Abstract

Light propagation through scattering samples involves long optical path lengths typically resulting in short spectral correlation widths. Here, we demonstrate experimentally and theoretically a chromato-axial memory effect through volumetric forward-scattering slabs of thickness lying in the range between the scattering mean free path and the transport mean free path: a spectral shift of the impinging collimated beam results in an axial homothetic dilation of the speckle pattern over large spectral widths. The center of the homothety is shown to lie at a virtual plane located at $ 1/(3n) $ of the slab thickness $ L $, before the output surface. These results provide new tools to blindly control wavefields through scattering samples.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]

2019 (3)

2018 (2)

2017 (3)

2016 (3)

L. Borcea and J. Garnier, “Derivation of a one-way radiative transfer equation in random media,” Phys. Rev. E 93, 022115 (2016).
[Crossref]

M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, “Spatiotemporal coherent control of light through a multiple scattering medium with the multispectral transmission matrix,” Phys. Rev. Lett. 116, 253901 (2016).
[Crossref]

H. B. de Aguiar, S. Gigan, and S. Brasselet, “Enhanced nonlinear imaging through scattering media using transmission-matrix-based wave-front shaping,” Phys. Rev. A 94, 043830 (2016).
[Crossref]

2015 (5)

B. Judkewitz, R. Horstmeyer, I. M. Vellekoop, I. N. Papadopoulos, and C. Yang, “Translation correlations in anisotropically scattering media,” Nat. Phys. 11, 684–689 (2015).
[Crossref]

J.-X. Cheng and X. S. Xie, “Vibrational spectroscopic imaging of living systems: an emerging platform for biology and medicine,” Science 350, aaa8870 (2015).
[Crossref]

R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9, 563–571 (2015).
[Crossref]

S. Schott, J. Bertolotti, J.-F. Léger, L. Bourdieu, and S. Gigan, “Characterization of the angular memory effect of scattered light in biological tissues,” Opt. Express 23, 13505–13516 (2015).
[Crossref]

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

2014 (1)

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8, 784–790 (2014).
[Crossref]

2012 (1)

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[Crossref]

2011 (1)

2010 (2)

N. V. Petrov, V. G. Bespalov, and A. A. Gorodetsky, “Phase retrieval method for multiple wavelength speckle patterns,” Proc. SPIE 7387, 73871T (2010).
[Crossref]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

2006 (1)

O. Agam, A. V. Andreev, and B. Spivak, “Long-range correlations in the speckle patterns of directed waves,” Phys. Rev. Lett. 97, 223901 (2006).
[Crossref]

2005 (1)

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2, 932–940 (2005).
[Crossref]

1995 (2)

D. J. Durian, “Accuracy of diffusing-wave spectroscopy theories,” Phys. Rev. E 51, 3350–3358 (1995).
[Crossref]

L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, “Time-dependent photon migration using path integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[Crossref]

1992 (1)

I. Freund and D. Eliyahu, “Surface correlations in multiple-scattering media,” Phys. Rev. A 45, 6133–6134 (1992).
[Crossref]

1990 (3)

K. M. Yoo and R. R. Alfano, “Time-resolved coherent and incoherent components of forward light scattering in random media,” Opt. Lett. 15, 320–322 (1990).
[Crossref]

D. Pine, D. Weitz, J. Zhu, and E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. France 51, 2101–2127 (1990).
[Crossref]

I. Freund, “Looking through walls and around corners,” Physica A 168, 49–65 (1990).
[Crossref]

1988 (2)

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[Crossref]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[Crossref]

1975 (1)

A. Ishimaru and S. T. Hong, “Multiple scattering effects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles,” Radio Sci. 10, 637–644 (1975).
[Crossref]

1970 (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).

1962 (1)

I. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory 8, 194–195 (1962).
[Crossref]

1949 (1)

M. Kac, “On distributions of certain Wiener functionals,” Trans. Am. Math. Soc. 65, 1–13 (1949).
[Crossref]

1948 (1)

R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
[Crossref]

Agam, O.

