Abstract

While subwavelength dielectric structures enclosed by a thin metallic nanoshell have found a wide range of applications, their wavelength-scale counterparts have not been addressed. Conventionally, a dielectric enclosed by a thick metallic shell is considered isolated as the fields attenuate to a negligible value. Here, we show that, due to the Mie resonances of the wavelength-scale dielectric core, the energy density in the core can be enhanced by six orders of magnitude as compared to the off-resonance case, despite the presence of a thick metallic shell. In contrast to the widely studied case of plasmonic core-shell subwavelength particles, where the field enhancement occurs at the boundary of the metallic shell, the thick metallic shell surrounding the wavelength-scale dielectric core provides a strong energy confinement at the center of the core at longer wavelengths, where plasmonic effects are negligible. The observed enhancement can find applications for the probing of shielded materials and designing structures with engineered electromagnetic responses.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2018 (1)

2017 (2)

Y. Yang, A. E. Miroshnichenko, S. V. Kostinski, M. Odit, P. Kapitanova, M. Qiu, and Y. S. Kivshar, “Multimode directionality in all-dielectric metasurfaces,” Phys. Rev. B 95(16), 165426 (2017).
[Crossref]

T. Shibanuma, G. Grinblat, P. Albella, and S. A. Maier, “Efficient third harmonic generation from metal–dielectric hybrid nanoantennas,” Nano Lett. 17(4), 2647–2651 (2017).
[Crossref]

2013 (2)

C. Argyropoulos, F. Monticone, G. D’Aguanno, and A. Alù, “Plasmonic nanoparticles and metasurfaces to realize fano spectra at ultraviolet wavelengths,” Appl. Phys. Lett. 103(14), 143113 (2013).
[Crossref]

D. Wu, S. Jiang, Y. Cheng, and X. Liu, “Three-layered metallodielectric nanoshells: plausible meta-atoms for metamaterials with isotropic negative refractive index at visible wavelengths,” Opt. Express 21(1), 1076–1086 (2013).
[Crossref]

2009 (1)

M. A. Ochsenkuhn, P. R. Jess, H. Stoquert, K. Dholakia, and C. J. Campbell, “Nanoshells for surface-enhanced raman spectroscopy in eukaryotic cells: cellular response and sensor development,” ACS Nano 3(11), 3613–3621 (2009).
[Crossref]

2008 (2)

F. Hao, Y. Sonnefraud, P. V. Dorpe, S. A. Maier, N. J. Halas, and P. Nordlander, “Symmetry breaking in plasmonic nanocavities: subradiant lspr sensing and a tunable fano resonance,” Nano Lett. 8(11), 3983–3988 (2008).
[Crossref]

M. W. Knight and N. J. Halas, “Nanoshells to nanoeggs to nanocups: optical properties of reduced symmetry core–shell nanoparticles beyond the quasistatic limit,” New J. Phys. 10(10), 105006 (2008).
[Crossref]

2006 (1)

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
[Crossref]

2004 (1)

C. Oubre and P. Nordlander, “Optical properties of metallodielectric nanostructures calculated using the finite difference time domain method,” J. Phys. Chem. B 108(46), 17740–17747 (2004).
[Crossref]

1999 (2)

1996 (1)

1989 (1)

1988 (1)

R. B. Schulz, V. Plantz, and D. Brush, “Shielding theory and practice,” IEEE Trans. Electromagn. Compat. 30(3), 187–201 (1988).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, vol. 55 (Courier Corporation, 1965).

Albella, P.

T. Shibanuma, G. Grinblat, P. Albella, and S. A. Maier, “Efficient third harmonic generation from metal–dielectric hybrid nanoantennas,” Nano Lett. 17(4), 2647–2651 (2017).
[Crossref]

Alù, A.

C. Argyropoulos, F. Monticone, G. D’Aguanno, and A. Alù, “Plasmonic nanoparticles and metasurfaces to realize fano spectra at ultraviolet wavelengths,” Appl. Phys. Lett. 103(14), 143113 (2013).
[Crossref]

Argyropoulos, C.

