Abstract

We study the surface plasmon modes in a silver double-nanowire system by employing the eigenmode analysis approach based on the finite element method. Calculated dispersion relations, surface charge distributions, field patterns and propagation lengths of ten lowest energy plasmon modes in the system are presented. These ten modes are categorized into three groups because they are found to originate from the monopole-monopole, dipole-dipole and quadrupole-quadrupole hybridizations between the two wires, respectively. Interestingly, in addition to the well studied gap mode (mode 1), the other mode from group 1 which is a symmetrically coupled charge mode (mode 2) is found to have a larger group velocity and a longer propagation length than mode 1, suggesting mode 2 to be another potential signal transporter for plasmonic circuits. Scenarios to efficiently excite (inject) group 1 modes in the two-wire system and also to convert mode 2 (mode 1) to mode 1 (mode 2) are demonstrated by numerical simulations.

© 2013 OSA

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References

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2013 (1)

Y. Ma, G. Farrell, Y. Semenova, H. P. Chan, H. Zhang, and Q. Wu, “Novel dielectric-loaded plasmonic waveguide for tight-confined hybrid plasmon mode,” Plasmonics 8(2), 1259–1263 (2013).
[Crossref]

2012 (4)

2011 (1)

2010 (7)

Z. X. Zhang, M. L. Hu, K. T. Chan, and C. Y. Wang, “Plasmonic waveguiding in a hexagonally ordered metal wire array,” Opt. Lett. 35(23), 3901–3903 (2010).
[Crossref] [PubMed]

K. Bao, H. Sobhani, and P. Nordlander, “Plasmon hybridization for real metals,” Chin. Sci. Bull. 55(24), 2629–2634 (2010).
[Crossref]

V. Klimov and G.-Y. Guo, “Bright and dark plasmon modes in three nanocylinder cluster,” J. Phys. Chem. C 114(51), 22398–22405 (2010).
[Crossref]

M. Bora, B. J. Fasenfest, E. M. Behymer, A. S. P. Chang, H. T. Nguyen, J. A. Britten, C. C. Larson, J. W. Chan, R. R. Miles, and T. C. Bond, “Plasmon resonant cavities in vertical nanowire arrays,” Nano Lett. 10(8), 2832–2837 (2010).
[Crossref] [PubMed]

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010).
[Crossref]

J. Tian, Z. Ma, Q. Li, Y. Song, Z. Liu, Q. Yang, C. Zha, J. Åkerman, L. Tong, and M. Qiu, “Nanowaveguides and couplers based on hybrid plasmonic modes,” Appl. Phys. Lett. 97(23), 231121 (2010).
[Crossref]

W. Cai, L. Wang, X. Zhang, J. Xu, and F. J. Garcia de Abajo, “Controllable excitation of gap plasmons by electron beams in metallic nanowire pairs,” Phys. Rev. B 82(12), 125454 (2010).
[Crossref]

2009 (4)

A. Manjavacas and F. J. García de Abajo, “Robust plasmon waveguides in strongly interacting nanowire arrays,” Nano Lett. 9(4), 1285–1289 (2009).
[Crossref] [PubMed]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[Crossref] [PubMed]

J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabry-Pérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. 9(6), 2372–2377 (2009).
[Crossref] [PubMed]

A. Manjavacas and F. J. García de Abajo, “Coupling of gap plasmons in multi-wire waveguides,” Opt. Express 17(22), 19401–19413 (2009).
[Crossref] [PubMed]

2008 (6)

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

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]

S. Kim, J. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453(7196), 757–760 (2008).
[Crossref] [PubMed]

D. K. Gramotnev, K. C. Vernon, and D. F. P. Pile, “Directional coupler using gap plasmon waveguides,” Appl. Phys. B 93(1), 99–106 (2008).
[Crossref]

G. Veronis and S. Fan, “Crosstalk between three-dimensional plasmonic slot waveguides,” Opt. Express 16(3), 2129–2140 (2008).
[Crossref] [PubMed]

G. B. Hoffman and R. M. Reano, “Vertical coupling between gap plasmon waveguides,” Opt. Express 16(17), 12677–12687 (2008).
[Crossref] [PubMed]

2007 (4)

K. H. Fung and C. T. Chan, “Plasmonic modes in periodic metal nanoparticle chains: a direct dynamic eigenmode analysis,” Opt. Lett. 32(8), 973–975 (2007).
[Crossref] [PubMed]

M. Davanco, Y. Urzhumov, and G. Shvets, “The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express 15(15), 9681–9691 (2007).
[Crossref] [PubMed]

Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: Analysis of optical properties,” Phys. Rev. B 75(3), 035411 (2007).
[Crossref]

M. Danckwerts and L. Novotny, “Optical frequency mixing at coupled gold nanoparticles,” Phys. Rev. Lett. 98(2), 026104 (2007).
[Crossref] [PubMed]

2005 (4)

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]

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95(25), 257403 (2005).
[Crossref] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005).
[Crossref] [PubMed]

L. Liu, Z. Han, and S. He, “Novel surface plasmon waveguide for high integration,” Opt. Express 13(17), 6645–6650 (2005).
[Crossref] [PubMed]

2003 (3)

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302(5644), 419–422 (2003).
[Crossref] [PubMed]

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003).
[Crossref] [PubMed]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

2002 (1)

F. J. Garcia de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B 65(11), 115418 (2002).
[Crossref]

2001 (1)

1998 (1)

1997 (2)

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering,” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[Crossref]

S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
[Crossref] [PubMed]

1989 (1)

M. Schmeits, “Surface-plasmon coupling in cylindrical pores,” Phys. Rev. B Condens. Matter 39(11), 7567–7577 (1989).
[Crossref] [PubMed]

1974 (1)

J. C. Ashley and L. C. Emerson, “Dispersion relations for non-radiative surface plasmons on cylinders,” Surf. Sci. 41(2), 615–618 (1974).
[Crossref]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Åkerman, J.

J. Tian, Z. Ma, Q. Li, Y. Song, Z. Liu, Q. Yang, C. Zha, J. Åkerman, L. Tong, and M. Qiu, “Nanowaveguides and couplers based on hybrid plasmonic modes,” Appl. Phys. Lett. 97(23), 231121 (2010).
[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]

Ashley, J. C.

J. C. Ashley and L. C. Emerson, “Dispersion relations for non-radiative surface plasmons on cylinders,” Surf. Sci. 41(2), 615–618 (1974).
[Crossref]

Atwater, H. A.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2(4), 229–232 (2003).
[Crossref] [PubMed]

Aussenegg, F. R.

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95(25), 257403 (2005).
[Crossref] [PubMed]

M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. 23(17), 1331–1333 (1998).
[Crossref] [PubMed]

Bao, K.

K. Bao, H. Sobhani, and P. Nordlander, “Plasmon hybridization for real metals,” Chin. Sci. Bull. 55(24), 2629–2634 (2010).
[Crossref]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Bartal, G.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[Crossref] [PubMed]

Behymer, E. M.

M. Bora, B. J. Fasenfest, E. M. Behymer, A. S. P. Chang, H. T. Nguyen, J. A. Britten, C. C. Larson, J. W. Chan, R. R. Miles, and T. C. Bond, “Plasmon resonant cavities in vertical nanowire arrays,” Nano Lett. 10(8), 2832–2837 (2010).
[Crossref] [PubMed]

Bond, T. C.

M. Bora, B. J. Fasenfest, E. M. Behymer, A. S. P. Chang, H. T. Nguyen, J. A. Britten, C. C. Larson, J. W. Chan, R. R. Miles, and T. C. Bond, “Plasmon resonant cavities in vertical nanowire arrays,” Nano Lett. 10(8), 2832–2837 (2010).
[Crossref] [PubMed]

Bora, M.

M. Bora, B. J. Fasenfest, E. M. Behymer, A. S. P. Chang, H. T. Nguyen, J. A. Britten, C. C. Larson, J. W. Chan, R. R. Miles, and T. C. Bond, “Plasmon resonant cavities in vertical nanowire arrays,” Nano Lett. 10(8), 2832–2837 (2010).
[Crossref] [PubMed]

Bozhevolnyi, S. I.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010).
[Crossref]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005).
[Crossref] [PubMed]

Britten, J. A.

M. Bora, B. J. Fasenfest, E. M. Behymer, A. S. P. Chang, H. T. Nguyen, J. A. Britten, C. C. Larson, J. W. Chan, R. R. Miles, and T. C. Bond, “Plasmon resonant cavities in vertical nanowire arrays,” Nano Lett. 10(8), 2832–2837 (2010).
[Crossref] [PubMed]

Cai, W.

W. Cai, L. Wang, X. Zhang, J. Xu, and F. J. Garcia de Abajo, “Controllable excitation of gap plasmons by electron beams in metallic nanowire pairs,” Phys. Rev. B 82(12), 125454 (2010).
[Crossref]

Chan, C. T.

Chan, H. P.

