Abstract

The rigorous analytical approach for the calculation of the spontaneous decay rate for a quantum emitter located in a cylindrical cavity of arbitrary diameter and length is developed. The approach is based on the dyadic Green’s function of the Helmholtz equation, which is obtained by introducing the fictitious surface current sheets at both ends of the nanocavity. The cases when an emitter is located on the cavity axis and when the cavity length exceeds essentially its diameter are considered in further detail. The general theory is illustrated by the calculations for the system, which models a quantum dot embedded in a GaAs nanowire.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  38. V. G. Bordo, “Reflection and diffraction at the end of a cylindrical dielectric nanowire: exact analytical solution,” Phys. Rev. B 78, 085318 (2008).
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2011 (5)

J. Bleuse, J. Claudon, M. Creasey, N. S. Malik, J.-M. Gérard, I. Maksymov, J.-P. Hugonin, and P. Lalanne, “Inhibition, enhancement, and control of spontaneous emission in photonic nanowires,” Phys. Rev. Lett. 106, 103601 (2011).
[CrossRef]

D. Vanmaekelbergh and L. K. van Vugt, “ZnO nanowire lasers,” Nanoscale 3, 2783–2800 (2011).
[CrossRef]

S. Strauf and F. Jahnke, “Single quantum dot nanolaser,” Laser Photon. Rev. 5, 607–633 (2011).

J. Barthes, G. Colas des Francs, A. Bouhelier, J.-C. Weeber, and A. Dereux, “Purcell factor for a point-like dipolar emitter coupled to a two-dimensional waveguide,” Phys. Rev. B 84, 073403 (2011).
[CrossRef]

V. G. Bordo, “Ab initio analytical model of light transmission through a cylindrical subwavelength hole in an optically thick film,” Phys. Rev. B 84, 075465 (2011).
[CrossRef]

2010 (7)

V. G. Bordo, “Model of Fabry-Pérot-type electromagnetic modes of a cylindrical nanowire,” Phys. Rev. B 81, 035420 (2010).
[CrossRef]

S. N. Dorenbos, H. Sasakura, M. P. van Kouwen, N. Akopian, S. Adachi, N. Namekata, M. Jo, J. Motohisa, Y. Kobayashi, K. Tomioka, T. Fukui, S. Inoue, H. Kumano, C. M. Natarajan, R. H. Hadfield, T. Zijlstra, T. M. Klapwijk, V. Zwiller, and I. Suemune, “Position controlled nanowires for infrared single photon emission,” Appl. Phys. Lett. 97, 171106 (2010).
[CrossRef]

R. Esteban, T. V. Teperik, and J. J. Greffet, “Optical patch antennas for single photon emission using surface plasmon resonances,” Phys. Rev. Lett. 104, 026802 (2010).
[CrossRef]

J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J.-M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photon. 4, 174–177 (2010).
[CrossRef]

A. Mohan, M. Felici, P. Gallo, B. Dwir, A. Rudra, J. Faist, and E. Kapon, “Polarization-entangled photons produced with high-symmetry site-controlled quantum dots,” Nat. Photon. 4, 302–306 (2010).
[CrossRef]

C. Z. Ning, “Semiconductor nanolasers,” Phys. Status Solidi B 247, 774–788 (2010).

N. Gregersen, T. R. Nielsen, J. Mørk, J. Claudon, and J.-M. Gérard, “Designs for high-efficiency electrically pumped photonic nanowire single-photon sources,” Opt. Express 18, 21204–21218 (2010).
[CrossRef]

2009 (5)

I. Friedler, C. Sauvan, J. P. Hugonin, P. Lalanne, J. Claudon, and J. M. Gérard, “Solid-state single photon sources: the nanowire antenna,” Opt. Express 17, 2095–2110 (2009).
[CrossRef]

I. D. Rukhlenko, D. Handapangoda, M. Premarante, A. V. Fedorov, A. V. Baranov, and C. Jagadish, “Spontaneous emission of guided polaritons by quantum dot coupled to metallic nanowire: beyond the dipole approximation,” Opt. Express 17, 17570–17581 (2009).
[CrossRef]

R. Singh, and G. Bester, “Nanowire quantum dots as an ideal source of entangled photon pairs,” Phys. Rev. Lett. 103, 063601 (2009).
[CrossRef]

V. G. Bordo, “Erratum: Reflection and diffraction at the end of a cylindrical dielectric nanowire: exact analytical solution,” Phys. Rev. B 79, 039901(E) (2009).
[CrossRef]

