Abstract

We show how time-resolved measurements of the diffuse light transmitted through a thick scattering slab can be performed with a standard CCD camera, thanks to an interferometric protocol. Time-resolved correlations measured at a fixed photon transit time are also presented. The high number of pixels of the camera allows us to attain a quite good sensitivity for a reasonably low acquisition time.

© 2010 OSA

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2009

L. Azizi, K. Zarychta, D. Ettori, E. Tinet, and J.-M. Tualle, “Ultimate spatial resolution with Diffuse Optical Tomography,” Opt. Express 17(14), 12132–12144 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-12132 .
[CrossRef] [PubMed]

2008

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source-detector separation and fast single-photon gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[CrossRef] [PubMed]

R. Esposito, S. De Nicola, M. Brambilla, A. Pifferi, L. Spinelli, and M. Lepore, “Depth dependence of estimated optical properties of a scattering inclusion by time-resolved contrast functions,” Opt. Express 16(22), 17667–17681 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-17667 .
[CrossRef] [PubMed]

A. Liebert, H. Wabnitz, N. Zołek, and R. Macdonald, “Monte Carlo algorithm for efficient simulation of time-resolved fluorescence in layered turbid media,” Opt. Express 16(17), 13188–13202 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13188 .
[CrossRef] [PubMed]

L. Gagnon, C. Gauthier, R. D. Hoge, F. Lesage, J. Selb, and D. A. Boas, “Double-layer estimation of intra- and extracerebral hemoglobin concentration with a time-resolved system,” J. Biomed. Opt. 13(5), 054019 (2008).
[CrossRef] [PubMed]

L. Gagnon, M. Desjardins, J. Jehanne-Lacasse, L. Bherer, and F. Lesage, “Investigation of diffuse correlation spectroscopy in multi-layered media including the human head,” Opt. Express 16(20), 15514–15530 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15514 .
[CrossRef] [PubMed]

2007

G. Dietsche, M. Ninck, C. Ortolf, J. Li, F. Jaillon, and T. Gisler, “Fiber-based multispeckle detection for time-resolved diffusing-wave spectroscopy: characterization and application to blood flow detection in deep tissue,” Appl. Opt. 46(35), 8506–8514 (2007).
[CrossRef] [PubMed]

B. Varghese, V. Rajan, T. G. Van Leeuwen, and W. Steenbergen, “Path-length-resolved measurements of multiple scattered photons in static and dynamic turbid media using phase-modulated low-coherence interferometry,” J. Biomed. Opt. 12(2), 024020 (2007).
[CrossRef] [PubMed]

2006

J. Li, F. Jaillon, G. Dietsche, G. Maret, and T. Gisler, “Pulsation-resolved deep tissue dynamics measured with diffusing-wave spectroscopy,” Opt. Express 14(17), 7841–7851 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-17-7841 .
[CrossRef] [PubMed]

J.-M. Tualle, H. L. Nghiêm, M. Cheikh, D. Ettori, E. Tinet, and S. Avrillier, “Time-resolved diffusing wave spectroscopy beyond 300 transport mean free paths,” J. Opt. Soc. Am. A 23(6), 1452 (2006).
[CrossRef]

M. Cheikh, H. L. Nghiêm, D. Ettori, E. Tinet, S. Avrillier, and J. M. Tualle, “Time-resolved diffusing wave spectroscopy applied to dynamic heterogeneity imaging,” Opt. Lett. 31(15), 2311–2313 (2006).
[CrossRef] [PubMed]

2005

J.-M. Tualle, H. L. Nghiêm, C. Schäfauer, P. Berthaud, É. Tinet, D. Ettori, and S. Avrillier, “Time-resolved measurements from speckle interferometry,” Opt. Lett. 30(1), 50–52 (2005).
[CrossRef] [PubMed]

D. Grosenick, K. T. Moesta, M. Möller, J. Mucke, H. Wabnitz, B. Gebauer, C. Stroszczynski, B. Wassermann, P. M. Schlag, and H. Rinneberg, “Time-domain scanning optical mammography: I. Recording and assessment of mammograms of 154 patients,” Phys. Med. Biol. 50(11), 2429–2449 (2005).
[CrossRef] [PubMed]

P. Taroni, A. Torricelli, L. Spinelli, A. Pifferi, F. Arpaia, G. Danesini, and R. Cubeddu, “Time-resolved optical mammography between 637 and 985 nm: clinical study on the detection and identification of breast lesions,” Phys. Med. Biol. 50(11), 2469–2488 (2005).
[CrossRef] [PubMed]

T. Durduran, R. Choe, G. Yu, C. Zhou, J. C. Tchou, B. J. Czerniecki, and A. G. Yodh, “Diffuse optical measurement of blood flow in breast tumors,” Opt. Lett. 30(21), 2915–2917 (2005).
[CrossRef] [PubMed]

B. Montcel, R. Chabrier, and P. Poulet, “Detection of cortical activation with time-resolved diffuse optical methods,” Appl. Opt. 44(10), 1942–1947 (2005).
[CrossRef] [PubMed]

J. Selb, J. J. Stott, M. A. Franceschini, A. G. Sorensen, and D. A. Boas, “Improved sensitivity to cerebral hemodynamics during brain activation with a time-gated optical system: analytical model and experimental validation,” J. Biomed. Opt. 10(1), 011013 (2005).
[CrossRef]

