Abstract

We investigate electromagnetic scattering from nanoscale wires and reveal the emergence of a family of exotic resonances for source waves close to grazing incidence. These grazing resonances have a much higher Q-bandwidth product and thus, a much higher Q factor and broader bandwidth than the pure plasmonic resonances found in metal nanowires. Furthermore, these grazing resonances are much less susceptible to material losses than surface plasmon resonances. Contrary to the process of exciting surface plasmon resonances, these grazing resonances can arise in both dielectric and metallic nanowires and appear near to the cutoff wavelength of the circular waveguide. This peculiar resonance effect originates from the excitation of long range guided surface waves through the interplay of coherently scattered continuum modes coupled with first-order azimuthal propagating modes of the cylindrical nanowire. These first-order cyclic Sommerfeld waves and associated cyclic Sommerfeld resonances revealed here opens up the possibility of an alternative scheme of enhanced fields with a better merit (higher Q-bandwidth product and lower loss) than conventional surface plasmon resonances in the nano-regime. This nanowire resonance phenomenon can be utilized in broad scientific areas, including: metamaterial designs, nanophotonic integration, nanoantennas, and nanosensors.

© 2009 Optical Society of America

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2008 (4)

K. R. Catchpole and A. Polman, "Plasmonic solar cells," Opt. Express 16, 21793 (2008).
[CrossRef] [PubMed]

C. Rockstuhl, S. Fahr, and F. Lederer, "Absorption enhancement in solar cells by localized plasmon polaritons," J. Appl. Phys. 104, 123102 (2008).
[CrossRef]

J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index." Nature (London) 455, 376 (2008).
[CrossRef]

E. Feigenbaum and M. Orenstein, "Ultrasmall Volume Plasmons, yet with Complete Retardation Effects," Phys. Rev. Lett. 101, 163902 (2008).
[CrossRef] [PubMed]

2007 (1)

P. Rousseau, H. Khemliche, A. G. Borisov, and P. Roncin, "Quantum Scattering of Fast Atoms and Molecules on Surfaces," Phys. Rev. Lett. 98, 016104 (2007)
[CrossRef] [PubMed]

2006 (3)

B. S. Luk’yanchuk and V. Ternovsky, "Light scattering by a thin wire with a surface-plasmon resonance: Bifurcations of the Poynting vector field," Phys. Rev. B 73, 235432 (2006).
[CrossRef]

S. Feng and J. M. Elson, "Diffraction-suppressed high-resolution imaging through metallodielectric nanofilms," Opt. Express 14, 216 (2006).
[CrossRef] [PubMed]

S. A. Maier, "Plasmonics: The Promise of Highly Integrated Optical Devices," IEEE J. Sel. Top. Quantum Electron. 12, 1671 (2006).
[CrossRef]

2005 (3)

K. Halterman, J. M. Elson, and S. Singh, "Plasmonic resonances and electromagnetic forces between coupled silver nanowires," Phys. Rev. B 72, 075429 (2005).
[CrossRef]

V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yua, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, "Negative index of refraction in optical metamaterials." Opt. Lett. 30, 3356 (2005).
[CrossRef]

F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Moreno, "Transmission of Light through a Single Rectangular Hole," Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef] [PubMed]

2003 (1)

D. R. Fredkin and I. D. Mayergoyz, "Resonant Behavior of Dielectric Objects (Electrostatic Resonances)," Phys. Rev. Lett. 91, 253902 (2003).
[CrossRef]

2002 (2)

M. M. J. Treacy, "Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings," Phys. Rev. B 66, 195105 (2002).
[CrossRef]

Z.-H. Gu, I. M. Fuks, and M. Ciftan, "Enhanced backscattering at grazing angles," Opt. Lett. 27, 2067 (2002).
[CrossRef]

2001 (1)

1999 (2)

H. Xu, E. J. Bjerneld, M. K'all, and L. B'orjesson, "Spectroscopy of Single Hemoglobin Molecules by Surface Enhanced Raman Scattering," Phys. Rev. Lett. 83, 4357 (1999).
[CrossRef]

E. Centeno and D. Felbacq, "Characterization of defect modes in finite bidimensional photonic crystals," J. Opt. Soc. Am. A 16, 2705 (1999).
[CrossRef]

1998 (1)

D. E. Barrick, "Grazing Behavior of Scatter and Propagation Above Any Rough Surface," IEEE Trans. Antennas Propag. 46, 73 (1998).
[CrossRef]

