Abstract

The optical memory effect is a well-known type of tilt/tilt wave correlation that is observed in coherent fields, allowing control over scattered light through thin and diffusive materials. Here we show that the optical memory effect is a special case of a more general class of combined shift/tilt correlations occurring in media of arbitrary geometry. We experimentally demonstrate the existence of these correlations, and provide an analytical framework that allows us to predict and understand this class of scattering correlations. This “generalized optical memory effect” can be utilized for maximizing the imaging field-of-view of deep tissue imaging techniques such as phase conjugation and adaptive optics.

© 2017 Optical Society of America

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References

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  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]
  2. O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photonics 6, 549–553 (2012).
    [Crossref]
  3. 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]
  4. X. Yang, Y. Pu, and D. Psaltis, “Imaging blood cells through scattering biological tissue using speckle scanning microscopy,” Opt. Express 22, 3405–3413 (2014).
    [Crossref]
  5. 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]
  6. 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]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2017 (2)

I. N. Papadopoulos, J.-S. 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]

C. W. Hsu, S. F. Liew, A. Goetschy, H. Cao, and A. D. Stone, “Correlation-enhanced control of wave focusing in disordered media,” Nat. Phys. 13, 497–502 (2017).
[Crossref]

2015 (5)

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]

X. Yang, Y. Pu, and D. Psaltis, “Imaging blood cells through scattering biological tissue using speckle scanning microscopy,” Opt. Express 22, 3405–3413 (2014).
[Crossref]

2012 (2)

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]

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photonics 6, 549–553 (2012).
[Crossref]

2000 (1)

C.-C. Cheng and M. Raymer, “Propagation of transverse optical coherence in random multiple-scattering media,” Phys. Rev. A 62, 023811 (2000).
[Crossref]

1998 (1)

1994 (1)

J. H. Li and A. Z. Genack, “Correlation in laser speckle,” Phys. Rev. E 49, 4530–4533 (1994).
[Crossref]

1988 (2)

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]

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]

1979 (1)

Bastiaans, M. J.

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]

Bifano, T. G.

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]

Bourdieu, L.

Cao, H.

C. W. Hsu, S. F. Liew, A. Goetschy, H. Cao, and A. D. Stone, “Correlation-enhanced control of wave focusing in disordered media,” Nat. Phys. 13, 497–502 (2017).
[Crossref]

Cheng, C.-C.

C.-C. Cheng and M. Raymer, “Propagation of transverse optical coherence in random multiple-scattering media,” Phys. Rev. A 62, 023811 (2000).
[Crossref]

Cui, M.

J.-H. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. USA 112, 9236–9241 (2015).
[Crossref]

Feng, S.

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]

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]

Freund, I.

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]

Genack, A. Z.

J. H. Li and A. Z. Genack, “Correlation in laser speckle,” Phys. Rev. E 49, 4530–4533 (1994).
[Crossref]

Gigan, S.

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]

Goetschy, A.

C. W. Hsu, S. F. Liew, A. Goetschy, H. Cao, and A. D. Stone, “Correlation-enhanced control of wave focusing in disordered media,” Nat. Phys. 13, 497–502 (2017).
[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]

Hennelly, B.

M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill Education, 2009).

Horstmeyer, R.

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]

Hsu, C. W.

C. W. Hsu, S. F. Liew, A. Goetschy, H. Cao, and A. D. Stone, “Correlation-enhanced control of wave focusing in disordered media,” Nat. Phys. 13, 497–502 (2017).
[Crossref]

Jonas, E.

Jouhanneau, J.-S.

I. N. Papadopoulos, J.-S. 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. 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]

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.

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]

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photonics 6, 549–553 (2012).
[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,” Nature 491, 232–234 (2012).
[Crossref]

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.

Levoy, M.

Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in IEEE International Conference on Computational Photography (ICCP) (IEEE, 2009), pp. 1–10.

Li, J. H.

J. H. Li and A. Z. Genack, “Correlation in laser speckle,” Phys. Rev. E 49, 4530–4533 (1994).
[Crossref]

Liew, S. F.

C. W. Hsu, S. F. Liew, A. Goetschy, H. Cao, and A. D. Stone, “Correlation-enhanced control of wave focusing in disordered media,” Nat. Phys. 13, 497–502 (2017).
[Crossref]

Liu, H.-Y.

Mertz, J.

Mosk, A. P.

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]

Ojeda-Castaneda, J.

M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill Education, 2009).

Papadopoulos, I. N.

I. N. Papadopoulos, J.-S. 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]

Park, J.-H.

J.-H. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. USA 112, 9236–9241 (2015).
[Crossref]

Paudel, H.

Poulet, J. F. A.

I. N. Papadopoulos, J.-S. 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]

Psaltis, D.

Pu, Y.

Raymer, M.

C.-C. Cheng and M. Raymer, “Propagation of transverse optical coherence in random multiple-scattering media,” Phys. Rev. A 62, 023811 (2000).
[Crossref]

Recht, B.

Rosenbluh, M.

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]

Schott, S.

Silberberg, Y.

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photonics 6, 549–553 (2012).
[Crossref]

Small, E.

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photonics 6, 549–553 (2012).
[Crossref]

Stone, A. D.

C. W. Hsu, S. F. Liew, A. Goetschy, H. Cao, and A. D. Stone, “Correlation-enhanced control of wave focusing in disordered media,” Nat. Phys. 13, 497–502 (2017).
[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]

Sun, W.

J.-H. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. USA 112, 9236–9241 (2015).
[Crossref]

Testorf, M.