A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, “Time-resolved reflectance at null source-detector separation: improving contrast and resolution in diffuse optical imaging,” Phys. Rev. Lett. 95(7), 078101 (2005).
[CrossRef] [PubMed]

T. D. Yates, J. C. Hebden, A. P. Gibson, N. L. Everdell, S. R. Arridge, and M. Douek, “Optical tomography of the breast using a multi-channel time-resolved imager,” Phys. Med. Biol. 50(11), 2503–2517 (2005).
[CrossRef] [PubMed]

M. Gross, P. Goy, B. C. Forget, M. Atlan, F. Ramaz, A. C. Boccara, and A. K. Dunn, “Heterodyne detection of multiply scattered monochromatic light with a multipixel detector,” Opt. Lett. 30(11), 1357–1359 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-11-1357 .
[CrossRef] [PubMed]

D. Grosenick, H. Wabnitz, K. T. Moesta, J. Mucke, P. M. Schlag, and H. Rinneberg, “Time-domain scanning optical mammography: II. Optical properties and tissue parameters of 87 carcinomas,” Phys. Med. Biol. 50(11), 2451–2468 (2005).
[CrossRef] [PubMed]

A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, R. Cubeddu, H. Wabnitz, D. Grosenick, M. Möller, R. Macdonald, J. Swartling, T. Svensson, S. Andersson-Engels, R. L. P. van Veen, H. J. C. M. Sterenborg, J.-M. Tualle, H. L. Nghiem, S. Avrillier, M. Whelan, and H. Stamm, “Performance assessment of photon migration instruments: the MEDPHOT protocol,” Appl. Opt. 44(11), 2104–2114 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ao-44-11-2104 .
[CrossRef] [PubMed]

2004

M. A. Webster, T. D. Gerke, A. M. Weiner, and K. J. Webb, “Spectral and temporal speckle field measurements of a random medium,” Opt. Lett. 29(13), 1491–1493 (2004), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-29-13-1491 .
[CrossRef] [PubMed]

A. Liebert, H. Wabnitz, J. Steinbrink, H. Obrig, M. Möller, R. Macdonald, A. Villringer, and H. Rinneberg, “Time-resolved multidistance near-infrared spectroscopy of the adult head: intracerebral and extracerebral absorption changes from moments of distribution of times of flight of photons,” Appl. Opt. 43(15), 3037–3047 (2004).
[CrossRef] [PubMed]

J.-M. Tualle, H. L. Nghiem, D. Ettori, R. Sablong, É. Tinet, and S. Avrillier, “Asymptotic behavior and inverse problem in layered scattering media,” J. Opt. Soc. Am. A 21(1), 24–34 (2004).
[CrossRef]

A. Pifferi, J. Swartling, E. Chikoidze, A. Torricelli, P. Taroni, A. Bassi, S. Andersson-Engels, and R. Cubeddu, “Spectroscopic time-resolved diffuse reflectance and transmittance measurements of the female breast at different interfiber distances,” J. Biomed. Opt. 9(6), 1143–1151 (2004).
[CrossRef] [PubMed]

2003

C. V. Zint, W. Uhring, M. Torregrossa, B. Cunin, and P. Poulet, “Streak camera: a multidetector for diffuse optical tomography,” Appl. Opt. 42(16), 3313–3320 (2003).
[CrossRef] [PubMed]

2002

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47(23), 4155–4166 (2002).
[CrossRef] [PubMed]

T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol. 47(16), 2847–2861 (2002).
[CrossRef] [PubMed]

2001

A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, “Dynamic imaging of cerebral blood flow using laser speckle,” J. Cereb. Blood Flow Metab. 21(3), 195–201 (2001).
[CrossRef] [PubMed]

E. M. C. Hillman, J. C. Hebden, M. Schweiger, H. Dehghani, F. E. W. Schmidt, D. T. Delpy, and S. R. Arridge, “Time resolved optical tomography of the human forearm,” Phys. Med. Biol. 46(4), 1117–1130 (2001).
[CrossRef] [PubMed]

V. Chernomordik, A. Gandjbakhche, M. Lepore, R. Esposito, and I. Delfino, “Depth dependence of the analytical expression for the width of the point spread function (spatial resolution) in time-resolved transillumination,” J. Biomed. Opt. 6(4), 441–445 (2001).
[CrossRef] [PubMed]

J.-M. Tualle, E. Tinet, and S. Avrillier, “A new and easy way to perform time-resolved measurements of the light scattered by a turbid medium,” Opt. Commun. 189(4–6), 211–220 (2001).
[CrossRef]

1999

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “Compact tissue oximeter based on dual-wavelength multichannel time-resolved reflectance,” Appl. Opt. 38(16), 3670–3680 (1999).
[CrossRef]

A. Kienle and T. Glanzmann, “In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model,” Phys. Med. Biol. 44(11), 2689–2702 (1999).
[CrossRef] [PubMed]

1998

A. Kienle, M. S. Patterson, N. Dögnitz, R. Bays, G. Wagniνres, and H. van den Bergh, “Noninvasive Determination of the Optical Properties of Two-Layered Turbid Media,” Appl. Opt. 37(4), 779–791 (1998).
[CrossRef]

1997

G. Le Tolguenec, F. Devaux, and E. Lantz, “Imaging through thick biological tissues by parametric image amplification and phase conjugation,” J. Opt. 28(5), 214–217 (1997).
[CrossRef]