O. Agam, A. V. Andreev, and B. Spivak, “Long-range correlations in the speckle patterns of directed waves,” Phys. Rev. Lett. 97, 223901 (2006).
[Crossref]

Aguiar, H. B. D.

Akkermans, E.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).

Alfano, R. R.

Alonso, M. A.

Andreev, A. V.

O. Agam, A. V. Andreev, and B. Spivak, “Long-range correlations in the speckle patterns of directed waves,” Phys. Rev. Lett. 97, 223901 (2006).
[Crossref]

Andreoli, D.

M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, “Spatiotemporal coherent control of light through a multiple scattering medium with the multispectral transmission matrix,” Phys. Rev. Lett. 116, 253901 (2016).
[Crossref]

Balla, N. K.

Bertolotti, J.

S. Schott, J. Bertolotti, J.-F. Léger, L. Bourdieu, and S. Gigan, “Characterization of the angular memory effect of scattered light in biological tissues,” Opt. Express 23, 13505–13516 (2015).
[Crossref]

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[Crossref]

Bespalov, V. G.

N. V. Petrov, V. G. Bespalov, and A. A. Gorodetsky, “Phase retrieval method for multiple wavelength speckle patterns,” Proc. SPIE 7387, 73871T (2010).
[Crossref]

Blochet, B.

Blum, C.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[Crossref]

Boccara, A. C.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Boniface, A.

Borcea, L.

L. Borcea and J. Garnier, “Derivation of a one-way radiative transfer equation in random media,” Phys. Rev. E 93, 022115 (2016).
[Crossref]

Bourdieu, L.

Brasselet, S.

A. G. Vesga, M. Hofer, N. K. Balla, H. B. D. Aguiar, M. Guillon, and S. Brasselet, “Focusing large spectral bandwidths through scattering media,” Opt. Express 27, 28384–28394 (2019).
[Crossref]

H. B. de Aguiar, S. Gigan, and S. Brasselet, “Enhanced nonlinear imaging through scattering media using transmission-matrix-based wave-front shaping,” Phys. Rev. A 94, 043830 (2016).
[Crossref]

Cai, J.

Carminati, R.

R. Savo, R. Pierrat, U. Najar, R. Carminati, S. Rotter, and S. Gigan, “Observation of mean path length invariance in light-scattering media,” Science 358, 765–768 (2017).
[Crossref]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Chaigne, T.

Chapman, H. N.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Cheng, J.-X.

J.-X. Cheng and X. S. Xie, “Vibrational spectroscopic imaging of living systems: an emerging platform for biology and medicine,” Science 350, aaa8870 (2015).
[Crossref]

Cohen, O.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Dasari, R. R.

L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, “Time-dependent photon migration using path integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[Crossref]

de Aguiar, H. B.

H. B. de Aguiar, S. Gigan, and S. Brasselet, “Enhanced nonlinear imaging through scattering media using transmission-matrix-based wave-front shaping,” Phys. Rev. A 94, 043830 (2016).
[Crossref]

Defienne, H.

M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, “Spatiotemporal coherent control of light through a multiple scattering medium with the multispectral transmission matrix,” Phys. Rev. Lett. 116, 253901 (2016).
[Crossref]

Denk, W.

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2, 932–940 (2005).
[Crossref]

Dong, J.

Durian, D. J.

D. J. Durian, “Accuracy of diffusing-wave spectroscopy theories,” Phys. Rev. E 51, 3350–3358 (1995).
[Crossref]

Eldar, Y. C.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Eliyahu, D.

I. Freund and D. Eliyahu, “Surface correlations in multiple-scattering media,” Phys. Rev. A 45, 6133–6134 (1992).
[Crossref]

Feld, M. S.

L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, “Time-dependent photon migration using path integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[Crossref]

Feng, S.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[Crossref]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[Crossref]

Feynman, R. P.

R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
[Crossref]

Fink, M.

