Abstract

Finite-thickness effects are analyzed theoretically for plasma frequency and the associated dielectric response of plasmonic films formed by periodically aligned, infinitely thin, identical metallic cylinders. The plasma frequency of the system is shown to have unidirectional square-root-of-momentum and quasi-linear momentum spatial dispersion for thick and ultrathin films, respectively. This spatial dispersion and the unidirectional dielectric response nonlocality associated with it can be adjusted not only by the film material composition but also by varying the film thickness, the cylinder length, the cylinder-radius-to-film-thickness ratio, and by choosing the substrates and superstrates of the film appropriately. Application of the theory developed to finite-thickness periodically aligned carbon nanotube films is discussed.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (4)

D. Shah, A. Catellani, H. Reddy, N. Kinsey, V. Shalaev, A. Boltasseva, and A. Calzolari, “Controlling the plasmonic properties of ultrathin TiN films at the atomic level,” ACS Phot. 5, 2816–2824 (2018).
[Crossref]

I. V. Bondarev, H. Mousavi, and V. M. Shalaev, “Optical response of finite-thickness ultrathin plasmonic films,” MRS Commun. 8, 1092–1097 (2018).
[Crossref]

G. T. Papadakis, D. Fleischman, A. Davoyan, P. Yeh, and H. A. Atwater, “Optical magnetism in planar metamaterial heterostructures,” Nature Commun. 9, 296 (2018).
[Crossref]

W. Gao, X. Li, M. Bamba, and J. Kono, “Continuous transition between weak and ultrastrong coupling through exceptional points in carbon nanotube microcavity exciton-polaritons,” Nature Phot. 12, 362–367 (2018).
[Crossref]

2017 (8)

K.-C. Chiu, A. L. Falk, P.-H. Ho, D. B. Farmer, G. Tulevski, Y.-H. Lee, Ph. Avouris, and S.-J. Han, “Strong and broadly tunable plasmon resonances in thick films of aligned carbon nanotubes,” Nano Lett. 17, 5641–5645 (2017).
[Crossref] [PubMed]

A. L. Falk, K.-C. Chiu, D. B. Farmer, Q. Cao, J. Tersoff, Y.-H. Lee, Ph. Avouris, and S.-J. Han, “Coherent plasmon and phonon-plasmon resonances in carbon nanotubes,” Phys. Rev. Lett. 118, 257401 (2017).
[Crossref] [PubMed]

I. V. Bondarev and V. M. Shalaev, “Universal features of the optical properties of ultrathin plasmonic films,” Optical Mater. Express 7, 3731–3740 (2017).
[Crossref]

D. Shah, H. Reddy, N. Kinsey, V. M. Shalaev, and A. Boltasseva, “Optical properties of plasmonic ultrathin TiN films,” Adv. Optical Mater. 5, 1700065 (2017).
[Crossref]

H. Reddy, U. Guler, K. Chaudhuri, A. Dutta, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Temperature-dependent optical properties of single crystalline and polycrystalline silver thin films,” ACS Phot. 4, 1083–1091 (2017).
[Crossref]

T. Gric and O. Hess, “Controlling hybrid-polarization surface plasmon polaritons in dielectric-transparent conducting oxides metamaterials via their effective properties,” J. Appl. Phys. 122, 193105 (2017).
[Crossref]

A. S. Kadochkin, S. G. Moiseev, Y. S. Dadoenkova, V. V. Svetukhin, and I. O. Zolotovskii, “Surface plasmon polariton amplification in a single-walled carbon nanotube,” Optics Express 25, 27165–27171 (2017).
[Crossref] [PubMed]

T. Stauber, G. G. Santos, and L. Brey, “Plasmonics in topological insulators: Spin-charge separation, the influence of the inversion layer, and phonon-plasmon coupling,” ACS Phot. 4, 2978–2988 (2017).
[Crossref]

2016 (4)

K. F. Mak and J. Shan, “Photonics and optoelectronics of 2D semiconductor transition metal dichalcogenides,” Nature Phot. 10, 216–226 (2016).
[Crossref]

