Abstract

A theory of scattering by a finite number of cylinders of arbitrary cross section is presented. This theory is based on a self-consistent approach that identifies incident and scattered fields around each cylinder and then uses the notion of a scattering matrix in order to get a linear system of equations. Special attention is paid to the simplified case of a sparse distribution of small cylinders for low frequencies. Surprisingly, it is found that the classical rules of homogenization must be modified in that case. The phenomenon of enhanced backscattering of light is investigated from numerical data for a dense distribution of cylinders.

© 1994 Optical Society of America

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References

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  1. D. Maystre, J. C. Dainty, Modern Analysis of Scattering Phenomena (Hilger, Bristol, UK, 1992).
  2. M. Abramovitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).
  3. J. Y. Suratteau, R. Petit, “The electromagnetic theory of the infinitely conducting wire grating using a Fourier–Bessel expansion of the field,” Int. J. Infrared Millimeter Waves 5, 1189–1200 (1984).
    [Crossref]
  4. R. C. McPhedran, “Transport properties of cylinder pairs and of the square array of cylinders,” Proc. R. Soc. London Ser. A 408, 31–43 (1986).
    [Crossref]
  5. R. C. McPhedran, L. Poladian, G. W. Milton, “Asymptotic studies of closely spaced, highly conducting cylinders,” Proc. R. Soc. London Ser. A 415, 185–196 (1988).
    [Crossref]
  6. H. A. Youssif, S. Köhler, “Scattering by two penetrable cylinders at oblique incidence. I. The analytical solution,” J. Opt. Soc. Am. A 5, 1085–1096 (1988).
    [Crossref]
  7. A. Z. Elsherbeni, A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,”IEEE Trans. Antennas Propag. 40, 96–99 (1992).
    [Crossref]
  8. J. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York, 1964).
  9. L. Schwartz, Mathematics for Physical Sciences (Addison-Wesley, London, 1967).
  10. D. Maystre, P. Vincent, “Diffraction d’une onde électromagnétique plane par un objet cylindrique non infiniment conducteur de section arbitraire,” Opt. Commun. 5, 327–330 (1972).
    [Crossref]
  11. F. Zolla, R. Petit, M. Cadilhac, “Electromagnetic theory of diffraction by a system of parallel rods: the method of fictitious sources,” J. Opt. Soc. Am. A 11, 1087–1096 (1994).
    [Crossref]
  12. K. A. O’Donnell, E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [Crossref]
  13. M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte Carlo simulations for scattering of electromagnetic waves from perfectly conductive random rough surfaces,” Opt. Lett. 12, 979–981 (1987).
    [Crossref] [PubMed]
  14. J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep reflection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
    [Crossref]
  15. A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
    [Crossref]
  16. A. A. Maradudin, E. R. Mendez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
    [Crossref] [PubMed]
  17. M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
    [Crossref]
  18. J. J. Greffet, “Backscattering of s-polarized light from a cloud of small particles above a dielectric substrate,” Waves Random Media 3, S65–S73 (1991).
    [Crossref]

1994 (1)

1992 (1)

A. Z. Elsherbeni, A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,”IEEE Trans. Antennas Propag. 40, 96–99 (1992).
[Crossref]

1991 (1)

J. J. Greffet, “Backscattering of s-polarized light from a cloud of small particles above a dielectric substrate,” Waves Random Media 3, S65–S73 (1991).
[Crossref]

1990 (1)

1989 (2)

1988 (2)

R. C. McPhedran, L. Poladian, G. W. Milton, “Asymptotic studies of closely spaced, highly conducting cylinders,” Proc. R. Soc. London Ser. A 415, 185–196 (1988).
[Crossref]

H. A. Youssif, S. Köhler, “Scattering by two penetrable cylinders at oblique incidence. I. The analytical solution,” J. Opt. Soc. Am. A 5, 1085–1096 (1988).
[Crossref]

1987 (2)

1986 (1)

R. C. McPhedran, “Transport properties of cylinder pairs and of the square array of cylinders,” Proc. R. Soc. London Ser. A 408, 31–43 (1986).
[Crossref]

1985 (1)

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[Crossref]

1984 (1)

J. Y. Suratteau, R. Petit, “The electromagnetic theory of the infinitely conducting wire grating using a Fourier–Bessel expansion of the field,” Int. J. Infrared Millimeter Waves 5, 1189–1200 (1984).
[Crossref]

1972 (1)

D. Maystre, P. Vincent, “Diffraction d’une onde électromagnétique plane par un objet cylindrique non infiniment conducteur de section arbitraire,” Opt. Commun. 5, 327–330 (1972).
[Crossref]

Abramovitz, M.

