Abstract

Needle beam is a guided beam with nanoscale beam size and significant power propagating in core area of a three-layer dielectric waveguide. Systematical numerical analyses of properties of the needle beam are presented. Properties of the fundamental mode of the needle beam, including field distribution, power distribution, and power concentration, are calculated for different waveguide parameters. It is shown that there is an optimum value of normalized frequency for maximum power concentration. Concentrated power is higher if the refractive index difference between the core and the middle layer is higher.

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References

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  1. V. Bondarenko and Y. Zhao, “Needle beam”: Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett. 89(14), 141103 (2006).
    [CrossRef]
  2. V. Bondarenko and Y. Zhao, “Addendum: The 'needle beam': Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett. 91(8), 089903 (2007).
    [CrossRef]
  3. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
    [CrossRef] [PubMed]
  4. R. Gordon, “Angle-dependent optical transmission through a narrow slit in a thick metal film,” Phys. Rev. B 75(19), 193401 (2007).
    [CrossRef]
  5. F. J. García-Vidal, L. Martin-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon, “Transmission of light through a single rectangular hole in a real metal,” Phys. Rev. B 74(15), 153411 (2006).
    [CrossRef]
  6. H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (2006).
    [CrossRef] [PubMed]
  7. K. Y. Kim, Y. K. Cho, H. S. Tae, and J. H. Lee, “Light transmission along dispersive plasmonic gap and its subwavelength guidance characteristics,” Opt. Express 14(1), 320–330 (2006).
    [CrossRef] [PubMed]
  8. S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(20), 205402 (2003).
    [CrossRef]
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    [CrossRef]
  10. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. 22(7), 475–477 (1997).
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    [CrossRef] [PubMed]
  12. V. R. Almeida, Q. F. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004).
    [CrossRef] [PubMed]
  13. L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
    [CrossRef] [PubMed]
  14. A. Yariv, Optical Electronics in Modern Communications (Oxford University Press, New York; Oxford, 1997).
  15. H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, “Optical-Potential for Atom Guidance in a Cylindrical-Core Hollow-Fiber,” Opt. Commun. 115(1-2), 57–64 (1995).
    [CrossRef]

2007 (2)

R. Gordon, “Angle-dependent optical transmission through a narrow slit in a thick metal film,” Phys. Rev. B 75(19), 193401 (2007).
[CrossRef]

V. Bondarenko and Y. Zhao, “Addendum: The 'needle beam': Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett. 91(8), 089903 (2007).
[CrossRef]

2006 (4)

V. Bondarenko and Y. Zhao, “Needle beam”: Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett. 89(14), 141103 (2006).
[CrossRef]

F. J. García-Vidal, L. Martin-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon, “Transmission of light through a single rectangular hole in a real metal,” Phys. Rev. B 74(15), 153411 (2006).
[CrossRef]

H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (2006).
[CrossRef] [PubMed]

K. Y. Kim, Y. K. Cho, H. S. Tae, and J. H. Lee, “Light transmission along dispersive plasmonic gap and its subwavelength guidance characteristics,” Opt. Express 14(1), 320–330 (2006).
[CrossRef] [PubMed]

2004 (2)

2003 (2)

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(20), 205402 (2003).
[CrossRef]

2002 (1)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[CrossRef] [PubMed]

1997 (1)

1995 (1)

H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, “Optical-Potential for Atom Guidance in a Cylindrical-Core Hollow-Fiber,” Opt. Commun. 115(1-2), 57–64 (1995).
[CrossRef]

1944 (1)

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66(7-8), 163–182 (1944).
[CrossRef]

Almeida, V. R.

Ashcom, J. B.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

Atwater, H. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(20), 205402 (2003).
[CrossRef]

Barrios, C. A.

Bethe, H. A.

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66(7-8), 163–182 (1944).
[CrossRef]

Bondarenko, V.

V. Bondarenko and Y. Zhao, “Addendum: The 'needle beam': Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett. 91(8), 089903 (2007).
[CrossRef]

V. Bondarenko and Y. Zhao, “Needle beam”: Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett. 89(14), 141103 (2006).
[CrossRef]

Cho, Y. K.

