Abstract

The two-dimensional vector plane wave spectrum (VPWS) is scattered from parallel circular cylinders using a boundary value solution with the T-matrix formalism. The VPWS allows us to define the incident, two- dimensional electromagnetic field with an arbitrary distribution and polarization, including both radiative and evanescent components. Using the fast Fourier transform, we can quickly compute the multiple scattering of fields that have any particular functional or numerical form. We perform numerical simulations to investigate a grating of cylinders that is capable of converting an evanescent field into a set of propagating beams. The direction of propagation of each beam is directly related to a spatial frequency component of the incident evanescent field.

© 2011 Optical Society of America

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References

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  1. V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46(1952).
    [CrossRef]
  2. J. E. Burke, D. Censor, and V. Twersky, “Exact inverse-separation series for multiple scattering in two dimensions,” J. Acoust. Soc. Am. 37, 5–13 (1965).
    [CrossRef]
  3. G. Olaofe, “Scattering of two cylinders,” Radio Sci. 5, 1351–1360(1970).
    [CrossRef]
  4. G. Olaofe, “Scattering by an arbitrary configuration of parallel circular cylinders,” J. Opt. Soc. Am. 60, 1233–1236(1970).
    [CrossRef]
  5. J. W. Young and J. C. Bertrand, “Multiple scattering by two cylinders,” J. Acoust. Soc. Am. 58, 1190–1195 (1975).
    [CrossRef]
  6. A. Z. Elsherbeni, “A comparative study of two-dimensional multiple scattering techniques,” Radio Sci. 29, 1023–1033(1994).
    [CrossRef]
  7. B. Peterson and S. Ström, “T matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661–3678 (1973).
    [CrossRef]
  8. T. Kojima, A. Ishikura, and M. Ieguchi, “Scattering of Hermite–Gaussian beams by two parallel conducting cylinders,” TGAP-83-36 Japan (IECE, 1983).
  9. H. Sugiyama and S. Kozaki, “Multiple scattering of a Gaussian beam by two cylinders having different radii,” IEICE Trans. E65-E, 173–174 (1982).
  10. M. Yokota, T. Takenaka, and O. Fukumitsu, “Scattering of a Hermite–Gaussian beam mode by parallel dielectric circular cylinders,” J. Opt. Soc. Am. A 3, 580–586 (1986).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  13. P. Pawliuk and M. Yedlin, “Gaussian beam scattering from a dielectric cylinder, including the evanescent region,” J. Opt. Soc. Am. A 26, 2558–2566 (2009).
    [CrossRef]
  14. 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]
  15. E. V. Jull, Aperture Antennas and Diffraction Theory(Peregrinus, 1981).
    [CrossRef]
  16. J. H. Kim and K. B. Song, “Recent progress of nano-technology with NSOM,” Micron 38, 409–426 (2007).
    [CrossRef]
  17. D. Marks and P. S. Carney, “Near-field diffractive elements,” Opt. Lett. 30, 1870–1872 (2005).
    [CrossRef] [PubMed]
  18. O. Malyuskin and V. Fusco, “Far field subwavelength source resolution using phase conjugating lens assisted with evanescent-to-propagating spectrum conversion,” IEEE Trans. Antennas Propag. 58, 459–468 (2010).
    [CrossRef]
  19. Y. V. Gulyaev, Y. N. Barabanenkov, M. Y. Barabanenkov, and S. Nikitov, “Optical theorem for electromagnetic field scattering by dielectric structures and energy emission from the evanescent wave,” Phys. Rev. E 72, 026602 (2005).
    [CrossRef]
  20. H. Guo, J. Chen, and S. Zhuang, “Vector plane wave spectrum of an arbitrary polarized electromagnetic wave,” Opt. Express 14, 2095–2100 (2006).
    [CrossRef] [PubMed]
  21. N. W. McLachlan, Bessel Functions for Engineers (Oxford University Press, 1934).
  22. B. G. Korenev, Bessel Functions and Their Applications (CRC Press, 2002).
  23. P. Pawliuk and M. Yedlin, “Truncating cylindrical wave modes in two-dimensional multiple scattering,” Opt. Lett. 35, 3997–3999(2010).
    [CrossRef] [PubMed]

2010 (2)

O. Malyuskin and V. Fusco, “Far field subwavelength source resolution using phase conjugating lens assisted with evanescent-to-propagating spectrum conversion,” IEEE Trans. Antennas Propag. 58, 459–468 (2010).
[CrossRef]