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8, 784–790 (2014).
[Crossref]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Freund, I.

I. Freund and D. Eliyahu, “Surface correlations in multiple-scattering media,” Phys. Rev. A 45, 6133–6134 (1992).
[Crossref]

I. Freund, “Looking through walls and around corners,” Physica A 168, 49–65 (1990).
[Crossref]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[Crossref]

Garnier, J.

L. Borcea and J. Garnier, “Derivation of a one-way radiative transfer equation in random media,” Phys. Rev. E 93, 022115 (2016).
[Crossref]

Gigan, S.

A. Boniface, B. Blochet, J. Dong, and S. Gigan, “Noninvasive light focusing in scattering media using speckle variance optimization,” Optica 6, 1381–1385 (2019).
[Crossref]

R. Savo, R. Pierrat, U. Najar, R. Carminati, S. Rotter, and S. Gigan, “Observation of mean path length invariance in light-scattering media,” Science 358, 765–768 (2017).
[Crossref]

H. B. de Aguiar, S. Gigan, and S. Brasselet, “Enhanced nonlinear imaging through scattering media using transmission-matrix-based wave-front shaping,” Phys. Rev. A 94, 043830 (2016).
[Crossref]

M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, “Spatiotemporal coherent control of light through a multiple scattering medium with the multispectral transmission matrix,” Phys. Rev. Lett. 116, 253901 (2016).
[Crossref]

S. Schott, J. Bertolotti, J.-F. Léger, L. Bourdieu, and S. Gigan, “Characterization of the angular memory effect of scattered light in biological tissues,” Opt. Express 23, 13505–13516 (2015).
[Crossref]

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8, 784–790 (2014).
[Crossref]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Glimm, J.

J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View (Springer, 1981).

Gorodetsky, A. A.

N. V. Petrov, V. G. Bespalov, and A. A. Gorodetsky, “Phase retrieval method for multiple wavelength speckle patterns,” Proc. SPIE 7387, 73871T (2010).
[Crossref]

Grésillon, S.

M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, “Spatiotemporal coherent control of light through a multiple scattering medium with the multispectral transmission matrix,” Phys. Rev. Lett. 116, 253901 (2016).
[Crossref]

Guillon, M.

Hanninen, A.

Heidmann, P.

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8, 784–790 (2014).
[Crossref]

Helmchen, F.

F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2, 932–940 (2005).
[Crossref]

Herbolzheimer, E.

D. Pine, D. Weitz, J. Zhu, and E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. France 51, 2101–2127 (1990).
[Crossref]

Hofer, M.

Hong, S. T.

A. Ishimaru and S. T. Hong, “Multiple scattering effects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles,” Radio Sci. 10, 637–644 (1975).
[Crossref]

Horstmeyer, R.

M. Kadobianskyi, I. N. Papadopoulos, T. Chaigne, R. Horstmeyer, and B. Judkewitz, “Scattering correlations of time-gated light,” Optica 5, 389–394 (2018).
[Crossref]

G. Osnabrugge, R. Horstmeyer, I. N. Papadopoulos, B. Judkewitz, and I. M. Vellekoop, “Generalized optical memory effect,” Optica 4, 886–892 (2017).
[Crossref]

B. Judkewitz, R. Horstmeyer, I. M. Vellekoop, I. N. Papadopoulos, and C. Yang, “Translation correlations in anisotropically scattering media,” Nat. Phys. 11, 684–689 (2015).
[Crossref]

R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9, 563–571 (2015).
[Crossref]

Ishimaru, A.

A. Ishimaru and S. T. Hong, “Multiple scattering effects on coherent bandwidth and pulse distortion of a wave propagating in a random distribution of particles,” Radio Sci. 10, 637–644 (1975).
[Crossref]

Itzkan, I.

L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, “Time-dependent photon migration using path integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[Crossref]

Jaffe, A.

J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View (Springer, 1981).

Judkewitz, B.

Kac, M.