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Cadilhac, M.

Celli, V.

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[Crossref]

Dainty, J. C.

D. Maystre, J. C. Dainty, Modern Analysis of Scattering Phenomena (Hilger, Bristol, UK, 1992).

Elsherbeni, A. Z.

A. Z. Elsherbeni, A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,”IEEE Trans. Antennas Propag. 40, 96–99 (1992).
[Crossref]

Greffet, J. J.

J. J. Greffet, “Backscattering of s-polarized light from a cloud of small particles above a dielectric substrate,” Waves Random Media 3, S65–S73 (1991).
[Crossref]

Kishk, A. A.

A. Z. Elsherbeni, A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,”IEEE Trans. Antennas Propag. 40, 96–99 (1992).
[Crossref]

Köhler, S.

Maradudin, A. A.

A. A. Maradudin, E. R. Mendez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
[Crossref] [PubMed]

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[Crossref]

Maystre, D.

M. Saillard, D. Maystre, “Scattering from metallic and dielectric rough surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
[Crossref]

D. Maystre, P. Vincent, “Diffraction d’une onde électromagnétique plane par un objet cylindrique non infiniment conducteur de section arbitraire,” Opt. Commun. 5, 327–330 (1972).
[Crossref]

D. Maystre, J. C. Dainty, Modern Analysis of Scattering Phenomena (Hilger, Bristol, UK, 1992).

McGurn, A. R.

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[Crossref]

McPhedran, R. C.

R. C. McPhedran, L. Poladian, G. W. Milton, “Asymptotic studies of closely spaced, highly conducting cylinders,” Proc. R. Soc. London Ser. A 415, 185–196 (1988).
[Crossref]

R. C. McPhedran, “Transport properties of cylinder pairs and of the square array of cylinders,” Proc. R. Soc. London Ser. A 408, 31–43 (1986).
[Crossref]

Mendez, E. R.

Michel, T.

Milton, G. W.

R. C. McPhedran, L. Poladian, G. W. Milton, “Asymptotic studies of closely spaced, highly conducting cylinders,” Proc. R. Soc. London Ser. A 415, 185–196 (1988).
[Crossref]

Nieto-Vesperinas, M.

O’Donnell, K. A.

Petit, R.

F. Zolla, R. Petit, M. Cadilhac, “Electromagnetic theory of diffraction by a system of parallel rods: the method of fictitious sources,” J. Opt. Soc. Am. A 11, 1087–1096 (1994).
[Crossref]

J. Y. Suratteau, R. Petit, “The electromagnetic theory of the infinitely conducting wire grating using a Fourier–Bessel expansion of the field,” Int. J. Infrared Millimeter Waves 5, 1189–1200 (1984).
[Crossref]

Poladian, L.

R. C. McPhedran, L. Poladian, G. W. Milton, “Asymptotic studies of closely spaced, highly conducting cylinders,” Proc. R. Soc. London Ser. A 415, 185–196 (1988).
[Crossref]

Saillard, M.

Schwartz, L.

L. Schwartz, Mathematics for Physical Sciences (Addison-Wesley, London, 1967).

Soto-Crespo, J. M.

Stegun, I.

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Suratteau, J. Y.

J. Y. Suratteau, R. Petit, “The electromagnetic theory of the infinitely conducting wire grating using a Fourier–Bessel expansion of the field,” Int. J. Infrared Millimeter Waves 5, 1189–1200 (1984).
[Crossref]

Van Bladel, J.

J. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York, 1964).

Vincent, P.

