Abstract

A new approach to optical diffraction tomography (ODT) based on intensity measurements is presented. By applying the Wolf transform directly to intensity measurements, we observed unexpected behavior in the 3D reconstruction of the sample. Such a reconstruction does not explicitly represent a quantitative measure of the refractive index of the sample; however, it contains interesting qualitative information. This 3D reconstruction exhibits edge enhancement and contrast enhancement for nanostructures compared with the conventional 3D refractive index reconstruction and thus could be used to localize nanoparticles such as lipids inside a biological sample.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2020 (2)

A. Matlock, A. Sentenac, P. C. Chaumet, J. Yi, and L. Tian, “Inverse scattering for reflection intensity phase microscopy,” Biomed. Opt. Express 11(2), 911–926 (2020).
[Crossref]

A. B. Ayoub, T.-A. Pham, J. Lim, M. Unser, and D. Psaltis, “A method for assessing the fidelity of optical diffraction tomography reconstruction methods using structured illumination,” Opt. Commun. 454, 124486 (2020).
[Crossref]

2019 (4)

S. Chowdhury, M. Chen, R. Eckert, D. Ren, F. Wu, N. Repina, and L. Waller, “High-resolution 3D refractive index microscopy of multiple-scattering samples from intensity images,” Optica 6(9), 1211–1219 (2019).
[Crossref]

J. Lim, A. B. Ayoub, E. E. Antoine, and D. Psaltis, “High-fidelity optical diffraction tomography of multiple scattering samples,” Light: Sci. Appl. 8(1), 82 (2019).
[Crossref]

A. Matlock and L. Tian, “High-throughput, volumetric quantitative phase imaging with multiplexed intensity diffraction tomography,” Biomed. Opt. Express 10(12), 6432–6448 (2019).
[Crossref]

J. Li, A. Matlock, Y. Li, Q. Chen, C. Zuo, and L. Tian, “High-speed in vitro intensity diffraction tomography,” Adv. Photonics 1, 33 (2019).
[Crossref]

2018 (3)

2016 (2)

R. Horstmeyer, J. Chung, X. Ou, G. Zheng, and C. Yang, “Diffraction tomography with Fourier ptychography,” Optica 3(8), 827–835 (2016).
[Crossref]

T. Kim, R. Zhou, L. L. Goddard, and G. Popescu, “Solving inverse scattering problems in biological samples by quantitative phase imaging,” Laser Photonics Rev. 10(1), 13–39 (2016).
[Crossref]

2015 (4)

2013 (1)

K. L. Cooper, S. Oh, Y. Sung, R. R. Dasari, M. W. Kirschner, and C. J. Tabin, “Multiple phases of chondrocyte enlargement underlie differences in skeletal proportions,” Nature 495(7441), 375–378 (2013).
[Crossref]

2012 (1)

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, “Stain-Free Quantification of Chromosomes in Live Cells Using Regularized Tomographic Phase Microscopy,” PLOS ONE 7(11), e49502 (2012).
[Crossref]

2010 (1)

O. Haeberlé, K. Belkebir, H. Giovaninni, and A. Sentenac, “Tomographic diffractive microscopy: basics, techniques and perspectives,” J. Mod. Opt. 57(9), 686–699 (2010).
[Crossref]

2009 (1)

2007 (1)

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref]

2006 (1)

2004 (2)

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102(1), 37–49 (2004).
[Crossref]

W. Zhou, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image Qualify Assessment: From Error Visibility to Structural Similarity,” IEEE Trans. on Image Process. 13(4), 600–612 (2004).
[Crossref]

1994 (1)

1993 (1)

1992 (1)

1984 (1)

M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of Imaging with First-Order Diffraction Tomography,” IEEE Trans. Microwave Theory Tech. 32(8), 860–874 (1984).
[Crossref]

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1(4), 153–156 (1969).
[Crossref]

1949 (1)

D. Gabor and W. L. Bragg, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. Lond. A 197(1051), 454–487 (1949).
[Crossref]

Antoine, E. E.

J. Lim, A. B. Ayoub, E. E. Antoine, and D. Psaltis, “High-fidelity optical diffraction tomography of multiple scattering samples,” Light: Sci. Appl. 8(1), 82 (2019).
[Crossref]

Ayoub, A. B.

A. B. Ayoub, T.-A. Pham, J. Lim, M. Unser, and D. Psaltis, “A method for assessing the fidelity of optical diffraction tomography reconstruction methods using structured illumination,” Opt. Commun. 454, 124486 (2020).
[Crossref]

J. Lim, A. B. Ayoub, E. E. Antoine, and D. Psaltis, “High-fidelity optical diffraction tomography of multiple scattering samples,” Light: Sci. Appl. 8(1), 82 (2019).
[Crossref]

Ayoub, A.B.