Y. N. Chen, G. Y. Chen, D. S. Chuu, and T. Brandes, “Quantum-dot exciton dynamics with a surface plasmon: band-edge quantum optics,” Phys. Rev. A 79, 033815 (2009).
[CrossRef]

2008 (2)

V. G. Bordo, “Reflection and diffraction at the end of a cylindrical dielectric nanowire: exact analytical solution,” Phys. Rev. B 78, 085318 (2008).
[CrossRef]

G. Y. Chen, Y. N. Chen, and D. S. Chuu, “Spontaneous emission of quantum dot excitons into surface plasmons in a nanowire,” Opt. Lett. 33, 2212–2214 (2008).
[CrossRef]

2007 (2)

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B 76, 035420 (2007).
[CrossRef]

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photon. 1, 449–458 (2007).
[CrossRef]

2006 (3)

S. Noda, “Seeking the ultimate nanolaser,” Science 314, 260–261 (2006).
[CrossRef]

P. J. Pauzauskie and P. Yang, “Nanowire photonics,” Mater. Today 9(10), 36–45 (2006).
[CrossRef]

A. V. Maslov, M. I. Bakunov, and C. Z. Ning, “Distribution of optical emission between guided modes and free space in a semiconductor nanowire,” J. Appl. Phys. 99, 024314(2006).
[CrossRef]

2005 (3)

D. P. Fussell, R. C. McPhedran, and C. Martijn de Sterke, “Decay rate and level shift in a circular dielectric waveguide,” Phys. Rev. A 71, 013815 (2005).
[CrossRef]

F. Le Kien, S. Dutta Gupta, V. I. Balykin, and K. Hakuta, “Spontaneous emission of a cesium atom near a nanofiber: efficient coupling of light to guided modes,” Phys. Rev. A 72, 032509 (2005).
[CrossRef]

B. Lounis and M. Oritt, “Single-photon sources,” Rep. Prog. Phys. 68, 1129–1179 (2005).
[CrossRef]

2004 (2)

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaard, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

V. V. Klimov, and M. Ducloy, “Spontaneous emission rate of an excited atom placed near a nanofiber,” Phys. Rev. A 69, 013812 (2004).
[CrossRef]

2003 (1)

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[CrossRef]

2002 (2)

M. Pelton, C. Santori, J. Vučković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient source of single photons: a single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89, 233602 (2002).
[CrossRef]

H. Mabuchi and A. C. Doherty, “Cavity quantum electrodynamics: coherence in context,” Science 298, 1372–1377 (2002).
[CrossRef]

2001 (2)

V. V. Klimov, M. Ducloy, and V. S. Letokhov, “Spontaneous emission of an atom in the presence of nanobodies,” Quantum Electron. 31, 569–586 (2001).
[CrossRef]

T. Søndergaard, and B. Tromborg, “General theory for spontaneous emission in active dielectric microstructures: example of a fiber amplifier,” Phys. Rev. A 64, 033812 (2001).
[CrossRef]

2000 (1)

W. Żakowicz, and M. Janowicz, “Spontaneous emission in the presence of a dielectric cylinder,” Phys. Rev. A 62, 013820 (2000).
[CrossRef]

1997 (2)

H. Nha, and W. Jhe, “Cavity quantum electrodynamics for a cylinder: inside a hollow dielectric and near a solid dielectric cylinder,” Phys. Rev. A 56, 2213–2220 (1997).
[CrossRef]

F. Wijnands, J. B. Pendry, F. J. Garcia-Vidal, P. M. Bell, P. J. Roberts, and L. Martín Moreno, “Green’s functions for Maxwell’s equations: application to spontaneous emission,” Opt. Quantum Electron. 29, 199–216 (1997).
[CrossRef]

1993 (1)

1987 (1)

1984 (1)

J. M. Wylie and J. E. Sipe, “Quantum electrodynamics near an interface,” Phys. Rev. A 30, 1185–1193 (1984).
[CrossRef]

1970 (1)

G. L. Yip, “Launching efficiency of the HE11 surface wave mode on a dielectric rod,” IEEE Trans. Microwave Theory Tech. MTT-18, 1033–1041 (1970).

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

1939 (1)

S. A. Schelkunoff, “On diffraction and radiation of electromagnetic waves,” Phys. Rev. 56, 308–316 (1939).
[CrossRef]

Adachi, S.