M. Kac, “On distributions of certain Wiener functionals,” Trans. Am. Math. Soc. 65, 1–13 (1949).
[Crossref]

Kadobianskyi, M.

Kane, C.

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

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Lerosey, G.

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[Crossref]

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M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, “Spatiotemporal coherent control of light through a multiple scattering medium with the multispectral transmission matrix,” Phys. Rev. Lett. 116, 253901 (2016).
[Crossref]

Najar, U.

R. Savo, R. Pierrat, U. Najar, R. Carminati, S. Rotter, and S. Gigan, “Observation of mean path length invariance in light-scattering media,” Science 358, 765–768 (2017).
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Papadopoulos, I. N.

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R. Savo, R. Pierrat, U. Najar, R. Carminati, S. Rotter, and S. Gigan, “Observation of mean path length invariance in light-scattering media,” Science 358, 765–768 (2017).
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Popoff, S. M.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
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R. Savo, R. Pierrat, U. Najar, R. Carminati, S. Rotter, and S. Gigan, “Observation of mean path length invariance in light-scattering media,” Science 358, 765–768 (2017).
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R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9, 563–571 (2015).
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R. Savo, R. Pierrat, U. Najar, R. Carminati, S. Rotter, and S. Gigan, “Observation of mean path length invariance in light-scattering media,” Science 358, 765–768 (2017).
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Schott, S.

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Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
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Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
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Spivak, B.

O. Agam, A. V. Andreev, and B. Spivak, “Long-range correlations in the speckle patterns of directed waves,” Phys. Rev. Lett. 97, 223901 (2006).
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S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[Crossref]

Thendiyammal, A.

van Putten, E. G.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
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Volpe, G.

M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, “Spatiotemporal coherent control of light through a multiple scattering medium with the multispectral transmission matrix,” Phys. Rev. Lett. 116, 253901 (2016).
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J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[Crossref]

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L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, “Time-dependent photon migration using path integrals,” Phys. Rev. E 51, 6134–6141 (1995).
[Crossref]

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D. Pine, D. Weitz, J. Zhu, and E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. France 51, 2101–2127 (1990).
[Crossref]

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L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, “Time-dependent photon migration using path integrals,” Phys. Rev. E 51, 6134–6141 (1995).
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Xie, X.

Xie, X. S.

J.-X. Cheng and X. S. Xie, “Vibrational spectroscopic imaging of living systems: an emerging platform for biology and medicine,” Science 350, aaa8870 (2015).
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Xu, X.

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B. Judkewitz, R. Horstmeyer, I. M. Vellekoop, I. N. Papadopoulos, and C. Yang, “Translation correlations in anisotropically scattering media,” Nat. Phys. 11, 684–689 (2015).
[Crossref]

R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9, 563–571 (2015).
[Crossref]

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Yu, X.

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D. Pine, D. Weitz, J. Zhu, and E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. France 51, 2101–2127 (1990).
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Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

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Nat. Phys. (1)

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Opt. Lett. (1)

Optica (3)

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O. Agam, A. V. Andreev, and B. Spivak, “Long-range correlations in the speckle patterns of directed waves,” Phys. Rev. Lett. 97, 223901 (2006).
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[Crossref]