D. Maystre, P. Vincent, “Diffraction d’une onde électromagnétique plane par un objet cylindrique non infiniment conducteur de section arbitraire,” Opt. Commun. 5, 327–330 (1972).
[Crossref]

Youssif, H. A.

Zolla, F.

IEEE Trans. Antennas Propag. (1)

A. Z. Elsherbeni, A. A. Kishk, “Modeling of cylindrical objects by circular dielectric and conducting cylinders,”IEEE Trans. Antennas Propag. 40, 96–99 (1992).
[Crossref]

Int. J. Infrared Millimeter Waves (1)

J. Y. Suratteau, R. Petit, “The electromagnetic theory of the infinitely conducting wire grating using a Fourier–Bessel expansion of the field,” Int. J. Infrared Millimeter Waves 5, 1189–1200 (1984).
[Crossref]

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

D. Maystre, P. Vincent, “Diffraction d’une onde électromagnétique plane par un objet cylindrique non infiniment conducteur de section arbitraire,” Opt. Commun. 5, 327–330 (1972).
[Crossref]

Opt. Lett. (2)

Phys. Rev. B (1)

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[Crossref]

Proc. R. Soc. London Ser. A (2)

R. C. McPhedran, “Transport properties of cylinder pairs and of the square array of cylinders,” Proc. R. Soc. London Ser. A 408, 31–43 (1986).
[Crossref]

R. C. McPhedran, L. Poladian, G. W. Milton, “Asymptotic studies of closely spaced, highly conducting cylinders,” Proc. R. Soc. London Ser. A 415, 185–196 (1988).
[Crossref]

Waves Random Media (1)

J. J. Greffet, “Backscattering of s-polarized light from a cloud of small particles above a dielectric substrate,” Waves Random Media 3, S65–S73 (1991).
[Crossref]

Other (4)

D. Maystre, J. C. Dainty, Modern Analysis of Scattering Phenomena (Hilger, Bristol, UK, 1992).

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

J. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York, 1964).

L. Schwartz, Mathematics for Physical Sciences (Addison-Wesley, London, 1967).

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Figures (15)

Fig. 1
Fig. 1

General description of the problem and notation. text for details.

Fig. 2
Fig. 2

Notation used for a change of the coordinate system of Bessel functions. The subscripts refer to the system of coordinates linked to a given cylinder. See text for details.

Fig. 3
Fig. 3

Domain of validity of the Fourier–Bessel expansion of the total field around one cylinder.

Fig. 4
Fig. 4

Simplified method for low frequencies.

Fig. 5
Fig. 5

Test of validity on a scattering object. (a) Scattering object: the central circular cylinder has diameter λ, and the ellipses have large and small axes, of 2λ and 2λ/3, respectively. The centers of the circle and of one ellipse are separated by 3λ. (b) Convergence of the intensity radiated by the scatterer shown in (a).

Fig. 6
Fig. 6

Comparison with another theory. (a) Scattering object: two dielectric circular cylinders (index 1.5) with diameter 60 mm, distance between centers 90 mm, and s-polarized light. (b) Scattered intensity versus diffraction angle for a wavelength of 30 mm, incidence α = 135°, and s-polarized light.

Fig. 7
Fig. 7

Scattering by a set of cylinders. (a) 30 cylinders of diameter λ/10, randomly placed in a rectangular box; L = 10λ, h = 5λ, s polarization. (b), (c) Scattered intensity D(θ) corresponding to cylinders of indices ν = 4 and ν = 1.5, with wavelength λ = 1 mm. The results are averaged over 1000 realizations.

Fig. 8
Fig. 8

Scattered intensity D(θ) produced by a set of 20 cylinders of diameter 1.2λ and p polarization. The results are averaged over 1000 realizations. Indices: ν = 1.5, λ = 1 mm. (a) Cylinders randomly placed in a circular box of diameter 2R = 28λ; angle of incidence, α = 90°. (b) Cylinders randomly placed on the x axis. The distance between the centers of two consecutive cylinders is a uniform random distribution between 1.2 and 4.8λ. Angle of incidence, α = 110°.

