Abstract

A solution has been obtained for the spatial and temporal distribution function for a pulsed fully collimated beam propagating through a homogeneous medium with Gaussian small-angle scattering. The solution was obtained first by separation of the general problem into two plane problems, which results in a partial differential equation in three variables. A Fourier transform on two projected variables (one angular and one spatial) and a Laplace transform on the projected temporal variable yielded a set of nonlinear differential equations, which were solved. A recursion relation for the moments of the distribution function was also obtained, and the software mathematica was used to evaluate these moments to high orders. The contractions on certain variables are also presented; they correspond to the solutions of less-general problems contained in the main problem. A change in the definition of the time-delay produces a remarkable change in the structure of the equations. These solutions should be quite useful for lidar studies in atmospheric and oceanic optics, x-ray and radio-wave scattering in the atmosphere and interstellar medium, and in medical physics.

© 1994 Optical Society of America

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References

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  1. I. P. Williamson, “Pulse broadening due to multiple scattering in the interstellar medium,” Mon. Not. R. Astronom. Soc. 157, 55–71 (1972).
  2. C. Alcock, S. Hatchett, “The effects of small-angle scattering on a pulse of radiation with an application of x-ray bursts and interstellar dust,” Astrophys. J. 222, 456–470 (1978).
    [CrossRef]
  3. E. A. Bucher, “Computer simulation of light pulse propagation for communication through thick clouds,” Appl. Opt. 12, 2391–2400 (1973).
    [CrossRef] [PubMed]
  4. E. A. Bucher, R. M. Lerner, “Experiments on light pulse communication and propagation through atmospheric clouds,” Appl. Opt. 12, 2401–2414 (1973).
    [CrossRef] [PubMed]
  5. L. B. Stotts, “The radiance produced by laser radiation transversing a particulate multiple-scattering medium,” J. Opt. Soc. Am. 67, 815–819, 1695(E) (1977).
    [CrossRef]
  6. A. C. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Chap. 15, pp. 321–325.
  7. R. M. Lerner, J. D. Summers, “Monte Carlo description of time- and space-resolved multiple forward scatter in natural waters,” Appl. Opt. 21, 861–869 (1982).
    [CrossRef] [PubMed]
  8. W. T. Scott, “The theory of small-angle scattering of fast charged particles,” Rev. Mod. Phys. 35, 231–313 (1963).
    [CrossRef]
  9. R. A. Elliott, “Multiple scattering of optical pulses in scale model clouds,” Appl. Opt. 22, 2670–2681 (1983).
    [CrossRef] [PubMed]
  10. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 1, Chap. 5, pp. 67–87.
  11. G. Zaccanti, P. Bruscaglioni, A. Ismaelli, L. Carraresi, M. Gurioli, Q. Wei, “Transmission of a pulsed thin light beam through thick turbid media: experimental results,” Appl. Opt. 31, 2141–2147 (1992).
    [CrossRef] [PubMed]
  12. J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 77, 7123–7128 (1972).
    [CrossRef]
  13. C. N. Yang, “Actual path length of electrons in foils,” Phys. Rev. 84, 599–600 (1951).
    [CrossRef]
  14. M. Kac, “On distributions of certain Wiener functionals,” Trans. Am. Math. Soc. 65, 1–13 (1949).
    [CrossRef]

1992 (1)

1983 (1)

1982 (1)

1978 (1)

C. Alcock, S. Hatchett, “The effects of small-angle scattering on a pulse of radiation with an application of x-ray bursts and interstellar dust,” Astrophys. J. 222, 456–470 (1978).
[CrossRef]

1977 (1)

1973 (2)

1972 (2)

I. P. Williamson, “Pulse broadening due to multiple scattering in the interstellar medium,” Mon. Not. R. Astronom. Soc. 157, 55–71 (1972).

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 77, 7123–7128 (1972).
[CrossRef]

1963 (1)

W. T. Scott, “The theory of small-angle scattering of fast charged particles,” Rev. Mod. Phys. 35, 231–313 (1963).
[CrossRef]

1951 (1)

C. N. Yang, “Actual path length of electrons in foils,” Phys. Rev. 84, 599–600 (1951).
[CrossRef]

1949 (1)

M. Kac, “On distributions of certain Wiener functionals,” Trans. Am. Math. Soc. 65, 1–13 (1949).
[CrossRef]

Alcock, C.

C. Alcock, S. Hatchett, “The effects of small-angle scattering on a pulse of radiation with an application of x-ray bursts and interstellar dust,” Astrophys. J. 222, 456–470 (1978).
[CrossRef]

Bruscaglioni, P.

Bucher, E. A.

Carraresi, L.

Elliott, R. A.

