Abstract

Surface plasmon excitation that is due to a single or a structured circular aperture in a flat metallic screen is investigated theoretically and numerically with a view to enhancing the electric field close to the metallic surface. A systematic study of the homogeneous solution of the electromagnetic scattering problem is made with cylindrical coordinates, expanding Maxwell equations on a Fourier–Bessel basis. A perturbation analysis devoted to simple physical analyses of different types of cylindrical nanostructure is developed for the optimization of plasmon excitation by a normally incident linearly polarized monochromatic plane wave. The conclusions drawn from this analysis agree well with the results of rigorous electromagnetic calculations obtained with the differential theory of diffraction in cylindrical coordinates.

© 2005 Optical Society of America

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  1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391, 667–669 (1998).
    [CrossRef]
  2. A. Moreau, G. Granet, F. I. Baida, D. Van Labeke, “Light transmission by subwavelength square coaxial aperture arrays in metallic films,” Opt. Express 11, 1131–1136 (2003).
    [CrossRef] [PubMed]
  3. F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett. 83, 4500–4502 (2003).
    [CrossRef]
  4. L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90, 167401 (2003).
    [CrossRef] [PubMed]
  5. R. Zakharian, M. Mansuripur, J. V. Moloney, “Transmission of light through small elliptical apertures,” Opt. Express 12, 2631–2648 (2004).
    [CrossRef] [PubMed]
  6. N. Bonod, E. Popov, M. Nevière, “Differential theory of diffraction by finite cylindrical objects,” J. Opt. Soc. Am. A 22, 481–490 (2005).
    [CrossRef]
  7. E. Popov, N. Bonod, M. Nevière, H. Rigneault, P.-F. Lenne, P. Chaumet, “Surface plasmon excitation on a single subwavelength hole in a metallic sheet,” Appl. Opt. 12, 2332–2337 (2005).
    [CrossRef]
  8. W. C. Chew, L. Gurel, “Reflection and transmission operators for strips or disks embedded in homogeneous and layered media,” IEEE Trans. Microwave Theory Technol. 36, 1488–1497 (1988).
    [CrossRef]
  9. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990).
  10. V. A. Kosobukin, “Polarization and resonance effects in optical initiation of cylindrical surface-polaritons and periodic structures,” Fizi. Tverd. Tela (Leningrad) 35, 884–898 (1993).
  11. P. J. Valle, E. M. Ortiz, J. M. Saiz, “Near field by subwavelength particles on metallic substrates with cylindrical surface plasmon excitation,” Opt. Commun. 137, 334–342 (1997).
    [CrossRef]

2005 (2)

E. Popov, N. Bonod, M. Nevière, H. Rigneault, P.-F. Lenne, P. Chaumet, “Surface plasmon excitation on a single subwavelength hole in a metallic sheet,” Appl. Opt. 12, 2332–2337 (2005).
[CrossRef]

N. Bonod, E. Popov, M. Nevière, “Differential theory of diffraction by finite cylindrical objects,” J. Opt. Soc. Am. A 22, 481–490 (2005).
[CrossRef]

2004 (1)

2003 (3)

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett. 83, 4500–4502 (2003).
[CrossRef]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90, 167401 (2003).
[CrossRef] [PubMed]

A. Moreau, G. Granet, F. I. Baida, D. Van Labeke, “Light transmission by subwavelength square coaxial aperture arrays in metallic films,” Opt. Express 11, 1131–1136 (2003).
[CrossRef] [PubMed]

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

1997 (1)

P. J. Valle, E. M. Ortiz, J. M. Saiz, “Near field by subwavelength particles on metallic substrates with cylindrical surface plasmon excitation,” Opt. Commun. 137, 334–342 (1997).
[CrossRef]

1993 (1)

V. A. Kosobukin, “Polarization and resonance effects in optical initiation of cylindrical surface-polaritons and periodic structures,” Fizi. Tverd. Tela (Leningrad) 35, 884–898 (1993).

