Abstract

Sub-diffraction limited waveguides have been studied as a means to manipulate light into nanoscale regions. Hybrid waveguides are popular candidates in optical regimes for subwavelength confinement and long range propagation. However, advances in the mid-IR are lacking due to high propagation losses and limited confinement. Here we present the first analysis of hybrid phononic waveguide using a hyperbolic material h-BN to generate surface phonon polaritons. The strong coupling between the photonic cylinder and phononic surface enhances the confined field up to 10−3 λo2o is free-space wavelength) and enables propagation distances up to 100 λo. Our work is fully compatible with integrated polaritonic devices in the mid-IR and provides a systematic approach to design hybrid phononic waveguides.

© 2016 Optical Society of America

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2016 (3)

F. Bigourdan, J. P. Hugonin, F. Marquier, C. Sauvan, and J. J. Greffet, “Nanoantenna for electrical generation of surface plasmon polaritons,” Phys. Rev. Lett. 116(10), 106803 (2016).
[Crossref] [PubMed]

T. J. Echtermeyer, S. Milana, U. Sassi, A. Eiden, M. Wu, E. Lidorikis, and A. C. Ferrari, “Surface plasmon polariton Graphene photodetectors,” Nano Lett. 16(1), 8–20 (2016).
[Crossref] [PubMed]

Y. Wu, L. Jiang, H. Xu, X. Dai, Y. Xiang, and D. Fan, “Hybrid nonlinear surface-phonon-plasmon-polaritons at the interface of nolinear medium and Graphene-covered hexagonal Boron Nitride crystal,” Opt. Express 24(3), 2109–2124 (2016).
[Crossref] [PubMed]

2015 (9)

J. D. Caldwell, L. Lindsay, V. Giannini, I. Vurgaftman, T. L. Reinecke, S. A. Maier, and O. J. Glembocki, “Low-loss, infrared and terahertz surface phonon polaritons,” Nanophotonics 4, 44–68 (2015).
[Crossref]

E. Yoxall, M. Schnell, A. Y. Nikitin, O. Txoperena, A. Woessner, M. B. Lundeberg, F. Casanova, L. E. Hueso, F. H. L. Koppens, and R. Hillenbrand, “Direct observation of ultraslow hyperbolic polariton propagation with negative phase velocity,” Nat. Photonics 9(10), 674–678 (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 (2015).
[Crossref] [PubMed]

S. Dai, Q. Ma, M. K. Liu, T. Andersen, Z. Fei, M. D. Goldflam, M. Wagner, K. Watanabe, T. Taniguchi, M. Thiemens, F. Keilmann, G. C. A. A. Janssen, S.-E. Zhu, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, “Graphene on hexagonal Boron Nitride as a tunable hyperbolic metamaterial,” Nat. Nanotechnol. 10(8), 682–686 (2015).
[Crossref] [PubMed]

Y. Jia, H. Zhao, Q. Guo, X. Wang, H. Wang, and F. Xia, “Tunable plasmon-phonon polaritons in layered Graphene-hexagonal Boron Nitride heterostructures,” ACS Photonics 2(7), 907–912 (2015).
[Crossref]

I. D. Barcelos, A. R. Cadore, L. C. Campos, A. Malachias, K. Watanabe, T. Taniguchi, F. C. B. Maia, R. Freitas, and C. Deneke, “Graphene/h-BN plasmon-phonon coupling and plasmon delocalization observed by infrared nano-spectroscopy,” Nanoscale 7(27), 11620–11625 (2015).
[Crossref] [PubMed]

Y. Fang and M. Sun, “Nanoplasmonic waveguides: towards applications in integrated nanophotonics circuits,” Light Sci. Appl. 4(6), e294 (2015).
[Crossref]

K. Feng, W. Streyer, S. M. Islam, J. Verma, D. Jena, D. Wasserman, and A. J. Hoffman, “Localized surface phonon polariton resonances in polar Gallium Nitride,” Appl. Phys. Lett. 107(8), 081108 (2015).
[Crossref]

Z. Shi, H. A. Bechtel, S. Berweger, Y. Sun, B. Zeng, C. Jin, H. Chang, M. C. Martin, M. B. Raschke, and F. Wang, “Amplitude- and phase- resolved nanospectral imaging of phonon polaritons in hexagonal Boron Nitride,” ACS Photonics 2(7), 790–796 (2015).
[Crossref]