S. N. Dorenbos, H. Sasakura, M. P. van Kouwen, N. Akopian, S. Adachi, N. Namekata, M. Jo, J. Motohisa, Y. Kobayashi, K. Tomioka, T. Fukui, S. Inoue, H. Kumano, C. M. Natarajan, R. H. Hadfield, T. Zijlstra, T. M. Klapwijk, V. Zwiller, and I. Suemune, “Position controlled nanowires for infrared single photon emission,” Appl. Phys. Lett. 97, 171106 (2010).
[CrossRef]

Akopian, N.

S. N. Dorenbos, H. Sasakura, M. P. van Kouwen, N. Akopian, S. Adachi, N. Namekata, M. Jo, J. Motohisa, Y. Kobayashi, K. Tomioka, T. Fukui, S. Inoue, H. Kumano, C. M. Natarajan, R. H. Hadfield, T. Zijlstra, T. M. Klapwijk, V. Zwiller, and I. Suemune, “Position controlled nanowires for infrared single photon emission,” Appl. Phys. Lett. 97, 171106 (2010).
[CrossRef]

Asano, T.

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photon. 1, 449–458 (2007).
[CrossRef]

Bakunov, M. I.

A. V. Maslov, M. I. Bakunov, and C. Z. Ning, “Distribution of optical emission between guided modes and free space in a semiconductor nanowire,” J. Appl. Phys. 99, 024314(2006).
[CrossRef]

Balykin, V. I.

F. Le Kien, S. Dutta Gupta, V. I. Balykin, and K. Hakuta, “Spontaneous emission of a cesium atom near a nanofiber: efficient coupling of light to guided modes,” Phys. Rev. A 72, 032509 (2005).
[CrossRef]

Baranov, A. V.

Barthes, J.

J. Barthes, G. Colas des Francs, A. Bouhelier, J.-C. Weeber, and A. Dereux, “Purcell factor for a point-like dipolar emitter coupled to a two-dimensional waveguide,” Phys. Rev. B 84, 073403 (2011).
[CrossRef]

Bazin, M.

J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J.-M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photon. 4, 174–177 (2010).
[CrossRef]

Bell, P. M.

F. Wijnands, J. B. Pendry, F. J. Garcia-Vidal, P. M. Bell, P. J. Roberts, and L. Martín Moreno, “Green’s functions for Maxwell’s equations: application to spontaneous emission,” Opt. Quantum Electron. 29, 199–216 (1997).
[CrossRef]

Bester, G.

R. Singh, and G. Bester, “Nanowire quantum dots as an ideal source of entangled photon pairs,” Phys. Rev. Lett. 103, 063601 (2009).
[CrossRef]

Bleuse, J.

J. Bleuse, J. Claudon, M. Creasey, N. S. Malik, J.-M. Gérard, I. Maksymov, J.-P. Hugonin, and P. Lalanne, “Inhibition, enhancement, and control of spontaneous emission in photonic nanowires,” Phys. Rev. Lett. 106, 103601 (2011).
[CrossRef]

J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J.-M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photon. 4, 174–177 (2010).
[CrossRef]

Bordo, V. G.

V. G. Bordo, “Ab initio analytical model of light transmission through a cylindrical subwavelength hole in an optically thick film,” Phys. Rev. B 84, 075465 (2011).
[CrossRef]

V. G. Bordo, “Model of Fabry-Pérot-type electromagnetic modes of a cylindrical nanowire,” Phys. Rev. B 81, 035420 (2010).
[CrossRef]

V. G. Bordo, “Erratum: Reflection and diffraction at the end of a cylindrical dielectric nanowire: exact analytical solution,” Phys. Rev. B 79, 039901(E) (2009).
[CrossRef]

V. G. Bordo, “Reflection and diffraction at the end of a cylindrical dielectric nanowire: exact analytical solution,” Phys. Rev. B 78, 085318 (2008).
[CrossRef]

Bouhelier, A.

J. Barthes, G. Colas des Francs, A. Bouhelier, J.-C. Weeber, and A. Dereux, “Purcell factor for a point-like dipolar emitter coupled to a two-dimensional waveguide,” Phys. Rev. B 84, 073403 (2011).
[CrossRef]

Brandes, T.

Y. N. Chen, G. Y. Chen, D. S. Chuu, and T. Brandes, “Quantum-dot exciton dynamics with a surface plasmon: band-edge quantum optics,” Phys. Rev. A 79, 033815 (2009).
[CrossRef]

Chang, D. E.

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B 76, 035420 (2007).
[CrossRef]

Chen, G. Y.