1988 (1)

W. C. Chew, L. Gurel, “Reflection and transmission operators for strips or disks embedded in homogeneous and layered media,” IEEE Trans. Microwave Theory Technol. 36, 1488–1497 (1988).
[CrossRef]

Baida, F. I.

Bonod, N.

N. Bonod, E. Popov, M. Nevière, “Differential theory of diffraction by finite cylindrical objects,” J. Opt. Soc. Am. A 22, 481–490 (2005).
[CrossRef]

E. Popov, N. Bonod, M. Nevière, H. Rigneault, P.-F. Lenne, P. Chaumet, “Surface plasmon excitation on a single subwavelength hole in a metallic sheet,” Appl. Opt. 12, 2332–2337 (2005).
[CrossRef]

Chaumet, P.

E. Popov, N. Bonod, M. Nevière, H. Rigneault, P.-F. Lenne, P. Chaumet, “Surface plasmon excitation on a single subwavelength hole in a metallic sheet,” Appl. Opt. 12, 2332–2337 (2005).
[CrossRef]

Chew, W. C.

W. C. Chew, L. Gurel, “Reflection and transmission operators for strips or disks embedded in homogeneous and layered media,” IEEE Trans. Microwave Theory Technol. 36, 1488–1497 (1988).
[CrossRef]

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990).

Degiron, A.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90, 167401 (2003).
[CrossRef] [PubMed]

Ebbesen, T. W.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90, 167401 (2003).
[CrossRef] [PubMed]

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett. 83, 4500–4502 (2003).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Garcia-Vidal, F. J.

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett. 83, 4500–4502 (2003).
[CrossRef]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90, 167401 (2003).
[CrossRef] [PubMed]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Granet, G.

Gurel, L.

W. C. Chew, L. Gurel, “Reflection and transmission operators for strips or disks embedded in homogeneous and layered media,” IEEE Trans. Microwave Theory Technol. 36, 1488–1497 (1988).
[CrossRef]

Kosobukin, V. A.

V. A. Kosobukin, “Polarization and resonance effects in optical initiation of cylindrical surface-polaritons and periodic structures,” Fizi. Tverd. Tela (Leningrad) 35, 884–898 (1993).

Lenne, P.-F.

E. Popov, N. Bonod, M. Nevière, H. Rigneault, P.-F. Lenne, P. Chaumet, “Surface plasmon excitation on a single subwavelength hole in a metallic sheet,” Appl. Opt. 12, 2332–2337 (2005).
[CrossRef]

Lezec, H. J.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90, 167401 (2003).
[CrossRef] [PubMed]

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett. 83, 4500–4502 (2003).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Mansuripur, M.

Martin-Moreno, L.

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett. 83, 4500–4502 (2003).
[CrossRef]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90, 167401 (2003).
[CrossRef] [PubMed]

Moloney, J. V.

Moreau, A.

Nevière, M.

N. Bonod, E. Popov, M. Nevière, “Differential theory of diffraction by finite cylindrical objects,” J. Opt. Soc. Am. A 22, 481–490 (2005).
[CrossRef]

E. Popov, N. Bonod, M. Nevière, H. Rigneault, P.-F. Lenne, P. Chaumet, “Surface plasmon excitation on a single subwavelength hole in a metallic sheet,” Appl. Opt. 12, 2332–2337 (2005).
[CrossRef]

Ortiz, E. M.

P. J. Valle, E. M. Ortiz, J. M. Saiz, “Near field by subwavelength particles on metallic substrates with cylindrical surface plasmon excitation,” Opt. Commun. 137, 334–342 (1997).
[CrossRef]

Popov, E.

