Abstract

We present calculations based on the Markov-chain technique of the intensity and direction of light traversing a multiply scattering medium in the isotropic scattering case for pulsed point-source illumination. From these calculations we derive expressions for the trade-off between spatial image resolution and detector integration time when the diffusion approximation is not valid. We show that image resolution better than the diffusion limit is theoretically possible use of multiply scattered light for samples thinner than ~35 scattering lengths. Monte Carlo simulations are used to extend these results to the anisotropic scattering case.

© 1994 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), pp. 1 ff.
  2. P. J. Heckman, R. T. Hodgson, IEEE J. Quantum Electron. QE-3, 445 (1967).
    [CrossRef]
  3. M. S. Patterson, B. Chance, B. C. Wilson, Appl. Opt. 28, 2331 (1989).
    [CrossRef] [PubMed]
  4. D. A. Benaron, D. K. Stevenson, Science 259, 1463 (1993).
    [CrossRef] [PubMed]
  5. L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, Science 253, 769 (1991).
    [CrossRef] [PubMed]
  6. M. D. Duncan, R. Mahon, L. L. Tankersley, J. Reintjes, Opt. Lett. 16, 1868 (1991).
    [CrossRef] [PubMed]
  7. M. R. Hee, J. A. Izatt, E. A. Swanson, J. G. Fujimoto, Opt. Lett. 18, 1107 (1993).
    [CrossRef] [PubMed]
  8. M. Bashkansky, J. Reintjes, Appl. Opt. 32, 3842 (1993).
    [PubMed]
  9. J. A. Moon, R. Mahon, M. D. Duncan, J. Reintjes, Opt. Lett. 18, 1591 (1993).
    [CrossRef] [PubMed]
  10. E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991), p. 12.
  11. J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979), p. 59.

1993 (4)

1991 (2)

1989 (1)

1967 (1)

P. J. Heckman, R. T. Hodgson, IEEE J. Quantum Electron. QE-3, 445 (1967).
[CrossRef]

Alfano, R. R.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, Science 253, 769 (1991).
[CrossRef] [PubMed]

Bashkansky, M.

Benaron, D. A.

D. A. Benaron, D. K. Stevenson, Science 259, 1463 (1993).
[CrossRef] [PubMed]

Chance, B.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), pp. 1 ff.

Duderstadt, J. J.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979), p. 59.

Duncan, M. D.

Fujimoto, J. G.

Heckman, P. J.

P. J. Heckman, R. T. Hodgson, IEEE J. Quantum Electron. QE-3, 445 (1967).
[CrossRef]

Hee, M. R.

Ho, P. P.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, Science 253, 769 (1991).
[CrossRef] [PubMed]

Hodgson, R. T.

P. J. Heckman, R. T. Hodgson, IEEE J. Quantum Electron. QE-3, 445 (1967).
[CrossRef]

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991), p. 12.

Izatt, J. A.

Katsev, I. L.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991), p. 12.

Liu, C.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, Science 253, 769 (1991).
[CrossRef] [PubMed]

Mahon, R.

Martin, W. R.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979), p. 59.

Moon, J. A.

Patterson, M. S.

Reintjes, J.

Stevenson, D. K.

D. A. Benaron, D. K. Stevenson, Science 259, 1463 (1993).
[CrossRef] [PubMed]

Swanson, E. A.

Tankersley, L. L.

Wang, L.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, Science 253, 769 (1991).
[CrossRef] [PubMed]

Wilson, B. C.

Zege, E. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991), p. 12.

Zhang, G.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, Science 253, 769 (1991).
[CrossRef] [PubMed]

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

P. J. Heckman, R. T. Hodgson, IEEE J. Quantum Electron. QE-3, 445 (1967).
[CrossRef]

Opt. Lett. (3)

Science (2)

D. A. Benaron, D. K. Stevenson, Science 259, 1463 (1993).
[CrossRef] [PubMed]

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, Science 253, 769 (1991).
[CrossRef] [PubMed]

Other (3)

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), pp. 1 ff.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Berlin, 1991), p. 12.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979), p. 59.

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Figures (3)

Fig. 1
Fig. 1

Imaging configuration and geometry used in the Markov-chain calculation. We consider imaging either the OP (with an apparent resolution OP) or the EP (with a point-spread function half-width of EP).

Fig. 2
Fig. 2

(a) Direction-integrated probability I 0 ( n ) ( r , t ) of a particle’s arriving at a point r = 1 at time t after a given number of collisions. (b) Average cosine r ^ · Ω ^ = I 1 ( n ) ( r , t ) / I 0 ( n ) ( r , t ) for various n values. The n = 3–6 curves have all been displaced vertically by 0.2 from one an other for clarity. In both (a) and (b) noisy lines are Monte Carlo simulations.

Fig. 3
Fig. 3

(a) Achievable resolution versus sample thickness for an attenuation for one photon collected in 1015. Curve OP is the optimum resolution (OPT) for imaging the OP. Curve EP is calculated for imaging the EP by use of Iqb. The points from μr ≃ 30–35 are calculated numerically from Eq. (7) directly, at thickness where the expansion used for Iqb begins to fail. The numbered curves are labeled with their constant time gates μcτg. Curve DL is the diffusion limit. (b) Resolution (upper graph) versus g calculated by means of Monte Carlo for a point source embedded in a turbid slab of overall thickness (1 − g) μd = 10. The solid curve is calculated from relation (8) with the assumption that the μ → (1 − g) μ. law holds. The received power (lower graph, points) drops much more quickly than this scaling law predicts (upper dashed curve) but not as quickly as the ballistic component (lower dashed curve).

Equations (8)

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P ( n + 1 ) ( r , t ; Ω ^ ) = d 3 r d t d Ω ^ h ( r - r , t - t , R ^ - Ω ^ ) × P ( n ) ( r , t ; Ω ^ ) δ ( R ^ - Ω ^ ) ,
P l m ( n + 1 ) ( r , t ) = d 3 r d t h ( R , t - t ) P 00 ( n ) ( r , t ) Y l m * ( R ^ ) ,
P l m ( n ) ( r , t ) = I l ( n ) ( r , t ) Y l m * ( r ^ ) .
h ( r , t ) = c μ 4 π exp ( - μ r ) δ ( r - c t ) r 2 ,
I 0 ( n ) ( r , t ) = c μ 4 ( μ r ) n - 4 exp ( - μ c t ) 2 n + 1 ( n - 3 ) ! 0 d y ( - y ) n - 3 ( y + 1 ) n - 1 L ( n ) ,
I 1 ( n ) ( r , t ) = c μ 4 ( μ r ) n - 4 exp ( - μ c t ) 2 n ( n - 3 ) ! 0 d y ( - y ) n - 3 ( y + 1 ) n - 1 × [ 1 + - y ( 1 + y ) ( n + 2 ) ] [ L ( n ) 2 ( 1 + y ) - L ( n - 1 ) ] ,
L ( n ) ( y ) = 1 π 2 { [ log ( 2 + y y ) ] 2 + π 2 } n / 2 × sin { n tan - 1 [ π log ( 2 + y y ) ] }
I qb ( r , τ g ) I 0 μ 3 4 π ( μ r ) 2 { [ μ c τ g 2 ln ( 2 r c τ g ) + 1 ] × [ ( 2 r c τ g ) μ c τ 2 - 1 ] + μ c τ g 2 ln ( 2 r c τ g ) - μ c τ g } exp ( - μ r ) ,

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