Abstract

The path-length-resolved power spectrum of a time-varying scattered light field measured by a time-of-flight method or low-coherence interferometry is evaluated by a new numerical simulation algorithm. The path-length-resolved power spectrum is theoretically derived by combining diffusing-wave-spectroscopy theory and radiative-transfer theory. The proposed algorithm, using the Monte Carlo method, is used to determine the scattering configurations and numerically calculate the power spectrum. The path-length distribution, path-length-dependent scattering order distribution, and path-length-resolved power spectrum are demonstrated numerically over all scattering orders. The resultant power spectra agree with experimental results measured by the low-coherence-dynamic-light-scattering method.

© 2008 Optical Society of America

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References

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  1. P. J. Berne and R. Pecora, Dynamic Light Scattering (Dover, 2000).
  2. D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134-1137 (1988).
    [CrossRef] [PubMed]
  3. A. G. Yodh, P. D. Kaplan, and D. J. Pine, “Pulsed diffusing-wave spectroscopy: High resolution through nonlinear optical gating,” Phys. Rev. B 40, 4744-4747 (1990).
    [CrossRef]
  4. D. A. Boas, K. K. Bizheva, and A. M. Siegel, “Using dynamic low-coherence interferometry to image Brownian motion within highly scattering media,” Opt. Lett. 23, 319-321 (1998).
    [CrossRef]
  5. K. Ishii, R. Yoshida, and T. Iwai, “Single-scattering spectroscopy for extremely dense colloidal suspensions by use of a low-coherence interferometer,” Opt. Lett. 30, 555-557 (2005).
    [CrossRef] [PubMed]
  6. R. Carminati, R. Elaloufi, and J.-J. Greffet, “Beyond the diffusing-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
    [CrossRef] [PubMed]
  7. A. Wax, C. Yang, R. R. Dasari, and M. S. Feld, “Path-length-resolved dynamic light scattering: modeling the transition from single to diffusive scattering,” Appl. Opt. 40, 4222-4227 (2001).
    [CrossRef]
  8. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

2005 (1)

2004 (1)

R. Carminati, R. Elaloufi, and J.-J. Greffet, “Beyond the diffusing-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
[CrossRef] [PubMed]

2001 (1)

1998 (1)

1990 (1)

A. G. Yodh, P. D. Kaplan, and D. J. Pine, “Pulsed diffusing-wave spectroscopy: High resolution through nonlinear optical gating,” Phys. Rev. B 40, 4744-4747 (1990).
[CrossRef]

1988 (1)

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134-1137 (1988).
[CrossRef] [PubMed]

Appl. Opt. (1)

Opt. Lett. (2)

Phys. Rev. B (1)

A. G. Yodh, P. D. Kaplan, and D. J. Pine, “Pulsed diffusing-wave spectroscopy: High resolution through nonlinear optical gating,” Phys. Rev. B 40, 4744-4747 (1990).
[CrossRef]

Phys. Rev. Lett. (2)

R. Carminati, R. Elaloufi, and J.-J. Greffet, “Beyond the diffusing-wave spectroscopy model for the temporal fluctuations of scattered light,” Phys. Rev. Lett. 92, 213903 (2004).
[CrossRef] [PubMed]

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60, 1134-1137 (1988).
[CrossRef] [PubMed]

Other (2)

P. J. Berne and R. Pecora, Dynamic Light Scattering (Dover, 2000).

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

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Figures (8)

Fig. 1
Fig. 1

Illustration of configurations of scattering points. The light is perpendicularly incident on a semi-infinite medium, propagates along a path with length L, undergoes n scattering events, and emerges from the scattering medium at the incident position traveling in the opposite direction of the incident light. r j and k j denote the scattering position of the j th scattering event and the wave vector of the j th -order scattered light, respectively.

Fig. 2
Fig. 2

Illustration of the scattering process. The specific intensity I ( r , s ̂ ) propagates to point r, is scattered in the direction s ̂ , and is converted to the specific intensity I ( r , s ̂ ) .

Fig. 3
Fig. 3

Illustration of the configurations of scattering points for single-scattered light. The incident light is scattered in the backward direction at a depth 1 2 and emerges from the scattering medium at the incident position traveling in the opposite direction of the incident light.

Fig. 4
Fig. 4

Illustration of the configurations of scattering points of fourth-order scattered light. The first and fourth scattering positions are located nearer than 1 2 on the axis of the incident light. The second-order scattered light is inside the ellipsoid that runs across a depth of 1 2 and whose focal points coincide with the first and final scattering positions. The third scattering position is on the ellipsoid whose focal point is the second and final scattering positions and which runs across the crossing point between the direction of the first-scattered light and the other ellipsoid.

Fig. 5
Fig. 5

Scattered light intensity obtained from numerical calculations. The vertical and horizontal axes denote the scattered light intensity normalized by the total scattered light intensity with zero total path length and the total path length normalized by the mean free path length, respectively. The thick curve denotes the total scattered light intensity. The thin curves correspond to light of each scattering order. Light scattered from the first to the 30th order is aligned along the arrow.