Y. N. Chen, G. Y. Chen, D. S. Chuu, and T. Brandes, “Quantum-dot exciton dynamics with a surface plasmon: band-edge quantum optics,” Phys. Rev. A 79, 033815 (2009).
[CrossRef]

G. Y. Chen, Y. N. Chen, and D. S. Chuu, “Spontaneous emission of quantum dot excitons into surface plasmons in a nanowire,” Opt. Lett. 33, 2212–2214 (2008).
[CrossRef]

Chen, Y. N.

Y. N. Chen, G. Y. Chen, D. S. Chuu, and T. Brandes, “Quantum-dot exciton dynamics with a surface plasmon: band-edge quantum optics,” Phys. Rev. A 79, 033815 (2009).
[CrossRef]

G. Y. Chen, Y. N. Chen, and D. S. Chuu, “Spontaneous emission of quantum dot excitons into surface plasmons in a nanowire,” Opt. Lett. 33, 2212–2214 (2008).
[CrossRef]

Chu, D. Y.

Chuu, D. S.

Y. N. Chen, G. Y. Chen, D. S. Chuu, and T. Brandes, “Quantum-dot exciton dynamics with a surface plasmon: band-edge quantum optics,” Phys. Rev. A 79, 033815 (2009).
[CrossRef]

G. Y. Chen, Y. N. Chen, and D. S. Chuu, “Spontaneous emission of quantum dot excitons into surface plasmons in a nanowire,” Opt. Lett. 33, 2212–2214 (2008).
[CrossRef]

Claudon, J.

J. Bleuse, J. Claudon, M. Creasey, N. S. Malik, J.-M. Gérard, I. Maksymov, J.-P. Hugonin, and P. Lalanne, “Inhibition, enhancement, and control of spontaneous emission in photonic nanowires,” Phys. Rev. Lett. 106, 103601 (2011).
[CrossRef]

N. Gregersen, T. R. Nielsen, J. Mørk, J. Claudon, and J.-M. Gérard, “Designs for high-efficiency electrically pumped photonic nanowire single-photon sources,” Opt. Express 18, 21204–21218 (2010).
[CrossRef]

J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J.-M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire,” Nat. Photon. 4, 174–177 (2010).
[CrossRef]

I. Friedler, C. Sauvan, J. P. Hugonin, P. Lalanne, J. Claudon, and J. M. Gérard, “Solid-state single photon sources: the nanowire antenna,” Opt. Express 17, 2095–2110 (2009).
[CrossRef]

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (IEEE, 1991).

Creasey, M.

J. Bleuse, J. Claudon, M. Creasey, N. S. Malik, J.-M. Gérard, I. Maksymov, J.-P. Hugonin, and P. Lalanne, “Inhibition, enhancement, and control of spontaneous emission in photonic nanowires,” Phys. Rev. Lett. 106, 103601 (2011).
[CrossRef]

de Sterke, C. Martijn

D. P. Fussell, R. C. McPhedran, and C. Martijn de Sterke, “Decay rate and level shift in a circular dielectric waveguide,” Phys. Rev. A 71, 013815 (2005).
[CrossRef]

Dereux, A.

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Figures (5)

Fig. 1.
Fig. 1.

Geometry of the problem.

Fig. 2.
Fig. 2.

Normalized spontaneous emission rates as a function of the normalized cavity diameter. An emitter is located at the nanocavity center and its transition moment is oriented perpendicularly to the z axis. ϵ 1 = 1 , ϵ 2 = 3.45 , and L / a = 20 . Dashed curve: normalized emission rate into the continuous spectrum of modes. Solid curve: the Purcell factor for the transition into the fundamental mode. Dashed–dotted curve: the same Purcell factor, but calculated in the limit of an infinite nanowire.

Fig. 3.
Fig. 3.

Dependence of the Purcell factor on the normalized cavity length. An emitter is located at the nanocavity center and its transition moment is oriented perpendicularly to the z axis. ϵ 1 = 1 , ϵ 2 = 3.45 , and 2 a / λ = 0.25 . Inset: the first peak shown on a larger scale.

Fig. 4.
Fig. 4.

Same as Fig. 3, but the position of the emitter is shifted along the cavity axis so that z 0 = 0.25 a . Inset: the first resonance shown on a larger scale.

Fig. 5.
Fig. 5.

Dependence of the Purcell factor on the emitter position in the vicinity of its center for different normalized cavity diameters. An emitter is located at the nanocavity axis and its transition moment is oriented perpendicularly to the z axis. ϵ 1 = 1 , ϵ 2 = 3.45 , and L = 20 a . Solid curve, 2 a / λ = 0.20 ; dashed curve, 2 a / λ = 0.25 ; dashed–dotted curve, 2 a / λ = 0.30 .

