Abstract

We present a new model of optical coherence tomography (OCT) taking into account multiple scattering. A theoretical analysis and experimental investigation reveals that in OCT, despite multiple scattering, the field backscattered from the sample is generally spatially coherent and that the resulting interference signal with the reference field is stationary relative to measurement time. On the basis of this result, we model an OCT signal as a sum of spatially coherent fields with random-phase arguments—constant during measurement time—caused by multiple scattering. We calculate the mean of such a random signal from classical results of statistical optics and a Monte Carlo simulation. OCT signals predicted by our model are in very good agreement with a depth scan measurement of a sample consisting of a mirror covered with an aqueous suspension of microspheres. We discuss other comprehensive OCT models based on the extended Huygens–Fresnel principle, which rest on the assumption of partially coherent interfering fields.

© 2005 Optical Society of America

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2005 (2)

2004 (3)

2001 (1)

2000 (1)

1999 (2)

G. Yao, L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[CrossRef] [PubMed]

C. Yang, K. An, L. T. Perelman, R. R. Dasari, M. S. Feld, “Spatial coherence of forward-scattered light in a turbid medium,” J. Opt. Soc. Am. A 16, 866–871 (1999).
[CrossRef]

1998 (2)

K. K. Bizheva, A. M. Siegel, D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: the transition to diffusing wave spectroscopy,” Phys. Rev. E 58, 7664–7667 (1998).
[CrossRef]

D. J. Smithies, T. Lindmo, Z. Chen, J. S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

1997 (2)

J. M. Schmitt, A. Knuettel, “Model of optical coherence tomography of heterogeneous tissue,” J. Opt. Soc. Am. A 14, 1231–1242 (1997).
[CrossRef]

W. Rudolph, M. Kempe, “Topical review: trends in optical biomedical imaging,” J. Mod. Opt. 44, 1617–1642 (1997).
[CrossRef]

1996 (1)

1995 (3)

1994 (1)

1993 (1)

1989 (2)

1983 (1)

1979 (1)

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar signal systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[CrossRef]

1978 (2)

1974 (1)

1967 (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967).
[CrossRef]

Aalders, M. C.

An, K.

Andersen, P. E.

Birngruber, R.

Bizheva, K. K.

K. K. Bizheva, A. M. Siegel, D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: the transition to diffusing wave spectroscopy,” Phys. Rev. E 58, 7664–7667 (1998).
[CrossRef]

Boas, D. A.

K. K. Bizheva, A. M. Siegel, D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: the transition to diffusing wave spectroscopy,” Phys. Rev. E 58, 7664–7667 (1998).
[CrossRef]

Bonner, R. F.

Bourquin, S.

Chance, B.

Chen, Z.

D. J. Smithies, T. Lindmo, Z. Chen, J. S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

Cheung, R.

Dasari, R. R.

de Mul, F. M.

Engelhardt, R.

Faber, D. J.

Feld, M. S.

Fried, D. L.

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967).
[CrossRef]

Fujimoto, J. G.

Gan, X.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 44–54.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 210–211.

Gu, M.

Hassler, K.

Hee, M. R.

Ishimaru, A.

Izatt, J. A.

Jacques, S. L.

L. H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

S. L. Jacques, “Time resolved propagation of ultrashort laser pulses within turbid tissues,” Appl. Opt. 28, 2223–2229 (1989).
[CrossRef] [PubMed]

Karamata, B.

Kempe, M.

W. Rudolph, M. Kempe, “Topical review: trends in optical biomedical imaging,” J. Mod. Opt. 44, 1617–1642 (1997).
[CrossRef]

Kinsinger, J. B.

Knuettel, A.

Knüttel, A.

Kuga, Y.

Kumar, G.

Lambelet, P.

Lasser, T.

Laubscher, M.

Leutenegger, M.

Lindmo, T.

D. J. Smithies, T. Lindmo, Z. Chen, J. S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

Lu, Q.

Luo, Q.

Lutomirski, R. F.

Mallick, S.

Milner, T. E.

D. J. Smithies, T. Lindmo, Z. Chen, J. S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

Nelson, J. S.

D. J. Smithies, T. Lindmo, Z. Chen, J. S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

Owen, G. M.

Pan, Y.

Patterson, M. S.

Perelman, L. T.

