Abstract

Based on the ability of plane structures to simultaneously optimize the propagation, confinement, and energy of surface plasmon-polaritons or surface phonon-polaritons, we develop the polaritonic figure of merit Z = βRΛ2/δ, where βR, Λ and δ are the longitudinal wave vector, propagation length, and penetration depth, respectively. Explicit and analytical expressions of Z are derived for a single interface and a suspended thin film, as functions of the material permittivities and the film thickness. Higher Z are obtained for thinner films and smaller energy losses. The application of the obtained results for a SiC-air interface and a SiC thin film suspended in air shows that both structures are able to maximize the presence of polaritons at a frequency near to, but different than that at which the real part of the SiC permittivity exhibits a dip. Furthermore, using the temperature change of this dip, we show that the propagation length, confinement and energy of polaritons increases with its deepness, which provides an effective way to enhance the overall Z of polaritonic structures.

© 2017 Optical Society of America

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References

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    [Crossref]

2016 (1)

Z. Chen, X. Zhao, C. Lin, S. Chen, L. Yin, and Y. Ding, “Figure of merit enhancement of surface plasmon resonance sensors using absentee layer,” Appl. Optics 55, 6832 (2016).
[Crossref]

2015 (5)

V. E. Babicheva, M. Y. Shalaginov, S. Ishii, A. Boltasseva, and A. V. Kildishev, “Long-range plasmonic waveguides with hyperbolic cladding,” Opt. Express 23, 31109 (2015).
[Crossref] [PubMed]

K. Joulain, Y. Ezzahri, J. Drevillon, B. Rousseau, and D. De Sousa Meneses, “Radiative thermal rectification between SiC and SiO2,” Opt. Express 23, A1388–A1397 (2015).
[Crossref] [PubMed]

J. Ordonez-Miranda, L. Tranchant, S. Gluchko, and S. Volz, “Energy transport of surface phonon polaritons propagating along a chain of spheroidal nanoparticles,” Phys. Rev. B 92, 115409 (2015).
[Crossref]

P. Li, M. Lewin, A. V. Kretinin, J. D. Caldwell, K. S. Novoselov, T. Taniguchi, K. Watanabe, F. Gaussmann, and T. Taubner, “Hyperbolic phonon-polaritons in boron nitride for near-field optical imaging and focusing,” Nat. Commun. 6, 7507–7515 (2015).
[Crossref] [PubMed]

S. Gluchko, J. Ordonez-Miranda, L. Tranchant, T. Antoni, and S. Volz, “Focusing of surface phonon-polaritons along conical and wedge polar nanostructures,” J. Appl. Phys. 118, 064301 (2015).
[Crossref]

2014 (4)

J. Ordonez-Miranda, L. Tranchant, B. Kim, Y. Chalopin, T. Antoni, and S. Volz, “Quantized thermal conductance of nanowires at room temperature due to Zenneck surface-phonon polaritons,” Phys. Rev. Lett. 112, 055901 (2014).
[Crossref] [PubMed]

J. Ordonez-Miranda, L. Tranchant, B. Kim, Y. Chalopin, T. Antoni, and S. Volz, “Effects of anisotropy and size of polar nano thin films on their thermal conductivity due to surface phonon-polaritons,” Appl. Phys. Express 7, 035201 (2014).
[Crossref]

J. Ordonez-Miranda, L. Tranchant, T. Antoni, Y. Chalopin, and S. Volz, “Thermal conductivity of nano-layered systems due to surface phonon-polaritons,” J. Appl. Phys. 115, 054311 (2014).
[Crossref]

X. G. Xu, B. G. Ghamsari, J.-H. Jiang, L. Gilburd, G. O. Andreev, C. Zhi, Y. Bando, D. Golberg, P. Berini, and G. C. Walker, “One-dimensional surface phonon polaritons in boron nitride nanotubes,” Nat. Commun. 5, 4782 (2014).
[Crossref] [PubMed]

2013 (3)

