Abstract

The optical memory effect is an interesting phenomenon that has attracted considerable attention in recent decades. Here, we present a new physical picture of the optical memory effect, in which the memory effect and the conventional spatial shift invariance are united. Based on this picture we depict the role of thickness, scattering times, and anisotropy factor and derive equations to calculate the ranges of the angular memory effect (AME) of different scattering components (ballistic light, singly scattered, doubly scattered, etc.), and hence a more accurate equation for the real AME ranges of volumetric turbid media. A conventional random phase mask model is modified according to the new picture. The self-consistency of the simulation model and its agreement with the experiment demonstrate the rationality of the model and the physical picture, which provide powerful tools for more sophisticated studies of the memory-effect-related phenomena and wavefront-sensitive techniques, such as wavefront shaping, optical phase conjugation, and optical trapping in/through scattering media.

© 2019 Chinese Laser Press

Full Article  |  PDF Article
OSA Recommended Articles
Modeling optical memory effects with phase screens

Malchiel Haskel and Adrian Stern
Opt. Express 26(22) 29231-29243 (2018)

Generalized optical memory effect

Gerwin Osnabrugge, Roarke Horstmeyer, Ioannis N. Papadopoulos, Benjamin Judkewitz, and Ivo M. Vellekoop
Optica 4(8) 886-892 (2017)

Characterization of the angular memory effect of scattered light in biological tissues

Sam Schott, Jacopo Bertolotti, Jean-Francois Léger, Laurent Bourdieu, and Sylvain Gigan
Opt. Express 23(10) 13505-13516 (2015)

References

  • View by:
  • |
  • |
  • |

  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]
  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]
  3. 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]
  4. 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]
  5. H. Zhuang, H. He, X. Xie, and J. Zhou, “High speed color imaging through scattering media with a large field of view,” Sci. Rep. 6, 032696 (2016).
    [Crossref]
  6. E. Edrei and G. Scarcelli, “Memory-effect based deconvolution microscopy for super-resolution imaging through scattering media,” Sci. Rep. 6, 33559 (2016).
    [Crossref]
  7. M. Cua, E. Zhou, and C. Yang, “Imaging moving targets through scattering media,” Opt. Express 25, 3935–3945 (2017).
    [Crossref]
  8. M. Qiao, H. Liu, G. Pang, and S. Han, “Non-invasive three-dimension control of light between turbid layers using a surface quasi-point light source for precorrection,” Sci. Rep. 7, 9792 (2017).
    [Crossref]
  9. L. Li, Q. Li, S. Sun, H. Lin, W. Liu, and P. Chen, “Imaging through scattering layers exceeding memory effect range with spatial-correlation-achieved point-spread-function,” Opt. Lett. 43, 1670–1673 (2018).
    [Crossref]
  10. C. Guo, J. Liu, T. Wu, L. Zhu, and X. Shao, “Tracking moving targets behind a scattering medium via speckle correlation,” Appl. Opt. 57, 905–913 (2018).
    [Crossref]
  11. W. Yang, G. Li, and G. Situ, “Imaging through scattering media with the auxiliary of a known reference object,” Sci. Rep. 8, 9614 (2018).
    [Crossref]
  12. Q. Chen, H. He, X. Xu, X. Xie, H. Zhuang, J. Ye, and Y. Guan, “Memory effect based filter to improve imaging quality through scattering layers,” IEEE Photon. J. 10, 6901010 (2018).
    [Crossref]
  13. S. Schott, J. Bertolotti, J. 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]
  14. 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]
  15. 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]
  16. G. Osnabrugge, R. Horstmeyer, I. N. Papadopoulos, B. Judkewitz, and I. M. Vellekoop, “Generalized optical memory effect,” Optica 4, 886–892 (2017).
    [Crossref]
  17. M. Qiao, H. Liu, and S. Han, “Bidirectional image transmission through physically thick scattering media using digital optical phase conjugation,” Opt. Express 26, 33066–33079 (2018).
    [Crossref]
  18. J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2015), p. 41.
  19. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).
  20. L. V. Wang and H.-I. Wu, Biomedical Optics: Principles and Imaging (Wiley, 2007).
  21. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
    [Crossref]
  22. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
    [Crossref]
  23. D. B. Conkey, A. M. Caravaca-Aguirre, and R. Piestun, “High-speed scattering medium characterization with application to focusing light through turbid media,” Opt. Express 20, 1733–1740 (2012).
    [Crossref]
  24. P. Lai, L. Wang, J. W. Tay, and L. V. Wang, “Photoacoustically guided wavefront shaping for enhanced optical focusing in scattering media,” Nat. Photonics 9, 126–132 (2015).
    [Crossref]
  25. D. Wang, E. H. Zhou, J. Brake, H. Ruan, M. Jang, and C. Yang, “Focusing through dynamic tissue with millisecond digital optical phase conjugation,” Optica 2, 728–735 (2015).
    [Crossref]
  26. C. Ma, X. Xu, Y. Liu, and L. V. Wang, “Time-reversed adapted-perturbation (TRAP) optical focusing onto dynamic objects inside scattering media,” Nat. Photonics 8, 931–936 (2014).
    [Crossref]
  27. T. R. Hillman, T. Yamauchi, W. Choi, R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci. Rep. 3, 1909 (2013).
    [Crossref]
  28. M.-C. Zhong, X.-B. Wei, J.-H. Zhou, Z.-Q. Wang, and Y.-M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. 4, 1768 (2013).
    [Crossref]
  29. J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. 92, 093903 (2004).
    [Crossref]
  30. S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
    [Crossref]
  31. Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, and T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
    [Crossref]
  32. Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
    [Crossref]
  33. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref]