Petoukhova, A. L.

Rosperich, J.

Rudolph, W.

W. Rudolph, M. Kempe, “Topical review: trends in optical biomedical imaging,” J. Mod. Opt. 44, 1617–1642 (1997).
[CrossRef]

Salathé, R. P.

Schmitt, J. M.

Shimizu, K.

Siegel, A. M.

K. K. Bizheva, A. M. Siegel, D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: the transition to diffusing wave spectroscopy,” Phys. Rev. E 58, 7664–7667 (1998).
[CrossRef]

Smithies, D. J.

D. J. Smithies, T. Lindmo, Z. Chen, J. S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

Steenbergen, W.

Swanson, E. A.

Thrane, L.

van der Meer, F. J.

van Leeuwen, T. G.

Wang, L. H.

L. H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Wang, L. V.

G. Yao, L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[CrossRef] [PubMed]

Wilson, B. C.

Yadlowsky, M. J.

Yang, C.

Yao, G.

G. Yao, L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[CrossRef] [PubMed]

Yura, H. T.

L. Thrane, H. T. Yura, P. E. Andersen, “Analysis of optical coherence tomography systems based on the extended Huygens–Fresnel principle,” J. Opt. Soc. Am. A 17, 484–490 (2000).
[CrossRef]

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar signal systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[CrossRef]

Zheng, L.-Q.

L. H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Appl. Opt. (7)

Comput. Methods Programs Biomed. (1)

L. H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

W. Rudolph, M. Kempe, “Topical review: trends in optical biomedical imaging,” J. Mod. Opt. 44, 1617–1642 (1997).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

Opt. Acta (1)

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar signal systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Med. Biol. (2)

D. J. Smithies, T. Lindmo, Z. Chen, J. S. Nelson, T. E. Milner, “Signal attenuation and localization in optical coherence tomography studied by Monte Carlo simulation,” Phys. Med. Biol. 43, 3025–3044 (1998).
[CrossRef] [PubMed]

G. Yao, L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[CrossRef] [PubMed]

Phys. Rev. E (1)

K. K. Bizheva, A. M. Siegel, D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: the transition to diffusing wave spectroscopy,” Phys. Rev. E 58, 7664–7667 (1998).
[CrossRef]

Proc. IEEE (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967).
[CrossRef]

Other (3)

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 44–54.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 210–211.

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Figures (6)

Fig. 1
Fig. 1

Scheme of a wide-field OCT setup based on a free-space Michelson interferometer.

Fig. 2
Fig. 2

Correlogram envelopes obtained in wide-field OCT with a sample consisting of a mirror covered with scattering solution [8 optical density (OD), g = 0.85 ] for a single scan (dashed curve) and for the average of 25 envelopes (thick solid curve). The thin solid curve represents the envelope of the source autocorrelation function (ACF) measured in water. Z 1 is the sample mirror position, and Z 2 is 60 μ m away from Z 1 . Speckle statistics are measured at Z 1 2 . All correlograms are normalized and plotted against the reference mirror position.

Fig. 3
Fig. 3

Histogram of the signal intensity distribution measured at position Z 2 (Experiment), indicated in Fig. 1, against the Rayleigh probability density function (Theory).

Fig. 4
Fig. 4

Time scale of fluctuations caused by Brownian motion for an OCT signal resulting from interference with multiply scattered light. In the main plot, T S = 500 ms is the time interval between two scans. For a much smaller time scale (zoomed portion of the main plot), T M = 150 μ s is the inverse of the Doppler modulation frequency.

Fig. 5
Fig. 5

Scheme of an amplitude-splitting interferometer with broadband spatially coherent illumination (SCI). E R and E S are the reference and sample fields, respectively. E j are components of E S , corresponding to ballistic light backscattered once (double line) and multiply scattered light (single line). Interference between E R and E S is considered on a single detector whose size matches the coherence area A c (see Subsection 2B) located at conjugated distance with the probe volume P v . The optical setup, not illustrated for simplicity, is shown in Fig. 1.

Fig. 6
Fig. 6

(a) Photon distribution as a function of the additional path length L j of multiply scattered photons relative to ballistic photons ( L 0 ) , obtained with Monte Carlo simulation, and (b) the corresponding normalized mean OCT signal obtained with our model (dashed curve) compared with experimental results (solid curve). The plots are shown against the actual reference mirror position. Z 1 is the sample mirror position, and Z 2 is 60 μ m away from Z 1 .

