Abstract

The statistics of an asymmetric correlated random walk are developed. The results generalize expressions for the distribution of intensity beyond a diffuser, introducing random telegraph-wave phase variations.

© 1987 Optical Society of America

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References

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  1. P. N. Pusey, “Statistical properties of scattered radiation,” in Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1977), pp. 45–141.
  2. P. N. Pusy, R. J. A. Tough, “Particle interactions,” in Dynamic Light Scattering, R. Pecora, ed. (Plenum, New York, 1985), pp. 85–179.
    [CrossRef]
  3. E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. 23, 453–461 (1984).
    [CrossRef]
  4. E. Jakeman, “Optical scattering experiments,” in Wave Propagation and Scattering, B. J. Uscinski, ed. (Clarendon, Oxford, 1986), pp. 241–259.
  5. B. B. Mandelbrot, Fractals (Freeman, San Francisco, Calif., 1977).
  6. E. Jakeman, “Scattering by multiscale systems,” in Wave Propagation and Scattering, B. J. Uscinski, ed. (Clarendon, Oxford, 1986), pp. 49–63.
  7. E. Jakeman, B. J. Hoenders, “Scattering by a surface of rectangular grooves,” Opt. Acta 29, 1587–1598 (1982).
    [CrossRef]
  8. J. F. Benzoni, S. Sarkar, D. Sherrington, “Statistical mechanical models for phase screens,”J. Phys. A. 19, 589–594 (1986); “Random phase screens,” J. Opt. Soc. Am. A 4, 17–26 (1987).
    [CrossRef]
  9. E. Jakeman, J. G. McWhirter, “Correlation function dependence of the scintillation behind a deep random phase screen,”J. Phys. A 10, 1599–1643 (1977).
    [CrossRef]
  10. E. Renshaw, R. Henderson, “The correlated random walk,”J. Appl. Prob. 18, 403–414 (1981).
    [CrossRef]
  11. S. Goldstein, “On diffusion by discontinuous movements, and on the telegraph equation,” Q. J. Mech. 4, 129–156 (1951).
    [CrossRef]
  12. A. Okubo, Diffusion and Ecological Problems: Mathematical Models (Springer-Verlag, Berlin, 1980).
  13. D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B 31, 179–186 (1983).
    [CrossRef]
  14. R. Henderson, E. Renshaw, D. Ford, “A correlated random walk model for two-dimensional diffusion,”J. Appl. Probabil. 21, 233–246 (1984).
    [CrossRef]
  15. A. D. Bruce, “Probability density functions for collective coordinates in Ising-like systems,”J. Phys. C 14, 3667–3688 (1981).
    [CrossRef]

1986 (1)

J. F. Benzoni, S. Sarkar, D. Sherrington, “Statistical mechanical models for phase screens,”J. Phys. A. 19, 589–594 (1986); “Random phase screens,” J. Opt. Soc. Am. A 4, 17–26 (1987).
[CrossRef]

1984 (2)

E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. 23, 453–461 (1984).
[CrossRef]

R. Henderson, E. Renshaw, D. Ford, “A correlated random walk model for two-dimensional diffusion,”J. Appl. Probabil. 21, 233–246 (1984).
[CrossRef]

1983 (1)

D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B 31, 179–186 (1983).
[CrossRef]

1982 (1)

E. Jakeman, B. J. Hoenders, “Scattering by a surface of rectangular grooves,” Opt. Acta 29, 1587–1598 (1982).
[CrossRef]

1981 (2)

A. D. Bruce, “Probability density functions for collective coordinates in Ising-like systems,”J. Phys. C 14, 3667–3688 (1981).
[CrossRef]

E. Renshaw, R. Henderson, “The correlated random walk,”J. Appl. Prob. 18, 403–414 (1981).
[CrossRef]

1977 (1)

E. Jakeman, J. G. McWhirter, “Correlation function dependence of the scintillation behind a deep random phase screen,”J. Phys. A 10, 1599–1643 (1977).
[CrossRef]

1951 (1)

S. Goldstein, “On diffusion by discontinuous movements, and on the telegraph equation,” Q. J. Mech. 4, 129–156 (1951).
[CrossRef]

Benzoni, J. F.

J. F. Benzoni, S. Sarkar, D. Sherrington, “Statistical mechanical models for phase screens,”J. Phys. A. 19, 589–594 (1986); “Random phase screens,” J. Opt. Soc. Am. A 4, 17–26 (1987).
[CrossRef]

Bruce, A. D.