1997 (2)

S. Nie and S. R. Emory, "Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering," Science 275, 1102 (1997).
[CrossRef] [PubMed]

L. G. Guimarães and J. P. R. Furtado de Mendonca, "Analysis of the resonant scattering of light by cylinders at oblique incidence," Appl. Opt. 36, 8010 (1997).
[CrossRef]

1996 (1)

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837 (1996).
[CrossRef]

1976 (1)

1974 (1)

1965 (2)

J. R. Wait, "The Long Wavelength Limit In Scattering From A Dielectric Cylinder At Oblique Incidence," Can. J. Phys. 43, 2212 (1965).
[CrossRef]

A. Lind and J. Greenberg, "Electromagnetic Scattering by Obliquely Oriented Cylinders," J. Appl. Phys. 37, 3195 (1965).
[CrossRef]

1962 (1)

M. J. King and J. C. Wiltse, "Surface-Wave Propagation on Coated or Uncoated Metal Wires at Millimeter Wavelengths," IEEE Trans. Antennas Propag. 10246 (1962).
[CrossRef]

1955 (1)

J. R. Wait, "Scattering Of A Plane Wave From A Circular Dielectric Cylinder At Oblique Incidence," Can. J. Phys. 33, 189 (1955).
[CrossRef]

1881 (1)

L. Rayleigh, "On the Electromagnetic Theory of Light," Philos. Mag. 12, 81-101 (1881).

Avila, E. U.

J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index." Nature (London) 455, 376 (2008).
[CrossRef]

Barrick, D. E.

D. E. Barrick, "Grazing Behavior of Scatter and Propagation Above Any Rough Surface," IEEE Trans. Antennas Propag. 46, 73 (1998).
[CrossRef]

Bartal, G.

J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index." Nature (London) 455, 376 (2008).
[CrossRef]

Bjerneld, E. J.

H. Xu, E. J. Bjerneld, M. K'all, and L. B'orjesson, "Spectroscopy of Single Hemoglobin Molecules by Surface Enhanced Raman Scattering," Phys. Rev. Lett. 83, 4357 (1999).
[CrossRef]

Borisov, A. G.

P. Rousseau, H. Khemliche, A. G. Borisov, and P. Roncin, "Quantum Scattering of Fast Atoms and Molecules on Surfaces," Phys. Rev. Lett. 98, 016104 (2007)
[CrossRef] [PubMed]

B'orjesson, L.

H. Xu, E. J. Bjerneld, M. K'all, and L. B'orjesson, "Spectroscopy of Single Hemoglobin Molecules by Surface Enhanced Raman Scattering," Phys. Rev. Lett. 83, 4357 (1999).
[CrossRef]

Cai, W.

Catchpole, K. R.

Centeno, E.

Chettiar, U. K.

Ciftan, M.

Drachev, V. P.

Elson, J. M.

S. Feng and J. M. Elson, "Diffraction-suppressed high-resolution imaging through metallodielectric nanofilms," Opt. Express 14, 216 (2006).
[CrossRef] [PubMed]

K. Halterman, J. M. Elson, and S. Singh, "Plasmonic resonances and electromagnetic forces between coupled silver nanowires," Phys. Rev. B 72, 075429 (2005).
[CrossRef]

Emory, S. R.

S. Nie and S. R. Emory, "Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering," Science 275, 1102 (1997).
[CrossRef] [PubMed]

Fahr, S.

C. Rockstuhl, S. Fahr, and F. Lederer, "Absorption enhancement in solar cells by localized plasmon polaritons," J. Appl. Phys. 104, 123102 (2008).
[CrossRef]

Fan, S.

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837 (1996).
[CrossRef]

Feigenbaum, E.

E. Feigenbaum and M. Orenstein, "Ultrasmall Volume Plasmons, yet with Complete Retardation Effects," Phys. Rev. Lett. 101, 163902 (2008).
[CrossRef] [PubMed]

Felbacq, D.

Feng, S.

Fredkin, D. R.

D. R. Fredkin and I. D. Mayergoyz, "Resonant Behavior of Dielectric Objects (Electrostatic Resonances)," Phys. Rev. Lett. 91, 253902 (2003).
[CrossRef]

Fuks, I. M.

García-Vidal, F. J.

F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Moreno, "Transmission of Light through a Single Rectangular Hole," Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef] [PubMed]

Genov, D. A.

J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index." Nature (London) 455, 376 (2008).
[CrossRef]

Greenberg, J.