C. Argyropoulos, F. Monticone, G. D’Aguanno, and A. Alù, “Plasmonic nanoparticles and metasurfaces to realize fano spectra at ultraviolet wavelengths,” Appl. Phys. Lett. 103(14), 143113 (2013).
[Crossref]

Averitt, R. D.

Bennink, R. S.

Bhatia, A. B.

M. Born, E. Wolf, A. B. Bhatia, and P. Clemmow et al., Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, vol. 4 (Pergamon, 1970).

Birnboim, M. H.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 2008).

Born, M.

M. Born, E. Wolf, A. B. Bhatia, and P. Clemmow et al., Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, vol. 4 (Pergamon, 1970).

Boyd, R. W.

Brandl, D. W.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
[Crossref]

Brush, D.

R. B. Schulz, V. Plantz, and D. Brush, “Shielding theory and practice,” IEEE Trans. Electromagn. Compat. 30(3), 187–201 (1988).
[Crossref]

Campbell, C. J.

M. A. Ochsenkuhn, P. R. Jess, H. Stoquert, K. Dholakia, and C. J. Campbell, “Nanoshells for surface-enhanced raman spectroscopy in eukaryotic cells: cellular response and sensor development,” ACS Nano 3(11), 3613–3621 (2009).
[Crossref]

Cheng, Y.

Clemmow et al., P.

M. Born, E. Wolf, A. B. Bhatia, and P. Clemmow et al., Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, vol. 4 (Pergamon, 1970).

D’Aguanno, G.

C. Argyropoulos, F. Monticone, G. D’Aguanno, and A. Alù, “Plasmonic nanoparticles and metasurfaces to realize fano spectra at ultraviolet wavelengths,” Appl. Phys. Lett. 103(14), 143113 (2013).
[Crossref]

Daniels, J. J.

N. N. Luxon, R. J. Milelli, and J. J. Daniels, “Radiation shielding and range extending antenna assembly”, (2000). U.S. Patent 6,095,820.

Dholakia, K.

M. A. Ochsenkuhn, P. R. Jess, H. Stoquert, K. Dholakia, and C. J. Campbell, “Nanoshells for surface-enhanced raman spectroscopy in eukaryotic cells: cellular response and sensor development,” ACS Nano 3(11), 3613–3621 (2009).
[Crossref]

Dorpe, P. V.

F. Hao, Y. Sonnefraud, P. V. Dorpe, S. A. Maier, N. J. Halas, and P. Nordlander, “Symmetry breaking in plasmonic nanocavities: subradiant lspr sensing and a tunable fano resonance,” Nano Lett. 8(11), 3983–3988 (2008).
[Crossref]

Fox, A.

A. Fox, Optical Properties of Solids, Oxford Master Series in Condensed Matter Physics (Oxford University Press, 2001).

Frezza, F.

Grinblat, G.

T. Shibanuma, G. Grinblat, P. Albella, and S. A. Maier, “Efficient third harmonic generation from metal–dielectric hybrid nanoantennas,” Nano Lett. 17(4), 2647–2651 (2017).
[Crossref]

Halas, N. J.

M. W. Knight and N. J. Halas, “Nanoshells to nanoeggs to nanocups: optical properties of reduced symmetry core–shell nanoparticles beyond the quasistatic limit,” New J. Phys. 10(10), 105006 (2008).
[Crossref]

F. Hao, Y. Sonnefraud, P. V. Dorpe, S. A. Maier, N. J. Halas, and P. Nordlander, “Symmetry breaking in plasmonic nanocavities: subradiant lspr sensing and a tunable fano resonance,” Nano Lett. 8(11), 3983–3988 (2008).
[Crossref]

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
[Crossref]

R. D. Averitt, S. L. Westcott, and N. J. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B 16(10), 1824–1832 (1999).
[Crossref]

Hao, F.

