Abstract

We theoretically investigate composite cylindrical nanowires for the waveguiding of the lowest-order surface plasmon-polariton (SPP) mode. We find that the confinement of the SPP fields in a metallic nanowire can be significantly improved by a dielectric cladding and show that by adjusting the thickness of the optically-pumped cladding, the gain required to compensate for the losses can be minimized. If this structure is coated with an additional metal layer to form a metal–dielectric–metal (MDM) nanowire, we show that the field can be predominantly confined within the dielectric layer, to have amplitudes of three orders of magnitude higher than those in the metallic regions. We also show that the propagation lengths of SPPs can be maximized by the proper selection of the geometrical parameters. We further demonstrate that the mode is strongly confined in subwavelength scale, e.g., ∼λ02/1220 for a 60-nm-thick nanowire, where λ 0 is the wavelength in vacuum. We also find that regardless of the size of nanowire, it is possible to carry over 98.5% of the mode energy within the nanowire. In addition, we demonstrate that by appropriate choice of the material thicknesses, the losses of an MDM nanowire can be compensated by a considerably low level of optical gain in the dielectric region. For example, the losses of a 260-nm-thick Ag–ZnO–Ag nanowire can be entirely compensated by a gain of ∼ 400 cm−1. Our results will be useful for the optimum design of nanowires as interconnects for high-density nanophotonic circuit integration.

©2011 Optical Society of America

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    [Crossref]

2011 (1)

J. Lo, W. Lien, C. Lin, and J. He, “Er-doped ZnO nanorod arrays with enhanced 1540 nm emission by employing Ag island films and high-temperature annealing,” ACS Appl. Mater. Interfaces 3, 1009–1014 (2011).
[Crossref] [PubMed]

2010 (5)

2009 (2)

2008 (5)

J. Chen, G. A. Smolyakov, S. R. J. Brueck, and K. J. Malloy, “Surface plasmon modes of finite, planar, metal-insulator-metal plasmonic waveguides,” Opt. Express 16, 14902–14909 (2008).
[Crossref] [PubMed]

M. A. Noginov, V. A. Podolskiy, G. Zhu, M. Mayy, M. Bahoura, J. A. Adegoke, B. A. Ritzo, and K. Reynolds, “Compensation of loss in propagating surface plasmon polariton by gain in adjacent dielectric medium,” Opt. Express 16, 1385–1392 (2008).
[Crossref] [PubMed]

V. Krishnamurthy and B. Klein, “Theoretical investigation of metal cladding for nanowire and cylindrical micro-post lasers,” IEEE J. Quantum Electron. 44, 67–74 (2008).
[Crossref]

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

R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwave-length plasmonic modes,” N. J. Phys. 10, 105018 (2008).
[Crossref]

2007 (2)

J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Theoretical analysis of square surface plasmon-polariton waveguides for long-range polarization-independent waveguiding,” Phys. Rev. B 76, 035434 (2007).
[Crossref]

H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296, 56–63 (2007).
[Crossref] [PubMed]

2006 (6)

S. A. Maier, “Plasmonics: The promise of highly integrated optical devices,” IEEE Sel. Top. Quantum Electron. 12, 1671–1677 (2006).
[Crossref]

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[Crossref] [PubMed]

S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258, 295–299 (2006).
[Crossref]

W. L. Barnes, “Surface plasmon–polariton length scales: A route to sub-wavelength optics,” J. Opt. A: Pure Appl. Opt. 8, S87–S93 (2006).
[Crossref]

J. Wang, S. K. Hark, and Q. Li, “Electronic structure and luminescence properties of Er doped ZnO nanowires,” Microsc. Microanal. 12, 748–749 (2006).
[Crossref]

Z. K. Tang, M. Kawasaki, A. Ohtomo, H. Koinuma, and Y. Segawa, “Self-assembled ZnO nano-crystals and exciton lasing at room temperature,” J. Cryst. Growth 287, 169–179 (2006).
[Crossref]

2005 (3)

D. F. P. Pile, T. Ogawa, D. K. Gramontnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimetnal investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. 87, 061106 (2005).
[Crossref]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by sub-wavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[Crossref] [PubMed]

S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
[Crossref]

2004 (2)

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–829 (2003).
[Crossref] [PubMed]

2002 (2)

E. H. K. Stelzer, “Beyond the diffraction limit?,” Nature 417, 806–807 (2002).
[Crossref] [PubMed]

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys.: Condens. Matter 14, R597–R624 (2002).
[Crossref]

2001 (2)

U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64, 125420(1–10) (2001).
[Crossref]

Y. Chen, N. T. Tuan, Y. Segawa, H. Ko, S. Hong, and T. Yao, “Stimulated emission and optical gain in ZnO epilayers grown by plasma-assisted molecular-beam epitaxy with buffers,” Appl. Phys. Lett. 78, 1469–1471 (2001).
[Crossref]

1999 (1)

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: Review,” Sens. Actuators B 54, 3–15 (1999).
[Crossref]

1998 (1)

1997 (3)