2014 (6)

Y. Bian and Q. Gong, “Tuning the hybridization of plasmonic and coupled dielectric nanowire modes for high-performance optical waveguiding at sub-diffraction-limited scale,” Sci. Rep. 4, 6617 (2014).
[Crossref] [PubMed]

A. Woessner, M. B. Lundeberg, Y. Gao, A. Principi, P. Alonso-González, M. Carrega, K. Watanabe, T. Taniguchi, G. Vignale, M. Polini, J. Hone, R. Hillenbrand, and F. H. L. Koppens, “Highly confined low-loss plasmons in Graphene-Boron Nitride heterostructures,” Nat. Mater. 14(4), 421–425 (2014).
[Crossref] [PubMed]

V. W. Brar, M. S. Jang, M. Sherrott, S. Kim, J. J. Lopez, L. B. Kim, M. Choi, and H. Atwater, “Hybrid surface-phonon-plasmon polariton modes in Graphene/monolayer h-BN heterostructures,” Nano Lett. 14(7), 3876–3880 (2014).
[Crossref] [PubMed]

J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014).
[Crossref] [PubMed]

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]

S. Dai, Z. Fei, Q. Ma, A. S. Rodin, M. Wagner, A. S. McLeod, M. K. Liu, W. Gannett, W. Regan, K. Watanabe, T. Taniguchi, M. Thiemens, G. Dominguez, A. H. Castro Neto, A. Zettl, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, “Tunable phonon polaritons in atomically thin van der Waals crystals of boron nitride,” Science 343(6175), 1125–1129 (2014).
[Crossref] [PubMed]

2013 (1)

J. D. Caldwell, O. J. Glembocki, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme sub-diffraction photon confinement via Silicon Carbide surface phonon polariton nanopillar resonators,” Nano Lett. 13, 3690–3697 (2013).
[Crossref] [PubMed]

2012 (3)

L. Chen, X. Li, G. Wang, W. Li, S. Chen, L. Xiao, and D. Gao, “A Silicon-based 3-D hybrid long-range plasmonic waveguide for nanophotonic integration,” J. of Light. Tech. 30(1), 163–168 (2012).
[Crossref]

L. Chen, T. Zhang, X. Li, and W. Huang, “Novel hybrid plasmonic waveguide consisting of two identical dielectric nanowires symmetrically placed on each side of a thin metal film,” Opt. Express 20(18), 20535–20544 (2012).
[Crossref] [PubMed]

S. A. Holmstrom, T. H. Stievater, M. W. Pruessner, D. Park, W. S. Rabinovich, J. B. Khurgin, C. J. K. Richardson, S. Kanakaraju, L. C. Calhoun, and R. Ghodssi, “Guided-mode phonon-polaritons in suspended waveguides,” Phys. Rev. B 86(16), 165120 (2012).
[Crossref]

2011 (2)

X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011).
[Crossref] [PubMed]

K. M. Pitman, A. M. Hofmeister, A. B. Corman, and A. K. Speck, “Optical properties and applications of Silicon Carbide for astrophysics,” Astron. Astrophys. 483, 1–12 (2011).

2009 (1)

2008 (1)

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[Crossref]

2007 (3)

R. Kirchain and L. Kimerling, “A roadmap for nanophotonics,” Nat. Photonics 1(6), 303–305 (2007).
[Crossref]

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz polaritonics,” Annu. Rev. Mater. Res. 37(1), 317–350 (2007).
[Crossref]

Y. Cai, L. M. Zhang, Q. Zeng, L. Cheng, and Y. Xu, “Infrared reflectance spectrum of BN calculated from first principles,” Solid State Commun. 141(5), 262–266 (2007).
[Crossref]

2005 (1)

U. N. Roy, R. H. Hawrami, Y. Cui, S. Morgan, A. Burger, K. C. Mandal, C. C. Noblitt, S. A. Speakman, K. Rademaker, and S. A. Payne, “Tb3+-doped KPb2Br5: low-energy phonon mid-infrared laser crystal,” Appl. Phys. Lett. 86(15), 151911 (2005).
[Crossref]

