Abstract

The diffraction of a Gaussian beam by an impedance half-plane is studied through the method of the modified theory of physical optics. An electric line source, which is defined in the complex space, is used to represent the Gaussian beam. The uniform evaluation of the diffraction integral is performed and the scattering patterns of the field are investigated for various numerical parameters of the incident wave.

© 2007 Optical Society of America

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References

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  1. J. E. Pearson, T. C. McGill, S. Kurtin, and A. Yariv, "Diffraction of Gaussian laser beams by a semi-infinite plane," J. Opt. Soc. Am. 59, 1440-1445 (1969).
    [CrossRef]
  2. G. Otis and J. W. Y. Lit, "Edge-on diffraction of a Gaussian laser beam by a semi-infinite plane," Appl. Opt. 14, 1156-1160 (1975).
    [CrossRef] [PubMed]
  3. K. Miyamoto and E. Wolf, "Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave--Part I," J. Opt. Soc. Am. 52, 615-625 (1962).
    [CrossRef]
  4. K. Miyamoto and E. Wolf, "Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave--Part II," J. Opt. Soc. Am. 52, 626-637 (1962).
    [CrossRef]
  5. A. C. Green, H. L. Bertoni, and L. B. Felsen, "Properties of the shadow cast by a half-screen when illuminated by a Gaussian beam," J. Opt. Soc. Am. 69, 1503-1508 (1979).
    [CrossRef]
  6. T. Takenaka and O. Fukumitsu, "Asymptotic representation of the boundary-diffraction wave for a three-dimensional Gaussian beam incident upon a Kirchoff half-screen," J. Opt. Soc. Am. 72, 331-336 (1982).
    [CrossRef]
  7. G. A. Suedan and E. V. Jull, "Two-dimensional beam diffraction by a half-plane and wide slit," IEEE Trans. Antennas Propag. 35, 1077-1083 (1987).
    [CrossRef]
  8. G. Pelosi and S. Selleri, "A numerical approach for the diffraction of a Gaussian beam from a perfectly conducting wedge," IEEE Trans. Antennas Propag. 47, 1555-1559 (1999).
    [CrossRef]
  9. L. E. R. Petersson and G. S. Smith, "Three-dimensional electromagnetic diffraction of a Gaussian beam by a perfectly conducting half-plane," J. Opt. Soc. Am. A 19, 2265-2280 (2002).
    [CrossRef]
  10. H. D. Cheung and V. Jull, "Beam scattering by a right-angled impedance wedge," IEEE Trans. Antennas Propag. 52, 497-504 (2004).
    [CrossRef]
  11. R. Tiberio, G. Pelosi, G. Manara, and P. H. Pathak, "High-frequency scattering from a wedge with impedance faces illuminated by a line source--Part I: diffraction," IEEE Trans. Antennas Propag. 37, 212-218 (1989).
    [CrossRef]
  12. R. Tiberio, G. Pelosi, G. Manara, and P. H. Pathak, "High-frequency scattering from a wedge with impedance faces illuminated by a line source--Part II: surface waves," IEEE Trans. Antennas Propag. 41, 877-883 (1993).
    [CrossRef]
  13. Y. Z. Umul, "Modified theory of physical optics," Opt. Eng. (Bellingham) 12, 4959-4972 (2004); http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-20-4959.
  14. Y. Z. Umul, "Modified theory of physical optics solution of impedance half-plane problem," IEEE Trans. Antennas Propag. 54, 2048-2053 (2006).
    [CrossRef]
  15. 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]
  16. S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and R. H. Tew, "On the theory of complex rays," SIAM Rev. 41, 417-509 (1999).
    [CrossRef]
  17. Y. Z. Umul, "Uniform theory for the diffraction of evanescent plane waves," J. Opt. Soc. Am. A 24, 2426-2430 (2007).
    [CrossRef]

2007 (2)

2006 (1)

Y. Z. Umul, "Modified theory of physical optics solution of impedance half-plane problem," IEEE Trans. Antennas Propag. 54, 2048-2053 (2006).
[CrossRef]

2004 (2)

H. D. Cheung and V. Jull, "Beam scattering by a right-angled impedance wedge," IEEE Trans. Antennas Propag. 52, 497-504 (2004).
[CrossRef]

Y. Z. Umul, "Modified theory of physical optics," Opt. Eng. (Bellingham) 12, 4959-4972 (2004); http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-20-4959.

