Abstract

We present a differential theory for solving Maxwell equations in cylindrical coordinates, projecting them onto a Fourier–Bessel basis. Numerical calculations require the truncation of that basis, so that correct rules of factorization have to be used. The convergence of the method is studied for different cases of dielectric and metallic cylinders of finite length. Applications of such a method are presented, with a special emphasis on the near-field map inside a hole pierced in a plane metallic film.

© 2005 Optical Society of America

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References

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  1. P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 4.
  2. M. Nevière, E. Popov, Light Propagation in Periodic Medias: Differential Theory and Design (Marcel Dekker, New York, 2003).
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  5. J. D. Jackson, Classical Electromagnetism, 3rd ed. (Wiley, New York, 1998), Chap. 3.
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    [CrossRef]
  7. E. Popov, M. Nevière, N. Bonod, “Factorization of products of discontinuous functions applied to Fourier–Bessel basis,” J. Opt. Soc. Am. A 21, 46–52 (2004).
    [CrossRef]
  8. L. W. Davis, “Theory of electromagnetism beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  9. M. Abramovitz, I. A. Stegun, eds., Handbook of Mathematical Functions [9th printing (1970).] National Bureau of Standards Applied Mathematics Series 55 (U.S. Government Printing Office, Washington, D.C., 1964).
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    [CrossRef]
  11. A. Degiron, H. J. Lezec, N. Yamamoto, T. W. Ebbesen, “Optical transmission properties of a single subwavelength aperture in a real metal,” Opt. Commun. 239, 61–66 (2004).
    [CrossRef]

2004 (3)

2001 (2)

1996 (1)

1979 (1)

L. W. Davis, “Theory of electromagnetism beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Barchiesi, D.

Bonod, N.

Chernov, B.

Davis, L. W.

L. W. Davis, “Theory of electromagnetism beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Degiron, A.

A. Degiron, H. J. Lezec, N. Yamamoto, T. W. Ebbesen, “Optical transmission properties of a single subwavelength aperture in a real metal,” Opt. Commun. 239, 61–66 (2004).
[CrossRef]

Ebbesen, T. W.

A. Degiron, H. J. Lezec, N. Yamamoto, T. W. Ebbesen, “Optical transmission properties of a single subwavelength aperture in a real metal,” Opt. Commun. 239, 61–66 (2004).
[CrossRef]

Felbacq, D.

Guizal, B.

Jackson, J. D.

J. D. Jackson, Classical Electromagnetism, 3rd ed. (Wiley, New York, 1998), Chap. 3.

Lezec, H. J.

A. Degiron, H. J. Lezec, N. Yamamoto, T. W. Ebbesen, “Optical transmission properties of a single subwavelength aperture in a real metal,” Opt. Commun. 239, 61–66 (2004).
[CrossRef]

Li, L.

Nevière, M.

Popov, E.

Vincent, P.

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 4.

Yamamoto, N.

A. Degiron, H. J. Lezec, N. Yamamoto, T. W. Ebbesen, “Optical transmission properties of a single subwavelength aperture in a real metal,” Opt. Commun. 239, 61–66 (2004).
[CrossRef]

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

Opt. Commun. (1)

A. Degiron, H. J. Lezec, N. Yamamoto, T. W. Ebbesen, “Optical transmission properties of a single subwavelength aperture in a real metal,” Opt. Commun. 239, 61–66 (2004).
[CrossRef]

Phys. Rev. A (1)

L. W. Davis, “Theory of electromagnetism beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Other (4)

M. Abramovitz, I. A. Stegun, eds., Handbook of Mathematical Functions [9th printing (1970).] National Bureau of Standards Applied Mathematics Series 55 (U.S. Government Printing Office, Washington, D.C., 1964).

J. D. Jackson, Classical Electromagnetism, 3rd ed. (Wiley, New York, 1998), Chap. 3.

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 4.

M. Nevière, E. Popov, Light Propagation in Periodic Medias: Differential Theory and Design (Marcel Dekker, New York, 2003).

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Figures (8)

Fig. 1
Fig. 1

Representation of the spatial cylindrical basis added to the Cartesian one. In our problem the angle θ 0 is taken equal to 0. The angle θ shows the possibility of conical incidence. The projection of the wave vector onto the Oxy plane is equal to k ir .

