Abstract

A new exact representation for point sources is given in terms of complex point sources. In the simplest configuration, a point source is equivalent to a distribution of sources on the surface of a sphere in complex space. The representation can be used to consider the propagation of point disturbances through inhomogeneous media and across interfaces. In the high-frequency limit, these solutions may be obtained by the use of complex ray-tracing methods, which are just the analytic extension of ordinary ray methods. It is shown that the additional use of the paraxial approximation yields a procedure that is similar to the Gaussian beam summation method. The latter technique is normally based on a matched asymptotics argument. However, the complex point-source representation now offers an exact basis for this widely used method.

© 1986 Optical Society of America

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References

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  1. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [CrossRef]
  2. L. B. Felsen, “Evanescent waves,” J. Opt. Soc. Am. 66, 751–760 (1976).
    [CrossRef]
  3. L. B. Felsen, “Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams,” J. R. Astron. Soc. 79, 77–88 (1984).
    [CrossRef]
  4. J. B. Keller, W. Streifer, “Complex rays with application to Gaussian beams,” J. Opt. Soc. Am. 61, 40–43 (1971).
    [CrossRef]
  5. G. A. Deschamps, S. W. Lee, E. Gowan, T. Fontana, “Diffraction of an evanescent plane wave by a half plane,” Wave Motion 1, 25–35 (1979).
    [CrossRef]
  6. J. R. Ra, H. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
    [CrossRef]
  7. M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion 4, 85–97 (1982).
    [CrossRef]
  8. V. Cerveny, M. M. Popov, I. Psencik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” J. R. Astron. Soc. 70, 109–128 (1982).
    [CrossRef]
  9. B. S. White, R. Burridge, A. N. Norris, A. Bayliss, “Some remarks on the Gaussian beam summation method,” J. R. Astron. Soc. (to be published).
  10. P. Hubral, “Wavefront curvatures in 3-D laterally inhomogeneous media with curved interfaces,” Geophysics 45, 905–913 (1980),
    [CrossRef]

1984 (1)

L. B. Felsen, “Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams,” J. R. Astron. Soc. 79, 77–88 (1984).
[CrossRef]

1982 (2)

M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion 4, 85–97 (1982).
[CrossRef]

V. Cerveny, M. M. Popov, I. Psencik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” J. R. Astron. Soc. 70, 109–128 (1982).
[CrossRef]

1980 (1)

P. Hubral, “Wavefront curvatures in 3-D laterally inhomogeneous media with curved interfaces,” Geophysics 45, 905–913 (1980),
[CrossRef]

1979 (1)

G. A. Deschamps, S. W. Lee, E. Gowan, T. Fontana, “Diffraction of an evanescent plane wave by a half plane,” Wave Motion 1, 25–35 (1979).
[CrossRef]

1976 (1)

1973 (1)

J. R. Ra, H. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[CrossRef]

1971 (2)

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

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

Bayliss, A.

B. S. White, R. Burridge, A. N. Norris, A. Bayliss, “Some remarks on the Gaussian beam summation method,” J. R. Astron. Soc. (to be published).

Bertoni, H.

J. R. Ra, H. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[CrossRef]

Burridge, R.

B. S. White, R. Burridge, A. N. Norris, A. Bayliss, “Some remarks on the Gaussian beam summation method,” J. R. Astron. Soc. (to be published).

Cerveny, V.

V. Cerveny, M. M. Popov, I. Psencik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” J. R. Astron. Soc. 70, 109–128 (1982).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, S. W. Lee, E. Gowan, T. Fontana, “Diffraction of an evanescent plane wave by a half plane,” Wave Motion 1, 25–35 (1979).
[CrossRef]

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

Felsen, L. B.

L. B. Felsen, “Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams,” J. R. Astron. Soc. 79, 77–88 (1984).
[CrossRef]

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

J. R. Ra, H. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[CrossRef]

Fontana, T.

G. A. Deschamps, S. W. Lee, E. Gowan, T. Fontana, “Diffraction of an evanescent plane wave by a half plane,” Wave Motion 1, 25–35 (1979).
[CrossRef]

Gowan, E.

G. A. Deschamps, S. W. Lee, E. Gowan, T. Fontana, “Diffraction of an evanescent plane wave by a half plane,” Wave Motion 1, 25–35 (1979).
[CrossRef]

Hubral, P.

P. Hubral, “Wavefront curvatures in 3-D laterally inhomogeneous media with curved interfaces,” Geophysics 45, 905–913 (1980),
[CrossRef]

Keller, J. B.

Lee, S. W.

