Abstract

Using the coupled-mode theory, we study the transmission of surface plasmon polaritons (SPPs) guided by a thin metal film through an array of N identical nanowires, which are parallel to each other and to the surface of the metal film. By varying the parameters of the nanowire array, one can control the intensity of the transmitted SPP. Furthermore, we propose a novel mechano-optical modulation technique. The intensity of the transmitted SPP is modulated by changing the distance between the nanowire array and the metal film. The modulation frequency is in the kilohertz or megahertz range, owing to the unique mechanical properties of nanowires.

© 2010 OSA

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2010

D. Yu. Fedyanin, A. V. Arsenin, V. G. Leiman, and A. D. Gladun, “Backward waves in planar insulator-metal-insulator waveguide structures,” J. Opt. 12(1), 015002 (2010).
[CrossRef]

2009

W. Cai, J. S. White, and M. L. Brongersma, “Compact, high-speed and power-efficient electrooptic plasmonic modulators,” Nano Lett. 9(12), 4403–4411 (2009).
[CrossRef] [PubMed]

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3(1), 55–58 (2009).
[CrossRef]

2008

K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol. 3(9), 533–537 (2008).
[CrossRef] [PubMed]

P. Berini, “Bulk and surface sensitivities of surface plasmon waveguides,” N. J. Phys. 10(10), 105010 (2008).
[CrossRef]

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[CrossRef] [PubMed]

2007

D. Pacifici, H. J. Lezec, and H. A. Atwater, “All-optical modulation by plasmonic excitation of CdSe quantum dots,” Nat. Photonics 1(7), 402–406 (2007).
[CrossRef]

2006

J. Zhou, C. Lao, P. Gao, W. Mai, W. Hughes, S. Deng, N. Xu, and Z. Wang, “Nanowire as pico-gram balance at workplace atmosphere,” Solid State Commun. 139(5), 222–226 (2006).
[CrossRef]

J. Gaillard, M. J. Skove, R. Ciocan, and A. M. Rao, “Electrical detection of oscillations in microcantilevers and nanocantilevers,” Rev. Sci. Instrum. 77(7), 073907 (2006).
[CrossRef]

2005

P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, “Characterization of long-range surface-plasmonpolariton waveguides,” J. Appl. Phys. 98(4), 043109 (2005).
[CrossRef]

T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “In-line extinction modulator based on long-range surface plasmon polaritons,” Opt. Commun. 244(1-6), 455–459 (2005).
[CrossRef]

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005).
[CrossRef]

2004

B. Ilica, H. G. Craighead, S. Krylov, W. Senaratne, C. Ober, and P. Neuzil, “Attogram detection using nanoelectromechanical oscillators,” J. Appl. Phys. 95(7), 3694 (2004).
[CrossRef]

2003

D. A. Dikin, X. Chen, W. Ding, G. Wagner, and R. S. Ruoff, “Resonance vibration of amorphous SiO2 nanowires driven by mechanical or electrical field excitation,” J. Appl. Phys. 93(1), 226–230 (2003).
[CrossRef]

2000

1994

1986

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B Condens. Matter 33(8), 5186–5201 (1986).
[CrossRef] [PubMed]

1981

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981).
[CrossRef]

Anker, J. N.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[CrossRef] [PubMed]

Arsenin, A. V.

D. Yu. Fedyanin, A. V. Arsenin, V. G. Leiman, and A. D. Gladun, “Backward waves in planar insulator-metal-insulator waveguide structures,” J. Opt. 12(1), 015002 (2010).
[CrossRef]

Atwater, H. A.

D. Pacifici, H. J. Lezec, and H. A. Atwater, “All-optical modulation by plasmonic excitation of CdSe quantum dots,” Nat. Photonics 1(7), 402–406 (2007).
[CrossRef]

Berini, P.

P. Berini, “Bulk and surface sensitivities of surface plasmon waveguides,” N. J. Phys. 10(10), 105010 (2008).
[CrossRef]

P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, “Characterization of long-range surface-plasmonpolariton waveguides,” J. Appl. Phys. 98(4), 043109 (2005).
[CrossRef]

R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, “Experimental observation of plasmon polariton waves supported by a thin metal film of finite width,” Opt. Lett. 25(11), 844–846 (2000).
[CrossRef]

Berolo, E.

Bozhevolnyi, S. I.

