Abstract

The diffusion equations whose derivatives in time have prompted controversy in recent papers are reexamined from a different point of view. The resulting equation is demonstrated to agree with the first-order differential equation with respect to the time that was given by Furutsu’s theory. It is also shown that the reduced light velocity of the diffuse pulse, predicted by Ishimaru’s theory, is a result carried beyond the range of the applicability of the diffusion approximation utilized and is not characteristic of the diffusion.

© 1984 Optical Society of America

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References

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  1. A. Ishimaru, “Diffusion of a pulse in densely distributed scatterers,”J. Opt. Soc. Am. 68, 1045–1050 (1978).
    [CrossRef]
  2. K. Furutsu, “Diffusion equation derived from space–time transport equation,”J. Opt. Soc. Am. 70, 360–366 (1980).
    [CrossRef]
  3. K. Shimizu, A. Ishimaru, L. Reynolds, A. P. Bruckner, “Backscattering of a picosecond pulse from densely distributed scatterers,” Appl. Opt. 18, 3484–3488 (1979).
    [CrossRef] [PubMed]
  4. S. Ito, K. Furutsu, “Theory of light pulse propagation through thick clouds,”J. Opt. Soc. Am. 70, 366–374 (1980).
    [CrossRef]
  5. K. Shimizu, A. Ishimaru, “Experimental test of the reduced effective velocity of light in a diffuse medium,” Opt. Lett. 5, 205–207 (1980).
    [CrossRef] [PubMed]
  6. S. Ito, “Theory of beam light pulse propagation through thick clouds: effects of beamwidth and scatterers behind the light source on pulse broadening,” Appl. Opt. 20, 2706–2715 (1981).
    [CrossRef] [PubMed]
  7. A. Ishimaru, “Theoretical and experimental study of transient phenomena in random media,” in Multiple Scattering and Waves in Random Media, P. L. Chow et al., eds. (North-Holland, Amsterdam, 1981), pp. 155–163.
  8. E. A. Bucher, “Computer simulation of light pulse propagation for communication through thick clouds,” Appl. Opt. 12, 2391–2400 (1973).
    [CrossRef] [PubMed]
  9. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  10. B. Davison, Neutron Transport Theory (Oxford U. Press, London, 1957), p. 133.
  11. For a diffusion equation in anisotropic random media, see K. Furutsu, “Diffusion equation derived from the space-time transport equation in anisotropicrandom media,” J. Math. Phys. 21, 765–777 (1980).
    [CrossRef]
  12. K. Furutsu, “Higher-order boundary condition of the space-time diffusion equation,”J. Opt. Soc. Am. 73, 117–118 (1983).
    [CrossRef]
  13. It is also noted that, for the very broad pulse given by Eq. (25), the reduced light velocity c/3 predicted in Ref. 1 can not be estimated by the arrival time of the peak value of pulse intensity, although Shimizu and Ishimaru “ measured ” the reduced velocity by that method.5
  14. Equation (7.4.18) of Ref. 9, which was heuristically presumed, does not seem to support Ishimaru’s equation.

1983 (1)

1981 (1)

1980 (4)

1979 (1)

1978 (1)

1973 (1)

Bruckner, A. P.

Bucher, E. A.

Davison, B.

B. Davison, Neutron Transport Theory (Oxford U. Press, London, 1957), p. 133.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Furutsu, K.

Ishimaru, A.

Ito, S.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Reynolds, L.

Shimizu, K.

Appl. Opt. (3)

J. Math. Phys. (1)

For a diffusion equation in anisotropic random media, see K. Furutsu, “Diffusion equation derived from the space-time transport equation in anisotropicrandom media,” J. Math. Phys. 21, 765–777 (1980).
[CrossRef]

J. Opt. Soc. Am. (4)

Opt. Lett. (1)

Other (5)

It is also noted that, for the very broad pulse given by Eq. (25), the reduced light velocity c/3 predicted in Ref. 1 can not be estimated by the arrival time of the peak value of pulse intensity, although Shimizu and Ishimaru “ measured ” the reduced velocity by that method.5

Equation (7.4.18) of Ref. 9, which was heuristically presumed, does not seem to support Ishimaru’s equation.

A. Ishimaru, “Theoretical and experimental study of transient phenomena in random media,” in Multiple Scattering and Waves in Random Media, P. L. Chow et al., eds. (North-Holland, Amsterdam, 1981), pp. 155–163.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

B. Davison, Neutron Transport Theory (Oxford U. Press, London, 1957), p. 133.

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Figures (2)

Fig. 1
Fig. 1

Intensity of a pulse wave in an infinite scattering medium. The normalized intensity (4πr)2t0G is shown as a function of t/t0 for several values of the optical distance τ = ρσtr. Solid and dashed curves indicate the values calculated by Eqs. (23) and (25), respectively. W0 is the single-scattering albedo. μ = 0.6. t0 = r/c.