X. He, W. Gao, L. Xie, B. Li, Q. Zhang, S. Lei, J. M. Robinson, E. H. Hároz, S. K. Doorn, W. Wang, R. Vajtai, P. M. Ajayan, W. W. Adams, R. H. Hauge, and J. Kono, “Wafer-scale monodomain films of spontaneously aligned single-walled carbon nanotubes,” Nature Nanotechn. 11, 633–638 (2016).
[Crossref]

A. Graf, L. Tropf, Y. Zakharko, J. Zaumseil, and M. C. Gather, “Near-infrared exciton-polaritons in strongly coupled single-walled carbon nanotube microcavities,” Nature Commun. 7, 13078 (2016).
[Crossref]

M. F. Gelin and I. V. Bondarev, “One-dimensional transport in hybrid metal-semiconductor nanotube systems,” Phys. Rev. B 93, 115422 (2016).
[Crossref]

2015 (4)

I. V. Bondarev, “Plasmon enhanced Raman scattering effect for an atom near a carbon nanotube,” Optics Express 23, 3971–3984 (2015).
[Crossref] [PubMed]

L. Martin-Moreno, F. J. Garcia de Abajo, and F. J. Garcia-Vidal, “Ultraefficient coupling of a quantum emitter to the tunable guided plasmons of a carbon nanotube,” Phys. Rev. Lett. 115, 173601 (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. M. Janssen, S-E. Zhu, P. Jarillo-Herrero, M. Fogler, and D. N. Basov, “Graphene on hexagonal boron nitride as a tunable hyperbolic metamaterial,” Nature Nanotechn. 10, 682–686 (2015).
[Crossref]

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349, 165–168 (2015).
[Crossref] [PubMed]

2014 (4)

F. H. L. Koppens, T. Mueller, Ph. Avouris, A. C. Ferrari, M. S. Vitiello, and M. Polini, “Photodetectors based on graphene, other two-dimensional materials and hybrid systems,” Nature Nanotechn. 9, 780–793 (2014).
[Crossref]

A. Manjavacas and F. J. García de Abajo, “Tunable plasmons in atomically thin gold nanodisks,” Nature Commun. 5, 3548 (2014).
[Crossref]

D. N. Basov, M. M. Fogler, A. Lanzara, F. Wang, and Y. Zhang, “Colloquium: Graphene spectroscopy,” Rev. Mod. Phys. 86, 959–994 (2014).
[Crossref]

I. V. Bondarev and A. V. Meliksetyan, “Possibility for exciton Bose-Einstein condensation in carbon nanotubes,” Phys. Rev. B 89, 045414 (2014).
[Crossref]

2013 (1)

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339, 1289–1295 (2013).
[Crossref]

2012 (4)

I. V. Bondarev, “Single-wall carbon nanotubes as coherent plasmon generators,” Phys. Rev. B 85, 035448 (2012).
[Crossref]

P. Fan, U. K. Chettiar, L. Cao, F. Afshinmanesh, N. Engheta, and M. L. Brongersma, “An invisible metal-semiconductor photodetector,” Nature Phot. 6, 380–385 (2012).
[Crossref]

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012).
[Crossref] [PubMed]

Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100, 181105 (2012).
[Crossref]

2011 (3)

A. Boltasseva and H. A. Atwater, “Low-loss plasmonic metamaterials,” Science 331, 290–291 (2011).
[Crossref] [PubMed]

M. W. Knight, H. Sobhani, P. Nordlander, and N. Halas, “Photodetection with active optical antennas,” Science 332, 702–704 (2011).
[Crossref] [PubMed]

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: A platform for strong light-matter interactions,” Nano Lett. 11, 3370–3377 (2011).
[Crossref] [PubMed]

2010 (4)

J-S. Huang, V. Callegari, P. Geisler, C. Brüning, J. Kern, J. C. Prangsma, X. Wu, T. Feichtner, J. Ziegler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, U. Sennhauser, and B. Hecht, “Atomically flat single-crystalline gold nanostructures for plasmonic nanocircuitry,” Nature Commun. 1, 150 (2010).
[Crossref]

I. V. Bondarev, “Surface electromagnetic phenomena in pristine and atomically doped carbon nanotubes,” J. Comp. Theor. Nanosci. 7, 1673–1687 (2010).
[Crossref]