T. Thomsen, A.B. Ayoub, D. Psaltis, and H.A. Klok, “Fluorescence-based and Fluorescent label-free Characterization of Polymer Nanoparticle Decorated T cells,” Biomacromolecules (2020). Advance online publication. https://doi.org/10.1021/acs.biomac.0c00969

Badizadegan, K.

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17(1), 266–277 (2009).
[Crossref]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref]

Beleggia, M.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102(1), 37–49 (2004).
[Crossref]

Belkebir, K.

O. Haeberlé, K. Belkebir, H. Giovaninni, and A. Sentenac, “Tomographic diffractive microscopy: basics, techniques and perspectives,” J. Mod. Opt. 57(9), 686–699 (2010).
[Crossref]

Boufounos, P. T.

H. Liu, D. Liu, H. Mansour, P. T. Boufounos, L. Waller, and U. S. Kamilov, “SEAGLE: Sparsity-Driven Image Reconstruction Under Multiple Scattering,” IEEE Trans. Comput. Imaging 4(1), 73–86 (2018).
[Crossref]

Bovik, A. C.

W. Zhou, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image Qualify Assessment: From Error Visibility to Structural Similarity,” IEEE Trans. on Image Process. 13(4), 600–612 (2004).
[Crossref]

Bragg, W. L.

D. Gabor and W. L. Bragg, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. Lond. A 197(1051), 454–487 (1949).
[Crossref]

Charrière, F.

Chaumet, P. C.

Chen, M.

Chen, Q.

J. Li, A. Matlock, Y. Li, Q. Chen, C. Zuo, and L. Tian, “High-speed in vitro intensity diffraction tomography,” Adv. Photonics 1, 33 (2019).
[Crossref]

Choi, W.

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, “Stain-Free Quantification of Chromosomes in Live Cells Using Regularized Tomographic Phase Microscopy,” PLOS ONE 7(11), e49502 (2012).
[Crossref]

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17(1), 266–277 (2009).
[Crossref]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref]

Chowdhury, S.

Chung, J.

Colomb, T.

Cooper, K. L.

K. L. Cooper, S. Oh, Y. Sung, R. R. Dasari, M. W. Kirschner, and C. J. Tabin, “Multiple phases of chondrocyte enlargement underlie differences in skeletal proportions,” Nature 495(7441), 375–378 (2013).
[Crossref]

Cuche, E.

Dasari, R. R.

K. L. Cooper, S. Oh, Y. Sung, R. R. Dasari, M. W. Kirschner, and C. J. Tabin, “Multiple phases of chondrocyte enlargement underlie differences in skeletal proportions,” Nature 495(7441), 375–378 (2013).
[Crossref]

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, “Stain-Free Quantification of Chromosomes in Live Cells Using Regularized Tomographic Phase Microscopy,” PLOS ONE 7(11), e49502 (2012).
[Crossref]

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17(1), 266–277 (2009).
[Crossref]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref]

Depeursinge, C.

Devaney, A. J.

Dong, B.-Z.

Eckert, R.

Ersoy, O. K.

Fang-Yen, C.

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17(1), 266–277 (2009).
[Crossref]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref]

Feld, M. S.

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17(1), 266–277 (2009).
[Crossref]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref]

Gabor, D.

D. Gabor and W. L. Bragg, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. Lond. A 197(1051), 454–487 (1949).
[Crossref]

Giovaninni, H.

O. Haeberlé, K. Belkebir, H. Giovaninni, and A. Sentenac, “Tomographic diffractive microscopy: basics, techniques and perspectives,” J. Mod. Opt. 57(9), 686–699 (2010).
[Crossref]

Goddard, L. L.

T. Kim, R. Zhou, L. L. Goddard, and G. Popescu, “Solving inverse scattering problems in biological samples by quantitative phase imaging,” Laser Photonics Rev. 10(1), 13–39 (2016).
[Crossref]

Goy, A.

Gu, B.-Y.

Haeberlé, O.

O. Haeberlé, K. Belkebir, H. Giovaninni, and A. Sentenac, “Tomographic diffractive microscopy: basics, techniques and perspectives,” J. Mod. Opt. 57(9), 686–699 (2010).
[Crossref]

Horstmeyer, R.

Jin, K. H.

Kak, A. C.

M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of Imaging with First-Order Diffraction Tomography,” IEEE Trans. Microwave Theory Tech. 32(8), 860–874 (1984).
[Crossref]

Kamilov, U. S.