F. Hao, Y. Sonnefraud, P. V. Dorpe, S. A. Maier, N. J. Halas, and P. Nordlander, “Symmetry breaking in plasmonic nanocavities: subradiant lspr sensing and a tunable fano resonance,” Nano Lett. 8(11), 3983–3988 (2008).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 2008).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 2012).

Jess, P. R.

M. A. Ochsenkuhn, P. R. Jess, H. Stoquert, K. Dholakia, and C. J. Campbell, “Nanoshells for surface-enhanced raman spectroscopy in eukaryotic cells: cellular response and sensor development,” ACS Nano 3(11), 3613–3621 (2009).
[Crossref]

Jiang, S.

Kapitanova, P.

Y. Yang, A. E. Miroshnichenko, S. V. Kostinski, M. Odit, P. Kapitanova, M. Qiu, and Y. S. Kivshar, “Multimode directionality in all-dielectric metasurfaces,” Phys. Rev. B 95(16), 165426 (2017).
[Crossref]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation: Physical Chemistry: A Series of Monographs, vol. 16 (Academic, 2013).

Kivshar, Y. S.

Y. Yang, A. E. Miroshnichenko, S. V. Kostinski, M. Odit, P. Kapitanova, M. Qiu, and Y. S. Kivshar, “Multimode directionality in all-dielectric metasurfaces,” Phys. Rev. B 95(16), 165426 (2017).
[Crossref]

Knight, M. W.

M. W. Knight and N. J. Halas, “Nanoshells to nanoeggs to nanocups: optical properties of reduced symmetry core–shell nanoparticles beyond the quasistatic limit,” New J. Phys. 10(10), 105006 (2008).
[Crossref]

Kostinski, S. V.

Y. Yang, A. E. Miroshnichenko, S. V. Kostinski, M. Odit, P. Kapitanova, M. Qiu, and Y. S. Kivshar, “Multimode directionality in all-dielectric metasurfaces,” Phys. Rev. B 95(16), 165426 (2017).
[Crossref]

Le, F.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
[Crossref]

Liu, X.

Luxon, N. N.

N. N. Luxon, R. J. Milelli, and J. J. Daniels, “Radiation shielding and range extending antenna assembly”, (2000). U.S. Patent 6,095,820.

Maier, S. A.

T. Shibanuma, G. Grinblat, P. Albella, and S. A. Maier, “Efficient third harmonic generation from metal–dielectric hybrid nanoantennas,” Nano Lett. 17(4), 2647–2651 (2017).
[Crossref]

F. Hao, Y. Sonnefraud, P. V. Dorpe, S. A. Maier, N. J. Halas, and P. Nordlander, “Symmetry breaking in plasmonic nanocavities: subradiant lspr sensing and a tunable fano resonance,” Nano Lett. 8(11), 3983–3988 (2008).
[Crossref]

Mangini, F.

Milelli, R. J.

N. N. Luxon, R. J. Milelli, and J. J. Daniels, “Radiation shielding and range extending antenna assembly”, (2000). U.S. Patent 6,095,820.

Miroshnichenko, A. E.

Y. Yang, A. E. Miroshnichenko, S. V. Kostinski, M. Odit, P. Kapitanova, M. Qiu, and Y. S. Kivshar, “Multimode directionality in all-dielectric metasurfaces,” Phys. Rev. B 95(16), 165426 (2017).
[Crossref]

Monticone, F.

C. Argyropoulos, F. Monticone, G. D’Aguanno, and A. Alù, “Plasmonic nanoparticles and metasurfaces to realize fano spectra at ultraviolet wavelengths,” Appl. Phys. Lett. 103(14), 143113 (2013).
[Crossref]

Neeves, A. E.

Nesnidal, R. C.

Nordlander, P.