P. Gleyzes, A. C. Boccara, and H. Saint-Jalmes, “Multichannel Nomarski microscope with polarization modulation: performance and applications,” Opt. Lett. 22(20), 1529–1531 (1997).
[CrossRef]

1996

J.-M. Tualle, B. Gélébart, E. Tinet, S. Avrillier, and J. P. Ollivier, “Real time optical coefficients evaluation from time and space resolved reflectance measurements in biological tissues,” Opt. Commun. 124(3–4), 216–221 (1996).
[CrossRef]

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41(4), 767–783 (1996).
[CrossRef] [PubMed]

1995

R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12(11), 2532–2539 (1995).
[CrossRef]

T. Spirig, P. Seitz, O. Vietze, and F. Heitger, “The lock-in CCD- two-dimensional synchronous detection of light,” IEEE J. Quantum Electron. 31(9), 1705–1708 (1995).
[CrossRef]

1990

X. L. Wu, D. J. Pine, P. M. Chaikin, J. S. Huang, and D. A. Weitz, “Diffusing-wave spectroscopy in a shear flow,” J. Opt. Soc. Am. B 7(1), 15–20 (1990).
[CrossRef]

S. Andersson-Engels, R. Berg, S. Svanberg, and O. Jarlman, “Time-resolved transillumination for medical diagnostics,” Opt. Lett. 15(21), 1179–1181 (1990).
[CrossRef] [PubMed]

1989

M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28(12), 2331–2336 (1989).
[CrossRef] [PubMed]

1988

M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B Condens. Matter 37(1), 1–5 (1988).
[CrossRef] [PubMed]

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60(12), 1134–1137 (1988).
[CrossRef] [PubMed]

1987

G. Maret and P. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B Condens. Matter 65(4), 409–413 (1987).
[CrossRef]

Andersson-Engels, S.

A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, R. Cubeddu, H. Wabnitz, D. Grosenick, M. Möller, R. Macdonald, J. Swartling, T. Svensson, S. Andersson-Engels, R. L. P. van Veen, H. J. C. M. Sterenborg, J.-M. Tualle, H. L. Nghiem, S. Avrillier, M. Whelan, and H. Stamm, “Performance assessment of photon migration instruments: the MEDPHOT protocol,” Appl. Opt. 44(11), 2104–2114 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ao-44-11-2104 .
[CrossRef] [PubMed]

A. Pifferi, J. Swartling, E. Chikoidze, A. Torricelli, P. Taroni, A. Bassi, S. Andersson-Engels, and R. Cubeddu, “Spectroscopic time-resolved diffuse reflectance and transmittance measurements of the female breast at different interfiber distances,” J. Biomed. Opt. 9(6), 1143–1151 (2004).
[CrossRef] [PubMed]

S. Andersson-Engels, R. Berg, S. Svanberg, and O. Jarlman, “Time-resolved transillumination for medical diagnostics,” Opt. Lett. 15(21), 1179–1181 (1990).
[CrossRef] [PubMed]

Aronson, R.

R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12(11), 2532–2539 (1995).
[CrossRef]

Arpaia, F.

P. Taroni, A. Torricelli, L. Spinelli, A. Pifferi, F. Arpaia, G. Danesini, and R. Cubeddu, “Time-resolved optical mammography between 637 and 985 nm: clinical study on the detection and identification of breast lesions,” Phys. Med. Biol. 50(11), 2469–2488 (2005).
[CrossRef] [PubMed]

Arridge, S. R.

T. D. Yates, J. C. Hebden, A. P. Gibson, N. L. Everdell, S. R. Arridge, and M. Douek, “Optical tomography of the breast using a multi-channel time-resolved imager,” Phys. Med. Biol. 50(11), 2503–2517 (2005).
[CrossRef] [PubMed]

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47(23), 4155–4166 (2002).
[CrossRef] [PubMed]

E. M. C. Hillman, J. C. Hebden, M. Schweiger, H. Dehghani, F. E. W. Schmidt, D. T. Delpy, and S. R. Arridge, “Time resolved optical tomography of the human forearm,” Phys. Med. Biol. 46(4), 1117–1130 (2001).
[CrossRef] [PubMed]

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41(4), 767–783 (1996).
[CrossRef] [PubMed]

Atlan, M.

M. Gross, P. Goy, B. C. Forget, M. Atlan, F. Ramaz, A. C. Boccara, and A. K. Dunn, “Heterodyne detection of multiply scattered monochromatic light with a multipixel detector,” Opt. Lett. 30(11), 1357–1359 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-11-1357 .
[CrossRef] [PubMed]

Austin, T.

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47(23), 4155–4166 (2002).
[CrossRef] [PubMed]

Avrillier, S.

