colorectal cancer cells,” Biomaterials 32(36), 9776–9784 (2011).
[Crossref]

Williams, P.

T. Lund, M. F. Callaghan, P. Williams, M. Turmaine, C. Bachmann, T. Rademacher, I. M. Roitt, and R. Bayford, “The influence of ligand organization on the rate of uptake of gold nanoparticles by colorectal cancer cells,” Biomaterials 32(36), 9776–9784 (2011).
[Crossref]

Wilson, L. J.

S. J. Corr, M. Raoof, Y. Mackeyev, S. Phounsavath, M. A. Cheney, B. T. Cisneros, M. Shur, M. Gozin, P. J. McNally, L. J. Wilson, and S. A. Curley, “Citrate-capped gold nanoparticle electrophoretic heat production in response to a time-varying radiofrequency electric-field,” J. Phys. Chem. C 116(45), 24380–24389 (2012).
[Crossref]

Appl. Phys. Lett. (1)

N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. 85(21), 5040–5042 (2004).
[Crossref]

Biomaterials (1)

T. Lund, M. F. Callaghan, P. Williams, M. Turmaine, C. Bachmann, T. Rademacher, I. M. Roitt, and R. Bayford, “The influence of ligand organization on the rate of uptake of gold nanoparticles by colorectal cancer cells,” Biomaterials 32(36), 9776–9784 (2011).
[Crossref]

Int. Rev. Phys. Chem. (1)

S. Link and M. A. El-Sayed, “Shape and size dependence of radiative non-radiative and photothermal properties of gold nanocrystals,” Int. Rev. Phys. Chem. 19(3), 409–453 (2000).
[Crossref]

J. Appl. Phys. (1)

G. W. Hanson, R. C. Monreal, and S. P. Apell, “Electromagnetic absorption mechanisms in metal nanospheres: Bulk and surface effects in radiofrequency-terahertz heating of nanoparticles,” J. Appl. Phys. 109(12), 124306 (2011).
[Crossref]

J. Nanophotonics (2)

M. Dalarsson, M. Norgren, and Z. Jakšić, “Lossy gradient index metamaterial with sinusoidal periodicity of refractive index: case of constant impedance throughout the structure,” J. Nanophotonics 5(1), 051804 (2011).
[Crossref]

M. Dalarsson, M. Norgren, T. Asenov, N. Dončov, and Z. Jakšić, “Exact analytical solution for fields in gradient index metamaterials with different loss factors in negative and positive refractive index segments,” J. Nanophotonics 7(1), 073086 (2013).
[Crossref]

J. Opt. (1)

M. Dalarsson, M. Norgren, N. Dončov, and Z. Jakšić, “Lossy gradient index transmission optics with arbitrary periodic permittivity and permeability and constant impedance throughout the structure,” J. Opt. 14(6), 065102 (2012).
[Crossref]

J. Phys. Chem. C (1)

S. J. Corr, M. Raoof, Y. Mackeyev, S. Phounsavath, M. A. Cheney, B. T. Cisneros, M. Shur, M. Gozin, P. J. McNally, L. J. Wilson, and S. A. Curley, “Citrate-capped gold nanoparticle electrophoretic heat production in response to a time-varying radiofrequency electric-field,” J. Phys. Chem. C 116(45), 24380–24389 (2012).
[Crossref]

J. Phys. D: Appl. Phys. (3)

E. Sassaroli, K. C. P. Li, and B. E. O’Neil, “Radio frequency absorption in gold nanoparticle suspensions: a phenomenological study,” J. Phys. D: Appl. Phys. 45(7), 075303 (2012).
[Crossref]

S. Nordebo, M. Dalarsson, Y. Ivanenko, D. Sjöberg, and R. Bayford, “On the physical limitations for radio frequency absorption in gold nanoparticle suspensions,” J. Phys. D: Appl. Phys. 50(15), 155401 (2017).
[Crossref]

M. Dalarsson, S. Nordebo, D. Sjöberg, and R. Bayford, “Absorption and optimal plasmonic resonances for small ellipsoidal particles in lossy media,” J. Phys. D: Appl. Phys. 50(34), 345401 (2017).
[Crossref]

Nanoscale (1)

