Abstract

Diffraction of a Gaussian beam with a circular spot size normally incident upon a Kirchhoff half-screen is investigated based on the boundary-diffraction-wave (BDW) theory. The evaluation of the boundary-diffraction wave by the steepest-descent method yields the uniform asymptotic representation of the total diffracted field consisting of the geometrical-optics and diffraction components. The use of complex rays to construct the diffraction component is also shown by applying the geometrical theory of diffraction (GTD) to the problem. Comparison of the representation of the diffraction component by the BDW theory with that by the GTD gives the diffraction coefficient for the Gaussian beam that ensures the continuity of the diffracted field at the shadow boundary.

© 1982 Optical Society of America

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References

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  1. J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957).
    [CrossRef]
  2. J. B. Keller, R. M. Lewis, and B. D. Seckler, “Diffraction by an aperture. II,” J. Appl. Phys. 28, 570–579 (1957).
    [CrossRef]
  3. R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448–1461 (1974).
    [CrossRef]
  4. S. Choudhary and L. B. Felsen, “Asymptotic theory for inhomogeneous waves,” IEEE Trans. Antennas Propag. AP-21, 827–842 (1973).
    [CrossRef]
  5. S. Choudhary and L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
    [CrossRef]
  6. J. B. Keller and W. Streifer, “Complex rays with an application to Gaussian beams,” J. Opt. Soc. Am. 61, 40–43 (1971).
    [CrossRef]
  7. W. D. Wang and G. A. Deschamps, “Application of complex ray tracing to scattering problems,” Proc. IEEE 62, 1541–1551 (1974).
    [CrossRef]
  8. G. Otis, “Application of the boundary-diffraction-wave theory to Gaussian beams,” J. Opt. Soc. Am. 64, 1545–1550 (1974).
    [CrossRef]
  9. G. Otis, J.-L. Lachambre, J. W. Y. Lit, and P. Lavigne, “Diffracted waves in the shadow boundary region,” J. Opt. Soc. Am. 67, 551–553 (1977).
    [CrossRef]
  10. T. Takenaka, M. Kakeya, and O. Fukumitsu, “Asymptotic representation of the boundary diffraction wave for a Gaussian beam incident on a circular aperture,” J. Opt. Soc. Am. 70, 1323–1328, (1980).
    [CrossRef]
  11. 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]
  12. 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]
  13. 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]
  14. 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]
  15. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [CrossRef]
  16. S. Y. Shin and L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67, 699–700 (1977).
    [CrossRef]
  17. H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt. 4, 1562–1569 (1965).
    [CrossRef]
  18. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice Hall, Englewood Cliffs, N.J., 1973), Chap. 4.

1980 (1)

1979 (1)

1977 (2)

1974 (4)

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

S. Choudhary and L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
[CrossRef]

W. D. Wang and G. A. Deschamps, “Application of complex ray tracing to scattering problems,” Proc. IEEE 62, 1541–1551 (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. Antennas Propag. AP-21, 827–842 (1973).
[CrossRef]

1971 (2)

J. B. Keller and W. Streifer, “Complex rays with an application to Gaussian beams,” J. Opt. Soc. Am. 61, 40–43 (1971).
[CrossRef]

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

1969 (1)

1965 (1)

1962 (2)

1957 (2)

J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957).
[CrossRef]

J. B. Keller, R. M. Lewis, and B. D. Seckler, “Diffraction by an aperture. II,” J. Appl. Phys. 28, 570–579 (1957).
[CrossRef]

Bertoni, H. L.

Choudhary, S.

S. Choudhary and L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
[CrossRef]

S. Choudhary and L. B. Felsen, “Asymptotic theory for inhomogeneous waves,” IEEE Trans. Antennas Propag. AP-21, 827–842 (1973).
[CrossRef]

Deschamps, G. A.