Degiron, A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[CrossRef] [PubMed]

Devaux, E.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[CrossRef] [PubMed]

Ebbesen, T. W.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[CrossRef] [PubMed]

Garcia-Vidal, F. J.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[CrossRef] [PubMed]

García-Vidal, F. J.

F. J. García-Vidal, L. Martin-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon, “Transmission of light through a single rectangular hole in a real metal,” Phys. Rev. B 74(15), 153411 (2006).
[CrossRef]

Gattass, R. R.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

Gordon, R.

R. Gordon, “Angle-dependent optical transmission through a narrow slit in a thick metal film,” Phys. Rev. B 75(19), 193401 (2007).
[CrossRef]

F. J. García-Vidal, L. Martin-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon, “Transmission of light through a single rectangular hole in a real metal,” Phys. Rev. B 74(15), 153411 (2006).
[CrossRef]

He, S. L.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

Ito, H.

H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, “Optical-Potential for Atom Guidance in a Cylindrical-Core Hollow-Fiber,” Opt. Commun. 115(1-2), 57–64 (1995).
[CrossRef]

Jhe, W.

H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, “Optical-Potential for Atom Guidance in a Cylindrical-Core Hollow-Fiber,” Opt. Commun. 115(1-2), 57–64 (1995).
[CrossRef]

Kik, P. G.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(20), 205402 (2003).
[CrossRef]

Kim, K. Y.

Kobayashi, T.

Kumar, L. K. S.

F. J. García-Vidal, L. Martin-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon, “Transmission of light through a single rectangular hole in a real metal,” Phys. Rev. B 74(15), 153411 (2006).
[CrossRef]

Kurokawa, Y.

H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (2006).
[CrossRef] [PubMed]

Lee, J. H.

Lezec, H. J.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[CrossRef] [PubMed]

Linke, R. A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[CrossRef] [PubMed]

Lipson, M.

Lou, J. Y.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

Maier, S. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(20), 205402 (2003).
[CrossRef]

Martin-Moreno, L.

F. J. García-Vidal, L. Martin-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon, “Transmission of light through a single rectangular hole in a real metal,” Phys. Rev. B 74(15), 153411 (2006).
[CrossRef]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[CrossRef] [PubMed]

Maxwell, I.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

Mazur, E.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

Miyazaki, H. T.

H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (2006).
[CrossRef] [PubMed]

Moreno, E.

F. J. García-Vidal, L. Martin-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon, “Transmission of light through a single rectangular hole in a real metal,” Phys. Rev. B 74(15), 153411 (2006).
[CrossRef]

Morimoto, A.

Nakata, T.

H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, “Optical-Potential for Atom Guidance in a Cylindrical-Core Hollow-Fiber,” Opt. Commun. 115(1-2), 57–64 (1995).
[CrossRef]

Ohtsu, M.

H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, “Optical-Potential for Atom Guidance in a Cylindrical-Core Hollow-Fiber,” Opt. Commun. 115(1-2), 57–64 (1995).
[CrossRef]

Panepucci, R. R.

Sakaki, K.

H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, “Optical-Potential for Atom Guidance in a Cylindrical-Core Hollow-Fiber,” Opt. Commun. 115(1-2), 57–64 (1995).
[CrossRef]

Shen, M. Y.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

Tae, H. S.

Takahara, J.

Taki, H.

Tong, L. M.

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

Xu, Q. F.

Yamagishi, S.

Zhao, Y.

V. Bondarenko and Y. Zhao, “Addendum: The 'needle beam': Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett. 91(8), 089903 (2007).
[CrossRef]

V. Bondarenko and Y. Zhao, “Needle beam”: Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett. 89(14), 141103 (2006).
[CrossRef]

Appl. Phys. Lett. (2)

V. Bondarenko and Y. Zhao, “Needle beam”: Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett. 89(14), 141103 (2006).
[CrossRef]

V. Bondarenko and Y. Zhao, “Addendum: The 'needle beam': Beyond-diffraction-limit concentration of field and transmitted power in dielectric waveguide,” Appl. Phys. Lett. 91(8), 089903 (2007).
[CrossRef]

Nature (1)

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, “Optical-Potential for Atom Guidance in a Cylindrical-Core Hollow-Fiber,” Opt. Commun. 115(1-2), 57–64 (1995).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. (1)

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66(7-8), 163–182 (1944).
[CrossRef]

