Abstract

Using nonparaxial vector diffraction theory derived using the Hertz vector formalism, integral expressions for the electric and magnetic field components of light within and beyond an apertured plane are obtained for an incident plane wave. For linearly polarized light incident on a circular aperture, the integrals for the field components and for the Poynting vector are numerically evaluated. By further two-dimensional integration of a Poynting vector component, the total transmission of a circular aperture is determined as a function of the aperture radius to wavelength ratio. The validity of using Kirchhoff boundary conditions in the aperture plane is also examined in detail.

© 2005 Optical Society of America

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References

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  1. G. R. Kirchhoff, �??Zur Theorie der Lichtstrahlen,�?? Ann. Phys. (Leipzig) 18, 663-695 (1883).
    [CrossRef]
  2. A. Sommerfeld, �??Zur mathematischen Theorie der Beugungsercheinungen,�?? Nachr. Kgl.Wiss. Göttingen 4, 338-342 (1894).
  3. Lord Rayleigh, �??On the passage of waves through apertures in plane screens, and allied problems,�?? Philos. Mag. 43, 259-272 (1897).
  4. O. Mitrofanov, M. Lee, J. W. P. Hsu, L. N. Pfeifer, K. W. West, J. D. Wynn and J. F. Federici, �??Terahertz pulse propagation through small apertures,�?? App. Phys. Lett. 79, 907�??909 (2001).
    [CrossRef]
  5. B. Lü and K. Duan, �??Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,�?? Opt. Lett. 28, 2440�??2442 (2003).
    [CrossRef] [PubMed]
  6. K. Duan and B. Lü, �??Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture,�?? J. Opt. Sec. Am. A 21, 1613�??1620 (2004).
    [CrossRef]
  7. G. D. Gillen and S. Guha, �??Modeling and propagation of near-field diffraction patterns: A more complete approach,�?? Am. J. Phys. 72, 1195�??1201 (2004).
    [CrossRef]
  8. W. Freude and G. K. Grau, �??Rayleigh-Sommerfeld and Helmholtz-Kirchhoff integrals: application to the scalar and vectorial theory of wave propagation and diffraction,�?? J. Lightwave Technol. 13, 24�??32, (1995).
    [CrossRef]
  9. J. A. Stratton and L.J. Chu, �??Diffraction theory of electromagnetic waves,�?? Phys. Rev. 56, 99�??107, (1939).
    [CrossRef]
  10. H. A. Bethe, �??Theory of diffraction by small holes,�?? Phys. Rev. 66, 163�??182 (1944).
    [CrossRef]
  11. R. K. Luneberg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964).
  12. G. Bekefi, �??Diffraction of electromagnetic waves by an aperture in a large screen,�?? J. App. Phys. 24, 1123�??1130 (1953).
    [CrossRef]
  13. C. L. Andrews, �??Diffraction pattern in a circular aperture measured in the microwave region,�?? J. Appl. Phys. 21, 761�??767 (1950).
    [CrossRef]
  14. M. J. Ehrlich, S. Silver and G. Held,�??Studies of the Diffraction of Electromagnetic waves by circular apertures and complementary obstacles: the near-zone field,�?? J. Appl. Phys. 26, 336�??345 (1955).
    [CrossRef]
  15. M. Born and E. Wolf Principles of Optics (Cambridge University Press, Cambridge, 2003.)
  16. A. Schoch, �??Betrachtungen über das Schallfeld einer Kolbenmembran,�??Akust. Z. 6, 318�??326 (1941).
  17. A. H. Carter and A. O.Williams, Jr., �??A New Expansion for the velocity potential of a piston source,�?? J. Accoust. Soc. Am. 23, 179�??184 (1951).
    [CrossRef]
  18. M. Mansuripur, A. R. Zakharian and J. V. Moloney, �??Interaction of light with subwavelength structures,�?? Opt. Photonics News, 56�??61 (March 2003).