F. Hao, Y. Sonnefraud, P. V. Dorpe, S. A. Maier, N. J. Halas, and P. Nordlander, “Symmetry breaking in plasmonic nanocavities: subradiant lspr sensing and a tunable fano resonance,” Nano Lett. 8(11), 3983–3988 (2008).
[Crossref]

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
[Crossref]

C. Oubre and P. Nordlander, “Optical properties of metallodielectric nanostructures calculated using the finite difference time domain method,” J. Phys. Chem. B 108(46), 17740–17747 (2004).
[Crossref]

Ochsenkuhn, M. A.

M. A. Ochsenkuhn, P. R. Jess, H. Stoquert, K. Dholakia, and C. J. Campbell, “Nanoshells for surface-enhanced raman spectroscopy in eukaryotic cells: cellular response and sensor development,” ACS Nano 3(11), 3613–3621 (2009).
[Crossref]

Odit, M.

Y. Yang, A. E. Miroshnichenko, S. V. Kostinski, M. Odit, P. Kapitanova, M. Qiu, and Y. S. Kivshar, “Multimode directionality in all-dielectric metasurfaces,” Phys. Rev. B 95(16), 165426 (2017).
[Crossref]

Oubre, C.

C. Oubre and P. Nordlander, “Optical properties of metallodielectric nanostructures calculated using the finite difference time domain method,” J. Phys. Chem. B 108(46), 17740–17747 (2004).
[Crossref]

Plantz, V.

R. B. Schulz, V. Plantz, and D. Brush, “Shielding theory and practice,” IEEE Trans. Electromagn. Compat. 30(3), 187–201 (1988).
[Crossref]

Qiu, M.

Y. Yang, A. E. Miroshnichenko, S. V. Kostinski, M. Odit, P. Kapitanova, M. Qiu, and Y. S. Kivshar, “Multimode directionality in all-dielectric metasurfaces,” Phys. Rev. B 95(16), 165426 (2017).
[Crossref]

Schulz, R. B.

R. B. Schulz, V. Plantz, and D. Brush, “Shielding theory and practice,” IEEE Trans. Electromagn. Compat. 30(3), 187–201 (1988).
[Crossref]

Shibanuma, T.

T. Shibanuma, G. Grinblat, P. Albella, and S. A. Maier, “Efficient third harmonic generation from metal–dielectric hybrid nanoantennas,” Nano Lett. 17(4), 2647–2651 (2017).
[Crossref]

Sipe, J.

Sonnefraud, Y.

F. Hao, Y. Sonnefraud, P. V. Dorpe, S. A. Maier, N. J. Halas, and P. Nordlander, “Symmetry breaking in plasmonic nanocavities: subradiant lspr sensing and a tunable fano resonance,” Nano Lett. 8(11), 3983–3988 (2008).
[Crossref]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, vol. 55 (Courier Corporation, 1965).

Stoquert, H.

M. A. Ochsenkuhn, P. R. Jess, H. Stoquert, K. Dholakia, and C. J. Campbell, “Nanoshells for surface-enhanced raman spectroscopy in eukaryotic cells: cellular response and sensor development,” ACS Nano 3(11), 3613–3621 (2009).
[Crossref]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (John Wiley & Sons, 2007).

Tedeschi, N.

Walker, T. G.

Wang, H.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
[Crossref]

Westcott, S. L.

Wolf, E.

M. Born, E. Wolf, A. B. Bhatia, and P. Clemmow et al., Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, vol. 4 (Pergamon, 1970).

Wu, D.

Yang, Y.

Y. Yang, A. E. Miroshnichenko, S. V. Kostinski, M. Odit, P. Kapitanova, M. Qiu, and Y. S. Kivshar, “Multimode directionality in all-dielectric metasurfaces,” Phys. Rev. B 95(16), 165426 (2017).
[Crossref]

Yoon, Y.-K.