A. D. Bruce, “Probability density functions for collective coordinates in Ising-like systems,”J. Phys. C 14, 3667–3688 (1981).
[CrossRef]

Ford, D.

R. Henderson, E. Renshaw, D. Ford, “A correlated random walk model for two-dimensional diffusion,”J. Appl. Probabil. 21, 233–246 (1984).
[CrossRef]

Goldstein, S.

S. Goldstein, “On diffusion by discontinuous movements, and on the telegraph equation,” Q. J. Mech. 4, 129–156 (1951).
[CrossRef]

Henderson, R.

R. Henderson, E. Renshaw, D. Ford, “A correlated random walk model for two-dimensional diffusion,”J. Appl. Probabil. 21, 233–246 (1984).
[CrossRef]

E. Renshaw, R. Henderson, “The correlated random walk,”J. Appl. Prob. 18, 403–414 (1981).
[CrossRef]

Hoenders, B. J.

E. Jakeman, B. J. Hoenders, “Scattering by a surface of rectangular grooves,” Opt. Acta 29, 1587–1598 (1982).
[CrossRef]

Hollins, R. C.

D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B 31, 179–186 (1983).
[CrossRef]

Jakeman, E.

E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. 23, 453–461 (1984).
[CrossRef]

D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B 31, 179–186 (1983).
[CrossRef]

E. Jakeman, B. J. Hoenders, “Scattering by a surface of rectangular grooves,” Opt. Acta 29, 1587–1598 (1982).
[CrossRef]

E. Jakeman, J. G. McWhirter, “Correlation function dependence of the scintillation behind a deep random phase screen,”J. Phys. A 10, 1599–1643 (1977).
[CrossRef]

E. Jakeman, “Scattering by multiscale systems,” in Wave Propagation and Scattering, B. J. Uscinski, ed. (Clarendon, Oxford, 1986), pp. 49–63.

E. Jakeman, “Optical scattering experiments,” in Wave Propagation and Scattering, B. J. Uscinski, ed. (Clarendon, Oxford, 1986), pp. 241–259.

Jordan, D. L.

D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B 31, 179–186 (1983).
[CrossRef]

Mandelbrot, B. B.

B. B. Mandelbrot, Fractals (Freeman, San Francisco, Calif., 1977).

McWhirter, J. G.

E. Jakeman, J. G. McWhirter, “Correlation function dependence of the scintillation behind a deep random phase screen,”J. Phys. A 10, 1599–1643 (1977).
[CrossRef]

Okubo, A.

A. Okubo, Diffusion and Ecological Problems: Mathematical Models (Springer-Verlag, Berlin, 1980).

Pusey, P. N.

P. N. Pusey, “Statistical properties of scattered radiation,” in Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1977), pp. 45–141.

Pusy, P. N.

P. N. Pusy, R. J. A. Tough, “Particle interactions,” in Dynamic Light Scattering, R. Pecora, ed. (Plenum, New York, 1985), pp. 85–179.
[CrossRef]

Renshaw, E.

R. Henderson, E. Renshaw, D. Ford, “A correlated random walk model for two-dimensional diffusion,”J. Appl. Probabil. 21, 233–246 (1984).
[CrossRef]

E. Renshaw, R. Henderson, “The correlated random walk,”J. Appl. Prob. 18, 403–414 (1981).
[CrossRef]

Sarkar, S.

J. F. Benzoni, S. Sarkar, D. Sherrington, “Statistical mechanical models for phase screens,”J. Phys. A. 19, 589–594 (1986); “Random phase screens,” J. Opt. Soc. Am. A 4, 17–26 (1987).
[CrossRef]

Sherrington, D.

J. F. Benzoni, S. Sarkar, D. Sherrington, “Statistical mechanical models for phase screens,”J. Phys. A. 19, 589–594 (1986); “Random phase screens,” J. Opt. Soc. Am. A 4, 17–26 (1987).
[CrossRef]

Tough, R. J. A.