Gurioli, M.

Hatchett, S.

C. Alcock, S. Hatchett, “The effects of small-angle scattering on a pulse of radiation with an application of x-ray bursts and interstellar dust,” Astrophys. J. 222, 456–470 (1978).
[CrossRef]

Ishimaru, A. C.

A. C. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Chap. 15, pp. 321–325.

Ismaelli, A.

Kac, M.

M. Kac, “On distributions of certain Wiener functionals,” Trans. Am. Math. Soc. 65, 1–13 (1949).
[CrossRef]

Lerner, R. M.

Scott, W. T.

W. T. Scott, “The theory of small-angle scattering of fast charged particles,” Rev. Mod. Phys. 35, 231–313 (1963).
[CrossRef]

Shipley, S. T.

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 77, 7123–7128 (1972).
[CrossRef]

Stotts, L. B.

Summers, J. D.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 1, Chap. 5, pp. 67–87.

Wei, Q.

Weinman, J. A.

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 77, 7123–7128 (1972).
[CrossRef]

Williamson, I. P.

I. P. Williamson, “Pulse broadening due to multiple scattering in the interstellar medium,” Mon. Not. R. Astronom. Soc. 157, 55–71 (1972).

Yang, C. N.

C. N. Yang, “Actual path length of electrons in foils,” Phys. Rev. 84, 599–600 (1951).
[CrossRef]

Zaccanti, G.

Appl. Opt. (5)

Astrophys. J. (1)

C. Alcock, S. Hatchett, “The effects of small-angle scattering on a pulse of radiation with an application of x-ray bursts and interstellar dust,” Astrophys. J. 222, 456–470 (1978).
[CrossRef]

J. Geophys. Res. (1)

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 77, 7123–7128 (1972).
[CrossRef]

J. Opt. Soc. Am. (1)

Mon. Not. R. Astronom. Soc. (1)

I. P. Williamson, “Pulse broadening due to multiple scattering in the interstellar medium,” Mon. Not. R. Astronom. Soc. 157, 55–71 (1972).

Phys. Rev. (1)

C. N. Yang, “Actual path length of electrons in foils,” Phys. Rev. 84, 599–600 (1951).
[CrossRef]

Rev. Mod. Phys. (1)

W. T. Scott, “The theory of small-angle scattering of fast charged particles,” Rev. Mod. Phys. 35, 231–313 (1963).
[CrossRef]

Trans. Am. Math. Soc. (1)

M. Kac, “On distributions of certain Wiener functionals,” Trans. Am. Math. Soc. 65, 1–13 (1949).
[CrossRef]

Other (2)

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 1, Chap. 5, pp. 67–87.

A. C. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Chap. 15, pp. 321–325.

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Figures (2)

Fig. 1
Fig. 1

Geometry used in the description of the projected beam-spread distribution function. The solid curve is the three-dimensional path; the dashed curve is its projection in the x1z plane.

Fig. 2
Fig. 2

Geometry used to show the invariance of the projected distribution function.

Tables (1)

Tables Icon

Table 1 Values of gmnk used in Eq. (33) to Give the Momentsa

Equations (82)