ACS Nano (1)

M. A. Ochsenkuhn, P. R. Jess, H. Stoquert, K. Dholakia, and C. J. Campbell, “Nanoshells for surface-enhanced raman spectroscopy in eukaryotic cells: cellular response and sensor development,” ACS Nano 3(11), 3613–3621 (2009).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

C. Argyropoulos, F. Monticone, G. D’Aguanno, and A. Alù, “Plasmonic nanoparticles and metasurfaces to realize fano spectra at ultraviolet wavelengths,” Appl. Phys. Lett. 103(14), 143113 (2013).
[Crossref]

IEEE Trans. Electromagn. Compat. (1)

R. B. Schulz, V. Plantz, and D. Brush, “Shielding theory and practice,” IEEE Trans. Electromagn. Compat. 30(3), 187–201 (1988).
[Crossref]

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

J. Opt. Soc. Am. B (2)

J. Phys. Chem. B (1)

C. Oubre and P. Nordlander, “Optical properties of metallodielectric nanostructures calculated using the finite difference time domain method,” J. Phys. Chem. B 108(46), 17740–17747 (2004).
[Crossref]

Nano Lett. (3)

F. Hao, Y. Sonnefraud, P. V. Dorpe, S. A. Maier, N. J. Halas, and P. Nordlander, “Symmetry breaking in plasmonic nanocavities: subradiant lspr sensing and a tunable fano resonance,” Nano Lett. 8(11), 3983–3988 (2008).
[Crossref]

T. Shibanuma, G. Grinblat, P. Albella, and S. A. Maier, “Efficient third harmonic generation from metal–dielectric hybrid nanoantennas,” Nano Lett. 17(4), 2647–2651 (2017).
[Crossref]

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett. 6(4), 827–832 (2006).
[Crossref]

New J. Phys. (1)

M. W. Knight and N. J. Halas, “Nanoshells to nanoeggs to nanocups: optical properties of reduced symmetry core–shell nanoparticles beyond the quasistatic limit,” New J. Phys. 10(10), 105006 (2008).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (1)

Y. Yang, A. E. Miroshnichenko, S. V. Kostinski, M. Odit, P. Kapitanova, M. Qiu, and Y. S. Kivshar, “Multimode directionality in all-dielectric metasurfaces,” Phys. Rev. B 95(16), 165426 (2017).
[Crossref]

Other (8)

M. Born, E. Wolf, A. B. Bhatia, and P. Clemmow et al., Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, vol. 4 (Pergamon, 1970).

N. N. Luxon, R. J. Milelli, and J. J. Daniels, “Radiation shielding and range extending antenna assembly”, (2000). U.S. Patent 6,095,820.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation: Physical Chemistry: A Series of Monographs, vol. 16 (Academic, 2013).

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 2012).

A. Fox, Optical Properties of Solids, Oxford Master Series in Condensed Matter Physics (Oxford University Press, 2001).

J. A. Stratton, Electromagnetic Theory (John Wiley & Sons, 2007).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, vol. 55 (Courier Corporation, 1965).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 2008).

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Figures (6)

Fig. 1.
Fig. 1. (a) Schematic of a plane wave incident on the cross-section of the core-shell structures. (b) Infinite cylindrical mesoshell. (c) Spherical mesoshell.
Fig. 2.
Fig. 2. Relative energy density at the center of the cylindrical mesoshell with two different thicknesses $\Delta = 2 \mu$m (blue) and $\Delta = 10 \mu$m (orange) for different core materials: (a) air core, (b) FR4 dielectric core with $\epsilon _3 = 4.28$. (c)–(h) Electric field intensity distributions in the core of radius $3.299$ cm and thickness $\Delta =10$ $\mu$m for air core (c)–(e) and FR4 core (f)–(h). The field intensity is normalized to the intensity of the incident plane wave. The values on the radial axis are given in centimeters. In all the calculations the dispersion of the conductivity is neglected.
Fig. 3.
Fig. 3. Spectral dependence of the transmission coefficient on the geometry of the mesoshell with an air core. (a) $\Delta$ is increased by decreasing the inner radius $r_2$ while keeping the outer radius $r_1$ constant. (b) $\Delta$ is increased by increasing the outer radius while $r_2=\rm {const}$. The permittivity of the surrounding is taken as $\epsilon _1=1$.
Fig. 4.
Fig. 4. Logarithm of electric field intensity in the conducting medium as the wave propagates to the core region. As the permittivity of the core materials increases the field on the inner surface increases, deviating from exponential attenutaion thoughout the conducting region. The solid line shows the distribution of electric fields at the resonant frequency and dashed represents fields away from resonance.
Fig. 5.
Fig. 5. The comparison between zeroth order scattering coefficients. The peak position of peak shifts by $0.5$ MHz.
Fig. 6.
Fig. 6. Relative energy density at the center of the spherical mesoshell proportional to transmission coefficients for two different core materials: Air (a), FR4 dielectric (b). The shell thickness $\Delta = 10 \mu$m and the outer radius $r_1=3.3$ cm.

