Abstract

We report the experimental implementation of optical diffraction tomography for quantitative 3D mapping of refractive index in live biological cells. Using a heterodyne Mach-Zehnder interferometer, we record complex field images of light transmitted through a sample with varying directions of illumination. To quantitatively reconstruct the 3D map of complex refractive index in live cells, we apply optical diffraction tomography based on the Rytov approximation. In this way, the effect of diffraction is taken into account in the reconstruction process and diffraction-free high resolution 3D images are obtained throughout the entire sample volume. The quantitative refractive index map can potentially serve as an intrinsic assay to provide the molecular concentrations without the addition of exogenous agents and also to provide a method for studying the light scattering properties of single cells.

© 2009 Optical Society of America

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  5. C. Fang-Yen, S. Oh, Y. Park, W. Choi, S. Song, H. S. Seung, R. R. Dasari, and M. S. Feld, "Imaging voltage-dependent cell motions with heterodyne Mach-Zehnder phase microscopy," Opt. Lett. 32, 1572-1574 (2007).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  24. W. Choi, C. C. Yu, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Field-based angle-resolved light-scattering study of single live cells," Opt. Lett. 33, 1596-1598 (2008).
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2008 (5)

2007 (3)

2006 (5)

2005 (2)

2002 (1)

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2002).
[CrossRef] [PubMed]

2000 (1)

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

1998 (1)

1983 (1)

1981 (2)

1969 (1)

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

1955 (1)

Q1. G. Nomarski, "Microinterféromètre différentiel à ondes polarisées," J. Phys. Radium 16, 9S-11S (1955).

1942 (1)

F. Zernike, "Phase-contrast, a new method for microscopic observation of transparent objects. Part I.," Physica 9, 686-698 (1942).
[CrossRef]

Badizadegan, K.

Barty, A.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

Best-Popescu, C. A.

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef] [PubMed]

Brody, W. R.

Charriere, F.

Choi, W.

Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, "Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum," Proc. Natl. Acad. Sci. USA 105, 13730-13735 (2008).
[CrossRef] [PubMed]

W. Choi, C. C. Yu, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Field-based angle-resolved light-scattering study of single live cells," Opt. Lett. 33, 1596-1598 (2008).
[CrossRef] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

C. Fang-Yen, S. Oh, Y. Park, W. Choi, S. Song, H. S. Seung, R. R. Dasari, and M. S. Feld, "Imaging voltage-dependent cell motions with heterodyne Mach-Zehnder phase microscopy," Opt. Lett. 32, 1572-1574 (2007).
[CrossRef] [PubMed]

Choi, W. S.

Colomb, T.

Cuche, E.

Dasari, R. R.

W. S. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Extended depth of focus in tomographic phase microscopy using a propagation algorithm," Opt. Lett. 33, 171-173 (2008).
[CrossRef] [PubMed]

W. Choi, C. C. Yu, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Field-based angle-resolved light-scattering study of single live cells," Opt. Lett. 33, 1596-1598 (2008).
[CrossRef] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

C. Fang-Yen, S. Oh, Y. Park, W. Choi, S. Song, H. S. Seung, R. R. Dasari, and M. S. Feld, "Imaging voltage-dependent cell motions with heterodyne Mach-Zehnder phase microscopy," Opt. Lett. 32, 1572-1574 (2007).
[CrossRef] [PubMed]

G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, "Diffraction phase microscopy for quantifying cell structure and dynamics," Opt. Lett. 31, 775-777 (2006).
[CrossRef] [PubMed]

Y. K. Park, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Diffraction phase and fluorescence microscopy," Opt. Express 14, 8263-8268 (2006).
[CrossRef] [PubMed]

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef] [PubMed]

N. Lue, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Live cell refractometry using microfluidic devices," Opt. Lett. 31, 2759-2761 (2006).
[CrossRef] [PubMed]

Debailleul, M.