2002 (1)

1999 (2)

G. Pelosi and S. Selleri, "A numerical approach for the diffraction of a Gaussian beam from a perfectly conducting wedge," IEEE Trans. Antennas Propag. 47, 1555-1559 (1999).
[CrossRef]

S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and R. H. Tew, "On the theory of complex rays," SIAM Rev. 41, 417-509 (1999).
[CrossRef]

1993 (1)

R. Tiberio, G. Pelosi, G. Manara, and P. H. Pathak, "High-frequency scattering from a wedge with impedance faces illuminated by a line source--Part II: surface waves," IEEE Trans. Antennas Propag. 41, 877-883 (1993).
[CrossRef]

1989 (1)

R. Tiberio, G. Pelosi, G. Manara, and P. H. Pathak, "High-frequency scattering from a wedge with impedance faces illuminated by a line source--Part I: diffraction," IEEE Trans. Antennas Propag. 37, 212-218 (1989).
[CrossRef]

1987 (1)

G. A. Suedan and E. V. Jull, "Two-dimensional beam diffraction by a half-plane and wide slit," IEEE Trans. Antennas Propag. 35, 1077-1083 (1987).
[CrossRef]

1982 (1)

1979 (1)

1975 (1)

1969 (1)

1962 (2)

Bertoni, H. L.

Chapman, S. J.

S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and R. H. Tew, "On the theory of complex rays," SIAM Rev. 41, 417-509 (1999).
[CrossRef]

Cheung, H. D.

H. D. Cheung and V. Jull, "Beam scattering by a right-angled impedance wedge," IEEE Trans. Antennas Propag. 52, 497-504 (2004).
[CrossRef]

Felsen, L. B.

Fukumitsu, O.

Green, A. C.

Jull, E. V.

G. A. Suedan and E. V. Jull, "Two-dimensional beam diffraction by a half-plane and wide slit," IEEE Trans. Antennas Propag. 35, 1077-1083 (1987).
[CrossRef]

Jull, V.

H. D. Cheung and V. Jull, "Beam scattering by a right-angled impedance wedge," IEEE Trans. Antennas Propag. 52, 497-504 (2004).
[CrossRef]

Kurtin, S.

Lawry, J. M. H.

S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and R. H. Tew, "On the theory of complex rays," SIAM Rev. 41, 417-509 (1999).
[CrossRef]

Lit, J. W. Y.

Manara, G.

R. Tiberio, G. Pelosi, G. Manara, and P. H. Pathak, "High-frequency scattering from a wedge with impedance faces illuminated by a line source--Part II: surface waves," IEEE Trans. Antennas Propag. 41, 877-883 (1993).
[CrossRef]

R. Tiberio, G. Pelosi, G. Manara, and P. H. Pathak, "High-frequency scattering from a wedge with impedance faces illuminated by a line source--Part I: diffraction," IEEE Trans. Antennas Propag. 37, 212-218 (1989).
[CrossRef]

McGill, T. C.

Miyamoto, K.

Ockendon, J. R.

S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and R. H. Tew, "On the theory of complex rays," SIAM Rev. 41, 417-509 (1999).
[CrossRef]

Otis, G.

Pathak, P. H.