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[Crossref]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
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M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, “Spatiotemporal coherent control of light through a multiple scattering medium with the multispectral transmission matrix,” Phys. Rev. Lett. 116, 253901 (2016).
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Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. Principle of the experiment (a). A collimated super-continuum laser beam is sent onto a diffuser through a monochromator. When tuning the wavelength and simultaneously performing an axial scan of the proper amount, the same speckle pattern is observed. The zero-mean cross-correlation products of speckles are shown in (b) for $ L $-thick paraffin films, with $ L = 125\,\,\unicode{x00B5}\text{m} $ and $ L = 250\,\,\unicode{x00B5}\text{m} $. In both cases, the reference speckle pattern was considered for $ {\lambda _{\text{ref}}} = 571 \pm 5 \text{nm} $, lying at a distance $ {z_{\text{ref}}} = 760\,\,\unicode{x00B5}\text{m} $ behind the diffuser output surface. The correlation is qualitatively observed to be maximum along a hyperbolic line. Despite the forward-multiple-scattering nature of the paraffin diffusers and larger scattering angles, a more than 200 nm spectral correlation width is obtained for both paraffin films with $ L = 125\,\,\unicode{x00B5}\text{m} $ ($ \text{scattering angle} = 1.9^ \circ $-FWHM) and $ L = 250\,\,\unicode{x00B5}\text{m} $ ($\text{scattering angle} = 2.7^ \circ $-FWHM) [(c), blue and cyan, respectively]. The correlation width for the 1°-FWHM surface holographic diffuser [(c), red] is similar to the thicker paraffin slab.
Fig. 2.
Fig. 2. Model of a thick forward-multiple-scattering medium (a). The medium is sliced into $ N $ random and independent thin refractive diffusers spaced by distances $ d $. Propagation in the volumetric diffuser is achieved by an angular spectrum (plane wave) decomposition. A series of propagation directions, represented by two-dimensional angle vectors $ {\{ {{{\boldsymbol \Theta }_p}} \}_{p \in [[1;N]]}} \in {({{\mathbb R}^2})^N} $, defines a trajectory $ t({{\boldsymbol \Theta }_N}) $. Intermediate thin diffusers imprint the same phase shifts to beams of different wavelengths (blue and a red) along trajectories satisfying a transverse homothety by $ k/{k^{\prime}} $ (see Section 3.C). In the Fourier (plane-wave) domain, each individual diffuser scatters light with an angular spread of zero mean and with variance $ \langle {\theta ^2} \rangle = \sigma _0^2 $ per transverse coordinate (b). In the paraxial approximation, forward volumetric multiple scattering is equivalent to a two-dimensional random walk in the transverse ${k}$-space (c).
Fig. 3.
Fig. 3. Example of joint probability density function $ {\rho _L}(\Delta ,{\boldsymbol \Theta }) $ obtained numerically for a 250-µm-thick diffuser inducing a 3° angular output spread (a). The marginal distribution $ {\rho _L}({\boldsymbol \Theta }) $ is shown in (b) and yields a Gaussian distribution function. The dashed black curve in (a) is the average optical path $ {\langle \Delta \rangle _t}({\boldsymbol \Theta }) = \frac{L}{6}( {\Theta _0^2 + {{\boldsymbol \Theta }^2}} ) $ (see Sections 3 and 4 in Supplement 1), and dashed white lines represents the optical path spread at $ \pm 1 $ standard deviation according to Eq. (22).
Fig. 4.
Fig. 4. Cross-correlation product of speckles as a function of $ k $ and $ z $ for a 250-µm-thick paraffin film, when considering three different reference speckles taken at $ {\lambda _{\text{ref}}} = 571 \pm 5 \text{nm} $, and lying at three different distances behind the diffuser output surface: $ {z_{\text{ref}}} = 253\,\,\unicode{x00B5}\text{m} $, $ 506\,\,\unicode{x00B5}\text{m} $, and $ 760\,\,\unicode{x00B5}\text{m} $, respectively. In each case, a correlation maximum is obtained along a line of equation $ \lambda (z + {z_0}) = \text{constant} $, with $ {z_0} = - 110 \pm 10\,\,\unicode{x00B5}\text{m} $.