Phys. Rev. B (3)

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(20), 205402 (2003).
[CrossRef]

R. Gordon, “Angle-dependent optical transmission through a narrow slit in a thick metal film,” Phys. Rev. B 75(19), 193401 (2007).
[CrossRef]

F. J. García-Vidal, L. Martin-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon, “Transmission of light through a single rectangular hole in a real metal,” Phys. Rev. B 74(15), 153411 (2006).
[CrossRef]

Phys. Rev. Lett. (1)

H. T. Miyazaki and Y. Kurokawa, “Squeezing visible light waves into a 3-nm-thick and 55-nm-long plasmon cavity,” Phys. Rev. Lett. 96(9), 097401 (2006).
[CrossRef] [PubMed]

Science (1)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[CrossRef] [PubMed]

Other (1)

A. Yariv, Optical Electronics in Modern Communications (Oxford University Press, New York; Oxford, 1997).

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Figures (7)

Fig. 1
Fig. 1

Structure and refractive index profile of a purely dielectric three-medium coaxial cylindrical waveguide

Fig. 2
Fig. 2

Normalized propagation constant β/k0 as a function of V parameter for some lowest-order modes of purely dielectric three-medium coaxial cylindrical waveguide with θ = 0.2, n = 1, n = 3.48, n3 = 1.48 (n1, n2, n3 respectively represents refractive index in core, middle, outside medium).

Fig. 3
Fig. 3

Field distribution characteristic in core medium under different n2 with fixed V = 3.8. θ = 0.512, n1 = 1, n3 = 1.48. (a) Ez, (b) Hz, (c) Er, (d) Eφ, (e) Hr, and (f) Hφ.

Fig. 4
Fig. 4

Power density in core medium under different n2 with fixed parameters V = 3.8, θ = 0.512, n1 = 1.00, and n3 = 1.48.

Fig. 5
Fig. 5

Power percentage of HE11 mode under different V with parameters: θ = 0.729, n1 = 1.00, n2 = 3.48, n3 = 1.48.

Fig. 6
Fig. 6

Power percentage in the core medium of HE11 Mode with different n2. (a) Power percentage versus V with fixed θ = 0.729. (b) Power percentage versus θ with fixed V = 3.8.

Fig. 7
Fig. 7

Power distribution of HE11 mode in three mediums with V = 3.8, n1 = 1.00 and n2 = 3.48, n3 = 1.48 when θ changes. (a). θ = 0.052. (b). θ = 0.152. (c). θ = 0.212. (d). θ = 0.532. (e). three dimensional map of (a).

Tables (1)

Tables Icon

Table 1 Power percentage in core (nanoscale hole) and middle layers of different modes of three dielectric medium cylindrical waveguide with V = 20, θ = 0.729, n1 = 1, n2 = 3.48, and n3 = 1.48

Equations (48)