C. Untiedt, G. Rubio, S. Vieira, and N. Agraït, “Fabrication and characterization of metallic nanowires,” Phys. Rev. B 56, 2154–2160 (1997).
[Crossref]

J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. 22, 475–477 (1997).
[Crossref] [PubMed]

J. Koglin, U. C. Fischer, and H. Fuchs, “Material contrast in scanning near-field optical microscopy at 1–10 nm resolution,” Phys. Rev. B 55, 7977–7984 (1997).
[Crossref]

1995 (1)

L. Novotny, D. W. Pohl, and B. Hecht, “Light confinement in scanning near-field optical microscopy,” Ultramicroscopy 61, 1–9 (1995).
[Crossref]

1981 (1)

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

1974 (1)

C. A. Pfeiffer and E. N. Economou, “Surface polaritons in a circularly cylindrical interface: surface plasmons,” Phys. Rev. B 10, 3038–3051 (1974).
[Crossref]

Adachi, S.

S. Adachi, Optical Constants of Crystalline and Amorphous Semiconductors - Numerical Data and Graphical Information (Springer, 1999).
[Crossref] [PubMed]

Adegoke, J. A.

Agraït, N.

C. Untiedt, G. Rubio, S. Vieira, and N. Agraït, “Fabrication and characterization of metallic nanowires,” Phys. Rev. B 56, 2154–2160 (1997).
[Crossref]

Agrawal, G. P.

A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18, 6191–6204 (2010).
[Crossref] [PubMed]

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

M. Premaratne and G. P. Agrawal, Light Propagation in Gain Media: Optical Amplifiers (Cambridge University Press, 2011).

Atwater, H. A.

H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296, 56–63 (2007).
[Crossref] [PubMed]

S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
[Crossref]

Aussenegg, F. R.

Ayyub, P.

S. Chatterjee, O. D. Jayakumar, A. K. Tyagi, and P. Ayyub, “Template-based fabrication of Ag–ZnO core–shell nanorod arrays,” J. Cryst. Growth 312, 2724–2728 (2010).
[Crossref]

Bahoura, M.

Baranov, A. V.

Barnes, W. L.

W. L. Barnes, “Surface plasmon–polariton length scales: A route to sub-wavelength optics,” J. Opt. A: Pure Appl. Opt. 8, S87–S93 (2006).
[Crossref]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–829 (2003).
[Crossref] [PubMed]

Bartal, G.

R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwave-length plasmonic modes,” N. J. Phys. 10, 105018 (2008).
[Crossref]

Bozhevolnyi, S. I.

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

J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Theoretical analysis of square surface plasmon-polariton waveguides for long-range polarization-independent waveguiding,” Phys. Rev. B 76, 035434 (2007).
[Crossref]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by sub-wavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[Crossref] [PubMed]

Brueck, S. R. J.

Chatterjee, S.

S. Chatterjee, O. D. Jayakumar, A. K. Tyagi, and P. Ayyub, “Template-based fabrication of Ag–ZnO core–shell nanorod arrays,” J. Cryst. Growth 312, 2724–2728 (2010).
[Crossref]

Chen, D.

Chen, J.

Chen, Y.

Y. Chen, N. T. Tuan, Y. Segawa, H. Ko, S. Hong, and T. Yao, “Stimulated emission and optical gain in ZnO epilayers grown by plasma-assisted molecular-beam epitaxy with buffers,” Appl. Phys. Lett. 78, 1469–1471 (2001).
[Crossref]

Dasari, R. R.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys.: Condens. Matter 14, R597–R624 (2002).
[Crossref]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–829 (2003).
[Crossref] [PubMed]

U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64, 125420(1–10) (2001).
[Crossref]

Devaux, E.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by sub-wavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[Crossref] [PubMed]

Ebbesen, T. W.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by sub-wavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[Crossref] [PubMed]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–829 (2003).
[Crossref] [PubMed]

Economou, E. N.

C. A. Pfeiffer and E. N. Economou, “Surface polaritons in a circularly cylindrical interface: surface plasmons,” Phys. Rev. B 10, 3038–3051 (1974).
[Crossref]

Fainman, Y.

Fedorov, A. V.

Feld, M. S.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys.: Condens. Matter 14, R597–R624 (2002).
[Crossref]

Fischer, U. C.

J. Koglin, U. C. Fischer, and H. Fuchs, “Material contrast in scanning near-field optical microscopy at 1–10 nm resolution,” Phys. Rev. B 55, 7977–7984 (1997).
[Crossref]

Fuchs, H.

J. Koglin, U. C. Fischer, and H. Fuchs, “Material contrast in scanning near-field optical microscopy at 1–10 nm resolution,” Phys. Rev. B 55, 7977–7984 (1997).
[Crossref]

Fukui, M.

D. F. P. Pile, T. Ogawa, D. K. Gramontnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimetnal investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. 87, 061106 (2005).
[Crossref]

Gauglitz, G.

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: Review,” Sens. Actuators B 54, 3–15 (1999).
[Crossref]

Genov, D. A.

