Abstract

In this paper, a two-dimensional high-frequency formalism is presented which describes the diffraction of arbitrary wavefronts incident on edges of an otherwise smooth surface. The diffracted field in all points of a predefined region of interest is expressed in terms of the generalized Huygens representation of the incident field and a limited set of translation coefficients that take into account the arbitrary nature of the incident wavefront and its diffraction. The method is based on the Uniform Theory of Diffraction (UTD) and can therefore be utilized for every canonical problem for which the UTD diffraction coefficient is known. Moreover, the proposed technique is easy to implement as only standard Fast Fourier Transform (FFT) routines are required. The technique’s validity is confirmed both theoretically and numerically. It is shown that for fields emitted by a discrete line source and diffracted by a perfectly conducting wedge, the method is in excellent agreement with the analytic solution over the entire simulation domain, including regions near shadow and reflection boundaries. As an application example, the diffraction in the presence of a perfectly conducting wedge illuminated by a complex light source is analyzed, demonstrating the appositeness of the method.

© 2013 Optical Society of America

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References

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  1. G. Pelosi and S. Selleri, “The wedge-type problem: The building brick in high-frequency scattering from complex objects,” IEEE Antennas Propag. Mag.55, 41–58 (2013).
    [CrossRef]
  2. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am.52, 116–130 (1961).
    [CrossRef]
  3. R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE62, 1448–1461 (1974).
    [CrossRef]
  4. P. Pathak, W. Burnside, and R. J. Marhefka, “A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface,” IEEE Trans. Antennas Propag.28, 631–642 (1980).
    [CrossRef]
  5. Y. Rahmat-Samii and R. Mittra, “Spectral analysis of high-frequency diffraction of an arbitrary incident field by a half plane-comparison with four asymptotic techniques,” Radio Sci.13, 31–48 (1978).
    [CrossRef]
  6. Y. Z. Umul, “Scattering of a line source by a cylindrical parabolic impedance surface,” J. Opt. Soc. Am. A25, 1652–1659 (2008).
    [CrossRef]
  7. “NIST digital library of mathematical functions,” http://dlmf.nist.gov/ , Release 1.0.5 of 2012-10-01. Online companion to [12].
  8. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, 1994).
    [CrossRef]
  9. Y. Z. Umul, “Scattering of a Gaussian beam by an impedance half-plane,” J. Opt. Soc. Am. A24, 3159–3167 (2007).
    [CrossRef]
  10. 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]
  11. G. Suedan and E. Jull, “Beam diffraction by half planes and wedges: Uniform and asymptotic solutions,” J Electromagnet Wave3, 17–26 (1989).
    [CrossRef]
  12. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010). Print companion to [7].

2013 (1)

G. Pelosi and S. Selleri, “The wedge-type problem: The building brick in high-frequency scattering from complex objects,” IEEE Antennas Propag. Mag.55, 41–58 (2013).
[CrossRef]

2008 (1)

2007 (1)

1989 (1)

G. Suedan and E. Jull, “Beam diffraction by half planes and wedges: Uniform and asymptotic solutions,” J Electromagnet Wave3, 17–26 (1989).
[CrossRef]

1980 (1)

P. Pathak, W. Burnside, and R. J. Marhefka, “A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface,” IEEE Trans. Antennas Propag.28, 631–642 (1980).
[CrossRef]

1979 (1)

1978 (1)

Y. Rahmat-Samii and R. Mittra, “Spectral analysis of high-frequency diffraction of an arbitrary incident field by a half plane-comparison with four asymptotic techniques,” Radio Sci.13, 31–48 (1978).
[CrossRef]

1974 (1)

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE62, 1448–1461 (1974).
[CrossRef]

1961 (1)

Bertoni, H. L.

Boisvert, R. F.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010). Print companion to [7].

Burnside, W.

P. Pathak, W. Burnside, and R. J. Marhefka, “A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface,” IEEE Trans. Antennas Propag.28, 631–642 (1980).
[CrossRef]

Clark, C. W.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010). Print companion to [7].

Felsen, L. B.

Green, A. C.

Jull, E.

G. Suedan and E. Jull, “Beam diffraction by half planes and wedges: Uniform and asymptotic solutions,” J Electromagnet Wave3, 17–26 (1989).
[CrossRef]

Keller, J. B.

Kouyoumjian, R. G.

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE62, 1448–1461 (1974).
[CrossRef]

Lozier, D. W.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010). Print companion to [7].

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, 1994).
[CrossRef]

Marhefka, R. J.

P. Pathak, W. Burnside, and R. J. Marhefka, “A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface,” IEEE Trans. Antennas Propag.28, 631–642 (1980).
[CrossRef]

Mittra, R.

Y. Rahmat-Samii and R. Mittra, “Spectral analysis of high-frequency diffraction of an arbitrary incident field by a half plane-comparison with four asymptotic techniques,” Radio Sci.13, 31–48 (1978).
[CrossRef]

Olver, F. W. J.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010). Print companion to [7].

Pathak, P.

P. Pathak, W. Burnside, and R. J. Marhefka, “A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface,” IEEE Trans. Antennas Propag.28, 631–642 (1980).
[CrossRef]

Pathak, P. H.

