Abstract

The scattering of a two-dimensional Gaussian beam from a homogeneous dielectric cylinder is analyzed using a plane-wave spectrum. Special attention is given to the computation of the evanescent field of the beam and its effect in the scattering. A comparison is made between the evanescent field in Cartesian coordinates and in cylindrical coordinates as a sum of cylindrical waves. The field given by the cylindrical wave equation is found to converge spatially as we include more Bessel modes. The evanescent field incident on the dielectric cylinder is found to cause radiating waves to form and propagate outward.

© 2009 Optical Society of America

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Errata

Peter Pawliuk and Matthew Yedlin, "Gaussian beam scattering from a dielectric cylinder, including the evanescent region: erratum," J. Opt. Soc. Am. A 27, 166-166 (2010)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-27-2-166

References

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  1. G. Mie, “Considerations on the optics of turbid media, especially colloidal metal sols,” Ann. Phys. 25, 377-442 (1908).
    [CrossRef]
  2. L. Lorenz, “Upon the light reflected and refracted by a transparent sphere,” Vidensk. Selsk. Shrifter 6, 1-62 (1890).
  3. G. Gouesbet, “Generalized Lorenz-Mie theories, the third decade: A perspective,” J. Quant. Spectrosc. Radiat. Transf. 110, 1223-1238 (2009).
    [CrossRef]
  4. G. Gouesbet, “Interaction between an infinite cylinder and an arbitrary-shaped beam,” Appl. Opt. 36, 4292-4304 (1997).
    [CrossRef] [PubMed]
  5. K. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz-Mie theory: formulation and numerical results,” J. Opt. Soc. Am. A 14, 3014-3025 (1997).
    [CrossRef]
  6. L. Mees, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation: numerical results,” Appl. Opt. 38, 1867-1876 (1999).
    [CrossRef]
  7. H. J. Eom, Electromagnetic Wave Theory for Boundary-Value Problems: an Advanced Course on Analytical Methods (Springer-Verlag, 2004).
    [PubMed]
  8. G. Agrawal and D. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575-578 (1979).
    [CrossRef]
  9. T. Kojima and Y. Yanagiuchi, “Scattering of an offset two-dimensional Gaussian beam wave by a cylinder,” J. Appl. Phys. 50, 41-46 (1979).
    [CrossRef]
  10. S. Kozaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195-7200 (1982).
    [CrossRef]
  11. T. Rao, “Scattering by a radially inhomogeneous cylindrical dielectric shell due to an incident Gaussian beam,” Can. J. Phys. 67, 471-475 (1989).
    [CrossRef]
  12. Z. Wu and L. Guo, “Electromagnetic scattering from a multilayered cylinder arbitrarily located in a Gaussian beam; a new recursive algorithm,” J. Electromagn. Waves Appl. 12, 725-726 (1998).
    [CrossRef]
  13. E. Zimmermann, R. Dändliker, N. Souli, and B. Krattiger, “Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach,” J. Opt. Soc. Am. A 12, 398-403 (1995).
    [CrossRef]
  14. A. Elsherbeni, M. Hamid, and G. Tian, “Iterative scattering of a Gaussian beam by an array of circular conducting and dielectric cylinders,” J. Electromagn. Waves Appl. 7, 1323-1342 (1993).
    [CrossRef]
  15. J. Yang, L. W. Li, K. Yasumoto, and C. H. Liang, “Two-dimensional scattering of a Gaussian beam by a periodic array of circular cylinders,” IEEE Trans. Geosci. Remote Sens. 43, 280-285 (2005).
    [CrossRef]
  16. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).
  17. P. Varga and P. Török, “The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108-118 (1998).
    [CrossRef]
  18. E. V. Jull, Aperture Antennas and Diffraction Theory (Peter Peregrinus, 1981).
    [CrossRef]
  19. J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974-980 (1979).
    [CrossRef]
  20. S. Kozaki and H. Sakurai, “Characteristics of a Gaussian beam at a dielectric interface,” J. Opt. Soc. Am. 68, 508-514 (1978).
    [CrossRef]
  21. N. McLachlan, Bessel Functions for Engineers (Oxford Univ. Press, 1955).
  22. J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin/Cummings, 1970).