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

Gramontnev, D. K.

D. F. P. Pile, T. Ogawa, D. K. Gramontnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimetnal investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. 87, 061106 (2005).
[Crossref]

Gramotnev, D. K.

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

Handapangoda, D.

Haraguchi, M.

D. F. P. Pile, T. Ogawa, D. K. Gramontnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimetnal investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. 87, 061106 (2005).
[Crossref]

Hark, S. K.

J. Wang, S. K. Hark, and Q. Li, “Electronic structure and luminescence properties of Er doped ZnO nanowires,” Microsc. Microanal. 12, 748–749 (2006).
[Crossref]

Hattori, H. T.

He, J.

J. Lo, W. Lien, C. Lin, and J. He, “Er-doped ZnO nanorod arrays with enhanced 1540 nm emission by employing Ag island films and high-temperature annealing,” ACS Appl. Mater. Interfaces 3, 1009–1014 (2011).
[Crossref] [PubMed]

Hecht, B.

L. Novotny, D. W. Pohl, and B. Hecht, “Light confinement in scanning near-field optical microscopy,” Ultramicroscopy 61, 1–9 (1995).
[Crossref]

Hecht, J.

J. Hecht, The Laser Guidebook (McGraw-Hill, 1992).

Homola, J.

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: Review,” Sens. Actuators B 54, 3–15 (1999).
[Crossref]

Hong, S.

Y. Chen, N. T. Tuan, Y. Segawa, H. Ko, S. Hong, and T. Yao, “Stimulated emission and optical gain in ZnO epilayers grown by plasma-assisted molecular-beam epitaxy with buffers,” Appl. Phys. Lett. 78, 1469–1471 (2001).
[Crossref]

Itzkan, I.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys.: Condens. Matter 14, R597–R624 (2002).
[Crossref]

Jagadish, C.

Jayakumar, O. D.

S. Chatterjee, O. D. Jayakumar, A. K. Tyagi, and P. Ayyub, “Template-based fabrication of Ag–ZnO core–shell nanorod arrays,” J. Cryst. Growth 312, 2724–2728 (2010).
[Crossref]

Jung, J.

J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Theoretical analysis of square surface plasmon-polariton waveguides for long-range polarization-independent waveguiding,” Phys. Rev. B 76, 035434 (2007).
[Crossref]

Kawasaki, M.

Z. K. Tang, M. Kawasaki, A. Ohtomo, H. Koinuma, and Y. Segawa, “Self-assembled ZnO nano-crystals and exciton lasing at room temperature,” J. Cryst. Growth 287, 169–179 (2006).
[Crossref]

Klein, B.

V. Krishnamurthy and B. Klein, “Theoretical investigation of metal cladding for nanowire and cylindrical micro-post lasers,” IEEE J. Quantum Electron. 44, 67–74 (2008).
[Crossref]

Kneipp, H.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys.: Condens. Matter 14, R597–R624 (2002).
[Crossref]

Kneipp, K.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys.: Condens. Matter 14, R597–R624 (2002).
[Crossref]

Ko, H.

Y. Chen, N. T. Tuan, Y. Segawa, H. Ko, S. Hong, and T. Yao, “Stimulated emission and optical gain in ZnO epilayers grown by plasma-assisted molecular-beam epitaxy with buffers,” Appl. Phys. Lett. 78, 1469–1471 (2001).
[Crossref]

Kobayashi, T.

Koglin, J.

J. Koglin, U. C. Fischer, and H. Fuchs, “Material contrast in scanning near-field optical microscopy at 1–10 nm resolution,” Phys. Rev. B 55, 7977–7984 (1997).
[Crossref]

Koinuma, H.

Z. K. Tang, M. Kawasaki, A. Ohtomo, H. Koinuma, and Y. Segawa, “Self-assembled ZnO nano-crystals and exciton lasing at room temperature,” J. Cryst. Growth 287, 169–179 (2006).
[Crossref]

Krenn, J. R.

Krishnamurthy, V.

V. Krishnamurthy and B. Klein, “Theoretical investigation of metal cladding for nanowire and cylindrical micro-post lasers,” IEEE J. Quantum Electron. 44, 67–74 (2008).
[Crossref]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).

Leitner, A.

Li, Q.

J. Wang, S. K. Hark, and Q. Li, “Electronic structure and luminescence properties of Er doped ZnO nanowires,” Microsc. Microanal. 12, 748–749 (2006).
[Crossref]

Lien, W.

J. Lo, W. Lien, C. Lin, and J. He, “Er-doped ZnO nanorod arrays with enhanced 1540 nm emission by employing Ag island films and high-temperature annealing,” ACS Appl. Mater. Interfaces 3, 1009–1014 (2011).
[Crossref] [PubMed]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).

Lin, C.

J. Lo, W. Lien, C. Lin, and J. He, “Er-doped ZnO nanorod arrays with enhanced 1540 nm emission by employing Ag island films and high-temperature annealing,” ACS Appl. Mater. Interfaces 3, 1009–1014 (2011).
[Crossref] [PubMed]

Lo, J.

