Abstract

A method to recover the image of an object behind thin turbid layers is developed by wavefront shaping technique. The optimized wavefront is generated by modulating the scattering light of a known object with a spatial light modulator. A Pearson Correlation Coefficient is introduced as a cost function for the optimization. A beam scanning method based on optical memory effect is proposed to further enlarge the Field-of-View (FOV). The experimental results show good fidelity and large FOV of the recovered image.

© 2013 OSA

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2012

2011

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics5(6), 372–377 (2011).
[CrossRef]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett.106(19), 193905 (2011).
[CrossRef] [PubMed]

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett.107(2), 023902 (2011).
[CrossRef] [PubMed]

2010

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat Commun1(6), 81 (2010).
[CrossRef] [PubMed]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics4(5), 320–322 (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(10), 100601 (2010).
[CrossRef] [PubMed]

2008

A. R. Kherlopian, T. Song, Q. Duan, M. A. Neimark, M. J. Po, J. K. Gohagan, and A. F. Laine, “A review of imaging techniques for systems biology,” BMC Syst. Biol.2(1), 74 (2008).
[CrossRef] [PubMed]

I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun.281(11), 3071–3080 (2008).
[CrossRef]

2007

1990

I. Freund, “Looking through walls and around corners,” Physica A168(1), 49–65 (1990).
[CrossRef]

1988

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett.61(20), 2328–2331 (1988).
[CrossRef] [PubMed]

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(7), 834–837 (1988).
[CrossRef] [PubMed]

Aegerter, C. M.

Akbulut, D.

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett.106(19), 193905 (2011).
[CrossRef] [PubMed]

Bertolotti, J.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature491(7423), 232–234 (2012).
[CrossRef] [PubMed]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett.106(19), 193905 (2011).
[CrossRef] [PubMed]

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,” Nature491(7423), 232–234 (2012).
[CrossRef] [PubMed]

Boccara, A. C.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat Commun1(6), 81 (2010).
[CrossRef] [PubMed]

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(10), 100601 (2010).
[CrossRef] [PubMed]

Bromberg, Y.

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics5(6), 372–377 (2011).
[CrossRef]

Caravaca-Aguirre, A. M.

Carminati, R.

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(10), 100601 (2010).
[CrossRef] [PubMed]

Choi, W.

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett.107(2), 023902 (2011).
[CrossRef] [PubMed]

Choi, Y.

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett.107(2), 023902 (2011).
[CrossRef] [PubMed]

Conkey, D. B.

Dasari, R. R.

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett.107(2), 023902 (2011).
[CrossRef] [PubMed]

Duan, Q.

A. R. Kherlopian, T. Song, Q. Duan, M. A. Neimark, M. J. Po, J. K. Gohagan, and A. F. Laine, “A review of imaging techniques for systems biology,” BMC Syst. Biol.2(1), 74 (2008).
[CrossRef] [PubMed]

Fang-Yen, C.

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett.107(2), 023902 (2011).
[CrossRef] [PubMed]

Feld, M. S.

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett.107(2), 023902 (2011).
[CrossRef] [PubMed]

Feng, S.

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett.61(20), 2328–2331 (1988).
[CrossRef] [PubMed]

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(7), 834–837 (1988).
[CrossRef] [PubMed]

Fink, 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(10), 100601 (2010).
[CrossRef] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat Commun1(6), 81 (2010).
[CrossRef] [PubMed]

Freund, I.

I. Freund, “Looking through walls and around corners,” Physica A168(1), 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(20), 2328–2331 (1988).
[CrossRef] [PubMed]

Ghielmetti, G.

Gigan, S.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat Commun1(6), 81 (2010).
[CrossRef] [PubMed]

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(10), 100601 (2010).
[CrossRef] [PubMed]

Gohagan, J. K.

A. R. Kherlopian, T. Song, Q. Duan, M. A. Neimark, M. J. Po, J. K. Gohagan, and A. F. Laine, “A review of imaging techniques for systems biology,” BMC Syst. Biol.2(1), 74 (2008).
[CrossRef] [PubMed]

Guan, Y.

Hsieh, C.-L.

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(7), 834–837 (1988).
[CrossRef] [PubMed]

Kang, P.

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett.107(2), 023902 (2011).
[CrossRef] [PubMed]

Katz, O.