E. Popov, N. Bonod, M. Nevière, H. Rigneault, P.-F. Lenne, P. Chaumet, “Surface plasmon excitation on a single subwavelength hole in a metallic sheet,” Appl. Opt. 12, 2332–2337 (2005).
[CrossRef]

N. Bonod, E. Popov, M. Nevière, “Differential theory of diffraction by finite cylindrical objects,” J. Opt. Soc. Am. A 22, 481–490 (2005).
[CrossRef]

Rigneault, H.

E. Popov, N. Bonod, M. Nevière, H. Rigneault, P.-F. Lenne, P. Chaumet, “Surface plasmon excitation on a single subwavelength hole in a metallic sheet,” Appl. Opt. 12, 2332–2337 (2005).
[CrossRef]

Saiz, J. M.

P. J. Valle, E. M. Ortiz, J. M. Saiz, “Near field by subwavelength particles on metallic substrates with cylindrical surface plasmon excitation,” Opt. Commun. 137, 334–342 (1997).
[CrossRef]

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Valle, P. J.

P. J. Valle, E. M. Ortiz, J. M. Saiz, “Near field by subwavelength particles on metallic substrates with cylindrical surface plasmon excitation,” Opt. Commun. 137, 334–342 (1997).
[CrossRef]

Van Labeke, D.

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Zakharian, R.

Appl. Opt. (1)

E. Popov, N. Bonod, M. Nevière, H. Rigneault, P.-F. Lenne, P. Chaumet, “Surface plasmon excitation on a single subwavelength hole in a metallic sheet,” Appl. Opt. 12, 2332–2337 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett. 83, 4500–4502 (2003).
[CrossRef]

Fizi. Tverd. Tela (Leningrad) (1)

V. A. Kosobukin, “Polarization and resonance effects in optical initiation of cylindrical surface-polaritons and periodic structures,” Fizi. Tverd. Tela (Leningrad) 35, 884–898 (1993).

IEEE Trans. Microwave Theory Technol. (1)

W. C. Chew, L. Gurel, “Reflection and transmission operators for strips or disks embedded in homogeneous and layered media,” IEEE Trans. Microwave Theory Technol. 36, 1488–1497 (1988).
[CrossRef]

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

Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Opt. Commun. (1)

P. J. Valle, E. M. Ortiz, J. M. Saiz, “Near field by subwavelength particles on metallic substrates with cylindrical surface plasmon excitation,” Opt. Commun. 137, 334–342 (1997).
[CrossRef]

Opt. Express (2)

Phys. Rev. Lett. (1)

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, T. W. Ebbesen, “Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations,” Phys. Rev. Lett. 90, 167401 (2003).
[CrossRef] [PubMed]

Other (1)

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990).

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Figures (19)

Fig. 1
Fig. 1

Schematic representation of a structured aperture: (a) general view, (b) cross section with the notation used in the text.

Fig. 2
Fig. 2

Spectral dependence of |Ez| calculated at a point (4 μm, 0, 0) for a silver film with t = 230 nm thickness, R1 = 50 nm, four equidistant channels with h = 30 nm, and λ = 500 nm.

Fig. 3
Fig. 3

Comparison of the computed kr dependence of | b ^ 1 H , +| and its approximation given by the phenomenological formula [relation (34)] in the vicinity of the plasmon propagation constant for a silver screen with 50 nm radius of aperture and λ = 500 nm.

Fig. 4
Fig. 4

Field map of a surface plasmon wave excited by a circular aperture of 50 nm radius in a silver film by an x-polarized normally incident plane wave with wavelength equal to 500 nm. Rigorous results, solid curve; corresponding Hankel function, squares.

Fig. 5
Fig. 5

Comparison of values of |Ez| at point (4 μm, 0, 0) computed from the rigorous theory and from the perturbative approach. Aperture radius R is varied; other parameters are the same as in Fig. 4.

Fig. 6
Fig. 6

Same as in Fig. 5 but with the wavelength varied and R = 50 nm.

Fig. 7
Fig. 7

Comparison of two expressions for the coupling integral given in Eqs. (39) and (40) as a function of wavelength for a Bessel-function-type corrugation with radii given in Table 1 and situated about a 50 nm radius aperture.