A. Lind and J. Greenberg, "Electromagnetic Scattering by Obliquely Oriented Cylinders," J. Appl. Phys. 37, 3195 (1965).
[CrossRef]

Gu, Z.-H.

Halterman, K.

K. Halterman, J. M. Elson, and S. Singh, "Plasmonic resonances and electromagnetic forces between coupled silver nanowires," Phys. Rev. B 72, 075429 (2005).
[CrossRef]

Joannopoulos, J. D.

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837 (1996).
[CrossRef]

K'all, M.

H. Xu, E. J. Bjerneld, M. K'all, and L. B'orjesson, "Spectroscopy of Single Hemoglobin Molecules by Surface Enhanced Raman Scattering," Phys. Rev. Lett. 83, 4357 (1999).
[CrossRef]

Khemliche, H.

P. Rousseau, H. Khemliche, A. G. Borisov, and P. Roncin, "Quantum Scattering of Fast Atoms and Molecules on Surfaces," Phys. Rev. Lett. 98, 016104 (2007)
[CrossRef] [PubMed]

Kildishev, A. V.

King, M. J.

M. J. King and J. C. Wiltse, "Surface-Wave Propagation on Coated or Uncoated Metal Wires at Millimeter Wavelengths," IEEE Trans. Antennas Propag. 10246 (1962).
[CrossRef]

Kottmann, J. P.

Lederer, F.

C. Rockstuhl, S. Fahr, and F. Lederer, "Absorption enhancement in solar cells by localized plasmon polaritons," J. Appl. Phys. 104, 123102 (2008).
[CrossRef]

Lind, A.

A. Lind and J. Greenberg, "Electromagnetic Scattering by Obliquely Oriented Cylinders," J. Appl. Phys. 37, 3195 (1965).
[CrossRef]

Luk’yanchuk, B. S.

B. S. Luk’yanchuk and V. Ternovsky, "Light scattering by a thin wire with a surface-plasmon resonance: Bifurcations of the Poynting vector field," Phys. Rev. B 73, 235432 (2006).
[CrossRef]

Maier, S. A.

S. A. Maier, "Plasmonics: The Promise of Highly Integrated Optical Devices," IEEE J. Sel. Top. Quantum Electron. 12, 1671 (2006).
[CrossRef]

Martin, O. J. F.

Martín-Moreno, L.

F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Moreno, "Transmission of Light through a Single Rectangular Hole," Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef] [PubMed]

Mayergoyz, I. D.

D. R. Fredkin and I. D. Mayergoyz, "Resonant Behavior of Dielectric Objects (Electrostatic Resonances)," Phys. Rev. Lett. 91, 253902 (2003).
[CrossRef]

Mitchell, D. J.

Moreno, E.

F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Moreno, "Transmission of Light through a Single Rectangular Hole," Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef] [PubMed]

Nie, S.

S. Nie and S. R. Emory, "Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering," Science 275, 1102 (1997).
[CrossRef] [PubMed]

Orenstein, M.

E. Feigenbaum and M. Orenstein, "Ultrasmall Volume Plasmons, yet with Complete Retardation Effects," Phys. Rev. Lett. 101, 163902 (2008).
[CrossRef] [PubMed]

Polman, A.

Porto, J. A.

F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Moreno, "Transmission of Light through a Single Rectangular Hole," Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef] [PubMed]

Rayleigh, L.

L. Rayleigh, "On the Electromagnetic Theory of Light," Philos. Mag. 12, 81-101 (1881).

Rockstuhl, C.

C. Rockstuhl, S. Fahr, and F. Lederer, "Absorption enhancement in solar cells by localized plasmon polaritons," J. Appl. Phys. 104, 123102 (2008).
[CrossRef]

Roncin, P.

P. Rousseau, H. Khemliche, A. G. Borisov, and P. Roncin, "Quantum Scattering of Fast Atoms and Molecules on Surfaces," Phys. Rev. Lett. 98, 016104 (2007)
[CrossRef] [PubMed]

Rousseau, P.

P. Rousseau, H. Khemliche, A. G. Borisov, and P. Roncin, "Quantum Scattering of Fast Atoms and Molecules on Surfaces," Phys. Rev. Lett. 98, 016104 (2007)
[CrossRef] [PubMed]

Sammut, R.

Sarychev, A. K.

Shalaev, V. M.

Singh, S.