J. Lo, W. Lien, C. Lin, and J. He, “Er-doped ZnO nanorod arrays with enhanced 1540 nm emission by employing Ag island films and high-temperature annealing,” ACS Appl. Mater. Interfaces 3, 1009–1014 (2011).
[Crossref] [PubMed]

Maier, S. A.

S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258, 295–299 (2006).
[Crossref]

S. A. Maier, “Plasmonics: The promise of highly integrated optical devices,” IEEE Sel. Top. Quantum Electron. 12, 1671–1677 (2006).
[Crossref]

S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
[Crossref]

S. A. Maier, Plasmonics: Fundamentals and Applications, (Springer, 2007).

Malloy, K. J.

Matsuo, S.

D. F. P. Pile, T. Ogawa, D. K. Gramontnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimetnal investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. 87, 061106 (2005).
[Crossref]

Mayy, M.

Morimoto, A.

Morkoç, H.

H. Morkoç and Ü. Özgür, Zinc Oxide: Fundamentals, Materials and Device Technology, (Wiley-VCH Verlag GmbH & Co. KGaA, 2009).

Nezhad, M. P.

Noginov, M. A.

Novotny, L.

L. Novotny, D. W. Pohl, and B. Hecht, “Light confinement in scanning near-field optical microscopy,” Ultramicroscopy 61, 1–9 (1995).
[Crossref]

Ogawa, T.

D. F. P. Pile, T. Ogawa, D. K. Gramontnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimetnal investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. 87, 061106 (2005).
[Crossref]

Ohtomo, A.

Z. K. Tang, M. Kawasaki, A. Ohtomo, H. Koinuma, and Y. Segawa, “Self-assembled ZnO nano-crystals and exciton lasing at room temperature,” J. Cryst. Growth 287, 169–179 (2006).
[Crossref]

Okamoto, T.

D. F. P. Pile, T. Ogawa, D. K. Gramontnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimetnal investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. 87, 061106 (2005).
[Crossref]

Oulton, R. F.

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

R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwave-length plasmonic modes,” N. J. Phys. 10, 105018 (2008).
[Crossref]

Ozbay, E.

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[Crossref] [PubMed]

Özgür, Ü.

H. Morkoç and Ü. Özgür, Zinc Oxide: Fundamentals, Materials and Device Technology, (Wiley-VCH Verlag GmbH & Co. KGaA, 2009).

Pannipitiya, A.

Pfeiffer, C. A.

C. A. Pfeiffer and E. N. Economou, “Surface polaritons in a circularly cylindrical interface: surface plasmons,” Phys. Rev. B 10, 3038–3051 (1974).
[Crossref]

Pile, D. F. P.

R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwave-length plasmonic modes,” N. J. Phys. 10, 105018 (2008).
[Crossref]

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

D. F. P. Pile, T. Ogawa, D. K. Gramontnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimetnal investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. 87, 061106 (2005).
[Crossref]

Podolskiy, V. A.

Pohl, D. W.

L. Novotny, D. W. Pohl, and B. Hecht, “Light confinement in scanning near-field optical microscopy,” Ultramicroscopy 61, 1–9 (1995).
[Crossref]

Pozar, D. M.

D. M. Pozar, Microwave Engineering, (Wiley, 1998).

Premaratne, M.

Quinten, M.

Reynolds, K.

Ritzo, B. A.

Rubio, G.

C. Untiedt, G. Rubio, S. Vieira, and N. Agraït, “Fabrication and characterization of metallic nanowires,” Phys. Rev. B 56, 2154–2160 (1997).
[Crossref]

Rukhlenko, I. D.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[Crossref]

Sarid, D.

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

Schröter, U.

U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64, 125420(1–10) (2001).
[Crossref]

Segawa, Y.

Z. K. Tang, M. Kawasaki, A. Ohtomo, H. Koinuma, and Y. Segawa, “Self-assembled ZnO nano-crystals and exciton lasing at room temperature,” J. Cryst. Growth 287, 169–179 (2006).
[Crossref]

Y. Chen, N. T. Tuan, Y. Segawa, H. Ko, S. Hong, and T. Yao, “Stimulated emission and optical gain in ZnO epilayers grown by plasma-assisted molecular-beam epitaxy with buffers,” Appl. Phys. Lett. 78, 1469–1471 (2001).
[Crossref]

Smolyakov, G. A.

Søndergaard, T.

J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Theoretical analysis of square surface plasmon-polariton waveguides for long-range polarization-independent waveguiding,” Phys. Rev. B 76, 035434 (2007).
[Crossref]

Sorger, V. J.

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

Stelzer, E. H. K.

E. H. K. Stelzer, “Beyond the diffraction limit?,” Nature 417, 806–807 (2002).
[Crossref] [PubMed]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory, (McGraw-Hill, 1941).

Su, W.

P. Zhao, W. Su, R. Wang, X. Xu, and F. Zhang, “Properties of thin silver films with different thickness,” Physica E 41, 387–390 (2009).
[Crossref]

Takahara, J.