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE62, 1448–1461 (1974).
[CrossRef]

Pelosi, G.

G. Pelosi and S. Selleri, “The wedge-type problem: The building brick in high-frequency scattering from complex objects,” IEEE Antennas Propag. Mag.55, 41–58 (2013).
[CrossRef]

Rahmat-Samii, Y.

Y. Rahmat-Samii and R. Mittra, “Spectral analysis of high-frequency diffraction of an arbitrary incident field by a half plane-comparison with four asymptotic techniques,” Radio Sci.13, 31–48 (1978).
[CrossRef]

Selleri, S.

G. Pelosi and S. Selleri, “The wedge-type problem: The building brick in high-frequency scattering from complex objects,” IEEE Antennas Propag. Mag.55, 41–58 (2013).
[CrossRef]

Suedan, G.

G. Suedan and E. Jull, “Beam diffraction by half planes and wedges: Uniform and asymptotic solutions,” J Electromagnet Wave3, 17–26 (1989).
[CrossRef]

Umul, Y. Z.

IEEE Antennas Propag. Mag. (1)

G. Pelosi and S. Selleri, “The wedge-type problem: The building brick in high-frequency scattering from complex objects,” IEEE Antennas Propag. Mag.55, 41–58 (2013).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

P. Pathak, W. Burnside, and R. J. Marhefka, “A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface,” IEEE Trans. Antennas Propag.28, 631–642 (1980).
[CrossRef]

J Electromagnet Wave (1)

G. Suedan and E. Jull, “Beam diffraction by half planes and wedges: Uniform and asymptotic solutions,” J Electromagnet Wave3, 17–26 (1989).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Proc. IEEE (1)

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE62, 1448–1461 (1974).
[CrossRef]

Radio Sci. (1)

Y. Rahmat-Samii and R. Mittra, “Spectral analysis of high-frequency diffraction of an arbitrary incident field by a half plane-comparison with four asymptotic techniques,” Radio Sci.13, 31–48 (1978).
[CrossRef]

Other (3)

“NIST digital library of mathematical functions,” http://dlmf.nist.gov/ , Release 1.0.5 of 2012-10-01. Online companion to [12].

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, 1994).
[CrossRef]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010). Print companion to [7].

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Figures (5)

Fig. 1
Fig. 1

Canonical problem geometry. A single line source illuminates the wedge, leading to a (diffracted) field at a single, discrete observation point. The shadow boundary and the reflection boundary are also indicated.

Fig. 2
Fig. 2

Canonical problem geometry. An arbitrary, spatially distributed light source (hatched light gray) illuminates the wedge, leading to a (diffracted) field within a user-defined, spatially distributed region of interest (hatched light gray).

Fig. 3
Fig. 3

Configuration for the numerical validation. The region of interest is held at a constant distance from the edge, while changing its angular position. Its trajectory is indicated by means of the dotted line.

Fig. 4
Fig. 4

Top panel: total field at the varying observation point obtained via the exact solution of Eq. (28) (black line) and via the proposed technique of Eq. (19), where the result indicated by crosses (x) is computed relying on the traditional diffraction coefficient, and whereas the result indicated by circles (o) is based on the equivalent UTD coefficient of Eq. (23). Bottom panel: Absolute error. Note that the maximum relative error (not shown in the figure) remains bounded to about 1% when the equivalent UTD coefficient is used.

Fig. 5
Fig. 5

The total field for an incoming Gaussian beam in the presence of a perfectly conducting wedge.

Equations (31)