R. Tiberio, G. Pelosi, G. Manara, and P. H. Pathak, "High-frequency scattering from a wedge with impedance faces illuminated by a line source--Part II: surface waves," IEEE Trans. Antennas Propag. 41, 877-883 (1993).
[CrossRef]

R. Tiberio, G. Pelosi, G. Manara, and P. H. Pathak, "High-frequency scattering from a wedge with impedance faces illuminated by a line source--Part I: diffraction," IEEE Trans. Antennas Propag. 37, 212-218 (1989).
[CrossRef]

Pearson, J. E.

Pelosi, G.

G. Pelosi and S. Selleri, "A numerical approach for the diffraction of a Gaussian beam from a perfectly conducting wedge," IEEE Trans. Antennas Propag. 47, 1555-1559 (1999).
[CrossRef]

R. Tiberio, G. Pelosi, G. Manara, and P. H. Pathak, "High-frequency scattering from a wedge with impedance faces illuminated by a line source--Part II: surface waves," IEEE Trans. Antennas Propag. 41, 877-883 (1993).
[CrossRef]

R. Tiberio, G. Pelosi, G. Manara, and P. H. Pathak, "High-frequency scattering from a wedge with impedance faces illuminated by a line source--Part I: diffraction," IEEE Trans. Antennas Propag. 37, 212-218 (1989).
[CrossRef]

Petersson, L. E. R.

Selleri, S.

G. Pelosi and S. Selleri, "A numerical approach for the diffraction of a Gaussian beam from a perfectly conducting wedge," IEEE Trans. Antennas Propag. 47, 1555-1559 (1999).
[CrossRef]

Smith, G. S.

Suedan, G. A.

G. A. Suedan and E. V. Jull, "Two-dimensional beam diffraction by a half-plane and wide slit," IEEE Trans. Antennas Propag. 35, 1077-1083 (1987).
[CrossRef]

Takenaka, T.

Tew, R. H.

S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and R. H. Tew, "On the theory of complex rays," SIAM Rev. 41, 417-509 (1999).
[CrossRef]

Tiberio, R.

R. Tiberio, G. Pelosi, G. Manara, and P. H. Pathak, "High-frequency scattering from a wedge with impedance faces illuminated by a line source--Part II: surface waves," IEEE Trans. Antennas Propag. 41, 877-883 (1993).
[CrossRef]

R. Tiberio, G. Pelosi, G. Manara, and P. H. Pathak, "High-frequency scattering from a wedge with impedance faces illuminated by a line source--Part I: diffraction," IEEE Trans. Antennas Propag. 37, 212-218 (1989).
[CrossRef]

Umul, Y. Z.

Y. Z. Umul, "Uniform theory for the diffraction of evanescent plane waves," J. Opt. Soc. Am. A 24, 2426-2430 (2007).
[CrossRef]

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]

Y. Z. Umul, "Modified theory of physical optics solution of impedance half-plane problem," IEEE Trans. Antennas Propag. 54, 2048-2053 (2006).
[CrossRef]

Y. Z. Umul, "Modified theory of physical optics," Opt. Eng. (Bellingham) 12, 4959-4972 (2004); http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-20-4959.

Wolf, E.

Yariv, A.

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (6)

H. D. Cheung and V. Jull, "Beam scattering by a right-angled impedance wedge," IEEE Trans. Antennas Propag. 52, 497-504 (2004).
[CrossRef]

R. Tiberio, G. Pelosi, G. Manara, and P. H. Pathak, "High-frequency scattering from a wedge with impedance faces illuminated by a line source--Part I: diffraction," IEEE Trans. Antennas Propag. 37, 212-218 (1989).
[CrossRef]

R. Tiberio, G. Pelosi, G. Manara, and P. H. Pathak, "High-frequency scattering from a wedge with impedance faces illuminated by a line source--Part II: surface waves," IEEE Trans. Antennas Propag. 41, 877-883 (1993).
[CrossRef]

G. A. Suedan and E. V. Jull, "Two-dimensional beam diffraction by a half-plane and wide slit," IEEE Trans. Antennas Propag. 35, 1077-1083 (1987).
[CrossRef]