H. Liu, D. Liu, H. Mansour, P. T. Boufounos, L. Waller, and U. S. Kamilov, “SEAGLE: Sparsity-Driven Image Reconstruction Under Multiple Scattering,” IEEE Trans. Comput. Imaging 4(1), 73–86 (2018).
[Crossref]

U. S. Kamilov, I. N. Papadopoulos, M. H. Shoreh, A. Goy, C. Vonesch, M. Unser, and D. Psaltis, “Learning approach to optical tomography,” Optica 2(6), 517–522 (2015).
[Crossref]

Kim, K.

Kim, T.

T. Kim, R. Zhou, L. L. Goddard, and G. Popescu, “Solving inverse scattering problems in biological samples by quantitative phase imaging,” Laser Photonics Rev. 10(1), 13–39 (2016).
[Crossref]

Kirschner, M. W.

K. L. Cooper, S. Oh, Y. Sung, R. R. Dasari, M. W. Kirschner, and C. J. Tabin, “Multiple phases of chondrocyte enlargement underlie differences in skeletal proportions,” Nature 495(7441), 375–378 (2013).
[Crossref]

Klok, H.A.

T. Thomsen, A.B. Ayoub, D. Psaltis, and H.A. Klok, “Fluorescence-based and Fluorescent label-free Characterization of Polymer Nanoparticle Decorated T cells,” Biomacromolecules (2020). Advance online publication. https://doi.org/10.1021/acs.biomac.0c00969

Kuehn, J.

Larsen, L. E.

M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of Imaging with First-Order Diffraction Tomography,” IEEE Trans. Microwave Theory Tech. 32(8), 860–874 (1984).
[Crossref]

Lee, H.

Lee, K.

Lee, S.

Li, J.

J. Li, A. Matlock, Y. Li, Q. Chen, C. Zuo, and L. Tian, “High-speed in vitro intensity diffraction tomography,” Adv. Photonics 1, 33 (2019).
[Crossref]

Li, Y.

J. Li, A. Matlock, Y. Li, Q. Chen, C. Zuo, and L. Tian, “High-speed in vitro intensity diffraction tomography,” Adv. Photonics 1, 33 (2019).
[Crossref]

Lim, J.

A. B. Ayoub, T.-A. Pham, J. Lim, M. Unser, and D. Psaltis, “A method for assessing the fidelity of optical diffraction tomography reconstruction methods using structured illumination,” Opt. Commun. 454, 124486 (2020).
[Crossref]

J. Lim, A. B. Ayoub, E. E. Antoine, and D. Psaltis, “High-fidelity optical diffraction tomography of multiple scattering samples,” Light: Sci. Appl. 8(1), 82 (2019).
[Crossref]

T.-A. Pham, E. Soubies, A. Goy, J. Lim, F. Soulez, D. Psaltis, and M. Unser, “Versatile reconstruction framework for diffraction tomography with intensity measurements and multiple scattering,” Opt. Express 26(3), 2749–2763 (2018).
[Crossref]

J. Lim, K. Lee, K. H. Jin, S. Shin, S. Lee, Y. Park, and J. C. Ye, “Comparative study of iterative reconstruction algorithms for missing cone problems in optical diffraction tomography,” Opt. Express 23(13), 16933–16948 (2015).
[Crossref]

Lin, H.-Y.

Ling, R.

Liu, D.

H. Liu, D. Liu, H. Mansour, P. T. Boufounos, L. Waller, and U. S. Kamilov, “SEAGLE: Sparsity-Driven Image Reconstruction Under Multiple Scattering,” IEEE Trans. Comput. Imaging 4(1), 73–86 (2018).
[Crossref]

Liu, H.

H. Liu, D. Liu, H. Mansour, P. T. Boufounos, L. Waller, and U. S. Kamilov, “SEAGLE: Sparsity-Driven Image Reconstruction Under Multiple Scattering,” IEEE Trans. Comput. Imaging 4(1), 73–86 (2018).
[Crossref]

Lue, N.

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, “Stain-Free Quantification of Chromosomes in Live Cells Using Regularized Tomographic Phase Microscopy,” PLOS ONE 7(11), e49502 (2012).
[Crossref]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref]

Maleki, M. H.

Mansour, H.

H. Liu, D. Liu, H. Mansour, P. T. Boufounos, L. Waller, and U. S. Kamilov, “SEAGLE: Sparsity-Driven Image Reconstruction Under Multiple Scattering,” IEEE Trans. Comput. Imaging 4(1), 73–86 (2018).
[Crossref]

Marian, A.