Fig. 2
Fig. 2

Same as Fig. 1, except that μ = 0.87.

Equations (30)

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d I r i / d s = - i ( K 1 - K 2 * ) I r i + ,
d I d / d s = - i ( K 1 - K 2 * ) I d + ρ d Ω f 1 f 2 * I d + r i ,
K i = k i + 2 π ρ f i ( o ^ , o ^ ) / k i ,
r i ( r , s ^ ) = ρ d Ω f 1 f 2 * I r i ,
I d ( r , s ^ ) ~ U d ( r ) + ( 3 / 4 π ) F d ( r ) · s ^ ,
U d ( r ) = ( 4 π ) - 1 d Ω I d ( r , s ^ ) ,
F d ( r ) = d Ω I d ( r , s ^ ) s ^ .
U d ( 3 / 4 π ) F d ,
· F d = 4 π ( - i K d - ρ σ α ) U d + 4 π ρ σ s U r i ,
K d = Re ( K 1 - K 2 * ) , σ t = - Im ( K 1 - K 2 * ) / ρ = σ a + σ s , U r i = ( 4 π ) - 1 d Ω I r i ( r , s ^ ) ,
s ^ · grad U d + ( 3 / 4 π ) s ^ · grad ( F d · s ^ ) = ( - i K d - ρ σ a ) U d + ( - i K d - ρ σ t + ρ σ s μ ) ( 3 / 4 π ) F d · s ^ + r i ,
grad U d = ( 3 / 4 π ) [ - i K d - ρ σ a - ρ σ s ( 1 - μ ) ] F d + ( 3 / 4 π ) d Ω r i s ^ .
d Ω r i s ^ = ρ σ s μ d Ω I r i ( r , s ^ ) s ^ = ρ σ s μ F r i .
grad U d = - ( 3 / 4 π ) ρ σ s ( 1 - μ ) × [ 1 ρ σ s ( 1 - μ ) c ( t + ρ σ a c ) + 1 ] F d .
1 ρ σ s ( 1 - μ ) c ( t + ρ σ a c ) 1 ,
F d ~ - D grad U d ,
D - 1 = ( 3 / 4 π ) ρ σ s ( 1 - μ ) .
l d grad U d U d ,
[ ( η ρ σ s ) - 1 2 - ( i K d + + ρ σ a ) ] U d = - ρ σ s U r i .
U d ( r s ) - 2 ( η ρ σ s ) - 1 n ^ · U d ( r s ) = 0 ,
( 2 - q 2 ) U d = - 3 ( i K d + ρ σ t r ) ρ σ s U r i + ( 3 / 4 π ) div d Ω r i s ^ ,
U d ( r s ) - ( 2 / 3 ) ( i K d + ρ σ t r ) - 1 n ^ · U d ( r s ) + 2 n ^ · Q 1 ( r s ) / 4 π = 0 ,
q 2 = 3 ( i K d + ρ σ a ) ( i K d + ρ σ t r ) ,             σ t r = ρ σ s ( 1 - μ ) + σ a , Q 1 ( r s ) = ( i K d + ρ σ t r ) - 1 d Ω r i s ^ .
G ( r , t ) = η ρ σ s 8 π 2 r - d ν exp { i ν t - [ η ρ σ s ( ρ σ a + i ν / c ) ] 1 / 2 r } = { 0 ,             t < 0 2 π ( 4 π r ) 2 t 0 ( η ρ σ s r ) 3 / 2 ( t 0 / t ) 3 / 2 × exp [ - ρ σ a c t - ( η ρ σ s r 2 / 4 c t ) ] ,             t > 0 ,
t m = ( 3 / 4 ρ σ a c ) { [ ( 4 / 9 ) ( η σ a / σ s ) ( ρ σ s r ) 2 + 1 ] 1 / 2 - 1 } ,
G I ( r , t ) = { 0 ,             t < 3 t 0 [ 3 exp ( - α t ) / 4 π r c ] { a δ ( t - 3 t 0 ) + δ ( t - 3 t 0 ) + 3 [ ( a 3 t 0 / Z ) I 1 ( Z ) + a 4 t t 0 I 2 ( Z ) / Z ] } ,             t 3 t 0
U d = 3 ( i K d + ρ σ t r ) exp ( - q r ) / 4 π r ,
F d = ( 4 π / 3 ) { 3 [ 1 + ρ σ s ( 1 - μ ) / i K d + ρ σ a ) ] - 1 / 2 + [ ( i K d + ρ σ t r ) r ] - 1 } U d .
| 1 ρ σ s ( 1 - μ ) ( i K d + ρ σ a ) | 1 ,
F d ~ ( 4 π / 3 ) U d ,

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