J. Ordonez-Miranda, L. Tranchant, T. Tokunaga, B. Kim, B. Palpant, Y. Chalopin, T. Antoni, and S. Volz, “Anomalous thermal conductivity by surface phonon-polaritons of polar nano thin films due to their asymmetric surrounding media,” J. Appl. Phys. 113, 084311 (2013).
[Crossref]

J. Kim, G. V. Naik, A. V. Gavrilenko, K. D. Vladimir, I. Gavrilenko, S. M. Prokes, O. J. Glembocki, V. M. Shalaev, and A. Boltasseva, “Optical properties of gallium-doped zinc oxide–a low-loss plasmonic material: first-principles theory and experiment,” Phys. Rev. X 3, 041037 (2013).

A. Babuty, K. Joulain, P. O. Chapuis, J. J. Greffet, and Y. De Wilde, “Blackbody spectrum revisited in the near field,” Phys. Rev. Lett. 110, 146103 (2013).
[Crossref] [PubMed]

2012 (1)

Y. Chalopin, H. Dammak, M. Hayoun, M. Besbes, and J. J. Greffet, “Size-dependent infrared properties of MgO nanoparticles with evidence of screening effect,” Appl. Phys. Lett. 100, 241904 (2012).
[Crossref]

2010 (5)

N. Stojanovic, D. H. S. Maithripala, J. M. Berg, and M. Holtz, “Thermal conductivity in metallic nanostructures at high temperature: electrons, phonons, and the Wiedemann-Franz law,” Phys. Rev. B 82, 075418 (2010).
[Crossref]

H. C. Kim and X. Cheng, “Infrared dipole antenna enhanced by surface phonon polaritons,” Opt. Lett. 35, 3748–3750 (2010).
[Crossref] [PubMed]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

E. Berglind, L. Thylen, and L. Liu, “Plasmonic/metallic passive waveguides and waveguide components for photonic dense integrated circuits: a feasibility study based on microwave engineering,” I. E. T. Optoelectron. 4, 1–16 (2010).

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photon. 4, 83–91 (2010).
[Crossref]

2009 (1)

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J. J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photon. 3, 514–517 (2009).
[Crossref]

2008 (4)

P. Ben-Abdallah, K. Joulain, J. Drevillon, and C. Le Goff, “Heat transport through plasmonic interactions in closely spaced metallic nanoparticle chains,” Phys. Rev. B 77, 075417 (2008).
[Crossref]

M. Francoeur, M. P. Menguc, and R. Vaillon, “Near-field radiative heat transfer enhancement via surface phonon polaritons coupling in thin films,” Appl. Phys. Lett. 93, 043109 (2008).
[Crossref]

P. O. Chapuis, M. Laroche, S. Volz, and J. J. Greffet, “Radiative heat transfer between metallic nanoparticles,” Appl. Phys. Lett. 92, 201906 (2008).
[Crossref]

P. O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J. J. Greffet, “Effects of spatial dispersion in near-field radiative heat transfer between two parallel metallic surfaces,” Phys. Rev. B 92, 035431 (2008).
[Crossref]

2007 (3)

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–50 (2007).
[Crossref]

D.-Z. A. Chen and G. Chen, “Measurement of silicon dioxide surface phonon-polariton propagation length by attenuated total reflection,” Appl. Phys. Lett. 91, 121906 (2007).
[Crossref]

R. Buckley and P. Berini, “Figures of merit for 2D surface plasmon waveguides and application to metal stripes,” Opt. Express 15, 12174 (2007).
[Crossref] [PubMed]

2006 (5)

L. Thylen and E. Berglind, “Nanophotonics and negative epsilon materials,” J. Zheijiang University: Science A 7, 41–44 (2006).
[Crossref]

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J. J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444, 740–743 (2006).
[Crossref] [PubMed]

S. A. Maier, “Effective mode volume of nanoscale plasmon cavities,” Opt. Quantum Electron. 38, 257–267 (2006).
[Crossref]

S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14, 1957 (2006).
[Crossref] [PubMed]

P. Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express 14, 13030 (2006).
[Crossref] [PubMed]

2005 (2)