Fig. 2
Fig. 2

Representations of the near field. The superstrate is glass with a relative dielectric permittivity r = 2.28 . The substrate is air. The device is illuminated in normal incidence with wavelength of 647 nm. The thickness of the cylinder is 100 nm, and its diameter is taken equal to the wavelength. (a) Near-electric-field map in the Oxz plane, (b) Poynting vector, (c) shows electric-field |E| map in the Oxy plane. Δ k r = 0.001   nm - 1 , k r , Max / k 0 = 6.001 , N = 4 .

Fig. 3
Fig. 3

Convergence of the near-field amplitude as a function of the truncation parameter k r , Max , normalized with respect to the incident wave number k 0 . The results represent the value of | E r | at a point on the z axis situated 15 nm below the hole opening. Solid curve, FNF method; dashed curve, without FNF. The working parameters ( Δ k r and m) are represented in the figure. R = 250   nm .

Fig. 4
Fig. 4

Near-field distribution of | E r |   ( θ = 0 ) in the x direction for different working parameters, calculated 15 nm below the hole opening. Δ k r = 0.0003   nm - 1 . (a) Max = 100 , k r , max / k 0 = 2.387 . Solid curve, FNF; dashed curve, without FNF. (b) dotted curve, Max = 200 ; dashed curve, Max = 400 ; solid curve, Max = 800 . FNF method. (c) Same as (b) but without FNF.

Fig. 5
Fig. 5

Radial distribution of | E z | 15 nm below the hole opening (FNF method).

Fig. 6
Fig. 6

Same as Fig. 3 but for R = 25   nm . Δ k r = 0.0003   nm - 1 .

Fig. 7
Fig. 7

| E r | as a function of x for θ = 0 calculated with Δ k r = 0.0003   nm - 1 15 nm below the hole with R = 25   nm . (a) Max = 400 (FNF), (b) Max = 800 (FNF), (c) Max = 800 without FNF.

Fig. 8
Fig. 8

Spectral dependence of the maximum of | E | 2 in the plane z = - 40   nm , R = 135   nm . The dispersion of the permittivity is not taken into account.

Equations (87)