Tables (1)

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Table 1. Resonance conditions for TM waves

Equations (36)

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( 2 + k j 2 ) { E j ( r , t ) , H j ( r , t ) } T = 0 ,
n ^ × [ E j + 1 ( r , t ) E j ( r , t ) ] = 0 ,
n ^ × [ H j + 1 ( r , t ) H j ( r , t ) ] = 0 ,
E inc ( r , t ) = z ^ E 0 e i ( k 1 x ω t ) = z ^ E 0 e i ω t n = i n J n ( k 1 r ) e i n ϕ .
E 1 , z ( r , ϕ ) = E 0 n = i n { J n ( k 1 r ) + a n H n ( 1 ) ( k 1 r ) } e i n ϕ ,
E 2 , z ( r , ϕ ) = E 0 n = i n { b n J n ( k 2 r ) + c n H n ( 1 ) ( k 2 r ) } e i n ϕ ,
E 3 , z ( r , ϕ ) = E 0 n = i n { d n J n ( k 3 r ) } e i n ϕ ,
[ H n ( ρ 11 ) J n ( ρ 21 ) H n ( ρ 21 ) 0 0 J n ( ρ 22 ) H n ( ρ 22 ) J n ( ρ 32 ) k 1 H n ˙ ( ρ 11 ) k 2 J n ˙ ( ρ 21 ) k 2 H n ˙ ( ρ 21 ) 0 0 k 2 J n ˙ ( ρ 22 ) k 2 H n ˙ ( ρ 22 ) k 3 J n ˙ ( ρ 32 ) ] [ a n b n c n d n ] = [ J n ( ρ 11 ) 0 k 1 J n ˙ ( ρ 11 ) 0 ] .
D = [ H n ( ρ 11 ) J n ( ρ 21 ) H n ( ρ 21 ) J n ( ρ 11 ) 0 J n ( ρ 22 ) H n ( ρ 22 ) 0 k 1 H n ˙ ( ρ 11 ) k 2 J n ˙ ( ρ 21 ) k 2 H n ˙ ( ρ 21 ) k 1 J n ˙ ( ρ 11 ) 0 k 2 J n ˙ ( ρ 22 ) k 2 H n ˙ ( ρ 22 ) 0 ] .
D e t ( D ) = 4 π 2 r 1 r 2 .
D e t ( M ) = k 2 2 H n ( ρ 11 ) J n ( ρ 32 ) [ J n ˙ ( ρ 21 ) H n ˙ ( ρ 22 ) J n ˙ ( ρ 22 ) H n ˙ ( ρ 21 ) ] k 3 k 2 H n ( ρ 11 ) J n ˙ ( ρ 32 ) [ J n ˙ ( ρ 21 ) H n ( ρ 22 ) J n ( ρ 22 ) H n ˙ ( ρ 21 ) ] k 1 k 2 H n ˙ ( ρ 11 ) J n ( ρ 32 ) [ J n ( ρ 21 ) H n ˙ ( ρ 22 ) J n ˙ ( ρ 22 ) H n ( ρ 21 ) ] + k 1 k 2 H n ˙ ( ρ 11 ) J n ˙ ( ρ 32 ) [ J n ( ρ 21 ) H n ( ρ 22 ) J n ( ρ 22 ) H n ( ρ 21 ) ] .
I = | E | 2 = | E 0 | 2 e 2 x δ .
ρ 21 ρ 22 = ( 1 + i ) × 2.2 × 10 4 .