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E z diff ( ρ d ) = J s ω μ 0 4 H 0 ( 2 ) ( k ρ d ) D U T D ( L ; ρ d , ρ d ) e j k ρ d ρ d ,
D U T D ( L ; ρ d , ρ d ) = 1 j 4 ( 2 π α ) π k [ cot ( π ( π + ψ o ψ o ) 2 ( 2 π α ) ) F ( k L a + ( ψ o ψ o ) ) + cot ( π ( π ψ o + ψ o ) 2 ( 2 π α ) ) F ( k L a ( ψ o ψ o ) ) cot ( π ( π + ψ o + ψ o ) 2 ( 2 π α ) ) F ( k L a + ( ψ o + ψ o ) ) cot ( π ( π ψ o ψ o ) 2 ( 2 π α ) ) F ( k L a ( ψ o + ψ o ) ) ] ,
F ( X ) = 2 j X e j X X + d τ e j τ 2
a ± ( β ) = 2 cos 2 ( 2 ( 2 π α ) N ± β 2 ) ,
2 ( 2 π α ) N ± β = ± π .
L = ρ d ρ d ρ d + ρ d ,
E z inc ( ρ ) = ω μ 0 4 q = a q H q ( 2 ) ( k ρ ) e j q ϕ , ρ > R ,
a q = 2 π ω μ 0 H q ( 2 ) ( k R ) π π d ϕ b E z inc ( R ) e j q ϕ b .
E z inc ( ρ ) = ω μ 0 4 π π R d ϕ b 𝒥 z ( R ) H 0 ( 2 ) ( k | ρ R | ) .
𝒥 z ( R ) = q = I q e j q ϕ b ,
H 0 ( 2 ) ( k | ρ R | ) = m = H m ( 2 ) ( k ρ ) J m ( k R ) e j m ( ϕ ϕ b ) , ρ > R ,
I q = a q 2 π R J q ( k R ) .
E z inc ( ρ ) = ω μ 0 4 q = a q 2 π J q ( k R ) π π d ϕ b H 0 ( 2 ) ( k | ρ R | ) e j q ϕ b ,
E z diff ( ρ ) = ω μ 0 4 e j k | ρ d + ρ | | ρ d + ρ | q = a q 2 π J q ( k R ) × π π d ϕ b D U T D ( L ; ρ d R , ρ d + ρ ) e j q ϕ b H 0 ( 2 ) ( k | ρ d R | ) .
E z diff ( ρ ) = ω μ 0 4 q = b q J q ( k ρ ) e j q ϕ ,
b q = 1 2 π J q ( k R ) q = a q 2 π J q ( k R ) π π d ϕ b e j k | ρ d + R | | ρ d + R | × π π d ϕ b D U T D ( L ; ρ d R , ρ d + R ) e j q ϕ b H 0 ( 2 ) ( k | ρ d R | ) .
H 0 ( 2 ) ( k | ρ d + R | ) ( 1 + j ) e j k | ρ d + R | π k | ρ d + R | , k | ρ d + R | 1 ,
b q 1 2 π J q ( k R ) q = a q 2 π J q ( k R ) 1 j 2 π k π π d ϕ b H 0 ( 2 ) ( k | ρ d + R | ) × π π d ϕ b D U T D ( L ; ρ d R , ρ d + R ) e j q ϕ b H 0 ( 2 ) ( k | ρ d + R | ) .
E z diff ( ρ ) ω μ 0 4 q = q = J q ( k ρ ) e j q ϕ J q ( k R ) T q , q a q J q ( k R ) ,
T q , q = n = H n ( 2 ) ( k ρ d ) J n ( k R ) e j n ϕ o m = 1 j 2 π k d n + q , m q H m ( 2 ) ( k ρ d ) J m ( k R ) e j m ϕ o .
d s , l = 1 4 π 2 π π d ϕ b e j s ϕ b π π d ϕ b e j l ϕ b D U T D ( L ; ρ d R , ρ d + R ) ,
E z tot ( ρ ; ρ ) = E diff ( ρ ; ρ ) + E refl ( ρ ; ρ ) + E direct ( ρ ; ρ ) ω μ 0 4 H 0 ( 2 ) ( k | ρ d ρ | ) D U T D ( L ; ρ d ρ , ρ d + ρ ) × 1 j 2 π k H 0 ( 2 ) ( k | ρ d + ρ | ) + ω μ 0 4 H 0 ( 2 ) ( k | ρ d ρ ρ d + ρ | ) u refl ω μ 0 4 H 0 ( 2 ) ( k | ρ d ρ ρ d + ρ | ) u direct ω μ 0 4 H 0 ( 2 ) ( k | ρ d ρ | ) D ( L ; ρ d ρ , ρ d + ρ ) × 1 j 2 π k H 0 ( 2 ) ( k | ρ d + ρ | ) .
D ( L ; ρ d ρ , ρ d + ρ ) = D U T D ( L ; ρ d ρ , ρ d + ρ ) + 1 + j π k 1 H 0 ( 2 ) ( k | ρ d ρ | ) H 0 ( 2 ) ( k | ρ d + ρ | ) × [ H 0 ( 2 ) ( k | ρ d ρ ρ d + ρ | ) u refl + H 0 ( 2 ) ( k | ρ d ρ ρ d + ρ | ) u direct ] .
d s , l ~ D U T D ( L ; ρ d , ρ d ) δ s , 0 δ l , 0 ,
E z diff ( ρ ) ~ ω μ 0 4 q = a q H q ( 2 ) ( k ρ d ) e j q ϕ o D U T D ( L ; ρ d , ρ d ) × 1 j 2 π k q = J q ( k ρ ) H q ( 2 ) ( k ρ d ) e j q ( ϕ ϕ o ) .
a q = J q ( k ρ ) e j q ϕ .
E z diff ( ρ ; ρ ) ~ ω μ 0 4 H 0 ( 2 ) ( k | ρ d ρ | ) D U T D ( L ; ρ d , ρ d ) e j k | ρ d + ρ | | ρ d + ρ | .
G ( ρ ; ρ ) = j π 2 π α l = 1 H μ ( 2 ) ( k r > ) J μ ( k r < ) sin ( μ ψ ) sin ( μ ψ ) ,
E z inc ( ρ ) = j 4 H 0 ( 2 ) ( k | ρ d ρ | 2 + r b 2 2 r b | ρ d ρ | cos ( ψ b ψ ) ) ,
r b = r 0 2 2 j b r 0 cos ( β ψ 0 ) b 2 , ( r b ) > 0 ,
cos ψ b = r 0 cos ψ 0 j b cos β r b ,

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