Y. Ma, G. Farrell, Y. Semenova, H. P. Chan, H. Zhang, and Q. Wu, “Novel dielectric-loaded plasmonic waveguide for tight-confined hybrid plasmon mode,” Plasmonics 8(2), 1259–1263 (2013).
[Crossref]

Chan, J. W.

M. Bora, B. J. Fasenfest, E. M. Behymer, A. S. P. Chang, H. T. Nguyen, J. A. Britten, C. C. Larson, J. W. Chan, R. R. Miles, and T. C. Bond, “Plasmon resonant cavities in vertical nanowire arrays,” Nano Lett. 10(8), 2832–2837 (2010).
[Crossref] [PubMed]

Chan, K. T.

Chang, A. S. P.

M. Bora, B. J. Fasenfest, E. M. Behymer, A. S. P. Chang, H. T. Nguyen, J. A. Britten, C. C. Larson, J. W. Chan, R. R. Miles, and T. C. Bond, “Plasmon resonant cavities in vertical nanowire arrays,” Nano Lett. 10(8), 2832–2837 (2010).
[Crossref] [PubMed]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Dai, L.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[Crossref] [PubMed]

Danckwerts, M.

M. Danckwerts and L. Novotny, “Optical frequency mixing at coupled gold nanoparticles,” Phys. Rev. Lett. 98(2), 026104 (2007).
[Crossref] [PubMed]

Dasari, R. R.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering,” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[Crossref]

Davanco, M.

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Devaux, E.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005).
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J. Tian, Z. Ma, Q. Li, Y. Song, Z. Liu, Q. Yang, C. Zha, J. Åkerman, L. Tong, and M. Qiu, “Nanowaveguides and couplers based on hybrid plasmonic modes,” Appl. Phys. Lett. 97(23), 231121 (2010).
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Y. Ma, G. Farrell, Y. Semenova, H. P. Chan, H. Zhang, and Q. Wu, “Novel dielectric-loaded plasmonic waveguide for tight-confined hybrid plasmon mode,” Plasmonics 8(2), 1259–1263 (2013).
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Zhang, X.

W. Cai, L. Wang, X. Zhang, J. Xu, and F. J. Garcia de Abajo, “Controllable excitation of gap plasmons by electron beams in metallic nanowire pairs,” Phys. Rev. B 82(12), 125454 (2010).
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R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
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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).
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Appl. Phys. B (1)

D. K. Gramotnev, K. C. Vernon, and D. F. P. Pile, “Directional coupler using gap plasmon waveguides,” Appl. Phys. B 93(1), 99–106 (2008).
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J. Tian, Z. Ma, Q. Li, Y. Song, Z. Liu, Q. Yang, C. Zha, J. Åkerman, L. Tong, and M. Qiu, “Nanowaveguides and couplers based on hybrid plasmonic modes,” Appl. Phys. Lett. 97(23), 231121 (2010).
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R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
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Nature (3)

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
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Opt. Express (10)

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Figures (12)

Fig. 1
Fig. 1

Illustration of the infinite long silver double-nanowire system studied here and also the supported hybridized SPP modes. Here D and w denote the wire diameter and the separation of the two nanowires, respectively.

Fig. 2
Fig. 2

(a) Dispersion relations and (b) propagation lengths of the SPP modes in the single silver nanowire, obtained via the eigenmode analysis based the FEM simulations. Here kz is the real part of the momentum of the SPP modes along the nanowire. Modes m = 0, 1, and 2 are monopole, dipole and quadrupole SPP modes, respectively. In (a), the dispersion relations from the Mie theory [Eq. (4)] are also displayed as hexagon dots. In the insets in (a) and (b), the electric field patterns and surface charge distributions are shown, respectively. In the insets in (b), the red and blue colors denote the positive and negative surface charges, respectively.

Fig. 3
Fig. 3

(a) Dispersion relations and (b) propagation lengths of the ten lowest energy SPP modes on the silver double-nanowire system with w = 50 nm.

Fig. 4
Fig. 4

Electric field patterns of the ten lowest energy SPP modes in the silver double-nanowire system. White arrows denote the directions of electric field. Symbols “+” and “-” indicate the positive and negative surface charges, respectively. The chosen frequency is 1200 THz.

Fig. 5
Fig. 5

(a, c) Dispersion relations and (b, d) propagation lengths of the ten SPP modes in the silver double-nanowire system. In (a, b) and (c, d), w = 100 and 200 nm, respectively. For comparison, the results of the m = 0, 1, 2 SPP modes in the single nanowire are also shown as dots in (c) and (d).