W. D. Wang and G. A. Deschamps, “Application of complex ray tracing to scattering problems,” Proc. IEEE 62, 1541–1551 (1974).
[CrossRef]

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

Felsen, L. B.

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]

S. Y. Shin and L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67, 699–700 (1977).
[CrossRef]

S. Choudhary and L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
[CrossRef]

S. Choudhary and L. B. Felsen, “Asymptotic theory for inhomogeneous waves,” IEEE Trans. Antennas Propag. AP-21, 827–842 (1973).
[CrossRef]

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice Hall, Englewood Cliffs, N.J., 1973), Chap. 4.

Fukumitsu, O.

Green, A. C.

Kakeya, M.

Keller, J. B.

J. B. Keller and W. Streifer, “Complex rays with an application to Gaussian beams,” J. Opt. Soc. Am. 61, 40–43 (1971).
[CrossRef]

J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957).
[CrossRef]

J. B. Keller, R. M. Lewis, and B. D. Seckler, “Diffraction by an aperture. II,” J. Appl. Phys. 28, 570–579 (1957).
[CrossRef]

Kogelnik, H.

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. IEEE 62, 1448–1461 (1974).
[CrossRef]

Kurtin, S.

Lachambre, J.-L.

Lavigne, P.

Lewis, R. M.

J. B. Keller, R. M. Lewis, and B. D. Seckler, “Diffraction by an aperture. II,” J. Appl. Phys. 28, 570–579 (1957).
[CrossRef]

Lit, J. W. Y.

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice Hall, Englewood Cliffs, N.J., 1973), Chap. 4.

McGill, T. C.

Miyamoto, K.

Otis, G.

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. IEEE 62, 1448–1461 (1974).
[CrossRef]

Pearson, J. E.

Seckler, B. D.

J. B. Keller, R. M. Lewis, and B. D. Seckler, “Diffraction by an aperture. II,” J. Appl. Phys. 28, 570–579 (1957).
[CrossRef]

Shin, S. Y.

Streifer, W.

Takenaka, T.

Wang, W. D.

W. D. Wang and G. A. Deschamps, “Application of complex ray tracing to scattering problems,” Proc. IEEE 62, 1541–1551 (1974).
[CrossRef]

Wolf, E.

Yariv, A.

Appl. Opt. (1)

Electron. Lett. (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

S. Choudhary and L. B. Felsen, “Asymptotic theory for inhomogeneous waves,” IEEE Trans. Antennas Propag. AP-21, 827–842 (1973).
[CrossRef]

J. Appl. Phys. (2)

J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957).
[CrossRef]

J. B. Keller, R. M. Lewis, and B. D. Seckler, “Diffraction by an aperture. II,” J. Appl. Phys. 28, 570–579 (1957).
[CrossRef]

J. Opt. Soc. Am. (9)

G. Otis, “Application of the boundary-diffraction-wave theory to Gaussian beams,” J. Opt. Soc. Am. 64, 1545–1550 (1974).
[CrossRef]

G. Otis, J.-L. Lachambre, J. W. Y. Lit, and P. Lavigne, “Diffracted waves in the shadow boundary region,” J. Opt. Soc. Am. 67, 551–553 (1977).
[CrossRef]

T. Takenaka, M. Kakeya, and O. Fukumitsu, “Asymptotic representation of the boundary diffraction wave for a Gaussian beam incident on a circular aperture,” J. Opt. Soc. Am. 70, 1323–1328, (1980).
[CrossRef]

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]

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]

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]

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]

S. Y. Shin and L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67, 699–700 (1977).
[CrossRef]

J. B. Keller and W. Streifer, “Complex rays with an application to Gaussian beams,” J. Opt. Soc. Am. 61, 40–43 (1971).
[CrossRef]

Proc. IEEE (3)

W. 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 geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448–1461 (1974).
[CrossRef]

S. Choudhary and L. B. Felsen, “Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking,” Proc. IEEE 62, 1530–1541 (1974).
[CrossRef]

Other (1)

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice Hall, Englewood Cliffs, N.J., 1973), Chap. 4.