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curl   E ( r ) = i ω μ 0 H ( r ) ,
curl   H ( r ) = - i ω ( r ) E ( r ) .
curl n = - E n ( r ,   z ) exp ( in   θ ) = i ω μ 0 n = - H n ( r ,   z ) × exp ( in   θ ) ,
curl n = - H n ( r ,   z ) exp ( in   θ ) = - i ω ( r ,   z ) n = - E n ( r ,   z ) × exp ( in   θ ) .
n = - [ curl   E n ( r ,   z ) + in θ ˆ     E n ( r ,   z ) ] exp ( in   θ )
= n = - i ω μ 0 H n ( r ,   z ) exp ( in   θ ) ,
n = - [ curl   H n ( r ,   z ) + in θ ˆ     H n ( r ,   z ) ] exp ( in   θ )
= - n = - i ω ( r ,   z ) E n ( r ,   z ) exp ( in   θ ) .
Δ E r , n - E r , n r 2 + ω 2 μ 0 ( r ) E r , n - 2 in r 2   E θ , n = 0 ,
Δ E θ , n - E θ , n r 2 + ω 2 μ 0 ( r ) E θ , n + 2 in r 2   E r , n = 0 .
E θ , n + iE r , n = E n + ,
E θ , n - iE r , n = E n - ,
Δ E n + - E n + r 2 + ω 2 μ 0 ( r ) E n + + 2 n r 2   E n + = 0 ,
Δ E n - - E n - r 2 + ω 2 μ 0 ( r ) E n - - 2 n r 2   E n - = 0 .
E n + = E θ , n + iE r , n = k r = 0 2 c n E ( k r ,   z ) J n - 1 ( k r r ) k r d k r ,
E n - = E θ , n - iE r , n = k r = 0 2 b n E ( k r ,   z ) J n + 1 ( k r r ) k r d k r ,
k r = 0 k r d k r m = 1 Max k m Δ k m ,
n = - + n = - N + N .
E r ( r ,   z ) = i n = - N N m = 1 Max k m Δ k m [ b n , m E ( z ) J n + 1 ( k m r ) - c n , m E ( z ) J n - 1 ( k m r ) ] exp ( in   θ ) ,
E θ ( r ,   z ) = n = - N N m = 1 Max k m Δ k m [ b n , m E ( z ) J n + 1 ( k m r ) + c n , m E ( z ) J n - 1 ( k m r ) ] exp ( in   θ ) .
H r ( r ,   z ) = i n = - N N m = 1 Max k m Δ k m [ b n , m H ( z ) J n + 1 ( k m r ) - c n , m H ( z ) J n - 1 ( k m r ) ] exp ( in   θ ) ,
H θ ( r ,   z ) = n = - N N m = 1 Max k m Δ k m [ b n , m H ( z ) J n + 1 ( k m r ) + c n , m H ( z ) J n - 1 ( k m r ) ] exp ( in   θ ) .
i ω μ 0 H z , n = m = 1 Max ( b n , m E - c n , m E ) J n ( k m r ) k m 2 Δ k m ,
- i ω ( r ) E z , n = m = 1 Max ( b n , m H - c n , m H ) J n ( k m r ) k m 2 Δ k m .
E z , n = m = 1 Max k m Δ k m E z , n , m J n ( k m r ) .
k m [ b n , m H - c n , m H ] = - i ω m = 1 Max k m Δ k m E z , n , m × r = 0 ( r ) J n ( k m r ) J n ( k m r ) r d r .
r = 0 ( r ) J n ( k m r ) J n ( k m r ) r d r [ ] m , m n , n ,
m , m [ 1 ,   Max ] .
[ ] m , m n , n ˜ = [ ] m , m n , n k m Δ k m and
[ ] m , m n ˜ , n = [ ] m , m n , n k m Δ k m
k m [ b n , m H - c n , m H ] = - i ω m = 1 Max E z , n , m [ ] m , m n , n ˜ .
E z , n = m = 1 Max   - k m i ω   Δ k m J n ( k m r ) m = 1 Max ( [ ] m , m n , n ˜ ) - 1 k m × [ b n , m H - c n , m H ] .
E i = E 0 n = - i n J n ( k i , r r ) exp [ in ( θ - θ 0 ) ] exp ( - ik i , z z ) .
E 0 = E 0 x x ˆ + E 0 y y ˆ + E 0 z z ˆ .
E 0 = E 0 x ( r ˆ cos θ - θ ˆ sin θ ) + E 0 y ( r ˆ sin θ + θ ˆ cos θ ) + E 0 z z ˆ ,