2009 (1)

G. Gouesbet, “Generalized Lorenz-Mie theories, the third decade: A perspective,” J. Quant. Spectrosc. Radiat. Transf. 110, 1223-1238 (2009).
[CrossRef]

2005 (1)

J. Yang, L. W. Li, K. Yasumoto, and C. H. Liang, “Two-dimensional scattering of a Gaussian beam by a periodic array of circular cylinders,” IEEE Trans. Geosci. Remote Sens. 43, 280-285 (2005).
[CrossRef]

1999 (1)

1998 (2)

P. Varga and P. Török, “The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108-118 (1998).
[CrossRef]

Z. Wu and L. Guo, “Electromagnetic scattering from a multilayered cylinder arbitrarily located in a Gaussian beam; a new recursive algorithm,” J. Electromagn. Waves Appl. 12, 725-726 (1998).
[CrossRef]

1997 (2)

1995 (1)

1993 (1)

A. Elsherbeni, M. Hamid, and G. Tian, “Iterative scattering of a Gaussian beam by an array of circular conducting and dielectric cylinders,” J. Electromagn. Waves Appl. 7, 1323-1342 (1993).
[CrossRef]

1989 (1)

T. Rao, “Scattering by a radially inhomogeneous cylindrical dielectric shell due to an incident Gaussian beam,” Can. J. Phys. 67, 471-475 (1989).
[CrossRef]

1982 (1)

S. Kozaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195-7200 (1982).
[CrossRef]

1979 (3)

G. Agrawal and D. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575-578 (1979).
[CrossRef]

T. Kojima and Y. Yanagiuchi, “Scattering of an offset two-dimensional Gaussian beam wave by a cylinder,” J. Appl. Phys. 50, 41-46 (1979).
[CrossRef]

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974-980 (1979).
[CrossRef]

1978 (1)

1908 (1)

G. Mie, “Considerations on the optics of turbid media, especially colloidal metal sols,” Ann. Phys. 25, 377-442 (1908).
[CrossRef]

1890 (1)

L. Lorenz, “Upon the light reflected and refracted by a transparent sphere,” Vidensk. Selsk. Shrifter 6, 1-62 (1890).

Agrawal, G.

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).

Dändliker, R.

Elsherbeni, A.

A. Elsherbeni, M. Hamid, and G. Tian, “Iterative scattering of a Gaussian beam by an array of circular conducting and dielectric cylinders,” J. Electromagn. Waves Appl. 7, 1323-1342 (1993).
[CrossRef]

Eom, H. J.

H. J. Eom, Electromagnetic Wave Theory for Boundary-Value Problems: an Advanced Course on Analytical Methods (Springer-Verlag, 2004).
[PubMed]

Gouesbet, G.

Gréhan, G.

Guo, L.

Z. Wu and L. Guo, “Electromagnetic scattering from a multilayered cylinder arbitrarily located in a Gaussian beam; a new recursive algorithm,” J. Electromagn. Waves Appl. 12, 725-726 (1998).
[CrossRef]

Hamid, M.

A. Elsherbeni, M. Hamid, and G. Tian, “Iterative scattering of a Gaussian beam by an array of circular conducting and dielectric cylinders,” J. Electromagn. Waves Appl. 7, 1323-1342 (1993).
[CrossRef]

Harvey, J. E.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974-980 (1979).
[CrossRef]

Jull, E. V.

E. V. Jull, Aperture Antennas and Diffraction Theory (Peter Peregrinus, 1981).
[CrossRef]

Kojima, T.

T. Kojima and Y. Yanagiuchi, “Scattering of an offset two-dimensional Gaussian beam wave by a cylinder,” J. Appl. Phys. 50, 41-46 (1979).
[CrossRef]

Kozaki, S.

S. Kozaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195-7200 (1982).
[CrossRef]

S. Kozaki and H. Sakurai, “Characteristics of a Gaussian beam at a dielectric interface,” J. Opt. Soc. Am. 68, 508-514 (1978).
[CrossRef]

Krattiger, B.

Li, L. W.