M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill Education, 2009).

Thomas, J. E.

Tian, L.

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).
[Crossref]

Vellekoop, I. M.

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]

Vos, W. L.

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]

Waller, L.

Wax, A.

Yang, C.

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]

Yang, X.

Zhang, Z.

Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in IEEE International Conference on Computational Photography (ICCP) (IEEE, 2009), pp. 1–10.

Zhong, J.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

Nat. Photonics (3)

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photonics 6, 549–553 (2012).
[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]

I. N. Papadopoulos, J.-S. 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]

Nat. Phys. (2)

C. W. Hsu, S. F. Liew, A. Goetschy, H. Cao, and A. D. Stone, “Correlation-enhanced control of wave focusing in disordered media,” Nat. Phys. 13, 497–502 (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]

Nature (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]

Opt. Express (3)

Phys. Rev. A (1)

C.-C. Cheng and M. Raymer, “Propagation of transverse optical coherence in random multiple-scattering media,” Phys. Rev. A 62, 023811 (2000).
[Crossref]

Phys. Rev. E (1)

J. H. Li and A. Z. Genack, “Correlation in laser speckle,” Phys. Rev. E 49, 4530–4533 (1994).
[Crossref]

Phys. Rev. Lett. (2)

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]

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]

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

J.-H. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. USA 112, 9236–9241 (2015).
[Crossref]

Other (2)

M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (McGraw-Hill Education, 2009).

Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in IEEE International Conference on Computational Photography (ICCP) (IEEE, 2009), pp. 1–10.

Supplementary Material (1)

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» Supplement 1       Mathematical supplement

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Figures (5)

Fig. 1.
Fig. 1.

Three different types of spatial correlations in disordered media. (a) The optical “tilt” memory effect [6], where an input wavefront tilt leads to a tilt at the target plane. (b) The anisotropic “shift” memory effect [9], where an input wavefront shift also shifts the target plane wavefront. (c) Our new generalized memory effect, relying on both tilts and shifts, can maximize correlations along the target plane for a maximum imaging/focus scanning area.

Fig. 2.
Fig. 2.

AO focus scanning/imaging inside a scattering medium uses different memory effects. (a) The “tilt” effect arises with the AO tilt plane (dashed line) conjugated to the input surface. (b) The “shift” effect arises with the AO tilt plane at infinity. (c) The optimal joint tilt/shift scheme requires the AO tilt plane to be located at a depth of L/3 inside the sample.

Fig. 3.
Fig. 3.

Schematic of the experimental setup used for measuring both the light field transmission function and generalized correlation functions. The pinhole and diffuser are used in the P and C experiments, respectively. Both the diffuser and the sample holder are placed on a translation stage.

Fig. 4.
Fig. 4.

Results of the light field transmission function (P) experiment. We compare our measurements, Pex, to the Fokker–Planck model prediction [PFP, from Eq. (8)] for samples with (a), (b) L=258  μm and (c), (d) L=520  μm. Color bar indicates the normalized transmitted intensity as function of xb and kb.

Fig. 5.
Fig. 5.

Results of the generalized correlation function (C) experiments. Measurements of Cex are compared to CPW and the Fokker–Planck correlation model CFP from Eq. (10) for samples with (a)–(c) L=258  μm and (d)–(f) L=520  μm. Dashed lines indicate the optimal scanning condition in Eq. (11). (g)–(i) Cross sections of the 2D correlation functions in (a)–(c); evaluated (g) along the horizontal axis at Δkb=0, (h) along vertical axis at Δxb=0, and (i) along the optimal scan line for the 258 μm thick sample. (j)–(l) Same cross sections for the 520 μm thick sample. The black dashed line denotes the CFP model, while red and blue stars denote the measured Cex and CPW, respectively.

Tables (1)

Tables Icon

Table 1. Comparison of the Performance of the Three Different Memory Effects in Terms of Adaptive Optics Scan Range

Equations (12)

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

Eb(rb)=T(rb,ra)Ea(ra)d2ra.
C(Δrb,Δkb;Δra,Δka)T(rb+,ra+)T*(rb,ra)ei(Δka·raΔkb·rb)d2rad2rb.
W(r,k)E(r+)E*(r)eik·Δrd2Δr.
Wb(rb,kb)=1(2π)2P(rb,kb;ra,ka)Wa(ra,ka)d2rad2ka.
P(rb,kb;ra,ka)T(rb+,ra+)T*(rb,ra)ei(Δra·kaΔrb·kb)d2Δrad2Δrb.
T(rb+,ra+)T*(rb,ra)=1(2π)4P(rb,kb;ra,ka)ei(Δra·ka+Δrb·kb)d2kad2kb,
C(Δrb,Δkb;Δra,Δka)=1(2π)4P(rb,kb;ra,ka)ei(Δra·ka+Δrb·kb+Δka·raΔkb·rb)d2kad2rad2kbd2rb.
PFP(r^,k^)=12tr2k02L4exp(6trL[|r^|2L2k^·r^k0L+|k^|23k02]).
C(Δrb,Δkb;Δra,Δka)=(2π)2  CFP(Δrb,Δkb)δ(ΔkaΔkb)δ(ΔrbΔraΔkaL/k0).
CFP(Δrb,Δkb)=exp(L3k022tr[|Δkb|23k02Δkb·Δrbk0L+|Δrb|2L2]).
Δkbopt=3k0Δrb2L.
Δkaopt=3k0Δrb2LandΔraopt=Δrb/2.