2004 (1)

2002 (1)

R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light matter interaction at the nanometre scale,” Nature 418(6894), 159–162 (2002).
[Crossref] [PubMed]

1999 (1)

T. E. Tiwald, J. A. Woolam, S. Zollner, J. Christiansen, R. B. Gregory, T. Wetteroth, S. R. Wilson, and A. R. Powell, “Carrier concentration and lattice absorption in bulk and epitaxial Silicon Carbide determined using infrared ellipsometry,” Phys. Rev. B 60(16), 11464–11474 (1999).
[Crossref]

1996 (1)

W. J. Moore and R. T. Holm, “Infrared dielectric constant of GaAs,” J. Appl. Phys. 80(12), 6939–6942 (1996).
[Crossref]

1990 (1)

M. Haraguchi, M. Fukui, and S. Muto, “Experimental observation of attenuated-total-reflection spectra of GaAs/AlAs superlattice,” Phys. Rev. B Condens. Matter 41(2), 1254–1257 (1990).
[Crossref] [PubMed]

1966 (1)

R. Geick, C. H. Perry, and G. Rupprecht, “Normal modes in hexagonal Boron Nitride,” Phys. Rev. B 146, 543 (1966).

Almeida, V. R.

Alonso-González, P.

A. Woessner, M. B. Lundeberg, Y. Gao, A. Principi, P. Alonso-González, M. Carrega, K. Watanabe, T. Taniguchi, G. Vignale, M. Polini, J. Hone, R. Hillenbrand, and F. H. L. Koppens, “Highly confined low-loss plasmons in Graphene-Boron Nitride heterostructures,” Nat. Mater. 14(4), 421–425 (2014).
[Crossref] [PubMed]

Andersen, T.

S. Dai, Q. Ma, M. K. Liu, T. Andersen, Z. Fei, M. D. Goldflam, M. Wagner, K. Watanabe, T. Taniguchi, M. Thiemens, F. Keilmann, G. C. A. A. Janssen, S.-E. Zhu, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, “Graphene on hexagonal Boron Nitride as a tunable hyperbolic metamaterial,” Nat. Nanotechnol. 10(8), 682–686 (2015).
[Crossref] [PubMed]

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]

Atwater, H.

V. W. Brar, M. S. Jang, M. Sherrott, S. Kim, J. J. Lopez, L. B. Kim, M. Choi, and H. Atwater, “Hybrid surface-phonon-plasmon polariton modes in Graphene/monolayer h-BN heterostructures,” Nano Lett. 14(7), 3876–3880 (2014).
[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]

Barcelos, I. D.

I. D. Barcelos, A. R. Cadore, L. C. Campos, A. Malachias, K. Watanabe, T. Taniguchi, F. C. B. Maia, R. Freitas, and C. Deneke, “Graphene/h-BN plasmon-phonon coupling and plasmon delocalization observed by infrared nano-spectroscopy,” Nanoscale 7(27), 11620–11625 (2015).
[Crossref] [PubMed]

Barrios, C. A.

Basov, D. N.

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S. Dai, Q. Ma, M. K. Liu, T. Andersen, Z. Fei, M. D. Goldflam, M. Wagner, K. Watanabe, T. Taniguchi, M. Thiemens, F. Keilmann, G. C. A. A. Janssen, S.-E. Zhu, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, “Graphene on hexagonal Boron Nitride as a tunable hyperbolic metamaterial,” Nat. Nanotechnol. 10(8), 682–686 (2015).
[Crossref] [PubMed]

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 (2015).
[Crossref] [PubMed]

I. D. Barcelos, A. R. Cadore, L. C. Campos, A. Malachias, K. Watanabe, T. Taniguchi, F. C. B. Maia, R. Freitas, and C. Deneke, “Graphene/h-BN plasmon-phonon coupling and plasmon delocalization observed by infrared nano-spectroscopy,” Nanoscale 7(27), 11620–11625 (2015).
[Crossref] [PubMed]

