Abstract

Diffraction of a two-dimensional electromagnetic beam wave by a thick slit pierced in a perfectly conducting screen is studied, based on Maxwell’s equations and their associated boundary conditions. The incident field is not a plane wave but a more-realistic beam. Special attention is devoted to the numerical aspect, which seems essential in such a study.

© 1983 Optical Society of America

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References

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  1. The address of the Laboratoire d’Astronomie Spatiale is Traverse du Siphon, Les Trois Lucs, 13012 Marseille, France.
  2. This is an assumption that, of course, is not realistic in the visible range.
  3. M. Wirgin, “Influence de l’épaisseur de l’écran sur la diffraction par une fente,” C. R. Acad. Sci. Paris 270, 1457–1460 (1970).
  4. F. L. Neerhoff and G. Mur, “Diffraction of a plane electromagnetic wave by a slit in a thick screen placed between two different media,” Appl. Sci. Res. 28, 73–88 (1973).
  5. J. L. Roumiguieres, D. Maystre, R. Petit, and M. Cadilhac, “Etude de la diffraction par une fente pratiquée dans un écran infiniment conducteur d’épaisseur quelconque,” Opt. Commun. 9, 402–405 (1973).
    [Crossref]
  6. H. Henke and H. Fruchting, “Irradiation in a slotted half space and diffraction by a slit in a thick screen,” Nachrichtentech. 29, 401–405 (1976).
  7. K. Hongo and G. Ishii, “Diffraction of an electromagnetic plane wave by a thick slit,” IEEE Trans. Antennas Propag. AP-26, 494–499 (1978).
    [Crossref]
  8. J. L. Roumiguieres, Motesim, La Bourridiere, Route Nationale 186, 92 350 Le Plessis-Robinson, France (personal communication).

1978 (1)

K. Hongo and G. Ishii, “Diffraction of an electromagnetic plane wave by a thick slit,” IEEE Trans. Antennas Propag. AP-26, 494–499 (1978).
[Crossref]

1976 (1)

H. Henke and H. Fruchting, “Irradiation in a slotted half space and diffraction by a slit in a thick screen,” Nachrichtentech. 29, 401–405 (1976).

1973 (2)

F. L. Neerhoff and G. Mur, “Diffraction of a plane electromagnetic wave by a slit in a thick screen placed between two different media,” Appl. Sci. Res. 28, 73–88 (1973).

J. L. Roumiguieres, D. Maystre, R. Petit, and M. Cadilhac, “Etude de la diffraction par une fente pratiquée dans un écran infiniment conducteur d’épaisseur quelconque,” Opt. Commun. 9, 402–405 (1973).
[Crossref]

1970 (1)

M. Wirgin, “Influence de l’épaisseur de l’écran sur la diffraction par une fente,” C. R. Acad. Sci. Paris 270, 1457–1460 (1970).

Cadilhac, M.

J. L. Roumiguieres, D. Maystre, R. Petit, and M. Cadilhac, “Etude de la diffraction par une fente pratiquée dans un écran infiniment conducteur d’épaisseur quelconque,” Opt. Commun. 9, 402–405 (1973).
[Crossref]

Fruchting, H.

H. Henke and H. Fruchting, “Irradiation in a slotted half space and diffraction by a slit in a thick screen,” Nachrichtentech. 29, 401–405 (1976).

Henke, H.

H. Henke and H. Fruchting, “Irradiation in a slotted half space and diffraction by a slit in a thick screen,” Nachrichtentech. 29, 401–405 (1976).

Hongo, K.

K. Hongo and G. Ishii, “Diffraction of an electromagnetic plane wave by a thick slit,” IEEE Trans. Antennas Propag. AP-26, 494–499 (1978).
[Crossref]

Ishii, G.

K. Hongo and G. Ishii, “Diffraction of an electromagnetic plane wave by a thick slit,” IEEE Trans. Antennas Propag. AP-26, 494–499 (1978).
[Crossref]

Maystre, D.

J. L. Roumiguieres, D. Maystre, R. Petit, and M. Cadilhac, “Etude de la diffraction par une fente pratiquée dans un écran infiniment conducteur d’épaisseur quelconque,” Opt. Commun. 9, 402–405 (1973).
[Crossref]

Mur, G.

F. L. Neerhoff and G. Mur, “Diffraction of a plane electromagnetic wave by a slit in a thick screen placed between two different media,” Appl. Sci. Res. 28, 73–88 (1973).

Neerhoff, F. L.

F. L. Neerhoff and G. Mur, “Diffraction of a plane electromagnetic wave by a slit in a thick screen placed between two different media,” Appl. Sci. Res. 28, 73–88 (1973).

Petit, R.

J. L. Roumiguieres, D. Maystre, R. Petit, and M. Cadilhac, “Etude de la diffraction par une fente pratiquée dans un écran infiniment conducteur d’épaisseur quelconque,” Opt. Commun. 9, 402–405 (1973).
[Crossref]

Roumiguieres, J. L.

J. L. Roumiguieres, D. Maystre, R. Petit, and M. Cadilhac, “Etude de la diffraction par une fente pratiquée dans un écran infiniment conducteur d’épaisseur quelconque,” Opt. Commun. 9, 402–405 (1973).
[Crossref]

J. L. Roumiguieres, Motesim, La Bourridiere, Route Nationale 186, 92 350 Le Plessis-Robinson, France (personal communication).

Wirgin, M.

M. Wirgin, “Influence de l’épaisseur de l’écran sur la diffraction par une fente,” C. R. Acad. Sci. Paris 270, 1457–1460 (1970).

Appl. Sci. Res. (1)

F. L. Neerhoff and G. Mur, “Diffraction of a plane electromagnetic wave by a slit in a thick screen placed between two different media,” Appl. Sci. Res. 28, 73–88 (1973).

