Abstract

We present a new wave-vector-space approach for solving inverse scattering problems. In our formulation, the theories for diffraction tomography and coherence gating are combined to explain 3D reconstruction with low-coherence light. Specifically, we apply this method to solve the scattering problem with broadband fields for transmission and reflection measurements. Our results can be applied to any interferometric measurements with low-coherence light, including optical coherence tomography, angle-resolved low-coherence interferometry, and white-light diffraction tomography.

© 2014 Optical Society of America

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  1. W. L. Bragg, Proc. R. Soc. A 89, 248 (1913).
    [CrossRef]
  2. J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics (Wiley, 2001).
  3. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2001).
  4. E. Wolf, Advances in Imaging and Electron Physics, P. W. E. Hawkes, ed. (Academic, 2011).
  5. D. Gabor, Nature 161, 777 (1948).
    [CrossRef]
  6. E. Wolf, Opt. Commun. 1, 153 (1969).
    [CrossRef]
  7. R. Fiolka, K. Wicker, R. Heintzmann, and A. Stemmer, Opt. Express 17, 12407 (2009).
    [CrossRef]
  8. O. Haeberle, K. Belkebir, H. Giovaninni, and A. Sentenac, J. Mod. Opt. 57, 686 (2010).
    [CrossRef]
  9. W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, Nat. Methods 4, 717 (2007).
    [CrossRef]
  10. Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, PloS ONE 7, e49502 (2012).
    [CrossRef]
  11. Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, Nat. Photonics 7, 113 (2013).
    [CrossRef]
  12. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
    [CrossRef]
  13. J. A. Izatt, M. D. Kulkami, S. Yazdanfar, J. K. Barton, and A. J. Welch, Opt. Lett. 22, 1439 (1997).
    [CrossRef]
  14. A. F. Fercher, J. Biomed. Opt. 1, 157 (1996).
    [CrossRef]
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    [CrossRef]
  16. T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, Nat. Phys. 3, 129 (2007).
    [CrossRef]
  17. M. Villiger and T. Lasser, J. Opt. Soc. Am. A 27, 2216 (2010).
    [CrossRef]
  18. T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, Nat. Photonics 8, 256 (2014).
    [CrossRef]
  19. Z. Wang, L. J. Millet, M. Mir, H. Ding, S. Unarunotai, J. A. Rogers, M. U. Gillette, and G. Popescu, Opt. Express 19, 1016 (2011).
    [CrossRef]
  20. G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, Opt. Lett. 31, 775 (2006).
    [CrossRef]
  21. B. Bhaduri, C. Edwards, H. Pham, R. Zhou, G. Popescu, and L. L. Goddard, Adv. Opt. Photon. 6, 57 (2014).
    [CrossRef]
  22. G. Popescu, Quantitative Phase Imaging of Cells and Tissues (McGraw-Hill, 2011).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  27. R. Zhu, S. Sridharan, K. Tangella, A. Balla, and G. Popescu, Opt. Lett. 36, 4209 (2011).
    [CrossRef]

2014 (2)

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, Nat. Photonics 8, 256 (2014).
[CrossRef]

B. Bhaduri, C. Edwards, H. Pham, R. Zhou, G. Popescu, and L. L. Goddard, Adv. Opt. Photon. 6, 57 (2014).
[CrossRef]

2013 (2)

T. Kim, R. Zhu, T. H. Nguyen, R. Zhou, C. Edwards, L. L. Goddard, and G. Popescu, Opt. Express 21, 20806 (2013).
[CrossRef]

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, Nat. Photonics 7, 113 (2013).
[CrossRef]

2012 (1)

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, PloS ONE 7, e49502 (2012).
[CrossRef]

2011 (2)

2010 (2)

M. Villiger and T. Lasser, J. Opt. Soc. Am. A 27, 2216 (2010).
[CrossRef]

O. Haeberle, K. Belkebir, H. Giovaninni, and A. Sentenac, J. Mod. Opt. 57, 686 (2010).
[CrossRef]

2009 (1)

2007 (2)

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, Nat. Methods 4, 717 (2007).
[CrossRef]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, Nat. Phys. 3, 129 (2007).
[CrossRef]

2006 (3)

2004 (1)

2002 (1)

1997 (1)

1996 (1)

A. F. Fercher, J. Biomed. Opt. 1, 157 (1996).
[CrossRef]

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef]

1969 (1)

E. Wolf, Opt. Commun. 1, 153 (1969).
[CrossRef]

1948 (1)

D. Gabor, Nature 161, 777 (1948).
[CrossRef]

1913 (1)

W. L. Bragg, Proc. R. Soc. A 89, 248 (1913).
[CrossRef]

Als-Nielsen, J.

J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics (Wiley, 2001).

Babacan, S. D.

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, Nat. Photonics 8, 256 (2014).
[CrossRef]

Backman, V.

Badizadegan, K.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, Nat. Methods 4, 717 (2007).
[CrossRef]

Balla, A.

Barton, J. K.

Belkebir, K.