P. N. Pusy, R. J. A. Tough, “Particle interactions,” in Dynamic Light Scattering, R. Pecora, ed. (Plenum, New York, 1985), pp. 85–179.
[CrossRef]

Appl. Phys. B (1)

D. L. Jordan, R. C. Hollins, E. Jakeman, “Experimental measurements of non-Gaussian scattering by a fractal diffuser,” Appl. Phys. B 31, 179–186 (1983).
[CrossRef]

J. Appl. Prob. (1)

E. Renshaw, R. Henderson, “The correlated random walk,”J. Appl. Prob. 18, 403–414 (1981).
[CrossRef]

J. Appl. Probabil. (1)

R. Henderson, E. Renshaw, D. Ford, “A correlated random walk model for two-dimensional diffusion,”J. Appl. Probabil. 21, 233–246 (1984).
[CrossRef]

J. Phys. A (1)

E. Jakeman, J. G. McWhirter, “Correlation function dependence of the scintillation behind a deep random phase screen,”J. Phys. A 10, 1599–1643 (1977).
[CrossRef]

J. Phys. A. (1)

J. F. Benzoni, S. Sarkar, D. Sherrington, “Statistical mechanical models for phase screens,”J. Phys. A. 19, 589–594 (1986); “Random phase screens,” J. Opt. Soc. Am. A 4, 17–26 (1987).
[CrossRef]

J. Phys. C (1)

A. D. Bruce, “Probability density functions for collective coordinates in Ising-like systems,”J. Phys. C 14, 3667–3688 (1981).
[CrossRef]

Opt. Acta (1)

E. Jakeman, B. J. Hoenders, “Scattering by a surface of rectangular grooves,” Opt. Acta 29, 1587–1598 (1982).
[CrossRef]

Opt. Eng. (1)

E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. 23, 453–461 (1984).
[CrossRef]

Q. J. Mech. (1)

S. Goldstein, “On diffusion by discontinuous movements, and on the telegraph equation,” Q. J. Mech. 4, 129–156 (1951).
[CrossRef]

Other (6)

A. Okubo, Diffusion and Ecological Problems: Mathematical Models (Springer-Verlag, Berlin, 1980).

E. Jakeman, “Optical scattering experiments,” in Wave Propagation and Scattering, B. J. Uscinski, ed. (Clarendon, Oxford, 1986), pp. 241–259.

B. B. Mandelbrot, Fractals (Freeman, San Francisco, Calif., 1977).

E. Jakeman, “Scattering by multiscale systems,” in Wave Propagation and Scattering, B. J. Uscinski, ed. (Clarendon, Oxford, 1986), pp. 49–63.

P. N. Pusey, “Statistical properties of scattered radiation,” in Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1977), pp. 45–141.

P. N. Pusy, R. J. A. Tough, “Particle interactions,” in Dynamic Light Scattering, R. Pecora, ed. (Plenum, New York, 1985), pp. 85–179.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Realization of the telegraph wave T and correlated random walk E.

Fig. 2
Fig. 2

The probability model.

Fig. 3
Fig. 3

Initial conditions and transition probabilities.

Fig. 4
Fig. 4

Distribution of scattered intensity in the forward direction when kh = (2n + 1)π/2.

Fig. 5
Fig. 5

Angular distribution of scattered intensity.

Fig. 6
Fig. 6

Second normalized intensity moment in the forward direction when kh = (2n + 1)π/2.

Equations (55)