J.-M. Tualle, H. L. Nghiêm, M. Cheikh, D. Ettori, E. Tinet, and S. Avrillier, “Time-resolved diffusing wave spectroscopy beyond 300 transport mean free paths,” J. Opt. Soc. Am. A 23(6), 1452 (2006).
[CrossRef]

M. Cheikh, H. L. Nghiêm, D. Ettori, E. Tinet, S. Avrillier, and J. M. Tualle, “Time-resolved diffusing wave spectroscopy applied to dynamic heterogeneity imaging,” Opt. Lett. 31(15), 2311–2313 (2006).
[CrossRef] [PubMed]

J.-M. Tualle, H. L. Nghiêm, C. Schäfauer, P. Berthaud, É. Tinet, D. Ettori, and S. Avrillier, “Time-resolved measurements from speckle interferometry,” Opt. Lett. 30(1), 50–52 (2005).
[CrossRef] [PubMed]

A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, R. Cubeddu, H. Wabnitz, D. Grosenick, M. Möller, R. Macdonald, J. Swartling, T. Svensson, S. Andersson-Engels, R. L. P. van Veen, H. J. C. M. Sterenborg, J.-M. Tualle, H. L. Nghiem, S. Avrillier, M. Whelan, and H. Stamm, “Performance assessment of photon migration instruments: the MEDPHOT protocol,” Appl. Opt. 44(11), 2104–2114 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ao-44-11-2104 .
[CrossRef] [PubMed]

J.-M. Tualle, H. L. Nghiem, D. Ettori, R. Sablong, É. Tinet, and S. Avrillier, “Asymptotic behavior and inverse problem in layered scattering media,” J. Opt. Soc. Am. A 21(1), 24–34 (2004).
[CrossRef]

J.-M. Tualle, E. Tinet, and S. Avrillier, “A new and easy way to perform time-resolved measurements of the light scattered by a turbid medium,” Opt. Commun. 189(4–6), 211–220 (2001).
[CrossRef]

J.-M. Tualle, B. Gélébart, E. Tinet, S. Avrillier, and J. P. Ollivier, “Real time optical coefficients evaluation from time and space resolved reflectance measurements in biological tissues,” Opt. Commun. 124(3–4), 216–221 (1996).
[CrossRef]

Azizi, L.

L. Azizi, K. Zarychta, D. Ettori, E. Tinet, and J.-M. Tualle, “Ultimate spatial resolution with Diffuse Optical Tomography,” Opt. Express 17(14), 12132–12144 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-12132 .
[CrossRef] [PubMed]

Bassi, A.

A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, R. Cubeddu, H. Wabnitz, D. Grosenick, M. Möller, R. Macdonald, J. Swartling, T. Svensson, S. Andersson-Engels, R. L. P. van Veen, H. J. C. M. Sterenborg, J.-M. Tualle, H. L. Nghiem, S. Avrillier, M. Whelan, and H. Stamm, “Performance assessment of photon migration instruments: the MEDPHOT protocol,” Appl. Opt. 44(11), 2104–2114 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ao-44-11-2104 .
[CrossRef] [PubMed]

A. Pifferi, J. Swartling, E. Chikoidze, A. Torricelli, P. Taroni, A. Bassi, S. Andersson-Engels, and R. Cubeddu, “Spectroscopic time-resolved diffuse reflectance and transmittance measurements of the female breast at different interfiber distances,” J. Biomed. Opt. 9(6), 1143–1151 (2004).
[CrossRef] [PubMed]

Bays, R.

A. Kienle, M. S. Patterson, N. Dögnitz, R. Bays, G. Wagniνres, and H. van den Bergh, “Noninvasive Determination of the Optical Properties of Two-Layered Turbid Media,” Appl. Opt. 37(4), 779–791 (1998).
[CrossRef]

Berg, R.

S. Andersson-Engels, R. Berg, S. Svanberg, and O. Jarlman, “Time-resolved transillumination for medical diagnostics,” Opt. Lett. 15(21), 1179–1181 (1990).
[CrossRef] [PubMed]

Berthaud, P.

J.-M. Tualle, H. L. Nghiêm, C. Schäfauer, P. Berthaud, É. Tinet, D. Ettori, and S. Avrillier, “Time-resolved measurements from speckle interferometry,” Opt. Lett. 30(1), 50–52 (2005).
[CrossRef] [PubMed]

Bherer, L.

L. Gagnon, M. Desjardins, J. Jehanne-Lacasse, L. Bherer, and F. Lesage, “Investigation of diffuse correlation spectroscopy in multi-layered media including the human head,” Opt. Express 16(20), 15514–15530 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15514 .
[CrossRef] [PubMed]

Boas, D. A.

L. Gagnon, C. Gauthier, R. D. Hoge, F. Lesage, J. Selb, and D. A. Boas, “Double-layer estimation of intra- and extracerebral hemoglobin concentration with a time-resolved system,” J. Biomed. Opt. 13(5), 054019 (2008).
[CrossRef] [PubMed]

J. Selb, J. J. Stott, M. A. Franceschini, A. G. Sorensen, and D. A. Boas, “Improved sensitivity to cerebral hemodynamics during brain activation with a time-gated optical system: analytical model and experimental validation,” J. Biomed. Opt. 10(1), 011013 (2005).
[CrossRef]

A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, “Dynamic imaging of cerebral blood flow using laser speckle,” J. Cereb. Blood Flow Metab. 21(3), 195–201 (2001).
[CrossRef] [PubMed]

Boccara, A. C.

M. Gross, P. Goy, B. C. Forget, M. Atlan, F. Ramaz, A. C. Boccara, and A. K. Dunn, “Heterodyne detection of multiply scattered monochromatic light with a multipixel detector,” Opt. Lett. 30(11), 1357–1359 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-11-1357 .
[CrossRef] [PubMed]

P. Gleyzes, A. C. Boccara, and H. Saint-Jalmes, “Multichannel Nomarski microscope with polarization modulation: performance and applications,” Opt. Lett. 22(20), 1529–1531 (1997).
[CrossRef]

Bolay, H.

