Abstract

The effect of the scatterer surface on the shadow region is examined by using the surface integrals of the modified theory of physical optics. It is shown that the shadow geometry has a considerable effect on the structure of the edge diffracted waves. The diffracted fields for the illuminated and shadowed surface of a half-plane are evaluated in terms of Fresnel integrals and plotted numerically.

© 2008 Optical Society of America

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References

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  1. G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Ann. Phys. 254, 663-695 (1883).
    [CrossRef]
  2. A. Rubinowicz, “Die beugungswelle in der Kirchhoffschen theorie der Beugungserscheinungen,” Ann. Phys. 358, 257-278 (1917).
    [CrossRef]
  3. A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160-162 (1957).
    [CrossRef]
  4. G. N. Cantor, “Was Thomas Young a wave theorist?” Am. J. Phys. 52, 305-308 (1984).
    [CrossRef]
  5. P. Langlois and A. Boivin, “Thomas Young's ideas on light diffraction in the context of electromagnetic theory,” Can. J. Phys. 63, 265-274 (1985).
    [CrossRef]
  6. P. Ya. Ufimtsev, Fundamentals of the Physical Theory of Diffraction (Wiley, 2007).
    [CrossRef]
  7. P. Ya. Ufimtsev, “Elementary edge waves and the physical theory of diffraction,” Electromagnetics 11, 125-160 (1991).
    [CrossRef]
  8. P. Ya. Ufimtsev, “Fast convergent integrals for nonuniform currents on wedge faces,” Electromagnetics 18, 289-313 (1998).
    [CrossRef]
  9. Y. Z. Umul, “Modified theory of physical optics,” Opt. Express 12, 4959-4972 (2004).
    [CrossRef] [PubMed]
  10. Y. Z. Umul, “Modified theory of physical optics approach to wedge diffraction problems,” Opt. Express 13, 216-224 (2005).
    [CrossRef] [PubMed]
  11. Y. Z. Umul, “Edge-dislocation waves in the diffraction process by an impedance half-plane,” J. Opt. Soc. Am. A 24, 507-511 (2007).
    [CrossRef]
  12. Y. Z. Umul, “Uniform line integral representation of edge-diffracted fields,” J. Opt. Soc. Am. A 25, 133-137 (2008).
    [CrossRef]
  13. Y. Z. Umul, “Diffraction of evanescent plane waves by a resistive half-plane,” J. Opt. Soc. Am. A 24, 3226-3232 (2007).
    [CrossRef]
  14. Y. Z. Umul, “Diffraction by a black half-plane: modified theory of physical optics approach,” Opt. Express 13, 7276-7287 (2005).
    [CrossRef] [PubMed]
  15. A. Sommerfeld, “Matematische Theorie der Diffraction,” Math. Ann. 47, 317-374 (1896).
    [CrossRef]
  16. R. G. Kouyoumjian and P. B. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448-1461 (1974).
    [CrossRef]

2008 (1)

2007 (2)

2005 (2)

2004 (1)

1998 (1)

P. Ya. Ufimtsev, “Fast convergent integrals for nonuniform currents on wedge faces,” Electromagnetics 18, 289-313 (1998).
[CrossRef]

1991 (1)

P. Ya. Ufimtsev, “Elementary edge waves and the physical theory of diffraction,” Electromagnetics 11, 125-160 (1991).
[CrossRef]

1985 (1)

P. Langlois and A. Boivin, “Thomas Young's ideas on light diffraction in the context of electromagnetic theory,” Can. J. Phys. 63, 265-274 (1985).
[CrossRef]

1984 (1)

G. N. Cantor, “Was Thomas Young a wave theorist?” Am. J. Phys. 52, 305-308 (1984).
[CrossRef]

1974 (1)

R. G. Kouyoumjian and P. B. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

1957 (1)

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160-162 (1957).
[CrossRef]

1917 (1)

A. Rubinowicz, “Die beugungswelle in der Kirchhoffschen theorie der Beugungserscheinungen,” Ann. Phys. 358, 257-278 (1917).
[CrossRef]

1896 (1)

A. Sommerfeld, “Matematische Theorie der Diffraction,” Math. Ann. 47, 317-374 (1896).
[CrossRef]

1883 (1)

G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Ann. Phys. 254, 663-695 (1883).
[CrossRef]

Boivin, A.