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E + ( R 1 t ) = i E 0 ( 1 + cos θ ) 2 λ R exp [ i ( k R - ω t ) ] × - d 2 r exp [ i k κ r 2 - i k r · r / R + i ϕ ( r ; t ) ] A ( r ) ,
E + = C L exp ( - i ω t ) - L / 2 L / 2 d x exp [ i ϕ ( x ) ] .
ϕ ( x ) = k h T ( x ) ,
T ( x ) = ± 1 ,
E + = C exp ( - i ω t ) [ cos ( k h ) + i L sin ( k h ) - L / 2 L / 2 d x T ( x ) ] .
I = E + 2 = C 2 [ cos 2 k h + sin 2 k h L 2 ( - L / 2 L / 2 d x T ( x ) ) 2 ] .
E = 1 L - L / 2 L / 2 d x T ( x ) .
p i ( n + 1 ) = p p i - 1 ( n ) + q q i + 1 ( n ) ,
q i ( n + 1 ) = q p i - 1 ( n ) + p q i + 1 ( n ) .
P ( z , s ) i = - n = 0 p i ( n ) z i s n , Q ( z , s ) i = - n = 0 q i ( n ) z i s n ,
p 0 ( 0 ) = α ,             q 0 ( 0 ) = β
P ( z , s ) [ ( 1 - s p z ) ( 1 - s p / z ) - s 2 q q ] = α ( 1 - p s / z ) + β q s / z
Q ( z , s ) [ ( 1 - s p z ) ( 1 - s p / z ) - s 2 q q ] = α q z s + β ( 1 - p z s ) .
R ( z , s ) = P ( z , s ) + Q ( z , s ) = n = 0 R n ( z ) s n = i = - n = 0 r i ( n ) z i s n ;
R ( z , s ) = 1 - s [ ( β p - α q ) z + ( α p + β q ) / z ] ( 1 - s p z ) ( 1 - s p / z ) - q q s 2 .
R n ( z ) = ( λ 1 - λ 2 ) - 1 { ( λ 1 n + 1 - λ 2 n + 1 ) - ( λ 1 n - λ 2 n ) × [ ( β p - α q ) z + ( α p - β q ) / z ] } .
p n = exp ( - μ ) ,             p n = exp ( - μ ) ,
r n ( n ) = Prob ( first step to right and no change of direction occurs ) = α p n - 1 ~ α e - μ ,
r - n ( n ) = Prob ( first step to left and no change of direction occurs ) = β ( p ) n - 1 ~ β e - μ .
p = exp ( - μ / n ) = 1 - μ / n + μ 2 / 2 n 2 -
p = exp ( - μ / n ) = 1 - μ / n + μ 2 / 2 n 2 - .
M ( θ ) = lim n R n ( exp θ / n ) = exp [ - 1 2 ( μ + μ ) ] × ( cosh { 1 2 [ ( μ - μ - 2 θ ) 2 + 4 μ μ ] 1 / 2 } + μ + μ + 2 ( α - β ) θ [ ( μ - μ - 2 θ ) 2 + 4 μ μ ] 1 / 2 × sinh { 1 2 [ ( μ - μ - 2 θ ) 2 + 4 μ μ ] 1 / 2 } ) .
T ¯ = ( μ - μ ) / ( μ + μ ) ,             R ¯ = 2 μ μ / ( μ + μ ) .
α = μ / ( μ + μ ) ,             β = μ ( μ + μ ) .
M ( θ ) = exp ( - N ¯ ) [ cosh ( N ¯ 2 + θ 2 ) 1 / 2 + N ¯ ( N ¯ 2 + θ 2 ) 1 / 2 sinh ( N ¯ 2 + θ 2 ) 1 / 2 ] .
P ( E ) = 1 2 exp ( - N ¯ ) ( N ¯ { I 0 [ N ¯ ( 1 - E 2 ) 1 / 2 ] + I 1 [ N ¯ ( 1 - E 2 ) 1 / 2 ] ( 1 - E 2 ) 1 / 2 } + δ ( E - 1 ) + δ ( E + 1 ) ) ,
M ( θ ) = exp ( - θ ) ,
P ( E ) = δ ( E + 1 ) ,
r i ( n ) - p r i - 1 ( n - 1 ) - p r i + 1 ( n - 1 ) + ( p p - q q ) r i ( n - 2 ) = 0 ,
P ( E , x ) - p P ( E - δ E , x - δ x ) - p P ( E + δ E , x - δ x ) + ( p p - q q ) P ( E , x - 2 δ x ) = 0.
2 P x 2 + μ + μ L P x = B 2 2 P E 2 + ( μ - μ ) B L P E .
2 P x 2 + ( μ + μ ) P x = 2 P E 2 + ( μ - μ ) P E .