Fig. 6
Fig. 6

Evolutions of (a) mode 6 and (b) mode 10 with the increasing gap width w. The chosen frequency is 1200 THz.

Fig. 7
Fig. 7

Classification of the ten hybridized SPP modes in the double-nanowire system into three groups (a-c) based on their origins, namely, (a) group 1 from the monopole-monopole, (b) group 2 from the dipole-dipole, and (c) group 3 from the quadrupole-quadrupole interactions between the two nanowires. The Ey field patterns of the ten modes are shown in (d). Modes 1-10 are denoted by 1-10, respectively. The chosen frequency is 1200 THz

Fig. 8
Fig. 8

Propagation lengths of modes 1 and 2 versus (a) the wire gap width w and (b) the permittivity ε of the surrounding material at the telecommunication wavelength λ = 1550 nm. In (a), the dashed line denotes the propagation length of the m = 0 SPP mode in the single nanowire. The normalized energy densities of modes 1 and 2 along the x = 0 line for w = 2, 50 and 200 nm are shown in (c-e), respectively. In (c-e), the percentage of the mode energy in the air region is also printed.

Fig. 9
Fig. 9

Normalized mode area Am/A0 versus (a) gap width w at the telecommunication wavelength λ = 1550 nm and (b) mode with a fixed gap width w = 50 nm at λ = 250 nm. In (a), the dashed line denotes the Am/A0 of the m = 0 SPP mode in the single silver nanowire.

Fig. 10
Fig. 10

Coupling length Lc of modes 1 and 2 on two parallel silver double-nanowire systems in (a) a side-by-side (see the inset) geometry and (b) an on-top (see the inset) geometry. Here the gap width w = 50 nm and the distance of the two double-nanowire systems is g. In (b), Lc becomes singularly large at g ≈95 nm (see the text for explanation).

Fig. 11
Fig. 11

Excitation of (a, b) mode 2, (c, d) mode 1 and (e, f) their mixed state by a dipole source (denoted by “P”) placed at the symmetric point 250 nm away from the left end of the two nanowires. The excited SPP waves are absorbed by the perfect match layer at the right boundary. The total energy flow of the excited SPP modes ISPP is given by an integral over the absorbing boundary. The relative injection efficiency ISPP/I0 as the dipole is rotated in the x-z plane (angle θ), the x-y plane (angle f) and the y-z plane (angle φ), is displayed in (a), (c) and (e), respectively. For comparison, the excitation efficiencies ISPP/I0 for injection using two dipoles are also plotted as green symbols in (a, c, e). Two wavelengths λSPP of 1242 and 1481 nm for mode 1 and 2 are measured based on the field patterns in the y-z plane displayed in (b, d, f). The white arrows represent the electric fields and the wavelength λ = 1550 nm. The electric field | E | patterns in the x-y plane of the excited SPP modes are shown in the insets in (a-f).

Fig. 12
Fig. 12

Proposed mode converters from mode 1 to mode 2 (a-b), and from mode 2 to mode 1 (c-f). The Ey field patterns and surface charge distributions are, respectively, displayed in (a, c, e) and (b, d, f). Here the length L is 430 nm in (a-d) and 1550 nm in (e-f). The refractive index of the glass is 1.5. The wavelength λ = 1550 nm. In (a), the color bar range of the Ey field on the left side of the box indicated by the dashed lines is set to be 10 times larger than that on the right side of the box in order to make clear the field patterns of both mode 1 and mode 2.

Equations (5)

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{ 2 ψ i + k i 2 ψ i =0 2 ψ o + k o 2 ψ o =0
{ ψ i = m=0 C m J m ( h i r) e imθ e i k z z ψ o = m=0 D m H m ( h o r) e imθ e i k z z .
[ 1 h o H m ( h o R) H m ( h o R) 1 h i J m ( h i R) J m ( h i R) ][ k o 2 h o H m ( h o R) H m ( h o R) k i 2 h i J m ( h i R) J m ( h i R) ]= m 2 k z 2 R 2 ( 1 h i 2 1 h o 2 ) 2 .
κ i 2 κ o 2 [ κ o ε i I m ( κ i R) I m ( κ i R) κ i ε o K m ( κ o R) K m ( κ o R) ][ κ o I m ( κ i R) I m ( κ i R) κ i K m ( κ o R) K m ( κ o R) ] m 2 k z 2 R 2 ( ε o ε i ) 2 ω 2 c 2 =0.
w(x,y)= 1 2 ( d( ε(x,y)ω ) dω | E (x,y) | 2 + μ 0 | H (x,y) | 2 ),

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