P. Pawliuk and M. Yedlin, “Truncating cylindrical wave modes in two-dimensional multiple scattering,” Opt. Lett. 35, 3997–3999(2010).
[CrossRef] [PubMed]

2009 (1)

2007 (1)

J. H. Kim and K. B. Song, “Recent progress of nano-technology with NSOM,” Micron 38, 409–426 (2007).
[CrossRef]

2006 (1)

2005 (3)

D. Marks and P. S. Carney, “Near-field diffractive elements,” Opt. Lett. 30, 1870–1872 (2005).
[CrossRef] [PubMed]

Y. V. Gulyaev, Y. N. Barabanenkov, M. Y. Barabanenkov, and S. Nikitov, “Optical theorem for electromagnetic field scattering by dielectric structures and energy emission from the evanescent wave,” Phys. Rev. E 72, 026602 (2005).
[CrossRef]

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]

1997 (1)

1994 (1)

A. Z. Elsherbeni, “A comparative study of two-dimensional multiple scattering techniques,” Radio Sci. 29, 1023–1033(1994).
[CrossRef]

1993 (1)

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

1986 (1)

1982 (1)

H. Sugiyama and S. Kozaki, “Multiple scattering of a Gaussian beam by two cylinders having different radii,” IEICE Trans. E65-E, 173–174 (1982).

1975 (1)

J. W. Young and J. C. Bertrand, “Multiple scattering by two cylinders,” J. Acoust. Soc. Am. 58, 1190–1195 (1975).
[CrossRef]

1973 (1)

B. Peterson and S. Ström, “T matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

1970 (2)

1965 (1)

J. E. Burke, D. Censor, and V. Twersky, “Exact inverse-separation series for multiple scattering in two dimensions,” J. Acoust. Soc. Am. 37, 5–13 (1965).
[CrossRef]

1952 (1)

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46(1952).
[CrossRef]

Barabanenkov, M. Y.

Y. V. Gulyaev, Y. N. Barabanenkov, M. Y. Barabanenkov, and S. Nikitov, “Optical theorem for electromagnetic field scattering by dielectric structures and energy emission from the evanescent wave,” Phys. Rev. E 72, 026602 (2005).
[CrossRef]

Barabanenkov, Y. N.

Y. V. Gulyaev, Y. N. Barabanenkov, M. Y. Barabanenkov, and S. Nikitov, “Optical theorem for electromagnetic field scattering by dielectric structures and energy emission from the evanescent wave,” Phys. Rev. E 72, 026602 (2005).
[CrossRef]

Bertrand, J. C.

J. W. Young and J. C. Bertrand, “Multiple scattering by two cylinders,” J. Acoust. Soc. Am. 58, 1190–1195 (1975).
[CrossRef]

Burke, J. E.

J. E. Burke, D. Censor, and V. Twersky, “Exact inverse-separation series for multiple scattering in two dimensions,” J. Acoust. Soc. Am. 37, 5–13 (1965).
[CrossRef]

Carney, P. S.

Censor, D.

J. E. Burke, D. Censor, and V. Twersky, “Exact inverse-separation series for multiple scattering in two dimensions,” J. Acoust. Soc. Am. 37, 5–13 (1965).
[CrossRef]

Chen, J.

Elsherbeni, A. Z.

A. Z. Elsherbeni, “A comparative study of two-dimensional multiple scattering techniques,” Radio Sci. 29, 1023–1033(1994).
[CrossRef]

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

Fukumitsu, O.

Fusco, V.

O. Malyuskin and V. Fusco, “Far field subwavelength source resolution using phase conjugating lens assisted with evanescent-to-propagating spectrum conversion,” IEEE Trans. Antennas Propag. 58, 459–468 (2010).
[CrossRef]

Gouesbet, G.

Gulyaev, Y. V.

Y. V. Gulyaev, Y. N. Barabanenkov, M. Y. Barabanenkov, and S. Nikitov, “Optical theorem for electromagnetic field scattering by dielectric structures and energy emission from the evanescent wave,” Phys. Rev. E 72, 026602 (2005).
[CrossRef]

Guo, H.

Hamid, M.

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

Ieguchi, M.

T. Kojima, A. Ishikura, and M. Ieguchi, “Scattering of Hermite–Gaussian beams by two parallel conducting cylinders,” TGAP-83-36 Japan (IECE, 1983).