Akust. Z. (1)

A. Schoch, �??Betrachtungen über das Schallfeld einer Kolbenmembran,�??Akust. Z. 6, 318�??326 (1941).

Am. J. Phys. (1)

G. D. Gillen and S. Guha, �??Modeling and propagation of near-field diffraction patterns: A more complete approach,�?? Am. J. Phys. 72, 1195�??1201 (2004).
[CrossRef]

Ann. Phys. (1)

G. R. Kirchhoff, �??Zur Theorie der Lichtstrahlen,�?? Ann. Phys. (Leipzig) 18, 663-695 (1883).
[CrossRef]

App. Phys. Lett. (1)

O. Mitrofanov, M. Lee, J. W. P. Hsu, L. N. Pfeifer, K. W. West, J. D. Wynn and J. F. Federici, �??Terahertz pulse propagation through small apertures,�?? App. Phys. Lett. 79, 907�??909 (2001).
[CrossRef]

J. Accoust. Soc. Am. (1)

A. H. Carter and A. O.Williams, Jr., �??A New Expansion for the velocity potential of a piston source,�?? J. Accoust. Soc. Am. 23, 179�??184 (1951).
[CrossRef]

J. App. Phys. (1)

G. Bekefi, �??Diffraction of electromagnetic waves by an aperture in a large screen,�?? J. App. Phys. 24, 1123�??1130 (1953).
[CrossRef]

J. Appl. Phys. (2)

C. L. Andrews, �??Diffraction pattern in a circular aperture measured in the microwave region,�?? J. Appl. Phys. 21, 761�??767 (1950).
[CrossRef]

M. J. Ehrlich, S. Silver and G. Held,�??Studies of the Diffraction of Electromagnetic waves by circular apertures and complementary obstacles: the near-zone field,�?? J. Appl. Phys. 26, 336�??345 (1955).
[CrossRef]

J. Lightwave Technol. (1)

W. Freude and G. K. Grau, �??Rayleigh-Sommerfeld and Helmholtz-Kirchhoff integrals: application to the scalar and vectorial theory of wave propagation and diffraction,�?? J. Lightwave Technol. 13, 24�??32, (1995).
[CrossRef]

J. Opt. Sec. Am. A (1)

K. Duan and B. Lü, �??Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture,�?? J. Opt. Sec. Am. A 21, 1613�??1620 (2004).
[CrossRef]

Nachr. Kgl.Wiss. Göttingen (1)

A. Sommerfeld, �??Zur mathematischen Theorie der Beugungsercheinungen,�?? Nachr. Kgl.Wiss. Göttingen 4, 338-342 (1894).

Opt. Lett. (1)

Opt. Photonics News (1)

M. Mansuripur, A. R. Zakharian and J. V. Moloney, �??Interaction of light with subwavelength structures,�?? Opt. Photonics News, 56�??61 (March 2003).

Philos. Mag. (1)

Lord Rayleigh, �??On the passage of waves through apertures in plane screens, and allied problems,�?? Philos. Mag. 43, 259-272 (1897).

Phys. Rev. (2)

J. A. Stratton and L.J. Chu, �??Diffraction theory of electromagnetic waves,�?? Phys. Rev. 56, 99�??107, (1939).
[CrossRef]

H. A. Bethe, �??Theory of diffraction by small holes,�?? Phys. Rev. 66, 163�??182 (1944).
[CrossRef]

Other (2)

R. K. Luneberg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964).

M. Born and E. Wolf Principles of Optics (Cambridge University Press, Cambridge, 2003.)

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Figures (18)

Fig. 1.
Fig. 1.

The real (solid lines) and the imaginary (dashed lines) x-components of the normalized electric field versus (a) the x-position and (b) the y-position in the aperture plane, calculated using the single integral HVDT for a/λ=5.

Fig. 2.
Fig. 2.

The real component, part (a), and the imaginary component, part (b), of the normalized electric field at the center of the aperture plane, (0,0,0), versus the aperture to wavelength ratio, a/λ.

Fig. 3.
Fig. 3.

The modulus square of the x-component of the electric field (red lines) and the z-component of the Poynting vector (black lines) versus (a) the x-position and (b) the y-position in the aperture plane, calculated using the single integral HVDT for a/λ=5.

Fig. 4.
Fig. 4.