Fig. 9
Fig. 9

Scattering by a set of 20 cylinders of diameter λ, randomly placed in a circular box of diameter 2R = 28λ, for p polarization, α = 90°, and λ = 30 mm. The results are averaged over 1000 realizations. (a) Scheme of the cylinders; (b), (c), (d) scattered intensity D(θ) corresponding to cylinders of indices ν = 1.5, ν = 1.1, and ν = 1.05, respectively.

Fig. 10
Fig. 10

Scattered intensity of a single perfectly conducting square cylinder of side 21 mm (one side lying on the x axis) illuminated for s polarization, α = 90°, and λ = 30 mm. The arrow indicates the backscattering direction.

Fig. 11
Fig. 11

Same as for Fig. 10 but with the square cylinder rotated 45° with respect to the z axis.

Fig. 12
Fig. 12

Scattered intensity produced by a set of 20 square cylinders similar to the cylinder of Fig. 10 and randomly placed in a circular box of diameter 2R = 28λ. The results are averaged over 1000 realizations.

Fig. 13
Fig. 13

Same as for Fig. 12 but with square cylinders similar to the cylinder of Fig. 11.

Fig. 14
Fig. 14

Homogenization of a set of five cylinders.

Fig. 15
Fig. 15

Homogenization of two sets of five cylinders. Curves are defined in text.

Equations (100)