2018 (5)

2017 (3)

2016 (3)

H. Zhuang, H. He, X. Xie, and J. Zhou, “High speed color imaging through scattering media with a large field of view,” Sci. Rep. 6, 032696 (2016).
[Crossref]

E. Edrei and G. Scarcelli, “Memory-effect based deconvolution microscopy for super-resolution imaging through scattering media,” Sci. Rep. 6, 33559 (2016).
[Crossref]

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
[Crossref]

2015 (4)

P. Lai, L. Wang, J. W. Tay, and L. V. Wang, “Photoacoustically guided wavefront shaping for enhanced optical focusing in scattering media,” Nat. Photonics 9, 126–132 (2015).
[Crossref]

D. Wang, E. H. Zhou, J. Brake, H. Ruan, M. Jang, and C. Yang, “Focusing through dynamic tissue with millisecond digital optical phase conjugation,” Optica 2, 728–735 (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]

S. Schott, J. Bertolotti, J. 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]

2014 (3)

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]

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. Ma, X. Xu, Y. Liu, and L. V. Wang, “Time-reversed adapted-perturbation (TRAP) optical focusing onto dynamic objects inside scattering media,” Nat. Photonics 8, 931–936 (2014).
[Crossref]

2013 (2)

T. R. Hillman, T. Yamauchi, W. Choi, R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci. Rep. 3, 1909 (2013).
[Crossref]

M.-C. Zhong, X.-B. Wei, J.-H. Zhou, Z.-Q. Wang, and Y.-M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. 4, 1768 (2013).
[Crossref]

2012 (2)

D. B. Conkey, A. M. Caravaca-Aguirre, and R. Piestun, “High-speed scattering medium characterization with application to focusing light through turbid media,” Opt. Express 20, 1733–1740 (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,” Nature 491, 232–234 (2012).
[Crossref]

2007 (1)

2004 (1)

J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. 92, 093903 (2004).
[Crossref]

1998 (1)

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, and T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[Crossref]

1997 (1)

1988 (3)

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]

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[Crossref]

1982 (1)

Bertolotti, J.

S. Schott, J. Bertolotti, J. 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]

Block, U.

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, and T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[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,” Nature 491, 232–234 (2012).
[Crossref]

Bourdieu, L.

S. Schott, J. Bertolotti, J. 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]

Brake, J.

Caravaca-Aguirre, A. M.

Chen, P.

Chen, Q.