Calculated modulus square of the x-component of the electric field (|Ex/Eo |2) versus x and y in the aperture plane using single integral HVDT, ( x a ) 2 + ( y a ) 2 < 1 and for a/λ=5.

Fig. 5.
Fig. 5.

Calculated z-component of the Poynting vector (Sz/So ) versus x and y in the aperture plane using single integral HVDT, ( x a ) 2 + ( y a ) 2 < 1 and for a/λ=5.

Fig. 6.
Fig. 6.

Calculated z-component of the Poynting vector (Sz/So ) versus x and y in the aperture plane using single integral HVDT,

Fig. 7.
Fig. 7.

Calculated z-component of the Poynting vector (Sz/So ) versus x and y in the aperture plane using single integral HVDT, ( x a ) 2 + ( y a ) 2 < 1 and for a/λ=2.

Fig. 8.
Fig. 8.

Calculated z-component of the Poynting vector (Sz/So ) versus x and y in the aperture plane using single integral HVDT, ( x a ) 2 + ( y a ) 2 < 1 and for a/λ=1.

Fig. 9.
Fig. 9.

Calculated z-component of the Poynting vector (Sz/So ) versus x and y in the aperture plane using single integral HVDT, ( x a ) 2 + ( y a ) 2 < 1 and for a/λ=5.

Fig. 10.
Fig. 10.

Calculated on-axis values for the modulus square of the x-component of the electric field (red line) using HVDT, the z-component of the Poynting vector (black line) using HVDT, and the z-component of the Poynting vector (blue line) using KVDT for: (a) a/λ=0.5, (b) a/λ=1, (c) a/λ=2.5 and (d) a/λ=5.

Fig. 11.
Fig. 11.

Calculated Sz/So for a/λ=0.5 with (a) z 1=10-4, (b) z 1=0.1, (c) z 1=1, and for a/λ=5 with (d) z 1=10-4, (e) z 1=0.1, and (f) z 1=1, using the single integral HVDT. The distance r 1 is either x or y normalized to the aperture radius, a.

Fig. 12.
Fig. 12.

Calculated z-component of the Poynting vector (Sz/So ) versus x and y for z 1=1/6π using the single integral HVDT, ( x a ) 2 + ( y a ) 2 < 1 and a/λ=5.

Fig. 13.
Fig. 13.

Calculated z-component of the Poynting vector (Sz/So ) versus x and y for z 1=1/4π using the single integral HVDT for a/π=5.

Fig. 14.
Fig. 14.

Calculated real, (a), and imaginary, (b), components of Ez/Eo for θ=0° (x-axis), 22.5°, 45°, 67.5° and 90° (y-axis), for a/λ=0.5 and z 1=0.05.

Fig. 15.
Fig. 15.

Calculated real, (a), and imaginary, (b), components of Ez/Eo for θ=0° (x-axis), 22.5°, 45°, 67.5° and 90° (y-axis), for a/λ=5 and z 1=0.05.

Fig. 16.
Fig. 16.

Calculated real, (a), and imaginary, (b), components of Ez/Eo along the x-axis for z 1=0.05, 0.1, 0.5 and 1, for a/λ=0.5.

Fig. 17.
Fig. 17.

Calculated real, (a), and imaginary, (b), components of Ez/Eo along the x-axis for z 1=0.05, 0.1, 0.5 and 1, for a/λ=5.

Fig. 18.
Fig. 18.

Calculated transmission function as a function of the aperture radius to wavelength of light ratio, calculated using HVDT.

Equations (137)