M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35, 1863–1865 (2010).
[Crossref] [PubMed]

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

2009 (3)

J. Deslippe, M. Dipoppa, D. Prendergast, M. V. O. Moutinho, R. B. Capaz, and S. G. Louie, “Electron-hole interaction in carbon nanotubes: novel screening and exciton excitation spectra,” Nano Lett. 9, 1330–1334 (2009).
[Crossref] [PubMed]

I. V. Bondarev, L. M. Woods, and K. Tatur, “Strong exciton-plasmon coupling in semiconducting carbon nanotubes,” Phys. Rev. B 80, 085407 (2009).
[Crossref]

T. Nakanishi and T. Ando, “Optical response of finite-length carbon nanotubes,” J. Phys. Soc. Jpn. 78, 114708 (2009).
[Crossref]

2007 (3)

Z. W. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[Crossref] [PubMed]

V. M. Shalaev, “Optical negative-index metamaterials,” Nature Phot. 1, 41–48 (2007).
[Crossref]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nature Mater. 6, 946–950 (2007).
[Crossref]

2006 (3)

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74, 075103 (2006).
[Crossref]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Optics Express 14, 8247–8256 (2006).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

2005 (1)

T. Ando, “Theory of electronic states and transport in carbon nanotubes,” J. Phys. Soc. Jpn. 74, 777–817 (2005).
[Crossref]

1979 (1)

L. V. Keldysh, “Coulomb interaction in thin semiconductor and semimetal films,” Pis’ma Zh. Eksp. Teor. Fiz. 29, 716–719 (1979) [Engl. translation: JETP Lett. 29, 658–661 (1980)]; N. S. Rytova, “Screened potential of a point charge in a thin film,” Mosc. Univ. Phys. Bull. 3, 30 (1967).

1952 (1)

D. Pines and D. Bohm, “A collective description of electron interactions: II. Collective vs individual particle aspects of the interactions,” Phys. Rev. 85, 338–353 (1952).
[Crossref]

Adams, W. W.

X. He, W. Gao, L. Xie, B. Li, Q. Zhang, S. Lei, J. M. Robinson, E. H. Hároz, S. K. Doorn, W. Wang, R. Vajtai, P. M. Ajayan, W. W. Adams, R. H. Hauge, and J. Kono, “Wafer-scale monodomain films of spontaneously aligned single-walled carbon nanotubes,” Nature Nanotechn. 11, 633–638 (2016).
[Crossref]

Afshinmanesh, F.

P. Fan, U. K. Chettiar, L. Cao, F. Afshinmanesh, N. Engheta, and M. L. Brongersma, “An invisible metal-semiconductor photodetector,” Nature Phot. 6, 380–385 (2012).
[Crossref]

Ajayan, P. M.

X. He, W. Gao, L. Xie, B. Li, Q. Zhang, S. Lei, J. M. Robinson, E. H. Hároz, S. K. Doorn, W. Wang, R. Vajtai, P. M. Ajayan, W. W. Adams, R. H. Hauge, and J. Kono, “Wafer-scale monodomain films of spontaneously aligned single-walled carbon nanotubes,” Nature Nanotechn. 11, 633–638 (2016).
[Crossref]

Alekseyev, L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nature Mater. 6, 946–950 (2007).
[Crossref]

Alekseyev, L. V.

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Optics Express 14, 8247–8256 (2006).
[Crossref] [PubMed]

Altug, H.

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349, 165–168 (2015).
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Sivco, D. L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nature Mater. 6, 946–950 (2007).
[Crossref]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

Smolyaninov, I. I.

Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100, 181105 (2012).
[Crossref]

Sobhani, H.

M. W. Knight, H. Sobhani, P. Nordlander, and N. Halas, “Photodetection with active optical antennas,” Science 332, 702–704 (2011).
[Crossref] [PubMed]

Stauber, T.

T. Stauber, G. G. Santos, and L. Brey, “Plasmonics in topological insulators: Spin-charge separation, the influence of the inversion layer, and phonon-plasmon coupling,” ACS Phot. 4, 2978–2988 (2017).
[Crossref]

Sun, C.

