Abstract

The scattering process of electromagnetic plane waves by a resistive half-screen is investigated for oblique incidence. First of all, it is shown that the existing solution in the literature is not correct, because the problem was solved by considering the normal components of the electromagnetic field, in terms of which the boundary conditions cannot be expressed. Instead of these, the components of the electric field, which is parallel to the edge discontinuity, are taken into account. The diffracted fields are obtained with the aid of the method of transition boundary. The uniform field expressions are obtained by using the Fresnel function. The behaviors of the total field and its subcomponents are analyzed numerically.

© 2019 Optical Society of America

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References

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  1. A. Sommerfeld, “Mathematische theorie der diffraction,” Math. Ann. 47, 317–374 (1896).
    [Crossref]
  2. C. V. Raman and H. S. Krishnan, “The diffraction of light by metallic screens,” Proc. R. Soc. London A 116, 254–267 (1927).
    [Crossref]
  3. Y. Z. Umul, “Scattering by an impedance half-plane: comparison of the solutions of Raman/Krishnan and Maliuzhinets/Senior,” Prog. Electromagn. Res. M 8, 39–50 (2009).
    [Crossref]
  4. T. B. A. Senior, “Diffraction by a semi-infinite metallic sheet,” Proc. R. Soc. London A 213, 436–458 (1952).
    [Crossref]
  5. G. D. Malyuzhinets, “Das Sommerfeldsche Integral und die Lösung von Beugungsaufgaben in Winkelgebieten,” Ann. Phys. (Leipzig) 461, 107–112 (1960).
    [Crossref]
  6. T. B. A. Senior, “Half plane edge diffraction,” Radio Sci. 10, 645–650 (1975).
    [Crossref]
  7. T. B. A. Senior, “Diffraction by a resistive half plane,” Electromagnetics 11, 183–192 (1991).
    [Crossref]
  8. P. C. Clemmow, “A method for the exact solution of a class of two-dimensional diffraction problems,” Proc. R. Soc. London A 205, 286–308 (1951).
    [Crossref]
  9. G. A. Deschamps, “Diffraction of an arbitrary plane electromagnetic wave by a half-plane,” IEEE Trans. Antennas Propag. 21, 126–127 (1973).
    [Crossref]
  10. T. B. A. Senior, “Diffraction tensors for imperfectly conducting edges,” Radio Sci. 10, 911–919 (1975).
    [Crossref]
  11. T. B. A. Senior, “Skew incidence on a material junction,” Radio Sci. 26, 305–311 (1991).
    [Crossref]
  12. T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics (IEE, 1995).
  13. G. A. Grünberg, “Theory of the coastal refraction of electromagnetic waves,” J. Phys. USSR 6, 185–209 (1942).
  14. G. Grünberg, “Suggestion for a theory of the coastal refraction,” Phys. Rev. 63, 185–189 (1943).
    [Crossref]
  15. C. Pfeiffer and A. Grbic, “Metamaterial Huygens’ surfaces: tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110, 197401 (2013).
    [Crossref]
  16. Y. Z. Umul, “Wave diffraction by a reflectionless half-plane,” Appl. Opt. 56, 9293–9300 (2017).
    [Crossref]
  17. Y. Z. Umul, “Edge diffraction in an anomalously reflecting half-plane,” Opt. Quantum Electron. 50, 334 (2018).
    [Crossref]
  18. Y. Z. Umul, “The method of transition boundary for the solution of diffraction problems,” Opt. Quantum Electron. 51, 181 (2019).
    [Crossref]
  19. E. V. Jull, Aperture Antennas and Diffraction Theory (Peter Peregrinus, 1981).
  20. Y. Z. Umul, “Three dimensional modified theory of physical optics,” Optik 127, 819–824 (2016).
    [Crossref]
  21. Y. Z. Umul, “Equivalent functions for the Fresnel integral,” Opt. Express 13, 8469–8482 (2005).
    [Crossref]

