Abstract

We report on the evolution of modes in cylindrical metal/dielectric systems. The transition between surface plasmon polaritons and localized modes is documented in terms of the real and imaginary parts of the effective refractive index as a function of geometric and optical parameters. We show the evolution process of SPP and localized modes. New phenomena of coupling between SPP and core-like modes, and of mode gap and super-long surface plasmon polaritons are found and discussed. We conclude that both superluminal light and slow light can be solutions of metallically coated dielectric fibers.

© 2011 OSA

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References

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2010

F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol. 5(1), 67–72 (2010).
[CrossRef] [PubMed]

H. Zhao, R. P. Zaccaria, P. Verma, J. F. Song, and H. B. Sun, “Validity of the V parameter for photonic quasi-crystal fibers,” Opt. Lett. 35(7), 1064–1066 (2010).
[CrossRef] [PubMed]

X. L. Zhang, J. F. Song, G. Q. Lo, and D. L. Kwong, “The observation of super-long range surface plasmon polaritons modes and its application as sensory devices,” Opt. Express 18(21), 22462–22470 (2010).
[CrossRef] [PubMed]

2009

U. Langbein, U. Trutschel, A. Unger, and M. Duguay, “Rigorous mode solver for multilayer cylindrical waveguide structures using constraints optimization,” Opt. Quantum Electron. 41(4), 223–233 (2009).
[CrossRef]

Y. Saito and P. Verma, “Imaging and spectroscopy through plasmonic nano-probe,” Eur. Phys. J. Appl. Phys. 46(2), 20101 (2009).
[CrossRef]

2008

F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A hybrid plasmonic-photonic nanodevice for label-free detection of a few molecules,” Nano Lett. 8(8), 2321–2327 (2008).
[CrossRef] [PubMed]

Y. Peng, X. Wang, and K. Kempa, “TEM-like optical mode of a coaxial nanowaveguide,” Opt. Express 16(3), 1758–1763 (2008).
[CrossRef] [PubMed]

2007

A. K. Sharma and B. D. Gupta, “Comparison of performance parameters of conventional and nano-plasmonic fiber optic sensors,” Plasmonics 2(2), 51–54 (2007).
[CrossRef]

R. Adato and J. Guo, “Characteristics of ultra-long range surface plasmon waves at optical frequencies,” Opt. Express 15(8), 5008–5017 (2007).
[CrossRef] [PubMed]

2005

2002

2001

T. Ito and K. Sakoda, “Photonic bands of metallic systems. II. Features of surface plasmon polaritons,” Phys. Rev. B 64(4), 045117(2001).
[CrossRef]

U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64(12), 125420 (2001).
[CrossRef]

2000

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000).
[CrossRef]

1997

1994

L. Novotny and C. Hafner, “Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(5), 4094–4106 (1994).
[CrossRef] [PubMed]

1978

1965

Adato, R.

Andreani, L. C.

F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol. 5(1), 67–72 (2010).
[CrossRef] [PubMed]

F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A hybrid plasmonic-photonic nanodevice for label-free detection of a few molecules,” Nano Lett. 8(8), 2321–2327 (2008).
[CrossRef] [PubMed]

Bek, A.

F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol. 5(1), 67–72 (2010).
[CrossRef] [PubMed]

Biswas, R.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000).
[CrossRef]

Businaro, L.

F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A hybrid plasmonic-photonic nanodevice for label-free detection of a few molecules,” Nano Lett. 8(8), 2321–2327 (2008).
[CrossRef] [PubMed]

Candeloro, P.

F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol. 5(1), 67–72 (2010).
[CrossRef] [PubMed]

Das, G.

F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol. 5(1), 67–72 (2010).
[CrossRef] [PubMed]

F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A hybrid plasmonic-photonic nanodevice for label-free detection of a few molecules,” Nano Lett. 8(8), 2321–2327 (2008).
[CrossRef] [PubMed]

De Angelis, F.