J n ( ρ ) 2 π ρ cos [ ρ ( n + 1 2 ) π 2 ] = 1 2 [ 2 π ρ e i [ ρ ( n + 1 2 ) π 2 ] + 2 π ρ e i [ ρ ( n + 1 2 ) π 2 ] ] .
D e t ( M ) = k 2 2 H n ( ρ 11 ) J n ( ρ 32 ) [ 8 i π 1 ρ 21 ρ 22 sin ( ρ 22 ρ 21 ) ] k 3 k 2 H n ( ρ 11 ) J n ˙ ( ρ 32 ) [ 4 i π 1 ρ 21 ρ 22 cos ( ρ 22 ρ 21 ) ] k 1 k 2 H n ˙ ( ρ 11 ) J n ( ρ 32 ) [ 4 i π 1 ρ 21 ρ 22 cos ( ρ 22 ρ 21 ) ] + k 1 k 2 H n ˙ ( ρ 11 ) J n ˙ ( ρ 32 ) [ 2 i π 1 ρ 21 ρ 22 sin ( ρ 22 ρ 21 ) ] .
d n = 4 i k 2 π r 1 r 2 { [ 4 k 2 2 H n ( ρ 11 ) J n ( ρ 32 ) + k 1 k 2 H n ˙ ( ρ 11 ) J n ˙ ( ρ 32 ) ] sin ( k 2 Δ ) + 2 [ k 3 k 2 H n ( ρ 11 ) J n ˙ ( ρ 32 ) k 1 k 2 H n ˙ ( ρ 11 ) J n ( ρ 32 ) ] cos ( k 2 Δ ) } 1 .
U ( r = 0 ) = 1 2 ϵ 0 ϵ 3 E 0 2 | d 0 | 2 .
E 2 , z ( r , ϕ ) = E 0 n = i n e i n ϕ { 2 i π 2 r 1 r 2 r [ ( k 3 J ˙ n ( ρ 32 ) 2 i k 2 J n ( ρ 32 ) ) e i k 2 ( r 2 r )                                                         + ( k 3 J ˙ n ( ρ 32 ) + 2 i k 2 J n ( ρ 32 ) ) e i k 2 ( r 2 r ) ] } × { [ 4 k 2 2 H n ( ρ 11 ) J n ( ρ 32 ) + k 1 k 2 H n ˙ ( ρ 11 ) J n ˙ ( ρ 32 ) ] sin ( k 2 Δ ) + 2 [ k 3 k 2 H n ( ρ 11 ) J n ˙ ( ρ 32 ) k 1 k 2 H n ˙ ( ρ 11 ) J n ( ρ 32 ) ] cos ( k 2 Δ ) } 1 .
a n , s o l i d σ = J n ( k 1 r 1 ) / H n ( k 1 r 1 ) ,
a n , s o l i d = i k 2 J n ( k 1 r 1 ) J ˙ n ( k 2 r 1 ) + k 1 J ˙ n ( k 1 r 1 ) J n ( k 2 r 1 ) i k 2 H n ( k 1 r 1 ) J ˙ n ( k 2 r 1 ) k 1 H ˙ n ( k 1 r 1 ) J n ( k 1 r 1 ) .
A = [ J n ( ρ 11 ) J n ( ρ 21 ) H n ( ρ 21 ) 0 0 J n ( ρ 22 ) H n ( ρ 22 ) J n ( ρ 32 ) k 1 J n ˙ ( ρ 11 ) k 2 J n ˙ ( ρ 21 ) k 2 H n ˙ ( ρ 21 ) 0 0 k 2 J n ˙ ( ρ 22 ) k 2 H n ˙ ( ρ 22 ) k 3 J n ˙ ( ρ 32 ) ] .