Taki, H.

Tang, Z. K.

Z. K. Tang, M. Kawasaki, A. Ohtomo, H. Koinuma, and Y. Segawa, “Self-assembled ZnO nano-crystals and exciton lasing at room temperature,” J. Cryst. Growth 287, 169–179 (2006).
[Crossref]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[Crossref]

Tetz, K.

Tuan, N. T.

Y. Chen, N. T. Tuan, Y. Segawa, H. Ko, S. Hong, and T. Yao, “Stimulated emission and optical gain in ZnO epilayers grown by plasma-assisted molecular-beam epitaxy with buffers,” Appl. Phys. Lett. 78, 1469–1471 (2001).
[Crossref]

Tyagi, A. K.

S. Chatterjee, O. D. Jayakumar, A. K. Tyagi, and P. Ayyub, “Template-based fabrication of Ag–ZnO core–shell nanorod arrays,” J. Cryst. Growth 312, 2724–2728 (2010).
[Crossref]

Untiedt, C.

C. Untiedt, G. Rubio, S. Vieira, and N. Agraït, “Fabrication and characterization of metallic nanowires,” Phys. Rev. B 56, 2154–2160 (1997).
[Crossref]

Vieira, S.

C. Untiedt, G. Rubio, S. Vieira, and N. Agraït, “Fabrication and characterization of metallic nanowires,” Phys. Rev. B 56, 2154–2160 (1997).
[Crossref]

Vogel, M. W.

M. W. Vogel, “Theoretical and numerical investigation of plasmon nanofocusing in metallic tapered rods and grooves,” PhD thesis (Queensland University of Technology, Australia, 2009).

Volkov, V. S.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by sub-wavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[Crossref] [PubMed]

Wang, B.

Wang, G. P.

Wang, J.

J. Wang, S. K. Hark, and Q. Li, “Electronic structure and luminescence properties of Er doped ZnO nanowires,” Microsc. Microanal. 12, 748–749 (2006).
[Crossref]

Wang, R.

P. Zhao, W. Su, R. Wang, X. Xu, and F. Zhang, “Properties of thin silver films with different thickness,” Physica E 41, 387–390 (2009).
[Crossref]

Xu, X.

P. Zhao, W. Su, R. Wang, X. Xu, and F. Zhang, “Properties of thin silver films with different thickness,” Physica E 41, 387–390 (2009).
[Crossref]

Yamagishi, S.

Yao, T.

Y. Chen, N. T. Tuan, Y. Segawa, H. Ko, S. Hong, and T. Yao, “Stimulated emission and optical gain in ZnO epilayers grown by plasma-assisted molecular-beam epitaxy with buffers,” Appl. Phys. Lett. 78, 1469–1471 (2001).
[Crossref]

Yee, S. S.

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: Review,” Sens. Actuators B 54, 3–15 (1999).
[Crossref]

Zhang, F.

P. Zhao, W. Su, R. Wang, X. Xu, and F. Zhang, “Properties of thin silver films with different thickness,” Physica E 41, 387–390 (2009).
[Crossref]

Zhang, X.

R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwave-length plasmonic modes,” N. J. Phys. 10, 105018 (2008).
[Crossref]

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

Zhao, P.

P. Zhao, W. Su, R. Wang, X. Xu, and F. Zhang, “Properties of thin silver films with different thickness,” Physica E 41, 387–390 (2009).
[Crossref]

Zhu, G.

ACS Appl. Mater. Interfaces (1)

J. Lo, W. Lien, C. Lin, and J. He, “Er-doped ZnO nanorod arrays with enhanced 1540 nm emission by employing Ag island films and high-temperature annealing,” ACS Appl. Mater. Interfaces 3, 1009–1014 (2011).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

D. F. P. Pile, T. Ogawa, D. K. Gramontnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimetnal investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. 87, 061106 (2005).
[Crossref]

Y. Chen, N. T. Tuan, Y. Segawa, H. Ko, S. Hong, and T. Yao, “Stimulated emission and optical gain in ZnO epilayers grown by plasma-assisted molecular-beam epitaxy with buffers,” Appl. Phys. Lett. 78, 1469–1471 (2001).
[Crossref]

IEEE J. Quantum Electron. (1)

V. Krishnamurthy and B. Klein, “Theoretical investigation of metal cladding for nanowire and cylindrical micro-post lasers,” IEEE J. Quantum Electron. 44, 67–74 (2008).
[Crossref]

IEEE Sel. Top. Quantum Electron. (1)

S. A. Maier, “Plasmonics: The promise of highly integrated optical devices,” IEEE Sel. Top. Quantum Electron. 12, 1671–1677 (2006).
[Crossref]

J. Appl. Phys. (1)

S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
[Crossref]

J. Cryst. Growth (2)

Z. K. Tang, M. Kawasaki, A. Ohtomo, H. Koinuma, and Y. Segawa, “Self-assembled ZnO nano-crystals and exciton lasing at room temperature,” J. Cryst. Growth 287, 169–179 (2006).
[Crossref]

