Abstract

Manipulating the propagation of light through scattering media remains a major challenge for many applications, including astronomy, biomedical imaging, and colloidal optics. Light can be focused through inhomogeneous media into any desired point with wavefront shaping techniques. However, scattering correlations, referred to as the optical memory effect, limit spatial range within the scattering medium, where a single wavefront correction remains accurate. Selecting only the weakly scattered light is a promising way to increase this correlation range. Here we investigate spatial scattering correlations by digitally time-gating the early arriving light in the spectral domain. We demonstrate that the range of the translational memory effect for the early arriving light is increased almost fourfold, paving the way for a range of scattering media imaging applications.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]

2017 (2)

I. N. Papadopoulos, J.-S. B. Jouhanneau, J. F. A. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11, 116–123 (2017).
[Crossref]

G. Osnabrugge, R. Horstmeyer, I. N. Papadopoulos, B. Judkewitz, and I. M. Vellekoop, “Generalized optical memory effect,” Optica 4, 886–887 (2017).
[Crossref]

2016 (1)

M. Mounaix, H. Defienne, and S. Gigan, “Deterministic light focusing in space and time through multiple scattering media with a time-resolved transmission matrix approach,” Phys. Rev. A 94, 041802 (2016).
[Crossref]

2015 (4)

D. Andreoli, G. Volpe, S. Popoff, O. Katz, S. Grésillon, and S. Gigan, “Deterministic control of broadband light through a multiply scattering medium via the multispectral transmission matrix,” Sci. Rep. 5, 1–8 (2015).
[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]

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, 13505–13512 (2015).
[Crossref]

2014 (2)

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]

C. Wang, R. Liu, D. E. Milkie, W. Sun, Z. Tan, A. Kerlin, T.-W. Chen, D. S. Kim, and N. Ji, “Multiplexed aberration measurement for deep tissue imaging in vivo,” Nat. Methods 11, 1037–1040 (2014).
[Crossref]

2012 (4)

J. Tang, R. N. Germain, and M. Cui, “Superpenetration optical microscopy by iterative multiphoton adaptive compensation technique,” Proc. Natl. Acad. Sci. USA 109, 8434–8439 (2012).
[Crossref]

Y. M. Wang, B. Judkewitz, C. A. DiMarzio, and C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. 3, 928 (2012).
[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,” Nat. Commun. 491, 232–234 (2012).
[Crossref]

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
[Crossref]

2011 (1)

X. Xu, H. Liu, and L. V. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics 5, 154–157 (2011).
[Crossref]

2010 (3)

V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods 7, 603–614 (2010).
[Crossref]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[Crossref]

C.-L. Hsieh, Y. Pu, R. Grange, G. Laporte, and D. Psaltis, “Imaging through turbid layers by scanning the phase conjugated second harmonic radiation from a nanoparticle,” Opt. Express 18, 20723–20731 (2010).
[Crossref]

2008 (1)

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008).
[Crossref]

2007 (1)

2005 (1)

2003 (1)

C. Dunsby and P. M. W. French, “Techniques for depth-resolved imaging through turbid media including coherence-gated imaging,” J. Phys. D: Appl. Phys. 36, R207–R227 (2003).
[Crossref]

1995 (1)

1993 (1)

1991 (1)

L. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[Crossref]

1990 (2)

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[Crossref]

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]

1988 (1)

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]

Alfano, R. R.

L. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[Crossref]

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]

Andreoli, D.

D. Andreoli, G. Volpe, S. Popoff, O. Katz, S. Grésillon, and S. Gigan, “Deterministic control of broadband light through a multiply scattering medium via the multispectral transmission matrix,” Sci. Rep. 5, 1–8 (2015).
[Crossref]

Arons, E.

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, 13505–13512 (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,” Nat. Commun. 491, 232–234 (2012).
[Crossref]

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,” Nat. Commun. 491, 232–234 (2012).
[Crossref]

Bourdieu, L.

Chen, T.-W.

C. Wang, R. Liu, D. E. Milkie, W. Sun, Z. Tan, A. Kerlin, T.-W. Chen, D. S. Kim, and N. Ji, “Multiplexed aberration measurement for deep tissue imaging in vivo,” Nat. Methods 11, 1037–1040 (2014).
[Crossref]

Cheong, W. F.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[Crossref]

Cui, M.

J. Tang, R. N. Germain, and M. Cui, “Superpenetration optical microscopy by iterative multiphoton adaptive compensation technique,” Proc. Natl. Acad. Sci. USA 109, 8434–8439 (2012).
[Crossref]

Defienne, H.

M. Mounaix, H. Defienne, and S. Gigan, “Deterministic light focusing in space and time through multiple scattering media with a time-resolved transmission matrix approach,” Phys. Rev. A 94, 041802 (2016).
[Crossref]

Dilworth, D.

DiMarzio, C. A.

Y. M. Wang, B. Judkewitz, C. A. DiMarzio, and C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. 3, 928 (2012).
[Crossref]

Dunsby, C.

