Abstract

An interferometric technique is utilized to measure both the time- and frequency-domain optical fields scattered by a random medium. The method uses a tunable continuous-wave laser source to make frequency-resolved measurements within a fixed-path-length interferometer. Measured frequency-domain field statistics, with a linearly polarized input, are shown to be zero-mean, circular complex Gaussian for both co- and cross-polarization states. With decreasing scatter, the extracted average impulse responses for co- and cross-polarized states show distinct differences, thereby providing insight into the scattering domain.

© 2005 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2004 (1)

2003 (3)

M. A. Webster, K. J. Webb, A. M. Weiner, J. Xu, H. Cao, “Temporal response of a random medium from speckle intensity frequency correlations,” J. Opt. Soc. Am. A 20, 2057–2070 (2003).
[CrossRef]

J. Pearce, Z. Jian, D. M. Mittleman, “Statistics of multiply scattered broadband terahertz pulses,” Phys. Rev. Lett. 91, 043903 (2003).
[CrossRef] [PubMed]

X. Wang, L. V. Wang, C.-W. Sun, C.-C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8, 608–617 (2003).
[CrossRef] [PubMed]

2002 (2)

M. A. Webster, K. J. Webb, A. M. Weiner, “Temporal response of a random medium from third order laser speckle frequency correlations,” Phys. Rev. Lett. 88, 033901 (2002).
[CrossRef]

A. B. Milstein, S. Oh, J. S. Reynolds, K. J. Webb, C. A. Bouman, R. P. Millane, “Three-dimensional Bayesian optical diffusion tomography with experimental data,” Opt. Lett. 27, 95–97 (2002).
[CrossRef]

2001 (2)

A. Ishimaru, S. Jaruwatanadilok, Y. Kuga, “Polarized pulse waves in random discrete scatterers,” Appl. Opt. 40, 5495–5502 (2001).
[CrossRef]

J.-M. Tualle, E. Tinet, S. Avrillier, “A new and easy way to perform time-resolved measurements of the light scattered by a turbid medium,” Opt. Commun. 189, 211–220 (2001).
[CrossRef]

2000 (2)

1999 (2)

1997 (4)

A. A. Chabanov, A. Z. Genack, “Field distributions in the crossover from ballistic to diffusive wave propagation,” Phys. Rev. E 56, R1338–R1341 (1997).
[CrossRef]

P. Sebbah, O. Lengrand, B. A. van Tiggelen, A. Z. Genack, “Statistics of the cumulative phase of microwave radiation in random media,” Phys. Rev. E 56, 3619–3623 (1997).
[CrossRef]

C. A. Thompson, K. J. Webb, A. M. Weiner, “Diffusive media characterization with laser speckle,” Appl. Opt. 36, 3726–3734 (1997).
[CrossRef] [PubMed]

C. A. Thompson, K. J. Webb, A. M. Weiner, “Imaging in scattering media by use of laser speckle,” J. Opt. Soc. Am. A 14, 2269–2277 (1997).
[CrossRef]

1995 (1)

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

1989 (2)

1985 (1)

1981 (1)

H. T. Shang, “Chromatic dispersion measurement by white-light interferometry on meter-length single-mode optical fibers,” Electron. Lett. 17, 603–605 (1981).
[CrossRef]

1970 (2)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

J. A. Leendertz, “Interferometric displacement measurement on scattering surfaces utilizing speckle effect,” J. Phys. E 3, 214–218 (1970).
[CrossRef]

Arridge, S. R.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R43 (1999).
[CrossRef]

Avrillier, S.

J.-M. Tualle, E. Tinet, S. Avrillier, “A new and easy way to perform time-resolved measurements of the light scattered by a turbid medium,” Opt. Commun. 189, 211–220 (2001).
[CrossRef]

Boppart, S. A.

Bouman, C. A.

Cao, H.

Chabanov, A. A.

A. A. Chabanov, A. Z. Genack, “Field distributions in the crossover from ballistic to diffusive wave propagation,” Phys. Rev. E 56, R1338–R1341 (1997).
[CrossRef]

Chance, B.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Chen, C.-T.

C.-T. Chen, Digital Signal Processing: Spectral Computation and Filter Design (Oxford U. Press, 2001).