G. A. Deschamps, S. W. Lee, E. Gowan, T. Fontana, “Diffraction of an evanescent plane wave by a half plane,” Wave Motion 1, 25–35 (1979).
[CrossRef]

Norris, A. N.

B. S. White, R. Burridge, A. N. Norris, A. Bayliss, “Some remarks on the Gaussian beam summation method,” J. R. Astron. Soc. (to be published).

Popov, M. M.

M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion 4, 85–97 (1982).
[CrossRef]

V. Cerveny, M. M. Popov, I. Psencik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” J. R. Astron. Soc. 70, 109–128 (1982).
[CrossRef]

Psencik, I.

V. Cerveny, M. M. Popov, I. Psencik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” J. R. Astron. Soc. 70, 109–128 (1982).
[CrossRef]

Ra, J. R.

J. R. Ra, H. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[CrossRef]

Streifer, W.

White, B. S.

B. S. White, R. Burridge, A. N. Norris, A. Bayliss, “Some remarks on the Gaussian beam summation method,” J. R. Astron. Soc. (to be published).

Electron. Lett. (1)

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

Geophysics (1)

P. Hubral, “Wavefront curvatures in 3-D laterally inhomogeneous media with curved interfaces,” Geophysics 45, 905–913 (1980),
[CrossRef]

J. Opt. Soc. Am. (2)

J. R. Astron. Soc. (2)

L. B. Felsen, “Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams,” J. R. Astron. Soc. 79, 77–88 (1984).
[CrossRef]

V. Cerveny, M. M. Popov, I. Psencik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” J. R. Astron. Soc. 70, 109–128 (1982).
[CrossRef]

SIAM J. Appl. Math. (1)

J. R. Ra, H. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[CrossRef]

Wave Motion (2)

M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion 4, 85–97 (1982).
[CrossRef]

G. A. Deschamps, S. W. Lee, E. Gowan, T. Fontana, “Diffraction of an evanescent plane wave by a half plane,” Wave Motion 1, 25–35 (1979).
[CrossRef]

Other (1)

B. S. White, R. Burridge, A. N. Norris, A. Bayliss, “Some remarks on the Gaussian beam summation method,” J. R. Astron. Soc. (to be published).

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Figures (8)

Fig. 1
Fig. 1

A complex point source x0 on the surface of the sphere of complex radius a and receiver at x. The exact distance is R(x, x0) = [(sa)2 + n2)1/2. The paraxial approximation is in Eq. (11).

Fig. 2
Fig. 2

Geometry of the point source at S, receiver at R, and spherical interface. Sθ represents a typical complex point source on the complex sphere about S.

Fig. 3
Fig. 3

The ray diagram for the spherical interface with a = 1, b = 2, c0 = 2c1.

Fig. 4
Fig. 4

The Gaussian beam geometrical parameter for the spherical interface.

Fig. 5
Fig. 5

Comparison of the Gaussian beam and complex point source methods for the spherical interface with c0 = c1, b = 2a, and receiver at x = 0. Four values of =− are considered.

Fig. 6
Fig. 6

The same as Fig. 5, except for c0 = 2c1.

Fig. 7
Fig. 7

Comparison of the Gaussian beam and complex point source methods for the spherical interface with c0 = 2c1, b = 2, a = 1, ka = 20, and variable receiver positions on the axis. The absolute magnitude of uRAY is 1/6π, independent of x.

Fig. 8
Fig. 8

The angle ψ of the point of refraction on the spherical interface (see Figs. 2 and 4). The upper-left-hand curve is for the Gaussian beams and is independent of . The other curves show the real and imaginary parts of ψ for the complex point sources with δ = i. The parameters are a = 1, b = 2, c0 = 2c1.

Equations (43)

