Abstract

When an inhomogeneous plane wave or other type of evanescent field in a lossless medium strikes a large opaque object, the boundary of the shadow zone is displaced from the familiar geometric optical location for nonevanescent fields. This change is attributable to the complex propagation direction of evanescent waves. To study the mechanism of shadow formation, and indeed the significance of a shadow zone when the incident evanescent field may be weaker than diffracted fields generated by the obstacle, the problem of inhomogeneous plane diffraction by a perfectly conducting semi-infinite screen is investigated. By a careful asymptotic treatment and error analysis of the known exact solution, the location of the shadow boundary is determined, as is the transition region surrounding it wherein the field cannot be separated into incident and diffracted constituents. It is found that for strongly evanescent fields, the shadow boundary and transition zones differ markedly from those for a homogeneous plane wave. Effects of losses in the medium are also considered.

© 1978 Optical Society of America

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Corrections

Henry L. Bertoni, Arthur C. Green, and Leopold B. Felsen, "Errata: Shadowing an Inhomogeneous Plane Wave by an Edge," J. Opt. Soc. Am. 68, 1787_2-1787 (1978)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-68-12-1787_2

References

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  1. G. A. Deschamps, “Ray Techniques in Electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
    [CrossRef]
  2. J. B. Keller, “Geometrical Theory of Diffraction,” J. Opt. Soc. Am. 52, 116–130 (1962).
    [CrossRef] [PubMed]
  3. R. G. Kouyoumjian and P. H. Pathak, “A Uniform Theory of Diffraction for an Edge in a Perfectly Conducting Surface,” Proc. IEEE 62, 1448–1461 (1974).
    [CrossRef]
  4. D. S. Ahluwalia, R. M. Lewis, and J. Boersma, “Uniform Asymptotic Theory of Diffraction by a Plane Screen,” Siam J. Appl. Math. 16, 783–807 (1968).
    [CrossRef]
  5. W. Y. D. Wang and G. A. Deschamps, “Application of Complex Ray Tracing to Scattering Problems,” Proc. IEEE 62, 1541–1551 (1974).
    [CrossRef]
  6. K. G. Budden and P. D. Terry, “Radio Ray Tracing in Complex Space,” Proc. R. Soc. Lond. A 321, 275–301 (1971).
    [CrossRef]
  7. Y. A. Kravtsov, “Complex Rays and Complex Caustics,” Bell Lab. Translation No. TR-69-109 (1969) of paper presented at Fourth All-Union Symposium on Diffraction of Waves (Kharkov, USSR, 1967).
  8. L. B. Felsen, “Evansecent Waves,” J. Opt. Soc. Am. 66, 751–760 (1976).
    [CrossRef]
  9. H. L. Bertoni, L. B. Felsen, and A. Hessel, “Local Properties of Radiation in Lossy Media,” IEEE Trans. Antennas Prop. AP-19, 226–237 (1971).
    [CrossRef]
  10. S. Choudhary and L. B. Felsen, “Asymptotic Theory for Inhomogeneous Waves,” IEEE Trans. Antenna Prop. AP-21, 827–842 (1973).
    [CrossRef]
  11. G. Otis, “Application of the Boundary-Diffraction-Wave Theory to Gaussian Beams,” J. Opt. Soc. Am. 64, 1545–1550 (1974).
    [CrossRef]
  12. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N. J., 1973), p. 673.
  13. (a)M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), pp. 297–298, gives the algebraic form in (7). (b)H. Jeffreys, Asymptotic Approximations (Clarendon, Oxford, 1962), p. 42, discusses the range of s over which to include the unit term in (7).
  14. H. Jeffreys, in Ref. 13(b), pp. 117–122.
  15. M. Abramowitz and I. A. Stegun, in Ref. 13(a), p. 257.
  16. H. Jeffreys, in Ref. 13(b), pp. 35 and 42.
  17. H. L. Bertoni, L. B. Felsen, and A. Green, “Shadowing of a Gaussian Beam by an Edge” (unpublished).

1976 (1)

1974 (3)

W. Y. D. Wang and G. A. Deschamps, “Application of Complex Ray Tracing to Scattering Problems,” Proc. IEEE 62, 1541–1551 (1974).
[CrossRef]

R. G. Kouyoumjian and P. H. Pathak, “A Uniform Theory of Diffraction for an Edge in a Perfectly Conducting Surface,” Proc. IEEE 62, 1448–1461 (1974).
[CrossRef]

G. Otis, “Application of the Boundary-Diffraction-Wave Theory to Gaussian Beams,” J. Opt. Soc. Am. 64, 1545–1550 (1974).
[CrossRef]

1973 (1)