a n = { 4 [ k 2 2 J n ( ρ 11 ) J n ( ρ 32 ) + k 1 k 2 J n ˙ ( ρ 11 ) J n ˙ ( ρ 32 ) ] sin ( k 2 Δ ) + 2 [ k 3 k 2 J n ( ρ 11 ) J n ˙ ( ρ 32 ) k 1 k 2 J n ˙ ( ρ 11 ) J n ( ρ 32 ) ] cos ( k 2 Δ ) } × { [ 4 k 2 2 H n ( ρ 11 ) J n ( ρ 32 ) + k 1 k 2 H n ˙ ( ρ 11 ) J n ˙ ( ρ 32 ) ] sin ( k 2 Δ ) + 2 [ k 3 k 2 H n ( ρ 11 ) J n ˙ ( ρ 32 ) k 1 k 2 H n ˙ ( ρ 11 ) J n ( ρ 32 ) ] cos ( k 2 Δ ) } 1 .
E inc = n = 1 ζ n [ M 1 n j + i N 1 n j ]
M 1 n j , h = z n j , h ( k r ) [ i π 1 n ( θ ) θ ^ τ 1 n ( θ ) ϕ ^ ] e i ϕ ,
N 1 n j , h = z n j , h ( k r ) r n ( n + 1 ) P n 1 ( cos θ ) r ^ + 1 r [ r z n ( k r ) ] r [ τ 1 n ( θ ) θ ^ + i π 1 n ( θ ) ϕ ^ ] e i ϕ .
π 1 n ( θ ) = P n 1 ( cos θ ) sin θ ,                 τ 1 n ( θ ) = θ P n 1 ( cos θ ) .
E 1 = n = 1 ζ n [ M 1 n j + a n N 1 n h i N 1 n j i b n N 1 n h ] ,
E 2 = n = 1 ζ n [ c n M 1 n j + d n M 1 n h i f n N 1 n j i g n N 1 n h ] ,
E 3 = n = 1 ζ n [ p n M 1 n j i q n N 1 n h ] ,
[ M 1 [ 0 0 0 0 ] [ 0 0 0 0 ] M 2 ] [ c n d n f n g n ] = [ i k 1 ρ 11 0 i k 2 ρ 11 0 ] ,
M 1 = [ { Γ h ( ρ 21 ) h ( ρ 11 ) Γ h ( ρ 11 ) h ( ρ 21 ) } { Γ j ( ρ 21 ) h ( ρ 11 ) Γ h ( ρ 11 ) j ( ρ 21 ) } { Γ h ( ρ 22 ) j ( ρ 32 ) Γ j ( ρ 32 ) h ( ρ 22 ) } { Γ j ( ρ 22 ) j ( ρ 32 ) Γ j ( ρ 32 ) j ( ρ 22 ) } ] ,
M 2 = [ { k 1 2 Γ h ( ρ 21 ) h ( ρ 11 ) k 2 2 Γ h ( ρ 11 ) h ( ρ 21 ) } { k 1 2 Γ j ( ρ 21 ) h ( ρ 11 ) k 2 2 Γ h ( ρ 11 ) j ( ρ 21 ) } { k 3 2 Γ h ( ρ 22 ) j ( ρ 32 ) k 2 2 Γ j ( ρ 32 ) h ( ρ 22 ) } { k 3 2 Γ j ( ρ 22 ) j ( ρ 32 ) k 2 2 Γ j ( ρ 32 ) j ( ρ 22 ) } ] .
p n = { ρ 11 ρ 22 D e t M 1 } 1 = i { ρ 11 ρ 22 h ( ρ 11 ) j n ( ρ 32 ) sin ( k 2 Δ ) } 1 ,
q n = { ρ 11 ρ 22 D e t M 2 } 1 = i ρ 21 ρ 11 { k 2 4 Γ h ( ρ 11 ) Γ j ( ρ 32 ) sin ( k 2 Δ ) } 1 .
| E | 2 = | q 1 E 0 N 11 j | 2 ,
| H | 2 = | p 1 H 0 N 11 j | 2 .