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E i = E i u z = exp [ - i k ( x cos α + y sin α ) ] u z .
E S = E - E i .
2 E + k ˜ 2 ( M ) E = 0 ,
k ˜ 2 ( M ) = k 2 ɛ ˜ ( M ) = { k 2 ɛ j if M C j ( j = 1 , 2 , N ) k 2 if M C j ( j = 1 , 2 , N ) ,
2 E + k 2 E = [ k 2 - k ˜ 2 ( M ) ] E ,
2 E i + k 2 E i = 0 ,
2 E S + k 2 E S = [ k 2 - k ˜ 2 ( M ) ] E ,
E S ( P ) = - i 4 k 2 H 0 ( 1 ) ( k P M ) [ 1 - ɛ ˜ ( M ) ] E ( M ) d x d y .
E S ( P ) = j = 1 , 2 , , N i k 2 ( ɛ j - 1 ) 4 C j H 0 ( 1 ) ( k P M ) E ( M ) d x d y H 0 ( 1 ) ( k P M ) E ( M ) d x d y .
E S = j = 1 , 2 , , N E j S ,
E j S ( P ) = i k 2 ( ɛ j - 1 ) 4 C j H 0 ( 1 ) ( k P M ) E ( M ) d x d y .
H 0 ( 1 ) ( k P M ) = m = - + exp [ - i m θ j ( M ) ] × J m [ k r j ( M ) ] H m ( 1 ) [ k r j ( P ) ] exp [ i m θ j ( P ) ] ,
E j S ( P ) = m = - + b j , m H m ( 1 ) [ k r j ( P ) ] exp [ i m θ j ( P ) ] ,
b j , m = i k 2 ( ɛ j - 1 ) 4 C j exp [ - i m θ j ( M ) ] × J m [ k r j ( M ) ] E ( M ) d x d y .
E S ( P ) = j = 1 , 2 , , N m = - + b j , m H m ( 1 ) [ k r j ( P ) ] exp [ i m θ j ( P ) ] .
H m ( 1 ) [ k r j ( P ) ] exp [ ( i m θ j ( P ) ] = q = - + exp [ i ( m - q ) θ l j ] H q - m ( 1 ) ( k r l j ) J q [ k r l ( P ) ] exp [ i q θ l ( P ) ]
E j s ( P ) = m = - + b j , m q = - + exp [ i ( m - q ) θ l j ] × H q - m ( 1 ) ( k r l j ) J q [ k r l ( P ) ] exp [ i q θ l ( P ) ] .
E i ( P ) = exp ( i k · OP ) = exp [ i k · ( O O l + O l P ) ] = exp [ - i k r l cos ( α - θ l ) ] × exp { - i k r l ( P ) cos [ α - θ l ( P ) ] } ,
exp ( i z cos u ) = n = - + ( i ) n J n ( z ) exp ( i n u ) ,
E i ( P ) = exp [ - i k r l cos ( α - θ l ) ] n = - , + ( - i ) n exp ( - i n α ) × J n [ k r l ( P ) ] exp [ i n θ l ( P ) ] .
E ( P ) = m = - , + a l , m J m [ k r l ( P ) ] exp [ i m θ l ( P ) ] + m = - , + b l , m H m ( 1 ) [ k r l ( P ) ] exp [ i m θ l ( P ) ] ,
a l , m = ( - i ) m exp [ - i k r l cos ( α - θ l ) - i m α ] + j 1 q = - , + b j , q exp [ i ( q - m ) θ l j ] H m - q ( 1 ) ( k r l j )
a ^ l = Q l + j l T l , j b ^ j ,
Q l , m = ( - i ) m exp [ - i k r l cos ( a - θ l ) - i m α ]
T l , j , m , q = exp [ i ( q - m ) θ l j ] H m - q ( 1 ) ( k r l j ) .
b ^ l = S l a ^ l ,
b ^ l - j l S l T l , j b ^ j = S l Q l .
[ I - S 1 T 1 , 2 - S 1 T 1 , 3 - S 2 T 2 , 1 I - S 2 T 2 , 3 - S 3 T 3 , 1 - S 3 T 3 , 2 I ] [ b ^ 1 b ^ 2 b ^ 3 ] = [ S 1 Q 1 S 2 Q 2 S 3 Q 3 ] ,
H m ( 1 ) [ k r j ( P ) ] exp [ i m θ j ( P ) ] = q = - , + exp [ i ( m - q ) θ j ] × J q - m ( k r j ) H q ( 1 ) ( k r ) exp ( i q θ ) ,