Q2. M. Debailleul, B. Simon, V. Georges, O. Haeberle, and V. Lauer, "Holographic microscopy and diffractive microtomography of transparent samples," Meas. Sci. Technol. 19, (2008).
[CrossRef]

Depeursinge, C.

Devaney, A. J.

Diez-Silva, M.

Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, "Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum," Proc. Natl. Acad. Sci. USA 105, 13730-13735 (2008).
[CrossRef] [PubMed]

Emery, Y.

Fang-Yen, C.

Feld, M. S.

W. S. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Extended depth of focus in tomographic phase microscopy using a propagation algorithm," Opt. Lett. 33, 171-173 (2008).
[CrossRef] [PubMed]

Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, "Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum," Proc. Natl. Acad. Sci. USA 105, 13730-13735 (2008).
[CrossRef] [PubMed]

W. Choi, C. C. Yu, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Field-based angle-resolved light-scattering study of single live cells," Opt. Lett. 33, 1596-1598 (2008).
[CrossRef] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

C. Fang-Yen, S. Oh, Y. Park, W. Choi, S. Song, H. S. Seung, R. R. Dasari, and M. S. Feld, "Imaging voltage-dependent cell motions with heterodyne Mach-Zehnder phase microscopy," Opt. Lett. 32, 1572-1574 (2007).
[CrossRef] [PubMed]

G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, "Diffraction phase microscopy for quantifying cell structure and dynamics," Opt. Lett. 31, 775-777 (2006).
[CrossRef] [PubMed]

Y. K. Park, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Diffraction phase and fluorescence microscopy," Opt. Express 14, 8263-8268 (2006).
[CrossRef] [PubMed]

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef] [PubMed]

N. Lue, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Live cell refractometry using microfluidic devices," Opt. Lett. 31, 2759-2761 (2006).
[CrossRef] [PubMed]

Georges, V.

Q2. M. Debailleul, B. Simon, V. Georges, O. Haeberle, and V. Lauer, "Holographic microscopy and diffractive microtomography of transparent samples," Meas. Sci. Technol. 19, (2008).
[CrossRef]

Goda, K.

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef] [PubMed]

Gorski, W.

Haeberle, O.

Q2. M. Debailleul, B. Simon, V. Georges, O. Haeberle, and V. Lauer, "Holographic microscopy and diffractive microtomography of transparent samples," Meas. Sci. Technol. 19, (2008).
[CrossRef]

Heger, T. J.

Ikeda, T.

Laposata, M.

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef] [PubMed]

Lauer, V.

Q2. M. Debailleul, B. Simon, V. Georges, O. Haeberle, and V. Lauer, "Holographic microscopy and diffractive microtomography of transparent samples," Meas. Sci. Technol. 19, (2008).
[CrossRef]

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2002).
[CrossRef] [PubMed]

Lue, N.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

N. Lue, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Live cell refractometry using microfluidic devices," Opt. Lett. 31, 2759-2761 (2006).
[CrossRef] [PubMed]

Lykotrafitis, G.

Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, "Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum," Proc. Natl. Acad. Sci. USA 105, 13730-13735 (2008).
[CrossRef] [PubMed]

Macovski, A.

Magistretti, P.

Magistretti, P. J.

Manley, S.

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef] [PubMed]

Marquet, P.

Medoff, B. P.

Mitchell, E. A. D.

Nassi, M.

Nomarski, G.

Q1. G. Nomarski, "Microinterféromètre différentiel à ondes polarisées," J. Phys. Radium 16, 9S-11S (1955).

Nugent, K. A.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

Oh, S.

Osten, W.

Paganin, D.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

Park, Y.

Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, "Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum," Proc. Natl. Acad. Sci. USA 105, 13730-13735 (2008).
[CrossRef] [PubMed]

C. Fang-Yen, S. Oh, Y. Park, W. Choi, S. Song, H. S. Seung, R. R. Dasari, and M. S. Feld, "Imaging voltage-dependent cell motions with heterodyne Mach-Zehnder phase microscopy," Opt. Lett. 32, 1572-1574 (2007).
[CrossRef] [PubMed]

Park, Y. K.