D.-Z. A. Chen, A. Narayanaswamy, and G. Chen, “Surface phonon-polariton mediated thermal conductivity enhancement of amorphous thin films,” Phys. Rev. B 72, 155435 (2005).
[Crossref]

L. J. Sherry, S.-H. Chang, G. C. Schatz, R. P. Van Duyne, B. J. Wiley, and Y. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett. 5, 2034–2038 (2005).
[Crossref] [PubMed]

2004 (3)

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 (2004).
[Crossref]

N. Ocelic and R. Hillenbrand, “Subwavelength-scale tailoring of surface phonon polaritons by focused ion-beam implantation,” Nat. Mater. 3, 606–609 (2004).
[Crossref] [PubMed]

T. Taubner, F. Keilmann, and R. Hillenbrand, “Nanomechanical resonance tuning and phase effects in optical near-field interaction,” Nano Lett. 4, 1669–1672 (2004).
[Crossref]

2003 (3)

T. Feurer, J. C. Vaughan, and K. A. Nelson, “Spatiotemporal coherent control of lattice vibrational waves,” Science 299, 374–377 (2003).
[Crossref] [PubMed]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, “Phonon-polariton excitations in photonic crystals,” Phys. Rev. B 68, 075209 (2003).
[Crossref]

2002 (1)

J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
[Crossref] [PubMed]

2000 (2)

P. Berini, “Plasmon polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000).
[Crossref]

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61, 10484 (2000).
[Crossref]

1993 (1)

E. L. Albuquerque and M. G. Cottam, “Superlattice plasmon-polaritons,” Phys. Reports 233, 67–135 (1993).
[Crossref]

1991 (2)

F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range surface modes supported by thin films,” Phys. Rev. B 44, 5855–5872 (1991).
[Crossref]

M. N. Zervas, “Surface plasmon-polariton waves guided by thin metal films,” Opt. Lett. 16, 720–722 (1991).
[Crossref] [PubMed]

1969 (1)

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969).
[Crossref]

Albuquerque, E. L.

E. L. Albuquerque and M. G. Cottam, “Superlattice plasmon-polaritons,” Phys. Reports 233, 67–135 (1993).
[Crossref]

Andreev, G. O.

X. G. Xu, B. G. Ghamsari, J.-H. Jiang, L. Gilburd, G. O. Andreev, C. Zhi, Y. Bando, D. Golberg, P. Berini, and G. C. Walker, “One-dimensional surface phonon polaritons in boron nitride nanotubes,” Nat. Commun. 5, 4782 (2014).
[Crossref] [PubMed]

Antoni, T.

S. Gluchko, J. Ordonez-Miranda, L. Tranchant, T. Antoni, and S. Volz, “Focusing of surface phonon-polaritons along conical and wedge polar nanostructures,” J. Appl. Phys. 118, 064301 (2015).
[Crossref]

J. Ordonez-Miranda, L. Tranchant, B. Kim, Y. Chalopin, T. Antoni, and S. Volz, “Quantized thermal conductance of nanowires at room temperature due to Zenneck surface-phonon polaritons,” Phys. Rev. Lett. 112, 055901 (2014).
[Crossref] [PubMed]

J. Ordonez-Miranda, L. Tranchant, B. Kim, Y. Chalopin, T. Antoni, and S. Volz, “Effects of anisotropy and size of polar nano thin films on their thermal conductivity due to surface phonon-polaritons,” Appl. Phys. Express 7, 035201 (2014).
[Crossref]

J. Ordonez-Miranda, L. Tranchant, T. Antoni, Y. Chalopin, and S. Volz, “Thermal conductivity of nano-layered systems due to surface phonon-polaritons,” J. Appl. Phys. 115, 054311 (2014).
[Crossref]

J. Ordonez-Miranda, L. Tranchant, T. Tokunaga, B. Kim, B. Palpant, Y. Chalopin, T. Antoni, and S. Volz, “Anomalous thermal conductivity by surface phonon-polaritons of polar nano thin films due to their asymmetric surrounding media,” J. Appl. Phys. 113, 084311 (2013).
[Crossref]

Atwater, H. A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

Babicheva, V. E.