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u G ( x , x 0 ) = e i k R 4 π R ,
2 u G + k 2 u G = δ ( x x 0 ) .
R ( x , x 0 ) = [ ( x x 0 ) 2 + ( y y 0 ) 2 + ( z z 0 ) 2 ] 1 / 2 ,
1 4 π j 0 ( k a ) 0 2 π d ϕ 0 π sin θ d θ u G ( x , a e r ) = u G ( x , 0 ) ,
u G ( x , x 0 ) = i 4 H 0 ( 1 ) ( k r ) ,
1 2 π J 0 ( k a ) 0 2 π d ϕ u G ( x , a e r ) = u G ( x , 0 ) .
u ( x 0 ) = S u ( x ) u 0 G n ( x , x 0 ) d S ( x ) ,
u 0 G n ( x , 0 ) = { [ 4 π a 2 j 0 ( k a ) ] 1 in 3 D [ 2 π a J 0 ( k a ) ] 1 in 2 D .
u 0 ( x ) = ( i k 8 π 2 ) 1 ( + s ) exp [ i k ( s + 1 2 n 2 + s ) ] ,
0 2 π d ϕ 0 sin θ d θ u θ ( x ) e i k z 4 π z
R ( x , a e r ) = s a + 1 2 n 2 s a + 0 ( θ 4 ) .
1 4 π j 0 ( k a ) u G ( x , a e r ) ~ [ e i k 16 π 2 j 0 ( k ) ] 1 ( + s ) × exp [ i k ( s + 1 2 n 2 + s ) ] ,
k Im 1 ,
e i k 16 π 2 j 0 ( k ) i k 8 π 2 .
u ϕ ( x ) = ( 4 π i ) 1 ( + s ) 1 / 2 exp [ i k ( s + 1 2 n 2 + s ) ] ,
u ϕ ( x ) d ϕ e i 3 π / 4 ( 8 π k x ) 1 / 2 e i k x
1 2 π J 0 ( k a ) u G ( x , a e r ) ~ [ exp [ i ( k 3 π / 4 ) ] 2 k ( 2 π ) 3 / 2 J 0 ( k ) ] 1 ( + s ) 1 / 2 × exp [ i k ( s + 1 2 n 2 + s ) ] .
u RAY = exp { i k [ b a + c 0 c 1 ( a + x ) ] } 2 π r ( 1 + c 0 / c 1 ) ,
r ( x ) = c 1 c 0 { b x a [ c 0 c 1 ( b a ) b ] } .
x c = b / [ ( b 2 a 2 1 ) 1 / 2 ( c 0 2 c 1 2 1 ) 1 / 2 1 ] .
ψ = θ b + c 0 c 1 x ( b + a ) ( a + x ) + 0 ( θ 2 ) .
x 2 + y 2 + z 2 = a 2 .
d 0 ( ψ ) = [ ( b + cos θ a cos ψ ) 2 + ( sin θ + a sin ψ ) 2 ] 1 / 2 ,
d 1 ( ψ ) = ( a 2 + x 2 + 2 a x cos ψ ) 1 / 2 .
1 d 0 [ b sin ψ + sin ( θ + ψ ) ] = c 0 c 1 x d 1 sin ψ .
U ( θ ) = T ( θ 0 ) exp [ i ω ( d 0 / c 0 + d 1 / c 1 ) ] 4 π d 0 ( 1 + d 1 / D 1 ) 1 / 2 ( 1 + d 1 / D 2 ) 1 / 2 ,
sin θ 1 = x d 1 sin ψ ,
sin θ 0 = c 0 c 1 sin θ .
T ( θ ) = 2 cos θ / [ cos θ + ( c 0 2 / c 1 2 sin 2 θ ) 1 / 2 ] ,
D 1 = cos 2 θ 1 / [ c 1 c 0 cos 2 θ 0 d 0 + 1 a ( c 1 c 0 cos θ 0 cos θ 1 ) ] ,
D 2 = [ c 1 c 0 d 0 + 1 a ( c 1 c 0 cos θ 0 cos θ 1 ) ] 1 .
u = 1 2 j 0 ( k ) 0 π sin θ U ( θ ) d θ .
θ m = sin 1 ( a / b ) .
sin ψ 0 = b a sin θ ,
sin ψ 1 = c 1 c 0 sin ψ 0 ,
ψ = ψ 0 θ ,
r 0 = b cos θ a cos ψ 0 ,
r 1 = a cos ψ 1 + x cos ( ψ 1 ψ ) ,
n = a sin ψ 1 + x sin ( ψ 1 ψ ) .
z 1 = cos 2 ψ 1 / [ c 1 c 0 cos 2 ψ 0 ( r 0 + ) + 1 a ( c 1 c 0 cos ψ 0 cos ψ 1 ) ] 1 ,
z 2 = [ c 1 c 0 1 ( r 0 + ) + 1 a ( c 1 c 0 cos ψ 0 cos ψ 1 ) ] 1 .
V ( θ ) = i k 8 π 2 T ( ψ 0 ) exp { i k [ r 0 + c 0 c 1 r 1 + 1 2 c 0 c 1 n 2 / ( r 1 + z 1 ) ] } ( 1 + r 0 / ) ( 1 + r 1 / z 1 ) 1 / 2 ( 1 + r 1 / z 2 ) 1 / 2 ,
u = 2 π 0 θ m V ( θ ) sin θ d θ .

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