Fig. 8
Fig. 8

Spectral dependence of the coupling integral given by Eq. (39) and its evolution when the number of channels is increased for the same structure as in Fig. 7.

Fig. 9
Fig. 9

Field enhancement as a function of channel depth for a four-channel structure surrounding a single 50 nm radius aperture at λ = 500 nm. Field calculated on the upper and lower film surfaces compared with the enhancement of | b ^ 1 H , +|.

Fig. 10
Fig. 10

Near-, transmitted-, and plasmon-field enhancement as a function of the number of channels for h = 50 nm and λ = 500 nm.

Fig. 11
Fig. 11

Spectral dependence of |Ez| at point (4 μm, 0, 0) computed from the rigorous theory (solid curves) and the perturbative approach (squares): (a) single channel and Eq. (40), (b) four channels and Eq. (39). Corrugation radii follow the values in Table 1.

Fig. 12
Fig. 12

Same as in Fig. 11(a) but for a single channel with R2 = 235 nm and R3 = 470 nm.

Fig. 13
Fig. 13

Staircase approximation of smoothly varying surface modulation described by the zeroth-order Bessel function.

Fig. 14
Fig. 14

Spectral dependence of the coupling integral obtained by use of two values of radius Rmax of the modulated region.

Fig. 15
Fig. 15

Spectral dependence of the rigorous (solid curve) and perturbative (squares) values of |Ez| at point (4 μm, 0, 0) for the structure presented in Fig. 13.

Fig. 16
Fig. 16

Dependence of | b ^ 1 H , +| at z = 0 (upper metallic interface) and of | b 1 H , -| at z = −200 nm (lower surface) on the values of kr/k0 for a 50 nm radius aperture in a silver film covered with a 50 nm thick dielectric layer with permittivity modulated according to Eq. (42), λ = 500 nm, ɛd/ɛ0 = 3.5, Δɛ/ɛ0 = 0.1, and mod k r mod = 0.02649 nm - 1.

Fig. 17
Fig. 17

Spectral dependence of | b ^ 1 H , +| for the structure described in Fig. 16.

Fig. 18
Fig. 18

| b ^ 1 H , +| as a function of kr/k0 in the vicinity of plasmon excitation on the interface metal-modulated dielectric layer for the structure described in Fig. 16 and presented for four wavelengths.

Fig. 19
Fig. 19

Spatial variation of the plasmon field for three wavelengths for the structure given in Fig. 16.

Tables (1)

Tables Icon

Table 1 Optimum Values of Rj

Equations (46)