C. R. Acad. Sci. Paris (1)

M. Wirgin, “Influence de l’épaisseur de l’écran sur la diffraction par une fente,” C. R. Acad. Sci. Paris 270, 1457–1460 (1970).

IEEE Trans. Antennas Propag. (1)

K. Hongo and G. Ishii, “Diffraction of an electromagnetic plane wave by a thick slit,” IEEE Trans. Antennas Propag. AP-26, 494–499 (1978).
[Crossref]

Nachrichtentech. (1)

H. Henke and H. Fruchting, “Irradiation in a slotted half space and diffraction by a slit in a thick screen,” Nachrichtentech. 29, 401–405 (1976).

Opt. Commun. (1)

J. L. Roumiguieres, D. Maystre, R. Petit, and M. Cadilhac, “Etude de la diffraction par une fente pratiquée dans un écran infiniment conducteur d’épaisseur quelconque,” Opt. Commun. 9, 402–405 (1973).
[Crossref]

Other (3)

The address of the Laboratoire d’Astronomie Spatiale is Traverse du Siphon, Les Trois Lucs, 13012 Marseille, France.

This is an assumption that, of course, is not realistic in the visible range.

J. L. Roumiguieres, Motesim, La Bourridiere, Route Nationale 186, 92 350 Le Plessis-Robinson, France (personal communication).

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Figures (4)

Fig. 1
Fig. 1

Slit (width l, depth h) is parallel to the z axis, i.e., perpendicular to the plane of the figure.

Fig. 2
Fig. 2

Integration paths in the complex plane. Lines α = k, γ ≥ 0, and α = − k, γ ≤ 0 are branch cuts linked with the presence of β = (k2α)1/2. Γ2 is the union of Δ, Δ′, and the half-circle. Notice that Γ1 bypasses the points γ = 0, α = ± (nπ/l).

Fig. 3
Fig. 3

Angle ϕ is used to describe the diffraction pattern.

Fig. 4
Fig. 4

Example of diffraction pattern: α = 4, l = 1, b = 0.5, h = 0.6, λ = 0.3. E, dashed line. H, solid line. The scale used for ordinates is arbitrary; a certain multiplying factor has been omitted.

Tables (4)

Tables Icon

Table 1 Variation of the Transmission Coefficient with the Thickness of the Conducting Screena

Tables Icon

Table 2 Variation of the Transmission Coefficient with the Thickness of the Screena

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Table 3 Variation of the Transmission Coefficient with the Thickness of the Conducting Screen (E, Gaussian Incident Spot)a

Tables Icon

Table 4 Variation of the Transmission Coefficient with the Thickness of the Conducting Screen (H, Gaussian Incident Spot)a

Equations (22)

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F ( x , y ) = ( 2 π ) 1 / 2 + F ̂ ( α , y ) exp ( i α x ) d α .
F ̂ ( α , y ) = A ( α ) exp [ i β ( α ) y ] + B ( α ) exp [ i β ( α ) y ] R 1 ( α , y ) for y > h 2 ( region 1 ) ,
F ̂ ( α , y ) = C ( α ) exp [ i β ( α ) y ] R 2 ( α , y ) for y < h 2 ( region 2 ) ,
β ( α ) = ( k 2 α 2 ) 1 / 2 , β 0 , β / i 0 .
F ( x , y ) = n = 1 [ a n cos ( μ n y ) + b n sin ( μ n y ) ] ϕ n ( x ) ,
ϕ n ( x ) = { sin ( n π x l ) for 0 x l 0 elsewhere ,
μ n = [ k 2 ( n π / l ) 2 ] 1 / 2 , μ n 0 , μ n / i 0 .
F ̂ ( α , y ) = n = 1 [ a n cos ( μ n y ) + b n sin ( μ n y ) ] ϕ ̂ n ( α ) n F n ( y ) ϕ ̂ n ( α ) ,
f ( x ) = 0 for 0 x l , + f ̂ ( α ) ϕ ̂ n ¯ ( α ) d α = 0 for n = 1 , 2 , ,
R 1 ( α , h 2 ) = n = 1 [ a n cos ( μ n h 2 ) + b n sin ( μ n h 2 ) ] ϕ ̂ n ( α ) = n = 1 F n ( h 2 ) ϕ ̂ n ( α ) .
f ̂ ( α ) = R 1 y ( α , h 2 ) m = 1 F m y ( h 2 ) ϕ ̂ m ( α )
R 1 y ( α , h 2 ) m = 1 F m y ( h 2 ) ϕ ̂ m ( α ) , ϕ ̂ n ( α ) = 0 .
N a + P b = S ,
N a P b = 0 .
I n , m = + G 1 ( α ) [ 1 ( 1 ) n cos ( α l ) ] d α ,
G 1 ( α ) = β ( α ) ( α 2 n 2 π 2 l 2 ) ( α 2 m 2 π 2 l 2 ) ,
I n , m = Γ 1 G 1 ( α ) [ 1 ( 1 ) n exp ( i α l ) ] d α .
I n , m = ρ n δ n , m + 2 Δ G 1 ( α ) [ 1 ( 1 ) n exp ( i α l ) ] d α ,
G 2 ( α ) = α 2 β ( α ) ( α 2 n 2 π 2 l 2 ) ( α 2 m 2 π 2 l 2 ) .
F i ( x , h 2 ) = exp [ 4 ( x b ) 2 / a 2 ] ,
A ( α ) = a 2 2 exp ( i α b ) exp ( a 2 α 2 / 16 ) .
τ = Im 0 l F ¯ ( x , h 2 ) F δ y ( x , h 2 ) d x Im + F ¯ i ( x , h 2 ) F i δ y ( x , h 2 ) d x .