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E = k 2 Π + ( . Π ) ,
H = k 2 i ω μ o × Π = i k ε o μ o × Π ,
E x = k 2 Π x + 2 Π x x 2 ,
E y = 2 Π x y x ,
E z = 2 Π x z x ,
H x = 0 ,
H y = k 2 i ω μ o Π x z = i k ε o μ o Π x z ,
H z = k 2 i ω μ o Π x y = i k ε o μ o Π x y .
S = Re ( E × H * )
= Re ( E y H z * E z H y * ) i ̂ + Re ( E x H z * ) j ̂ + Re ( E x H y * ) k ̂ ,
Π x ( x , y , z ) = i E o 2 π k e i k ρ ρ d x o d y o ,
ρ = ( x x 0 ) 2 + ( y y 0 ) 2 + z 2 ,
x 1 x a , y 1 y a , z 1 z z 0 , x 01 x 0 a , y 01 y 0 a ,
ρ 1 ( x 1 x 01 ) 2 + ( y 1 y 01 ) 2 + p 1 2 z 1 2 .
E x ( r 1 ) = i E o 2 π [ p 1 A 1 ( x 1 , y 1 , z 1 ) + 1 p 1 2 A 1 ( x 1 , y 1 , z 1 ) x 1 2 ] ,
E y ( r 1 ) = i E o 2 π p 1 2 A 1 ( x 1 , y 1 , z 1 ) y 1 x 1 ,
E z ( r 1 ) = i E o 2 π p 1 2 2 A 1 ( x 1 , y 1 , z 1 ) z 1 x 1 ,
H x ( r 1 ) = 0 ,
H y ( r 1 ) = H o 2 π p 1 A 1 ( x 1 , y 1 , z 1 ) z 1 ,
H z ( r 1 ) = H o 2 π A 1 ( x 1 , y 1 , z 1 ) y 1 ,
A 1 ( r 1 ) e i p 1 ρ 1 ρ 1 d x 01 d y 01
E x ( x 1 , y 1 , z 1 ) = i E 0 p 1 2 π f 1 [ ( 1 + s 1 ) ( 1 + 3 s 1 ) ( x 1 x 01 ) 2 ρ 1 2 ] d x 01 d y 01 ,
E y ( x 1 , y 1 , z 1 ) = i E o p 1 2 π f 1 ( 1 + 3 s 1 ) ( x 1 x 01 ) ( y 1 y 01 ) ρ 1 2 d x 01 d y 01 ,
E z ( x 1 , y 1 , z 1 ) = i E o p 1 2 z 1 2 π f 1 ( 1 + 3 s 1 ) ( x 1 x 01 ) ρ 1 2 d x 01 d y 01 ,
H x ( x 1 , y 1 , z 1 ) = 0 ,
H y ( x 1 , y 1 , z 1 ) = H o p 1 3 z 1 2 π f 1 s 1 d x 01 d y 01 ,
H z ( x 1 , y 1 , z 1 ) = H o p 1 2 2 π f 1 s 1 ( y 1 y 01 ) d x 01 d y 01 ,
f 1 e i p 1 ρ 1 ρ 1
s 1 1 i p 1 ρ 1 ( 1 + 1 i p 1 ρ 1 ) .
1 1 1 y 01 2 1 y 01 2 d x 01 d y 01 .
Π x ( x 1 , y 1 , z 1 ) = E o a 2 p 1 2 [ e i p 1 2 z 1 1 2 π 0 2 π e i p 1 q L 2 ( 1 r 1 cos ϕ ) d ϕ ] ,
q 2 ( x 1 , y 1 , z 1 , ϕ ) = L 2 + p 1 2 z 1 2 ,
L 2 ( x 1 , y 1 , ϕ ) = 1 + r 1 2 2 r 1 cos ϕ
r 1 ( x 1 , y 1 ) = x 1 2 + y 1 2 .
E x ( x 1 , y 1 , z 1 ) = E o ( e i p 1 2 z 1 1 2 π 0 2 π f 2 a d ϕ ) ,
E y ( x 1 , y 1 , z 1 ) = E o 2 π p 1 2 0 2 π f 2 b d ϕ ,
E z ( x 1 , y 1 , z 1 ) = E o 2 π p 1 3 0 2 π f 2 c d ϕ ,