C. B. Collins, R. S. McCoy, B. J. Ackerson, G. J. Collins, and C. J. Ackerson, “Radiofrequency heating pathways for gold nanoparticles,” Nanoscale 6(15), 8459–8472 (2014).
[Crossref]

Opt. Express (2)

Opt. Quantum Electron. (1)

M. Dalarsson and Z. Jakšić, “Exact analytical solution for fields in a lossy cylindrical structure with hyperbolic tangent gradient index metamaterials,” Opt. Quantum Electron. 48(3), 209 (2016).
[Crossref]

Phys. Med. Biol. (1)

S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz,” Phys. Med. Biol. 41(11), 2251–2269 (1996).
[Crossref]

Phys. Rev. A (1)

M. Dalarsson, “General theory of wave propagation through graded interfaces between positive- and negative-refractive-index media,” Phys. Rev. A 96(4), 043848 (2017).
[Crossref]

Prog. Electromagn. Res. (1)

M. Dalarsson, M. Norgren, and Z. Jakšić, “Exact analytical solution for fields in a lossy cylindrical structure with linear gradient index metamaterials,” Prog. Electromagn. Res. 151, 109–117 (2015).
[Crossref]

Other (4)

Y. Ivanenko, M. Gustafsson, B. L. G. Jonsson, A. Luger, B. Nilsson, S. Nordebo, and J. Toft, “Passive approximation and optimization using B-splines,” arXiv 1711.07937 (2017).

Y. Ivanenko, M. Dalarsson, S. Nordebo, and R. Bayford, “On the Plasmonic Resonances in a Layered Waveguide Structure,” in Proceedings of The 12th International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials’ 2018 (2018).

S. A. Maier, “Plasmonics: Fundamentals and Applications,” (Springer-Verlag, 2007).

M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables,” (Dover Books, New York, 1965).

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Figures (3)

Fig. 1.
Fig. 1. Hollow waveguide with a dielectric layer
Fig. 2.
Fig. 2. Three examples of permittivity functions changing from $\Re [\epsilon _{\mathrm {G}}] = 2$ to $\Re [\epsilon _{\mathrm {L}}] = 4$ and back for $z_0 = 0.1$ (black line), $z_0 = 0.2$ (red line) and $z_0 = 0.3$ (blue line). Here $\Re [\epsilon _{\mathrm {L}}\left (\omega \right )]\;>\;\Re [\epsilon _{\mathrm {G}}\left (\omega \right )]$ . Note, however, that this assumption is not essential for the present approach, and is used only for graphical illustration.
Fig. 3.
Fig. 3. Cross section of a rectangular waveguide with dimensions $a$ and $b$ such that $a\;>\;b$ .

Equations (43)