G. Pelosi and S. Selleri, "A numerical approach for the diffraction of a Gaussian beam from a perfectly conducting wedge," IEEE Trans. Antennas Propag. 47, 1555-1559 (1999).
[CrossRef]

Y. Z. Umul, "Modified theory of physical optics solution of impedance half-plane problem," IEEE Trans. Antennas Propag. 54, 2048-2053 (2006).
[CrossRef]

J. Opt. Soc. Am. (5)

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

Opt. Eng. (Bellingham) (1)

Y. Z. Umul, "Modified theory of physical optics," Opt. Eng. (Bellingham) 12, 4959-4972 (2004); http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-20-4959.

SIAM Rev. (1)

S. J. Chapman, J. M. H. Lawry, J. R. Ockendon, and R. H. Tew, "On the theory of complex rays," SIAM Rev. 41, 417-509 (1999).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Geometry of the problem of diffraction by an impedance half-plane.

Fig. 2
Fig. 2

Stationary phase geometry for reflected scattered fields.

Fig. 3
Fig. 3

Total field, scattered by a line source illuminated PEC half-plane.

Fig. 4
Fig. 4

Reflected scattered field.

Fig. 5
Fig. 5

Incident scattered field.

Fig. 6
Fig. 6

Total scattered and diffracted fields.

Fig. 7
Fig. 7

Total scattered fields for different values of b.

Fig. 8
Fig. 8

Total diffracted fields for different values of b.

Fig. 9
Fig. 9

Total scattered fields for different values of the surface impedance.

Fig. 10
Fig. 10

Total scattered field for η i , r is equal to (a) π π 6 (b) π π 3 (c) π 7 π 12 .

Fig. 11
Fig. 11

Geometry of a complex line source.

Fig. 12
Fig. 12

Geometry of the complex image source.

Equations (74)