S. Choudhary and L. B. Felsen, “Asymptotic Theory for Inhomogeneous Waves,” IEEE Trans. Antenna Prop. AP-21, 827–842 (1973).
[CrossRef]

1972 (1)

G. A. Deschamps, “Ray Techniques in Electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

1971 (2)

K. G. Budden and P. D. Terry, “Radio Ray Tracing in Complex Space,” Proc. R. Soc. Lond. A 321, 275–301 (1971).
[CrossRef]

H. L. Bertoni, L. B. Felsen, and A. Hessel, “Local Properties of Radiation in Lossy Media,” IEEE Trans. Antennas Prop. AP-19, 226–237 (1971).
[CrossRef]

1968 (1)

D. S. Ahluwalia, R. M. Lewis, and J. Boersma, “Uniform Asymptotic Theory of Diffraction by a Plane Screen,” Siam J. Appl. Math. 16, 783–807 (1968).
[CrossRef]

1962 (1)

Abramowitz, M.

(a)M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), pp. 297–298, gives the algebraic form in (7). (b)H. Jeffreys, Asymptotic Approximations (Clarendon, Oxford, 1962), p. 42, discusses the range of s over which to include the unit term in (7).

M. Abramowitz and I. A. Stegun, in Ref. 13(a), p. 257.

Ahluwalia, D. S.

D. S. Ahluwalia, R. M. Lewis, and J. Boersma, “Uniform Asymptotic Theory of Diffraction by a Plane Screen,” Siam J. Appl. Math. 16, 783–807 (1968).
[CrossRef]

Bertoni, H. L.

H. L. Bertoni, L. B. Felsen, and A. Hessel, “Local Properties of Radiation in Lossy Media,” IEEE Trans. Antennas Prop. AP-19, 226–237 (1971).
[CrossRef]

H. L. Bertoni, L. B. Felsen, and A. Green, “Shadowing of a Gaussian Beam by an Edge” (unpublished).

Boersma, J.

D. S. Ahluwalia, R. M. Lewis, and J. Boersma, “Uniform Asymptotic Theory of Diffraction by a Plane Screen,” Siam J. Appl. Math. 16, 783–807 (1968).
[CrossRef]

Budden, K. G.

K. G. Budden and P. D. Terry, “Radio Ray Tracing in Complex Space,” Proc. R. Soc. Lond. A 321, 275–301 (1971).
[CrossRef]

Choudhary, S.

S. Choudhary and L. B. Felsen, “Asymptotic Theory for Inhomogeneous Waves,” IEEE Trans. Antenna Prop. AP-21, 827–842 (1973).
[CrossRef]

Deschamps, G. A.

W. Y. D. Wang and G. A. Deschamps, “Application of Complex Ray Tracing to Scattering Problems,” Proc. IEEE 62, 1541–1551 (1974).
[CrossRef]

G. A. Deschamps, “Ray Techniques in Electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

Felsen, L. B.

L. B. Felsen, “Evansecent Waves,” J. Opt. Soc. Am. 66, 751–760 (1976).
[CrossRef]

S. Choudhary and L. B. Felsen, “Asymptotic Theory for Inhomogeneous Waves,” IEEE Trans. Antenna Prop. AP-21, 827–842 (1973).
[CrossRef]

H. L. Bertoni, L. B. Felsen, and A. Hessel, “Local Properties of Radiation in Lossy Media,” IEEE Trans. Antennas Prop. AP-19, 226–237 (1971).
[CrossRef]

H. L. Bertoni, L. B. Felsen, and A. Green, “Shadowing of a Gaussian Beam by an Edge” (unpublished).

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N. J., 1973), p. 673.

Green, A.

H. L. Bertoni, L. B. Felsen, and A. Green, “Shadowing of a Gaussian Beam by an Edge” (unpublished).

Hessel, A.

H. L. Bertoni, L. B. Felsen, and A. Hessel, “Local Properties of Radiation in Lossy Media,” IEEE Trans. Antennas Prop. AP-19, 226–237 (1971).
[CrossRef]

Jeffreys, H.

H. Jeffreys, in Ref. 13(b), pp. 117–122.

H. Jeffreys, in Ref. 13(b), pp. 35 and 42.

Keller, J. B.

Kouyoumjian, R. G.

R. G. Kouyoumjian and P. H. Pathak, “A Uniform Theory of Diffraction for an Edge in a Perfectly Conducting Surface,” Proc. IEEE 62, 1448–1461 (1974).
[CrossRef]

Kravtsov, Y. A.

Y. A. Kravtsov, “Complex Rays and Complex Caustics,” Bell Lab. Translation No. TR-69-109 (1969) of paper presented at Fourth All-Union Symposium on Diffraction of Waves (Kharkov, USSR, 1967).

Lewis, R. M.