A. Woessner, M. B. Lundeberg, Y. Gao, A. Principi, P. Alonso-González, M. Carrega, K. Watanabe, T. Taniguchi, G. Vignale, M. Polini, J. Hone, R. Hillenbrand, and F. H. L. Koppens, “Highly confined low-loss plasmons in Graphene-Boron Nitride heterostructures,” Nat. Mater. 14(4), 421–425 (2014).
[Crossref] [PubMed]

J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014).
[Crossref] [PubMed]

S. Dai, Z. Fei, Q. Ma, A. S. Rodin, M. Wagner, A. S. McLeod, M. K. Liu, W. Gannett, W. Regan, K. Watanabe, T. Taniguchi, M. Thiemens, G. Dominguez, A. H. Castro Neto, A. Zettl, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, “Tunable phonon polaritons in atomically thin van der Waals crystals of boron nitride,” Science 343(6175), 1125–1129 (2014).
[Crossref] [PubMed]

Wetteroth, T.

T. E. Tiwald, J. A. Woolam, S. Zollner, J. Christiansen, R. B. Gregory, T. Wetteroth, S. R. Wilson, and A. R. Powell, “Carrier concentration and lattice absorption in bulk and epitaxial Silicon Carbide determined using infrared ellipsometry,” Phys. Rev. B 60(16), 11464–11474 (1999).
[Crossref]

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J. D. Caldwell, O. J. Glembocki, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme sub-diffraction photon confinement via Silicon Carbide surface phonon polariton nanopillar resonators,” Nano Lett. 13, 3690–3697 (2013).
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T. E. Tiwald, J. A. Woolam, S. Zollner, J. Christiansen, R. B. Gregory, T. Wetteroth, S. R. Wilson, and A. R. Powell, “Carrier concentration and lattice absorption in bulk and epitaxial Silicon Carbide determined using infrared ellipsometry,” Phys. Rev. B 60(16), 11464–11474 (1999).
[Crossref]

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E. Yoxall, M. Schnell, A. Y. Nikitin, O. Txoperena, A. Woessner, M. B. Lundeberg, F. Casanova, L. E. Hueso, F. H. L. Koppens, and R. Hillenbrand, “Direct observation of ultraslow hyperbolic polariton propagation with negative phase velocity,” Nat. Photonics 9(10), 674–678 (2015).
[Crossref]

A. Woessner, M. B. Lundeberg, Y. Gao, A. Principi, P. Alonso-González, M. Carrega, K. Watanabe, T. Taniguchi, G. Vignale, M. Polini, J. Hone, R. Hillenbrand, and F. H. L. Koppens, “Highly confined low-loss plasmons in Graphene-Boron Nitride heterostructures,” Nat. Mater. 14(4), 421–425 (2014).
[Crossref] [PubMed]

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J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014).
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T. E. Tiwald, J. A. Woolam, S. Zollner, J. Christiansen, R. B. Gregory, T. Wetteroth, S. R. Wilson, and A. R. Powell, “Carrier concentration and lattice absorption in bulk and epitaxial Silicon Carbide determined using infrared ellipsometry,” Phys. Rev. B 60(16), 11464–11474 (1999).
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Xiao, L.

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S. Dai, Z. Fei, Q. Ma, A. S. Rodin, M. Wagner, A. S. McLeod, M. K. Liu, W. Gannett, W. Regan, K. Watanabe, T. Taniguchi, M. Thiemens, G. Dominguez, A. H. Castro Neto, A. Zettl, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, “Tunable phonon polaritons in atomically thin van der Waals crystals of boron nitride,” Science 343(6175), 1125–1129 (2014).
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Y. Cai, L. M. Zhang, Q. Zeng, L. Cheng, and Y. Xu, “Infrared reflectance spectrum of BN calculated from first principles,” Solid State Commun. 141(5), 262–266 (2007).
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Zhang, X.