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photonics6(8), 549–553 (2012).
[CrossRef]

E. Small, O. Katz, Y. Guan, and Y. Silberberg, “Spectral control of broadband light through random media by wavefront shaping,” Opt. Lett.37(16), 3429–3431 (2012).
[CrossRef] [PubMed]

Y. Guan, O. Katz, E. Small, J. Zhou, and Y. Silberberg, “Polarization control of multiply scattered light through random media by wavefront shaping,” Opt. Lett.37(22), 4663–4665 (2012).
[CrossRef] [PubMed]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics5(6), 372–377 (2011).
[CrossRef]

Kherlopian, A. R.

A. R. Kherlopian, T. Song, Q. Duan, M. A. Neimark, M. J. Po, J. K. Gohagan, and A. F. Laine, “A review of imaging techniques for systems biology,” BMC Syst. Biol.2(1), 74 (2008).
[CrossRef] [PubMed]

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,” Nature491(7423), 232–234 (2012).
[CrossRef] [PubMed]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett.106(19), 193905 (2011).
[CrossRef] [PubMed]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics4(5), 320–322 (2010).
[CrossRef]

Laine, A. F.

A. R. Kherlopian, T. Song, Q. Duan, M. A. Neimark, M. J. Po, J. K. Gohagan, and A. F. Laine, “A review of imaging techniques for systems biology,” BMC Syst. Biol.2(1), 74 (2008).
[CrossRef] [PubMed]

Lee, K. J.

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett.107(2), 023902 (2011).
[CrossRef] [PubMed]

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(7), 834–837 (1988).
[CrossRef] [PubMed]

Lerosey, G.

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(10), 100601 (2010).
[CrossRef] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat Commun1(6), 81 (2010).
[CrossRef] [PubMed]

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,” Nature491(7423), 232–234 (2012).
[CrossRef] [PubMed]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett.106(19), 193905 (2011).
[CrossRef] [PubMed]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics4(5), 320–322 (2010).
[CrossRef]

I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun.281(11), 3071–3080 (2008).
[CrossRef]

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett.32(16), 2309–2311 (2007).
[CrossRef] [PubMed]

Neimark, M. A.

A. R. Kherlopian, T. Song, Q. Duan, M. A. Neimark, M. J. Po, J. K. Gohagan, and A. F. Laine, “A review of imaging techniques for systems biology,” BMC Syst. Biol.2(1), 74 (2008).
[CrossRef] [PubMed]

Piestun, R.

Po, M. J.

A. R. Kherlopian, T. Song, Q. Duan, M. A. Neimark, M. J. Po, J. K. Gohagan, and A. F. Laine, “A review of imaging techniques for systems biology,” BMC Syst. Biol.2(1), 74 (2008).
[CrossRef] [PubMed]

Popoff, S.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat Commun1(6), 81 (2010).
[CrossRef] [PubMed]

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(10), 100601 (2010).
[CrossRef] [PubMed]

Psaltis, D.

Pu, Y.

Rosenbluh, M.

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett.61(20), 2328–2331 (1988).
[CrossRef] [PubMed]

Silberberg, Y.

Y. Guan, O. Katz, E. Small, J. Zhou, and Y. Silberberg, “Polarization control of multiply scattered light through random media by wavefront shaping,” Opt. Lett.37(22), 4663–4665 (2012).
[CrossRef] [PubMed]

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photonics6(8), 549–553 (2012).
[CrossRef]

E. Small, O. Katz, Y. Guan, and Y. Silberberg, “Spectral control of broadband light through random media by wavefront shaping,” Opt. Lett.37(16), 3429–3431 (2012).
[CrossRef] [PubMed]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics5(6), 372–377 (2011).
[CrossRef]

Small, E.

E. Small, O. Katz, Y. Guan, and Y. Silberberg, “Spectral control of broadband light through random media by wavefront shaping,” Opt. Lett.37(16), 3429–3431 (2012).
[CrossRef] [PubMed]

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photonics6(8), 549–553 (2012).
[CrossRef]

Y. Guan, O. Katz, E. Small, J. Zhou, and Y. Silberberg, “Polarization control of multiply scattered light through random media by wavefront shaping,” Opt. Lett.37(22), 4663–4665 (2012).
[CrossRef] [PubMed]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics5(6), 372–377 (2011).
[CrossRef]

Song, T.