K. Halterman, J. M. Elson, and S. Singh, "Plasmonic resonances and electromagnetic forces between coupled silver nanowires," Phys. Rev. B 72, 075429 (2005).
[CrossRef]

Snyder, A. W.

Ternovsky, V.

B. S. Luk’yanchuk and V. Ternovsky, "Light scattering by a thin wire with a surface-plasmon resonance: Bifurcations of the Poynting vector field," Phys. Rev. B 73, 235432 (2006).
[CrossRef]

Treacy, M. M. J.

M. M. J. Treacy, "Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings," Phys. Rev. B 66, 195105 (2002).
[CrossRef]

Valentine, J.

J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index." Nature (London) 455, 376 (2008).
[CrossRef]

Villeneuve, P. R.

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837 (1996).
[CrossRef]

Wait, J. R.

J. R. Wait, "The Long Wavelength Limit In Scattering From A Dielectric Cylinder At Oblique Incidence," Can. J. Phys. 43, 2212 (1965).
[CrossRef]

J. R. Wait, "Scattering Of A Plane Wave From A Circular Dielectric Cylinder At Oblique Incidence," Can. J. Phys. 33, 189 (1955).
[CrossRef]

Wiltse, J. C.

M. J. King and J. C. Wiltse, "Surface-Wave Propagation on Coated or Uncoated Metal Wires at Millimeter Wavelengths," IEEE Trans. Antennas Propag. 10246 (1962).
[CrossRef]

Xu, H.

H. Xu, E. J. Bjerneld, M. K'all, and L. B'orjesson, "Spectroscopy of Single Hemoglobin Molecules by Surface Enhanced Raman Scattering," Phys. Rev. Lett. 83, 4357 (1999).
[CrossRef]

Yua, H.-K.

Zentgraf, T.

J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index." Nature (London) 455, 376 (2008).
[CrossRef]

Zhang, S.

J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index." Nature (London) 455, 376 (2008).
[CrossRef]

Zhang, X.

J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index." Nature (London) 455, 376 (2008).
[CrossRef]

Appl. Opt. (2)

Can. J. Phys. (2)

J. R. Wait, "Scattering Of A Plane Wave From A Circular Dielectric Cylinder At Oblique Incidence," Can. J. Phys. 33, 189 (1955).
[CrossRef]

J. R. Wait, "The Long Wavelength Limit In Scattering From A Dielectric Cylinder At Oblique Incidence," Can. J. Phys. 43, 2212 (1965).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

S. A. Maier, "Plasmonics: The Promise of Highly Integrated Optical Devices," IEEE J. Sel. Top. Quantum Electron. 12, 1671 (2006).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

M. J. King and J. C. Wiltse, "Surface-Wave Propagation on Coated or Uncoated Metal Wires at Millimeter Wavelengths," IEEE Trans. Antennas Propag. 10246 (1962).
[CrossRef]

D. E. Barrick, "Grazing Behavior of Scatter and Propagation Above Any Rough Surface," IEEE Trans. Antennas Propag. 46, 73 (1998).
[CrossRef]

J. Appl. Phys. (2)

A. Lind and J. Greenberg, "Electromagnetic Scattering by Obliquely Oriented Cylinders," J. Appl. Phys. 37, 3195 (1965).
[CrossRef]

C. Rockstuhl, S. Fahr, and F. Lederer, "Absorption enhancement in solar cells by localized plasmon polaritons," J. Appl. Phys. 104, 123102 (2008).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Nature (London) (1)

J. Valentine, S. Zhang, T. Zentgraf, E. U. Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index." Nature (London) 455, 376 (2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Philos. Mag. (1)

L. Rayleigh, "On the Electromagnetic Theory of Light," Philos. Mag. 12, 81-101 (1881).

Phys. Rev. B (4)

K. Halterman, J. M. Elson, and S. Singh, "Plasmonic resonances and electromagnetic forces between coupled silver nanowires," Phys. Rev. B 72, 075429 (2005).
[CrossRef]

B. S. Luk’yanchuk and V. Ternovsky, "Light scattering by a thin wire with a surface-plasmon resonance: Bifurcations of the Poynting vector field," Phys. Rev. B 73, 235432 (2006).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837 (1996).
[CrossRef]

M. M. J. Treacy, "Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings," Phys. Rev. B 66, 195105 (2002).
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F. J. García-Vidal, E. Moreno, J. A. Porto, and L. Martín-Moreno, "Transmission of Light through a Single Rectangular Hole," Phys. Rev. Lett. 95, 103901 (2005).
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Figures (12)

Fig. 1.
Fig. 1.