Q. Chen, H. He, X. Xu, X. Xie, H. Zhuang, J. Ye, and Y. Guan, “Memory effect based filter to improve imaging quality through scattering layers,” IEEE Photon. J. 10, 6901010 (2018).
[Crossref]

Cheng, C.-F.

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, and T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[Crossref]

Cheng, J.

J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. 92, 093903 (2004).
[Crossref]

Choi, W.

T. R. Hillman, T. Yamauchi, W. Choi, R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci. Rep. 3, 1909 (2013).
[Crossref]

Conkey, D. B.

Cua, M.

Dasari, R.

T. R. Hillman, T. Yamauchi, W. Choi, R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci. Rep. 3, 1909 (2013).
[Crossref]

Edrei, E.

E. Edrei and G. Scarcelli, “Memory-effect based deconvolution microscopy for super-resolution imaging through scattering media,” Sci. Rep. 6, 33559 (2016).
[Crossref]

Feld, M. S.

T. R. Hillman, T. Yamauchi, W. Choi, R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci. Rep. 3, 1909 (2013).
[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]

Fienup, J. R.

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]

Garoff, S.

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[Crossref]

Gigan, S.

S. Schott, J. Bertolotti, J. 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]

Goodman, J. W.

J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2015), p. 41.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

Guan, Y.

Q. Chen, H. He, X. Xu, X. Xie, H. Zhuang, J. Ye, and Y. Guan, “Memory effect based filter to improve imaging quality through scattering layers,” IEEE Photon. J. 10, 6901010 (2018).
[Crossref]

Guo, C.

Han, S.

M. Qiao, H. Liu, and S. Han, “Bidirectional image transmission through physically thick scattering media using digital optical phase conjugation,” Opt. Express 26, 33066–33079 (2018).
[Crossref]

M. Qiao, H. Liu, G. Pang, and S. Han, “Non-invasive three-dimension control of light between turbid layers using a surface quasi-point light source for precorrection,” Sci. Rep. 7, 9792 (2017).
[Crossref]

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
[Crossref]

J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. 92, 093903 (2004).
[Crossref]

He, H.

Q. Chen, H. He, X. Xu, X. Xie, H. Zhuang, J. Ye, and Y. Guan, “Memory effect based filter to improve imaging quality through scattering layers,” IEEE Photon. J. 10, 6901010 (2018).
[Crossref]

H. Zhuang, H. He, X. Xie, and J. Zhou, “High speed color imaging through scattering media with a large field of view,” Sci. Rep. 6, 032696 (2016).
[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]

Hillman, T. R.

T. R. Hillman, T. Yamauchi, W. Choi, R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci. Rep. 3, 1909 (2013).
[Crossref]

Horstmeyer, R.

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]

Jang, M.

Judkewitz, B.

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]

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]

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]

Lai, P.

P. Lai, L. Wang, J. W. Tay, and L. V. Wang, “Photoacoustically guided wavefront shaping for enhanced optical focusing in scattering media,” Nat. Photonics 9, 126–132 (2015).
[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.

S. Schott, J. Bertolotti, J. 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]

Li, E.

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
[Crossref]

Li, G.

W. Yang, G. Li, and G. Situ, “Imaging through scattering media with the auxiliary of a known reference object,” Sci. Rep. 8, 9614 (2018).
[Crossref]

Li, L.

Li, Q.

Li, Y.-M.

M.-C. Zhong, X.-B. Wei, J.-H. Zhou, Z.-Q. Wang, and Y.-M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. 4, 1768 (2013).
[Crossref]

Lin, H.

Liu, H.

M. Qiao, H. Liu, and S. Han, “Bidirectional image transmission through physically thick scattering media using digital optical phase conjugation,” Opt. Express 26, 33066–33079 (2018).
[Crossref]

M. Qiao, H. Liu, G. Pang, and S. Han, “Non-invasive three-dimension control of light between turbid layers using a surface quasi-point light source for precorrection,” Sci. Rep. 7, 9792 (2017).
[Crossref]

Liu, J.

Liu, W.

Liu, Y.