D. S. Ahluwalia, R. M. Lewis, and J. Boersma, “Uniform Asymptotic Theory of Diffraction by a Plane Screen,” Siam J. Appl. Math. 16, 783–807 (1968).
[CrossRef]

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N. J., 1973), p. 673.

Otis, G.

Pathak, P. H.

R. G. Kouyoumjian and P. H. Pathak, “A Uniform Theory of Diffraction for an Edge in a Perfectly Conducting Surface,” Proc. IEEE 62, 1448–1461 (1974).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, in Ref. 13(a), p. 257.

(a)M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), pp. 297–298, gives the algebraic form in (7). (b)H. Jeffreys, Asymptotic Approximations (Clarendon, Oxford, 1962), p. 42, discusses the range of s over which to include the unit term in (7).

Terry, P. D.

K. G. Budden and P. D. Terry, “Radio Ray Tracing in Complex Space,” Proc. R. Soc. Lond. A 321, 275–301 (1971).
[CrossRef]

Wang, W. Y. D.

W. Y. D. Wang and G. A. Deschamps, “Application of Complex Ray Tracing to Scattering Problems,” Proc. IEEE 62, 1541–1551 (1974).
[CrossRef]

IEEE Trans. Antenna Prop. (1)

S. Choudhary and L. B. Felsen, “Asymptotic Theory for Inhomogeneous Waves,” IEEE Trans. Antenna Prop. AP-21, 827–842 (1973).
[CrossRef]

IEEE Trans. Antennas Prop. (1)

H. L. Bertoni, L. B. Felsen, and A. Hessel, “Local Properties of Radiation in Lossy Media,” IEEE Trans. Antennas Prop. AP-19, 226–237 (1971).
[CrossRef]

J. Opt. Soc. Am. (3)

Proc. IEEE (3)

R. G. Kouyoumjian and P. H. Pathak, “A Uniform Theory of Diffraction for an Edge in a Perfectly Conducting Surface,” Proc. IEEE 62, 1448–1461 (1974).
[CrossRef]

W. Y. D. Wang and G. A. Deschamps, “Application of Complex Ray Tracing to Scattering Problems,” Proc. IEEE 62, 1541–1551 (1974).
[CrossRef]

G. A. Deschamps, “Ray Techniques in Electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

Proc. R. Soc. Lond. A (1)

K. G. Budden and P. D. Terry, “Radio Ray Tracing in Complex Space,” Proc. R. Soc. Lond. A 321, 275–301 (1971).
[CrossRef]

Siam J. Appl. Math. (1)

D. S. Ahluwalia, R. M. Lewis, and J. Boersma, “Uniform Asymptotic Theory of Diffraction by a Plane Screen,” Siam J. Appl. Math. 16, 783–807 (1968).
[CrossRef]

Other (7)

Y. A. Kravtsov, “Complex Rays and Complex Caustics,” Bell Lab. Translation No. TR-69-109 (1969) of paper presented at Fourth All-Union Symposium on Diffraction of Waves (Kharkov, USSR, 1967).

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N. J., 1973), p. 673.

(a)M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), pp. 297–298, gives the algebraic form in (7). (b)H. Jeffreys, Asymptotic Approximations (Clarendon, Oxford, 1962), p. 42, discusses the range of s over which to include the unit term in (7).

H. Jeffreys, in Ref. 13(b), pp. 117–122.

M. Abramowitz and I. A. Stegun, in Ref. 13(a), p. 257.

H. Jeffreys, in Ref. 13(b), pp. 35 and 42.

H. L. Bertoni, L. B. Felsen, and A. Green, “Shadowing of a Gaussian Beam by an Edge” (unpublished).

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Figures (4)

FIG. 1
FIG. 1

Shadow formation and diffraction for a homogeneous plane wave incident on a conducting half-screen.

FIG. 2
FIG. 2

Shadow boundaries for an inhomogeneous plane wave incident on a conducting half-screen at a complex angle θ0 = u + iv for the case v > 0. The phase paths (Ref. 10) for the incident field is the family of lines parallel to θ = u.

FIG. 3
FIG. 3

Transition regions outside of which asymptotic approximation is valid for the case v > 0.

FIG. 4
FIG. 4

Transition region and shadow boundary for a plane wave traveling in a medium with dissipative loss at a complex angle θ0 with respect to the z axis.