Marquet, P.

Matlock, A.

Montfort, F.

Oh, S.

K. L. Cooper, S. Oh, Y. Sung, R. R. Dasari, M. W. Kirschner, and C. J. Tabin, “Multiple phases of chondrocyte enlargement underlie differences in skeletal proportions,” Nature 495(7441), 375–378 (2013).
[Crossref]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref]

Ou, X.

Papadopoulos, I. N.

Park, Y.

Pham, T.-A.

A. B. Ayoub, T.-A. Pham, J. Lim, M. Unser, and D. Psaltis, “A method for assessing the fidelity of optical diffraction tomography reconstruction methods using structured illumination,” Opt. Commun. 454, 124486 (2020).
[Crossref]

T.-A. Pham, E. Soubies, A. Goy, J. Lim, F. Soulez, D. Psaltis, and M. Unser, “Versatile reconstruction framework for diffraction tomography with intensity measurements and multiple scattering,” Opt. Express 26(3), 2749–2763 (2018).
[Crossref]

Popescu, G.

T. Kim, R. Zhou, L. L. Goddard, and G. Popescu, “Solving inverse scattering problems in biological samples by quantitative phase imaging,” Laser Photonics Rev. 10(1), 13–39 (2016).
[Crossref]

Psaltis, D.

A. B. Ayoub, T.-A. Pham, J. Lim, M. Unser, and D. Psaltis, “A method for assessing the fidelity of optical diffraction tomography reconstruction methods using structured illumination,” Opt. Commun. 454, 124486 (2020).
[Crossref]

J. Lim, A. B. Ayoub, E. E. Antoine, and D. Psaltis, “High-fidelity optical diffraction tomography of multiple scattering samples,” Light: Sci. Appl. 8(1), 82 (2019).
[Crossref]

T.-A. Pham, E. Soubies, A. Goy, J. Lim, F. Soulez, D. Psaltis, and M. Unser, “Versatile reconstruction framework for diffraction tomography with intensity measurements and multiple scattering,” Opt. Express 26(3), 2749–2763 (2018).
[Crossref]

U. S. Kamilov, I. N. Papadopoulos, M. H. Shoreh, A. Goy, C. Vonesch, M. Unser, and D. Psaltis, “Learning approach to optical tomography,” Optica 2(6), 517–522 (2015).
[Crossref]

T. Thomsen, A.B. Ayoub, D. Psaltis, and H.A. Klok, “Fluorescence-based and Fluorescent label-free Characterization of Polymer Nanoparticle Decorated T cells,” Biomacromolecules (2020). Advance online publication. https://doi.org/10.1021/acs.biomac.0c00969

Ren, D.

Repina, N.

Schatzberg, A.

Schofield, M. A.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102(1), 37–49 (2004).
[Crossref]

Sentenac, A.

A. Matlock, A. Sentenac, P. C. Chaumet, J. Yi, and L. Tian, “Inverse scattering for reflection intensity phase microscopy,” Biomed. Opt. Express 11(2), 911–926 (2020).
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T. Thomsen, A.B. Ayoub, D. Psaltis, and H.A. Klok, “Fluorescence-based and Fluorescent label-free Characterization of Polymer Nanoparticle Decorated T cells,” Biomacromolecules (2020). Advance online publication. https://doi.org/10.1021/acs.biomac.0c00969

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Appl. Opt. (1)

Biomed. Opt. Express (3)

IEEE Trans. Comput. Imaging (1)

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IEEE Trans. on Image Process. (1)

W. Zhou, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image Qualify Assessment: From Error Visibility to Structural Similarity,” IEEE Trans. on Image Process. 13(4), 600–612 (2004).
[Crossref]

J. Mod. Opt. (1)

O. Haeberlé, K. Belkebir, H. Giovaninni, and A. Sentenac, “Tomographic diffractive microscopy: basics, techniques and perspectives,” J. Mod. Opt. 57(9), 686–699 (2010).
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J. Opt. Soc. Am. A (2)

Laser Photonics Rev. (1)

T. Kim, R. Zhou, L. L. Goddard, and G. Popescu, “Solving inverse scattering problems in biological samples by quantitative phase imaging,” Laser Photonics Rev. 10(1), 13–39 (2016).
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J. Lim, A. B. Ayoub, E. E. Antoine, and D. Psaltis, “High-fidelity optical diffraction tomography of multiple scattering samples,” Light: Sci. Appl. 8(1), 82 (2019).
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Opt. Lett. (2)

Optica (4)

PLOS ONE (1)