T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “In-line extinction modulator based on long-range surface plasmon polaritons,” Opt. Commun. 244(1-6), 455–459 (2005).
[CrossRef]

Brongersma, M. L.

W. Cai, J. S. White, and M. L. Brongersma, “Compact, high-speed and power-efficient electrooptic plasmonic modulators,” Nano Lett. 9(12), 4403–4411 (2009).
[CrossRef] [PubMed]

Burke, J. J.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B Condens. Matter 33(8), 5186–5201 (1986).
[CrossRef] [PubMed]

Cai, W.

W. Cai, J. S. White, and M. L. Brongersma, “Compact, high-speed and power-efficient electrooptic plasmonic modulators,” Nano Lett. 9(12), 4403–4411 (2009).
[CrossRef] [PubMed]

Charbonneau, R.

P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, “Characterization of long-range surface-plasmonpolariton waveguides,” J. Appl. Phys. 98(4), 043109 (2005).
[CrossRef]

R. Charbonneau, P. Berini, E. Berolo, and E. Lisicka-Shrzek, “Experimental observation of plasmon polariton waves supported by a thin metal film of finite width,” Opt. Lett. 25(11), 844–846 (2000).
[CrossRef]

Chen, X.

D. A. Dikin, X. Chen, W. Ding, G. Wagner, and R. S. Ruoff, “Resonance vibration of amorphous SiO2 nanowires driven by mechanical or electrical field excitation,” J. Appl. Phys. 93(1), 226–230 (2003).
[CrossRef]

Ciocan, R.

J. Gaillard, M. J. Skove, R. Ciocan, and A. M. Rao, “Electrical detection of oscillations in microcantilevers and nanocantilevers,” Rev. Sci. Instrum. 77(7), 073907 (2006).
[CrossRef]

Craighead, H. G.

B. Ilica, H. G. Craighead, S. Krylov, W. Senaratne, C. Ober, and P. Neuzil, “Attogram detection using nanoelectromechanical oscillators,” J. Appl. Phys. 95(7), 3694 (2004).
[CrossRef]

Deng, S.

J. Zhou, C. Lao, P. Gao, W. Mai, W. Hughes, S. Deng, N. Xu, and Z. Wang, “Nanowire as pico-gram balance at workplace atmosphere,” Solid State Commun. 139(5), 222–226 (2006).
[CrossRef]

Dikin, D. A.

D. A. Dikin, X. Chen, W. Ding, G. Wagner, and R. S. Ruoff, “Resonance vibration of amorphous SiO2 nanowires driven by mechanical or electrical field excitation,” J. Appl. Phys. 93(1), 226–230 (2003).
[CrossRef]

Ding, W.

D. A. Dikin, X. Chen, W. Ding, G. Wagner, and R. S. Ruoff, “Resonance vibration of amorphous SiO2 nanowires driven by mechanical or electrical field excitation,” J. Appl. Phys. 93(1), 226–230 (2003).
[CrossRef]

Fedyanin, D. Yu.

D. Yu. Fedyanin, A. V. Arsenin, V. G. Leiman, and A. D. Gladun, “Backward waves in planar insulator-metal-insulator waveguide structures,” J. Opt. 12(1), 015002 (2010).
[CrossRef]

Gaillard, J.

J. Gaillard, M. J. Skove, R. Ciocan, and A. M. Rao, “Electrical detection of oscillations in microcantilevers and nanocantilevers,” Rev. Sci. Instrum. 77(7), 073907 (2006).
[CrossRef]

Gao, P.

J. Zhou, C. Lao, P. Gao, W. Mai, W. Hughes, S. Deng, N. Xu, and Z. Wang, “Nanowire as pico-gram balance at workplace atmosphere,” Solid State Commun. 139(5), 222–226 (2006).
[CrossRef]

Gladun, A. D.

D. Yu. Fedyanin, A. V. Arsenin, V. G. Leiman, and A. D. Gladun, “Backward waves in planar insulator-metal-insulator waveguide structures,” J. Opt. 12(1), 015002 (2010).
[CrossRef]

Hall, W. P.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[CrossRef] [PubMed]

Huang, W.-P.

Hughes, W.

J. Zhou, C. Lao, P. Gao, W. Mai, W. Hughes, S. Deng, N. Xu, and Z. Wang, “Nanowire as pico-gram balance at workplace atmosphere,” Solid State Commun. 139(5), 222–226 (2006).
[CrossRef]

Ilica, B.