Schematic: A plane wave of TM mode is incident onto a long cylinder. The grazing angle θ is the angle between the k-vector of the incident wave and the symmetry axis of the rod.

Fig. 2.
Fig. 2.

Comparison of the exact and grazing results: plots of the electric field coefficients versus the grazing angle for the n=0 azimuthal mode. Blue: Exact solution [Eq. (4)]. Green: Grazing solution [Eq. (21)]. Top: Real part. Bottom: Imaginary part. Clearly, the grazing solutions asymptotically match the exact solutions as the grazing angle approaches zero.

Fig. 3.
Fig. 3.

Electric field coefficients as in Fig. 2, except now for the n=1 azimuthal mode.

Fig. 4.
Fig. 4.

Comparison of the exact and grazing results: Magnetic field coefficients in Eq. (21) for the n=1 azimuthal mode. Blue: Exact solution. Green: Grazing solution. Top: Real part. Bottom: Imaginary part.

Fig. 5.
Fig. 5.

Comparison of SPR and CSR in a silver nanorod. Left: TE mode at normal incidence and the excitation of SPR at the wavelength about 309 nm. Right: TE mode incident at the grazing angle of 0.01° and the corresponding resonance (CSR) at about 380 nm. Solid Blue: without material loss. The Q of CSR is about 10 times higher than that of SPR. Dashed Green: with material loss. The Q of CSR is then about 23 times higher than that of SPR. Moreover, due to the higher QBW of CSR, CSR has a broader bandwidth than SPR. Without the material loss, QBW≈0.04 for SPR and QBW≈0.7 for CSR.

Fig. 6.
Fig. 6.

CSRs and corresponding waveguide modes in a silver nanorod at the low grazing angles. The resonant peaks are associated with the excitation of a family of guided cyclic periodic (n=1) surface waves, i.e. cyclic Sommerfeld waves. Left plot: the peaks from low to high correspond to θ=2°, θ=1°, θ=0.5°, θ=0.1°, θ=0.05°, θ=0.02°, and θ=0.01°. The Qθ value and the wavelength of grazing resonances increase continuously as the grazing angle approaches the minimum angle. Right plot: the n=1 waveguide modes calculated from Eq. (26) with the same kz values of the CSRs for the silver nanorod. These modes are related to the first-order CSRs shown on the left plot.

Fig. 7.
Fig. 7.

CSRs for a dielectric nanorod (solid curves) with the relative permittivity ε 1r =4 and permeability µ 1r =1 and for a metallic rod (dashed curves) with the relative permittivity ε 1r =-4 and permeability µ 1r =1: These grazing resonances correspond to the excitation of the first-order (n=1) cyclic Sommerfeld waves. Upper-left panel: the peaks from low to high correspond to θ=2°, θ=1°, θ=0.5°, θ=0.1°, θ=0.05°, θ=0.02°, and θ=0.01°. The value and the wavelength of grazing resonances increases continuously as the grazing angle approaches the minimum angle. Lower-left panel: the corresponding n=1 waveguide modes calculated from Eq. (26) with the same kz values of the CSRs. These modes are related to the first-order CSRs shown on the Upper-left panel. Upper-right panel: the QBW of the corresponding CSRs vs. grazing angle. Lower-right panel: the true bandwidth ΔF of the corresponding CSRs vs. grazing angle.

Fig. 8.
Fig. 8.

Resonant wavelength, QBW, , and bandwidth at grazing incidences: Upper-left: Wavelength of CSR vs. grazing angle calculated from the maximum Qθ (Solid Blue) and from the eigenmode Eq. (26) with n=1 (Dashed Green) for the silver nanorod. The two curves coincide with each other. Upper-right: The QBW of the CSRs vs. grazing angle. Lower-left: Total factor of the CSRs vs. grazing angle. Lower-right: Bandwidth of the CSRs vs. grazing angle. The bandwidth decreases as the increases in the grazing region.

Fig. 9.
Fig. 9.

Various Q factors of the CSRs at near-zero grazing angles for a silver nanowire: Upper-left: The Qc z vs. grazing angle. Negative value indicates the backward propagation inside the silver nanowire. Upper-right: The Qs z vs. grazing angle. Lower-left: Ratio of the Qc z and the Qs z . Lower-right: Ratio of the Qs z and the .

Fig. 10.
Fig. 10.

