Abstract

We present analytic expressions for the achievable resolution for imaging through a turbid medium with multiply scattered light in the diffusion limit. We find that for detectable levels of light the spatial resolution ℛ (the half-width of the point-spread function) scales with thickness d of the sample as ℛ ≃ (0.2 ± 0.04)d over 10 orders of magnitude in input intensity and transport length. Experiments with a time-gated stimulated Raman amplifier are in good agreement with the calculations.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. S. Patterson, B. Chance, B. C. Wilson, Appl. Opt. 28, 2331 (1989).
    [CrossRef] [PubMed]
  2. D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, J. Phys. (Paris) 51, 2101 (1990).
    [CrossRef]
  3. J. Fishkin, E. Gratton, J. Opt. Soc. Am. A 10, 127 (1993).
    [CrossRef] [PubMed]
  4. A. Knüttel, J. M. Schmitt, J. R. Knutsen, Appl. Opt. 32, 381 (1993).
    [CrossRef] [PubMed]
  5. S. Andersson-Engels, R. Berg, S. Svanberg, Opt. Lett. 15, 1179 (1990).
    [CrossRef] [PubMed]
  6. J. C. Hebden, R. A. Kruger, Med. Phys. 17, 351 (1990).
    [CrossRef] [PubMed]
  7. D. A. Benaron, D. K. Stevenson, Science 259, 1463 (1993).
    [CrossRef] [PubMed]
  8. D. T. Delpy, M. Cope, P. van der Zee, S. Arrige, S. Wroy, J. Wyatt, Phys. Med. Biol. 33, 1433 (1988).
    [CrossRef] [PubMed]
  9. K. M. Yoo, F. Liu, R. R. Alfano, J. Opt. Soc. Am. B 7, 1685 (1990).
    [CrossRef]
  10. L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, Science 253, 769 (1991).
    [CrossRef] [PubMed]
  11. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), p. 175.
  12. M. D. Duncan, R. Mahon, L. L. Tankersley, J. Reintjes, Opt. Lett. 16, 1868 (1991).
    [CrossRef] [PubMed]
  13. V. G. Peters, D. R. Wyman, M. S. Paterson, G. L. Frank, Phys. Med. Biol. 35, 1317 (1990).
    [CrossRef] [PubMed]
  14. A. Fetter, J. Walecka, Theoretical Mechanics of Particles and Continua (McGraw-Hill, New York, 1980), p. 430.
  15. D. Sliney, M. Wolbarsht, Safety with Lasers and Other Optical Sources (Plenum, New York, 1982), p. 263.

1993 (3)

1991 (2)

1990 (4)

V. G. Peters, D. R. Wyman, M. S. Paterson, G. L. Frank, Phys. Med. Biol. 35, 1317 (1990).
[CrossRef] [PubMed]

K. M. Yoo, F. Liu, R. R. Alfano, J. Opt. Soc. Am. B 7, 1685 (1990).
[CrossRef]

S. Andersson-Engels, R. Berg, S. Svanberg, Opt. Lett. 15, 1179 (1990).
[CrossRef] [PubMed]

J. C. Hebden, R. A. Kruger, Med. Phys. 17, 351 (1990).
[CrossRef] [PubMed]

1989 (1)

1988 (1)

D. T. Delpy, M. Cope, P. van der Zee, S. Arrige, S. Wroy, J. Wyatt, Phys. Med. Biol. 33, 1433 (1988).
[CrossRef] [PubMed]

Alfano, R. R.

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

K. M. Yoo, F. Liu, R. R. Alfano, J. Opt. Soc. Am. B 7, 1685 (1990).
[CrossRef]

Andersson-Engels, S.

Arrige, S.

D. T. Delpy, M. Cope, P. van der Zee, S. Arrige, S. Wroy, J. Wyatt, Phys. Med. Biol. 33, 1433 (1988).
[CrossRef] [PubMed]

Benaron, D. A.

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

Berg, R.

Chance, B.

Cope, M.

D. T. Delpy, M. Cope, P. van der Zee, S. Arrige, S. Wroy, J. Wyatt, Phys. Med. Biol. 33, 1433 (1988).
[CrossRef] [PubMed]

Delpy, D. T.

D. T. Delpy, M. Cope, P. van der Zee, S. Arrige, S. Wroy, J. Wyatt, Phys. Med. Biol. 33, 1433 (1988).
[CrossRef] [PubMed]

Duncan, M. D.

Fetter, A.

A. Fetter, J. Walecka, Theoretical Mechanics of Particles and Continua (McGraw-Hill, New York, 1980), p. 430.

Fishkin, J.

Frank, G. L.

V. G. Peters, D. R. Wyman, M. S. Paterson, G. L. Frank, Phys. Med. Biol. 35, 1317 (1990).
[CrossRef] [PubMed]

Gratton, E.

Hebden, J. C.

J. C. Hebden, R. A. Kruger, Med. Phys. 17, 351 (1990).
[CrossRef] [PubMed]

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, J. Phys. (Paris) 51, 2101 (1990).
[CrossRef]

Ho, P. P.

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

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), p. 175.

Knutsen, J. R.

Knüttel, A.

Kruger, R. A.

J. C. Hebden, R. A. Kruger, Med. Phys. 17, 351 (1990).
[CrossRef] [PubMed]

Liu, C.

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

Liu, F.