P. Langlois and A. Boivin, “Thomas Young's ideas on light diffraction in the context of electromagnetic theory,” Can. J. Phys. 63, 265-274 (1985).
[CrossRef]

Cantor, G. N.

G. N. Cantor, “Was Thomas Young a wave theorist?” Am. J. Phys. 52, 305-308 (1984).
[CrossRef]

Kirchhoff, G.

G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Ann. Phys. 254, 663-695 (1883).
[CrossRef]

Kouyoumjian, R. G.

R. G. Kouyoumjian and P. B. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

Langlois, P.

P. Langlois and A. Boivin, “Thomas Young's ideas on light diffraction in the context of electromagnetic theory,” Can. J. Phys. 63, 265-274 (1985).
[CrossRef]

Pathak, P. B.

R. G. Kouyoumjian and P. B. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

Rubinowicz, A.

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160-162 (1957).
[CrossRef]

A. Rubinowicz, “Die beugungswelle in der Kirchhoffschen theorie der Beugungserscheinungen,” Ann. Phys. 358, 257-278 (1917).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, “Matematische Theorie der Diffraction,” Math. Ann. 47, 317-374 (1896).
[CrossRef]

Ufimtsev, P. Ya.

P. Ya. Ufimtsev, “Fast convergent integrals for nonuniform currents on wedge faces,” Electromagnetics 18, 289-313 (1998).
[CrossRef]

P. Ya. Ufimtsev, “Elementary edge waves and the physical theory of diffraction,” Electromagnetics 11, 125-160 (1991).
[CrossRef]

P. Ya. Ufimtsev, Fundamentals of the Physical Theory of Diffraction (Wiley, 2007).
[CrossRef]

Umul, Y. Z.

Am. J. Phys. (1)

G. N. Cantor, “Was Thomas Young a wave theorist?” Am. J. Phys. 52, 305-308 (1984).
[CrossRef]

Ann. Phys. (2)

G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Ann. Phys. 254, 663-695 (1883).
[CrossRef]

A. Rubinowicz, “Die beugungswelle in der Kirchhoffschen theorie der Beugungserscheinungen,” Ann. Phys. 358, 257-278 (1917).
[CrossRef]

Can. J. Phys. (1)

P. Langlois and A. Boivin, “Thomas Young's ideas on light diffraction in the context of electromagnetic theory,” Can. J. Phys. 63, 265-274 (1985).
[CrossRef]

Electromagnetics (2)

P. Ya. Ufimtsev, “Elementary edge waves and the physical theory of diffraction,” Electromagnetics 11, 125-160 (1991).
[CrossRef]

P. Ya. Ufimtsev, “Fast convergent integrals for nonuniform currents on wedge faces,” Electromagnetics 18, 289-313 (1998).
[CrossRef]

J. Opt. Soc. Am. A (3)

Math. Ann. (1)

A. Sommerfeld, “Matematische Theorie der Diffraction,” Math. Ann. 47, 317-374 (1896).
[CrossRef]

Nature (1)

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160-162 (1957).
[CrossRef]

Opt. Express (3)

Proc. IEEE (1)

R. G. Kouyoumjian and P. B. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448-1461 (1974).
[CrossRef]

Other (1)

P. Ya. Ufimtsev, Fundamentals of the Physical Theory of Diffraction (Wiley, 2007).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Geometry of the conducting half-plane.

Fig. 2
Fig. 2

Scattering geometry of the shadow plane.

Fig. 3
Fig. 3

Contribution of the shadow geometry to the reflected diffracted wave.

Fig. 4
Fig. 4

Contribution of the shadow geometry to the reflected diffracted wave.

Fig. 5
Fig. 5

Reflected diffracted fields.

Fig. 6
Fig. 6

Incident diffracted fields.

Fig. 7
Fig. 7

Total diffracted fields.