Pavillon, N.

Perezmendez, V.

Popescu, G.

Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, "Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum," Proc. Natl. Acad. Sci. USA 105, 13730-13735 (2008).
[CrossRef] [PubMed]

N. Lue, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Live cell refractometry using microfluidic devices," Opt. Lett. 31, 2759-2761 (2006).
[CrossRef] [PubMed]

Y. K. Park, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Diffraction phase and fluorescence microscopy," Opt. Express 14, 8263-8268 (2006).
[CrossRef] [PubMed]

G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, "Diffraction phase microscopy for quantifying cell structure and dynamics," Opt. Lett. 31, 775-777 (2006).
[CrossRef] [PubMed]

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef] [PubMed]

Rappaz, B.

Roberts, A.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

Seung, H. S.

Simon, B.

Q2. M. Debailleul, B. Simon, V. Georges, O. Haeberle, and V. Lauer, "Holographic microscopy and diffractive microtomography of transparent samples," Meas. Sci. Technol. 19, (2008).
[CrossRef]

Song, S.

Suresh, S.

Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, "Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum," Proc. Natl. Acad. Sci. USA 105, 13730-13735 (2008).
[CrossRef] [PubMed]

Tam, K. C.

Wolf, E.

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

Yu, C. C.

Zernike, F.

F. Zernike, "Phase-contrast, a new method for microscopic observation of transparent objects. Part I.," Physica 9, 686-698 (1942).
[CrossRef]

J. Microsc. (1)

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (2)

J. Phys. Radium (1)

Q1. G. Nomarski, "Microinterféromètre différentiel à ondes polarisées," J. Phys. Radium 16, 9S-11S (1955).

Meas. Sci. Technol. (1)

Q2. M. Debailleul, B. Simon, V. Georges, O. Haeberle, and V. Lauer, "Holographic microscopy and diffractive microtomography of transparent samples," Meas. Sci. Technol. 19, (2008).
[CrossRef]

Nature Methods (1)

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic phase microscopy," Nature Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Opt. Commun. (2)

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

E. Wolf, "Three-dimensional structure determination of semi-transparent objects from holographic data," Opt. Commun. 1, 153-156 (1969).
[CrossRef]

Opt. Express (3)

Opt. Lett. (10)

W. S. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Extended depth of focus in tomographic phase microscopy using a propagation algorithm," Opt. Lett. 33, 171-173 (2008).
[CrossRef] [PubMed]

G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, "Diffraction phase microscopy for quantifying cell structure and dynamics," Opt. Lett. 31, 775-777 (2006).
[CrossRef] [PubMed]

C. Fang-Yen, S. Oh, Y. Park, W. Choi, S. Song, H. S. Seung, R. R. Dasari, and M. S. Feld, "Imaging voltage-dependent cell motions with heterodyne Mach-Zehnder phase microscopy," Opt. Lett. 32, 1572-1574 (2007).
[CrossRef] [PubMed]

P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, "Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy," Opt. Lett. 30, 468-470 (2005).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

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Phys. Rev. Lett. (1)

G. Popescu, T. Ikeda, K. Goda, C. A. Best-Popescu, M. Laposata, S. Manley, R. R. Dasari, K. Badizadegan, and M. S. Feld, "Optical measurement of cell membrane tension," Phys. Rev. Lett. 97, 218101 (2006).
[CrossRef] [PubMed]

Physica (1)

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

Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, "Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum," Proc. Natl. Acad. Sci. USA 105, 13730-13735 (2008).
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Figures (6)

Fig. 1.
Fig. 1.

Tomographic phase microscope. GM: galvanometer scanning mirror; L1: lens (f = 250 mm); C: condenser lens (NA 1.4); OL: objective lens (NA 1.4); Tube: tube lens (f = 200 mm); BS1 and BS2: Beamsplitters; AOMs: acousto-optic modulators. The frequency shifted reference laser beam is shown in blue.

Fig. 2.
Fig. 2.