Equations (33)

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E inc ( x , z ) = exp [ ( i k ρ cos ( θ - θ 0 ) ] = exp ( i k · ρ ) ,
k = x 0 k sin θ 0 + z 0 k cos θ 0
Re k = k cosh v [ x 0 sin u + z 0 cos u ] , Im k = k sinh v [ x 0 cos u - z 0 sin u ] .
E ( ρ , θ ) = exp [ i k ρ cos ( θ - θ 0 ) ] Q ( p ) - exp [ - i k ρ cos ( θ - θ 0 ) ] Q ( q ) .
p = e - i π / 4 ( 2 k ρ ) 1 / 2 sin [ 1 2 ( θ 0 - θ ) ] , q = e - i π / 4 ( 2 k ρ ) 1 / 2 cos [ 1 2 ( θ 0 + θ ) ] ,
Q ( s ) = 1 π s e - τ 2 d τ
Q ( s ) ~ ( e - s 2 / 2 π s ) I ( s ) ,             Re ( s ) > 0 ,
Q ( s ) ~ 1 + ( e - s 2 / 2 π s ) I ( s ) ,             Re ( s ) < 0 ,
I ( s ) = 1 + n = 1 N ( 1 · 3 · 5 2 n - 1 ) ( - 2 ) n s 2 n .
E = E GO + E D .
E GO = exp [ i k ρ cos ( θ - θ 0 ) ] U ( θ - θ s i ) - exp [ - i k ρ cos ( θ + θ 0 ) ] U ( θ - θ s r ) ,
0 = sin ( u - θ s i 2 ) cosh v 2 + cos ( u - θ s i 2 ) sinh v 2 , 0 = cos ( u + θ s r 2 ) cosh v 2 - sin ( u + θ s r 2 ) sinh v 2 .
θ s i = u + tan - 1 ( sinh v ) , θ s r = π - θ s i .
E D = e i ( k ρ + π / 4 ) 2 ( 2 π k ρ ) 1 / 2 ( I ( p ) sin [ 1 2 ( θ 0 - θ ) ] - I ( q ) cos [ 1 2 ( θ 0 + θ ) ] ) .
( 2 k ρ ) 1 / 2 sin [ 1 2 ( θ 0 - θ ) ] A .
ρ [ cosh v - cos ( θ - u ) ] = A / k .
L = 2 A cosh v / ( k sinh 2 v ) , W = 2 A / k sinh v .
( N ) = 0 ( ( 1 · 3 · 5 2 N + 1 ) 2 ( 2 π k ρ ) 1 / 2 sin [ 1 2 ( θ 0 - θ ) ] ( 2 N + 1 ) p 2 ( N + 1 ) ) ,
1 · 3 · 5 2 N + 1 2 N + 1 = Γ ( N + 3 2 ) Γ ( 1 2 ) ~ 2 e - N ( N ) N + 1 ,
( N ) = 0 ( e - N ( N ) N + 1 ( 2 π ) 1 / 2 p p 2 ( N + 1 ) ) .
1 = ( 2 N ¯ + 1 ) / 2 p 2
N N ¯ = p 2 - 1 2 .
( p 2 - 1 2 ) p 2 ~ p 2 p 2 e - 1 / 2 ,
( N ) = 0 ( 1 ( 2 π ) 1 / 2 e - p 2 p ) = 0 ( exp { - 2 k ρ sin [ 1 2 ( θ - θ 0 ) ] 2 } 2 ( π k ρ ) 1 / 2 sin [ 1 2 ( θ - θ 0 ) ] ) .
E D E G T D = e i ( k ρ + π / 4 ) 2 ( 2 π k ρ ) 1 / 2 × ( 1 sin [ 1 2 ( θ 0 - θ ) ] - 1 cos [ 1 2 ( θ 0 + θ ) ] ) .
( 0 ) = 0 ( 1 2 π p 3 ) .
( N ) = 0 [ 1 / 2 ( π k ρ ) 1 / 2 sin [ 1 2 ( θ - θ 0 ) ] exp ( - k ρ cos ( θ s i - u ) × [ 1 - 1 2 cos ( θ s i - θ ) - 1 2 cos ( θ s i + θ - 2 u ) ] ) ] .
E inc = exp ( - k ρ cos ( θ s i - u ) × [ 1 2 cos ( θ s i - θ ) - 1 2 cos ( θ s i + θ - 2 u ] ) .
( N ) E inc = 0 [ 1 / 2 ( π k ρ ) 1 / 2 sin [ 1 2 ( θ - θ 0 ) ] × exp ( - k ρ cos ( θ s i - u ) [ 1 - cos ( θ s i - θ ) ] ) ] .
k ρ [ 1 - cos ( θ s i - θ ) ] = B ,
k = β + i α .
θ s i = u + tan - 1 ( β sinh v k + α cosh v ) ,
( N ) e - α ρ = 0 ( 1 / 2 ( π k ρ ) 1 / 2 sin [ 1 2 ( θ - θ 0 ) ] exp { - 2 k ρ sin [ 1 2 ( θ - θ 0 ) ] 2 } ) .