Radially symmetric scattering and backward propagation in the silver nanorod of radius a=15 nm: θ≈0.001° at the resonance λ=406 nm. Upper-left: Poynting vector (Sρ ) in the ρ-direction. The Sρ is nearly uniform in the x-y plane. Lower-left: Vector plot of the Poynting vector (Sx ,Sy ). Power in the x-y plane flows radially towards the center of the rod. Upper-right: The z-direction Poynting (Sz ) inside and outside the rod. The Sz is asymmetric and flows backward inside the rod. Lower-right: Contour plot of the Sz further showing the asymmetric distribution in the x-y plane.

Fig. 11.
Fig. 11.

Intensity of the total electric field at the surface inside and outside the silver nanorod of radius a=15 nm: θ≈0.001° at the resonance λ=406 nm. Showing a 2D standing-wave pattern in the azimuthal and propagation directions. Upper plot: Just outside the surface ρ=a +. Lower plot: Just inside the surface ρ=a -. Vertical axis: arc length along the circumference of the rod. Horizontal axis: distance along the axis of the rod.

Fig. 12.
Fig. 12.

Influence of the radius of a lossless silver nanorod on CSRs at the grazing angle θ=0.01°. Upper-left: CSR wavelength and the corresponding QBW vs. the radius of the rod. Left y-axis for the solid-blue curve; Right y-axis for the dashed-green curve. Upper-right: The Qc z of CSRs vs. the radius of the rod. Negative value indicates the backward propagation inside the nanowire. Lower-left: The total Qθ factor and the corresponding bandwidth ΔF of CSRs vs. the radius of the rod. Left y-axis for the solid-blue curve; Right y-axis for the dashed-green curve. A decrease of the Qθ factor implies an increase of the bandwidth of CSRs. Lower-right: The ratio Qc z /Qs z of the CSRs vs. the radius of the rod.

Equations (112)