A. R. Kherlopian, T. Song, Q. Duan, M. A. Neimark, M. J. Po, J. K. Gohagan, and A. F. Laine, “A review of imaging techniques for systems biology,” BMC Syst. Biol.2(1), 74 (2008).
[CrossRef] [PubMed]

Stone, A. D.

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(7), 834–837 (1988).
[CrossRef] [PubMed]

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,” Nature491(7423), 232–234 (2012).
[CrossRef] [PubMed]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett.106(19), 193905 (2011).
[CrossRef] [PubMed]

Vellekoop, I. M.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics4(5), 320–322 (2010).
[CrossRef]

I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun.281(11), 3071–3080 (2008).
[CrossRef]

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett.32(16), 2309–2311 (2007).
[CrossRef] [PubMed]

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,” Nature491(7423), 232–234 (2012).
[CrossRef] [PubMed]

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett.106(19), 193905 (2011).
[CrossRef] [PubMed]

Yang, T. D.

Y. Choi, T. D. Yang, C. Fang-Yen, P. Kang, K. J. Lee, R. R. Dasari, M. S. Feld, and W. Choi, “Overcoming the diffraction limit using multiple light scattering in a highly disordered medium,” Phys. Rev. Lett.107(2), 023902 (2011).
[CrossRef] [PubMed]

Yang, X.

Zhou, J.

BMC Syst. Biol.

A. R. Kherlopian, T. Song, Q. Duan, M. A. Neimark, M. J. Po, J. K. Gohagan, and A. F. Laine, “A review of imaging techniques for systems biology,” BMC Syst. Biol.2(1), 74 (2008).
[CrossRef] [PubMed]

Nat Commun

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat Commun1(6), 81 (2010).
[CrossRef] [PubMed]

Nat. Photonics

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photonics6(8), 549–553 (2012).
[CrossRef]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics4(5), 320–322 (2010).
[CrossRef]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics5(6), 372–377 (2011).
[CrossRef]

Nature

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature491(7423), 232–234 (2012).
[CrossRef] [PubMed]

Opt. Commun.

I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun.281(11), 3071–3080 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett.106(19), 193905 (2011).
[CrossRef] [PubMed]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett.61(20), 2328–2331 (1988).
[CrossRef] [PubMed]

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(7), 834–837 (1988).
[CrossRef] [PubMed]

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(10), 100601 (2010).
[CrossRef] [PubMed]

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[CrossRef]

Other

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Figures (4)

Fig. 1
Fig. 1

Experimental setup for scattered light correction based on a reference object. The back surface of the scattering medium is precisely imaged on the SLM. The focal length of lens 1, lens 2 and lens 3 is 45 mm, 130 mm and 75 mm, respectively.

Fig. 2
Fig. 2

Optimization process and results. (a) Constructed image of the reference object: a letter “S” with the size of about 1 mm. (b) Before correction, the reference object “S” is directly imaged through scattering system (with a flat pattern on SLM). (c) The evolution of correlation coefficient in the optimization process. (d) The optimized result. Inset: the optimized phase pattern (0~2π). (e),(f)Images of replacing the letter “S” with letters “O” and “E”, respectively. Scale bars, 1mm.

Fig. 3
Fig. 3

Pinhole tracking experiment. (a) Images of a pinhole shifting laterally on the object plane (green), overlapped by the optimized image of the reference letter “S” (red). (b) Normalized intensity of images of the pinhole (blue line). The red line indicates the intensity that drops to 1/2.

Fig. 4
Fig. 4

Experiment for FOV enlargement. (a) Experimental setup of scanning mechanism. The parts (not shown here) after the scattering medium are the same as those in Fig. 1. (b) Tested object placed on the object plane. (c) Optimization result of the reference object. (d) Image of directly illuminating the whole tested object. (e) Image of tilting the incident light to illuminate the letter “O”. (f) Image of tilting the incident light to illuminate the letter “A”. (g) Combination of the figures (c), (e) and (f). Scale bars, 1mm.

Equations (2)

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γ= s=1 P t=1 Q [ f( s,t ) f ¯ ][ w( s,t ) w ¯ ] { s=1 P t=1 Q [ f( s,t ) f ¯ ] 2 s=1 P t=1 Q [ w( s,t ) w ¯ ] 2 } 1 2 .
C( qL )= [ qL / sinh( qL ) ] 2 ,

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