Ishikura, A.

T. Kojima, A. Ishikura, and M. Ieguchi, “Scattering of Hermite–Gaussian beams by two parallel conducting cylinders,” TGAP-83-36 Japan (IECE, 1983).

Jull, E. V.

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

Kim, J. H.

J. H. Kim and K. B. Song, “Recent progress of nano-technology with NSOM,” Micron 38, 409–426 (2007).
[CrossRef]

Kojima, T.

T. Kojima, A. Ishikura, and M. Ieguchi, “Scattering of Hermite–Gaussian beams by two parallel conducting cylinders,” TGAP-83-36 Japan (IECE, 1983).

Korenev, B. G.

B. G. Korenev, Bessel Functions and Their Applications (CRC Press, 2002).

Kozaki, S.

H. Sugiyama and S. Kozaki, “Multiple scattering of a Gaussian beam by two cylinders having different radii,” IEICE Trans. E65-E, 173–174 (1982).

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]

Malyuskin, O.

O. Malyuskin and V. Fusco, “Far field subwavelength source resolution using phase conjugating lens assisted with evanescent-to-propagating spectrum conversion,” IEEE Trans. Antennas Propag. 58, 459–468 (2010).
[CrossRef]

Marks, D.

McLachlan, N. W.

N. W. McLachlan, Bessel Functions for Engineers (Oxford University Press, 1934).

Nikitov, S.

Y. V. Gulyaev, Y. N. Barabanenkov, M. Y. Barabanenkov, and S. Nikitov, “Optical theorem for electromagnetic field scattering by dielectric structures and energy emission from the evanescent wave,” Phys. Rev. E 72, 026602 (2005).
[CrossRef]

Olaofe, G.

Pawliuk, P.

Peterson, B.

B. Peterson and S. Ström, “T matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

Song, K. B.

J. H. Kim and K. B. Song, “Recent progress of nano-technology with NSOM,” Micron 38, 409–426 (2007).
[CrossRef]

Ström, S.

B. Peterson and S. Ström, “T matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

Sugiyama, H.

H. Sugiyama and S. Kozaki, “Multiple scattering of a Gaussian beam by two cylinders having different radii,” IEICE Trans. E65-E, 173–174 (1982).

Takenaka, T.

Tian, G.

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

Twersky, V.

J. E. Burke, D. Censor, and V. Twersky, “Exact inverse-separation series for multiple scattering in two dimensions,” J. Acoust. Soc. Am. 37, 5–13 (1965).
[CrossRef]

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46(1952).
[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]

Yedlin, M.

Yokota, M.

Young, J. W.

J. W. Young and J. C. Bertrand, “Multiple scattering by two cylinders,” J. Acoust. Soc. Am. 58, 1190–1195 (1975).
[CrossRef]

Zhuang, S.

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

O. Malyuskin and V. Fusco, “Far field subwavelength source resolution using phase conjugating lens assisted with evanescent-to-propagating spectrum conversion,” IEEE Trans. Antennas Propag. 58, 459–468 (2010).
[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]

IEICE Trans. (1)

H. Sugiyama and S. Kozaki, “Multiple scattering of a Gaussian beam by two cylinders having different radii,” IEICE Trans. E65-E, 173–174 (1982).

J. Acoust. Soc. Am. (3)

V. Twersky, “Multiple scattering of radiation by an arbitrary configuration of parallel cylinders,” J. Acoust. Soc. Am. 24, 42–46(1952).
[CrossRef]

J. E. Burke, D. Censor, and V. Twersky, “Exact inverse-separation series for multiple scattering in two dimensions,” J. Acoust. Soc. Am. 37, 5–13 (1965).
[CrossRef]

J. W. Young and J. C. Bertrand, “Multiple scattering by two cylinders,” J. Acoust. Soc. Am. 58, 1190–1195 (1975).
[CrossRef]

J. Elect. Waves Appl. (1)

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

J. Opt. Soc. Am. (1)

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

Micron (1)

J. H. Kim and K. B. Song, “Recent progress of nano-technology with NSOM,” Micron 38, 409–426 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. D (1)

B. Peterson and S. Ström, “T matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

Phys. Rev. E (1)