S. Chatterjee, O. D. Jayakumar, A. K. Tyagi, and P. Ayyub, “Template-based fabrication of Ag–ZnO core–shell nanorod arrays,” J. Cryst. Growth 312, 2724–2728 (2010).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

W. L. Barnes, “Surface plasmon–polariton length scales: A route to sub-wavelength optics,” J. Opt. A: Pure Appl. Opt. 8, S87–S93 (2006).
[Crossref]

J. Phys.: Condens. Matter (1)

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Surface-enhanced Raman scattering and biophysics,” J. Phys.: Condens. Matter 14, R597–R624 (2002).
[Crossref]

Microsc. Microanal. (1)

J. Wang, S. K. Hark, and Q. Li, “Electronic structure and luminescence properties of Er doped ZnO nanowires,” Microsc. Microanal. 12, 748–749 (2006).
[Crossref]

N. J. Phys. (1)

R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwave-length plasmonic modes,” N. J. Phys. 10, 105018 (2008).
[Crossref]

Nat. Photonics (2)

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

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

Nature (2)

E. H. K. Stelzer, “Beyond the diffraction limit?,” Nature 417, 806–807 (2002).
[Crossref] [PubMed]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–829 (2003).
[Crossref] [PubMed]

Opt. Commun. (1)

S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258, 295–299 (2006).
[Crossref]

Opt. Express (5)

Opt. Lett. (4)

Phys. Rev. B (5)

U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64, 125420(1–10) (2001).
[Crossref]

C. Untiedt, G. Rubio, S. Vieira, and N. Agraït, “Fabrication and characterization of metallic nanowires,” Phys. Rev. B 56, 2154–2160 (1997).
[Crossref]

C. A. Pfeiffer and E. N. Economou, “Surface polaritons in a circularly cylindrical interface: surface plasmons,” Phys. Rev. B 10, 3038–3051 (1974).
[Crossref]

J. Jung, T. Søndergaard, and S. I. Bozhevolnyi, “Theoretical analysis of square surface plasmon-polariton waveguides for long-range polarization-independent waveguiding,” Phys. Rev. B 76, 035434 (2007).
[Crossref]

J. Koglin, U. C. Fischer, and H. Fuchs, “Material contrast in scanning near-field optical microscopy at 1–10 nm resolution,” Phys. Rev. B 55, 7977–7984 (1997).
[Crossref]

Phys. Rev. Lett. (2)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by sub-wavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[Crossref] [PubMed]

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

Physica E (1)

P. Zhao, W. Su, R. Wang, X. Xu, and F. Zhang, “Properties of thin silver films with different thickness,” Physica E 41, 387–390 (2009).
[Crossref]

Sci. Am. (1)

H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296, 56–63 (2007).
[Crossref] [PubMed]

Science (1)

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[Crossref] [PubMed]

Sens. Actuators B (1)

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: Review,” Sens. Actuators B 54, 3–15 (1999).
[Crossref]

Ultramicroscopy (1)

L. Novotny, D. W. Pohl, and B. Hecht, “Light confinement in scanning near-field optical microscopy,” Ultramicroscopy 61, 1–9 (1995).
[Crossref]

Other (11)

S. A. Maier, Plasmonics: Fundamentals and Applications, (Springer, 2007).

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

D. M. Pozar, Microwave Engineering, (Wiley, 1998).

J. A. Stratton, Electromagnetic Theory, (McGraw-Hill, 1941).

M. W. Vogel, “Theoretical and numerical investigation of plasmon nanofocusing in metallic tapered rods and grooves,” PhD thesis (Queensland University of Technology, Australia, 2009).

S. Adachi, Optical Constants of Crystalline and Amorphous Semiconductors - Numerical Data and Graphical Information (Springer, 1999).
[Crossref] [PubMed]

M. Premaratne and G. P. Agrawal, Light Propagation in Gain Media: Optical Amplifiers (Cambridge University Press, 2011).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[Crossref]

J. Hecht, The Laser Guidebook (McGraw-Hill, 1992).

H. Morkoç and Ü. Özgür, Zinc Oxide: Fundamentals, Materials and Device Technology, (Wiley-VCH Verlag GmbH & Co. KGaA, 2009).

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).

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Figures (10)

Fig. 1
Fig. 1

Composite cylindrical nanowire consisting of (n – 2) shells (left) and its cross section (right). The jth medium is characterized by the permittivity ɛj and the outer radius Rj .

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Fig. 2
Fig. 2

(a) Dispersion of pure silver (Ag) nanowire and Ag–ZnO nanowire (blue curves), the corresponding SPP wavelength (red curves), and the attenuation coefficient (green curves). Black is the light line in air. Density plots of the electric field amplitudes along (b) Ag nanowire with R 1 = 100 nm, and along (c) Ag–ZnO nanowire with R 1 = 20 nm, R 2 = 100 nm, and ɛ 2 = 3.725 [39]. In (b) and (c), the dashed-dotted white lines represent the axes of the nanowires, and the solid white lines denote the material interfaces. In all three panels, it is assumed that λ 0 = 1.55 μm, ɛ ∞ = 9.6, ω p = 3.76 eV, and δ = 13 meV. The surrounding medium is assumed to be air.