Equations (103)

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2 Π e μ ϵ c 2 2 Π e t 2 = 4 π ϵ P 0 ,
2 Π m μ ϵ c 2 2 Π m t 2 = 4 π M 0 ,
E = ( · Π e ) ϵ c 2 2 Π e t 2 1 c × Π m t ,
H = ϵ c × Π e t + ( · Π m ) ϵ c 2 2 Π m t 2 ,
F ( t ) = 1 2 π F ( ω ) e i ω t d ω ,
E = ( · Π e ) + k 2 Π e + i ω c × Π m ,
H = i ϵ ω c × Π e + ( · Π m ) + k 2 Π m ,
2 Π e + k 2 Π e = 4 π ϵ P 0 ,
2 Π m + k 2 Π m = 0 ,
k = ω c ϵ = 2 π λ ϵ .
Π e ( R ; ω ) = 4 π ϵ G ¯ ( R , R ; ω ) P 0 ( R ; ω ) d R ,
2 G ¯ ( R , R 0 ; ω ) + k 2 G ¯ ( R , R 0 ; ω ) = I ¯ δ ( R R 0 ) ,
I ¯ = e x e x + e y e y + e z e z
G ¯ ( R , R 0 ; ω ) = e x G x ( R , R 0 ; ω ) + e y G y ( R , R 0 ; ω ) + e z G z ( R , R 0 ; ω ) ,
2 G α ( R , R 0 ; ω ) + k 2 G α ( R , R 0 ; ω ) = e α δ ( R R 0 ) ,
G α ( R , R 0 ; ω ) = e i k | R R 0 | 4 π | R R 0 | e α .
G 0 ( R , R 0 ; ω ) = e i k 2 | R R 0 | 4 π | R R 0 | .
G 2 α 0 ( R , R 0 ; ω ) = G 0 ( R , R 0 ; ω ) e α + Ψ 2 e ( R ; ω ) e z + Ψ 2 m ( R ; ω ) e z ,
G 1 0 ( R ; ω ) = Ψ 1 e ( R ; ω ) e z + Ψ 1 m ( R ; ω ) e z ,
Ψ j σ ( r , θ , z ) = 1 2 π C Ψ ˜ j σ ( r , θ ; β ) e i β z d β ,
Ψ ˜ 1 σ ( r , θ ; β ) = 1 q 1 2 n = a 1 n σ ( β ) H n ( 1 ) ( q 1 r ) e i n θ ,
Ψ ˜ 2 σ ( r , θ ; β ) = 1 q 2 2 n = a 2 n σ ( β ) J n ( q 2 r ) e i n θ ,
E j t ± + E j t ± = E j t ± ,
H j t ± + H j t ± = H j t ± ,
K j e ± = ± c 4 π e z × H j t ± ,
K j m ± = c 4 π e z × E j t ± ,
Φ j σ ± ( R ) = i τ j σ ω S j ± K j σ ± ( R ) e i k j | R R | | R R | d R ,
G 2 α ( R , R 0 ; ω ) = G 2 α 0 ( R , R 0 ; ω ) + Φ 2 e ( R ; ω ) + Φ 2 e + ( R ; ω ) + Φ 2 m ( R ; ω ) + Φ 2 m + ( R ; ω ) ,
G 1 ( R ; ω ) = G 1 0 ( R ; ω ) + Φ 1 e ( R ; ω ) + Φ 1 e + ( R ; ω ) + Φ 1 m ( R ; ω ) + Φ 1 m + ( R ; ω ) .
M ^ n ( β ) A⃗ n α ( β ) 1 2 π C [ e i ( β β ) ( L / 2 z 0 ) e i ( β β ) ( L / 2 + z 0 ) ] N ^ n ( β , β ) A⃗ n α ( β ) d β = B⃗ n α ( β ) ,
A⃗ n α ( β ) = ( a ^ 2 n e α ( β ) a ^ 2 n m α ( β ) a ^ 1 n e α ( β ) a ^ 1 n m α ( β ) ) ,
B⃗ n α ( β ) = ( E ˜ θ n 0 α ( β ) E ˜ z n 0 α ( β ) H ˜ θ n 0 α ( β ) H ˜ z n 0 α ( β ) ) ,
a ^ j n σ α ( β ) = e i β z 0 a j n σ α ( β ) ,
F μ 0 α ( a , θ , z ) = 1 2 π C n = F ˜ μ n 0 α ( β ) e i n θ e i β z d β ,