Equations (29)

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C ( λ , z ) = | E k ( r , z ) E k ref ( r , z ref ) r | 2 .
I k ( r , z ) = 1 ( λ z ) 2 | r E k ( r , 0 ) e 2 i π λ z ( r r ) 2 d 2 r | 2 ,
E k ( p ) ( r , z p ) = E ~ k ( p ) ( Θ p ) exp { i k [ r Θ p + z p ( 1 Θ p 2 2 ) ] } d 2 Θ p ,
E ~ k ( N ) ( Θ ) = Θ 0 , , Θ N E ~ k ( 0 ) ( Θ 0 ) d 2 Θ 0 δ ( Θ N Θ ) × p = 1 N T ~ ( p ) [ k ( Θ p Θ p 1 ) ] h k ( Θ p ) d 2 Θ p ,
E ~ k ( N ) ( Θ ) = t ( Θ N = Θ ) p = 1 N T ~ ( p ) ( k Δ Θ p ) h k ( Θ p ) D t ,
ϕ k ( t ) = N k d + p = 1 N { Arg [ T ~ ( p ) ( k Δ Θ p ) ] k d 2 Θ p 2 } ,
A k ( t ) = p = 1 N | T ~ ( p ) ( k Δ Θ p ) | .
E ~ k ( Θ , z ) = exp [ i k z ( 1 Θ 2 2 ) ] t ( Θ ) A k ( t ) e i ϕ k ( t ) D t .
E k ( r , z ) E k ( r , z ) = E ~ k ( Θ , z ) E ~ k ( Θ , z ) e i ( k r Θ k r Θ ) d 2 Θ d 2 Θ .
E k ( r , z ) E k ( r , z ) r = E k ( 0 , z ) E k ( 0 , z ) ens = E ~ k E ~ k ens d 2 Θ d 2 Θ .
E ~ k E ~ k = exp { i [ k z ( 1 Θ 2 / 2 ) k z ( 1 Θ 2 / 2 ) ] } × t t A k ( t ) A k ( t ) e i Δ ϕ D t D t .
Δ ϕ = ( k k ) d ( N + k k p = 1 N Θ p 2 2 ) .
ρ N ( Δ , Θ ) = E 0 { δ ( Θ N Θ ) δ ( Δ N Δ ) } .
W N ( k 0 , Θ ) = ρ N ( Δ , Θ ) e i k 0 Δ d Δ ,
L W z z = Θ 0 2 2 2 W z Θ 2 i k 0 L 2 Θ 2 W z ,
W L ( k 0 , Θ ) = 1 2 π Θ 0 i k 0 L sinh ( Θ 0 i k 0 L ) × exp [ i k 0 L 2 Θ 0 tanh ( Θ 0 i k 0 L ) Θ 2 ] .
E ~ k E ~ k = exp { i [ k z ( 1 Θ 2 / 2 ) k z ( 1 Θ 2 / 2 ) ] } × δ ( k Θ k Θ ) A k A k t e i Δ ϕ t .
E ~ k E ~ k = δ ( k Θ k Θ ) A k A k t × exp { i k k Θ 2 2 [ k ( z + L 3 ) k ( z + L 3 ) ] + i k 0 L Θ 0 2 6 ( k 0 L ) 2 Θ 0 2 90 ( Θ 2 + Θ 0 2 2 ) } .
C = 1 1 + ( k 0 L Θ 0 2 ) 2 18 + ( Θ 0 2 k k ) 2 [ k ( z + L 3 ) k ( z + L 3 ) ] 2 .
k ( z + L 3 ) = k ( z + L 3 ) .
C max = 1 1 + ( L Θ 0 2 ) 2 18 k 0 2 .
Δ k = 3 2 L Θ 0 2 .
σ Δ 2 = Δ 2 t Δ t 2 = L 2 Θ 0 2 45 ( Θ 0 2 2 + Θ 2 ) ,
Δ ϕ = k d 2 p = 1 N [ ( Θ p Θ inc ) 2 Θ p 2 ] ,
= k d 2 ( N Θ inc 2 2 Θ inc p = 1 N Θ p ) ,
Δ ϕ = k L 2 ( Θ inc 2 Θ inc Θ ) .
( Δ ϕ Δ ϕ ) 2 = 2 ( k L ) 2 Θ inc 2 1 N 2 Var ( X 2 )
= ( k L ) 2 3 Θ inc 2 Θ 0 2 .
Θ inc 3 2 λ L Θ 0 .

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