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Fig. 3
Fig. 3

The metallic nanowire with dielectric cladding (left) and its cross-section (right).

Download Full Size | PPT Slide | PDF

Fig. 4
Fig. 4

(a) Effect of gain on the propagation length of SPPs. The quantity γc is defined as the limit at which L SPP → ∞. (b) Variation of critical gain with relative cladding thickness. The critical gain is minimal (γc = γ 0) at q = q 0.

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Fig. 5
Fig. 5

(a) Minimum gain required for loss compensation (blue) and normalized SPP wavelength (red), as a function of nanowire size. (b) Optimum values of relative cladding thickness (blue) and the dielectric thickness (red). (c) Variation of the power containment factor with nanowire size.

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Fig. 6
Fig. 6

Metal–dielectric–metal (MDM) nanowire and the notations employed.

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Fig. 7
Fig. 7

(a) Dispersion of SPPs in a Ag–ZnO–Ag nanowire surrounded by air, with R 1 = 40 nm, R 2 = 80 nm, and R 3 = 100 nm; the dotted line in the inset represents the wavelength λ 0 = 1.55 μm. (b) Variation of Poynting vector in the radial direction, corresponding to the points A and B in (a).

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Fig. 8
Fig. 8

Absolute value of the electric field in the longitudinal cross-section of (a) a Ag–SiO2–Si nanowire and (b) a Ag–ZnO–Ag nanowire. (c) Radial distribution of the normalized electric field amplitude, for the cases shown in (a) (blue curve) and (b) (red curve). The geometrical parameters of the nanowires are the same as in Fig. 7.

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Fig. 9
Fig. 9

Variation of the propagation length of SPPs when (a) R 1 = 50 nm, (b) d = 40 nm, and (c) R 3 = 100 nm. Effect of nanowire size on (d) maximum values of L SPP, (e) normalized effective mode area (blue) and relative dielectric area (red), and (f) power containment factor.

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Fig. 10
Fig. 10

Variation of (a) γc for a fixed nanowire size and (b) min{γc } with nanowire size. The blue and red curves in (b) correspond to the MDM nanowires and metallic nanowires with dielectric cladding, respectively.

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Equations (20)