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Ezi=ψ (z)sinθ[J0(x)+2n=1+(i)nJn(x)cos(nϕ)],
Eρi=iψ(z)cosθ[J1(x)+2n=1+(i)nJ'n(x)cos(nϕ)],
Eϕi=2iψ(z)cosθn=1+(i)nnJn(x)xsin(nϕ),
Hzi=0 ,
Hρi=2 i 𝒴0 ψ (z)n=1+(i)nnJn(x)xsin(nϕ),
Hϕi=i 𝒴0 ψ (z)[J1(x)+2n=1+(i)nJ'n(x)cos(nϕ)],
E=iβj2 [kzEzωμêz×Hz],
H=iβj2[kzHz+ωεêz×Ez],
Evt=ψ (z) [Av,0t(ρ)+n=1Av,nt(ρ)cos(nϕ)] ,
Eϕt=ψ (z) n=1Aϕ,nt(ρ)sin(nϕ),
Hvt=ψ (z)n=1Bv,nt(ρ)sin(nϕ),
Hϕt=ψ (z) [Bϕ,0t(ρ)+n=1Bϕ,nt(ρ)cos(nϕ)] ,
Az,0s(ρ)=a0H0(x),Az,ns(ρ)=2 an Hn (x),
Aρ,0s(ρ)=icosθsinθa0H1(x),
Aρ,ns(ρ)=2sinθ [icosθanH'n(x)bn𝒴0nHn(x)x] ,
Aϕ,ns(ρ)=2sinθ [icosθannHn(x)x+bn𝒴0H'n(x)] ,
Bz,ns(ρ)=2 i bn Hn(x), Bϕ,0s(ρ)=i𝒴0sinθa0H1(x),
Bρ,ns(ρ)=2sinθ [i𝒴0annHn(x)xcosθbnH'n(x)] ,
Bϕ,ns(ρ)=2sinθ[i𝒴0anH'n(x)cosθbnnHn(x)x],
an=AnPn+QnInμAn2+InεInμ,
bn=AnQnPnInεAn2+InεInμ,
An=(1sin2θ1η2)cosθk0anHn(x0),
Inε=i 𝒴0 [εrηJ'n(x1)Jn(x1)Hn(x0)H'n(x0)sinθ] ,
Inμ=i𝒴0[μrηJ'n(x1)Jn(x1)Hn(x0)H'n(x0)sinθ],
Pn=(i)nnJn(x0)sin(2θ)2k0a (1η21sin2θ),
Qn=i (i)n 𝒴0 [J'n(x0)εrηJ'n(x1)Jn(x0)Jn(x1)sinθ] .
[μ1rβ1J'n(x1)Jn(x1)μ0rβ0H'n(x0)Hn(x0)][ε1rβ1J'n(x1)Jn(x1)ε0rβ0H'n(x0)Hn(x0)]=n2kz2k2a2[1β021β12]2,
Az,0c(ρ)=c0J0(y),Az,nc(ρ)=2 cn Jn (y),
Aρ,0c(ρ)=icosθηc0J1(y),
Aρ,nc(ρ)=2η[icosθcnJ'n(y)μrdn𝒴0nJn(y)y],
Aφ,nc(ρ)=2η[icosθcnnJn(y)y+μr𝒴0dnJ'n(y)],
Bz,nc(ρ)=2idnJn(y),Bφ,0c(ρ)=i𝒴0εrηc0J1(y),
Bρ,nc(ρ)=2η[i𝒴0εrcnnJn(y)ycosθdnJ'n(y)],
Bφ,nc(ρ)=2η[i𝒴0εrcnJ'n(y)cosθdnnJn(y)y],
cn=Hn(x0)Jn(x1)an+(i)nJn(x0)Jn(x1)sinθ,
dn=Hn(x0)Jn(x1)bn,n=0, 1 , 2 , ,
An2+InεInμn2Hn2(x0)r02δ2[ξn(0)(x0)+ξn(2)(x0)δ2] ,
AnPn+QnInμ(i)nn2Jn(x0)Hn(x0)r02δ3[2ξn(1)(x0)δ2] ,
AnQnPnInεi (i)n 𝒴0n2Jn(x0)Hn(x0)r02δ3 [2ξn(3)(x0)δ2] ,
ξn(0)(x0)1+2η02+2r02αn(1)(x0)+χn(+),
ξn(1)(x0)1+2η02+r022n(n+1)r02αn(1)(x0)+χn(),
ξn(2)(x0)r04[αn(1)(x0)]2+2r04αn(2)(x0)+r02αn(1)(x0)χn(+)
χn(+)η02+(εr+μr)qnη02+εrμrBn02η04(3η04+2η02),
ξn(3)(x0)ξn(1)(x0)+χn(),
χn(±)(εr±μr)Bn0η02.
anUn(x0)[1gn(1)(x0)δ2] ,
bni𝒴0Un(x0)[1gn(2)(x0)δ2],
cnVn(x0)[1gn(3)(x0)δ2],
dni𝒴0Vn(x0)[1gn(2)(x0)δ2],
Un(x0)(i)n2Jn(x0)δ1Hn(x0)ξn(0)(x0),
Vn(x0)(i)n2Jn(x0)δ1Jn(x1)ξn(0)(x0),
gn(1)(x0)12ξn(1)(x0)+ξn(2)(x0)ξn(0)(x0),
gn(2)(x0)12ξn(3)(x0)+ξn(2)(x0)ξn(0)(x0),
gn(3)(x0)12[ξn(1)(x0)ξn(0)(x0)]+ξn(2)(x0)ξn(0)(x0).
a0=Q0I0ε=ηJ0(x1)J1(x0)sinθ+εrJ1(x1)J0(x0)sin2θηJ0(x1)H1(x0)εrJ1(x1)H0(x0)sinθ