C. Ma, X. Xu, Y. Liu, and L. V. Wang, “Time-reversed adapted-perturbation (TRAP) optical focusing onto dynamic objects inside scattering media,” Nat. Photonics 8, 931–936 (2014).
[Crossref]

Liu, Z.

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
[Crossref]

Lu, T.-M.

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, and T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[Crossref]

Ma, C.

C. Ma, X. Xu, Y. Liu, and L. V. Wang, “Time-reversed adapted-perturbation (TRAP) optical focusing onto dynamic objects inside scattering media,” Nat. Photonics 8, 931–936 (2014).
[Crossref]

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]

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

Osnabrugge, G.

Pang, G.

M. Qiao, H. Liu, G. Pang, and S. Han, “Non-invasive three-dimension control of light between turbid layers using a surface quasi-point light source for precorrection,” Sci. Rep. 7, 9792 (2017).
[Crossref]

Papadopoulos, I. N.

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]

Park, Y.

T. R. Hillman, T. Yamauchi, W. Choi, R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci. Rep. 3, 1909 (2013).
[Crossref]

Piestun, R.

Psaltis, D.

Pu, Y.

Qiao, M.

M. Qiao, H. Liu, and S. Han, “Bidirectional image transmission through physically thick scattering media using digital optical phase conjugation,” Opt. Express 26, 33066–33079 (2018).
[Crossref]

M. Qiao, H. Liu, G. Pang, and S. Han, “Non-invasive three-dimension control of light between turbid layers using a surface quasi-point light source for precorrection,” Sci. Rep. 7, 9792 (2017).
[Crossref]

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]

Ruan, H.

Scarcelli, G.

E. Edrei and G. Scarcelli, “Memory-effect based deconvolution microscopy for super-resolution imaging through scattering media,” Sci. Rep. 6, 33559 (2016).
[Crossref]

Schott, S.

S. Schott, J. Bertolotti, J. 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]

Shao, X.

Shen, X.

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
[Crossref]

Sinha, S. K.

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[Crossref]

Sirota, E. B.

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[Crossref]

Situ, G.

W. Yang, G. Li, and G. Situ, “Imaging through scattering media with the auxiliary of a known reference object,” Sci. Rep. 8, 9614 (2018).
[Crossref]

Stanley, H. B.

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[Crossref]

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, 834–837 (1988).
[Crossref]

Sun, S.

Tan, S.

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
[Crossref]

Tay, J. W.

P. Lai, L. Wang, J. W. Tay, and L. V. Wang, “Photoacoustically guided wavefront shaping for enhanced optical focusing in scattering media,” Nat. Photonics 9, 126–132 (2015).
[Crossref]

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.

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]

Wang, D.

Wang, G.-C.

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, and T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[Crossref]

Wang, L.

P. Lai, L. Wang, J. W. Tay, and L. V. Wang, “Photoacoustically guided wavefront shaping for enhanced optical focusing in scattering media,” Nat. Photonics 9, 126–132 (2015).
[Crossref]

Wang, L. V.

P. Lai, L. Wang, J. W. Tay, and L. V. Wang, “Photoacoustically guided wavefront shaping for enhanced optical focusing in scattering media,” Nat. Photonics 9, 126–132 (2015).
[Crossref]

C. Ma, X. Xu, Y. Liu, and L. V. Wang, “Time-reversed adapted-perturbation (TRAP) optical focusing onto dynamic objects inside scattering media,” Nat. Photonics 8, 931–936 (2014).
[Crossref]

L. V. Wang and H.-I. Wu, Biomedical Optics: Principles and Imaging (Wiley, 2007).

Wang, Z.-Q.

M.-C. Zhong, X.-B. Wei, J.-H. Zhou, Z.-Q. Wang, and Y.-M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. 4, 1768 (2013).
[Crossref]

Wei, X.-B.

M.-C. Zhong, X.-B. Wei, J.-H. Zhou, Z.-Q. Wang, and Y.-M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. 4, 1768 (2013).
[Crossref]

Wu, H.-I.

L. V. Wang and H.-I. Wu, Biomedical Optics: Principles and Imaging (Wiley, 2007).

Wu, I.

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, and T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[Crossref]

Wu, J.