Cheriaux, G.

Creath, K.

Drexler, W.

Duderstadt, J. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, 1976).

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G.

Genack, A. Z.

P. Sebbah, R. Pnini, A. Z. Genack, “Field and intensity correlation in random media,” Phys. Rev. E 62, 7348–7352 (2000).
[CrossRef]

P. Sebbah, O. Lengrand, B. A. van Tiggelen, A. Z. Genack, “Statistics of the cumulative phase of microwave radiation in random media,” Phys. Rev. E 56, 3619–3623 (1997).
[CrossRef]

A. A. Chabanov, A. Z. Genack, “Field distributions in the crossover from ballistic to diffusive wave propagation,” Phys. Rev. E 56, R1338–R1341 (1997).
[CrossRef]

Gerke, T. D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, 1984).

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hamilton, L. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, 1976).

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Ippen, E. P.

W. Drexler, U. Morgner, F. X. Kartner, C. Pitris, S. A. Boppart, X. D. Li, E. P. Ippen, J. G. Fujimoto, “ In vivo ultrahigh-resolution optical coherence tomography,” Opt. Lett. 24, 1221–1223 (1999).
[CrossRef]

E. P. Ippen, C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses: Picosecond Techniques and Applications, S. H. Shapiro, ed., (Springer-Verlag, 1984), Vol. 18, pp. 83–122.
[CrossRef]

Ishimaru, A.

Ito, S.

Jaruwatanadilok, S.

Jian, Z.

J. Pearce, Z. Jian, D. M. Mittleman, “Statistics of multiply scattered broadband terahertz pulses,” Phys. Rev. Lett. 91, 043903 (2003).
[CrossRef] [PubMed]

Joffre, M.

Kartner, F. X.

Kuga, Y.

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Leendertz, J. A.

J. A. Leendertz, “Interferometric displacement measurement on scattering surfaces utilizing speckle effect,” J. Phys. E 3, 214–218 (1970).
[CrossRef]

Lengrand, O.

P. Sebbah, O. Lengrand, B. A. van Tiggelen, A. Z. Genack, “Statistics of the cumulative phase of microwave radiation in random media,” Phys. Rev. E 56, 3619–3623 (1997).
[CrossRef]

Lepetit, L.

Li, X. D.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

McKinney, J. D.

Millane, R. P.

Milstein, A. B.

Mittleman, D. M.

J. Pearce, Z. Jian, D. M. Mittleman, “Statistics of multiply scattered broadband terahertz pulses,” Phys. Rev. Lett. 91, 043903 (2003).
[CrossRef] [PubMed]

Morgner, U.

Oguchi, T.

Oh, S.

Patterson, M. S.

Pearce, J.

J. Pearce, Z. Jian, D. M. Mittleman, “Statistics of multiply scattered broadband terahertz pulses,” Phys. Rev. Lett. 91, 043903 (2003).
[CrossRef] [PubMed]

Pitris, C.

Pnini, R.

P. Sebbah, R. Pnini, A. Z. Genack, “Field and intensity correlation in random media,” Phys. Rev. E 62, 7348–7352 (2000).
[CrossRef]

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Reynolds, J. S.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Sebbah, P.

P. Sebbah, R. Pnini, A. Z. Genack, “Field and intensity correlation in random media,” Phys. Rev. E 62, 7348–7352 (2000).
[CrossRef]

P. Sebbah, O. Lengrand, B. A. van Tiggelen, A. Z. Genack, “Statistics of the cumulative phase of microwave radiation in random media,” Phys. Rev. E 56, 3619–3623 (1997).
[CrossRef]

Shang, H. T.

H. T. Shang, “Chromatic dispersion measurement by white-light interferometry on meter-length single-mode optical fibers,” Electron. Lett. 17, 603–605 (1981).
[CrossRef]

Shank, C. V.

E. P. Ippen, C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses: Picosecond Techniques and Applications, S. H. Shapiro, ed., (Springer-Verlag, 1984), Vol. 18, pp. 83–122.
[CrossRef]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Sun, C.-W.