E S ( P ) = q = - , + b q H q ( 1 ) ( k r ) exp ( i q θ ) ,
b q = j = 1 , N m = - + b j , m exp [ i ( m - q ) θ j ] J q - m ( k r j ) .
H q ( 1 ) ( k r ) 2 π k r exp [ i ( k r - q π 2 - π 4 ) ] ,
E S ( P ) g ( θ ) exp ( i k r ) r ,
g ( θ ) = 2 π k exp ( - i π 4 ) × q = - , + b q exp ( - i q π 2 ) exp ( i q θ ) .
D ( θ ) = 2 π g ( θ ) 2 ;
0 2 π g ( θ ) 2 d θ + 2 λ Re [ exp ( i π 4 ) g ( α + π ) ] = 0.
E S ( P ) = - i k 2 4 C H 0 ( 1 ) ( k P M ) [ 1 - ɛ ˜ ( M ) ] E ( M ) d x d y .
P M = P O + q ( P , M ) ,
H 0 ( 1 ) ( k P M ) = H 0 ( 1 ) [ k P O + k q ( P , M ) ] H 0 ( 1 ) ( k P O ) - k q ( P , M ) H 1 ( 1 ) ( k P O ) .
E S ( P ) = - i 4 k 2 ( I 1 + I 2 ) ,
I 1 = H 0 ( 1 ) ( k P O ) C [ 1 - ɛ ˜ ( M ) ] E ( M ) d x d y ,
I 2 = H 1 ( 1 ) ( k P O ) C k q ( P , M ) [ 1 - ɛ ˜ ( M ) ] E ( M ) d x d y .
E S ( P ) A k 2 log ( k ) + k 2 B ( P ) ,
2 H + k ˜ 2 H = [ d H / d n ] δ S ,
[ d H / d n ] δ S , φ = S [ d H / d n ] ( M ) φ ( M ) d l ,
2 H S + k 2 H S = [ k 2 - k ˜ 2 ( M ) ] H + [ d H / d n ] δ S ,
H S ( P ) = - i 4 k 2 C H 0 ( 1 ) ( k P M ) [ 1 - ɛ ˜ ( M ) ] × H ( M ) d x d y - i 4 S H 0 ( 1 ) ( k P M ) [ d H d n ] ( M ) d l H 0 ( 1 ) ( k P M ) [ 1 - ɛ ˜ ( M ) ] × H ( M ) d x d y - i 4 S H 0 ( 1 ) ( k P M ) [ d H d n ] ( M ) d l .
H S ( P ) = - i 4 k 2 H 0 ( 1 ) ( k P O ) C [ 1 - ɛ ˜ ( M ) ] H ( M ) d x d y - i 4 H 0 ( 1 ) ( k P O ) S [ d H d n ] d l + i 4 k 3 H 1 ( 1 ) ( k P O ) C q ( P , M ) × [ 1 - ɛ ˜ ( M ) ] H ( M ) d x d y + i 4 k H 1 ( 1 ) ( k P O ) S q ( P , M ) [ d H d n ] d l .
C k 2 [ 1 - ɛ ˜ ( M ) ] H ( M ) d x d y + S [ d H d n ] d l = 0.
div ( grad H ) + k 2 ɛ j H = 0 ,
S j d H d n - d l + k 2 C j ɛ j H d x d y = 0 ,
d H d n - = ɛ j d H d n + ,
S j d H d n + d l + k 2 C j H d x d y = 0 ,
S j [ d H d n ( M ) ] d l + k 2 C j ( 1 - ɛ j ) H ( M ) d x d y = 0.
q - O P · O M O P
H S - i 4 O P O P · V H 1 ( 1 ) ( k O P ) ,
V = C k 3 [ 1 - ɛ ˜ ( M ) ] H ( M ) O M d x d y + S k [ d H d n ] O M d l .
H S ( P ) = m = - + b m H m ( 1 ) ( k r ) exp ( i m θ ) ,
b ± 1 = i 8 ( V x i V y ) .
E j S ( P ) b j , 0 H 0 ( 1 ) [ k r j ( P ) ] .
E i ( P ) exp [ - i k r cos ( α - θ l ) ] .
E j S ( P ) b j , 0 H 0 ( 1 ) ( k r j l ) .
E ( P ) = exp [ - i k r l cos ( α - θ l ) ] + j l b j , 0 H 0 ( 1 ) ( k r l j ) + b l , 0 H 0 ( 1 ) [ k r l ( P ) ]
a l , 0 = Q l , 0 + j l T l , j , 0 , 0 b j , 0 ,
b l , 0 = S l , 0 , 0 a l , 0 .
b l , 0 - j 1 S l , 0 , 0 T l , j , 0 , 0 b j , 0 = S l , 0 , 0 Q l , 0 .