B. Ilica, H. G. Craighead, S. Krylov, W. Senaratne, C. Ober, and P. Neuzil, “Attogram detection using nanoelectromechanical oscillators,” J. Appl. Phys. 95(7), 3694 (2004).
[CrossRef]

Jensen, K.

K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol. 3(9), 533–537 (2008).
[CrossRef] [PubMed]

Kim, K.

K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol. 3(9), 533–537 (2008).
[CrossRef] [PubMed]

Krylov, S.

B. Ilica, H. G. Craighead, S. Krylov, W. Senaratne, C. Ober, and P. Neuzil, “Attogram detection using nanoelectromechanical oscillators,” J. Appl. Phys. 95(7), 3694 (2004).
[CrossRef]

Lahoud, N.

P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, “Characterization of long-range surface-plasmonpolariton waveguides,” J. Appl. Phys. 98(4), 043109 (2005).
[CrossRef]

Lao, C.

J. Zhou, C. Lao, P. Gao, W. Mai, W. Hughes, S. Deng, N. Xu, and Z. Wang, “Nanowire as pico-gram balance at workplace atmosphere,” Solid State Commun. 139(5), 222–226 (2006).
[CrossRef]

Leiman, V. G.

D. Yu. Fedyanin, A. V. Arsenin, V. G. Leiman, and A. D. Gladun, “Backward waves in planar insulator-metal-insulator waveguide structures,” J. Opt. 12(1), 015002 (2010).
[CrossRef]

Leosson, K.

T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “In-line extinction modulator based on long-range surface plasmon polaritons,” Opt. Commun. 244(1-6), 455–459 (2005).
[CrossRef]

Lezec, H. J.

D. Pacifici, H. J. Lezec, and H. A. Atwater, “All-optical modulation by plasmonic excitation of CdSe quantum dots,” Nat. Photonics 1(7), 402–406 (2007).
[CrossRef]

Lisicka-Shrzek, E.

Lyandres, O.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[CrossRef] [PubMed]

MacDonald, K. F.

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3(1), 55–58 (2009).
[CrossRef]

Mai, W.

J. Zhou, C. Lao, P. Gao, W. Mai, W. Hughes, S. Deng, N. Xu, and Z. Wang, “Nanowire as pico-gram balance at workplace atmosphere,” Solid State Commun. 139(5), 222–226 (2006).
[CrossRef]

Maradudin, A. A.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005).
[CrossRef]

Mattiussi, G.

P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, “Characterization of long-range surface-plasmonpolariton waveguides,” J. Appl. Phys. 98(4), 043109 (2005).
[CrossRef]

Neuzil, P.

B. Ilica, H. G. Craighead, S. Krylov, W. Senaratne, C. Ober, and P. Neuzil, “Attogram detection using nanoelectromechanical oscillators,” J. Appl. Phys. 95(7), 3694 (2004).
[CrossRef]

Nikolajsen, T.

T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “In-line extinction modulator based on long-range surface plasmon polaritons,” Opt. Commun. 244(1-6), 455–459 (2005).
[CrossRef]

Ober, C.

B. Ilica, H. G. Craighead, S. Krylov, W. Senaratne, C. Ober, and P. Neuzil, “Attogram detection using nanoelectromechanical oscillators,” J. Appl. Phys. 95(7), 3694 (2004).
[CrossRef]

Pacifici, D.

D. Pacifici, H. J. Lezec, and H. A. Atwater, “All-optical modulation by plasmonic excitation of CdSe quantum dots,” Nat. Photonics 1(7), 402–406 (2007).
[CrossRef]

Rao, A. M.

J. Gaillard, M. J. Skove, R. Ciocan, and A. M. Rao, “Electrical detection of oscillations in microcantilevers and nanocantilevers,” Rev. Sci. Instrum. 77(7), 073907 (2006).
[CrossRef]

Ruoff, R. S.

D. A. Dikin, X. Chen, W. Ding, G. Wagner, and R. S. Ruoff, “Resonance vibration of amorphous SiO2 nanowires driven by mechanical or electrical field excitation,” J. Appl. Phys. 93(1), 226–230 (2003).
[CrossRef]

Samson, Z. L.

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3(1), 55–58 (2009).
[CrossRef]

Sarid, D.

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981).
[CrossRef]

Senaratne, W.