Fig. 6
Fig. 6

Scattering order distribution functions for scattered light with six different path lengths of (a) l , (b) 3 l , (c) 5 l , (d) 10 l , (e) 15 l , and (f) 20 l . The vertical and horizontal axes denote the fraction of the n th -order scattered-light intensity I n ( L ) I ( L ) and the scattering order, respectively. The circles denote the numerical results. The dashed curves lines express Poisson distributions.

Fig. 7
Fig. 7

Dependence of width of power spectrum on scattering order, normalized by that of single-scattered light. The vertical and horizontal axes denote the HWHM normalized by that for single-scattered light and the scattering order, respectively. The symbols ●, ◼, and ▴ correspond to numerical results for polystyrene latex suspensions with diameters of 100 nm , 200 nm , and 300 nm , respectively. The dashed curve represents the HWHM for single-scattered light. The solid curves denote the DWS theory prediction.

Fig. 8
Fig. 8

Dependence of power spectrum of scattered light field on path length. The vertical and horizontal axes denote HWHM normalized by that for single-scattered light and the total path length normalized by the mean free path length, respectively. The symbols ◼, ●, and ◆ denote the experimental results measured by low-coherence dynamic light scattering [5] for 10 vol.% polystyrene latex suspensions with diameters of 150 nm , 225 nm , and 300 nm , respectively. The solid, dashed, and dotted curves denote the numerical results for polystyrene latex suspensions with diameters of 150 nm , 225 nm , and 300 nm , respectively. The dashed–dotted curve denotes the HWHM for single-scattered light.

Equations (19)

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γ ( τ ; L ) E ( τ ; L ) E * ( 0 ; L ) I ( L ) = n = 1 [ I n ( L ) I ( L ) ] γ n ( τ ) ,
I ( f ; L ) = F [ γ ( τ ; L ) ] = n = 1 [ I n ( L ) I ( L ) ] P n ( f ) ,
γ n ( τ ) = exp i [ j = 1 n q j Δ r j ( τ ) ] q , Δ r ,
γ n ( τ ) = exp [ j = 1 n q j 2 D τ ] q ,
d I ( r , s ̂ ) d l = ρ σ t I ( r , s ̂ ) + ρ σ s 4 π p ( s ̂ , s ̂ ) I ( r , s ̂ ) d ω + ε ( r , s ̂ ) ,
I ( r , s ̂ ) = ρ σ s 0 exp [ ρ σ t l ] [ 4 π p ( s ̂ , s ̂ ) I ( r , s ̂ ) d ω ] d l ,
I n ( 0 , k 0 ; L ) = I 0 ( 0 , k 0 ) ( ρ σ s ) n exp ( ρ σ t L ) l 1 l n ω 1 ω n [ j = 0 n 1 p ( k j + 1 , k j ) ] d ω 1 d ω n d l 1 d l n ,
I n ( 0 , k 0 ; L ) = I 0 ( 0 , k 0 ) ( ρ σ s ) n L n 1 exp ( ρ σ t L ) × A n ,
A n = l ̂ 1 l ̂ n ω 1 ω n [ j = 0 n 1 p ( k j + 1 , k j ) ] d ω 1 d ω n d l ̂ 1 d l ̂ n .
I 1 ( 0 , k 0 ; L ) = I 0 ( 0 , k 0 ) ( ρ σ s ) exp ( ρ σ t L ) p ( k 0 , k 0 ) .
I 2 ( 0 , k 0 ; L ) = A 2 I 0 ( 0 , k 0 ) ( ρ σ s ) 2 L exp ( ρ σ t L ) ,
A 2 = p ( k 0 , k 0 ) p ( k 0 , k 0 ) 0 1 2 d l ̂ 1 + p ( k 0 , k 0 ) p ( k 0 , k 0 ) 0 1 2 d l ̂ 2 = p ( k 0 , k 0 ) p ( k 0 , k 0 ) .
I 3 ( 0 , k 0 ; L ) = A 3 I 0 ( 0 , k 0 ) ( ρ σ s ) 3 L 2 exp ( ρ σ t L ) ,
A 3 = 0 1 2 0 1 2 ω 1 p ( k 0 , k 2 ) p ( k 2 , k 1 ) p ( k 1 , k 0 ) d ω 1 d l ̂ 1 d l ̂ 3 .
I 4 ( 0 , k 0 ; L ) = A 4 I 0 ( 0 , k 0 ) ( ρ σ s ) 4 L 3 exp ( ρ σ t L ) ,
A 4 = 0 1 2 0 s 2 ̱ max 0 1 2 ω 1 ω 2 p ( k 0 , k 3 ) p ( k 3 , k 2 ) p ( k 2 , k 1 ) p ( k 1 , k 0 ) d ω 1 d ω 2 d l ̂ 1 d l ̂ 2 d l ̂ 4 ,
p ( l ̂ ) = n ( 1 l ̂ ) n 1 .
p ( l ̂ ) = ( n 2 ) ( 1 l ̂ ) n 3 .
P n ( l ̂ ) = exp ( K ) ( K ) n n ! ,

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