J. Yang, L. W. Li, K. Yasumoto, and C. H. Liang, “Two-dimensional scattering of a Gaussian beam by a periodic array of circular cylinders,” IEEE Trans. Geosci. Remote Sens. 43, 280-285 (2005).
[CrossRef]

Liang, C. H.

J. Yang, L. W. Li, K. Yasumoto, and C. H. Liang, “Two-dimensional scattering of a Gaussian beam by a periodic array of circular cylinders,” IEEE Trans. Geosci. Remote Sens. 43, 280-285 (2005).
[CrossRef]

Lorenz, L.

L. Lorenz, “Upon the light reflected and refracted by a transparent sphere,” Vidensk. Selsk. Shrifter 6, 1-62 (1890).

Mathews, J.

J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin/Cummings, 1970).

McLachlan, N.

N. McLachlan, Bessel Functions for Engineers (Oxford Univ. Press, 1955).

Mees, L.

Mie, G.

G. Mie, “Considerations on the optics of turbid media, especially colloidal metal sols,” Ann. Phys. 25, 377-442 (1908).
[CrossRef]

Pattanayak, D.

Rao, T.

T. Rao, “Scattering by a radially inhomogeneous cylindrical dielectric shell due to an incident Gaussian beam,” Can. J. Phys. 67, 471-475 (1989).
[CrossRef]

Ren, K.

Sakurai, H.

Souli, N.

Tian, G.

A. Elsherbeni, M. Hamid, and G. Tian, “Iterative scattering of a Gaussian beam by an array of circular conducting and dielectric cylinders,” J. Electromagn. Waves Appl. 7, 1323-1342 (1993).
[CrossRef]

Török, P.

P. Varga and P. Török, “The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108-118 (1998).
[CrossRef]

Varga, P.

P. Varga and P. Török, “The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108-118 (1998).
[CrossRef]

Walker, R. L.

J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin/Cummings, 1970).

Wu, Z.

Z. Wu and L. Guo, “Electromagnetic scattering from a multilayered cylinder arbitrarily located in a Gaussian beam; a new recursive algorithm,” J. Electromagn. Waves Appl. 12, 725-726 (1998).
[CrossRef]

Yanagiuchi, Y.

T. Kojima and Y. Yanagiuchi, “Scattering of an offset two-dimensional Gaussian beam wave by a cylinder,” J. Appl. Phys. 50, 41-46 (1979).
[CrossRef]

Yang, J.

J. Yang, L. W. Li, K. Yasumoto, and C. H. Liang, “Two-dimensional scattering of a Gaussian beam by a periodic array of circular cylinders,” IEEE Trans. Geosci. Remote Sens. 43, 280-285 (2005).
[CrossRef]

Yasumoto, K.

J. Yang, L. W. Li, K. Yasumoto, and C. H. Liang, “Two-dimensional scattering of a Gaussian beam by a periodic array of circular cylinders,” IEEE Trans. Geosci. Remote Sens. 43, 280-285 (2005).
[CrossRef]

Zimmermann, E.

Am. J. Phys. (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974-980 (1979).
[CrossRef]

Ann. Phys. (1)

G. Mie, “Considerations on the optics of turbid media, especially colloidal metal sols,” Ann. Phys. 25, 377-442 (1908).
[CrossRef]

Appl. Opt. (2)

Can. J. Phys. (1)

T. Rao, “Scattering by a radially inhomogeneous cylindrical dielectric shell due to an incident Gaussian beam,” Can. J. Phys. 67, 471-475 (1989).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

J. Yang, L. W. Li, K. Yasumoto, and C. H. Liang, “Two-dimensional scattering of a Gaussian beam by a periodic array of circular cylinders,” IEEE Trans. Geosci. Remote Sens. 43, 280-285 (2005).
[CrossRef]

J. Appl. Phys. (2)

T. Kojima and Y. Yanagiuchi, “Scattering of an offset two-dimensional Gaussian beam wave by a cylinder,” J. Appl. Phys. 50, 41-46 (1979).
[CrossRef]

S. Kozaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195-7200 (1982).
[CrossRef]

J. Electromagn. Waves Appl. (2)

Z. Wu and L. Guo, “Electromagnetic scattering from a multilayered cylinder arbitrarily located in a Gaussian beam; a new recursive algorithm,” J. Electromagn. Waves Appl. 12, 725-726 (1998).
[CrossRef]

A. Elsherbeni, M. Hamid, and G. Tian, “Iterative scattering of a Gaussian beam by an array of circular conducting and dielectric cylinders,” J. Electromagn. Waves Appl. 7, 1323-1342 (1993).
[CrossRef]

J. Opt. Soc. Am. (2)

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

J. Quant. Spectrosc. Radiat. Transf. (1)

G. Gouesbet, “Generalized Lorenz-Mie theories, the third decade: A perspective,” J. Quant. Spectrosc. Radiat. Transf. 110, 1223-1238 (2009).
[CrossRef]

Opt. Commun. (1)

P. Varga and P. Török, “The Gaussian wave solution of Maxwell's equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108-118 (1998).
[CrossRef]

Vidensk. Selsk. Shrifter (1)

L. Lorenz, “Upon the light reflected and refracted by a transparent sphere,” Vidensk. Selsk. Shrifter 6, 1-62 (1890).

Other (5)

H. J. Eom, Electromagnetic Wave Theory for Boundary-Value Problems: an Advanced Course on Analytical Methods (Springer-Verlag, 2004).
[PubMed]

E. V. Jull, Aperture Antennas and Diffraction Theory (Peter Peregrinus, 1981).
[CrossRef]

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).

N. McLachlan, Bessel Functions for Engineers (Oxford Univ. Press, 1955).

J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin/Cummings, 1970).

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Figures (13)

Fig. 1
Fig. 1

Coordinate system and layout for a Gaussian beam emitted from ( y 0 , z 0 ) at an angle θ in . The dielectric cylinder with radius a 0 and permittivity ϵ 2 is centered at the origin. The exterior region has permittivity ϵ 1 . The x coordinate is perpendicular to the y - z plane.

Fig. 2
Fig. 2

Cylindrical wave coefficients for the evanescent field of a Gaussian beam centered about the origin with β = 5 at a frequency of 1 GHz . The coefficients are purely real.

Fig. 3
Fig. 3

Spatial convergence of the evanescent field with increasing Bessel modes N. The field is taken along the y axis at z = 0 for a Gaussian beam with β = 5 at a frequency of 1 GHz .

Fig. 4
Fig. 4

Incident radiated part of a Gaussian beam with β = 2.5 at a frequency of 1 GHz scattering from a dielectric cylinder with radius a 0 = 0.3 m and relative permittivity ϵ = 10 . Displayed are the (a) incident field, (b) scattered field, (c) transmitted field inside the cylinder, (d) total outside field ( a ) + ( b ) , and (e) a scaled version of the transmitted field. (a), (b), (c), and (d) have the same scaling.

Fig. 5
Fig. 5

Incident radiated part of a Gaussian beam with β = 2.5 at a frequency of 1 GHz scattering from a dielectric cylinder with radius a 0 = 0.3 m and relative permittivity ϵ = 1000 . Displayed are the (a) incident field, (b) scattered field, (c) transmitted field inside the cylinder, (d) total outside field ( a ) + ( b ) , and (e) a scaled version of the transmitted field. (a), (b), (c), and (d) have the same scaling.

Fig. 6
Fig. 6

Incident radiated part of a Gaussian beam with β = 2.5 at a frequency of 1 GHz scattering from a dielectric cylinder with radius a 0 = 0.03 m and relative permittivity ϵ = 1000 . Displayed are the (a) incident field, (b) scattered field, (c) transmitted field inside the cylinder, (d) total outside field ( a ) + ( b ) , and (e) a scaled version of the transmitted field. (a), (b), (c), and (d) have the same scaling.

Fig. 7
Fig. 7

Evanescent field along the y axis at z = 0 for a Gaussian beam with β = 5 at a frequency of 1 GHz .

Fig. 8
Fig. 8

Evanescent field for a Gaussian beam with β = 5 at a frequency of 1 GHz generated from the Cartesian formula (23).