Z. W. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[Crossref] [PubMed]

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A. S. Kadochkin, S. G. Moiseev, Y. S. Dadoenkova, V. V. Svetukhin, and I. O. Zolotovskii, “Surface plasmon polariton amplification in a single-walled carbon nanotube,” Optics Express 25, 27165–27171 (2017).
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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. M. Janssen, S-E. Zhu, P. Jarillo-Herrero, M. Fogler, and D. N. Basov, “Graphene on hexagonal boron nitride as a tunable hyperbolic metamaterial,” Nature Nanotechn. 10, 682–686 (2015).
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I. V. Bondarev, L. M. Woods, and K. Tatur, “Strong exciton-plasmon coupling in semiconducting carbon nanotubes,” Phys. Rev. B 80, 085407 (2009).
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A. L. Falk, K.-C. Chiu, D. B. Farmer, Q. Cao, J. Tersoff, Y.-H. Lee, Ph. Avouris, and S.-J. Han, “Coherent plasmon and phonon-plasmon resonances in carbon nanotubes,” Phys. Rev. Lett. 118, 257401 (2017).
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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. M. Janssen, S-E. Zhu, P. Jarillo-Herrero, M. Fogler, and D. N. Basov, “Graphene on hexagonal boron nitride as a tunable hyperbolic metamaterial,” Nature Nanotechn. 10, 682–686 (2015).
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A. Graf, L. Tropf, Y. Zakharko, J. Zaumseil, and M. C. Gather, “Near-infrared exciton-polaritons in strongly coupled single-walled carbon nanotube microcavities,” Nature Commun. 7, 13078 (2016).
[Crossref]

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K.-C. Chiu, A. L. Falk, P.-H. Ho, D. B. Farmer, G. Tulevski, Y.-H. Lee, Ph. Avouris, and S.-J. Han, “Strong and broadly tunable plasmon resonances in thick films of aligned carbon nanotubes,” Nano Lett. 17, 5641–5645 (2017).
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A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nature Mater. 6, 946–950 (2007).
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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. M. Janssen, S-E. Zhu, P. Jarillo-Herrero, M. Fogler, and D. N. Basov, “Graphene on hexagonal boron nitride as a tunable hyperbolic metamaterial,” Nature Nanotechn. 10, 682–686 (2015).
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J-S. Huang, V. Callegari, P. Geisler, C. Brüning, J. Kern, J. C. Prangsma, X. Wu, T. Feichtner, J. Ziegler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, U. Sennhauser, and B. Hecht, “Atomically flat single-crystalline gold nanostructures for plasmonic nanocircuitry,” Nature Commun. 1, 150 (2010).
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X. He, W. Gao, L. Xie, B. Li, Q. Zhang, S. Lei, J. M. Robinson, E. H. Hároz, S. K. Doorn, W. Wang, R. Vajtai, P. M. Ajayan, W. W. Adams, R. H. Hauge, and J. Kono, “Wafer-scale monodomain films of spontaneously aligned single-walled carbon nanotubes,” Nature Nanotechn. 11, 633–638 (2016).
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A. Graf, L. Tropf, Y. Zakharko, J. Zaumseil, and M. C. Gather, “Near-infrared exciton-polaritons in strongly coupled single-walled carbon nanotube microcavities,” Nature Commun. 7, 13078 (2016).
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D. N. Basov, M. M. Fogler, A. Lanzara, F. Wang, and Y. Zhang, “Colloquium: Graphene spectroscopy,” Rev. Mod. Phys. 86, 959–994 (2014).
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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. M. Janssen, S-E. Zhu, P. Jarillo-Herrero, M. Fogler, and D. N. Basov, “Graphene on hexagonal boron nitride as a tunable hyperbolic metamaterial,” Nature Nanotechn. 10, 682–686 (2015).
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J-S. Huang, V. Callegari, P. Geisler, C. Brüning, J. Kern, J. C. Prangsma, X. Wu, T. Feichtner, J. Ziegler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, U. Sennhauser, and B. Hecht, “Atomically flat single-crystalline gold nanostructures for plasmonic nanocircuitry,” Nature Commun. 1, 150 (2010).
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H. Reddy, U. Guler, K. Chaudhuri, A. Dutta, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Temperature-dependent optical properties of single crystalline and polycrystalline silver thin films,” ACS Phot. 4, 1083–1091 (2017).
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D. Shah, A. Catellani, H. Reddy, N. Kinsey, V. Shalaev, A. Boltasseva, and A. Calzolari, “Controlling the plasmonic properties of ultrathin TiN films at the atomic level,” ACS Phot. 5, 2816–2824 (2018).
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Figures (3)