2019 (1)

Y. Z. Umul, “The method of transition boundary for the solution of diffraction problems,” Opt. Quantum Electron. 51, 181 (2019).
[Crossref]

2018 (1)

Y. Z. Umul, “Edge diffraction in an anomalously reflecting half-plane,” Opt. Quantum Electron. 50, 334 (2018).
[Crossref]

2017 (1)

2016 (1)

Y. Z. Umul, “Three dimensional modified theory of physical optics,” Optik 127, 819–824 (2016).
[Crossref]

2013 (1)

C. Pfeiffer and A. Grbic, “Metamaterial Huygens’ surfaces: tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110, 197401 (2013).
[Crossref]

2009 (1)

Y. Z. Umul, “Scattering by an impedance half-plane: comparison of the solutions of Raman/Krishnan and Maliuzhinets/Senior,” Prog. Electromagn. Res. M 8, 39–50 (2009).
[Crossref]

2005 (1)

1991 (2)

T. B. A. Senior, “Skew incidence on a material junction,” Radio Sci. 26, 305–311 (1991).
[Crossref]

T. B. A. Senior, “Diffraction by a resistive half plane,” Electromagnetics 11, 183–192 (1991).
[Crossref]

1975 (2)

T. B. A. Senior, “Half plane edge diffraction,” Radio Sci. 10, 645–650 (1975).
[Crossref]

T. B. A. Senior, “Diffraction tensors for imperfectly conducting edges,” Radio Sci. 10, 911–919 (1975).
[Crossref]

1973 (1)

G. A. Deschamps, “Diffraction of an arbitrary plane electromagnetic wave by a half-plane,” IEEE Trans. Antennas Propag. 21, 126–127 (1973).
[Crossref]

1960 (1)

G. D. Malyuzhinets, “Das Sommerfeldsche Integral und die Lösung von Beugungsaufgaben in Winkelgebieten,” Ann. Phys. (Leipzig) 461, 107–112 (1960).
[Crossref]

1952 (1)

T. B. A. Senior, “Diffraction by a semi-infinite metallic sheet,” Proc. R. Soc. London A 213, 436–458 (1952).
[Crossref]

1951 (1)

P. C. Clemmow, “A method for the exact solution of a class of two-dimensional diffraction problems,” Proc. R. Soc. London A 205, 286–308 (1951).
[Crossref]

1943 (1)

G. Grünberg, “Suggestion for a theory of the coastal refraction,” Phys. Rev. 63, 185–189 (1943).
[Crossref]

1942 (1)

G. A. Grünberg, “Theory of the coastal refraction of electromagnetic waves,” J. Phys. USSR 6, 185–209 (1942).

1927 (1)

C. V. Raman and H. S. Krishnan, “The diffraction of light by metallic screens,” Proc. R. Soc. London A 116, 254–267 (1927).
[Crossref]

1896 (1)

A. Sommerfeld, “Mathematische theorie der diffraction,” Math. Ann. 47, 317–374 (1896).
[Crossref]

Clemmow, P. C.

P. C. Clemmow, “A method for the exact solution of a class of two-dimensional diffraction problems,” Proc. R. Soc. London A 205, 286–308 (1951).
[Crossref]

Deschamps, G. A.

G. A. Deschamps, “Diffraction of an arbitrary plane electromagnetic wave by a half-plane,” IEEE Trans. Antennas Propag. 21, 126–127 (1973).
[Crossref]

Grbic, A.

C. Pfeiffer and A. Grbic, “Metamaterial Huygens’ surfaces: tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110, 197401 (2013).
[Crossref]

Grünberg, G.

G. Grünberg, “Suggestion for a theory of the coastal refraction,” Phys. Rev. 63, 185–189 (1943).
[Crossref]

Grünberg, G. A.

G. A. Grünberg, “Theory of the coastal refraction of electromagnetic waves,” J. Phys. USSR 6, 185–209 (1942).

Jull, E. V.