Fig. 9
Fig. 9

Evanescent field for a Gaussian beam with β = 5 at a frequency of 1 GHz , generated from the cylindrical wave formula (22).

Fig. 10
Fig. 10

Evanescent part of a Gaussian beam with β = 2.5 at a frequency of 1 GHz scattering from a dielectric cylinder with radius a 0 = 0.03 m and relative permittivity ϵ = 1000 . Displayed are the (a) incident field, (b) scattered field, (c) transmitted field inside the cylinder, (d) total outside field ( a ) + ( b ) , and (e) a scaled version of the transmitted field. (a), (b), (c), and (d) have the same scaling.

Fig. 11
Fig. 11

Evanescent part of the scattered field from Fig. 10 showing three time stamps (a)–(c).

Fig. 12
Fig. 12

Evanescent part of the scattered field from Fig. 10 displaying real and imaginary parts. The sectional view is taken along the z axis on the far side of the cylinder.

Fig. 13
Fig. 13

Evanescent part of a Gaussian beam with β = 2.5 at a frequency of 1 GHz scattering from a dielectric cylinder with radius a 0 = 0.3 m and relative permittivity ϵ = 1000 . Displayed are the (a) incident field, (b) scattered field, (c) transmitted field inside the cylinder, (d) total outside field ( a ) + ( b ) , and (e) a scaled version of the transmitted field. (a), (b), (c), and (d) have the same scaling.

Tables (1)

Tables Icon

Table 1 Numerical Comparison between the Cartesian Formula (23) and the Cylindrical Wave Representation (22) for the Evanescent Field

Equations (38)