A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, “Dynamic imaging of cerebral blood flow using laser speckle,” J. Cereb. Blood Flow Metab. 21(3), 195–201 (2001).
[CrossRef] [PubMed]

Brambilla, M.

R. Esposito, S. De Nicola, M. Brambilla, A. Pifferi, L. Spinelli, and M. Lepore, “Depth dependence of estimated optical properties of a scattering inclusion by time-resolved contrast functions,” Opt. Express 16(22), 17667–17681 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-17667 .
[CrossRef] [PubMed]

Chabrier, R.

B. Montcel, R. Chabrier, and P. Poulet, “Detection of cortical activation with time-resolved diffuse optical methods,” Appl. Opt. 44(10), 1942–1947 (2005).
[CrossRef] [PubMed]

Chaikin, P. M.

X. L. Wu, D. J. Pine, P. M. Chaikin, J. S. Huang, and D. A. Weitz, “Diffusing-wave spectroscopy in a shear flow,” J. Opt. Soc. Am. B 7(1), 15–20 (1990).
[CrossRef]

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing wave spectroscopy,” Phys. Rev. Lett. 60(12), 1134–1137 (1988).
[CrossRef] [PubMed]

Chance, B.

T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol. 47(16), 2847–2861 (2002).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28(12), 2331–2336 (1989).
[CrossRef] [PubMed]

Cheikh, M.

M. Cheikh, H. L. Nghiêm, D. Ettori, E. Tinet, S. Avrillier, and J. M. Tualle, “Time-resolved diffusing wave spectroscopy applied to dynamic heterogeneity imaging,” Opt. Lett. 31(15), 2311–2313 (2006).
[CrossRef] [PubMed]

J.-M. Tualle, H. L. Nghiêm, M. Cheikh, D. Ettori, E. Tinet, and S. Avrillier, “Time-resolved diffusing wave spectroscopy beyond 300 transport mean free paths,” J. Opt. Soc. Am. A 23(6), 1452 (2006).
[CrossRef]

Chernomordik, V.

V. Chernomordik, A. Gandjbakhche, M. Lepore, R. Esposito, and I. Delfino, “Depth dependence of the analytical expression for the width of the point spread function (spatial resolution) in time-resolved transillumination,” J. Biomed. Opt. 6(4), 441–445 (2001).
[CrossRef] [PubMed]

Chikoidze, E.

A. Pifferi, J. Swartling, E. Chikoidze, A. Torricelli, P. Taroni, A. Bassi, S. Andersson-Engels, and R. Cubeddu, “Spectroscopic time-resolved diffuse reflectance and transmittance measurements of the female breast at different interfiber distances,” J. Biomed. Opt. 9(6), 1143–1151 (2004).
[CrossRef] [PubMed]

Choe, R.

T. Durduran, R. Choe, G. Yu, C. Zhou, J. C. Tchou, B. J. Czerniecki, and A. G. Yodh, “Diffuse optical measurement of blood flow in breast tumors,” Opt. Lett. 30(21), 2915–2917 (2005).
[CrossRef] [PubMed]

T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol. 47(16), 2847–2861 (2002).
[CrossRef] [PubMed]

Contini, D.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source-detector separation and fast single-photon gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[CrossRef] [PubMed]

Cova, S.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source-detector separation and fast single-photon gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[CrossRef] [PubMed]

Cubeddu, R.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source-detector separation and fast single-photon gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[CrossRef] [PubMed]

P. Taroni, A. Torricelli, L. Spinelli, A. Pifferi, F. Arpaia, G. Danesini, and R. Cubeddu, “Time-resolved optical mammography between 637 and 985 nm: clinical study on the detection and identification of breast lesions,” Phys. Med. Biol. 50(11), 2469–2488 (2005).
[CrossRef] [PubMed]

A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, R. Cubeddu, H. Wabnitz, D. Grosenick, M. Möller, R. Macdonald, J. Swartling, T. Svensson, S. Andersson-Engels, R. L. P. van Veen, H. J. C. M. Sterenborg, J.-M. Tualle, H. L. Nghiem, S. Avrillier, M. Whelan, and H. Stamm, “Performance assessment of photon migration instruments: the MEDPHOT protocol,” Appl. Opt. 44(11), 2104–2114 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ao-44-11-2104 .
[CrossRef] [PubMed]

A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, “Time-resolved reflectance at null source-detector separation: improving contrast and resolution in diffuse optical imaging,” Phys. Rev. Lett. 95(7), 078101 (2005).
[CrossRef] [PubMed]

A. Pifferi, J. Swartling, E. Chikoidze, A. Torricelli, P. Taroni, A. Bassi, S. Andersson-Engels, and R. Cubeddu, “Spectroscopic time-resolved diffuse reflectance and transmittance measurements of the female breast at different interfiber distances,” J. Biomed. Opt. 9(6), 1143–1151 (2004).
[CrossRef] [PubMed]

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “Compact tissue oximeter based on dual-wavelength multichannel time-resolved reflectance,” Appl. Opt. 38(16), 3670–3680 (1999).
[CrossRef]

Culver, J. P.

T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol. 47(16), 2847–2861 (2002).
[CrossRef] [PubMed]

Cunin, B.

C. V. Zint, W. Uhring, M. Torregrossa, B. Cunin, and P. Poulet, “Streak camera: a multidetector for diffuse optical tomography,” Appl. Opt. 42(16), 3313–3320 (2003).
[CrossRef] [PubMed]

Czerniecki, B. J.