Mapping of the complex E-field onto the 3D Fourier space of the object function. (a) Quantitative phase image of a 6 μm polystyrene bead at zero degree illumination. The color bar indicates phase in radians. Scale bar, 5 μm. (b) Amplitude of the Fourier transform of the complex E-field image at zero degree illumination on a logarithmic scale. (c) Amplitude distribution in KxKy plane after mapping all the angular E-field images. (d) Amplitude distribution in the Kx -Kz plane. The color bar indicates base-10 logarithm of the amplitude of E-field. The scale bars in (b-d) indicate 2 μm-1.

Fig. 3.
Fig. 3.

Iterative constraint algorithm. (a) Slice image of a 6 μm bead before application of the constraint algorithm. (b) Same slice image as in (a) after application of the non-negative constraint. (c) Same slice image as in (b) after 100 iterations. The color bar indicates the refractive indices at 633 nm wavelength. The scale bar, 5 μm. (d) Amplitude distribution in Kx -Ky plane before application of the constraint algorithm. (e) 3D Fourier transform of tomogram after non-negative constraint. (f) 3D Fourier transform of tomogram after 100 iterations. The color bar indicates base-10 logarithm of E-field amplitude. Scale bar, 2 μm-1.

Fig. 4.
Fig. 4.

Comparison between filtered back-projection algorithm and diffraction tomography with the Rytov approximation. (a) and (b): Slice images of tomogram in the middle of a 6 μm bead reconstructed by the filtered back-projection algorithm when the objective focus is in the middle of the bead (a) and 4 μm above the center of the bead (b). (c) and (d): Same slice images as (a) and (b) after reconstructed by the diffraction tomography based on the Rytov approximation at objective focus in the middle of the bead (c) and 4 μm above the center of the bead (d). The color bar indicates refractive indices at 633 nm wavelength. Scale bar, 5 μm.

Fig. 5.
Fig. 5.

Comparison of the Born and Rytov approximations in diffraction tomography. (a) and (c): Slice images of a 6 μm bead reconstructed based on the Born and Rytov approximations, respectively. Line profiles across the center of the bead are presented next to the slice images. (b) and (d): Same as (a) and (c) for a 10 μm bead. The color bar indicates refractive indices at 633 nm wavelength.

Fig. 6.
Fig. 6.

Bright field images (a-d), and 3D tomogram of HT29 cells reconstructed by the filtered back-projection algorithm (e-h) and diffraction tomography based on the Rytov approximation (i-l). (f) and (j) are slice images corresponding to the objective focus. (e) and (i) are slice images 1.7 μm above the original focus. (g) and (k) are slice images 2.9 μm below the focus. (a-c) are bright field images at the same foci as (e-g). (d), (h) and (l) are zoom-in images of the rectangular boxes in (c), (g) and (k), respectively. The color bar indicates refractive indices at 633 nm wavelength. Scale bar, 10 μm.

Equations (11)

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2U(R)+k02n(R)2U(R)=0.
U(R)=U(I)(R)+U(S)(R),
(2+k02nm2)U(S)(R)=F(R)U(R),
U(S)(R)=G(RR)F(R)U(R)d3R,
U(S)(R)G(RR)F(R)U(I)(R)d3R,
F̂(Kx,Ky,Kz)=ikzπ Û(S) (kx,ky;z+=0) .
nδ>>(φ(S)λ2π)2,withφ(S)=ln (U(R)U(I)(R)) ,
URytov(S)=U(I)(R)ln(U(R)U(I)(R)).
F̂(Kx,Ky,Kz)=i(Kz+kz0)π Û(S) (Kx+kx0,Ky+ky0;z+=0) .
Û(S)(Kx+kx0,Ky+ky0;θ)=(U(x,y;θ)Ubg(x,y;θ)) / Ubg (x,y;θ) eiKxxiKyy dxdy
ÛRytov(S)(Kx+kx0,Ky+ky0;θ)=ln(U(x,y;θ)/Ubg(x,y;θ)) eiKxxiKyy dxdy

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