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E x i ( y , z ) = exp ( j ω t ) 2 π F ( α ) exp [ j ( α y + γ z ) ] d α .
γ = { k 2 α 2 , | α | k j α 2 k 2 , | α | > k , z 0 j α 2 k 2 , | α | > k , z 0 } .
F ( α ) = f ( y ) exp ( j α y ) d y .
f ( y ) = E x i ( y , 0 ) .
f ( y ) = E 0 exp [ ( β y ) 2 ] .
F ( α ) = E 0 π β exp ( α 2 4 β 2 )
E x i ( y , z ) = E 0 exp ( j ϖ t ) 2 π β exp [ α 2 4 β 2 j ( y α + z k 2 α 2 ) ] d α .
sin ( ϕ ) = α k ,
cos ( ϕ ) = { 1 sin 2 ( ϕ ) , sin 2 ( ϕ ) 1 j sin 2 ( ϕ ) 1 , sin 2 ( ϕ ) > 1 , z 0 j sin 2 ( ϕ ) 1 , sin 2 ( ϕ ) > 1 , z 0 } ,
E x i ( y , z ) = E 0 k exp ( j ϖ t ) 2 π β π 2 j π 2 + j exp { k 2 sin 2 ( ϕ ) 4 β 2 j k [ y sin ( ϕ ) + z cos ( ϕ ) ] } cos ( ϕ ) d ϕ .
y = [ ρ sin ( θ ) y 0 ] cos ( θ in ) + [ ρ cos ( θ ) z 0 ] sin ( θ in ) ,
z = [ ρ sin ( θ ) y 0 ] sin ( θ in ) [ ρ cos ( θ ) z 0 ] cos ( θ in ) .
E x i ( ρ , θ ) = E 0 k exp ( j ϖ t ) 2 π β π 2 j π 2 + j exp { k 2 sin 2 ( ϕ ) 4 β 2 + j k [ ρ cos ( θ θ in ϕ ) ρ 0 cos ( θ 0 θ in ϕ ) ] } cos ( ϕ ) d ϕ .
exp [ j k ρ cos ( θ θ in ϕ ) ] = n = j n exp [ j n ( θ θ in ϕ ) ] J n ( k ρ ) .
E x i ( ρ , θ ) = exp ( j ω t ) n = j n exp ( j n θ ) J n ( k ρ ) A n ,
A n = E 0 k exp ( j n θ in ) 2 π β π 2 j π 2 + j exp { k 2 sin 2 ( ϕ ) 4 β 2 j [ n ϕ + k ρ 0 cos ( θ 0 θ in ϕ ) ] } cos ( ϕ ) d ϕ .
A n rad = E 0 k exp ( j n θ in ) 2 π β π 2 π 2 exp { k 2 sin 2 ( ϕ ) 4 β 2 j [ n ϕ + k ρ 0 cos ( θ 0 θ in ϕ ) ] } cos ( ϕ ) d ϕ .
A n evan = E 0 k exp ( j n θ in ) 2 π β π 2 π 2 + j exp { k 2 sin 2 ( ϕ ) 4 β 2 j [ n ϕ + k ρ 0 cos ( θ 0 θ in ϕ ) ] } cos ( ϕ ) d ϕ + E 0 k exp ( j n θ in ) 2 π β π 2 j π 2 exp { k 2 sin 2 ( ϕ ) 4 β 2 j [ n ϕ + k ρ 0 cos ( θ 0 θ in ϕ ) ] } cos ( ϕ ) d ϕ .
ϕ = π 2 + j u ,
ϕ = π 2 j u .
A n evan = E 0 k exp ( j n θ in ) 2 π β π 2 π 2 + j exp [ k 2 sin 2 ( ϕ ) 4 β 2 ] sinh ( u ) { j n exp [ n u j k ρ 0 sin ( θ 0 θ in j u ) ] + j n exp [ n u + j k ρ 0 sin ( θ 0 θ in j u ) ] } d u .
E x evan ( ρ , θ ) = exp ( j ω t ) n = N N j n exp ( j n θ ) J n ( k ρ ) A n evan .
E x evan ( y , z ) = E 0 exp ( j ω t ) π β k exp ( α 2 4 β 2 | z | α 2 k 2 ) cos ( y α ) d α .
y = ( y y 0 ) cos ( θ in ) + ( z z 0 ) sin ( θ in ) ,
z = ( y y 0 ) sin ( θ in ) ( z z 0 ) cos ( θ in ) .
E = exp ( ± j ω t ) exp ( ± j n θ ) [ R J n ( k ρ ) + S Y n ( k ρ ) ] .
E x t = exp ( j ω t ) n = j n exp ( j n θ ) J n ( ϵ k ρ ) C n ,
ϵ = ϵ 2 ϵ 1 .
E x s = exp ( j ω t ) n = j n exp ( j n θ ) H n ( 2 ) ( k ρ ) B n .
× E = d B d t .
× E = ( 1 ρ d E x d θ ) ρ ̂ + ( d E x d ρ ) θ ̂ .
H i = exp ( j ω t ) ρ ω μ n = j n n exp ( j n θ ) J n ( k ρ ) A n ρ ̂ exp ( j ω t ) k ω μ n = j n + 1 exp ( j n θ ) J n ( k ρ ) A n θ ̂ ,
H s = exp ( j ω t ) ρ ω μ n = j n n exp ( j n θ ) H n ( 2 ) ( k ρ ) B n ρ ̂ exp ( j ω t ) k ω μ n = j n + 1 exp ( j n θ ) H n ( 2 ) ( k ρ ) B n θ ̂ ,
H t = exp ( j ω t ) ρ ω μ n = j n n exp ( j n θ ) J n ( ϵ k ρ ) C n ρ ̂ exp ( j ω t ) ϵ k ω μ n = j n + 1 exp ( j n θ ) J n ( ϵ k ρ ) C n θ ̂ .
E x i ( a 0 , θ ) + E x s ( a 0 , θ ) = E x t ( a 0 , θ ) ,
H θ i ( a 0 , θ ) + H θ s ( a 0 , θ ) = H θ t ( a 0 , θ ) .
B n = J n ( ϵ k a 0 ) J n ( k a 0 ) J n ( k a 0 ) J n ( ϵ k a 0 ) ϵ ϵ J n ( ϵ k a 0 ) H n ( 2 ) ( k a 0 ) J n ( ϵ k a 0 ) H n ( 2 ) ( k a 0 ) A n ,
C n = 2 j π k a 0 ϵ J n ( ϵ k a 0 ) H n ( 2 ) ( k a 0 ) J n ( ϵ k a 0 ) H n ( 2 ) ( k a 0 ) A n .

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