F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol. 5(1), 67–72 (2010).
[CrossRef] [PubMed]

F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A hybrid plasmonic-photonic nanodevice for label-free detection of a few molecules,” Nano Lett. 8(8), 2321–2327 (2008).
[CrossRef] [PubMed]

Dereux, A.

U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64(12), 125420 (2001).
[CrossRef]

Di Fabrizio, E.

F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol. 5(1), 67–72 (2010).
[CrossRef] [PubMed]

F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A hybrid plasmonic-photonic nanodevice for label-free detection of a few molecules,” Nano Lett. 8(8), 2321–2327 (2008).
[CrossRef] [PubMed]

Duguay, M.

U. Langbein, U. Trutschel, A. Unger, and M. Duguay, “Rigorous mode solver for multilayer cylindrical waveguide structures using constraints optimization,” Opt. Quantum Electron. 41(4), 223–233 (2009).
[CrossRef]

El-Kady, I.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000).
[CrossRef]

Galli, M.

F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol. 5(1), 67–72 (2010).
[CrossRef] [PubMed]

F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A hybrid plasmonic-photonic nanodevice for label-free detection of a few molecules,” Nano Lett. 8(8), 2321–2327 (2008).
[CrossRef] [PubMed]

Guo, J.

Gupta, B. D.

A. K. Sharma and B. D. Gupta, “Comparison of performance parameters of conventional and nano-plasmonic fiber optic sensors,” Plasmonics 2(2), 51–54 (2007).
[CrossRef]

Hafner, C.

L. Novotny and C. Hafner, “Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(5), 4094–4106 (1994).
[CrossRef] [PubMed]

Ho, K. M.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000).
[CrossRef]

Ito, T.

T. Ito and K. Sakoda, “Photonic bands of metallic systems. II. Features of surface plasmon polaritons,” Phys. Rev. B 64(4), 045117(2001).
[CrossRef]

Jeong, H. S.

Kempa, K.

Kobayashi, T.

Kuramochi, E.

Kwong, D. L.

Kyong, C. S.

Langbein, U.

U. Langbein, U. Trutschel, A. Unger, and M. Duguay, “Rigorous mode solver for multilayer cylindrical waveguide structures using constraints optimization,” Opt. Quantum Electron. 41(4), 223–233 (2009).
[CrossRef]

Lapchuk, A. S.

Lazzarino, M.

F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol. 5(1), 67–72 (2010).
[CrossRef] [PubMed]

Liberale, C.

F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol. 5(1), 67–72 (2010).
[CrossRef] [PubMed]

Lo, G. Q.

Maksymov, I.

F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol. 5(1), 67–72 (2010).
[CrossRef] [PubMed]

F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A hybrid plasmonic-photonic nanodevice for label-free detection of a few molecules,” Nano Lett. 8(8), 2321–2327 (2008).
[CrossRef] [PubMed]

Malitson, I. H.

Marom, E.

Mitsugi, S.

Morimoto, A.

Notomi, M.

Novotny, L.

L. Novotny and C. Hafner, “Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(5), 4094–4106 (1994).
[CrossRef] [PubMed]

Ouyang, G.

Patrini, M.

F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol. 5(1), 67–72 (2010).
[CrossRef] [PubMed]

F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A hybrid plasmonic-photonic nanodevice for label-free detection of a few molecules,” Nano Lett. 8(8), 2321–2327 (2008).
[CrossRef] [PubMed]

Peng, Y.

Saito, Y.

Y. Saito and P. Verma, “Imaging and spectroscopy through plasmonic nano-probe,” Eur. Phys. J. Appl. Phys. 46(2), 20101 (2009).
[CrossRef]

Sakoda, K.

T. Ito and K. Sakoda, “Photonic bands of metallic systems. II. Features of surface plasmon polaritons,” Phys. Rev. B 64(4), 045117(2001).
[CrossRef]

Schröter, U.