Mahon, R.

Paterson, M. S.

V. G. Peters, D. R. Wyman, M. S. Paterson, G. L. Frank, Phys. Med. Biol. 35, 1317 (1990).
[CrossRef] [PubMed]

Patterson, M. S.

Peters, V. G.

V. G. Peters, D. R. Wyman, M. S. Paterson, G. L. Frank, Phys. Med. Biol. 35, 1317 (1990).
[CrossRef] [PubMed]

Pine, D. J.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, J. Phys. (Paris) 51, 2101 (1990).
[CrossRef]

Reintjes, J.

Schmitt, J. M.

Sliney, D.

D. Sliney, M. Wolbarsht, Safety with Lasers and Other Optical Sources (Plenum, New York, 1982), p. 263.

Stevenson, D. K.

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

Svanberg, S.

Tankersley, L. L.

van der Zee, P.

D. T. Delpy, M. Cope, P. van der Zee, S. Arrige, S. Wroy, J. Wyatt, Phys. Med. Biol. 33, 1433 (1988).
[CrossRef] [PubMed]

Walecka, J.

A. Fetter, J. Walecka, Theoretical Mechanics of Particles and Continua (McGraw-Hill, New York, 1980), p. 430.

Wang, L.

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

Weitz, D. A.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, J. Phys. (Paris) 51, 2101 (1990).
[CrossRef]

Wilson, B. C.

Wolbarsht, M.

D. Sliney, M. Wolbarsht, Safety with Lasers and Other Optical Sources (Plenum, New York, 1982), p. 263.

Wroy, S.

D. T. Delpy, M. Cope, P. van der Zee, S. Arrige, S. Wroy, J. Wyatt, Phys. Med. Biol. 33, 1433 (1988).
[CrossRef] [PubMed]

Wyatt, J.

D. T. Delpy, M. Cope, P. van der Zee, S. Arrige, S. Wroy, J. Wyatt, Phys. Med. Biol. 33, 1433 (1988).
[CrossRef] [PubMed]

Wyman, D. R.

V. G. Peters, D. R. Wyman, M. S. Paterson, G. L. Frank, Phys. Med. Biol. 35, 1317 (1990).
[CrossRef] [PubMed]

Yoo, K. M.

Zhang, G.

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

Zhu, J. X.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, J. Phys. (Paris) 51, 2101 (1990).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Med. Phys. (1)

J. C. Hebden, R. A. Kruger, Med. Phys. 17, 351 (1990).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Med. Biol. (2)

V. G. Peters, D. R. Wyman, M. S. Paterson, G. L. Frank, Phys. Med. Biol. 35, 1317 (1990).
[CrossRef] [PubMed]

D. T. Delpy, M. Cope, P. van der Zee, S. Arrige, S. Wroy, J. Wyatt, Phys. Med. Biol. 33, 1433 (1988).
[CrossRef] [PubMed]

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 (4)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), p. 175.

A. Fetter, J. Walecka, Theoretical Mechanics of Particles and Continua (McGraw-Hill, New York, 1980), p. 430.

D. Sliney, M. Wolbarsht, Safety with Lasers and Other Optical Sources (Plenum, New York, 1982), p. 263.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, J. Phys. (Paris) 51, 2101 (1990).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Geometry used in the calculation of the PSF.

Fig. 2
Fig. 2

(a) Resolution ℛ/lt (left scale) and fraction of received power N(τi) (right scale) as a function of integration time (in units of transport time) for several thicknesses. Arrows show which curve corresponds to which scale. Thicknesses Z = d/lt are Z = 10 (curve a), Z = 15 (curve b), Z = 20 (curve c), Z = 25 (curve d), and Z = 30 (curve e). (b) Minimum integration time (in transport times) versus sample thickness needed to reach a given fraction of received photons. N0 = 10−10 (curve a), N0 = 10−15 (curve b), and N0 = 10−20 (curve c). Curve d is τi = 4(ℛ/lt)2/3, the integration time versus resolution.

Fig. 3
Fig. 3

Image resolution versus thickness of turbid material measured with the experimental setup for three different transport lengths. Squares, lt = 0.33 mm; crosses, lt = 0.5 mm; diamonds, lt = 1 mm. The dotted line is the derived scaling law from the text, ℛ = 0.2d.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

2 Ψ = 1 D Ψ t , Ψ = 0 on z = 0 , d ,
P ( t ) = 4 π E 0 D [ 1 exp ( R 2 / 4 D t ) ] ( 4 π D t ) 3 / 2 × n = 0 [ f n ( t ) f n + ( t ) ] ,
f n ± ( t ) = [ ( 2 n + 1 ) d ± l t ] exp { [ ( 2 n + 1 ) d ± l t ] 2 4 D t } .
N ( τ i , ρ , Z ) = n = 0 [ n ( τ i , 0 , Z ) n + ( τ i , 0 , Z ) n ( τ i , ρ , Z ) + n + ( τ i , ρ , Z ) ] ,
n ± ( τ i , ρ , Z ) = erfc ( 3 τ i [ ( 2 n + 1 ) Z ± 1 ] 2 { 1 + [ ρ ( 2 n + 1 ) Z ± 1 ] 2 } 1 / 2 ) { 1 + [ ρ ( 2 n + 1 ) Z ± 1 ] 2 } 1 / 2

Metrics