B. Ilica, H. G. Craighead, S. Krylov, W. Senaratne, C. Ober, and P. Neuzil, “Attogram detection using nanoelectromechanical oscillators,” J. Appl. Phys. 95(7), 3694 (2004).
[CrossRef]

Shah, N. C.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[CrossRef] [PubMed]

Skove, M. J.

J. Gaillard, M. J. Skove, R. Ciocan, and A. M. Rao, “Electrical detection of oscillations in microcantilevers and nanocantilevers,” Rev. Sci. Instrum. 77(7), 073907 (2006).
[CrossRef]

Smolyaninov, I. I.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005).
[CrossRef]

Stegeman, G. I.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B Condens. Matter 33(8), 5186–5201 (1986).
[CrossRef] [PubMed]

Stockman, M. I.

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3(1), 55–58 (2009).
[CrossRef]

Tamir, T.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B Condens. Matter 33(8), 5186–5201 (1986).
[CrossRef] [PubMed]

Van Duyne, R. P.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[CrossRef] [PubMed]

Wagner, G.

D. A. Dikin, X. Chen, W. Ding, G. Wagner, and R. S. Ruoff, “Resonance vibration of amorphous SiO2 nanowires driven by mechanical or electrical field excitation,” J. Appl. Phys. 93(1), 226–230 (2003).
[CrossRef]

Wang, Z.

J. Zhou, C. Lao, P. Gao, W. Mai, W. Hughes, S. Deng, N. Xu, and Z. Wang, “Nanowire as pico-gram balance at workplace atmosphere,” Solid State Commun. 139(5), 222–226 (2006).
[CrossRef]

White, J. S.

W. Cai, J. S. White, and M. L. Brongersma, “Compact, high-speed and power-efficient electrooptic plasmonic modulators,” Nano Lett. 9(12), 4403–4411 (2009).
[CrossRef] [PubMed]

Xu, N.

J. Zhou, C. Lao, P. Gao, W. Mai, W. Hughes, S. Deng, N. Xu, and Z. Wang, “Nanowire as pico-gram balance at workplace atmosphere,” Solid State Commun. 139(5), 222–226 (2006).
[CrossRef]

Zayats, A. V.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005).
[CrossRef]

Zettl, A.

K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol. 3(9), 533–537 (2008).
[CrossRef] [PubMed]

Zhao, J.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[CrossRef] [PubMed]

Zheludev, N. I.

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3(1), 55–58 (2009).
[CrossRef]

Zhou, J.

J. Zhou, C. Lao, P. Gao, W. Mai, W. Hughes, S. Deng, N. Xu, and Z. Wang, “Nanowire as pico-gram balance at workplace atmosphere,” Solid State Commun. 139(5), 222–226 (2006).
[CrossRef]

J. Appl. Phys.

P. Berini, R. Charbonneau, N. Lahoud, and G. Mattiussi, “Characterization of long-range surface-plasmonpolariton waveguides,” J. Appl. Phys. 98(4), 043109 (2005).
[CrossRef]

D. A. Dikin, X. Chen, W. Ding, G. Wagner, and R. S. Ruoff, “Resonance vibration of amorphous SiO2 nanowires driven by mechanical or electrical field excitation,” J. Appl. Phys. 93(1), 226–230 (2003).
[CrossRef]

B. Ilica, H. G. Craighead, S. Krylov, W. Senaratne, C. Ober, and P. Neuzil, “Attogram detection using nanoelectromechanical oscillators,” J. Appl. Phys. 95(7), 3694 (2004).
[CrossRef]

J. Opt.

D. Yu. Fedyanin, A. V. Arsenin, V. G. Leiman, and A. D. Gladun, “Backward waves in planar insulator-metal-insulator waveguide structures,” J. Opt. 12(1), 015002 (2010).
[CrossRef]

J. Opt. Soc. Am. A

N. J. Phys.

P. Berini, “Bulk and surface sensitivities of surface plasmon waveguides,” N. J. Phys. 10(10), 105010 (2008).
[CrossRef]

Nano Lett.

W. Cai, J. S. White, and M. L. Brongersma, “Compact, high-speed and power-efficient electrooptic plasmonic modulators,” Nano Lett. 9(12), 4403–4411 (2009).
[CrossRef] [PubMed]

Nat. Mater.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[CrossRef] [PubMed]

Nat. Nanotechnol.