O. Haeberle, K. Belkebir, H. Giovaninni, and A. Sentenac, J. Mod. Opt. 57, 686 (2010).
[CrossRef]

Bhaduri, B.

Boppart, S. A.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, Nat. Phys. 3, 129 (2007).
[CrossRef]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, J. Opt. Soc. Am. A 23, 1027 (2006).
[CrossRef]

Boss, D.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, Nat. Photonics 7, 113 (2013).
[CrossRef]

Bragg, W. L.

W. L. Bragg, Proc. R. Soc. A 89, 248 (1913).
[CrossRef]

Carney, P. S.

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, Nat. Photonics 8, 256 (2014).
[CrossRef]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, Nat. Phys. 3, 129 (2007).
[CrossRef]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, J. Opt. Soc. Am. A 23, 1027 (2006).
[CrossRef]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef]

Choi, W.

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, PloS ONE 7, e49502 (2012).
[CrossRef]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, Nat. Methods 4, 717 (2007).
[CrossRef]

Cotte, Y.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, Nat. Photonics 7, 113 (2013).
[CrossRef]

Dasari, R. R.

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, PloS ONE 7, e49502 (2012).
[CrossRef]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, Nat. Methods 4, 717 (2007).
[CrossRef]

G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, Opt. Lett. 31, 775 (2006).
[CrossRef]

A. Wax, C. H. Yang, V. Backman, M. Kalashnikov, R. R. Dasari, and M. S. Feld, J. Opt. Soc. Am. A 19, 737 (2002).
[CrossRef]

Depeursinge, C.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, Nat. Photonics 7, 113 (2013).
[CrossRef]

Ding, H.

Edwards, C.

Fang-Yen, C.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, Nat. Methods 4, 717 (2007).
[CrossRef]

Feld, M. S.

Fercher, A. F.

A. F. Fercher, J. Biomed. Opt. 1, 157 (1996).
[CrossRef]

Fiolka, R.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef]

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef]

Gabor, D.

D. Gabor, Nature 161, 777 (1948).
[CrossRef]

Gillette, M. U.

Giovaninni, H.

O. Haeberle, K. Belkebir, H. Giovaninni, and A. Sentenac, J. Mod. Opt. 57, 686 (2010).
[CrossRef]

Goddard, L. L.

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef]

Haeberle, O.

O. Haeberle, K. Belkebir, H. Giovaninni, and A. Sentenac, J. Mod. Opt. 57, 686 (2010).
[CrossRef]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef]

Heintzmann, R.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef]

Ikeda, T.

Itoh, M.

Izatt, J. A.

Jourdain, P.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, Nat. Photonics 7, 113 (2013).
[CrossRef]

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2001).

Kalashnikov, M.

Kim, T.

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, Nat. Photonics 8, 256 (2014).
[CrossRef]

T. Kim, R. Zhu, T. H. Nguyen, R. Zhou, C. Edwards, L. L. Goddard, and G. Popescu, Opt. Express 21, 20806 (2013).
[CrossRef]

Kulkami, M. D.

Lasser, T.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef]

Lue, N.

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, PloS ONE 7, e49502 (2012).
[CrossRef]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, Nat. Methods 4, 717 (2007).
[CrossRef]

Magistretti, P.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, Nat. Photonics 7, 113 (2013).
[CrossRef]

Makita, S.

Marks, D. L.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, Nat. Phys. 3, 129 (2007).
[CrossRef]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, J. Opt. Soc. Am. A 23, 1027 (2006).
[CrossRef]

Marquet, P.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, Nat. Photonics 7, 113 (2013).
[CrossRef]

McMorrow, D.

J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics (Wiley, 2001).

Millet, L. J.

Mir, M.

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, Nat. Photonics 8, 256 (2014).
[CrossRef]

Z. Wang, L. J. Millet, M. Mir, H. Ding, S. Unarunotai, J. A. Rogers, M. U. Gillette, and G. Popescu, Opt. Express 19, 1016 (2011).
[CrossRef]

Nakamura, Y.

Nguyen, T. H.

Oh, S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, Nat. Methods 4, 717 (2007).
[CrossRef]

Pavillon, N.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, Nat. Photonics 7, 113 (2013).
[CrossRef]

Pham, H.

Popescu, G.

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef]

Pyhtila, J. W.

Ralston, T. S.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, Nat. Phys. 3, 129 (2007).
[CrossRef]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, J. Opt. Soc. Am. A 23, 1027 (2006).
[CrossRef]

Rogers, J. A.

Sando, Y.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef]

Sentenac, A.

O. Haeberle, K. Belkebir, H. Giovaninni, and A. Sentenac, J. Mod. Opt. 57, 686 (2010).
[CrossRef]

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2001).

Sridharan, S.

Stemmer, A.

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef]

Sugisaka, J.

Sung, Y.

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, PloS ONE 7, e49502 (2012).
[CrossRef]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef]

Tangella, K.

Toy, F.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, Nat. Photonics 7, 113 (2013).
[CrossRef]

Unarunotai, S.

Villiger, M.