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Figures (3)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Complex y0 plane and contours of integration.

Fig. 3
Fig. 3

Definition of the angles α and θ.

Equations (49)

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U i ( x , y , z ) = q ( 0 ) q ( z ) exp [ i k ϕ ( x , y , z ) ] ,
ϕ ( x , y , z ) = z + ( x 2 + y 2 ) / 2 q ( z ) , q ( z ) = z - i b / 2 , b = k w s 2 ,             ( k = 2 π / λ ) ,
U K ( P ) = j F j ( P ) + U B ( P ) .
W ( P , Q ) = U i ( Q ) e i k s 4 π s ŝ × Q ϕ 1 + ŝ · Q ϕ ,
Q ϕ = x 0 q ( z 0 ) x ^ + y 0 q ( z 0 ) ŷ + [ 1 - x 0 2 + y 0 2 2 q 2 ( z 0 ) ] z ^ .
s Z + [ ( x 0 - x ) 2 + ( y 0 - y ) 2 ] / 2 Z ( for phase term ) , s Z ( for amplitude term ) , ŝ [ ( x 0 - x ) x ^ + ( y 0 - y ) ŷ - Z z ^ ] / Z ,
s ( 1 + ŝ · Q ϕ ) = 1 2 Z { [ x 0 q ( z ) q ( z 0 ) - x ] 2 + ( y 0 q ( z ) q ( z 0 ) - y ) 2 } = 0.
x 0 1 = x - γ y 1 + γ 2 ,             y 0 1 = γ x + y 1 + γ 2 ,
x 0 2 = x + γ y 1 + γ 2 ,             y 0 2 = - γ x + y 1 + γ 2 ,
j = 1 2 F j ( P ) = [ E ( x h 1 - x ) + E ( x h 2 - x ) ] U i ( P ) / 2 ,
x h 1 = a ( 1 + γ 2 ) + γ y ,
x h 2 = a ( 1 + γ 2 ) - γ y .
U B ( P ) = Γ W ( P , Q ) · l ^ d l ,
U B ( P ) = - G ( y 0 ) exp [ i k d ( y 0 ) ] d y 0 ,
d ( y 0 ) = q ( z 0 ) + a 2 + y 2 2 q ( z 0 ) + Z + ( a - x ) 2 + ( y 0 - y ) 2 2 Z , G ( y 0 ) = q ( 0 ) q ( z o ) [ x - a - a Z / q ( z 0 ) ] 2 π q 2 ( z ) ( y 0 - y p 1 ) ( y 0 - y p 2 ) e k b / 2 ,
y p 1 = q ( z 0 ) q ( z ) y - i a [ 1 - q ( z 0 ) a q ( z ) x ] ,
y p 2 = q ( z 0 ) q ( z ) y + i a [ 1 - q ( z 0 ) a q ( z ) x ] .
d ( y 0 ) [ a 2 + y 0 2 + ( z 0 - i b / 2 ) 2 ] 1 / 2 + [ ( a - x ) 2 + ( y 0 - y ) 2 + ( z 0 - z ) 2 ] 1 / 2 .
y s = [ q ( z 0 ) / q ( z ) ] y .
x s = a [ 4 z 2 + b 2 4 z 0 z + b 2 + 2 b ( z - z 0 ) 4 z 0 z + b 2 ( 4 z 2 + b 2 4 z 0 2 + b 2 ) 1 / 2 ] .
U B ( P ) = [ E ( x - x h 1 ) + E ( x - x h 2 ) - 2 E ( x - x s ) ] 2 π i κ exp [ i k d ( y p 1 ) ] + S D P G ( y 0 ) exp [ i k d ( y 0 ) ] d y 0 ,