E. V. Jull, Aperture Antennas and Diffraction Theory (Peter Peregrinus, 1981).

Krishnan, H. S.

C. V. Raman and H. S. Krishnan, “The diffraction of light by metallic screens,” Proc. R. Soc. London A 116, 254–267 (1927).
[Crossref]

Malyuzhinets, G. D.

G. D. Malyuzhinets, “Das Sommerfeldsche Integral und die Lösung von Beugungsaufgaben in Winkelgebieten,” Ann. Phys. (Leipzig) 461, 107–112 (1960).
[Crossref]

Pfeiffer, C.

C. Pfeiffer and A. Grbic, “Metamaterial Huygens’ surfaces: tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110, 197401 (2013).
[Crossref]

Raman, C. V.

C. V. Raman and H. S. Krishnan, “The diffraction of light by metallic screens,” Proc. R. Soc. London A 116, 254–267 (1927).
[Crossref]

Senior, T. B. A.

T. B. A. Senior, “Diffraction by a resistive half plane,” Electromagnetics 11, 183–192 (1991).
[Crossref]

T. B. A. Senior, “Skew incidence on a material junction,” Radio Sci. 26, 305–311 (1991).
[Crossref]

T. B. A. Senior, “Diffraction tensors for imperfectly conducting edges,” Radio Sci. 10, 911–919 (1975).
[Crossref]

T. B. A. Senior, “Half plane edge diffraction,” Radio Sci. 10, 645–650 (1975).
[Crossref]

T. B. A. Senior, “Diffraction by a semi-infinite metallic sheet,” Proc. R. Soc. London A 213, 436–458 (1952).
[Crossref]

T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics (IEE, 1995).

Sommerfeld, A.

A. Sommerfeld, “Mathematische theorie der diffraction,” Math. Ann. 47, 317–374 (1896).
[Crossref]

Umul, Y. Z.

Y. Z. Umul, “The method of transition boundary for the solution of diffraction problems,” Opt. Quantum Electron. 51, 181 (2019).
[Crossref]

Y. Z. Umul, “Edge diffraction in an anomalously reflecting half-plane,” Opt. Quantum Electron. 50, 334 (2018).
[Crossref]

Y. Z. Umul, “Wave diffraction by a reflectionless half-plane,” Appl. Opt. 56, 9293–9300 (2017).
[Crossref]

Y. Z. Umul, “Three dimensional modified theory of physical optics,” Optik 127, 819–824 (2016).
[Crossref]

Y. Z. Umul, “Scattering by an impedance half-plane: comparison of the solutions of Raman/Krishnan and Maliuzhinets/Senior,” Prog. Electromagn. Res. M 8, 39–50 (2009).
[Crossref]

Y. Z. Umul, “Equivalent functions for the Fresnel integral,” Opt. Express 13, 8469–8482 (2005).
[Crossref]

Volakis, J. L.

T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics (IEE, 1995).

Ann. Phys. (Leipzig) (1)

G. D. Malyuzhinets, “Das Sommerfeldsche Integral und die Lösung von Beugungsaufgaben in Winkelgebieten,” Ann. Phys. (Leipzig) 461, 107–112 (1960).
[Crossref]

Appl. Opt. (1)

Electromagnetics (1)

T. B. A. Senior, “Diffraction by a resistive half plane,” Electromagnetics 11, 183–192 (1991).
[Crossref]

IEEE Trans. Antennas Propag. (1)

G. A. Deschamps, “Diffraction of an arbitrary plane electromagnetic wave by a half-plane,” IEEE Trans. Antennas Propag. 21, 126–127 (1973).
[Crossref]

J. Phys. USSR (1)

G. A. Grünberg, “Theory of the coastal refraction of electromagnetic waves,” J. Phys. USSR 6, 185–209 (1942).

Math. Ann. (1)

A. Sommerfeld, “Mathematische theorie der diffraction,” Math. Ann. 47, 317–374 (1896).
[Crossref]

Opt. Express (1)