X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011).
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T. E. Tiwald, J. A. Woolam, S. Zollner, J. Christiansen, R. B. Gregory, T. Wetteroth, S. R. Wilson, and A. R. Powell, “Carrier concentration and lattice absorption in bulk and epitaxial Silicon Carbide determined using infrared ellipsometry,” Phys. Rev. B 60(16), 11464–11474 (1999).
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[Crossref]

Y. Jia, H. Zhao, Q. Guo, X. Wang, H. Wang, and F. Xia, “Tunable plasmon-phonon polaritons in layered Graphene-hexagonal Boron Nitride heterostructures,” ACS Photonics 2(7), 907–912 (2015).
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X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011).
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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).
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A. Woessner, M. B. Lundeberg, Y. Gao, A. Principi, P. Alonso-González, M. Carrega, K. Watanabe, T. Taniguchi, G. Vignale, M. Polini, J. Hone, R. Hillenbrand, and F. H. L. Koppens, “Highly confined low-loss plasmons in Graphene-Boron Nitride heterostructures,” Nat. Mater. 14(4), 421–425 (2014).
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E. Yoxall, M. Schnell, A. Y. Nikitin, O. Txoperena, A. Woessner, M. B. Lundeberg, F. Casanova, L. E. Hueso, F. H. L. Koppens, and R. Hillenbrand, “Direct observation of ultraslow hyperbolic polariton propagation with negative phase velocity,” Nat. Photonics 9(10), 674–678 (2015).
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Science (1)

S. Dai, Z. Fei, Q. Ma, A. S. Rodin, M. Wagner, A. S. McLeod, M. K. Liu, W. Gannett, W. Regan, K. Watanabe, T. Taniguchi, M. Thiemens, G. Dominguez, A. H. Castro Neto, A. Zettl, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, “Tunable phonon polaritons in atomically thin van der Waals crystals of boron nitride,” Science 343(6175), 1125–1129 (2014).
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Y. Cai, L. M. Zhang, Q. Zeng, L. Cheng, and Y. Xu, “Infrared reflectance spectrum of BN calculated from first principles,” Solid State Commun. 141(5), 262–266 (2007).
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Figures (8)

Fig. 1
Fig. 1 (a) Real and imaginary parts of the dielectric function of flat h-BN near the phonon resonance. (b) Dispersion of the SPhP mode of the flat h-BN-air boundary.
Fig. 2
Fig. 2 The hybrid mid-IR waveguide which includes a dielectric cylinder of diameter d placed at a gap height h above an h-BN slab on a dielectric substrate.
Fig. 3
Fig. 3 (a) Modal area as a function of cylinder diameter d for different gap height h. (b) Hybrid propagation distance as a function of cylinder diameter d for different gap height h.
Fig. 4
Fig. 4 Electromagnetic energy density distribution for (a) [d, h] = [2.5, 1] μm, (b) [d, h] = [1.4, 0.25] μm, (c) [d, h] = [1.4, 0.1] μm and (d) [d, h] = [1.4, 0.01] μm.
Fig. 5
Fig. 5 Effective index of the hybrid waveguide for different gap height h versus cylinder diameter d. The dashed line indicates the index of the SPhP mode n SPhP .
Fig. 6
Fig. 6 The dependence of coupling strength κ on cylinder diameter d and gap height h.
Fig. 7
Fig. 7 Normalized energy density along z axis at wavenumber (a) 1400 cm−1 (b) 1426 cm−1 and (c) 1483 cm−1. (d) Normalized energy density at z = -d/2 for different wavenumbers.
Fig. 8
Fig. 8 (a) Critical angle θ1 of the hyperbolic polaritons propagating inside the h-BN slab at 1400 cm−1 for [d, h] = [1.4, 0.1] μm. (b) Frequency-dependent directional angles of the hyperbolic polaritons, where θ1 is at 1400 cm−1, θ2 at 1426 cm−1 and θ3 at 1483 cm−1.

Equations (8)

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

L m = [2Im( k hyb (d,h))] 1
A m = W m max{ W( r ) }
W( r )= 1 2 Re{ d(ωε( r )) dω } | E( r ) | 2 + 1 2 μ 0 | H( r ) | 2
ψ hyb ( d,h )=a( d,h ) ψ cyl ( d )+b(d,h) ψ SPhP
n cyl (d)a(d,h)+κ(d,h)b(d,h)= n hyb (d,h)a(d,h)
κ(d,h)a( d,h )+ n SPhP b( d,h )= n hyb (d,h)b(d,h)
κ( d,h )= ( n hyb (d,h) n SPhP )( n hyb ( d,h ) n cyl (d))
θ= π 2 arctan( ε z (ω) i ε xy (ω) )

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