K. Jensen, K. Kim, and A. Zettl, “An atomic-resolution nanomechanical mass sensor,” Nat. Nanotechnol. 3(9), 533–537 (2008).
[CrossRef] [PubMed]

Nat. Photonics

D. Pacifici, H. J. Lezec, and H. A. Atwater, “All-optical modulation by plasmonic excitation of CdSe quantum dots,” Nat. Photonics 1(7), 402–406 (2007).
[CrossRef]

K. F. MacDonald, Z. L. Samson, M. I. Stockman, and N. I. Zheludev, “Ultrafast active plasmonics,” Nat. Photonics 3(1), 55–58 (2009).
[CrossRef]

Opt. Commun.

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A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3-4), 131–314 (2005).
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Phys. Rev. B Condens. Matter

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Figures (6)

Fig. 1
Fig. 1

Schematic operation of the mechano-optic modulator, β is the SPP wavevector, h 1>h 2. (a). If the distance h between the nanowire array and the metal film is very large, there is no interaction between the SPP and the nanowires and we do not have any effect. (b). When the distance decreases, the effect of the nanowire array may be considered as a perturbation Δε of the dielectric constant of the waveguide and the coupling between guided and radiation modes occurs. Changing the distance h, one can control the intensity of the transmitted SPP.

Fig. 2
Fig. 2

(a) Sketch of a thin metal film waveguide. The guide axis is chosen to coincide with the x-axis, the core is the metal with the dielectric function ε 1. Also shown is a spatial distribution of the longitudinal electric field magnitude of guided and radiation modes (notations “S” and “AS” correspond to the symmetric and antisymmetric modes, respectively). (b). Dispersion curves of the SPP. The blue line corresponds to the symmetric guided mode, the red line to the antisymmetric guided mode. Radiation modes are placed in the region “RR”. The metal is assumed to be described by the general Drude model ε 1 = ε r ω p 2 / ω 2 , where ε r is the high frequency dielectric constant and ω p is the plasma frequency.

Fig. 3
Fig. 3

(a). Sketch of the structure under investigation (the nanobelt array near the Au film). (b). Spatial spectrum of the function f(x), F ( β ) = | f ^ ( β ) | , β g is the wavenumber of the guided antisymmetric mode (superscript “AS” is omitted for the sake of brevity). (c). Dependence of the coupling coefficients on β. (d). Radiation loss power spectrum, β is assumed to be positive, so all curves are located in the right-hand part of the plot. For all figures, the following parameters are used: h = 500 nm, u = b = 80 nm, l = 500 nm, N = 10, 2πc/ω = 800 nm.

Fig. 4
Fig. 4

(a). Sketch of the nanowire array above the Au film. (b). Spatial spectrum of the radiation loss power for the array of nanowires with a circular cross-section, r = 40 nm, h = 500 nm, l = 500 nm, N = 10, 2πc/ω = 800 nm. The normalized total radiation loss power is equal to 4.3%.

Fig. 5
Fig. 5

Dependence of the normalized total radiation loss power on the gap between the nanobelt array and the metal film for different values of the nanobelt transverse size, l = 500 nm, N = 10, 2πc/ω = 800 nm.

Fig. 6
Fig. 6

(a). Dependence of the normalized total radiation loss power on the number of nanobelts, h = 500 nm, l = 500 nm, N = 10, 2πc/ω = 800 nm. (b). Dependence of the normalized total radiation loss power on the permittivity of the nanobelts, h = 500 nm, l = 500 nm, N = 5, 2πc/ω = 800 nm.

Equations (19)