E 0 = r ˆ E 0 x 2 + E 0 y 2 i exp ( i θ ) + E 0 x 2 - E 0 y 2 i exp ( - i θ ) + θ ˆ   - E 0 x 2 i + E 0 y 2 exp ( i θ ) + E 0 x 2 i + E 0 y 2 exp ( - i θ ) + z ˆ E 0 z .
E r i = n = - N N E 0 x 2 + E 0 y 2 i i n exp ( - in   θ 0 ) × J n ( k i , r r ) exp [ i ( n + 1 ) θ ] exp ( - ik i , z z ) + n = - N N E 0 x 2 - E 0 y 2 i i n exp ( - in   θ 0 ) × J n ( k i , r r ) exp [ i ( n - 1 ) θ ] exp ( - ik i , z z ) .
E r i = n = - N + 1 N + 1 i E 0 x 2 i - E 0 y 2 i n - 1 exp [ - i ( n - 1 ) θ 0 ] × J n - 1 ( k i , r r ) exp ( in   θ ) exp ( - ik i , z z ) + n = - N - 1 N - 1 i E 0 x 2 i + E 0 y 2 i n + 1 exp ( - i ( n + 1 ) × θ 0 ) J n + 1 ( k i , r r ) exp ( in   θ ) exp ( - ik i , z z ) ,
E θ i = n = - N + 1 N + 1 - E 0 x 2 i + E 0 y 2 i n - 1 exp ( - i ( n - 1 ) θ 0 ) × J n - 1 ( k i , r r ) exp ( in   θ ) exp ( - ik i , z z ) + n = - N - 1 N - 1 E 0 x 2 i + E 0 y 2 i n + 1 exp ( - i ( n + 1 ) θ 0 ) × J n + 1 ( k i , r r ) exp ( in   θ ) exp ( - ik i , z z ) .
k i , r = k m i , k i , z = k z , m i ,
b ˜ n , m i E = k m i b n , m i E = E 0 x 2 i + E 0 y 2   i n + 1 Δ k m i × exp [ - i ( n + 1 ) θ 0 ] exp ( - ik z , m i z ) ,
c ˜ n , m i E = k m i c n , m i E = - E 0 x 2 i + E 0 y 2   i n - 1 Δ k m i × exp [ - i ( n - 1 ) θ 0 ] exp ( - ik z , m i z ) .
E x i = E 0 Q exp ( - Qh 2 ) exp ( - ik z , m i z ) ,
E z i = - E 0 2 i   Q 2 x l exp ( - Qh 2 ) exp ( - ik z , m i z ) = - 2 i   Qx l   E x i ,
h 2 = ( x 2 + y 2 ) / w 0 2 ,
Q = i / ( 2 z / l + i ) ,
E x i = E 0 exp ( - r 2 / w 0 2 ) exp ( - ik z , m i z ) ,
E z i = - E 0   ix z R exp ( - r 2 / w 0 2 ) exp ( - ik z , m i z ) ,
E i = E x i ( r ˆ cos θ - θ ˆ sin θ ) + E z i z ˆ ,
E i = E 0 2 exp ( - r 2 / w 0 2 ) exp ( - ik z , m i z ) ( exp ( i θ ) + exp ( - i θ ) ) r ˆ - E 0 2 i exp ( - r 2 / w 0 2 ) × exp ( - ik z , m i z ) [ exp ( i θ ) - exp ( - i θ ) ] × θ ˆ - i   E 0 x z R exp ( - r 2 / w 0 2 ) exp ( - ik z , m i z ) × [ exp ( i θ ) + exp ( - i θ ) ] z ˆ .
E θ i + iE r i = iE 0 exp ( - r 2 / w 0 2 ) exp ( - ik z , m i z ) exp ( i θ ) ,
E θ i - iE r i = - iE 0 exp ( - r 2 / w 0 2 ) exp ( - ik z , m i z ) exp ( - i θ ) .
iE 0 exp ( - r 2 / w 0 2 ) exp ( - ik z , m i z ) = 2 c 1 , m i E J 0 ( k m i r )
×   k m i Δ k m i ,
- iE 0 exp ( - r 2 / w 0 2 ) exp ( - ik z , m i z ) = 2 b - 1 , m i E J 0 ( k m i r )
×   k m i Δ k m i .
c 1 , m i E = iE 0 2 Δ k m i exp ( - ik z , m i z )   w 0 2 2 exp ( - k m i 2 w 0 2 / 4 ) = - b - 1 , m i E ,
d F n ( z ) d z = M n F n ( z ) , n [ - N ,   N ] ,
F n = b ˜ n , m E c ˜ n , m E b ˜ n , m H c ˜ n , m H .
d d z   ( E θ , n - iE r , n ) = m = 1 Max 2 k m Δ k m b n , m E J n + 1 ( k m r ) = in r   E z , n - i ω μ 0 H r , n - i   E z , n r + ω μ 0 H θ , n .
b ˜ n , m E = ω μ 0 b ˜ n , m H - k m 2 ω   m = 1 max ( [ ˜ ] m ˜ , m n ˜ , n ) - 1 k m × [ b ˜ n , m H - c ˜ n , m H ] .