Equations (33)

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u s ( P ) = u i ( P ) k exp ( j π 4 ) 2 π 0 u i ( Q ) ( sin β + α 2 sin β α 2 ) exp ( j k R ) k R d x ,
sin β + α 2 = 1 2 cos β cos α sin β α 2 ,
sin β + α 2 = 1 4 ( cos β cos α ) ( cot g β α 4 + t g β α 4 ) .
sin β + α 2 = 1 4 ( cos β cos α ) ( cot g β α 4 cot g β 2 π α 4 ) .
sin β α 2 = 1 4 ( cos β cos α ) ( cot g β + α 4 cot g β + 2 π + α 4 ) .
u s ( P ) = u rs 1 ( P ) + u rs 2 ( P ) + u is 1 ( P ) + u is 2 ( P ) ,
u rs 1 ( P ) = k exp ( j π 4 ) 4 2 π 0 u i ( Q ) ( cos β cos α ) cot g ( β α 4 ) exp ( j k R ) k R d x ,
u rs 2 ( P ) = k exp ( j π 4 ) 4 2 π 0 u i ( Q ) [ cos β cos ( 2 π + α ) ] cot g ( β 2 π α 4 ) exp ( j k R ) k R d x ,
u is 1 ( P ) = u i ( P ) k exp ( j π 4 ) 4 2 π 0 u i ( Q ) ( cos β cos α ) cot g ( β + α 4 ) exp ( j k R ) k R d x ,
u is 2 ( P ) = k exp ( j π 4 ) 2 π 0 u i ( Q ) [ cos β cos ( 2 π + α ) ] cot g ( β + 2 π + α 4 ) exp ( j k R ) k R d x .
u i ( P ) = u 0 exp [ j k ( x cos ϕ 0 + y sin ϕ 0 ) ] ,
u i ( Q ) = u 0 exp ( j k x cos ϕ 0 ) ,
I = exp ( j π 4 ) π x e f ( x ) exp [ j k g ( x ) ] d x ,
I = exp [ j k g ( x s ) ] 2 k { j f ( x s ) g ( x s ) U ( t e ) 2 k t e f ( x e ) g ( x e ) sgn ( t e ) F [ t e ] } ,
F [ x ] = exp ( j π 4 ) π x exp ( j t 2 ) d t .
g ( x ) = x cos ϕ 0 R .
u rs = u r 1 GO + u rd 1 + u rd 2 ,
u is = u i 1 GO + u i d 1 + u i d 2 ,
u r 1 GO = u 0 exp [ j k ρ cos ( ϕ + ϕ 0 ) ] U ( ξ r ) ,
u i 1 GO = u 0 exp [ j k ρ cos ( ϕ ϕ 0 ) ] U ( ξ i ) ,
ξ r = 2 k ρ cos ϕ + ϕ 0 2 ,
ξ i = 2 k ρ cos ϕ ϕ 0 2 .
u rd 1 = u 0 2 exp [ j k ρ cos ( ϕ + ϕ 0 ) ] cos ϕ + ϕ 0 2 cot g π ϕ ϕ 0 4 sgn ( ξ r ) F [ ξ r ] ,
u rd 2 = u 0 2 exp [ j k ρ cos ( ϕ + ϕ 0 ) ] cos ϕ + ϕ 0 2 cot g π + ϕ + ϕ 0 4 sgn ( ξ r ) F [ ξ r ] ,
u i d 1 = u 0 2 exp [ j k ρ cos ( ϕ ϕ 0 ) ] cos ϕ ϕ 0 2 cot g π ϕ + ϕ 0 4 sgn ( ξ i ) F [ ξ i ] ,
u i d 2 = u 0 2 exp [ j k ρ cos ( ϕ ϕ 0 ) ] cos ϕ ϕ 0 2 cot g 3 π ϕ + ϕ 0 4 sgn ( ξ i ) F [ ξ i ] ,
u rs 2 = k exp ( j π 4 ) 4 2 π 0 [ cos β cos ( 2 π + ϕ 0 ) ] cot g ( β 2 π ϕ 0 4 ) exp [ j k ( x cos ϕ 0 R ) ] k R d x ,
u is 2 = k exp ( j π 4 ) 2 π 0 [ cos β cos ( 2 π + ϕ 0 ) ] cot g ( β + 2 π + ϕ 0 4 ) exp [ j k ( x cos ϕ 0 R ) ] k R d x .
u i ( P ) = u 0 exp { j k [ x cos ( 2 π + ϕ 0 ) + y sin ( 2 π + ϕ 0 ) ] } ,
u s e = u i s e + u r s e ,
u i s e = u 0 exp [ j k ρ cos ( ϕ ϕ 0 ) ] F [ ξ i ] ,
u r s e = u 0 exp [ j k ρ cos ( ϕ + ϕ 0 ) ] F [ ξ r ] ,
F [ x ] = U ( x ) + sgn ( x ) F [ x ] .

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