P ( E , x ) = exp { - 1 2 ( μ + μ ) x - 1 2 ( μ - μ ) E } Q ( E , x )
2 Q x 2 - μ μ Q = 2 Q E 2 ,
K 0 I 0 { [ μ μ ( x 2 - E 2 ) ] 1 / 2 } ,
K 1 x I 1 { [ μ μ ( x 2 - E 2 ) ] 1 / 2 } / [ ( x 2 - E 2 ) ] 1 / 2 ,
K 2 E I 1 { [ μ μ ( x 2 - E 2 ) ] 1 / 2 } / [ ( x 2 - E 2 ) ] 1 / 2 ,
r n ( n - 2 ) = P r [ ( 0 , 0 ) ( - 1 , 1 ) ( 0 , 2 ) ( 1 , 3 ) ] + P r [ ( 0 , 0 ) ( 1 , 1 ) ( j , j ) ( j - 1 , j + 1 ) ( j , j + 2 ) for j = 1 , n - 2 ] + P r [ 0 , 0 ( 1 , 1 ) ( n - 1 , n - 1 ) ( n - 2 , n ) ] ;
r n ( n - 2 ) = p n - 3 [ α p q + β p q + α ( n - 2 ) q q ] .
r - n ( n - 2 ) = p ( n - 3 ) [ α p q + β p q + β ( n - 2 ) q q ] .
x - 3 δ x x - δ x P ( E , x ) d E = 2 δ x P ( E , x ) E = x - ;
P ( E , x ) E = x - = ½ exp ( - μ x ) ( α μ + β μ + α μ μ x ) ,
P ( E , x ) E = - x + = ½ exp ( - μ x ) ( α μ + β μ + β μ μ x ) .
r n ( n ) = α p n - 1 ~ α exp ( - μ x ) , r - n ( n ) = β p n - 1 ~ β exp ( - μ x ) .
K 0 = ½ ( α μ + β μ ) ,             K 1 = ½ ( α + β ) μ μ , K 2 = ½ ( α - β ) μ μ .
P ( E , x ) = ½ exp [ - ½ ( μ + μ ) x - ½ ( μ - μ ) E ] × { ( α μ + β μ ) I 0 ( y ) + ( μ μ ) [ ( α + β ) x + ( α - β ) E ] I 1 ( y ) / y } + α e - μ x δ ( E - x ) + β e - μ x δ ( E + x ) .
P ( E ) = P ( E , 1 ) .
P ( E ) = exp [ N ¯ ( T ¯ E - 1 ) 1 - T ¯ 2 ] ( N ¯ / 2 { I 0 [ N ¯ ( 1 - E 2 1 - T ¯ 2 ) 1 / 2 ] 1 + ( 1 + T ¯ E ) [ ( 1 - E 2 ) ( 1 - T ¯ 2 ) ] 1 / 2 I 1 [ N ¯ ( 1 - E 2 1 - T ¯ 2 ) 1 / 2 ] } + ½ ( 1 + T ¯ ) δ ( E - 1 ) + ½ ( 1 - T ¯ ) δ ( E + 1 ) ) .
P ( I ) = { 2 I exp [ N ¯ / ( 1 - T ¯ 2 ) ] } - 1 ( N ¯ cosh ( T ¯ I 1 - T ¯ 2 ) × { I 0 [ N ¯ ( 1 - I 1 - T ¯ 2 ) 1 / 2 ] + I 1 [ N ¯ ( 1 - I 1 - T ¯ 2 ) 1 / 2 ] / × [ ( 1 - I ) ( 1 - T ¯ 2 ) ] 1 / 2 } + N T ¯ I sinh ( T ¯ I 1 - T ¯ 2 ) × I 1 [ N ¯ ( 1 - I 1 - T ¯ 2 ) 1 / 2 ] / [ ( 1 - I ) ( 1 - T ¯ 2 ) ] 1 / 2 + 2 [ cosh ( N T ¯ 1 - T ¯ 2 ) + T ¯ sinh ( N T ¯ 1 - T ¯ 2 ) ] δ ( I - 1 ) ) .
E 2 = T ¯ 2 + 1 2 ( 1 - T ¯ 2 ) 1 γ [ 2 - 1 γ + 1 γ exp ( - 2 γ ) ] ,
1 2 d 2 d L 2 ( L 2 E 2 ) T ( 0 ) T ( L ) ,
T ( 0 ) T ( x ) = T ¯ 2 + ( 1 - T ¯ 2 ) exp ( - 2 γ x / L ) .
I ( θ ) = C 2 - 1 / 2 1 / 2 d x - 1 / 2 1 / 2 d x [ cos 2 k h + T ( x L ) T ( x L ) sin 2 k h ] exp [ i k L ( x - x ) sin θ ] .
I ( θ ) = 2 ( cos 2 k h + T ¯ 2 sin 2 k h ) ( 1 - cos ψ ) / ψ 2 + 2 ( 1 - T ¯ 2 ) sin 2 k h ( 4 γ 2 + ψ 2 ) 2 { 2 γ ( 4 γ 2 + ψ 2 ) + exp ( - 2 γ ) × [ ( 4 γ 2 - ψ 2 ) cos ψ - 4 γ ψ sin ψ ] - 4 γ 2 + ψ 2 } ,
E 4 = T ¯ 4 + 3 ( 1 - T ¯ 2 ) 2 γ 4 { 2 γ 2 ( 1 - 6 T ¯ 2 ) + 4 γ 3 T ¯ 2 + ( 1 - 5 T ¯ 2 ) ( 3 - 4 γ ) + exp ( - 2 γ ) [ 2 γ 2 T ¯ 2 - ( 1 - 5 T ¯ 2 ) ( 2 γ + 3 ) ] } .

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