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
[Crossref]

Wu, T.

Xie, X.

Q. Chen, H. He, X. Xu, X. Xie, H. Zhuang, J. Ye, and Y. Guan, “Memory effect based filter to improve imaging quality through scattering layers,” IEEE Photon. J. 10, 6901010 (2018).
[Crossref]

H. Zhuang, H. He, X. Xie, and J. Zhou, “High speed color imaging through scattering media with a large field of view,” Sci. Rep. 6, 032696 (2016).
[Crossref]

Xu, X.

Q. Chen, H. He, X. Xu, X. Xie, H. Zhuang, J. Ye, and Y. Guan, “Memory effect based filter to improve imaging quality through scattering layers,” IEEE Photon. J. 10, 6901010 (2018).
[Crossref]

C. Ma, X. Xu, Y. Liu, and L. V. Wang, “Time-reversed adapted-perturbation (TRAP) optical focusing onto dynamic objects inside scattering media,” Nat. Photonics 8, 931–936 (2014).
[Crossref]

Yamaguchi, I.

Yamauchi, T.

T. R. Hillman, T. Yamauchi, W. Choi, R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci. Rep. 3, 1909 (2013).
[Crossref]

Yang, C.

Yang, W.

W. Yang, G. Li, and G. Situ, “Imaging through scattering media with the auxiliary of a known reference object,” Sci. Rep. 8, 9614 (2018).
[Crossref]

Yang, X.

Yaqoob, Z.

T. R. Hillman, T. Yamauchi, W. Choi, R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci. Rep. 3, 1909 (2013).
[Crossref]

Ye, J.

Q. Chen, H. He, X. Xu, X. Xie, H. Zhuang, J. Ye, and Y. Guan, “Memory effect based filter to improve imaging quality through scattering layers,” IEEE Photon. J. 10, 6901010 (2018).
[Crossref]

Zhang, T.

Zhao, Y.-P.

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, and T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[Crossref]

Zhong, M.-C.

M.-C. Zhong, X.-B. Wei, J.-H. Zhou, Z.-Q. Wang, and Y.-M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. 4, 1768 (2013).
[Crossref]

Zhou, E.

Zhou, E. H.

Zhou, J.

H. Zhuang, H. He, X. Xie, and J. Zhou, “High speed color imaging through scattering media with a large field of view,” Sci. Rep. 6, 032696 (2016).
[Crossref]

Zhou, J.-H.

M.-C. Zhong, X.-B. Wei, J.-H. Zhou, Z.-Q. Wang, and Y.-M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. 4, 1768 (2013).
[Crossref]

Zhu, L.

Zhuang, H.

Q. Chen, H. He, X. Xu, X. Xie, H. Zhuang, J. Ye, and Y. Guan, “Memory effect based filter to improve imaging quality through scattering layers,” IEEE Photon. J. 10, 6901010 (2018).
[Crossref]

H. Zhuang, H. He, X. Xie, and J. Zhou, “High speed color imaging through scattering media with a large field of view,” Sci. Rep. 6, 032696 (2016).
[Crossref]

Appl. Opt. (2)

IEEE Photon. J. (1)

Q. Chen, H. He, X. Xu, X. Xie, H. Zhuang, J. Ye, and Y. Guan, “Memory effect based filter to improve imaging quality through scattering layers,” IEEE Photon. J. 10, 6901010 (2018).
[Crossref]

J. Appl. Phys. (1)

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, and T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[Crossref]

Nat. Commun. (1)

M.-C. Zhong, X.-B. Wei, J.-H. Zhou, Z.-Q. Wang, and Y.-M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. 4, 1768 (2013).
[Crossref]

Nat. Photonics (3)

C. Ma, X. Xu, Y. Liu, and L. V. Wang, “Time-reversed adapted-perturbation (TRAP) optical focusing onto dynamic objects inside scattering media,” Nat. Photonics 8, 931–936 (2014).
[Crossref]

P. Lai, L. Wang, J. W. Tay, and L. V. Wang, “Photoacoustically guided wavefront shaping for enhanced optical focusing in scattering media,” Nat. Photonics 9, 126–132 (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]