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exp ( 2 κ 1 a ) = ± κ 1 + κ 2 ε 1 κ 1 κ 2 ε 1 .
{ H y ( z ) = D exp ( κ 2 z ) , | z | > a H y ( z ) = C [ exp ( κ 1 z ) + exp ( κ 1 z ) ] , | z | a D = 2 C cosh ( κ 1 a ) exp ( κ 2 a )
{ H y ( z ) = sgn ( z ) D exp ( κ 2 z ) , | z | > a H y ( z ) = C [ exp ( κ 1 z ) exp ( κ 1 z ) ] , | z | a D = 2 C sinh ( κ 1 a ) exp ( κ 2 a )
{ H y ( z ) = D exp ( i χ 2 | z | ) + D * exp ( i χ 2 | z | ) , | z | > a H y ( z ) = C [ exp ( κ 1 z ) + exp ( κ 1 z ) ] , | z | a D = C [ cosh ( κ 1 a ) + i κ 1 χ 2 ε 1 sinh ( κ 1 a ) ] exp ( i χ 2 a )
{ H y ( z ) = sgn ( z ) [ D exp ( i χ 2 | z | ) + D * exp ( i χ 2 | z | ) ] , | z | > a H y ( z ) = C [ exp ( κ 1 z ) exp ( κ 1 z ) ] , | z | a D = C [ sinh ( κ 1 a ) + i κ 1 χ 2 ε 1 cosh ( κ 1 a ) ] exp ( i χ 2 a )
D = G g AS = { c 2 π β k exp ( 2 κ 2 a ) ( 1 κ 2 + 1 κ 1 ε 1 [ tanh ( κ 1 a ) + κ 1 a cosh 2 ( κ 1 a ) ] ) } 1 / 2 ,
D = G g S = { c 2 π β k exp ( 2 κ 2 a ) ( 1 κ 2 + 1 κ 1 ε 1 [ 1 tanh ( κ 1 a ) κ 1 a sinh 2 ( κ 1 a ) ] ) } 1 / 2 ,
D = G r AS = ω e 2 i χ 2 a 2 β c 2 [ cosh 2 ( κ 1 a ) + κ 1 2 χ 2 2 ε 1 2 sinh 2 ( κ 1 a ) ] 1 2 [ cosh ( κ 1 a ) + i κ 1 χ 2 ε 1 sinh ( κ 1 a ) ]
D = G r S = ω e 2 i χ 2 a 2 β c 2 [ sinh 2 ( κ 1 a ) + κ 1 2 χ 2 2 ε 1 2 cosh 2 ( κ 1 a ) ] 1 2 [ sinh ( κ 1 a ) + i κ 1 χ 2 ε 1 cosh ( κ 1 a ) ] .
E z = μ ( A μ exp ( i β μ x ) + B μ exp ( i β μ x ) ) E z μ ,
H y = μ ( A μ exp ( i β μ x ) B μ exp ( i β μ x ) ) H y μ .
{ d A μ d x = i ν [ A ν ( K ν μ z + K ν μ x ) exp ( i ( β ν β μ ) x ) + B ν ( K ν μ z K ν μ x ) exp ( i ( β ν + β μ ) x ) ] d B μ d x = i ν [ A ν ( K ν μ z K ν μ x ) exp ( i ( β ν + β μ ) x ) + B ν ( K ν μ z + K ν μ x ) exp ( i ( β ν β μ ) x ) ]
{ d A μ ( x ) d x = i A g AS [ K μ z ( x ) + K μ x ( x ) ] exp ( i ( β g AS β μ ) x ) d B μ ( x ) d x = i A g AS [ K μ z ( x ) K μ x ( x ) ] exp ( i ( β g AS + β μ ) x )
{ A μ ( N l ) = 0 N l ( d A μ ( x ) d x )   d x B μ ( 0 ) = N l 0 ( d B μ ( x ) d x )   d x
{ A r AS ( β , N l ) = i A g AS ( 0 ) 0 N l [ K rAS z ( β , x ) + K rAS x ( β , x ) ] exp ( i ( β g AS β ) x ) d x B r AS ( β , 0 ) = i A g AS ( 0 ) N l 0 [ K rAS z ( β , x ) K rAS x ( β , x ) ] exp ( i ( β g AS + β ) x ) d x A r S ( β , N l ) = i A g AS ( 0 ) 0 N l [ K rS z ( β , x ) + K rS x ( β , x ) ] exp ( i ( β g AS β ) x ) d x B r S ( β , 0 ) = i A g AS ( 0 ) N l 0 [ K rS z ( β , x ) K rS x ( β , x ) ] exp ( i ( β g AS + β ) x ) d x
f ^ ( β ) = e i β u 1 i β e i β N l 1 e i β l 1 ,
K η x ( β ) = ω 4 π + Δ ε Δ ε +1 E x gAS ( z ) E x η * ( β , z)   d z ,
K η z ( β ) = ω 4 π + Δ ε E z gAS ( z ) E z η * ( β , z)   d z .
K η z ( h , β ) = ω ( ε d 1 ) β g AS β e κ 2 gAS h 4 π k 2 { G g AS G r η * e + i χ 2 η h ( 1 e ( κ 2 gAS i χ 2 η ) b ) κ 2 gAS i χ 2 η + + G g AS G r η e i χ 2 η h ( 1 e ( κ 2 gAS + i χ 2 η ) b ) κ 2 gAS + i χ 2 η } .

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