d d z   ( H θ , n - iH r , n ) = 2 m = 1 k m Δ k m b n , m H J n + 1 ( k m r ) = 1 r   H z , n θ + i ω ( r ) × E r , n - i   H z , n r - ω ( r ) E θ , n .
2 b ˜ n , m H = k m 2 ω μ 0   ( b ˜ n , m E - c ˜ n , m E ) + i ω k m r = 0 ( r ) E r , n ( r ) J n + 1 ( k m r ) r d r - ω k m r = 0 ( r ) E θ , n ( r ) J n + 1 ( k m r ) r d r .
ik m r = 0 ( r ) E r , n ( r ) J n + 1 ( k m r ) r d r = - m = 1 Max m = 1 Max ( [ Ψ ] n - 1 ˜ , n + 1 ) m , m 1 n + 1 ˜ , n + 1 m , m - 1 × b ˜ n , m E + m = 1 Max 1 n - 1 ˜ , n - 1 m , m - 1 c ˜ n , m E ,
[ Ψ ] m , m n - 1 ˜ , n + 1 = k m Δ k m r = 0 J n - 1 ( k m r ) J n + 1 ( k m r ) r d r .
r = 0 k m ( r ) E θ , n J n + 1 ( k m r ) r d r = m = 1 Max [ ] m , m n + 1 ˜ , n + 1 b ˜ n , m E + m = 1 max [ ] m , m n + 1 ˜ , n - 1 c ˜ n , m E .
M n = M n , 11 M n , 12 M n , 21 M n , 22 ,
M n , 11 = 0 ,
M n , 12 = - 1 2 ω   K [ n ˜ , n ] - 1 K + ω μ 0 I - 1 2 ω   K [ n ˜ , n ] - 1 K - 1 2 ω   K [ n ˜ , n ] - 1 K 1 2 ω   K [ n ˜ , n ] - 1 K - ω μ 0 I ,
M n , 21 = K 2 2 ω μ 0   I - ω 2   1 n + 1 ˜ , n + 1 - 1 + [ n + 1 ˜ , n + 1 ] - K 2 2 ω μ 0   I + ω 2   [ Ψ ] n + 1 ˜ , n - 1 1 n - 1 ˜ , n - 1 - 1 - [ n + 1 ˜ , n - 1 ] K 2 2 ω μ 0   I - ω 2   [ Ψ ] n - 1 ˜ , n + 1 1 n + 1 ˜ , n + 1 - 1 - [ n - 1 ˜ , n + 1 ] - K 2 2 ω μ 0   I + ω 2   1 n - 1 ˜ , n - 1 - 1 + [ n - 1 ˜ , n - 1 ] ,
M n , 22 = 0 ,
1 n + 1 ˜ , n + 1 - 1 [ n + 1 ˜ , n + 1 ] , 1 n - 1 ˜ , n - 1 - 1 [ n - 1 ˜ , n - 1 ] .
r = 0 k m ( r ) E θ , n J n + 1 ( k m r ) r d r .
m = 1 Max b ˜ n , m E k m Δ k m r = 0 ( r ) J n + 1 ( k m r ) J n + 1 ( k m r ) r d r
+ m = 1 Max c ˜ n , m E k m Δ k m
× r = 0 ( r ) J n - 1 ( k m r ) J n + 1 ( k m r ) r d r
= m = 1 Max b ˜ n , m E k m Δ k m [ ] m , m n + 1 , n + 1
+ m = 1 Max c ˜ n , m E k m Δ k m [ ] m , m n + 1 , n - 1 .
r = 0 k m ( r ) E r , n J n - 1 ( k m r ) r d r = D r , n , m .
D r , n = m = 1 Max D r , n , m J n - 1 ( k m r ) k m Δ k m .
E r , n = m = 1 Max   1 ( r )   D r , n , m J n - 1 ( k m r ) k m Δ k m .
E r , n , m = r = 0 E r , n J n - 1 ( k m r ) r d r = m = 1 Max D r , n , m k m Δ k m r = 0   1 ( r )   J n - 1 ( k m r ) J n - 1 ( k m r ) rdr ,
k m E r , n , m = m = 1 Max 1 m , m n - 1 ˜ , n - 1 k m D r , n , m .
k m D r , n , m = m = 1 Max 1 n - 1 ˜ , n - 1 m , m - 1 k m E r , n , m .
ik m E r , n , m = k m m = 1 Max ( - b ˜ n , m E Δ k m [ Ψ ] m , m n + 1 ˜ , n - 1 ) + c ˜ n , m E .
ik m D r , n , m = - m = 1 Max m = 0 Max 1 n - 1 ˜ , n - 1 m , m - 1 × ( [ Ψ ] n - 1 ˜ , n + 1 ) m , m b ˜ n , m E + m = 1 Max 1 n - 1 ˜ , n - 1 m , m - 1 c ˜ n , m E ,
ik m D r , n , m = - m = 1 Max m = 0 Max ( [ Ψ ] n - 1 ˜ , n + 1 ) m , m × 1 n + 1 ˜ , n + 1 m , m - 1 b ˜ n , m E + m = 1 Max 1 n - 1 ˜ , n - 1 m , m - 1 c ˜ n , m E .

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