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E j ( r , θ , z ) = n = - + E j , n ( r , z ) exp ( i n θ ) ,             j = r , θ , z ,
Δ E θ , n - E θ , n r 2 + 2 i n r 2 E r , n + k 2 E θ , n = 0 , Δ E r , n - E r , n r 2 - 2 i n r 2 E θ , n + k 2 E r , n = 0 ,
Δ = 2 r 2 + 1 r r + 1 r 2 2 θ 2 + 2 z 2 .
E , n = E θ , n ± i E r , n
[ 2 r 2 + 1 r r - ( n ± 1 ) 2 r 2 + k 2 + 2 z 2 ] E ± , n ( r , z ) = 0.
E ± , n ( r , z ) = 0 e ± , n ( k r , z ) Z n ± 1 ( k r r ) k r d k r .
0 Z n ± 1 ( k r r ) k r d k r ( k r 2 - k 2 - 2 z 2 ) e ± , n ( k r , z ) = 0.
0 Z n ± 1 ( k r r ) Z n ± 1 ( k r r ) r d r = δ ( k r - k r ) k r ,
- 2 z 2 e ± , n ( k r , z ) = ( k 2 - k r 2 ) e ± , n ( k r , z ) .
e + , n ± ( k r , z ) = b n E , ± ( k r ) exp ( ± i k z z ) , e - , n ± ( k r , z ) = c n E , ± ( k r ) exp ( ± i k z z ) ,
k z = k 2 - k r 2 .
i ω μ 0 H z = ( rot E ) z , i ω ɛ E z = - ( rot H ) z .
H z , n ( r , z ) = 1 i ω μ 0 0 Z n ( k r r ) k r d k r × [ b n E , ± ( k r ) - c n E , ± ( k r ) ] exp ( ± i k z z ) , E z , n ( r , z ) = - 1 i ω ɛ 0 Z n ( k r r ) k r d k r × [ b n H , ± ( k r ) - c n H , ± ( k r ) ] exp ( ± i k z z ) .
b n H , ± = c n H , ± = ± i k z ω μ 0 b n E , ± , b n E , ± = - c n E , ± ;
b n E , ± = c n E , ± = i k z ω ɛ b n H , ± , b n H , ± = - c n H , ± .
b ^ n E , + + b ^ n E , - = b n E , - b ^ n H , + + b ^ n H , - = b n H - { TE polarization , Eqs . ( 14 ) | b ^ n E , + + b ^ n E , - = b n E , - k ^ z b ^ n E , + - k ^ z b ^ n E , - = - k z b n E , - TM polarization , Eqs . ( 15 ) | - k ^ z ɛ ^ b ^ n H , + + k ^ z ɛ ^ b ^ n H , - = k z ɛ b n H , - b ^ n H , + + b ^ n H , - = b n H , -
r TE b ^ n E , + b ^ n E , - = k ^ z - k z k ^ z + k z ,             r TM b ^ n H , + b ^ n H , - = k ^ z / ɛ ^ - k z / ɛ k ^ z / ɛ ^ + k z / ɛ , t TE b n E , - b ^ n E , - = 2 k ^ z k ^ z + k z ,             t TM b n H , - b ^ n H , - = 2 k ^ z / ɛ ^ k ^ z / ɛ ^ + k z / ɛ .
[ k ^ 2 - ( k r p ) 2 ] 1 / 2 ɛ ^ + [ k 2 - ( k r p ) 2 ] 1 / 2 ɛ = 0 ,
k ^ z p = [ k ^ 2 - ( k r p ) 2 ] 1 / 2 , k z p = [ k 2 - ( k r p ) 2 ] 1 / 2 ,
E ^ r , n , k r p p ( r , θ , z ) = - i 2 ( E ^ - . n , k r p p - E ^ + , n , k r p p ) exp ( i n θ ) = k ^ z p 2 ω ɛ ^ b ^ n H , + ( k ^ r p ) exp ( i n θ + i k ^ z p z ) × [ Z n - 1 ( k ^ r p r ) - Z n + 1 ( k r p r ) ] , E ^ θ , n , k r p p ( r , θ , z ) = 1 2 ( E ^ - , n , k r p p + E ^ + , n , k r p p ) exp ( i n θ ) = i k ^ z p 2 ω ɛ ^ b ^ n H , + ( k r p ) exp ( i n θ + i k ^ z p z ) × [ Z n - 1 ( k r p r ) + Z n + 1 ( k r p r ) ] , E ^ z , n , k r p p ( r , θ , z ) = 2 i k r p ω ɛ ^ b ^ n H , + ( k r p ) exp ( i n θ + i k ^ z p z ) Z n ( k r p r ) .