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, “Stain-Free Quantification of Chromosomes in Live Cells Using Regularized Tomographic Phase Microscopy,” PLOS ONE 7(11), e49502 (2012).
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Ultramicroscopy (1)

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102(1), 37–49 (2004).
[Crossref]

Other (1)

T. Thomsen, A.B. Ayoub, D. Psaltis, and H.A. Klok, “Fluorescence-based and Fluorescent label-free Characterization of Polymer Nanoparticle Decorated T cells,” Biomacromolecules (2020). Advance online publication. https://doi.org/10.1021/acs.biomac.0c00969

Supplementary Material (6)

NameDescription
» Supplement 1       Supplement 1
» Visualization 1       3D Mapping of spherical caps for the principal image in XY view
» Visualization 2       3D Mapping of spherical caps for the twin image in XY view
» Visualization 3       Axial fly-through of a HCT-116 cell for the Born reconstruction
» Visualization 4       Axial fly-through of a HCT-116 cell for the Rytov reconstruction
» Visualization 5       Axial fly-through of a HCT-116 cell for the proposed technique

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Figures (5)

Fig. 1.
Fig. 1. Experimental tomographic setup. (M: Mirror, L: Lens, OBJ: Objective lens, BS: Beam splitter) An iris diaphragm is used to block the extra orders generated by the SLM due to its limited fill factor. 4f systems represented by the dotted rectangular blocks are used to obtain an imaging plane at the best plane of focus of the immersion objective lenses.
Fig. 2.
Fig. 2. The amplitude of measured total field and associated 2D Fourier transforms (logarithmic scale). The middle column shows the Fourier transform of the raw intensity map where the principal image (red circle) and the twin image (orange circle) appear concentric for normal illumination and symmetrically shifted around the origin for different oblique illuminations. The right-most column shows the Fourier transform of the raw intensity map after shifting by ${k_i}$ in the frequency domain as a result of multiplying by the incident wave in the spatial domain. Scale bar = 9 µm.
Fig. 3.
Fig. 3. (A), The spectrum of the different field components as a function of (kx, ky) of the scattered field (left) and its twin image (right) for different illumination angles, (B) 3D Fourier transform of the estimated scattering potential as a function of the spatial frequency components (${\kappa _x},{\kappa _y},{\kappa _z}$) of the object plotted in 2D at ${\kappa _z} = 0$ for (I) the scattered fields and (II) the twin images. Panel (B) shows how the 2D projections are mapped inside the 3D Fourier space of the object where the colored circles in Panel (A) are mapped according to the incident k-vector ${\vec{k}_i}$. Note that the size of the scattering potential in B(I) at ${\kappa _z} = 0$ is not (4k*NA) as shown by the green dotted line since we are not illuminating with the maximum possible angle allowed by the numerical aperture of the objective lens.
Fig. 4.
Fig. 4. HCT116 cell XY slices of 3D reconstructions at different depths obtained from (left) holographic measurements of the complex field and (right) intensity-only measurements. Columns (A), (B) and (C) were retrieved using the holographic mode in our optical setup, while column (D) represents the intensity-only measurement, Columns (C) and (D) show high structural similarity index which validate the proposed study. Scale bar = 9 µm. Colorbar shows the estimated RI map for (A) and (B) while it shows the modified refractive index map for (C) and (D) using the proposed method.
Fig. 5.
Fig. 5. HCT116 cell XY slices of 3D reconstructions at different depths obtained using Born, Rytov, and intensity-based reconstruction methods. While Born and Rytov provide a quantitative estimate of the 3D refractive index map, intensity-based reconstruction shows a modified refractive index map where only the high frequency components inside the cell (i.e. cell membrane and lipid structures). Scale bar = 9 µm.

Equations (4)

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| E t | 2 = | E i | 2 + | E s | 2 + 2 | E i | | E s | cos ( ϕ s ϕ i )
| E t | 2 1 + 2 | E s | cos ( Δ ϕ ) | E t | = 1 + 2 | E s | cos ( Δ ϕ )
| E t | = 1 + 2 | E s | cos ( Δ ϕ ) | E t | 1 + 1 2 ( 2 | E s | cos ( Δ ϕ ) ) = 1 + 1 2 | E s | e j Δ ϕ + 1 2 | E s | e j Δ ϕ
| E t | e j ϕ i = 1 2 | E s | e j ϕ s + e j ϕ i + 1 2 | E s | e j ϕ s e 2 j ϕ i = 1 2 E s + e j ϕ i + 1 2 E s e 2 j ϕ i = 1 2 E s + e j ϕ i ( 1 + 1 2 E s e j ϕ i )

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