H x ( x 1 , y 1 , z 1 ) = 0 ,
H y ( x 1 , y 1 , z 1 ) = H o ( e i p 1 2 z 1 i 2 π p 1 2 0 2 π f 2 d d ϕ ) ,
H z ( x 1 , y 1 , z 1 ) = i H o 2 π p 1 0 2 π f 2 e d ϕ ,
f 2 a α β γ + 1 p 1 2 ( α 11 β γ + β 11 α γ + γ 11 α β ) + 2 p 1 2 ( α 1 β 1 γ + β 1 γ 1 α + γ 1 α 1 β ) ,
f 2 b α 12 β γ + β 12 α γ + γ 12 α β
+ α ( β 1 γ 2 + γ 1 β 2 ) + β ( α 1 γ 2 + γ 1 α 2 ) + γ ( α 1 β 2 + β 1 α 2 ) ,
f 2 c α 13 β γ + α 3 ( β γ 1 + γ β 1 ) ,
f 2 d α 3 β γ ,
f 2 e α 2 β γ + β 2 α γ + γ 2 α β .
α e i p 1 q ,
β 1 L 2 ,
γ 1 r 1 cos ϕ .
α 1 = α x 1 , α 2 = α y 1 , α 3 = α z 1 ,
α 11 = 2 α x 1 2 , α 12 = 2 α y 1 x 1 , etc .
α 1 i p 1 α q 1 , α 2 i p 1 α q 2 , α 3 i p 1 α q 3 ,
α 11 i p 1 ( α 1 q 1 + α q 11 ) , α 12 i p 1 ( α 2 q 1 + α q 12 ) , α 13 i p 1 ( α 3 q 1 + α q 13 ) ,
q 1 x 1 δ q , q 2 y 1 δ q , q 3 p 1 2 z 1 q ,
q 11 δ q + x 1 2 q ( cos ϕ r 1 3 δ 2 q 2 ) ,
q 12 x 1 y 1 q ( cos ϕ r 1 3 δ 2 q 2 ) , q 13 p 1 2 x 1 z 1 δ q 3
δ r 1 cos ϕ r 1 .
β 1 2 L 1 L 3 , β 2 2 L 2 L 3 ,
β 11 6 L 1 2 L 4 2 L 11 L 3 , β 12 6 L 1 L 2 L 4 2 L 12 L 3 ,
L 1 x 1 δ L , L 2 y 1 δ L ,
L 11 δ L + x 1 2 L ( cos ϕ r 1 3 δ 2 L 2 ) , L 2 x 1 y 1 L ( cos ϕ r 1 3 δ 2 L 2 ) .
γ 1 x 1 cos ϕ r 1 , γ 2 y 1 cos ϕ r 1 ,
γ 11 cos ϕ r 1 ( 1 x 1 2 r 1 2 ) , γ 12 x 1 y 1 cos ϕ r 1 3 .
Π x ( x 1 , y 1 , z 1 ) = a 2 E o π p 1 2 0 π 2 u v cos ψ d ψ ,
u e i p 1 g e i p 1 h , v r 1 2 sin 2 ψ ,
g p 1 2 z 1 2 + ( v + cos ψ ) 2 , h p 1 2 z 1 2 + ( v cos ψ ) 2 .
E x ( x 1 , y 1 , z 1 ) = E o π 0 π 2 f 3 a d ψ ,
E y ( x 1 , y 1 , z 1 ) = E o π p 1 2 0 π 2 f 3 b d ψ ,
E z ( x 1 , y 1 , z 1 ) = E o π p 1 3 0 π 2 f 3 c d ψ ,
H x ( x 1 , y 1 , z 1 ) = 0 ,
H y ( x 1 , y 1 , z 1 ) = i H o π p 1 2 0 π 2 f 3 d d ψ ,
H z ( x 1 , y 1 , z 1 ) = i H o π p 1 0 π 2 f 3 e d ψ ,
f 3 a cos ψ v [ u + 1 p 1 2 ( u 11 2 x 1 u 1 v 2 + 3 u x 1 2 v 4 u v 2 ) ] ,
f 3 b cos ψ v ( u 12 y 12 u 1 v 2 + 3 x 1 y 1 u v 4 x 1 u 2 v 2 ) ,
f 3 c cos ψ v ( u 13 x 1 u 3 v 2 ) ,
f 3 d cos ψ v u 3 ,