Babuty, A.

A. Babuty, K. Joulain, P. O. Chapuis, J. J. Greffet, and Y. De Wilde, “Blackbody spectrum revisited in the near field,” Phys. Rev. Lett. 110, 146103 (2013).
[Crossref] [PubMed]

Bando, Y.

X. G. Xu, B. G. Ghamsari, J.-H. Jiang, L. Gilburd, G. O. Andreev, C. Zhi, Y. Bando, D. Golberg, P. Berini, and G. C. Walker, “One-dimensional surface phonon polaritons in boron nitride nanotubes,” Nat. Commun. 5, 4782 (2014).
[Crossref] [PubMed]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

Ben-Abdallah, P.

P. Ben-Abdallah, K. Joulain, J. Drevillon, and C. Le Goff, “Heat transport through plasmonic interactions in closely spaced metallic nanoparticle chains,” Phys. Rev. B 77, 075417 (2008).
[Crossref]

Berg, J. M.

N. Stojanovic, D. H. S. Maithripala, J. M. Berg, and M. Holtz, “Thermal conductivity in metallic nanostructures at high temperature: electrons, phonons, and the Wiedemann-Franz law,” Phys. Rev. B 82, 075418 (2010).
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S. Gluchko, J. Ordonez-Miranda, L. Tranchant, T. Antoni, and S. Volz, “Focusing of surface phonon-polaritons along conical and wedge polar nanostructures,” J. Appl. Phys. 118, 064301 (2015).
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J. Ordonez-Miranda, L. Tranchant, B. Kim, Y. Chalopin, T. Antoni, and S. Volz, “Effects of anisotropy and size of polar nano thin films on their thermal conductivity due to surface phonon-polaritons,” Appl. Phys. Express 7, 035201 (2014).
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J. Ordonez-Miranda, L. Tranchant, T. Antoni, Y. Chalopin, and S. Volz, “Thermal conductivity of nano-layered systems due to surface phonon-polaritons,” J. Appl. Phys. 115, 054311 (2014).
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J. Ordonez-Miranda, L. Tranchant, B. Kim, Y. Chalopin, T. Antoni, and S. Volz, “Quantized thermal conductance of nanowires at room temperature due to Zenneck surface-phonon polaritons,” Phys. Rev. Lett. 112, 055901 (2014).
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L. J. Sherry, S.-H. Chang, G. C. Schatz, R. P. Van Duyne, B. J. Wiley, and Y. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett. 5, 2034–2038 (2005).
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Sherry, L. J.