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E i = e z E i H 0 ( 2 ) ( k R 0 ) ,
E r s = e z j E i π 0 sin β + α 2 Γ ( β , α , θ ) e j k ( R 0 + R ) R R 0 d x ,
E i s = e z j E i π 0 sin β α 2 Γ ( β , α , θ ) e j k ( R 0 + R ) R R 0 d x ,
Γ ( β , α , θ ) = sin β + α 2 sin θ sin β + α 2 + sin θ ,
I = e j π 4 π 0 f ( u ) e j k g ( u ) d u
I e j k g ( u s ) G ( u s ) U ( t e ) + e j k g ( u s ) G ( 0 ) sgn ( t e ) F [ t e ]
F [ x ] = e j π 4 π x e j t 2 d t .
p ( u ) = k g ( u ) 2 t .
E r s = e j π 4 π 0 f ( x ) e j k g ( x ) d x ,
f ( x ) = E i e j π 4 π sin β + α 2 Γ ( β , α , θ ) R R 0 ,
g ( x ) = R R 0 ,
g ( x ) = ρ cos γ ρ 0 cos σ + x ( cos α cos β ) ,
g ( x ) = cos α cos β ,
f ( x s ) = E i e j π 4 π sin α s Γ ( α s , θ ) l l 0
Γ ( α s , θ ) = sin α s sin θ sin α s + sin θ .
t = k ( R + R 0 l l 0 ) .
t e = 2 k ρ ρ 0 ρ + ρ 0 + R 1 cos ϕ + ϕ 0 2 ,
p ( x s ) = k R 1 2 l l 0 sin α s ,
p ( 0 ) = k ( ρ + ρ 0 + R 1 ) 4 ρ ρ 0 cos ϕ ϕ 0 2 ,
E r s = e j π 4 2 π e j k R 1 k R 1 { U ( t e 1 ) Γ ( α s , θ ) + 2 R 1 ρ + ρ 0 + R 1 Γ ( π ϕ , ϕ 0 , θ ) sgn ( t e 1 ) F [ t e 1 ] }
E i s = e j π 4 2 π e j k R 2 k R 2 { U ( t e 2 ) 2 R 2 ρ + ρ 0 + R 2 Γ ( π ϕ , ϕ 0 , θ ) sgn ( t e 2 ) F [ t e 2 ] }
t e 2 = 2 k ρ ρ 0 ρ + ρ 0 + R 1 cos ϕ ϕ 0 2 ,
E r s = e j π 4 2 π e j k R 1 k R 1 { U ( t e 1 ) [ Γ ( α s , θ ) 2 R 1 ρ + ρ 0 + R 1 Γ ( π ϕ , ϕ 0 , θ ) ] + 2 R 1 ρ + ρ 0 + R 1 Γ ( π ϕ , ϕ 0 , θ ) F [ t e 1 ] } ,
E i s = e j π 4 2 π e j k R 2 k R 2 { U ( t e 2 ) [ 1 + 2 R 2 ρ + ρ 0 + R 2 Γ ( π ϕ , ϕ 0 , θ ) ] 2 R 2 ρ + ρ 0 + R 2 Γ ( π ϕ , ϕ 0 , θ ) F [ t e 2 ] } ,
E z = { 2 n = 1 J ϑ n ( k ρ ) H ϑ n ( 2 ) ( k ρ 0 ) sin ϑ n ϕ 0 sin ϑ n ϕ , ρ < ρ 0 2 n = 1 J ϑ n ( k ρ 0 ) H ϑ n ( 2 ) ( k ρ ) sin ϑ n ϕ 0 sin ϑ n ϕ , ρ > ρ 0 }
t e i = ρ + ρ 0 i R i ,
t e r = ρ + ρ 0 r R r .
ϕ RB = cos 1 1 cosh ϕ r 2 ϕ r 1 ,
ϕ SB = cos 1 1 cosh ϕ i 2 + ϕ i 1 ,
E t = E t GO + E t diff ,
E r GO = e j π 4 2 π e j k Re [ R r ] k R r e k Im [ R r ] Γ ( α s r , θ ) U ( ϕ R B ϕ ) .
E i GO = e j π 4 2 π e j k Re [ R i ] k R i e k Im [ R i ] U ( ϕ S B ϕ ) .
E r diff = e j π 4 2 π e j k Re [ R r ] k R r e k Im [ R r ] D r ,
D r = 2 R r ρ + ρ 0 r + R r Γ ( π ϕ , ϕ 0 r , θ ) { F [ t e r ] U ( ϕ R B ϕ ) } .
E i diff = e j π 4 2 π e j k Re [ R i ] k R i e k Im [ R i ] D i
D i = 2 R i ρ + ρ 0 i + R i Γ ( π ϕ , ϕ 0 i , θ ) { F [ t e i ] U ( ϕ S B ϕ ) } .
R i = ρ 0 i cos σ s i + ρ cos γ s i ,
σ s i = α s i ϕ 0 i ,