Y. V. Gulyaev, Y. N. Barabanenkov, M. Y. Barabanenkov, and S. Nikitov, “Optical theorem for electromagnetic field scattering by dielectric structures and energy emission from the evanescent wave,” Phys. Rev. E 72, 026602 (2005).
[CrossRef]

Radio Sci. (2)

A. Z. Elsherbeni, “A comparative study of two-dimensional multiple scattering techniques,” Radio Sci. 29, 1023–1033(1994).
[CrossRef]

G. Olaofe, “Scattering of two cylinders,” Radio Sci. 5, 1351–1360(1970).
[CrossRef]

Other (4)

T. Kojima, A. Ishikura, and M. Ieguchi, “Scattering of Hermite–Gaussian beams by two parallel conducting cylinders,” TGAP-83-36 Japan (IECE, 1983).

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

N. W. McLachlan, Bessel Functions for Engineers (Oxford University Press, 1934).

B. G. Korenev, Bessel Functions and Their Applications (CRC Press, 2002).

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Figures (10)

Fig. 1
Fig. 1

Geometry and coordinate systems for the cylinders and the incident field. The x direction is perpendicular to the page.

Fig. 2
Fig. 2

Triangle showing the coordinates of transformation in the Graf addition theorem [Eq. (24)].

Fig. 3
Fig. 3

Setup for the numerical simulations. The grating is made up of 41 conducting cylinders with a = 0.005 m . The dimensions are not drawn to scale.

Fig. 4
Fig. 4

Far-field intensity ( W / m ) plot for scattering of the incident field, with aperture distribution E x i ( 0 , y ) = exp ( j 1.1 k y ) , from the grating.

Fig. 5
Fig. 5

Far-field intensity ( W / m ) plot for scattering of the incident field, with aperture distribution E x i ( 0 , y ) = exp ( j 1.4 k y ) , from the grating.

Fig. 6
Fig. 6

Far-field intensity ( W / m ) plot for scattering of the incident field, with aperture distribution E x i ( 0 , y ) = exp ( j 1.7 k y ) , from the grating.

Fig. 7
Fig. 7

Far-field intensity ( W / m ) plot for scattering of the incident field, with aperture distribution E x i ( 0 , y ) = exp ( j 2 k y ) , from the grating.

Fig. 8
Fig. 8

Far-field intensity ( W / m ) polar plot for scattering of the evanescent field of a square shaped beam.

Fig. 9
Fig. 9

Far-field intensity ( W / m ) plot for scattering of the evanescent field of a square beam for π / 2 θ 0 .

Fig. 10
Fig. 10

Absolute value of the spatial frequency content of the incident evanescent field from k to 2 k .

Equations (53)