E ^ r , p + , k r p p ( r , θ , z ) = i k ^ z p ω ɛ ^ b ^ 1 H , + ( k r p ) exp ( i k ^ z p z ) × [ Z 0 ( k r p r ) - Z 2 ( k r p r ) ] sin ( θ ) , E ^ θ , p + , k r p p ( r , θ , z ) = i k ^ z p ω ɛ ^ b ^ 1 H , + ( k r p ) exp ( i k ^ z p z ) × [ Z 0 ( k r p r ) + Z 2 ( k r p r ) 0 ] cos ( θ ) , E ^ z , p + , k r p p ( r , θ , z ) = - 4 k ^ r p ω ɛ ^ b ^ 1 H , + ( k r p ) exp ( i k ^ z p z ) Z 1 ( k r p r ) sin ( θ ) .
E ^ r , p + , k r p p ( r , θ , z ) = 2 i k ^ z p ω ɛ ^ b ^ 1 H , + ( k r p ) exp ( i k ^ z p z ) Z 1 ( k r p r ) sin ( θ ) , E ^ θ , p + , k r p p ( r , θ , z ) = i k ^ z p ω ɛ ^ b ^ 1 H , + ( k r p ) × exp ( i k ^ z p z ) 2 k r p r Z 1 ( k r p r ) cos ( θ ) , E ^ z , p + , k r p p ( r , θ , z ) = - 4 k ^ r p ω ɛ ^ b ^ 1 H , + ( k r p ) × exp ( i k ^ z p z ) Z 1 ( k r p r ) sin ( θ ) ,
J 1 ( k r p r ) r 2 / π k r p r cos ( k r p r - 3 4 π ) , H n + ( k r p r ) r 2 / π k r p r exp ( i k r p r - i 3 4 π ) .
E ^ r , p - , k r p p ( r , θ , z ) = 2 k ^ z p ω ɛ ^ k r p b ^ 1 H , + ( k r p ) exp ( i k ^ z p z ) Z 1 ( k r p r ) cos ( θ ) , E ^ θ , p - , k r p p ( r , θ , z ) = - k ^ z p ω ɛ ^ k r p b ^ 1 H , + ( k r p ) exp ( i k ^ z p z ) 2 k r p r Z 1 ( k r p r ) × sin ( θ ) , E ^ z , p - , k r p p ( r , θ , z ) = 4 i ( k r p ) 2 ω ɛ ^ b ^ 1 H , + ( k r p ) exp ( i k ^ z p z ) Z 1 ( k r p r ) cos ( θ ) ,
z S = h f ( r ) ,
E ± , n i ( r , z ) = b ± , n i ( k r i ) exp ( - i k ^ z i z ) J n ± 1 ( k r i r )
b ^ n H , - ( k r ) = k 0 2 b ^ n H , i δ ( k r - k r i ) k r i .
| 1 ɛ ^ 0 k ^ z ( k r ) exp ( i k ^ z z s ) b ^ n H , + ( k r ) Z n + 1 ( k r r ) k r d k r - k 0 2 k ^ z i ɛ ^ exp ( - i k ^ z i z s ) b ^ n H , i J n + 1 ( k r i r ) = - 1 ɛ 0 k z ( k r ) exp ( - i k z z s ) b n H , - ( k r ) Z n + 1 ( k r r ) k r d k r , 0 exp ( i k ^ z z s ) b ^ n H , + ( k r ) Z n + 1 ( k r r ) k r d k r + k 0 2 exp ( - i k ^ z i z s ) b ^ n H , i J n + 1 ( k r i r ) = 0 exp ( - i k z z s ) b n H , - ( k r ) Z n + 1 ( k r r ) k r d k r .
exp ( ± i k z z S ) 1 ± i h k z f ( r ) .
k ^ z ɛ ^ b ^ n H , + ( k r ) + i k ^ z i ɛ ^ b ^ n H , i F n + 1 ( k r , k r i ) = - k z ɛ b n H , - ( k r ) , b ^ n H , + ( k r ) - i b ^ n H , i F n + 1 ( k r , k r i ) = b n H , - ( k r ) ,