C. Dunsby and P. M. W. French, “Techniques for depth-resolved imaging through turbid media including coherence-gated imaging,” J. Phys. D: Appl. Phys. 36, R207–R227 (2003).
[Crossref]

Feld, M. S.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008).
[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]

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]

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
[Crossref]

French, P. M. W.

C. Dunsby and P. M. W. French, “Techniques for depth-resolved imaging through turbid media including coherence-gated imaging,” J. Phys. D: Appl. Phys. 36, R207–R227 (2003).
[Crossref]

Germain, R. N.

J. Tang, R. N. Germain, and M. Cui, “Superpenetration optical microscopy by iterative multiphoton adaptive compensation technique,” Proc. Natl. Acad. Sci. USA 109, 8434–8439 (2012).
[Crossref]

Gigan, S.

M. Mounaix, H. Defienne, and S. Gigan, “Deterministic light focusing in space and time through multiple scattering media with a time-resolved transmission matrix approach,” Phys. Rev. A 94, 041802 (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, 13505–13512 (2015).
[Crossref]

D. Andreoli, G. Volpe, S. Popoff, O. Katz, S. Grésillon, and S. Gigan, “Deterministic control of broadband light through a multiply scattering medium via the multispectral transmission matrix,” Sci. Rep. 5, 1–8 (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]

Grange, R.

Grésillon, S.

D. Andreoli, G. Volpe, S. Popoff, O. Katz, S. Grésillon, and S. Gigan, “Deterministic control of broadband light through a multiply scattering medium via the multispectral transmission matrix,” Sci. Rep. 5, 1–8 (2015).
[Crossref]

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]

Ho, P. P.

L. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[Crossref]

Horstmeyer, R.

G. Osnabrugge, R. Horstmeyer, I. N. Papadopoulos, B. Judkewitz, and I. M. Vellekoop, “Generalized optical memory effect,” Optica 4, 886–887 (2017).
[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]

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]

Hsieh, C.-L.

Ji, N.

C. Wang, R. Liu, D. E. Milkie, W. Sun, Z. Tan, A. Kerlin, T.-W. Chen, D. S. Kim, and N. Ji, “Multiplexed aberration measurement for deep tissue imaging in vivo,” Nat. Methods 11, 1037–1040 (2014).
[Crossref]

Jouhanneau, J.-S. B.

I. N. Papadopoulos, J.-S. B. Jouhanneau, J. F. A. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11, 116–123 (2017).
[Crossref]

Judkewitz, B.

I. N. Papadopoulos, J.-S. B. Jouhanneau, J. F. A. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11, 116–123 (2017).
[Crossref]

G. Osnabrugge, R. Horstmeyer, I. N. Papadopoulos, B. Judkewitz, and I. M. Vellekoop, “Generalized optical memory effect,” Optica 4, 886–887 (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]

Y. M. Wang, B. Judkewitz, C. A. DiMarzio, and C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. 3, 928 (2012).
[Crossref]

Kane, C.

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]

Katz, O.

D. Andreoli, G. Volpe, S. Popoff, O. Katz, S. Grésillon, and S. Gigan, “Deterministic control of broadband light through a multiply scattering medium via the multispectral transmission matrix,” Sci. Rep. 5, 1–8 (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]

Kerlin, A.

C. Wang, R. Liu, D. E. Milkie, W. Sun, Z. Tan, A. Kerlin, T.-W. Chen, D. S. Kim, and N. Ji, “Multiplexed aberration measurement for deep tissue imaging in vivo,” Nat. Methods 11, 1037–1040 (2014).
[Crossref]

Kim, D. S.

C. Wang, R. Liu, D. E. Milkie, W. Sun, Z. Tan, A. Kerlin, T.-W. Chen, D. S. Kim, and N. Ji, “Multiplexed aberration measurement for deep tissue imaging in vivo,” Nat. Methods 11, 1037–1040 (2014).
[Crossref]

Lagendijk, A.

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

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
[Crossref]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[Crossref]

Laporte, G.

Lee, P. A.

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]

Léger, J.-F.

Lerosey, G.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
[Crossref]

Liu, C.

L. Wang, P. P. Ho, C. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[Crossref]

Liu, H.

X. Xu, H. Liu, and L. V. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics 5, 154–157 (2011).
[Crossref]

Liu, R.

C. Wang, R. Liu, D. E. Milkie, W. Sun, Z. Tan, A. Kerlin, T.-W. Chen, D. S. Kim, and N. Ji, “Multiplexed aberration measurement for deep tissue imaging in vivo,” Nat. Methods 11, 1037–1040 (2014).
[Crossref]

Mätzler, C.

C. Mätzler, “MATLAB functions for Mie scattering and absorption, version 2,” IAP Research Report (2002).

Milkie, D. E.

C. Wang, R. Liu, D. E. Milkie, W. Sun, Z. Tan, A. Kerlin, T.-W. Chen, D. S. Kim, and N. Ji, “Multiplexed aberration measurement for deep tissue imaging in vivo,” Nat. Methods 11, 1037–1040 (2014).
[Crossref]

Mosk, A. P.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
[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,” Nat. Commun. 491, 232–234 (2012).
[Crossref]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[Crossref]

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
[Crossref]

Mounaix, M.