U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64(12), 125420 (2001).
[CrossRef]

Sharma, A. K.

A. K. Sharma and B. D. Gupta, “Comparison of performance parameters of conventional and nano-plasmonic fiber optic sensors,” Plasmonics 2(2), 51–54 (2007).
[CrossRef]

Shin, D.

Shin, D. I.

Shinya, A.

Sigalas, M. M.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000).
[CrossRef]

Song, J. F.

Soukoulis, C. M.

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000).
[CrossRef]

Sun, H. B.

Takahara, J.

Taki, H.

Trutschel, U.

U. Langbein, U. Trutschel, A. Unger, and M. Duguay, “Rigorous mode solver for multilayer cylindrical waveguide structures using constraints optimization,” Opt. Quantum Electron. 41(4), 223–233 (2009).
[CrossRef]

Unger, A.

U. Langbein, U. Trutschel, A. Unger, and M. Duguay, “Rigorous mode solver for multilayer cylindrical waveguide structures using constraints optimization,” Opt. Quantum Electron. 41(4), 223–233 (2009).
[CrossRef]

Verma, P.

H. Zhao, R. P. Zaccaria, P. Verma, J. F. Song, and H. B. Sun, “Validity of the V parameter for photonic quasi-crystal fibers,” Opt. Lett. 35(7), 1064–1066 (2010).
[CrossRef] [PubMed]

Y. Saito and P. Verma, “Imaging and spectroscopy through plasmonic nano-probe,” Eur. Phys. J. Appl. Phys. 46(2), 20101 (2009).
[CrossRef]

Wang, X.

Xu, Y.

Yamagishi, S.

Yariv, A.

Yeh, P.

Zaccaria, R. P.

Zhang, X. L.

Zhao, H.

Appl. Opt.

Eur. Phys. J. Appl. Phys.

Y. Saito and P. Verma, “Imaging and spectroscopy through plasmonic nano-probe,” Eur. Phys. J. Appl. Phys. 46(2), 20101 (2009).
[CrossRef]

J. Opt. Soc. Am.

Nano Lett.

F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A hybrid plasmonic-photonic nanodevice for label-free detection of a few molecules,” Nano Lett. 8(8), 2321–2327 (2008).
[CrossRef] [PubMed]

Nat. Nanotechnol.

F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nat. Nanotechnol. 5(1), 67–72 (2010).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

U. Langbein, U. Trutschel, A. Unger, and M. Duguay, “Rigorous mode solver for multilayer cylindrical waveguide structures using constraints optimization,” Opt. Quantum Electron. 41(4), 223–233 (2009).
[CrossRef]

Phys. Rev. B

U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64(12), 125420 (2001).
[CrossRef]

T. Ito and K. Sakoda, “Photonic bands of metallic systems. II. Features of surface plasmon polaritons,” Phys. Rev. B 64(4), 045117(2001).
[CrossRef]

I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000).
[CrossRef]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics

L. Novotny and C. Hafner, “Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 50(5), 4094–4106 (1994).
[CrossRef] [PubMed]

Plasmonics

A. K. Sharma and B. D. Gupta, “Comparison of performance parameters of conventional and nano-plasmonic fiber optic sensors,” Plasmonics 2(2), 51–54 (2007).
[CrossRef]

Other

J. A. Buck, Fundamentals of Optical Fibers (Wiley-Interscience, 2004).

Supplementary Material (3)

» Media 1: MOV (678 KB)     
» Media 2: MOV (798 KB)     
» Media 3: MOV (548 KB)     

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Figures (16)

Fig. 1
Fig. 1

Sketch of a multi-layer fiber. The cylindrical coordinates (ρ,φ,z) are shown.

Fig. 2
Fig. 2

Sketch of three kinds of metallic fibers: (A), dieletric core covered by an infinitely extended layer of metal; (B), metal covered by infinitely extended layer of dielectric; (C), thin metal film is located in between two dielectrics.