f 3 e cos ψ v ( u 2 y 1 u v 2 ) .
u 1 i p 1 ( g 1 e i p 1 g h 1 e i p 1 h ) , u 2 i p 1 ( g 2 e i p 1 g h 2 e i p 1 h ) ,
u 3 i p 1 3 z 1 ( e i p 1 g g e i p 1 h h ) ,
u 11 i p 1 ( g 11 e i p 1 g h 11 e i p 1 h ) p 1 2 ( g 1 2 e i p 1 g h 1 2 e i p 1 h ) ,
u 12 i p 1 ( g 12 e i p 1 g h 12 e i p 1 h ) p 1 2 ( g 1 g 2 e i p 1 g h 1 h 2 e i p 1 h ) ,
u 13 i p 1 ( g 13 e i p 1 g h 13 e i p 1 h ) p 1 2 ( g 1 g 3 e i p 1 g h 1 h 3 e i p 1 h ) ,
g 1 x 1 g a , g 2 y 1 g a , g 3 z 1 p 1 2 g ,
g 11 g a x 1 2 g ( g a 2 + cos ψ v 3 ) , g 12 x 1 y 1 g ( g a 2 + cos ψ v 3 ) , g 13 x 1 z 1 p 1 2 g 2 g a ,
h 1 x 1 h a , h 2 y 1 h a , h 3 z 1 p 1 2 h ,
h 11 h a x 1 2 h ( h a 2 cos ψ v 3 ) , h 12 x 1 y 1 h ( h a 2 cos ψ v 3 ) , h 13 x 1 z 1 p 1 2 h a g 2 ,
g a v + cos ψ v g , h a v cos ψ v h .
E x ( r ) = 1 2 π E x ( r 0 ) G ( r , r 0 ) z d x 0 d y 0 ,
E y ( r ) = 1 2 π E y ( r 0 ) G ( r , r 0 ) z d x 0 d y 0 ,
G ( r , r 0 ) = e i k ρ ρ ,
E z z = ( E x x + E y y ) ,
2 π E z z = ( E x ( r 0 ) 2 G x z + E y ( r 0 ) 2 G y z ) d x 0 d y 0
= z [ ( E x ( r 0 ) G x + E y ( r 0 ) G y ) d x 0 d y 0 ] .
E z ( r ) = 1 2 π ( E x ( r 0 ) G x + E y ( r 0 ) G y ) d x 0 d y 0 + F 1 ( x , y ) ,
E x ( r 1 ) = p 1 3 z 1 2 π E x ( r 01 ) f 1 s 1 d x 01 d y 01 ,
E y ( r 1 ) = p 1 3 z 1 2 π E y ( r 01 ) f 1 s 1 d x 01 d y 01 ,
E z ( r 1 ) = p 1 2 2 π [ ( E x ( r 01 ) ( x 1 x 01 )
+ E y ( r 01 ) ( y 1 y 01 ) ) f 1 s 1 d x 01 d y 01 ] + F 1 ( x 1 , y 1 ) ,
H x ( r 1 ) = p 1 3 z 1 2 π H x ( r 01 ) f 1 s 1 d x 01 d y 01 ,
H y ( r 1 ) = p 1 3 z 1 2 π H y ( r 01 ) f 1 s 1 d x 01 d y 01 ,
H z ( r 1 ) = p 1 2 2 π [ ( H x ( r 01 ) ( x 1 x 01 )
+ H y ( r 01 ) ( y 1 y 01 ) ) f 1 s 1 d x 01 d y 01 ] + F 1 ( x 1 , y 1 ) ,
E x ( x 01 , y 01 , 0 ) = { E 0 if x 01 2 + y 01 2 < 1 0 otherwise
E y ( x 01 , y 01 , 0 ) = 0
E z ( x 01 , y 01 , 0 ) = 0 ,
H x ( x 01 , y 01 , 0 ) = 0
H y ( x 01 , y 01 , 0 ) = { H 0 if x 01 2 + y 01 2 < 1 0 otherwise
H z ( x 01 , y 01 , 0 ) = 0 .