M. Mounaix, H. Defienne, and S. Gigan, “Deterministic light focusing in space and time through multiple scattering media with a time-resolved transmission matrix approach,” Phys. Rev. A 94, 041802 (2016).
[Crossref]

Ntziachristos, V.

V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods 7, 603–614 (2010).
[Crossref]

Oron, D.

Osnabrugge, G.

Papadopoulos, I. N.

G. Osnabrugge, R. Horstmeyer, I. N. Papadopoulos, B. Judkewitz, and I. M. Vellekoop, “Generalized optical memory effect,” Optica 4, 886–887 (2017).
[Crossref]

I. N. Papadopoulos, J.-S. B. Jouhanneau, J. F. A. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11, 116–123 (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]

Popoff, S.

D. Andreoli, G. Volpe, S. Popoff, O. Katz, S. Grésillon, and S. Gigan, “Deterministic control of broadband light through a multiply scattering medium via the multispectral transmission matrix,” Sci. Rep. 5, 1–8 (2015).
[Crossref]

Poulet, J. F. A.

I. N. Papadopoulos, J.-S. B. Jouhanneau, J. F. A. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11, 116–123 (2017).
[Crossref]

Prahl, S. A.

W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[Crossref]

Psaltis, D.

C.-L. Hsieh, Y. Pu, R. Grange, G. Laporte, and D. Psaltis, “Imaging through turbid layers by scanning the phase conjugated second harmonic radiation from a nanoparticle,” Opt. Express 18, 20723–20731 (2010).
[Crossref]

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008).
[Crossref]

Pu, Y.

Ruan, H.

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Optica (1)

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Proc. Natl. Acad. Sci. USA (1)

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D. Andreoli, G. Volpe, S. Popoff, O. Katz, S. Grésillon, and S. Gigan, “Deterministic control of broadband light through a multiply scattering medium via the multispectral transmission matrix,” Sci. Rep. 5, 1–8 (2015).
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Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. Principal optical setup. The output of a tunable laser source is split into two arms of a Mach–Zehnder interferometer by the beamsplitter BS1. The combination of lens L1 and microscope objective OBJ1 projects a plane wave at the input plane of the scattering medium. A focus spot can be formed and precisely positioned at the same plane by removing L1 and focusing using a reflection imaging system (lens L3 and CMOS1). The scattered light at the output plane of the medium is imaged by the transmission imaging system (OBJ2 and L2) onto the CMOS2, where it is combined with the reference beam, forming a digital hologram. Global phase delay introduced by imaging elements in the object arm is corrected as described in the Methods section. Inset: The translational memory effect can be measured in two ways: (a) as the width of speckle autocorrelation resulting from plane wave input, and (b) as the cross-correlation width between fields produced by physically translating the sample under focused light excitation.
Fig. 2.
Fig. 2. Digital time gating of the scattered light in the spectral domain. (a) Sketch of digitally time-gated electric field reconstruction. The amplitudes and phases of the complex speckle fields at the output of a scattering medium are measured for a set of equally spaced frequencies (left), which are then brought to the same global phase (middle). The scatterer’s temporal response is reconstructed by taking a Fourier transform across the frequency dimension (right). (b) Amplitude and phase (color) of the complex speckle pattern at different time delays for the thinnest sample (scale bar, 10 μm). (c) Spatial Fourier transform of each resulting complex speckle pattern (scale bar, absolute value of k that corresponds to a mean single scattering angle, 13°). At larger time delays, photons emerge from the medium at an increasing angle. All images were normalized to unitary power.
Fig. 3.
Fig. 3. Speckle autocorrelation of digitally time-gated light. (a) Speckle autocorrelation as a function of time delay for three samples (under plane wave illumination) with varying thickness, 360 μm (left), 720 μm (center), and 1080 μm (right). (b) Speckle autocorrelation for the earliest time delay compared to the non-gated speckle (CW illumination, 815 nm). The translational memory effect range is wider for earlier-arriving light as compared to CW illumination.
Fig. 4.
Fig. 4. Influence of bandwidth on the translational memory effect range and comparison of this range for early-arriving “snake” photons versus CW response based on two independent measurements. (a) Dependence of the speckle autocorrelation FWHM (mean speckle grain size) on the bandwidth of the synthesized pulse for the thinnest sample. A short (broadband) pulse is essential for observing an increased memory effect for early-arriving snake photons. (b) Speckle autocorrelation value (autocorrelation function, ACF) measured for an input plane wave illumination (solid lines) and the “shift/shift” memory effect (cross-correlation function, XCF) correlation value measured for an input focus (dotted lines). Here we show ACF and XCF measurements for both CW illumination (central wavelength 815 nm, blue curves) and for digitally gated early-arriving “snake” photons (red curves). Both measurements indicate an increase in the FOV of the translational memory effect. Data for the early-arriving “snake” photons presented as mean ± standard error of the mean.

Equations (3)

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uout(x,y,t)Δνn=1Nuout(x,y,νn)ei2πνnt.
klukl*(ν2)ukl(ν1)=kl|u|kl2eiΔϕ=eiΔϕ.
Cshift/shift(nΔx)=1mn+1ij=nCij,n=1m.

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