F n + 1 ( k r , k r i ) = k 0 2 k ^ z i h 0 f ( r ) Z n + 1 ( k r r ) J n + 1 ( k r i r ) r d r .
b ^ n H , + ( k r ) = - i F n + 1 ( k r , k r i ) k ^ z i / ɛ ^ - k z / ɛ k ^ z / ɛ ^ + k z / ɛ b ^ n H , i , b n H , - ( k r ) = i F n + 1 ( k r , k r i ) k ^ z i / ɛ ^ + k z / ɛ k ^ z / ɛ ^ + k z / ɛ b ^ n H , i .
b ^ n H , + ( k r ) , b n H , - ( k r ) ~ 1 k ^ z / ɛ ^ + k z / ɛ F n + 1 ( k r , k r i ) b ^ n H , i C k r 2 - ( k r p ) 2 F n + 1 ( k r , k r i ) b ^ n H , i .
F 0 ( k r , k r i ) = k 0 2 k ^ z i h 0 f ( r ) Z 0 ( k r r ) r d r .
f ( r ) = J 0 ( k r r ) F 0 ( k r , k r i ) = k 0 2 k z i h δ ( k r - k r ) k r .
0 J 1 ( k r r ) k r d k r k r 2 - ( k r p ) 2 = i π 2 H 1 + ( k r p r ) .
F 0 ( k r p , 0 ) = k 0 2 k ^ z i h 0 R J 0 ( k r p r ) r d r = k 0 2 k ^ z i h R k r p J 1 ( k r R ) .
F 0 ( k r p , 0 ) = k 0 2 [ k ^ z i h 0 R 1 J 0 ( k r p r ) r d r + k z i h R 1 R 2 J 0 ( k r p r ) r d r + k ^ z i h R 2 R 3 J 0 ( k r p r ) r d r + ] = k 0 2 ( k ^ z i - k z i ) h k r p j = 1 j max ( - 1 ) j R j J 1 ( k r p R j ) .
F 0 ( k r p , 0 ) = k 0 2 ( k ^ z i - k z i ) h k r p j = 1 j max ( - 1 ) j R j × [ J 1 ( k r p R j ) - J 3 ( k r p R j ) ] .
k r 2 e ± , n ( k r , z ) - 2 z 2 e ± , n ( k r , z ) = ω 2 μ 0 0 ɛ ( r ) Z n ± 1 ( k r r ) J n ± 1 ( k r i r ) r d r e ± , n ( k r , z ) .
ɛ ( r ) = ɛ d + Δ ɛ J 0 ( k r mod r ) ,
1 r E z θ - E θ z = i ω μ 0 H r , E r z - E z r = i ω μ 0 H θ , E θ r + E θ r - 1 r E r θ = i ω μ 0 H z , 1 r H z θ - H θ z = - i ω ɛ E r , H r z - H z r = - i ω ɛ E θ , H θ r + H θ r - 1 r H r θ = - i ω ɛ E z , E θ , n z = i n r E z , n - i ω μ 0 H r , n , E r , n z = E z , n r + i ω μ 0 H θ , n , i ω μ 0 H z , n = E θ , n r + E θ , n r - i n r E r , n , H θ , n z = i n r H z , n + i ω ɛ E r , n , H r , n z = H z , n r - i ω ɛ E θ , n , - i ω ɛ E z , n = H θ , n r + H θ , n r - i n r H r , n ,
i ω μ 0 H z , n = 0 ( e + , n - e - , n ) Z n ( k r r ) k r 2 d k r , - i ω ɛ E z , n = 0 ( h + , n - h - , n ) Z n ( k r r ) k r 2 d k r .
z E + , n = ω μ 0 H + , n ± i k z b n E , ± = ω μ 0 b n H , ± , z E - , n = - ω μ 0 H - , n ± i k z c n E , ± = - ω μ 0 c n H , ± ,
z H + , n = - ω ɛ E + , n ± i k z b n H , ± = - ω ɛ b n E , ± , z H - , n = ω ɛ E - , n ± i k z c n H , ± = ω ɛ c n E , ± ,
E r i = E x i cos θ , E θ i = - E x i sin θ , E z i = 0 ,

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