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ϵ ( ω , z ) = ϵ 0 ϵ R ( z ) = ϵ 0 { ϵ L ( ω ) [ ϵ L ( ω ) ϵ G ( ω ) ] tanh 2 ( z z 0 ) } ,
tanh 2 ( z z 0 ) 1 ϵ ( ω , ± ) = ϵ 0 ϵ G ( ω ) ,
tanh 2 ( z z 0 ) 0 ϵ ( ω , 0 ) = ϵ 0 ϵ L ( ω ) ,
× E = j ω μ 0 H , [ ϵ ( z ) E ] = 0 , × H = j ω ϵ ( z ) E , H = 0 .
2 E + ( 1 ϵ R d ϵ R d z E z ) + k 2 ϵ R ( z ) E = 0 ,
2 H + 1 ϵ R d ϵ R d z ( H z H z ) + k 2 ϵ R ( z ) H = 0 ,
2 E + k 2 ϵ R ( z ) E = 0 , 2 H z + k 2 ϵ R ( z ) H z = 0 .
H = j ω μ 0 × E .
2 E j + k 2 ϵ R ( z ) E j = 0 , j { x , y } .
( 2 x 2 + 2 y 2 ) F j + k T 2 F j = 0 , j { x , y } ,
d 2 Z d z 2 + [ k 2 ϵ R ( z ) k T 2 ] Z = 0 ,
F x = A ( n π b ) cos ( m π x a ) sin ( n π y b ) ,
F y = A ( m π a ) sin ( m π x a ) cos ( n π y b ) ,
k T 2 = ( m π a ) 2 + ( n π b ) 2 ,
k 2 { ϵ R ( z ) } k T 2 > 0 k 2 [ { ϵ R ( z ) } ] min k T 2 > 0 .
k 2 { ϵ G } k T 2 > 0 k 2 { ϵ G } > k T 2 ω 2 > k T 2 c 2 { ϵ G } ,
f > c 2 π k T { ϵ G } = f c , max .
d 2 Z d w 2 + ( D B tanh 2 w ) Z = 0 ,
D = ( k 2 ϵ L k T 2 ) z 0 2 , B = k 2 z 0 2 ( ϵ L ϵ G ) .
Z ( z ) = T exp ( 2 p z z 0 ) [ 1 + exp ( 2 z z 0 ) ] 2 p
2 F 1 { 2 p + 1 2 + r 2 + 1 4 , 2 p + 1 2 r 2 + 1 4 , 2 p + 1 ; [ 1 + exp ( 2 z z 0 ) ] 1 } ,
F ( a , b , c ; u ) = Γ ( c ) Γ ( a ) Γ ( b ) n = 0 Γ ( a + n ) Γ ( b + n ) Γ ( c + n ) u n n ! ,
p = j z 0 2 k 2 ϵ G k T 2 = j k z G z 0 2 , r = k z 0 ϵ L ϵ G ,
u = [ 1 + exp ( 2 z z 0 ) ] 1 0 for z + .
Z ( z ) T exp ( j k z G z ) for z + ,
u = [ 1 + exp ( 2 z z 0 ) ] 1 1 for z .
F ( a , b , c ; u ) = Γ ( c ) Γ ( c a b ) Γ ( c a ) Γ ( c b ) F ( a , b , a + b c + 1 ; 1 u )
+ ( 1 u ) c a b Γ ( c ) Γ ( a + b c ) Γ ( a ) Γ ( b ) F ( c a , c b , c a b + 1 ; 1 u ) ,
Z ( z ) T Γ ( c ) Γ ( c a b ) Γ ( c a ) Γ ( c b ) exp ( + j k z G z )
+ T Γ ( c ) Γ ( a + b c ) Γ ( a ) Γ ( b ) exp ( j k z G z ) for z .
Z ( z ) exp ( j k z G z ) + R exp ( + j k z G z ) for z ,
a = 2 p + 1 2 + r 2 + 1 4 , b = 2 p + 1 2 r 2 + 1 4 , c = 2 p + 1 .
T = Γ ( a ) Γ ( b ) Γ ( c ) Γ ( a + b c ) , R = Γ ( a ) Γ ( b ) Γ ( c a ) Γ ( c b ) Γ ( c a b ) Γ ( a + b c ) .
E x = A T ( n π b ) cos ( m π x a ) sin ( n π y b ) exp ( 2 p z z 0 ) [ 1 + exp ( 2 z z 0 ) ] 2 p
2 F 1 { 2 p + 1 2 + r 2 + 1 4 , 2 p + 1 2 r 2 + 1 4 , 2 p + 1 ; [ 1 + exp ( 2 z z 0 ) ] 1 } ,
E y = A T ( m π a ) sin ( m π x a ) cos ( n π y b ) exp ( 2 p z z 0 ) [ 1 + exp ( 2 z z 0 ) ] 2 p
2 F 1 { 2 p + 1 2 + r 2 + 1 4 , 2 p + 1 2 r 2 + 1 4 , 2 p + 1 ; [ 1 + exp ( 2 z z 0 ) ] 1 } ,
T = 1 + j k z L z 0 ϵ L ϵ G ϵ L k T 2 / k 2 ϵ G k T 2 / k 2 + O { z 0 2 } ,
R = j k z L z 0 ϵ L ϵ G ϵ L k T 2 / k 2 ϵ G k T 2 / k 2 + O { z 0 2 } ,
T 11 ( 2 ) = 2 j k z L ( 2 z 0 ) S 11 ( 1 ) 1 ( S 11 ( 1 ) ) 2 + O { z 0 2 } ,
S 11 ( 1 ) = k z G k z L k z G + k z L .
T 11 ( 2 ) = 2 j k z L ( 2 z 0 ) k z G 2 k z L 2 4 k z G k z L + O { z 0 2 } ,
T 11 ( 2 ) = R = j k z L z 0 ϵ L ϵ G ϵ L k T 2 / k 2 ϵ G k T 2 / k 2 + O { z 0 2 } .

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