A⃗ n α ( β ) = M ^ n 1 ( β ) B⃗ n α ( β ) .
Π e = G 0 e α + Ψ 2 e e z + Φ 2 e + Φ 2 e + ,
Π m = Ψ 2 m e z + Φ 2 m + Φ 2 m + .
E ( R , ω ) = F ¯ ( R , R 0 ; ω ) · μ .
P 0 ( R ) = μ δ ( R R 0 ) ,
Π ( R ; ω ) = 4 π ϵ 2 ( ω ) [ μ x G 2 x ( R , R 0 ; ω ) + μ y G 2 y ( R , R 0 ; ω ) + μ z G 2 z ( R , R 0 ; ω ) ] ,
E ( R ; ω ) = 4 π ϵ 2 ( ω ) [ μ x E x ( R , R 0 ; ω ) + μ y E y ( R , R 0 ; ω ) + μ z E z ( R , R 0 ; ω ) ] ,
F α β ( R , R 0 ; ω ) = 4 π ϵ 2 ( ω ) E α β ( R , R 0 ; ω ) ,
γ f i 0 = 4 ω 3 3 c 3 ϵ 2 α | μ α f i | 2 ,
γ f i c = 2 α β μ α f i μ β i f Im F α β c ( R 0 , R 0 ; ω ) = 8 π α β μ α f i μ β i f Im [ 1 ϵ 2 ( ω ) E α c β ( R 0 , R 0 ; ω ) ] ,
E c α = C E ˜ c α ( β ) d β ,
E c α = 0 k 1 [ E ˜ c α ( β ) + E ˜ c α ( β ) ] d β + P k 1 [ E ˜ c α ( β ) + E ˜ c α ( β ) ] d β + i π a [ Res E ˜ c α ( β a ) Res E ˜ c α ( β a ) ] ,
γ f i c = γ f i r + γ f i e v + a γ f i a ,
γ f i cs = γ f i 0 + γ f i r + γ f i e v .
F P = 1 γ f i 0 a γ f i a ,
F β a = γ f i a γ f i 0 + γ f i c ,
Ψ 2 σ ( x , y , z ) = 1 2 π π π C b σ ( β , ψ ) e i q 2 x cos ψ e i q 2 y sin ψ e i β z d β d ψ ,
e i q 2 x cos ψ e i q 2 y sin ψ = e i q 2 r cos ( θ ψ ) = n = i n J n ( q 2 r ) e i n ( θ ψ ) .
π π b σ ( β , ψ ) e i n ψ d ψ = 1 i n q 2 2 a 2 n σ ( β ) .
b σ ( β , ψ ) = k = c k σ ( β ) e i k ψ ,
c k σ ( β ) = 1 2 π i k q 2 2 a 2 k σ ( β ) .
F x z c ( z 0 , z 0 ; ω ) = F y z c ( z 0 , z 0 ; ω ) = F z x c ( z 0 , z 0 ; ω ) = F z y c ( z 0 , z 0 ; ω ) = 0 .
B⃗ ± 1 y ( β ) = ± i B⃗ ± 1 x ( β ) .
a j , ± 1 σ y ( β ) = ± i a j , ± 1 σ x ( β ) .
F x x c ( z 0 , z 0 ; ω ) = F y y c ( z 0 , z 0 ; ω ) F c ( z 0 , z 0 ; ω ) , F x y c ( z 0 , z 0 ; ω ) = F y x c ( z 0 , z 0 ; ω ) .
γ f i c ( z 0 ; ω ) = 2 [ ( μ i f ) 2 Im F c ( z 0 , z 0 ; ω ) + ( μ z i f ) 2 Im F z z c ( z 0 , z 0 ; ω ) ] ,
( μ i f ) 2 = ( μ x i f ) 2 + ( μ y i f ) 2 ,
A⃗ n α ( β ) = M ^ n 1 ( β ) C⃗ n α ( β ) .
C⃗ n α ( β ) 1 2 π [ C + e i ( β β ) ( L / 2 z 0 ) N ^ n ( β , β ) M ^ n 1 ( β ) C⃗ n α ( β ) d β C e i ( β β ) ( L / 2 + z 0 ) N ^ n ( β , β ) M ^ n 1 ( β ) C⃗ n α ( β ) d β ] = B⃗ n α ( β ) ,