T. Durduran, R. Choe, G. Yu, C. Zhou, J. C. Tchou, B. J. Czerniecki, and A. G. Yodh, “Diffuse optical measurement of blood flow in breast tumors,” Opt. Lett. 30(21), 2915–2917 (2005).
[CrossRef] [PubMed]

Dalla Mora, A.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source-detector separation and fast single-photon gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[CrossRef] [PubMed]

Danesini, G.

P. Taroni, A. Torricelli, L. Spinelli, A. Pifferi, F. Arpaia, G. Danesini, and R. Cubeddu, “Time-resolved optical mammography between 637 and 985 nm: clinical study on the detection and identification of breast lesions,” Phys. Med. Biol. 50(11), 2469–2488 (2005).
[CrossRef] [PubMed]

De Nicola, S.

R. Esposito, S. De Nicola, M. Brambilla, A. Pifferi, L. Spinelli, and M. Lepore, “Depth dependence of estimated optical properties of a scattering inclusion by time-resolved contrast functions,” Opt. Express 16(22), 17667–17681 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-17667 .
[CrossRef] [PubMed]

Dehghani, H.

E. M. C. Hillman, J. C. Hebden, M. Schweiger, H. Dehghani, F. E. W. Schmidt, D. T. Delpy, and S. R. Arridge, “Time resolved optical tomography of the human forearm,” Phys. Med. Biol. 46(4), 1117–1130 (2001).
[CrossRef] [PubMed]

Del Bianco, S.

A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, “Time-resolved reflectance at null source-detector separation: improving contrast and resolution in diffuse optical imaging,” Phys. Rev. Lett. 95(7), 078101 (2005).
[CrossRef] [PubMed]

Delfino, I.

V. Chernomordik, A. Gandjbakhche, M. Lepore, R. Esposito, and I. Delfino, “Depth dependence of the analytical expression for the width of the point spread function (spatial resolution) in time-resolved transillumination,” J. Biomed. Opt. 6(4), 441–445 (2001).
[CrossRef] [PubMed]

Delpy, D. T.

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. 47(23), 4155–4166 (2002).
[CrossRef] [PubMed]

E. M. C. Hillman, J. C. Hebden, M. Schweiger, H. Dehghani, F. E. W. Schmidt, D. T. Delpy, and S. R. Arridge, “Time resolved optical tomography of the human forearm,” Phys. Med. Biol. 46(4), 1117–1130 (2001).
[CrossRef] [PubMed]

M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41(4), 767–783 (1996).
[CrossRef] [PubMed]

Desjardins, M.

L. Gagnon, M. Desjardins, J. Jehanne-Lacasse, L. Bherer, and F. Lesage, “Investigation of diffuse correlation spectroscopy in multi-layered media including the human head,” Opt. Express 16(20), 15514–15530 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15514 .
[CrossRef] [PubMed]

Devaux, F.

G. Le Tolguenec, F. Devaux, and E. Lantz, “Imaging through thick biological tissues by parametric image amplification and phase conjugation,” J. Opt. 28(5), 214–217 (1997).
[CrossRef]

Dietsche, G.

G. Dietsche, M. Ninck, C. Ortolf, J. Li, F. Jaillon, and T. Gisler, “Fiber-based multispeckle detection for time-resolved diffusing-wave spectroscopy: characterization and application to blood flow detection in deep tissue,” Appl. Opt. 46(35), 8506–8514 (2007).
[CrossRef] [PubMed]

J. Li, F. Jaillon, G. Dietsche, G. Maret, and T. Gisler, “Pulsation-resolved deep tissue dynamics measured with diffusing-wave spectroscopy,” Opt. Express 14(17), 7841–7851 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-17-7841 .
[CrossRef] [PubMed]

Dögnitz, N.

A. Kienle, M. S. Patterson, N. Dögnitz, R. Bays, G. Wagniνres, and H. van den Bergh, “Noninvasive Determination of the Optical Properties of Two-Layered Turbid Media,” Appl. Opt. 37(4), 779–791 (1998).
[CrossRef]

Douek, M.

T. D. Yates, J. C. Hebden, A. P. Gibson, N. L. Everdell, S. R. Arridge, and M. Douek, “Optical tomography of the breast using a multi-channel time-resolved imager,” Phys. Med. Biol. 50(11), 2503–2517 (2005).
[CrossRef] [PubMed]

Dunn, A. K.

M. Gross, P. Goy, B. C. Forget, M. Atlan, F. Ramaz, A. C. Boccara, and A. K. Dunn, “Heterodyne detection of multiply scattered monochromatic light with a multipixel detector,” Opt. Lett. 30(11), 1357–1359 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-11-1357 .
[CrossRef] [PubMed]

A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, “Dynamic imaging of cerebral blood flow using laser speckle,” J. Cereb. Blood Flow Metab. 21(3), 195–201 (2001).
[CrossRef] [PubMed]

Durduran, T.

T. Durduran, R. Choe, G. Yu, C. Zhou, J. C. Tchou, B. J. Czerniecki, and A. G. Yodh, “Diffuse optical measurement of blood flow in breast tumors,” Opt. Lett. 30(21), 2915–2917 (2005).
[CrossRef] [PubMed]

T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol. 47(16), 2847–2861 (2002).
[CrossRef] [PubMed]

Esposito, R.