Nat. Phys. (1)

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 (4)

Opt. Express. (1)

S. Schott, J. Bertolotti, J. 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]

Opt. Lett. (3)

Optica (2)

Phys. Rev. B (1)

S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[Crossref]

Phys. Rev. Lett. (3)

J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. 92, 093903 (2004).
[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).
[Crossref]

Sci. Rep. (6)

H. Zhuang, H. He, X. Xie, and J. Zhou, “High speed color imaging through scattering media with a large field of view,” Sci. Rep. 6, 032696 (2016).
[Crossref]

E. Edrei and G. Scarcelli, “Memory-effect based deconvolution microscopy for super-resolution imaging through scattering media,” Sci. Rep. 6, 33559 (2016).
[Crossref]

M. Qiao, H. Liu, G. Pang, and S. Han, “Non-invasive three-dimension control of light between turbid layers using a surface quasi-point light source for precorrection,” Sci. Rep. 7, 9792 (2017).
[Crossref]

W. Yang, G. Li, and G. Situ, “Imaging through scattering media with the auxiliary of a known reference object,” Sci. Rep. 8, 9614 (2018).
[Crossref]

T. R. Hillman, T. Yamauchi, W. Choi, R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci. Rep. 3, 1909 (2013).
[Crossref]

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
[Crossref]

Other (3)

J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2015), p. 41.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

L. V. Wang and H.-I. Wu, Biomedical Optics: Principles and Imaging (Wiley, 2007).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1. Comparison of laser light incident upon an aperture (System 1) and a random phase mask (System 2) with no thickness. (a) Tilting the laser beam an angle θ (a shift of dθ along the y axis correspondingly) results in the same amount of tilt (a shift dθ on the observation plane) in the output, as in a typical shift-invariant phenomenon. (b) Tilting the laser beam leads to a corresponding shift in the entire speckle pattern. The red and yellow arrows call attention to similar characteristics and their shifts.
Fig. 2.
Fig. 2. Comparison of light propagating through two screens in sequence. The second screen is (a) an aperture, referred to as System 3, and (b) a random phase mask, referred to as System 4.
Fig. 3.
Fig. 3. Simulation and experimental results. (a) Comparison of the normalized angular PSDs obtained from the MC simulation and Eq. (17) with different anisotropy factors g. As g decreases, the PSD broadens. Both curves peak at θ=0. The value of P(θ) is slightly smaller than that obtained from the MC simulation at larger angles, which might be due to the truncation at m=6 when calculating P(θ). (b) AME curves obtained from the experiment (magenta curve, square markers), simulations with the conventional model (green solid curve), and our new model (black solid curve). (c) A comparison of AME curves with different scattering components including only singly scattered light (green solid curve), i.e., the conventional model, all scattering components integrated (black solid curve), i.e., the sPSD is obtained from the standard MC simulation, ballistic and the first six scattering components (magenta solid curve), and ballistic and only singly scattered light (blue solid curve).
Fig. 4.
Fig. 4. Workflow for generating the real-domain phase-only masks using the G-S algorithm. M1(fx,fy) and M2(fx,fy) are the Fourier-domain expressions of the mask. M1(x,y) and M2(x,y) are the real-domain expressions of the mask. The algorithm starts from M1(fx,fy) with an initial value of I0exp(iϑ0), where I0 is the envelope of its amplitude and ϑ0 is a random phase distribution. After several iterations, the algorithm will converge; then, M2(x,y) is the desired phase-only mask.
Fig. 5.
Fig. 5. Comparison between the conventional phase mask model and our new phase mask model for a scattering medium with g=0.95. (a) Phase map of the conventional mask. (b) and (c) Phase maps of our new masks when the interval d is set to be the MFP and 0.5 times the MFP, respectively. (d) Phase distributions of the three masks. Compared to the conventional mask in (a), which has a full 2π phase modulation depth, the new masks in (b) and (c) have shallower modulation depths, which allow more ballistic light to pass through. When the interval between adjacent masks is smaller, the modulation depth will be shallower. (e) Spatial frequency distributions of the three masks. The conventional mask has a very smooth spatial frequency distribution, but for the new masks, there is a sharp peak at zero frequency, which is caused by the delta function of the ballistic light. (f) and (g) Memory effect curves for 0.5 mm and 1 mm thick scattering media with an MFP of 0.1 mm and g=0.95 modelled by the conventional phase mask model (purple curve) and our new model (blue curve for d=MFP, orange curve for d=0.5×MFP).
Fig. 6.
Fig. 6. Principle of scanning a time-reversed focus. (a) A point source is placed in front of a scattering medium, and a wavefront of the transmitted light through the medium is recorded by the digital optical phase conjugation (DOPC) system. (b) The point source is removed, and a phase-conjugated wavefront is generated by the DOPC system to create a time-reversed focus at the original position of the point source. By adding a phase ramp to the phase-conjugated wavefront, we can scan the focus along a desired direction.