γ s i = π + ϕ α s i ,
ϕ 0 i = tan 1 y 0 j b sin η i x 0 j b cos η i .
ϕ i 1 = 1 2 tan 1 2 A i 1 A i 2 B i 2 ,
ϕ i 2 = 1 2 tanh 1 2 B i 1 + A i 2 + B i 2 ,
A i = x 0 y 0 + b 2 sin η i cos η i x 0 2 + b 2 cos 2 η i ,
B i = b ( y 0 cos η i x 0 sin η i ) x 0 2 + b 2 cos 2 η i .
α s i = tan 1 y 0 j b sin η i x 0 j b cos η i x s i ,
α s i = tan 1 y x s i x .
α i 1 = 1 2 tan 1 2 C i 1 C i 2 D i 2 ,
α i 2 = 1 2 tanh 1 2 D i 1 + C i 2 + D i 2 ,
tan α s i = y + y 0 j b sin η i x 0 x j b cos η i .
C i = ( y + y 0 ) ( x 0 x ) + b 2 sin η i cos η i ( x 0 x ) 2 + b 2 cos 2 η i ,
D i = b [ ( y + y 0 ) cos η i + ( x x 0 ) sin η i ] ( x 0 x ) 2 + b 2 cos 2 η i ,
ρ 0 i = ρ 0 cos ( ϕ 0 ϕ i 1 ) cosh ϕ i 2 + b sin ( η i ϕ i 1 ) sinh ϕ i 2 + j [ ρ 0 sin ( ϕ 0 ϕ i 1 ) sinh ϕ i 2 b cos ( η i ϕ i 1 ) cosh ϕ i 2 ] .
R i = Re [ R i ] + j Im [ R i ]
Re [ R i ] = Re [ ρ 0 i ] cos ( α i 1 ϕ i 1 ) cosh ( α i 2 ϕ i 2 ) + Im [ ρ 0 i ] sin ( α i 1 ϕ i 1 ) sinh ( α i 2 ϕ i 2 ) ρ cos ( ϕ α i 1 ) cosh α i 2 ,
Im [ R i ] = Im [ ρ 0 i ] cos ( α i 1 ϕ i 1 ) cosh ( α i 2 ϕ i 2 ) Re [ ρ 0 i ] sin ( α i 1 ϕ i 1 ) sinh ( α i 2 ϕ i 2 ) ρ sin ( ϕ α i 1 ) sinh α i 2 .
R r = ρ 0 r cos σ s r + ρ cos γ s r .
σ s r = α s r ϕ 0 r ,
γ s r = π ( ϕ + α s r ) ,
ϕ 0 r = tan 1 y 0 j b sin η r x 0 j b cos η r .
ϕ r 1 = 1 2 tan 1 2 A r 1 A r 2 B r 2 ,
ϕ r 2 = 1 2 tanh 1 2 B r 1 + A r 2 + B r 2 ,
A r = x 0 y 0 + b 2 sin η r cos η r x 0 2 + b 2 cos 2 η r
B r = b ( y 0 cos η r + x 0 sin η r ) x 0 2 + b 2 cos 2 η r .
α s r = tan 1 y 0 j b sin η r x 0 j b cos η r x s r ,
α s r = tan 1 y x s r x .
α r 1 = 1 2 tan 1 2 C r 1 C r 2 D r 2 ,
α r 2 = 1 2 tanh 1 2 D r 1 + C r 2 + D r 2 ,
tg α s r = y y 0 j b sin η r x 0 x j b cos η r .
C r = ( y y 0 ) ( x 0 x ) + b 2 sin η r cos η r ( x 0 x ) 2 + b 2 cos 2 η r ,
D r = b [ ( y y 0 ) cos η r + ( x x 0 ) sin η r ] ( x 0 x ) 2 + b 2 cos 2 η r ,
ρ 0 r = ρ 0 cos ( ϕ 0 + ϕ r 1 ) cosh ϕ r 2 + b sin ( η r ϕ r 1 ) sinh ϕ r 2 j [ ρ 0 sin ( ϕ 0 + ϕ r 1 ) sinh ϕ r 2 + b cos ( η r ϕ r 1 ) cosh ϕ r 2 ] .
R r = Re [ R r ] + j Im [ R r ]
Re [ R r ] = Re [ ρ 0 r ] cos ( α r 1 ϕ r 1 ) cosh ( α r 2 ϕ r 2 ) + Im [ ρ 0 r ] sin ( α r 1 ϕ r 1 ) sinh ( α r 2 ϕ r 2 ) ρ cos ( ϕ + α r 1 ) c h α r 2 ,
Im [ R r ] = Im [ ρ 0 r ] cos ( α r 1 ϕ r 1 ) cosh ( α r 2 ϕ r 2 ) Re [ ρ 0 r ] sin ( α r 1 ϕ r 1 ) sinh ( α r 2 ϕ r 2 ) + ρ sin ( ϕ + α r 1 ) sinh α r 2 .

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