Opt. Quantum Electron. (2)

Y. Z. Umul, “Edge diffraction in an anomalously reflecting half-plane,” Opt. Quantum Electron. 50, 334 (2018).
[Crossref]

Y. Z. Umul, “The method of transition boundary for the solution of diffraction problems,” Opt. Quantum Electron. 51, 181 (2019).
[Crossref]

Optik (1)

Y. Z. Umul, “Three dimensional modified theory of physical optics,” Optik 127, 819–824 (2016).
[Crossref]

Phys. Rev. (1)

G. Grünberg, “Suggestion for a theory of the coastal refraction,” Phys. Rev. 63, 185–189 (1943).
[Crossref]

Phys. Rev. Lett. (1)

C. Pfeiffer and A. Grbic, “Metamaterial Huygens’ surfaces: tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110, 197401 (2013).
[Crossref]

Proc. R. Soc. London A (3)

P. C. Clemmow, “A method for the exact solution of a class of two-dimensional diffraction problems,” Proc. R. Soc. London A 205, 286–308 (1951).
[Crossref]

C. V. Raman and H. S. Krishnan, “The diffraction of light by metallic screens,” Proc. R. Soc. London A 116, 254–267 (1927).
[Crossref]

T. B. A. Senior, “Diffraction by a semi-infinite metallic sheet,” Proc. R. Soc. London A 213, 436–458 (1952).
[Crossref]

Prog. Electromagn. Res. M (1)

Y. Z. Umul, “Scattering by an impedance half-plane: comparison of the solutions of Raman/Krishnan and Maliuzhinets/Senior,” Prog. Electromagn. Res. M 8, 39–50 (2009).
[Crossref]

Radio Sci. (3)

T. B. A. Senior, “Half plane edge diffraction,” Radio Sci. 10, 645–650 (1975).
[Crossref]

T. B. A. Senior, “Diffraction tensors for imperfectly conducting edges,” Radio Sci. 10, 911–919 (1975).
[Crossref]

T. B. A. Senior, “Skew incidence on a material junction,” Radio Sci. 26, 305–311 (1991).
[Crossref]

Other (2)

T. B. A. Senior and J. L. Volakis, Approximate Boundary Conditions in Electromagnetics (IEE, 1995).

E. V. Jull, Aperture Antennas and Diffraction Theory (Peter Peregrinus, 1981).

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Figures (7)

Fig. 1.
Fig. 1. Diffraction of an obliquely incident wave from the edge of a resistive half-screen.
Fig. 2.
Fig. 2. Bird’s eye view of the incident wave for perpendicular and skew incidence.
Fig. 3.
Fig. 3. $x$ components of the total GO and diffracted electric fields.
Fig. 4.
Fig. 4. $x$ component of the total electric field.
Fig. 5.
Fig. 5. $z$ components of the total GO and diffracted electric fields.
Fig. 6.
Fig. 6. $z$ component of the total electric field.
Fig. 7.
Fig. 7. $z$ component of the total electric field for different values of ${\theta_0}$.

Equations (76)