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× E = B t ,
× H = D t .
E x i ( z , y ) = 1 2 π E ˜ TM ( k y ) exp [ j ( k y y + k z z ) ] d k y ,
H x i ( z , y ) = 1 2 π H ˜ TE ( k y ) exp [ j ( k y y + k z z ) ] d k y ,
E ˜ TM ( k y ) = E x i ( 0 , y ) exp ( j k y y ) d y ,
H ˜ TE ( k y ) = H x i ( 0 , y ) exp ( j k y y ) d y .
y = ρ v sin ( θ v ) ρ v 0 sin ( θ v 0 ) ,
z = ρ v cos ( θ v ) ρ v 0 cos ( θ v 0 ) ,
k y = k sin ( ϕ ) .
E x i ( ρ v , θ v ) = k 2 π π / 2 j π / 2 + j E ˜ TM [ k sin ( ϕ ) ] cos ( ϕ ) exp { j k [ ρ v cos ( θ v ϕ ) ρ v 0 cos ( θ v 0 ϕ ) ] } d ϕ ,
H x i ( ρ v , θ v ) = k 2 π π / 2 j π / 2 + j H ˜ TE [ k sin ( ϕ ) ] cos ( ϕ ) exp { j k [ ρ v cos ( θ v ϕ ) ρ v 0 cos ( θ v 0 ϕ ) ] } d ϕ .
exp [ j k ρ v cos ( θ v ϕ ) ] = n = j n exp [ j n ( θ v ϕ ) ] J n ( k ρ v ) .
E x i = n = j n exp ( j n θ v ) J n ( k ρ v ) A n v ,
H x i = n = j n exp ( j n θ v ) J n ( k ρ v ) Q n v ,
A n v = k 2 π π / 2 j π / 2 + j E ˜ TM [ k sin ( ϕ ) ] exp { j [ k ρ v 0 cos ( θ v 0 ϕ ) n ϕ ] } cos ( ϕ ) d ϕ ,
Q n v = k 2 π π / 2 j π / 2 + j H ˜ TE [ k sin ( ϕ ) ] exp { j [ k ρ v 0 cos ( θ v 0 ϕ ) n ϕ ] } cos ( ϕ ) d ϕ .
ϕ = π / 2 j u ,
ϕ = π / 2 + j u ,
A n evan v = k 2 π 0 sinh ( u ) { S n + v E ˜ TM [ k cosh ( u ) ] + S n v E ˜ TM [ k cosh ( u ) ] } d u ,
Q n evan v = k 2 π 0 sinh ( u ) { S n + v H ˜ TE [ k cosh ( u ) ] + S n v H ˜ TE [ k cosh ( u ) ] } d u ,
S n ± v = j n exp [ ± n u ± j k ρ v 0 sin ( θ v 0 j u ) ] .
E x s v = n = j n exp ( j n θ v ) H n ( 2 ) ( k ρ v ) b n v ,
H x s v = n = j n exp ( j n θ v ) H n ( 2 ) ( k ρ v ) g n v .
Z v ( z ) exp ( j v ψ ) = m = Z v + m ( x ) J m ( y ) exp ( j m δ ) ,
| exp ( ± j δ ) y | < | x | .
H n ( 2 ) ( k ρ w ) exp ( j n θ w ) = m = H n m ( 2 ) ( k d w v ) J m ( k ρ v ) exp { j [ ( n m ) ϕ w v + m θ v ] } .
E x s w = n = j n exp ( j n θ v ) J n ( k ρ v ) B n w v ,
B n w v = m = j n m exp [ j ( m n ) ϕ w v ] H m n ( 2 ) ( k d w v ) b m w ,
H x s w = n = j n exp ( j n θ v ) J n ( k ρ v ) G n w v ,
G n w v = m = j n m exp [ j ( m n ) ϕ w v ] H m n ( 2 ) ( k d w v ) g m w .
E x t v = n = j n exp ( j n θ v ) J n ( ε r v k ρ v ) d n v ,
H x t v = n = j n exp ( j n θ v ) J n ( ε r v k ρ v ) h n v .
E x t v ( a v , θ v ) = E x i ( a v , θ v ) + E x s v ( a v , θ v ) + w v E x s w ( a v , θ v ) ,
H θ t v ( a v , θ v ) = H θ i ( a v , θ v ) + H θ s v ( a v , θ v ) + w v H θ s w ( a v , θ v ) ,
E θ t v ( a v , θ v ) = E θ i ( a v , θ v ) + E θ s v ( a v , θ v ) + w v E θ s w ( a v , θ v ) ,
H x t v ( a v , θ v ) = H x i ( a v , θ v ) + H x s v ( a v , θ v ) + w v H x s w ( a v , θ v ) .
b n v = f n v A n v + f n v w v B n w v ,
g n v = f n v Q n v + f n v w v G n w v .
f n v = J n ( k a v ) H n ( 2 ) ( k a v ) ,
f n v = J n ( k a v ) J n ( ε r v k a v ) ε r v J n ( k a v ) J n ( ε r v k a v ) ε r v H n ( 2 ) ( k a v ) J n ( ε r v k a v ) H n ( 2 ) ( k a v ) J n ( ε r v k a v ) .
L TM = [ b N 1 1 b N 1 1 b N 2 2 ] ,
L TE = [ g N 1 1 g N 1 1 g N 2 2 ] ,
P TM = [ A N 1 1 A N 1 1 A N 2 2 ] ,
P TE = [ Q N 1 1 Q N 1 1 Q N 2 2 ] ,
L TM = T P TM ,
L TE = T P TE .
L = F P + F C L ,
F = diag [ f N 1 1 f N 1 1 f N 2 2 ] ,
C = [ 0 C 12 C 1 M C 21 0 C 2 M C M 1 C M 2 0 ] .
c n , m = j n m H m n ( 2 ) ( k d w v ) exp [ j ( m n ) ϕ w v ] .
L = ( F 1 C ) 1 P = T P .
U = ρ | E x | 2 2 η + ρ η | H x | 2 2 W / m ,
E x i ( 0 , y ) = 1 | y | 1 0 | y | > 1 .

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