Equations (27)

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Ω = k 2 k B T Z 3 π 2 η d l S ( 1 g ) ,
E R ( k ) = Re { I ( k ) exp [ j ( k z ω t ) ] } ,
E S ( k ) = Re ( r ( k ) I ( k ) exp { j [ φ ( k ) ω t ] } ) ,
I ( k ) = 2 [ E R ( k ) + E S ( k ) ] 2 = I ( k ) [ 1 + r 2 ( k ) + 2 r ( k ) Re ( exp { j [ φ ( k ) k z ] } ) ] ,
I = 0 I ( k ) d k = 0 I ( k ) [ 1 + r 2 ( k ) ] d k + 2 Re { 0 I ( k ) r ( k ) × exp [ j φ ( k ) ] exp ( j k z ) d k } .
I = I C + 2 Re ( F { I ( k ) r ( k ) exp [ j φ ( k ) ] } ) .
E S ( k ) = Re [ j = 1 N E j ( k ) ] = Re [ I ( k ) j = 1 N u j ( k ) exp ( j k L j ω t ) ] .
I = I 0 + 2 Re { F [ I ( k ) j = 1 N u j ( k ) exp ( j k L j ) ] } ,
I = I 0 + 2 Re { F [ I ( k ) ] j = 1 N F [ u j ( k ) exp ( j k L j ) ] } .
g ( z ) = F [ I ( k ) ] = F [ I ( Δ k ) ] = exp ( j k 0 z ) F [ I ( k ) ] = exp ( j k 0 z ) g 0 ( z ) ,
F [ exp ( j k L j ) ] = δ ( z L j ) .
I = I 0 + 2 Re [ j = 1 N g 0 ( z ) exp ( j k 0 z ) u j δ ( z L j ) ] .
I = I 0 + 2 Re [ exp ( j k 0 z ) j = 1 N u j g 0 ( z L j ) exp ( j k 0 L j ) ] .
i ( z ) j = 1 N u j g 0 ( z L j ) exp ( j k 0 L j ) .
j = 1 N u j g 0 ( z L j ) exp ( j k 0 L j ) j = 1 N u j g 0 ( z L j ) exp ( j k 0 L j ) = j = 1 N α j ( z ) exp ( j θ j ) = A ( z ) exp [ j Φ ( z ) ] .
A ( z ) ¯ = ( π 2 ) 1 2 σ ( z ) ,
σ 2 ( z ) = α j 2 ( z ) ¯ 2 .
σ 2 ( z ) = u j 2 g 0 ( z L j ) 2 ¯ 2 = 1 2 N j = 1 N U j [ g 0 ( z L j ) ] 2 ,
σ ( z ) = [ 1 2 N U j g 0 2 ( z ) ] 1 2 ,
σ 2 ( z ) = 1 2 N j = 1 N U j [ g 0 ( z L j ) ] 2 = 1 2 v = 1 V 1 m v + 1 m v j = m v + 1 m v + 1 U j [ g 0 ( z L j ) ] 2 = 1 2 v = 1 V I v [ g 0 ( z L v ) ] 2 ,
σ ( z ) = [ 1 2 I v g 0 2 ( z ) ] 1 2 ,
A ( z ) ¯ = ( π 2 ) 1 2 σ ( z ) = [ π 4 I v g 0 2 ( z ) ] 1 2 .
S ( z ) + A ( z ) exp [ j Φ ( z ) ] = u 0 g 0 ( z L 0 ) exp ( j k 0 L 0 ) + j = 1 N u j g 0 ( z L j ) exp ( j k 0 L j ) ,
A ( z ) ¯ = ( π 2 ) 1 2 σ ( z ) exp [ β 2 ( z ) 4 ] { [ 1 + β 2 ( z ) 2 ] I 0 [ β 2 ( z ) 4 ] + β 2 ( z ) 2 I 1 [ β 2 ( z ) 4 ] } ,
β ( z ) = S ( z ) σ ( z ) = u 0 g 0 ( z ) σ ( z )
u 0 = U 0 = I B ,
Γ = U ( r , z ) U * ( r + ρ , z ) ,

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