[ 1 - S 1 , 0 , 0 H 0 ( 1 ) ( k r 1 2 ) - S 1 , 0 , 0 H 0 ( 1 ) ( k r 1 3 ) - S 2 , 0 , 0 H 0 ( 1 ) ( k r 2 1 ) 1 ] [ b 1 , 0 b 2 , 0 ] = [ S 1 , 0 , 0 exp [ - i k r 1 cos ( α - θ 1 ) ] S 2 , 0 , 0 exp [ - i k r 2 cos ( α - θ 2 ) ] ] .
b m = j = 1 , N b j , 0 exp ( - i m θ j ) J m ( k r j ) .
b 0 = j = 1 , N b j , 0 .
E S ( P ) = b 0 H 0 ( 1 ) ( k r ) = ( j = 1 , N b j , 0 ) H 0 ( 1 ) ( k r ) .
E S ( P ) = b ˜ 0 H 0 ( 1 ) ( k r ) ,
( ɛ ˜ - 1 ) R 2 = ( ɛ - 1 ) N ρ 2 .
b ˜ 0 = J 1 ( k R ˜ ) J 0 ( k ν ˜ R ˜ ) - ν ˜ J 0 ( k R ˜ ) J 1 ( k ν ˜ R ˜ ) ν ˜ H 0 ( 1 ) ( k R ˜ ) J 1 ( k ν ˜ R ˜ ) - H 1 ( 1 ) ( k R ˜ ) J 0 ( k ν ˜ R ˜ ) .
b ˜ 0 = 1 - 1 + i X ˜ ,
X ˜ = ν ˜ Y 0 ( k R ˜ ) J 1 ( k ν ˜ R ˜ ) - Y 1 ( k R ˜ ) J 0 ( k ν ˜ R ˜ ) J 1 ( k R ˜ ) J 0 ( k ν ˜ R ˜ ) - ν ˜ J 0 ( k R ˜ ) J 1 ( k ν ˜ R ˜ ) .
X ˜ = - 4 π ( ɛ ˜ - 1 ) ( k R ˜ ) 2 - 2 π log ( k R ˜ ) + 1 - 4 γ + 4 log 2 2 π - 1 π ( ɛ ˜ - 1 ) + O ( k R ˜ ) 2 ,
Q l , 0 = exp [ - i k r l cos ( α - θ l ) ] 1 ,
b l , 0 - j l S T l , j , 0 , 0 b j , 0 = S .
N b l , 0 - S b l , 0 l = 1 , N j l T l , j , 0 , 0 - S l = 1 , N j l δ j T l , j , 0 , 0 = N S ,
b l , 0 = S 1 - D 1 1 - D 2 ,
D 1 = 1 N l = 1 , N j l δ j T l , j , 0 , 0 , D 2 = S N l = 1 , N j l T l , j , 0 , 0 .
T l , j , 0 , 0 = H 0 ( 1 ) ( k r l j ) 2 i π log ( k r l j ) + [ 1 + 2 i π ( γ - log 2 ) ] ,
D 1 D 2 .
b l , 0 = S 1 - S N l = 1 , N j l T l , j , 0 , 0 .
1 N l j l T l , j , 0 , 0 = ( N - 1 ) [ 1 + 2 i π ( γ - log 2 ) + 2 i π log ( k ) ] ,
= ( i j 1 r i j ) 1 / N ( N - 1 ) .
b 0 = 1 / { - 1 + 1 S N + 1 N - 2 i π ( N - 1 ) N × [ γ + log ( k 2 ) ] } = 1 - 1 + i X ,
1 S = - i + i [ - 4 π ( ɛ - 1 ) ( k ρ ) 2 - 2 π log ( k ρ ) + 1 - 4 γ + 4 log 2 2 π - 1 π ( ɛ - 1 ) ] ,
X = 1 N π [ - 4 ( ɛ - 1 ) ( k ρ ) 2 - 2 log ( k ρ ) + 1 - 4 γ + 4 log 2 2 - 1 ( ɛ - 1 ) ] - 2 ( N - 1 ) N π [ γ + log ( k 2 ) ] .
1 ɛ ˜ - 1 [ 1 4 + 1 ( k R ˜ ) 2 ] = 1 N ( ɛ - 1 ) [ 1 4 + 1 ( k ρ ) 2 ] + 1 2 log [ ( k ρ ) 1 / N ( k / 2 ) 1 - 1 / N k R ˜ ] + 1 + 4 log 2 8 ( 1 - 1 N ) .
ɛ ˜ - 1 = ( ɛ - 1 ) N ρ 2 R ˜ 2 1 / 1 + ( ɛ - 1 ) N ( k ρ ) 2 2 × [ log ( ρ 1 / N 1 - 1 / N R ˜ ) + 1 4 ( 1 - 1 N ) ] .
ɛ ˜ - 1 = ( ɛ - 1 ) N ρ 2 / R ˜ 2 ,
b ˜ 0 = 1 - 1 + i X ˜ 1 - 1 + i [ - 4 / π ( ɛ ˜ - 1 ) ( k R ˜ ) 2 ] i ( ɛ ˜ - 1 ) k 2 π R ˜ 2 4 i ( ɛ - 1 ) k 2 N π ρ 2 4 .
ɛ ˜ - 1 = ( ɛ - 1 ) ( N ρ 2 / R ˜ 2 ) 1 / ( 1 + C ) ,
C = ( ɛ - 1 ) N ( k ρ 2 ) 2 log ( ρ 1 / N R 1 - 1 / N R ) .
R ˜ = R 0 = ρ 1 / N R 1 - 1 / N ,
ɛ ˜ - 1 = ( ɛ - 1 ) ( N ρ 2 / R ˜ 2 ) .
( ɛ ˜ - 1 ) R ˜ 2 = ( ɛ - 1 ) N ρ 2 ,
σ ˜ = σ ( N ρ 2 / R ˜ 2 ) ,
ɛ ˜ = 1 + i σ ˜ / ɛ 0 ω .

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