L. J. Sherry, S.-H. Chang, G. C. Schatz, R. P. Van Duyne, B. J. Wiley, and Y. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett. 5, 2034–2038 (2005).
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E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J. J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photon. 3, 514–517 (2009).
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J. Ordonez-Miranda, L. Tranchant, S. Gluchko, and S. Volz, “Energy transport of surface phonon polaritons propagating along a chain of spheroidal nanoparticles,” Phys. Rev. B 92, 115409 (2015).
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J. Ordonez-Miranda, L. Tranchant, B. Kim, Y. Chalopin, T. Antoni, and S. Volz, “Effects of anisotropy and size of polar nano thin films on their thermal conductivity due to surface phonon-polaritons,” Appl. Phys. Express 7, 035201 (2014).
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J. Ordonez-Miranda, L. Tranchant, T. Antoni, Y. Chalopin, and S. Volz, “Thermal conductivity of nano-layered systems due to surface phonon-polaritons,” J. Appl. Phys. 115, 054311 (2014).
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J. Ordonez-Miranda, L. Tranchant, T. Tokunaga, B. Kim, B. Palpant, Y. Chalopin, T. Antoni, and S. Volz, “Anomalous thermal conductivity by surface phonon-polaritons of polar nano thin films due to their asymmetric surrounding media,” J. Appl. Phys. 113, 084311 (2013).
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S. Gluchko, J. Ordonez-Miranda, L. Tranchant, T. Antoni, and S. Volz, “Focusing of surface phonon-polaritons along conical and wedge polar nanostructures,” J. Appl. Phys. 118, 064301 (2015).
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J. Ordonez-Miranda, L. Tranchant, S. Gluchko, and S. Volz, “Energy transport of surface phonon polaritons propagating along a chain of spheroidal nanoparticles,” Phys. Rev. B 92, 115409 (2015).
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J. Ordonez-Miranda, L. Tranchant, T. Antoni, Y. Chalopin, and S. Volz, “Thermal conductivity of nano-layered systems due to surface phonon-polaritons,” J. Appl. Phys. 115, 054311 (2014).
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J. Ordonez-Miranda, L. Tranchant, B. Kim, Y. Chalopin, T. Antoni, and S. Volz, “Effects of anisotropy and size of polar nano thin films on their thermal conductivity due to surface phonon-polaritons,” Appl. Phys. Express 7, 035201 (2014).
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J. Ordonez-Miranda, L. Tranchant, B. Kim, Y. Chalopin, T. Antoni, and S. Volz, “Quantized thermal conductance of nanowires at room temperature due to Zenneck surface-phonon polaritons,” Phys. Rev. Lett. 112, 055901 (2014).
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L. J. Sherry, S.-H. Chang, G. C. Schatz, R. P. Van Duyne, B. J. Wiley, and Y. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett. 5, 2034–2038 (2005).
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Appl. Optics (1)

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J. Ordonez-Miranda, L. Tranchant, B. Kim, Y. Chalopin, T. Antoni, and S. Volz, “Effects of anisotropy and size of polar nano thin films on their thermal conductivity due to surface phonon-polaritons,” Appl. Phys. Express 7, 035201 (2014).
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J. Appl. Phys. (3)

J. Ordonez-Miranda, L. Tranchant, T. Tokunaga, B. Kim, B. Palpant, Y. Chalopin, T. Antoni, and S. Volz, “Anomalous thermal conductivity by surface phonon-polaritons of polar nano thin films due to their asymmetric surrounding media,” J. Appl. Phys. 113, 084311 (2013).
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J. Ordonez-Miranda, L. Tranchant, T. Antoni, Y. Chalopin, and S. Volz, “Thermal conductivity of nano-layered systems due to surface phonon-polaritons,” J. Appl. Phys. 115, 054311 (2014).
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P. Li, M. Lewin, A. V. Kretinin, J. D. Caldwell, K. S. Novoselov, T. Taniguchi, K. Watanabe, F. Gaussmann, and T. Taubner, “Hyperbolic phonon-polaritons in boron nitride for near-field optical imaging and focusing,” Nat. Commun. 6, 7507–7515 (2015).
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Nat. Mater. (2)

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
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E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J. J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photon. 3, 514–517 (2009).
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J. Ordonez-Miranda, L. Tranchant, B. Kim, Y. Chalopin, T. Antoni, and S. Volz, “Quantized thermal conductance of nanowires at room temperature due to Zenneck surface-phonon polaritons,” Phys. Rev. Lett. 112, 055901 (2014).
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[Crossref]

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[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Scheme of the thin-film nanostructure under consideration. The red line represents the profile of polaritions propagating along the film interfaces.
Fig. 2
Fig. 2 Real and imaginary parts of the relative permittivity ε2 = εR + I of SiC at two temperatures, as functions of frequency.
Fig. 3
Fig. 3 Frequency dependence of the (a) longitudinal wave vector, (b) propagation length, and (c) FOM Ze of SPhPs propagation along an interface of SiC and a dielectric of relative permittivity ε1 = 1, 2, 4.
Fig. 4
Fig. 4 Figures of merit (a) Zp and (b) Z as functions of the excitation frequency. Calculations are done for two temperatures and three dielectric permittivities.
Fig. 5
Fig. 5 (a) Wave vector and (b) propagation length of SPhPs propagating along the interface of a thin film suspended in air (ε1 = 1). Calculations are done for two temperatures and three film thicknesses.
Fig. 6
Fig. 6 (a) Figure of merit Ze and (b) quality factor Q as functions of the excitation frequency. Calculations are done for two temperatures and three film thicknesses.
Fig. 7
Fig. 7 Figures of merit (a) Zp and (b) Z as functions of the excitation frequency of polaritons propagating along a suspended SiC thin film. Calculations are done for two temperatures and three film thicknesses.
Fig. 8
Fig. 8 Optimal frequencies at which the figures of merit Ze, Zp, and Z take their maxima, as functions of the SiC thin film thickness. Calculations are done for T = 295 K.