Fig. 3
Fig. 3

The transmission constants for type-A, B and C structures. The real and imaginary parts of the refractive index are showed in the left and right columns, respectively. The azimuthal number m is equal to 1. The refractive index of silicon dioxide is shown in type-A figure (dashed line). Different colors are associated to different radial numbers, namely different modes. From right to left we move from the fundamental to higher order modes.

Fig. 4
Fig. 4

z-component of the Poynting vector associated to the fundamental mode for the type-A structure at (a) λ = 0.5 μm. It is an internal SPP-like mode; (b) λ = 2.0 μm. It is a core mode. (c): type-B structure at λ = 1.0 μm. The SPP mode is mostly localized in the dielectric (external SPP mode).

Fig. 5
Fig. 5

Poynting vector along z direction (Sz) of the mode is denoted by red line in Fig. 3C (Media 1). Shown is the evolution of the mode distribution by changing the wavelength from 400 nm to 930 nm. Both core-like and SPP-like shapes are taken by the mode.

Fig. 6
Fig. 6

The model was the type-C structure with thickness of the silver ring equal to 0.5 μm and wavelength fixed at 1.55 μm. (a) and (b) show the real- and imaginary- part of the axial effective refractive index vs. core radius, respectively. (c) and (d) are the close-up graphs of (a) and (b) in the range from 0.84 μm to 0.89 μm. (e) and (f) show the relation between 1.24 μm to 1.246 μm. See Media 2.

Fig. 7
Fig. 7

(a) and (b) are the dispersion relation and the propagation length versus the thickness of the silver ring, respectively (the type-C structure is considered). Same color in the figures means same mode. (c) a particular of the blue line in (b). The radius of the SiO2 core was taken equal to 0.30 μm and λ = 1.55 μm. The dots indicate the calculated mode profiles as in Fig. 8. See Media 3.

Fig. 8
Fig. 8

Distribution of the Poynting vector Sz associated with the modes in Fig. 7: (a) h = 10 nm, green line; (b) h = 20 nm, green line; (c) h = 2 nm, blue line; (d) h = 30 nm, blue line. These fields were calculated by considering the terms with azimuthal number m = + 1 and m = −1.

Fig. 9
Fig. 9

(a) and (b) are the dispersion relation and the propagation length versus the thickness of the silver ring, respectively (the type-C structure is considered). Same color in the figures means same mode. The radius of the SiO2 core was taken equal to 0.5μm and λ = 1.55 μm. The dots indicate the calculated mode profiles as in Fig. 10.

Fig. 10
Fig. 10

(a) and (b) correspond to the Sz distribution for the green and blue line in Fig. 9, respectively. Silver thickness h = 20 nm is taken.

Fig. 11
Fig. 11

(a) real and (b) imaginary part of the effective refractive index versus background index. The chosen parameters are: Rc = 0.30 μm, h = 50 nm and λ = 1.55 μm. The green and blue lines are associated to external SPP and core-like mode, respectively.

Fig. 12
Fig. 12

(a) real and (b) imaginary part of the effective refractive index versus background index. The chosen parameters are: Rc = 0.50 μm, h = 50 nm and λ = 1.55 μm. The insets are the close up view of the crossing points. The dots on the lines represent the calculated points.

Fig. 13
Fig. 13

(a) Relation between SiO2 core radius and thickness of the silver ring; (b) real part of nz vs. thickness of the silver ring. The core radius is Rc = 0.3 μm. Both the graphs show only no-loss modes. The wavelength λ = 1.55 μm. The dots represent the calculated modes as in Figs. 14 and 15.

Fig. 14
Fig. 14

z component of the electric- and magnetic- field distributions. (a) and (b) are associated to the blue circle of Figs. 13(c) and (d) correspond to the red circle of the same figure. The azimuthal number m = 1.