R. Esposito, S. De Nicola, M. Brambilla, A. Pifferi, L. Spinelli, and M. Lepore, “Depth dependence of estimated optical properties of a scattering inclusion by time-resolved contrast functions,” Opt. Express 16(22), 17667–17681 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-17667 .
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A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, R. Cubeddu, H. Wabnitz, D. Grosenick, M. Möller, R. Macdonald, J. Swartling, T. Svensson, S. Andersson-Engels, R. L. P. van Veen, H. J. C. M. Sterenborg, J.-M. Tualle, H. L. Nghiem, S. Avrillier, M. Whelan, and H. Stamm, “Performance assessment of photon migration instruments: the MEDPHOT protocol,” Appl. Opt. 44(11), 2104–2114 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ao-44-11-2104 .
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[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef] [PubMed]

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A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source-detector separation and fast single-photon gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[CrossRef] [PubMed]

A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, R. Cubeddu, H. Wabnitz, D. Grosenick, M. Möller, R. Macdonald, J. Swartling, T. Svensson, S. Andersson-Engels, R. L. P. van Veen, H. J. C. M. Sterenborg, J.-M. Tualle, H. L. Nghiem, S. Avrillier, M. Whelan, and H. Stamm, “Performance assessment of photon migration instruments: the MEDPHOT protocol,” Appl. Opt. 44(11), 2104–2114 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ao-44-11-2104 .
[CrossRef] [PubMed]

P. Taroni, A. Torricelli, L. Spinelli, A. Pifferi, F. Arpaia, G. Danesini, and R. Cubeddu, “Time-resolved optical mammography between 637 and 985 nm: clinical study on the detection and identification of breast lesions,” Phys. Med. Biol. 50(11), 2469–2488 (2005).
[CrossRef] [PubMed]

A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, “Time-resolved reflectance at null source-detector separation: improving contrast and resolution in diffuse optical imaging,” Phys. Rev. Lett. 95(7), 078101 (2005).
[CrossRef] [PubMed]

A. Pifferi, J. Swartling, E. Chikoidze, A. Torricelli, P. Taroni, A. Bassi, S. Andersson-Engels, and R. Cubeddu, “Spectroscopic time-resolved diffuse reflectance and transmittance measurements of the female breast at different interfiber distances,” J. Biomed. Opt. 9(6), 1143–1151 (2004).
[CrossRef] [PubMed]

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “Compact tissue oximeter based on dual-wavelength multichannel time-resolved reflectance,” Appl. Opt. 38(16), 3670–3680 (1999).
[CrossRef]

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A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. Dalla Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source-detector separation and fast single-photon gating,” Phys. Rev. Lett. 100(13), 138101 (2008).
[CrossRef] [PubMed]

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M. Cheikh, H. L. Nghiêm, D. Ettori, E. Tinet, S. Avrillier, and J. M. Tualle, “Time-resolved diffusing wave spectroscopy applied to dynamic heterogeneity imaging,” Opt. Lett. 31(15), 2311–2313 (2006).
[CrossRef] [PubMed]

Tualle, J.-M.

L. Azizi, K. Zarychta, D. Ettori, E. Tinet, and J.-M. Tualle, “Ultimate spatial resolution with Diffuse Optical Tomography,” Opt. Express 17(14), 12132–12144 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-12132 .
[CrossRef] [PubMed]

J.-M. Tualle, H. L. Nghiêm, M. Cheikh, D. Ettori, E. Tinet, and S. Avrillier, “Time-resolved diffusing wave spectroscopy beyond 300 transport mean free paths,” J. Opt. Soc. Am. A 23(6), 1452 (2006).
[CrossRef]

A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, R. Cubeddu, H. Wabnitz, D. Grosenick, M. Möller, R. Macdonald, J. Swartling, T. Svensson, S. Andersson-Engels, R. L. P. van Veen, H. J. C. M. Sterenborg, J.-M. Tualle, H. L. Nghiem, S. Avrillier, M. Whelan, and H. Stamm, “Performance assessment of photon migration instruments: the MEDPHOT protocol,” Appl. Opt. 44(11), 2104–2114 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ao-44-11-2104 .
[CrossRef] [PubMed]

J.-M. Tualle, H. L. Nghiêm, C. Schäfauer, P. Berthaud, É. Tinet, D. Ettori, and S. Avrillier, “Time-resolved measurements from speckle interferometry,” Opt. Lett. 30(1), 50–52 (2005).
[CrossRef] [PubMed]

J.-M. Tualle, H. L. Nghiem, D. Ettori, R. Sablong, É. Tinet, and S. Avrillier, “Asymptotic behavior and inverse problem in layered scattering media,” J. Opt. Soc. Am. A 21(1), 24–34 (2004).
[CrossRef]

J.-M. Tualle, E. Tinet, and S. Avrillier, “A new and easy way to perform time-resolved measurements of the light scattered by a turbid medium,” Opt. Commun. 189(4–6), 211–220 (2001).
[CrossRef]

J.-M. Tualle, B. Gélébart, E. Tinet, S. Avrillier, and J. P. Ollivier, “Real time optical coefficients evaluation from time and space resolved reflectance measurements in biological tissues,” Opt. Commun. 124(3–4), 216–221 (1996).
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http://www.me.rochester.edu/courses/ME241/SE3.html

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Figures (6)

Fig. 1
Fig. 1

Experimental setup: two weak reflectivity (15%) beamsplitters (BS) constitute a two-arms interferometer; a lens (L1, f1 = −12mm) allows uniform illumination of the CCD camera; a lens (L2, f2 = + 50mm) project the transmitted scattered light on the camera, which registers the interferometric signal with a typical grain size of 3μm, that is of about half the pixel size; the value of the BS reflectivity was chosen in order to have both a correct detection of the reference beam and a maximal illumination of the scattering sample; a (3:1) anamorphic prisms pair allows to correct the ellipticity of the laser beam; an acousto-optic modulator (AOM) allows lock-in detection through the multiplication by the positive function ξ(t).