Equations (38)

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

U(x,y)=U0(x1,y1)A1(x1,y1)h1(xx1,yy1)dx1dy1,
A1(x1,y1)={1x1,y1Σ0otherwise,
A1(x1,y1)={exp[iφ(x1,y1)]x1,y1Σ0otherwise,
U(x,y)=U0(x1,y1)A1(x1,y1)h1(x2x1,y2y1)×A2(x2,y2)h2(xx2,yy2)dx1dy1dx2dy2,
U(fx,fy)={[U0(fx,fy)A1(fx,fy)]exp[ikd1(λfx)2(λfy)2]}A2(fx,fy)exp[ikl21(λfx)2(λfy)2],
U(fx,fy)=[U0(fx,fy)A1(fx,fy)]exp[ik(d+l2)1(λfx)2(λfy)2].
U(fx,fy)=[U0(fx,fy)A1(fx,fy)]exp[ikd1(λfx)2(λfy)2].
U(fx,fy)={{U0(fx,fy)exp[ikd1(λfx)2(λfy)2]}A2(fx,fy)}exp[ikl21(λfx)2(λfy)2].
C(2,2)(x,y,x,y)=I(x,y)I(x,y),
A2(x2,y2)A2*(x2,y2)=exp{[2π(n1)]2σ2λ2κ2[(x2x2)2+(y2y2)2]},
Δθ=λ2πd·κ2(n1)σ.
A2(x2,y2)A2*(x2,y2)=exp{[2π(n1)]2σ2λ2κ2(Δx22+Δy22)}.
S(fx,fy)=exp{[2π(n1)]2σ2λ2κ2(Δx22+Δy22)}exp[i2π(fxΔx2+fyΔy2)]dΔx2dΔy2=λκ2π(n1)σexp{λ2κ2[2(n1)]2σ2(fx2+fy2)},
P1(θ)=1g22(1+g22gcosθ)3/2.
Wm,d=1m!(μsd)mexp(μsd),m=0,1,2.
Pm(θ)=P1(θ)P1(θ)m,m=1,2,3.
P(θ)=mWm,dPm(θ).
S(θ,φ)=λκ2π(n1)σexp{κ2sin2θ[2(n1)]2σ2}.
λκ2π(n1)σexp{κ2sin2θ[2(n1)]2σ2}=nWd,nPn(θ).
λκ2π(n1)σexp{κ2sin2θ[2(n1)]2σ}=δ(θ),
λκ2π(n1)σexp{κ2sin2θ[2(n1)]2σ2}=e1δ(θ)+e11g22(1+g22gcosθ)3/2+e121g22(1+g22gcosθ)3/21g22(1+g22gcosθ)3/2+.
C(2,2)(x,y,x,y)=dx1dy1dx1dy1dx1dy1dx1dy1A1(x1,y1)A1*(x1,y1)A1(x1,y1)A1*(x1,y1)×U0(x1,y1)U0*(x1,y1)Uθ(x1,y1)Uθ*(x1,y1)dx2dy2dx2dy2dx2dy2dx2dy2h1(x2x1,y2y1)h1*(x2x1,y2y1)h1(x2x1,y2y1)h1*(x2x1,y2y1)A2(x2,y2)A2*(x2,y2)A2(x2,y2)A2*(x2,y2)h2(xx2,yy2)h2*(xx2,yy2)h2(xx2,yy2)h2*(xx2,yy2),