X. Wang, L. V. Wang, C.-W. Sun, C.-C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8, 608–617 (2003).
[CrossRef] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Thompson, C. A.

Tinet, E.

J.-M. Tualle, E. Tinet, S. Avrillier, “A new and easy way to perform time-resolved measurements of the light scattered by a turbid medium,” Opt. Commun. 189, 211–220 (2001).
[CrossRef]

Tualle, J.-M.

J.-M. Tualle, E. Tinet, S. Avrillier, “A new and easy way to perform time-resolved measurements of the light scattered by a turbid medium,” Opt. Commun. 189, 211–220 (2001).
[CrossRef]

van Tiggelen, B. A.

P. Sebbah, O. Lengrand, B. A. van Tiggelen, A. Z. Genack, “Statistics of the cumulative phase of microwave radiation in random media,” Phys. Rev. E 56, 3619–3623 (1997).
[CrossRef]

Wang, L. V.

X. Wang, L. V. Wang, C.-W. Sun, C.-C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8, 608–617 (2003).
[CrossRef] [PubMed]

Wang, X.

X. Wang, L. V. Wang, C.-W. Sun, C.-C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8, 608–617 (2003).
[CrossRef] [PubMed]

Webb, K. J.

Webster, M. A.

Weiner, A. M.

Wilson, B. C.

Xu, J.

Yang, C.-C.

X. Wang, L. V. Wang, C.-W. Sun, C.-C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8, 608–617 (2003).
[CrossRef] [PubMed]

Appl. Opt. (4)

Astron. Astrophys. (1)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Electron. Lett. (1)

H. T. Shang, “Chromatic dispersion measurement by white-light interferometry on meter-length single-mode optical fibers,” Electron. Lett. 17, 603–605 (1981).
[CrossRef]

Inverse Probl. (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R43 (1999).
[CrossRef]

J. Biomed. Opt. (1)

X. Wang, L. V. Wang, C.-W. Sun, C.-C. Yang, “Polarized light propagation through scattering media: time-resolved Monte Carlo simulations and experiments,” J. Biomed. Opt. 8, 608–617 (2003).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (3)

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

J. Phys. E (1)

J. A. Leendertz, “Interferometric displacement measurement on scattering surfaces utilizing speckle effect,” J. Phys. E 3, 214–218 (1970).
[CrossRef]

Opt. Commun. (1)

J.-M. Tualle, E. Tinet, S. Avrillier, “A new and easy way to perform time-resolved measurements of the light scattered by a turbid medium,” Opt. Commun. 189, 211–220 (2001).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. E (3)

A. A. Chabanov, A. Z. Genack, “Field distributions in the crossover from ballistic to diffusive wave propagation,” Phys. Rev. E 56, R1338–R1341 (1997).
[CrossRef]

P. Sebbah, O. Lengrand, B. A. van Tiggelen, A. Z. Genack, “Statistics of the cumulative phase of microwave radiation in random media,” Phys. Rev. E 56, 3619–3623 (1997).
[CrossRef]

P. Sebbah, R. Pnini, A. Z. Genack, “Field and intensity correlation in random media,” Phys. Rev. E 62, 7348–7352 (2000).
[CrossRef]

Phys. Rev. Lett. (2)

J. Pearce, Z. Jian, D. M. Mittleman, “Statistics of multiply scattered broadband terahertz pulses,” Phys. Rev. Lett. 91, 043903 (2003).
[CrossRef] [PubMed]

M. A. Webster, K. J. Webb, A. M. Weiner, “Temporal response of a random medium from third order laser speckle frequency correlations,” Phys. Rev. Lett. 88, 033901 (2002).
[CrossRef]

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Other (6)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, 1984).

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, 1976).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

C.-T. Chen, Digital Signal Processing: Spectral Computation and Filter Design (Oxford U. Press, 2001).

E. P. Ippen, C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses: Picosecond Techniques and Applications, S. H. Shapiro, ed., (Springer-Verlag, 1984), Vol. 18, pp. 83–122.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Experimental setup; HWP is a half-wave plate, LP is a linear polarizer, PBS is a polarizing beam splitting cube, and the attenuator is composed of a HWP-LP pair.

Fig. 2
Fig. 2

Example of the temporal features of i ( t ) .