C⃗ n α ( β ) = i { e i ( β β 0 ) ( L / 2 z 0 ) N ^ n ( β , β 0 ) Res [ M ^ n 1 ( β 0 ) ] C⃗ n α ( β 0 ) + e i ( β + β 0 ) ( L / 2 + z 0 ) N ^ n ( β , β 0 ) Res [ M ^ n 1 ( β 0 ) ] C⃗ n α ( β 0 ) } B⃗ n α ( β ) ,
C⃗ n α ( β 0 ) i { N ^ n ( β 0 , β 0 ) Res [ M ^ n 1 ( β 0 ) ] C⃗ n α ( β 0 ) + e i β 0 ( L + 2 z 0 ) N ^ n ( β 0 , β 0 ) Res [ M ^ n 1 ( β 0 ) ] C⃗ n α ( β 0 ) } = B⃗ n α ( β 0 ) ,
C⃗ n α ( β 0 ) i { e i β 0 ( L 2 z 0 ) N ^ n ( β 0 , β 0 ) Res [ M ^ n 1 ( β 0 ) ] C⃗ n α ( β 0 ) + N ^ n ( β 0 , β 0 ) Res [ M ^ n 1 ( β 0 ) ] C⃗ n α ( β 0 ) } = B⃗ n α ( β 0 ) ,
β 0 ( d m ) L = 2 π m ψ ,
β 0 L m = 2 π m ϕ ,
γ a = 2 π ϵ 2 q 20 ( μ i f ) 2 × Im { β 0 [ Res a ^ 21 e a , x ( β 0 ) + Res a ^ 21 e a , x ( β 0 ) ] i ω c [ Res a ^ 21 m s , x ( β 0 ) Res a ^ 21 m s , x ( β 0 ) ] } ,
β 0 Δ z 0 = π ,
| R R 0 | = d 2 + ( z z 0 ) 2 ,
d = r 2 + r 0 2 2 r r 0 cos ( θ θ 0 ) ,
e i k 2 d 2 + ( z z 0 ) 2 d 2 + ( z z 0 ) 2 = i 2 H 0 ( 1 ) ( q 2 d ) e i β ( z z 0 ) d β ,
H 0 ( 1 ) ( q 2 d ) = n = J n ( q 2 r 0 ) H n ( 1 ) ( q 2 r ) e i n ( θ θ 0 ) , r > r 0 ,
E ˜ θ n 0 x = 1 8 { e i ( n + 1 ) θ 0 J n + 1 ( q 2 r 0 ) [ n q 2 a H n ( 1 ) ( q 2 a ) k 2 2 H n + 1 ( 1 ) ( q 2 a ) ] e i ( n 1 ) θ 0 J n 1 ( q 2 r 0 ) [ n q 2 a H n ( 1 ) ( q 2 a ) k 2 2 H n 1 ( 1 ) ( q 2 a ) ] } ,
E ˜ z n 0 x = β q 2 8 [ e i ( n + 1 ) θ 0 J n + 1 ( q 2 r 0 ) e i ( n 1 ) θ 0 J n 1 ( q 2 r 0 ) ] H n ( 1 ) ( q 2 a ) ,
H ˜ θ n 0 x = i ϵ 2 β ω 8 c [ e i ( n + 1 ) θ 0 J n + 1 ( q 2 r 0 ) H n + 1 ( 1 ) ( q 2 a ) + e i ( n 1 ) θ 0 J n 1 ( q 2 r 0 ) H n 1 ( 1 ) ( q 2 a ) ] ,
H ˜ z n 0 x = i ϵ 2 q 2 ω 8 c [ e i ( n + 1 ) θ 0 J n + 1 ( q 2 r 0 ) + e i ( n 1 ) θ 0 J n 1 ( q 2 r 0 ) ] H n ( 1 ) ( q 2 a ) .
E ˜ θ n 0 y = i 8 { e i ( n + 1 ) θ 0 J n + 1 ( q 2 r 0 ) [ n q 2 a H n ( 1 ) ( q 2 a ) k 2 2 H n + 1 ( 1 ) ( q 2 a ) ] + e i ( n 1 ) θ 0 J n 1 ( q 2 r 0 ) [ n q 2 a H n ( 1 ) ( q 2 a ) k 2 2 H n 1 ( 1 ) ( q 2 a ) ] } ,
E ˜ z n 0 y = i β q 2 8 [ e i ( n + 1 ) θ 0 J n + 1 ( q 2 r 0 ) + e i ( n 1 ) θ 0 J n 1 ( q 2 r 0 ) ] H n ( 1 ) ( q 2 a ) ,
H ˜ θ n 0 y = ϵ 2 β ω 8 c [ e i ( n + 1 ) θ 0 J n + 1 ( q 2 r 0 ) H n + 1 ( 1 ) ( q 2 a ) e i ( n 1 ) θ 0 J n 1 ( q 2 r 0 ) H n 1 ( 1 ) ( q 2 a ) ] ,