a0J1(x0)δH1(x0)[1εrηJ1(x1)J0(x0)δJ0(x1)J1(x0)]D0δ3,
D0iπ2[r022+εrB0η02],B0z0J1(z0)J0(z0).
c0δJ0(z0)[1γ(x0)δ2] ,
γ(x0)D0[γ00+2iπlnx0]+B02η02+r022.
Az,0s(ρ)D0δ3[2iπln(x)+γ00],
Az,ns(ρ)2Dninξn(0)(aρ)n [1+(Rngn(1))δ2] ,
Aρ,0s(ρ)2D0δπk0a (aρ) [1+(R112)δ2] ,
Aσ,ns(ρ)Dnk0ρ (aρ)n 𝒜σ,n(1)ξn(0) [1+(Rn+𝒜σ,n(2)𝒜σ,n(1))δ2] ,
Bz,ns(ρ)2𝒴0Dninξn(0) (aρ)n [1+(Rngn(2))δ2] ,
Bσ,ns(ρ)=𝒴0Dnk0ρ(aρ)n 𝓑σ,n(1)ξn(0) [1+(Rn+𝓑σ,n(2)𝓑σ,n(1))δ2],
Bϕ,0s(ρ)𝒴02D0δπk0a (aρ) [1+R1δ2] ,
R1=12[r02ln(x0)r2ln(x)+iπγ11(r2r02)] ,
Rn=r2r024(n1), n>1 ,
Dn=i (i)n 2 n Jn (x0) δ1 , n=1,2,3,.
𝒜ρ,n(1)=1+χn()2r2αn(1)(x),
𝒜ρ,n(2)=υn()(x)gn(1)+r2[2gn(1)+1]αn(1)(x),
𝒜ϕ,n(1)=1+χn()+2r2αn(1)(x),
𝒜ϕ,n(2)=υn(+)(x)gn(1)2r2gn(2)αn(1)(x),
𝓑ρ,n(1)=1χn()2r2αn(1)(x),
𝓑ρ,n(2)=υn()(x)gn(2)+r2[2gn(2)+1]αn(1)(x),
𝓑ϕ,n(2)=[1χn()+2r2αn(1)(x)],
𝓑ϕ,n(2)=υn(+)(x)gn(2)2r2gn(1)αn(1)(x),
υn(±)(x)0.25 ± 2 r4 αn(2) (x).
2αn(1)(x0)+1(k0a)2(1+2η02)+(εr+μr)Bn0(k0a)2η020 ,
QθQρssinθ+Qzscosθ,
Qρs=R2πaI0 02πSρs(R,ϕ)dϕ,
Qzs=1π(R2a2)I0aR 02πSzs(ρ,ϕ)ρdρdϕ.
Qzc=1πa2I0 0a 02πSzc(ρ,ϕ)ρdρdϕ,
B W(θ)(2πλ12πλ2)a,
ε=4.054.0λ2+iλ(0.38+0.71λ2),
𝒥n(x)=()n𝒥n(x),
𝒥n1(x)+𝒥n+1(x)=2nx𝒥n(x),
𝒥n1(x)𝒥n+1(x)=2 𝒥n (x),
[x𝒥n(x)]=(1n)𝒥n(x)+x𝒥n1(x),
[x𝒥n(x)]=(n2xx)𝒥n(x),
𝒥0(x)=𝒥1(x),
Jn(x)1Γ(n+1)(x2)n [11n+1(x2)2+12(n+1)(n+2)(x2)4] ,
J'n(x)nxΓ(n+1)(x2)n [1n+2n(n+1)(x2)2+n+42n(n+1)(n+2)(x2)4]
nxJn(x)[1x22n(n+1)x48n(n+1)2(n+2)].
H0(x)2iπlnx[1x24+x464]+γ00+γ01x2+γ02x4,
H1(x)2iπx[1x22lnx+x416lnx+iπ2γ11x2+iπ2γ12x4],
H'1(x)2iπx2[1+x22lnx+x22iπ2γ11x23x216lnxx4163iπ2γ12x4],
H2(x)=4iπx2[1+x24x416lnx+iπ4γ22x4],
H'2(x)=8iπx2[1+x416lnx+x432iπ4γ22x4],
γ00=2iγu+πi2ln2π,γ01=2iγuπ+i2ln2+2i4π,γ02=2iγu+πi2ln23i64π,γ11=2iγu+πi2ln2i2π,γ12=4iγu2π+i4ln2+5i32π,γ22=4iγu+2πi4ln23i16π,
Hn(x)Γ(n)iπ(2x)n[1+1n1(x2)2+12(n2)(n1)(x2)4], n3,
H'n(x)=Γ(n+1)iπx(2x)n[1+(n2)x24n(n1)+(n4)x432n(n1)(n2)],n3,
H'n(x)nxHn(x)[1+αn(1)x2+αn(2)x4],n1 ,
α1(1)(x)=lnx2+γuiπ2,α1(2)(x)=12ln2xiπγ11lnxπ22γ112+18,α2(1)(x)=14,α2(2)(x)=18lnxiπ16γ00,αn(1)(x)=12n(n1),αn(2)(x)=18n(n1)2(n2),n3.
Bn(z)zJ'n(z)nJn(z),n0.
dBn(z)dz=nz[1Bn2(z)(zn)2].
η(δ)εrμrcos2θη0[1+δ22η02δ48η04],
Bn(z)Bn0+qnδ2,
qnn2η02[1Bn02(z0n)2].
Hn(x)Hn(x0)(x0x)n[1+Rn(x,x0)],
R1(x,x0)=12[x02lnx0x2lnx+iπγ11(x2x02)],
Rn (x,x0) =x2x024(n1),n>1.

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