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

E ( ρ , z )   =   E ( ρ )   exp ( i β z )   =   [ E ρ   ( ρ )   e ρ   +   E z   ( ρ )   e z ]   exp ( i β z ) ,
H φ   ( ρ , z )   =   1 k   [ β E ρ   ( ρ )   +   i d E z d ρ ]   exp ( i β z ) ,
E ( j )   ( ρ )   =   − i ( β / α j ) χ j ( α j ρ )   e ρ   +   ψ j   ( α j ρ )   e z ,
H φ ( j )   ( ρ )   =   − i k ( ɛ j / α j ) χ j ( α j ρ ) ,
ψ j ( x )   =   A j I 0 ( x )   −   i B j K 0 ( x ) ,
E z ( j )   ( R j )   =   E z ( j + 1 )   ( R j ) ,   H φ ( j )   ( R j )   =   H φ ( j + 1 )   ( R j ) .
η   =   2 k 2 / π max [ w j ( ρ ) ]   ∑ j = 1 n ∫ R j − 1 R j w j ( ρ ) ρ d ρ ,
w j ( ρ )   ≈   1 8 π   ×   { ɛ j   ( | E ρ ( j ) | 2   +   | E z ( j ) | 2 )   +   | H φ ( j ) | 2 for   dielectric   layers ɛ ∞   [ 1   +   ( ω p / ω ) 2 ]   ( | E ρ ( j ) | 2   +   | E z ( j ) | 2 )   +   | H φ ( j ) | 2 for   metallic   layers .
ζ   =   Σ j = 1 n − 1   ∫ R j − 1 R j Re   ( E ρ ( j )   H φ * ( j ) )   ρ d ρ Σ j = 1 n   ∫ R j − 1 R j Re   ( E ρ ( j )   H φ * ( j ) )   ρ d ρ .
E ( ρ )   =   { A 1   [ − i ( k / α 1 ) I 1 ( α 1 ρ ) e ρ   +   I 0 ( α 1 ρ )   e z ] , ρ   ≤   R 1 , A 2   [ ( k / α 2 ) K 1   ( α 2 ρ )   e ρ   −   i K 0   ( α 2 ρ )   e z ] , ρ   >   R 1 ,
H φ   ( ρ )   =   k   ×   { − i A 1   ( ɛ 1 / α 1 ) I 1   ( α 1 ρ ) , ρ   ≤   R 1 , A 2   ( ɛ 2 / α 2 ) K 1   ( α 2 ρ ) , ρ   >   R 1 ,
ɛ 1 α 2 I 1   ( α 1 R 1 ) K 0 ( α 2 R 1 )   =   − ɛ 2 α 1 I 0 ( α 1 R 1 ) K 1 ( α 2 R 1 ) .
E ( ρ )   =   { A 1   [ − i ( β / α 1 ) I 1   ( α 1 ρ )   e ρ   +   I 0 ( α 1 ρ )   e z ] , ρ   ≤   R 1 , ( β / α 2 )   [ − i A 2 I 1   ( α 2 ρ )   +   B 2 K 1   ( α 2 ρ ) ]   e ρ +   [ A 2 I 0 ( α 2 ρ )   −   i B 2 K 0   ( α 2 ρ ) ]   e z , R 1   ≤   ρ   ≤   R 2 , A 3   [ ( β / α 3 ) K 1   ( α 3 ρ )   e ρ   −   i K 0   ( α 3 ρ )   e z ] , ρ   ≥   R 2 ,
H φ   ( ρ )   =   k   ×   { A 1   [ − i ( ɛ 1 / α 1 ) I 1   ( α 1 ρ ) ] , ρ   ≤   R 1 , ( ɛ 2 / α 2 )   [ − i A 2 I 1   ( α 2 ρ )   +   B 2 K 1   ( α 2 ρ ) ] , R 1   ≤   ρ   ≤   R 2 , A 3   [ ( ɛ 3 / α 3 ) K 1   ( α 3 ρ ) ] , ρ   ≥   R 2 ,
ɛ 1 α 2 I 1   ( α 1 R 1 )   [ α 2 ɛ 3 K 1   ( α 3 R 2 ) M 00   +   α 3 ɛ 2 K 0 ( α 3 R 2 ) M 10 ]                                                                                                             =   − ɛ 2 α 1 I 0   ( α 1 R 1 )   [ α 2 ɛ 3 K 1   ( α 3 R 2 ) M 01   +   α 3 ɛ 2 K 0   ( α 3 R 2 ) M 11 ] ,
M a b   =   I a   ( α 2 R 2 ) K b ( α 2 R 1 )   −   ( − 1 ) a + b   I b   ( α 2 R 1 ) K a ( α 2 R 2 ) .
ɛ 2   =   ɛ 2 ′   −   i ɛ 2 ″   =   ɛ 2 ′   −   i ɛ 2 ′ k γ .
E ( ρ )   =   { A 1   [ − i ( β / α 1 ) I 1 ( α 1 ρ )   e ρ   +   I 0 ( α 1 ρ )   e z ] , ρ   ≤   R 1 , ( β / α 2 )   [ − i A 2 I 1   ( α 2 ρ )   +   B 2 K 1   ( α 1 ρ ) ]   e ρ +   [ A 2 I 0   ( α 2 ρ )   −   i B 2 K 0 ( α 2 ρ ) ]   e z , R 1   ≤   ρ   ≤   R 2 , ( β / α 3 )   [ − i A 3 I 1   ( α 3 ρ )   +   B 3 K 1   ( α 3 ρ ) ]   e ρ +   [ A 3 I 0 ( α 3 ρ )   −   i B 3 K 0   ( α 3 ρ ) ]   e z , R 2   ≤   ρ   ≤   R 3 , A 4   [ ( β / α 4 ) K 1   ( α 4 ρ )   e ρ   −   i K 0   ( α 4 ρ )   e z ] , ρ   ≥   R 3 ,
H φ   ( ρ )   =   k   ×   { A 1   [ − i ( ɛ 1 / α 1 ) I 1 ( α 1 ρ ) ] , ρ   ≤   R 1 , ( ɛ 2 / α 2 )   [ − i A 2 I 1   ( α 2 ρ )   +   B 2 K 1   ( α 2 ρ ) ] , R 1   ≤   ρ   ≤   R 2 , ( ɛ 3 / α 3 )   [ − i A 3 I 1   ( α 3 ρ )   +   B 3 K 1   ( α 3 ρ ) ] , R 2   ≤   ρ   ≤   R 3 , A 4   [ ( ɛ 4 / α 4 ) K 1   ( α 4 ρ ) ] , ρ   ≥   R 3 ,
| i K 0 ( 43 ) I 0 ( 33 ) − i K 0 ( 33 ) 0 0 0 ɛ 4 α 4 K 1 ( 43 ) i ɛ 3 α 3 I 1 ( 33 ) − ɛ 3 α 3 K 1 ( 33 ) 0 0 0 0 − I 0 ( 32 ) i K 0 ( 32 ) I 0 ( 22 ) − i K 0 ( 22 ) 0 0 − i ɛ 3 α 3 I 1 ( 32 ) ɛ 3 α 3 K 1 ( 32 ) i ɛ 2 α 2 I 1 ( 22 ) − ɛ 2 α 2 K 1 ( 22 ) 0 0 0 0 − I 0 ( 21 ) i K 0 ( 21 ) I 0 ( 11 ) 0 0 0 − i ɛ 2 α 2 I 1 ( 21 ) ɛ 2 α 2 K 1 ( 21 ) i ɛ 1 α 1 I 1 ( 11 ) |   =   0 ,

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