E x ( x 1 , y 1 , z 1 ) = E 0 p 1 3 z 1 2 π B 1 ( x 1 , y 1 , z 1 ) ,
E y ( x 1 , y 1 , z 1 ) = 0 ,
E z ( x 1 , y 1 , z 1 ) = E 0 p 1 2 2 π [ B 2 ( x 1 , y 1 , z 1 ) B 2 ( x 1 , y 1 , 0 ) ] ,
H x ( x 1 , y 1 , z 1 ) = 0 ,
H y ( x 1 , y 1 , z 1 ) = H 0 p 1 2 z 1 2 π B 1 ( x 1 , y 1 , z 1 ) ,
H z ( x 1 , y 1 , z 1 ) = H 0 p 1 2 2 π [ B 2 ( x 1 , y 1 , z 1 ) B 3 ( x 1 , y 1 , 0 ) ] ,
B 1 ( x 1 , y 1 , z 1 ) = 1 1 1 y 01 2 1 y 01 2 f 1 s 1 d x 01 d y 01 ,
B 2 ( x 1 , y 1 , z 1 ) = 1 1 1 y 01 2 1 y 01 2 f 1 s 1 ( x 1 x 01 ) d x 01 d y 01 ,
B 3 ( x 1 , y 1 , z 1 ) = 1 1 1 y 01 2 1 y 01 2 f 1 s 1 ( y 1 y 01 ) d x 01 d y 01 .
E x ( 0 , 0 , z 1 ) = E o i 2 π [ ( p 1 e i p 1 r 2 r 2 i e i p 1 r 2 r 2 2 e i p 1 r 2 p 1 r 2 3 p 1 x 01 2 e i p 1 r 2 r 2 2
+ 3 i x 01 2 e i p 1 r 2 r 2 4 + 3 x 01 2 e i p 1 r 2 p 1 r 2 5 ) d x 01 d y 01 ] ,
r 2 = x 01 2 + y 01 2 + p 1 2 z 1 2 .
E x ( 0 , 0 , z 1 ) = E o i 2 [ i ( e 1 e 2 ) 1 p 1 ( e 1 d 1 e 2 d 2 )
+ i p 1 2 z 1 2 ( e 1 d 1 2 e 2 d 2 2 ) + p 1 z 1 2 ( e 1 d 1 3 e 2 d 2 3 ) ] ,
e 1 = e i p 1 1 + p 1 2 z 1 2 ,
e 2 = e i p 1 2 z 1 ,
d 1 = 1 + p 1 2 z 1 2 ,
d 2 = p 1 z 1 .
H y ( 0 , 0 , z 1 ) = H o p 1 z 1 2 π ( i p 1 e i p 1 r 2 r 2 2 + e i p 1 r 2 r 2 3 ) d x 01 d y 01 ,
H y ( 0 , 0 , z 1 ) = H o ( e i p 1 2 z 1 p 1 z 1 e i p 1 1 + p 1 2 z 1 2 1 + p 1 2 z 1 2 ) .
Real [ E x ( 0 , 0 , 0 ) ] = E o [ 1 1 2 ( cos p 1 + sin p 1 p 1 ) ]
Imaginary [ E x ( 0 , 0 , 0 ) ] = E 0 2 ( cos p 1 p 1 sin p 1 ) ,
E x ( 0 , 0 , z 1 ) = E o ( e i p 1 2 z 1 p 1 z 1 e i p 1 1 + p 1 2 z 1 2 1 + p 1 2 z 1 2 ) ,
H y ( 0 , 0 , z 1 ) = H o ( e i p 1 2 z 1 p 1 z 1 e i p 1 1 + p 1 2 z 1 2 1 + p 1 2 z 1 2 ) .
S ( 0 , 0 , z 1 ) = S 0 ( 1 + 2 p 1 2 z 1 2 1 + p 1 2 z 1 2 2 p 1 z 1 1 + p 1 2 z 1 2 cos [ p 1 2 z 1 ( 1 + 1 p 1 2 z 1 2 1 ) ] ) k ̂ ,
S o E o H o * = ε o μ o E o 2
z 1 m = p 1 2 m 2 π 2 2 m π p 1 2 ,
P z ( z 1 ) = S z ( x 1 , y 1 , z 1 ) d x 1 d y 1 .
T P z P o = 1 P o S z ( x 1 , y 1 , z 1 ) d x 1 d y 1 ,

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