Equations (27)

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κ = 1 4 π d ω Λ Re ( β ) f T d ω ,
Z e = Λ Re ( β ) = 2 π Λ λ pol = Re ( β ) 2 Im ( β ) .
Z p = Λ δ = Re ( p 2 ) Im ( β ) ,
Z = Z e Z p = Re ( β ) Λ 2 δ = Re ( β ) Re ( β 2 ε 2 k 0 2 ) 2 Im ( β ) 2 .
Q = ω τ = ω Λ Re ( β ) ω ,
β = k 0 ε 1 ε 2 ε 1 + ε 2 ,
p 1 = k 0 ε 1 ε 1 ε 2 ,
p 2 = k 0 ε 2 ε 1 ε 2 .
Z e = χ 2 2 ε 1 ε I ,
Z p = χ ε 1 ε 1 ε I ( ε I η ε R η + ) ,
Z = χ 3 2 ε 1 ( ε 1 ε I ) 2 ( ε I η ε R η + ) .
p 1 ε 1 + p 2 ε 2 tanh ( p 2 d 2 ) = 0 .
ε ε 1 ε 1 + ε ε 2 ε 2 ρ ( 1 ε ε 2 3 ρ 2 ) = 0 ,
ε = ε 1 + ε ( 2 ) ρ 2 + ε ( 4 ) ρ 4 .
β = k 0 ε 1 ( 1 + β ( 2 ) ρ 2 + β ( 4 ) ρ 4 ) ,
p 1 = k 0 ε 1 ( 1 ε 12 ) ( ρ + p 1 ( 3 ) ρ 3 ) ,
p 2 = k 0 ε 1 ε 2 ( 1 + p 2 ( 2 ) ρ 2 + p 2 ( 4 ) ρ 4 ) ,
Z e = Z e ( 0 ) ( 1 ρ 2 + ξ e ( 2 ) + ξ e ( 4 ) ρ 2 ) ,
Z p = Z p ( 0 ) ( 1 ρ 2 + ξ p ( 2 ) + ξ p ( 4 ) ρ 2 ) ,
Z = Z ( 0 ) ( 1 ρ 4 + ξ ( 2 ) ρ 2 + ξ ( 4 ) ) .
Z e ( 0 ) = 1 2 ε I ε 1 2 | ε 2 | 6 | ε 2 | 4 ε R ε 1 3 ,
Z p ( 0 ) = | ε 1 ε 2 | + ε 1 ε R 2 ε 1 ε I ε 1 2 | ε 2 | 6 | ε 2 | 4 ε R ε 1 3 ,
ξ e ( 2 ) = Re ( β ( 2 ) ) Im ( β ( 4 ) ) Im ( β ( 2 ) ) ,
ξ e ( 4 ) = Re ( β ( 4 ) ) Im ( β ( 4 ) ) Im ( β ( 2 ) ) ξ e ( 2 ) ,
ξ p ( 2 ) = Re ( p 2 ( 2 ) ε 1 ε 2 ) Re ( ε 1 ε 2 ) Im ( β ( 4 ) ) Im ( β ( 2 ) ) ,
ξ p ( 4 ) = Re ( p 2 ( 4 ) ε 1 ε 2 ) Re ( ε 1 ε 2 ) Im ( β ( 4 ) ) Im ( β ( 2 ) ) ξ p ( 2 ) .
ε 2 ( ω ) = ε ( 1 + ω L 2 ω T 2 ω T 2 ω 2 i Γ ω ) ,

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