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( 2 + k 2 ) { E z H z } = 0
{ E z H z } = ψ (r) exp [ i ( w t + l φ β z ) ]
ψ ( r ) = a 1 I l ( p r ) + a 2 K l ( p r ) r < r 1
ψ ( r ) = b J l ( h r ) + c Y l ( h r ) r 1 < r < r 2
ψ ( r ) = d 1 I l ( q r ) + d 2 K l ( q r ) r > r 2
E z = { A 1 I l ( p r ) exp [ i ( w t + l φ β z ) ] r < r 1 [ B 1 J l ( h r ) + C 1 Y l ( h r ) ] exp [ i ( w t + l φ β z ) ] r 1 < r < r 2 D 1 K l ( q r ) exp [ i ( w t + l φ β z ) ] r > r 2
H z = { A 2 I l ( p r ) exp [ i ( w t + l φ β z ) ] r < r 1 [ B 2 J l ( h r ) + C 2 Y l ( h r ) ] exp [ i ( w t + l φ β z ) ] r 1 < r < r 2 D 2 K l ( q r ) exp [ i ( w t + l φ β z ) ] r > r 2
E r = i β p 2 [ A 1 p I l ' ( p r ) + i w μ l β r A 2 I l ( p r ) ] exp [ i ( w t + l φ β z ) ]
E φ = i β p 2 [ i l r A 1 I l ( p r ) w μ β A 2 p I l ' ( p r ) ] exp [ i ( w t + l φ β z ) ]
H r = i β p 2 [ A 2 p I l ' ( p r ) i w ε 1 l β r A 1 I l ( p r ) ] exp [ i ( w t + l φ β z ) ]
H φ = i β p 2 [ i l r A 2 I l ( p r ) + w ε 1 β A 1 p I l ' ( p r ) ] exp [ i ( w t + l φ β z ) ]
E r = i β h 2 { h [ B 1 J l ' ( h r ) + C 1 Y l ' ( h r ) ] + i w μ l β r [ B 2 J l ( h r ) + C 2 Y l ( h r ) ] } × exp [ i ( w t + l φ β z ) ]
E φ = i β h 2 { i l r [ B 1 J l ( h r ) + C 1 Y l ( h r ) ] h w μ β [ B 2 J l ' ( h r ) + C 2 Y l ' ( h r ) ] } × exp [ i ( w t + l φ β z ) ]
H r = i β h 2 { h [ B 2 J l ' ( h r ) + C 2 Y l ' ( h r ) ] i w ε 2 l β r [ B 1 J l ( h r ) + C 1 Y l ( h r ) ] } × exp [ i ( w t + l φ β z ) ]
H φ = i β h 2 { i l r [ B 2 J l ( h r ) + C 2 Y l ( h r ) ] + h w ε 2 β [ B 1 J l ' ( h r ) + C 1 Y l ' ( h r ) ] } × exp [ i ( w t + l φ β z ) ]
E r = i β q 2 [ q D 1 K l ' ( q r ) + i w μ l β r D 2 K l ( q r ) ] exp [ i ( w t + l φ β z ) ]
E φ = i β q 2 [ i l r D 1 K l ( q r ) w μ β q D 2 K l ' ( q r ) ] exp [ i ( w t + l φ β z ) ]
H r = i β q 2 [ q D 2 K l ' ( q r ) i w ε 3 l β r D 1 K l ( q r ) ] exp [ i ( w t + l φ β z ) ]
H φ = i β q 2 [ i l r D 2 K l ( q r ) + w ε 3 β q D 1 K l ' ( q r ) ] exp [ i ( w t + l φ β z ) ]
[ M 11 M 18 M 21 M 81 M 88 ] [ A 1 A 2 B 1 C 1 B 2 C 2 D 1 D 2 ] = [ 0 0 0 0 0 0 0 0 ]
M 11 = M 32 = I l ( p r 1 )
M 13 = M 35 = J l ( h r 1 )
M 14 = M 36 = Y l ( h r 1 )
M 21 = M 42 = i l p 2 r 1 I l ( p r 1 )
M 22 = w u β p I l ' ( p r 1 )
M 23 = M 45 = i l h 2 r 1 J l ( h r 1 )
M 24 = M 46 = i l h 2 r 1 Y l ( h r 1 )
M 25 = w u β h J l ' ( h r 1 )
M 26 = w u β h Y l ' ( h r 1 )
M 41 = w ε 1 β p I l ' ( p r 1 )
M 43 = w ε 2 β h J l ' ( h r 1 )
M 44 = w ε 2 β h Y l ' ( h r 1 )
M 53 = M 75 = J l ( h r 2 )
M 54 = M 76 = Y l ( h r 2 )
M 57 = M 78 = K l ( q r 2 )
M 63 = M 85 = i l h 2 r 2 J l ( h r 2 )
M 64 = M 86 = i l h 2 r 2 Y l ( h r 2 )
M 65 = w u β h J l ' ( h r 2 )
M 66 = w u β h Y l ' ( h r 2 )
M 67 = M 88 = i l q 2 r 2 K l ( q r 2 )
M 68 = w u β q K l ' ( q r 2 )
M 83 = w ε 2 β h J l ' ( h r 2 )
M 84 = w ε 2 β h Y l ' ( h r 2 )
M 87 = w ε 3 β q I l ' ( q r 2 ) .
S z = 1 2 R e a l [ E r H φ * - E φ H r * ]
P c o r e = 0 2 π 0 r 1 S z r d r d φ
P m i d = 0 2 π r 1 r 2 S z r d r d φ
P o u t = 0 2 π r 2 S z r d r d φ

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