Wang, Z.

Wax, A.

Welch, A. J.

Wicker, K.

Wolf, E.

E. Wolf, Opt. Commun. 1, 153 (1969).
[CrossRef]

E. Wolf, Advances in Imaging and Electron Physics, P. W. E. Hawkes, ed. (Academic, 2011).

Yang, C. H.

Yaqoob, Z.

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, PloS ONE 7, e49502 (2012).
[CrossRef]

Yasuno, Y.

Yatagai, T.

Yazdanfar, S.

Zhou, R.

Zhu, R.

Adv. Opt. Photon. (1)

J. Biomed. Opt. (1)

A. F. Fercher, J. Biomed. Opt. 1, 157 (1996).
[CrossRef]

J. Mod. Opt. (1)

O. Haeberle, K. Belkebir, H. Giovaninni, and A. Sentenac, J. Mod. Opt. 57, 686 (2010).
[CrossRef]

J. Opt. Soc. Am. A (3)

Nat. Methods (1)

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, Nat. Methods 4, 717 (2007).
[CrossRef]

Nat. Photonics (2)

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, Nat. Photonics 8, 256 (2014).
[CrossRef]

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, Nat. Photonics 7, 113 (2013).
[CrossRef]

Nat. Phys. (1)

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, Nat. Phys. 3, 129 (2007).
[CrossRef]

Nature (1)

D. Gabor, Nature 161, 777 (1948).
[CrossRef]

Opt. Commun. (1)

E. Wolf, Opt. Commun. 1, 153 (1969).
[CrossRef]

Opt. Express (5)

Opt. Lett. (3)

PloS ONE (1)

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, PloS ONE 7, e49502 (2012).
[CrossRef]

Proc. R. Soc. A (1)

W. L. Bragg, Proc. R. Soc. A 89, 248 (1913).
[CrossRef]

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef]

Other (4)

J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics (Wiley, 2001).

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2001).

E. Wolf, Advances in Imaging and Electron Physics, P. W. E. Hawkes, ed. (Academic, 2011).

G. Popescu, Quantitative Phase Imaging of Cells and Tissues (McGraw-Hill, 2011).

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Figures (2)

Fig. 1.
Fig. 1.

Schematic of the scattering problem.

Fig. 2.
Fig. 2.

Interpretation of sectioning effects on the Ewald sphere. (a) Forward scattering wave-vector relations. (b) Backward scattering wave-vector relations.

Equations (25)

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

2U(r,ω)+n2(r)β02(ω)U(r,ω)=0,
2Us(r,ω)+β2(ω)Us(r,ω)=βo2(ω)χ(r)U(r,ω),
Us(k,ω)=β02(ω)A(ω)χ[k,kzβ(ω)]12q(1kzq1kz+q),
Us(k,z,ω)=βo2(ω)2A(ω){χ(k,z)eiβ(ω)zze±iqzq},
Uf(k,z,ω)=βo2(ω)A(ω)eiqz2qχ[k,qβ(ω)],
Ub(k,z,ω)=βo2(ω)A(ω)eiqz2qχ[k,qβ(ω)],
W(k,z,ω)=Us(k,z,ω)Ur*(zR,ω),
χ(k,qβ)=2qWf(k,z,β)βo2S(β)ei(qzβ·zR),
χ(k,qβ)=2qWb(k,z,β)βo2S(β)ei(qz+β·zR),
q=β2k2β(1k22β2),thus,β2/qβ(1+k22β2).
χ(k,k22β)2n¯2Wf(k,z,β)βS(β)eiβ(zzR)eik2z2β,
χ(k,k22β2β)2n¯2Wb(k,z,β)βS(β)eiβ(z+zR)eik2z2β.
k2=|β+kz||βkz|2β|βkz|,
|βkz|=k22β.
Wf(k,z,β)12n¯2βS(β)χ(k,k22β)eik2z2β.
S(β)=S(Ω)dΩdβ=S(Ω)k22β2=2Ω2S(Ω)k2.
Wf(k,z,Ω)12n¯2ΩS(Ω)χ(k,Ω)eiΩz.
Γf(k,z,cτ)=i2n¯2Γ0(k,cτ)(cτ)cτχ(k,cτz),
Γ0(k,cτ)=FT1{S(Ω)}=|Γ0(k,cτ)|eiΩ¯cτ.
Γ0(k,cτ)(cτ)=iΩ¯Γ0(k,cτ)+|Γ0(cτ)|(cτ)eiΩ¯cτiΩ¯Γ0(k,cτ).
Γf(k,z,cτ)i2n¯2Ω¯Γ0(k,cτ)cτχ(k,cτz).
Wb(k,z,β)12n¯2βS(β)χ(k,k22β2β)ei(k22β2β)z.
S(β)=S(Q)dQdβ=S(Q)4Q2+4k2QQ2+4k2,
Wb(k,z,β)2n¯2Q2+4k2S(Q)χ(k,Q)eiQz.
Wb(k,z,β)2n¯2QS(Q)χ(k,Q)eiQz.

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