Fig. 1
Fig. 1 Schematic to show the geometry notations for the model of the finite-thickness plasmonic film with periodic cylindrical anisotropy. See text for details.
Fig. 2
Fig. 2 (a) The ratio ω p ( q ) / ω p 3 D given by Eq. (7) with R/d = 0.5 as a function of the dimensionless variables qd and (ε1 + ε2)/ε. (b) The contour plot obtained by cutting the graph in (a) with parallel vertical planes of constant (ε1 + ε2)/ε taken to be equal to 0.01j, 0.1 + 0.03j and 0.4 + 0.05j (j =0, 1, 2, ...) over the intervals [0, 0.1], [0.1, 0.4] and [0.4, 1], respectively. The vertical blue arrow shows the direction of the (ε1 + ε2)/ε increase.
Fig. 3
Fig. 3 (a) Real (red) and imaginary (green) parts of Eq. (12) as functions of the dimensionless variables ω / ω p 3 D and qd. (b) The contour plot one obtains by cutting the graph in (a) with parallel vertical planes of constant qd taken to be equal to 0.03 + 0.25j (j =0, 1, 2, ...) over the interval [0.03, 2.8]. The thick horizontal blue arrow shows the direction of the qd decrease.

Equations (16)

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V ( ρ n ρ l ) 8 π e 2 ε L L k q R I 0 ( q R ) K 0 ( q R ) k [ k d + ( ε 1 + ε 2 ) / ε ] exp [ i k ( ρ n ρ l ) ] .
V ( ρ n ) = 8 π e 2 ε L L l , k q R I 0 ( q R ) K 0 ( q R ) k [ k d + ( ε 1 + ε 2 ) / ε ] exp [ i k ( ρ n ρ l ) ] ,
ρ ¨ n = 8 π e 2 ε m * L L l , k ( i k ) q R I 0 ( q R ) K 0 ( q R ) k [ k d + ( ε 1 + ε 2 ) / ε ] exp [ i k ( ρ n ρ l ) ] .
n ( ρ ) = l δ ( ρ ρ l ) = k n k exp ( i k ρ )
n k = 1 L L l exp ( i k ρ l ) , n k = 0 = N 2 D
n ¨ k = 8 π e 2 ε m * k ( k k ) n k n k k q R I 0 ( q R ) K 0 ( q R ) k [ k d + ( ε 1 + ε 2 ) / ε ] 1 L L l ( k ρ ˙ l ) 2 exp ( i k ρ l ) .
n ¨ k + ω p 2 ( k ) n k = 1 L L l ( k ρ ˙ l ) 2 exp ( i k ρ l )
ω p ( k ) = ω p ( q , k ) = 8 π e 2 N 2 D q R I 0 ( q R ) K 0 ( q R ) ε m * d [ 1 + ( ε 1 + ε 2 ) / ε k d ] .
ω p ( q ) = ω p 3 D 2 q R I 0 ( q R ) K 0 ( q R ) 1 + ( ε 1 + ε 2 ) / ε q d ,
ω p ( q ) = ω p 3 D 2 q R I 0 ( q R ) K 0 ( q R ) ,
ω p ( q ) = ω p 2 D ( q ) = q 8 π e 2 N 2 D R I 0 ( q R ) K 0 ( q R ) ( ε 1 + ε 2 ) m *
ε ^ ( ω , q ) = [ ε 0 0 ε ( ω , q ) ] ,
ε = ε
ε ( ω , q ) = ε ( 1 ω p 2 ( q ) ω 2 + i γ ω ) .
ω σ ( ω ) = 4 π i R [ q 2 ( ω / c ) 2 ] I p [ q 2 ( ω / c ) 2 R ] K p [ q 2 ( ω / c ) 2 R ] .
ω σ ( ω ) = 4 π i q 2 R I 0 ( q R ) K 0 ( q R ) ,

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