Fig. 3
Fig. 3

Example of the real part of the measured temporal features of the co-polarized i ( t ) for a 12 mm sample with μ s = 14 cm 1 and τ = 3.3 ns .

Fig. 4
Fig. 4

Real parts of four individual measurements of the (a) time, (b) frequency-domain scattered fields, offset for clarity. The sample has μ s = 4 cm 1 and a thickness of 12 mm. These data are for co-polarized light.

Fig. 5
Fig. 5

Measured probability distribution for (a) co-polarized, (b) cross-polarized frequency-domain scattered fields for a 9 mm sample ( μ s = 4 cm 1 ) . For this sample, the DOP is P = 0.4 .

Fig. 6
Fig. 6

(a) Field autocorrelation peak (6 mm sample only) for co-polarized light and bandwidths of 60, 40, 20 GHz. (b) Total intensity (co-polarized plus cross-polarized light) autocorrelation for 3, 6, 12 mm thick samples of μ s = 14 cm 1 , averaged over 100 measurements with a bandwidth of 60 GHz, and with the autocorrelation of the diffusion model simulated results overlaid in dotted curves.

Fig. 7
Fig. 7

Degree of polarization plotted as a function of sample thickness for two samples with differing reduced scattering coefficients.

Fig. 8
Fig. 8

Ensemble averages of 100 measurements of the temporal response of scattering samples ( μ s = 14 cm 1 ) of thickness (a) 3 mm and (b) 6 mm for both co-polarized and cross-polarization states and the total response, along with a diffusion model simulation for comparison.

Fig. 9
Fig. 9

Total and co- and cross-polarized responses according to sample thickness ( μ s = 4 cm 1 ) : (a) 9 mm, (b) 12 mm, (c) 18 mm.

Fig. 10
Fig. 10

Total temporal response for 12 mm sample ( μ s = 4 cm 1 ) compared with the diffusion model simulated result.

Equations (17)

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

e out ( t , ν ) = E o ( ν ) exp [ j 2 π ( ν o + ν ) t ] + c.c. ,
e out ( t , ν ) = x ̂ E ( ν ) exp [ j 2 π ( ν o + ν ) t ] + y ̂ E ( ν ) exp [ j 2 π ( ν o + ν ) t ] + c.c. ,
e in ( t ) = x ̂ E in exp [ j 2 π ( ν o + ν ) t ] + c.c.
e 1 ( t ) = p ̂ E 1 exp [ j 2 π ( ν o + ν ) t ] + c.c. ,
e out ( t , ν ) = x ̂ E ( ν ) exp [ j 2 π ( ν o + ν ) ( t τ ) ] + y ̂ E ( ν ) exp [ j 2 π ( ν o + ν ) ( t τ ) ] + c.c. ,
I ( ν ) = E 1 + E ( ν ) exp [ j 2 π ( ν o + ν ) τ ] 2 = E 1 2 + E ( ν ) 2 + E 1 * E ( ν ) exp [ j 2 π ν τ ] + E 1 E * ( ν ) exp [ j 2 π ν τ ] ,
i ( t ) = δ ( t ) + g ( t ) + e ( t τ ) + e * ( t τ ) ,
e ( t τ ) = i ( t ) u ( t τ + δ ) ,
E ( ν ) = i ( t + τ ) u ( t + δ ) exp ( j 2 π ν t ) d t .
p n ( t ) = e ( t ) 2 I t = i ( t + τ ) u ( t + δ ) 2 I t ,
p ( t ) = 1 N n = 1 N p n ( t ) .
E ( ν ) = E ( ν ) W R ( ν ) = i ( t + τ ) u ( t + δ ) exp ( j 2 π ν t ) d t ,
e H ( t ) = e ( t ) w H ( t ) = E ( ν ) W H ( ν ) exp ( j 2 π ν t ) d ν ,
p n H ( t ) = e H ( t ) 2 I t = e ( t ) w H ( t ) 2 I t ,
G 1 ( η ) = e H ( t + η ) e H * ( t ) d t s ,
G 2 ( η ) = p n ( t + η ) p n ( t ) d t s ,
P = I s I s I s + I s ,

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