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E i = E 0 p ei e i
e i = e j k ( x sin θ 0 cos ϕ 0 + y sin θ 0 sin ϕ 0 + z cos θ 0 ) ,
s i = sin θ 0 cos ϕ 0 e x sin θ 0 sin ϕ 0 e y cos θ 0 e z ,
n × ( E + E ) | S = 0
n × ( n × E + ) | S = R e n × ( H + H ) | S ,
E i = E 0 p ei e i ,
p ei = cos θ 0 cos ϕ 0 e x cos θ 0 sin ϕ 0 e y + sin θ 0 e z
H i = E 0 Z 0 p h i e i ,
p h i = sin ϕ 0 e x + cos ϕ 0 e y ,
E r = E r p er e r ,
H r = E r Z 0 p h r e r ,
E t = E t p ei e i ,
H t = E t Z 0 p h i e i ,
p er = cos θ 0 cos ϕ 0 e x + cos θ 0 sin ϕ 0 e y + sin θ 0 e z ,
p h r = sin ϕ 0 e x + cos ϕ 0 e y ,
e r = e j k ( x sin θ 0 cos ϕ 0 y sin θ 0 sin ϕ 0 + z cos θ 0 ) ,
E 0 + E r = E t ,
cos θ 0 cos ϕ 0 ( E 0 + E r ) = 0
sin θ 0 ( E 0 + E r ) = R e sin ϕ 0 ( H i H r H t )
p ei = sin α e x + cos α e z ,
s i = cos β cos α e x sin β e y cos β sin α e z ,
α = tan 1 cos θ 0 sin θ 0 cos ϕ 0
β = tan 1 sin θ 0 sin ϕ 0 cos 2 θ 0 + sin 2 θ 0 cos 2 ϕ 0
sin β = sin θ 0 sin ϕ 0
cos β = cos 2 θ 0 + sin 2 θ 0 cos 2 ϕ 0 ,
H i = E 0 Z 0 p h i e i ,
p h i = sin β cos α e x + cos β e y sin β sin α e z ,
E i = E 0 p ei e i
H i = E 0 Z 0 p h i e i ,
E r = E r p er e r
H r = E r Z 0 p h r e r ,
p er = sin α e x + cos α e z
p h r = sin β cos α e x + cos β e y + sin β sin α e z ,
E t = E t p ei e i
H t = E t Z 0 p h i e i
E T G O = E i U ( ξ ) + E r U ( ξ + ) + E t U ( ξ )
H T G O = H i U ( ξ ) + H r U ( ξ + ) + H t U ( ξ )
ξ ± = 2 k ρ sin θ 0 cos ϕ ± ϕ 0 2 ,
E S G O = E r U ( ξ + ) + ( E t E i ) U ( ξ )
H S G O = H r U ( ξ + ) + ( H t H i ) U ( ξ )
E r = E 0 sin γ sin β + sin γ
E t = E 0 sin β sin β + sin γ
E r = E 0 sin γ sin θ 0 sin ϕ 0 + sin γ
E t = E 0 sin θ 0 sin ϕ 0 sin θ 0 sin ϕ 0 + sin γ
E S G O = E 0 sin γ sin θ 0 sin ϕ 0 + sin γ × [ p er e r U ( ξ + ) + p ei e i U ( ξ ) ]
H S G O = E 0 Z 0 sin γ sin θ 0 sin ϕ 0 + sin γ × [ p h r e r U ( ξ + ) + p h i e i U ( ξ ) ] ,
E d = e j π 4 2 π f ( ϕ , ϕ 0 ) cos ϕ + cos ϕ 0 e j k ρ sin θ 0 k ρ sin θ 0 e j k z cos θ 0
f ( π ϕ 0 , ϕ 0 ) = sin ϕ 0 A [ E S G O ]
E S G O x = E 0 cos θ 0 sin γ sin θ 0 sin ϕ 0 + sin γ [ e r U ( ξ + ) + e i U ( ξ ) ] cos 2 θ 0 + sin 2 θ 0 cos 2 ϕ 0