Fig. 15
Fig. 15

z component of the electric- and magnetic- field distributions. (a) and (b) are associated to the blue circle of Figs. 13. (c) and (d) correspond to the red circle of the same figure. The azimuthal number m = −1.

Fig. 16
Fig. 16

Group refractive index vs. wavelength. (a), (b) and (c) are associated to the blue, green and red lines in Fig. 3(C), respectively.

Equations (18)

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( e z ( j ) h ¯ z ( j ) )= m,l= + ( e m,l,z ( j ) ( ρ ¯ n m,l,t ( j ) ) h ¯ m,l,z ( j ) ( ρ ¯ n m,l,t ( j ) ) ) exp( imφ )exp( i n m,l,z z ¯ iωt )
( e m,l,z ( j ) ( ρ ¯ n m,l,t ( j ) ) h ¯ m,l,z ( j ) ( ρ ¯ n m,l,t ( j ) ) )= H m ( 1 ) ( ρ ¯ n m,l,t ( j ) ) Α m,l ( j ) + H m ( 2 ) ( ρ ¯ n m,l,t ( j ) ) B m,l ( j )
( n m,t ( j ) ) 2 + n m,z 2 = ε ( j ) μ ( j )
( e m,ρ (j) h ¯ m,ρ (j) )= 1 ( n m,t (j) ) 2 [ i n m,z ρ ¯ +( 0 μ (j) ε (j) 0 ) m ρ ¯ ]( e m,z (j) h ¯ m,z (j) )
( e m,φ (j) h ¯ m,φ (j) )= 1 ( n m,t (j) ) 2 [ m n m,z ρ ¯ +i( 0 μ (j) ε (j) 0 ) ρ ¯ ]( e m,z (j) h ¯ m,z (j) )
M(j,j)( A m (j) B m (j) )=M(j+1,j)( A m (j+1) B m (j+1) )
M 11 ( q,p )= H m (1) ( n m,t (q) R ¯ p )I
M 12 ( q,p )= H m (2) ( n m,t (q) R ¯ p )I
M 21 ( q,p )= 1 n m,t (q) R ¯ p [ n z H m (1)(+) ( n m,t (q) R ¯ p )I+ H m (1)() ( n m,t (q) R ¯ p )i( 0 μ (q) ε (q) 0 ) ]
M 22 ( q,p )= 1 n m,t (q) R ¯ p [ n z H m (2)(+) ( n m,t (q) R ¯ p )I+ H m (2)() ( n m,t (q) R ¯ p )i( 0 μ (q) ε (q) 0 ) ]
M 1 ( j+1,j )M( j,j )( A m ( j ) B m ( j ) )= T j j+1 ( A m ( j ) B m ( j ) )=( A m ( j+1 ) B m ( j+1 ) )
T N1 N T j1 j T 1 2 ( A m ( 1 ) B m ( 1 ) )= j=1 N1 T j j+1 ( A m ( 1 ) B m ( 1 ) )=T( A m ( 1 ) B m ( 1 ) )=( A m ( N ) B m ( N ) )
B m ( N ) =0
( T 21 + T 22 ) A m ( 1 ) =0
T 21 + T 22 =0
{ Re 2 ( n m,t ( N ) )I m 2 ( n m,t ( N ) )= ε ( N ) μ ( N ) Re 2 ( n m,z )+I m 2 ( n m,z ) Re( n m,t ( N ) )Im( n m,t ( N ) )=Re( n m,z )Im( n m,z )
H m ( 1 ) ( ρ ¯ n m,t ( N ) ) [ ρ ¯ | n m,t ( N ) | ] 1/2 exp( i ρ ¯ | Re( n m,t ( N ) ) | )exp( ρ ¯ Im( n m,t ( N ) ) )
N | H m ( 1 ) ( ρ ¯ n m,t ( N ) ) | 2 ρdρdφ 1 k 0 Im( n m,t ( N ) )

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