Fig. 2
Fig. 2

Temporal response of the setup, which has the expected position (δτ = 2060 ps) and width (270 ps).

Fig. 3
Fig. 3

Raw data recorded with the breast phantom illuminated by a 5 mW laser beam. The shot noise level (SNL) is indicated by a dotted line. The transmittance at its maximum is 15 times lower than the SNL. A spurious peak is surrounded by a dotted line.

Fig. 4
Fig. 4

(a)- Recorded transmittance as a function of the transit time, together with a theoretical fit (red line) based on diffusion approximation with extrapolated boundary conditions (the setup temporal response was included in the fitting; we have used an isotropic source at a depth z0 = 1/μ’s , and an extrapolated boundary condition at a distance zs = 2/μ’s [47]). (b)-same as (a) in a logarithmic scale.

Fig. 5
Fig. 5

Acquisition protocol for correlation measurements: the red curve symbolizes the wavelength modulation, and the blue curve represents the modulation function ξ(t), which is zero excepted on two modulation half-periods separated by a time interval pT.

Fig. 6
Fig. 6

Experimental values of ln[g1(t,τ)] for a transit time τ = 1,5ns and a correlation time t = pT (T = 2ms) with p running from 1 to 5. The red line is a fit, weighted according to the statistical error, by the function α t with α = −0.35 ± −0.015 ms −1 .

Equations (25)

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Ref ( t , τ ) = sin 4 ( 2 π f t ) cos [ Δ Ω τ cos ( 2 π f t ) ]
S D C , i ( τ ) S D C , i + p ( τ ) g 2 ( 0 ) η e i 0 4 f Π ( τ τ ' ) φ ( τ ' ) g 1 ( p T / 2 , τ ' ) d τ '
g ( τ ) = 2 f 0 T / 2 Ref(t, τ )dt
ξ ( t ) = Ref( t , τ ) + Ref( t , 0 )
Q = ξ ( t ) i ( t ) d t
Δ Q = Q ( a ) Q ( b ) S D C ( a ) ( τ ) S D C ( b ) ( τ )
Δ 2 Q 2 S D C 2 ( τ ) = g 2 ( 0 ) η e i 0 2 f Π ( τ τ ' ) φ ( τ ' ) d τ ' = g 2 ( 0 ) η e i 0 2 f Π φ ( τ )
t 0 = d τ   130 p s =   1930 p s .
δ 2 S = δ i ( t ) δ i ( t ' ) d t d t ' = e i 0 0 T / 2 ξ ( t ) d t e i 0 g ( 0 ) T / 2
Δ 2 Q 2 S D C 2 ( τ ) + 2 δ 2 S = 2 e Q 0 { 1 + η g ( 0 ) 2     Π φ ( τ ) }
σ = 2 δ 2 S 2 N = 2 e Q 0 2 N
i x i Δ 2 x i i x i 2 2 α { 1 + η g ( 0 ) 2     Π φ ( τ ) }
Δ Q S D C , i ( a ) ( τ ) + δ S i ( a ) + S D C , i + p ( a ) ( τ ) + δ S i + p ( a ) S D C , j ( b ) ( τ ) δ S j ( a ) S D C , j + p ( b ) ( τ ) δ S j + p ( a )
Δ 2 Q = 4 ( δ 2 S + S D C 2 ( τ ) + S D C , i ( τ ) S D C , p ( τ ) )
i x i Δ 2 x i i x i 2 4 α { 1 + η g ( 0 ) 2 Π φ ( τ ) [ 1 + g 1 ( p T , τ ) ] }
ln g 1 ( t , τ ) = 2 μ ' s c τ t t 0
t 0 = 1 k 2 D B = λ 2 ( 2 π n ) 2 D B
D B = k B T a 6 π η g a
s 0 ω ( t ) = K ( ω ) s ˜ 0 [ ω ( t ) ]
s ω ( t ) = K ( ω ) s ˜ [ ω ( t ) , t ]
K 2 ( ω ) 2 f ( ω ) T
s ˜ [ ω 1 , 0 ] s ˜ * [ ω 2 , t ] ϕ ˜ ( ω , Ω , t )
i int ( t ) = S A K ( ω ) {     s 0 s ˜ * [ ω ( t ) , t ] + s 0 * s ˜ [ ω ( t ) , t ]     }
S D C , i S D C , i + p = S i 0 T d t 1 d t 2 d τ ' φ ( τ ' ) g 1 ( t 2 t 1 + p T / 2 , τ ' ) ×                                                                 [ Ref ( t 2 , τ τ ' ) + Ref ( t 2 , τ + τ ' ) ] [ Ref ( t 2 , τ τ ' ) + Ref ( t 2 , τ + τ ' ) ]
S D C , i S D C , i + p = S i 0 4 f φ ( τ ' ) g 1 ( p T / 2 , τ ' ) [ g ( τ τ ' ) + g ( τ + τ ' ) ] 2 d τ '

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