E S G O z = E 0 sin θ 0 cos ϕ 0 sin γ sin θ 0 sin ϕ 0 + sin γ [ e r U ( ξ + ) + e i U ( ξ ) ] cos 2 θ 0 + sin 2 θ 0 cos 2 ϕ 0 ,
f x ( π ϕ 0 , ϕ 0 ) = E 0 cos θ 0 cos 2 θ 0 + sin 2 θ 0 cos 2 ϕ 0 × sin ϕ 0 sin η sin ϕ 0 + sin η
f z ( π ϕ 0 , ϕ 0 ) = E 0 sin θ 0 cos ϕ 0 cos 2 θ 0 + sin 2 θ 0 cos 2 ϕ 0 × sin ϕ 0 sin η sin ϕ 0 + sin η
sin η = sin γ sin θ 0 .
f x ( π ϕ 0 , ϕ 0 ) = E 0 cos θ 0 K + ( π ϕ 0 , η ) K + ( ϕ 0 , η ) cos 2 θ 0 + g 2 ( π ϕ 0 , ϕ 0 ) sin 2 θ 0
f z ( π ϕ 0 , ϕ 0 ) = E 0 g ( π ϕ 0 , ϕ 0 ) sin θ 0 K + ( π ϕ 0 , η ) K + ( ϕ 0 , η ) cos 2 θ 0 + g 2 ( π ϕ 0 , ϕ 0 ) sin 2 θ 0 ,
K + ( π x , y ) K + ( x , y ) = sin x sin y sin x + sin y
g ( π ± ϕ 0 , ϕ 0 ) = 1 4 sin 2 π ± ϕ 0 2 sin 2 ϕ 0 2 .
f ( ϕ , ϕ 0 ) = f x ( ϕ , ϕ 0 ) e x + f z ( ϕ , ϕ 0 ) e z ,
f x ( ϕ , ϕ 0 ) = E 0 cos θ 0 K + ( ϕ , η ) K + ( ϕ 0 , η ) cos 2 θ 0 + g 2 ( ϕ , ϕ 0 ) sin 2 θ 0
f z ( ϕ , ϕ 0 ) = E 0 g ( ϕ , ϕ 0 ) sin θ 0 K + ( ϕ , η ) K + ( ϕ 0 , η ) cos 2 θ 0 + g 2 ( ϕ , ϕ 0 ) sin 2 θ 0 ,
g ( ϕ , ϕ 0 ) = 1 4 sin 2 ϕ 2 sin 2 ϕ 0 2 .
E d = e j π 4 2 π f x ( ϕ , ϕ 0 ) e x + f z ( ϕ , ϕ 0 ) e z cos ϕ + cos ϕ 0 × e j k ρ sin θ 0 k ρ sin θ 0 e j k z cos θ 0 ,
f x ( ϕ , ϕ 0 ) = 0
f z ( ϕ , ϕ 0 ) = E 0 K + ( ϕ , γ ) K + ( ϕ 0 , γ ) ,
E d = e j π 4 2 π K + ( ϕ , γ ) K + ( ϕ 0 , γ ) cos ϕ + cos ϕ 0 e j k ρ 0 k ρ e z ,
s i g n ( x ) F [ | x | ] e j ( π 4 + x 2 ) 2 x π
F [ x ] = e j π 4 π x e j v 2 d v .
q S e j π 4 2 π 2 sin ϕ 2 sin ϕ 0 2 cos ϕ + cos ϕ 0 e j k ρ k ρ ,
q S = e j k ρ cos ( ϕ ϕ 0 ) s i g n ( ξ ) F [ | ξ | ] e j k ρ cos ( ϕ + ϕ 0 ) s i g n ( ξ + ) F [ | ξ + | ] ,
E d = f x ( ϕ , ϕ 0 ) e x + f z ( ϕ , ϕ 0 ) e z 2 sin ϕ 2 sin ϕ 0 2 q S e j k z cos θ 0 ,
q S = e j k ρ sin θ 0 cos ( ϕ ϕ 0 ) s i g n ( ξ ) F [ | ξ | ] e j k ρ cos sin θ 0 ( ϕ + ϕ 0 ) s i g n ( ξ + ) F [ | ξ + | ]
E T G O = E i U ( ξ ) + E r U ( ξ + ) + E t U ( ξ ) ,
E d = f x ( ϕ , ϕ 0 ) e x + f z ( ϕ , ϕ 0 ) e z 2 sin ϕ 2 sin ϕ 0 2 q S e j k z cos θ 0 ,
E T = E T G O + E d ,
ρ = r sin θ
z = r cos θ ,

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