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θ 2 = 4 s z ,
1 2 0 π Φ ( θ ) d ( cos θ ) = 1 ,
1 2 0 π Φ ( θ ) cos θ d ( cos θ ) = g ,
θ 2 = 2 1 - cos θ = 2 ( 1 - g ) .
4 s = 2 ( 1 - g ) σ .
s z 1.
t = f 001 / c = s z 2 6 c ,
Δ z sec θ = Δ z [ 1 + ( θ 1 2 + θ 2 2 ) / 2 ] + ,
θ 2 = θ 1 2 + θ 2 2 ,
θ 1 2 = θ 2 2 = 1 2 θ 2 .
t 1 ( Q ) - t 1 ( P ) = 1 2 α 1 2 Δ z ,
t ( Q ) - t ( P ) = 1 2 α 2 Δ z ,
t = t 1 + t 2 .
f ( x 1 , x 2 , θ 1 , θ 2 , t ; z ) = 0 t f 1 ( x 1 , θ 1 , t - t 2 ; z ) f 1 ( x 2 , θ 2 , t 2 ; z ) d t 2 .
f 1 [ x 1 + θ 1 Δ z , θ 1 , t 1 + 1 2 ( θ 1 - x 1 / z ) 2 Δ z ; z + Δ z ] = f 1 [ x 1 , θ 1 , t 1 ; z ] ,
f ( θ , z ) = 1 4 π s z exp ( - θ 2 / 4 s z ) ,
2 π 0 f ( θ , z ) θ d θ = 1
f ( θ , z ) = f 1 ( θ 1 , z ) f 2 ( θ 2 , z ) , f 1 ( θ 1 , z ) = 1 ( 4 π s z ) 1 / 2 exp ( - θ 1 2 / 4 s z )
f 1 z - s 2 f 1 θ 1 2 = 0 ,
f 1 z + θ 1 f 1 x 1 - s 2 f 1 θ 1 2 + 1 2 ( θ 1 - x 1 z ) 2 f 1 t 1 = 0.
f 1 ( x 1 , θ 1 , t 1 ; 0 ) = δ ( x 1 ) δ ( θ 1 ) δ ( t 1 ) .
F ( u , v , w ; z ) = - d x - d θ 0 d t exp ( i u x + i v θ - w t ) × f ( x , θ , t ; z ) ,
f ( x , θ , t ; z ) = 1 ( 2 π ) 2 - d u - d v exp ( - i u x - i v θ ) × L - 1 { F ( u , v , w ; z ) }
n F v n = T { ( i θ ) n f } ,             n F u n = T { ( i x ) n f } , n F w n = T { ( - t ) n f } ,
w n F = T { n f t n } ,             ( - i u ) n F = T { n f x n } , ( - i v ) n F = T { n f θ n } ,
w n n F v n = T { ( i θ ) n n f t n } .
F z - u F v + s v 2 F - 1 2 w ( 1 z 2 2 F u 2 - 2 z 2 F u v + 2 F v 2 ) = 0.
θ ~ v - 1 ~ ( s z ) 1 / 2 ,             x ~ u - 1 ~ z θ ~ s 1 / 2 z 3 / 2 , t ~ w - 1 ~ z θ 2 ~ s z 2 ,
v ˜ = ( s z ) 1 / 2 v ,             u ˜ = s 1 / 2 z 3 / 2 u ,             w ˜ = s z 2 w .
F ( u , v , w ; z ) = G ( u ˜ , v ˜ , w ˜ ) ,
v ˜ 2 G + 3 2 u ˜ G u ˜ + ( 1 2 v ˜ - u ˜ ) G v ˜ + 2 w ˜ G w ˜ - 1 2 w ˜ 2 2 G u ˜ 2 + w ˜ 2 G u ˜ v ˜ - 1 2 w ˜ 2 G v ˜ 2 = 0.
G ( u , v , w ) = exp ( - f ( w ) - a ( w ) u 2 - b ( w ) u v - c ( w ) v 2 ) .
- 2 w d a d w = 3 a - b + 1 2 w ( 2 a - b ) 2 ,
- 2 w d b d w = 2 b - 2 c - w ( 2 a - b ) ( 2 c - b ) ,
- 2 w d c d w = c - 1 + 1 2 w ( 2 c - b ) 2 ,
d f d w = 1 2 ( a - b + c ) .
- 2 w d h d w = 3 h - 1 + 2 w h 2 .
d f d w = 1 2 h ,
f m n k = - d x - d θ 0 d t x m θ n t k f ( x , θ , t ; z ) ,
f ( x , θ , t ; z ) = f ( - x , - θ , t ; z ) ,
f m n k = s ( 1 / 2 ) ( m + n ) + k z ( 1 / 2 ) ( 3 m + n ) + 2 k g m n k ,
g m n k = ( - 1 ) ( 1 / 2 ) ( m + n ) + k [ m + n + k G u m v n w k ] u = v = w = 0 .
f m n k z = m f m - 1 , n + 1 , k + n ( n - 1 ) s f m , n - 2 , k + k 2 f m , n + 2 , k - 1 - k z f m + 1 , n + 1 , k - 1 + k 2 z 2 f m + 2 , n , k - 1 ,
[ ( 3 m + n ) 2 + 2 k ] g m n k = m g m - 1 , n + 1 , k + n ( n - 1 ) g m , n - 2 , k + k 2 g m , n + 2 , k - 1 - k g m + 1 , n + 1 , k - 1 + k 2 g m + 2 , n , k - 1 .