κ = - i q ( 0 ) 4 π q ( z ) e - k b / 2 .
d ( y p 1 ) = ϕ ( x , y , z ) - i b / 2 ,
2 π i κ exp [ i k d ( y p 1 ) ] = U i ( P ) / 2.
SDP G ( y 0 ) exp [ i k d ( y 0 ) ] d y 0 ~ 2 π i κ exp [ i k d ( y p 1 ) erfc ( ± i { i k [ d ( y s ) - d ( y p 1 ) ] } 1 / 2 ) + π / k exp [ i k d ( y s ) ] ( G ( y s ) [ i 2 q ( z 0 ) Z q ( z ) ] 1 / 2 + 2 κ { i [ d ( y s ) - d ( y p 1 ) ] } 1 / 2 ) ,             ( x x s ) ,
erfc ( τ ) = 2 π τ e - x 2 d x .
U K ( P ) ~ U G ( P ) + U D ( P ) ,
U G ( P ) = E ( x s - x ) U i ( P ) ,
U D ( P ) = D u [ q ( z 0 ) q ( z ) Z ] 1 / 2 exp [ i k s ( Q d ) ] U i ( Q d ) ,
s ( Q d ) = Z + ( x - a ) 2 + ( y - y s ) 2 2 Z .
D u 1 2 erfc ( ± i { i k [ d ( y s ) - d ( y p 1 ) ] } 1 / 2 × [ q ( z 0 ) Z q ( z ) ] 1 / 2 exp { i k [ d ( y p 1 ) - d ( y s ) ] } ,             ( x x s ) .
erfc ( ± τ ) ~ - e - τ 2 π τ ,             ( τ 1 ) ,
D = e i π / 4 2 q ( z 0 ) Z 2 2 π k [ q ( z ) a - q ( z 0 ) x ] .
Ū D ( P ) = D ¯ [ r ( r + ρ ) ρ ] e i k ρ U i ( Q ¯ d ) ,
D ¯ = e i π / 4 ( cos θ - cos α ) 2 2 π k sin β ( sin θ + sin α ) = - e i π / 4 tan [ ( θ - α ) / 2 ] 2 2 π k sin β .
cos β = y d / q ( z 0 ) = ( y - y d ) / Z .
y d = [ q ( z 0 ) / q ( z ) ] y .
tan ( θ - α 2 ) 2 q ( z 0 ) [ q ( z ) - q ( z 0 ) ] q ( z ) a - q ( z 0 ) x .
r q ( z 0 ) , ρ q ( z ) - q ( z 0 ) + ( x - a ) 2 + ( y - y d ) 2 2 [ q ( z ) - q ( z 0 ) ] ( for phase term ) , ρ q ( z ) - q ( z 0 ) ( for amplitude term ) .
[ r ( r + ρ ) ρ ] 1 / 2 [ q ( z 0 ) q ( z ) [ q ( z ) - q ( z 0 ) ] ] 1 / 2 .
Ū D D [ q ( z 0 ) q ( z ) Z ] 1 / 2 exp [ i k s ( Q d ) ] U i ( Q d ) .
x p = [ q ( z 0 ) / q ( z ) ] x ,             y p = [ q ( z 0 ) / q ( z ) ] y .
x = [ q ( z ) / q ( z 0 ) ] a .
ζ 2 = i [ d ( y s ) - d ( y 0 ) ] ,
ζ p = { i [ d ( y s ) - d ( y p 1 ) ] } 1 / 2 .
SDP G ( y 0 ) exp [ i k d ( y 0 ) ] d y 0 = exp [ i k d ( y s ) ] - F ( ζ ) e - k ζ 2 d ζ ,
F ( ζ ) = G ( y 0 ) d y 0 d ζ .
F ( ζ ) = κ ζ - ζ p - κ ζ + ζ p + H ( ζ ) ,
- e - Ω ξ 2 ξ 2 - α 2 d ξ = i π α erfc ( ± i α Ω ) e - Ω α 2 ,             ( Im α 0 )