C(2,2)(x1,y1,x1,y1,x1,y1,x1,y1)=A1(x1,y1)A1*(x1,y1)A1(x1,y1)A1*(x1,y1)=C(1,1)(x1,y1,x1,y1)C(1,1)(x1,y1,x1,y1)+C(1,1)(x1,y1,x1,y1)C(1,1)(x1,y1,x1,y1),
C(2,2)(x2,y2,x2,y2,x2,y2,x2,y2)=A2(x2,y2)A2*(x2,y2)A2(x2,y2)A2*(x2,y2)=C(1,1)(x2,y2,x2,y2)C(1,1)(x2,y2,x2,y2)+C(1,1)(x2,y2,x2,y2)C(1,1)(x2,y2,x2,y2),
C(1,1)(x1,y1,x1,y1)=δ(x1x1,y1y1),
C(1,1)(x1,y1,x1,y1)=δ(x1x1,y1y1),
C(1,1)(x1,y1,x1,y1)=δ(x1x1,y1y1),
C(1,1)(x1,y1,x1,y1)=δ(x1x1,y1y1).
C(2,2)(x,y,x,y)=dx1dy1dx1dy1dx2dy2dx2dy2dx2dy2dx2dy2h1(x2x1,y2y1)h1*(x2x1,y2y1)h1(x2x1,y2y1)h1*(x2x1,y2y1)×A2(x2,y2)A2*(x2,y2)A2(x2,y2)A2*(x2,y2)h2(xx2,yy2)h2*(xx2,yy2)h2(xx2,yy2)h2*(xx2,yy2)+dx1dy1dx1dy1exp(iky1  sinθiky1sinθ)dx2dy2dx2dy2dx2dy2dx2dy2h1(x2x1,y2y1)h1*(x2x1,y2y1)×h1(x2x1,y2y1)h1*(x2x1,y2y1)×A2(x2,y2)A2*(x2,y2)A2(x2,y2)A2*(x2,y2)×h2(xx2,yy2)h2*(xx2,yy2)h2(xx2,yy2)h2*(xx2,yy2).
C(2,2)(x,y,x,y)=1λ8d4l24dx2dy2dx2dy2+1λ4d4dx2dy2dx2dy2exp[iksinθ(y2y2)]×A2(x2,y2)A2*(x2,y2)A2(x2,y2dsinθ)A2*(x2,y2dsinθ)h2(xx2,yy2)h2*(xx2,yy2)h2(xx2,yy2+dsinθ)h2*(xx2,yy2+dsinθ).
C(2,2)(x,y,x,y)=1λ8d4l24dx2dy2dx2dy2+δ(0,0)λ8d4l24dx2dy2+δ2(0,dsinθ)λ8d4l24|dx2dy2exp{ikl2[(xx)x2+(yydsinθl2sinθ)y2]}|2.
C(2,2)(x,y,x,y)=dx2dy2dx2dy2+δ(0,0)dx2dy2+δ2(0,dsinθ)|dx2dy2exp{ikl2[(xx)x2+(yydsinθl2sinθ)y2]}|2.
C(2,2)(x,y,x,y)=π2r4+δ(0,0)πr2+16r4δ2(0,dsinθ)sinc2[2rλl2(xx)]sinc2[2rλl2(yydsinθl2sinθ)].
ΔC(2,2)(x,y,x,y)=16r4δ2(0,dsinθ)sinc2[2rλl2(xx)]×sinc2[2rλl2(yydsinθl2sinθ)],
ΔC(2,2)(x,y,x,y)=16r4exp{2[2π(n1)]2σ2λ2κ2d2sin2θ}sinc2[2rλl2(xx)]×sinc2[2rλl2(yydsinθl2sinθ)].
2[2π(n1)]2σ2λ2κ2d2sinθ2=1.
22π(n1)σdθλκ=1.
Δθ=λπd·κ2(n1)σ.