H ˜ z n 0 y = ϵ 2 q 2 ω 8 c [ e i ( n + 1 ) θ 0 J n + 1 ( q 2 r 0 ) e i ( n 1 ) θ 0 J n 1 ( q 2 r 0 ) ] H n ( 1 ) ( q 2 a ) .
E ˜ θ n 0 z = i n β 4 a e i n θ 0 J n ( q 2 r 0 ) H n ( 1 ) ( q 2 a ) ,
E ˜ z n 0 z = i q 2 2 4 e i n θ 0 J n ( q 2 r 0 ) H n ( 1 ) ( q 2 a ) ,
H ˜ θ n 0 z = ϵ 2 ω q 2 4 c e i n θ 0 J n ( q 2 r 0 ) H n ( 1 ) ( q 2 a ) ,
H ˜ z n 0 z = 0 .
E c α ( z 0 , z 0 ; ω ) = E ψ α ( z 0 , z 0 ; ω ) + E ϕ , α ( z 0 , z 0 ; ω ) + E ϕ + , α ( z 0 , z 0 ; ω ) ,
E x ψ α ( z 0 , z 0 ; ω ) = i 4 π C [ β a ^ 21 e a , α ( β ) i ω c a ^ 21 m s , α ( β ) ] d β q 2 ,
E y ψ y ( z 0 , z 0 ; ω ) = E x ψ x ( z 0 , z 0 ; ω ) ,
E y ψ x ( z 0 , z 0 ; ω ) = E x ψ y ( z 0 , z 0 ; ω ) ,
E z ψ α ( z 0 , z 0 ; ω ) = 0 .
E γ ϕ ± , α ( z 0 , z 0 ; ω ) = ± i c 8 π ϵ 2 ω C E ^ γ ± , α ( z 0 , z 0 ; ω , β ) e ± i β ( L / 2 z 0 ) d β , γ = x , y , z ,
E ^ x ± , α ( z 0 , z 0 ; ω , β ) = 1 q 2 0 a e i k 2 d ± d ± 3 { 3 r 2 4 d ± 2 [ 3 ( i k 2 d ± 1 ) + k 2 2 d ± 2 ] [ i ω c ϵ 2 J 1 ( q 2 r ) a ^ 21 e a , α ( β ) + β q 2 r J 1 ( q 2 r ) a ^ 21 m s , α ( β ) ] + ( 1 i k 2 d ± k 2 2 d ± 2 ) J 0 ( q 2 r ) [ i ω c ϵ 2 a ^ 21 e a , α ( β ) + β a ^ 21 m s , α ( β ) ] + i ω c ϵ 2 ( i k 2 d ± 1 ) ( z 0 L 2 ) J 0 ( q 2 r ) [ i β a ^ 21 e a , α ( β ) + ω c a ^ 21 m s , α ( β ) ] } r d r ,
E ^ y ± , y ( z 0 , z 0 ; ω , β ) = E ^ x ± , x ( z 0 , z 0 ; ω , β ) ,
E ^ y ± , x ( z 0 , z 0 ; ω , β ) = E ^ x ± , y ( z 0 , z 0 ; ω , β ) ,
E ^ z ± , α ( z 0 , z 0 ; ω , β ) = 0 .
a ^ 21 σ a , α ( β ) = a ^ 21 σ α ( β ) a ^ 2 , 1 σ α ( β ) ,
a ^ 21 σ s , α ( β ) = a ^ 21 σ α ( β ) + a ^ 2 , 1 σ α ( β ) ,
d ± ( r ) = r 2 + ( z 0 L 2 ) 2 .
E x ψ z ( z 0 , z 0 ; ω ) = E y ψ z ( z 0 , z 0 ; ω ) = 0 ,
E z ψ z ( z 0 , z 0 ; ω ) = 1 2 π C a 20 e z ( β ) e i β z 0 d β ,
E ^ x ± , z ( z 0 , z 0 ; ω , β ) = E ^ y ± , z ( z 0 , z 0 ; ω , β ) = 0 ,
E ^ z ± , z ( z 0 , z 0 ; ω , β ) = i ω ϵ 2 2 c q 2 a ^ 20 e z ( β ) e i β z 0 0 a e i k 2 d ± d ± 5 { [ 3 ( i k 2 d ± 1 ) + k 2 2 d ± 2 ] ( z 0 L 2 ) i β ( i k 2 d ± 1 ) d ± 2 } J 1 ( q 2 r ) r 2 d r .

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