a ( w ) = h ( w ) = 1 3 - 2 45 w + 8 945 w 2 - 8 4725 w 3 + 32 93555 w 4 - 44224 6385121875 w 5 + ,
b ( w ) = c ( w ) = 1 - 1 6 w + 1 30 w 2 - 17 2520 w 3 + 31 22680 w 4 - 691 2494800 w 5 + ,
f ( w ) = 1 6 w - 1 90 w 2 + 4 2835 w 3 - 1 4725 w 4 + 16 467775 w 5 - 11056 1915538625 w 6 + .
G ( u , v , w ) = exp [ - ( 1 3 u 2 + u v + v 2 ) ] [ 1 + ( - 1 6 + 2 45 × u 2 + 1 6 u v + 1 6 v 2 ) w + ( 1 40 - 1 63 u 2 - 11 180 u v - 11 180 v 2 + 2 2025 u 4 + 1 135 u 3 v + 23 1080 u 2 v 2 + 1 36 u v 3 + 1 72 v 4 ) w 2 + ] = 1 - ( 1 3 u 2 + u v + v 2 ) + ( - 1 6 + 1 10 u 2 + 1 3 u v + 1 3 v 2 ) w + .
a ( w ) = h ( w ) = coth y y - 1 y 2 ,
b ( w ) = c ( w ) = 2 ( cosh y - 1 ) y sinh y ,
exp [ - f ( w ) ] = ( y sinh y ) 1 / 2 ,
y = ( 2 s z 2 w ) 1 / 2 .
f 1 ( x 1 , θ 1 ; z ) = 0 f 1 ( x 1 , θ 1 , t 1 ; z ) d t 1
G ( u , v , 0 ) = exp ( - 1 3 u 2 - u v - v 2 ) ,
F ( u , v , 0 ; z ) = exp ( - 1 3 s z 3 u 2 - s z 2 u v - s z v 2 ) .
f ( x , θ ; z ) = 3 1 / 2 2 π s z 2 exp ( - 3 x 2 s z 3 + 3 x θ s z 2 - θ 2 s z ) .
f ( z , α ; z ) = 3 1 / 2 2 π s z 2 exp ( - x 2 s z 3 + x α s z 2 - α 2 s z ) .
α = 0 ,             α 2 = θ 2 - 2 x θ / z + x 2 / z 2 = 2 s z / 3.
G ( 0 , 0 , w ) = exp [ - f ( w ) ] = ( y sinh y ) 1 / 2 .
t = f 001 = s z 2 g 001 = - s z 2 [ G ( 0 , 0 , w ) w ] w = 0 = 1 6 s z 2 ,
t 2 = f 002 = s 2 z 4 g 002 = s z 2 [ 2 G ( 0 , 0 , w ) w 2 ] w = 0 = 1 20 s 2 z 2 .
α 2 = β 2 ,
( θ - x z ) 2 = ( x z ) 2 ,
2 θ x = z θ 2 .
f ( β , α , t ; z ) = | ( x , θ ) ( α , β ) | f ( x , θ , t ; z ) = z f ( x , θ , t ; z ) .
F ( u , v , w ; z ) = - d β - d α 0 d t exp ( i u β + i v α - w t ) f ( β , α , t ; z ) .
G ( u , v , w ; z ) = G ( u , v , w ; z ) ,
- ln G = f ( w ) + p ( w ) u 2 + q ( w ) u v + p ( w ) v 2 .
- ln G = f ( w ) + p ( w ) u 2 + [ 2 p ( w ) + q ( w ) ] u v + [ 2 p ( w ) + q ( w ) ] v 2 ,
p ( w ) = a ( w ) ,             q ( w ) = b ( w ) - 2 a ( w ) .
f 1 z + θ 1 f 1 x 1 - s 2 f 1 θ 1 2 + 1 2 θ 1 2 f 1 t 1 = 0 ,
F z - u F v + s v 2 F - 1 2 w 2 F v 2 = 0 ,
v 2 G + 3 2 u G u + ( 1 2 v - u ) G v + 2 w G w - 1 2 w 2 G v 2 = 0 ,
- 2 w d a d w = 3 a - b + 1 2 w b 2 ,
- 2 w d b d w = 2 b - 2 c + 2 w b c ,
- 2 w d c d w = c - 1 + 2 w c 2 .
a ( w ) = 1 y 2 - tanh y y 3 = 1 3 - 4 15 w + 68 415 w 2 - 496 2835 w 2 + 22112 155925 w 4 + ,
b ( w ) = 1 - sech y 2 y 2 = 1 - 5 6 w + 61 90 w 2 - 277 504 × w 3 + 50521 113400 w 4 + ,
c ( w ) = tanh y y = 1 - 2 3 w + 8 15 w 2 - 136 315 w 3 + 992 2835 w 4 + ,
exp [ - f ( w ) ] = G ( 0 , 0 , w ) = ( cosh y ) - 1 / 2 = 1 - 1 2 w + 7 24 w 2 - 139 720 w 3 + 5473 40320 w 4 .
t 3 = 139 120 s 3 z 6 ,
t 3 = t 3 + 3 2 x 2 t 2 z + 3 4 x 4 t z 2 + 1 8 x 6 z 3 = s 3 z 6 ( 61 2520 + 183 1260 + 13 30 + 5 9 ) = 139 120 s 3 z 6 ,

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