Harmonically matched grating-based full-field quantitative high-resolution phase microscope for observing dynamics of transparent biological samples

Jigang Wu; Zahid Yaqoob; Xin Heng; Xiquan Cui; Changhuei Yang

doi:10.1364/OE.15.018141

Harmonically matched grating-based full-field quantitative high-resolution phase microscope for observing dynamics of transparent biological samples

Jigang Wu, Zahid Yaqoob, Xin Heng, Xiquan Cui, and Changhuei Yang

Optics Express
Vol. 15,
Issue 26,
pp. 18141-18155
(2007)
•https://doi.org/10.1364/OE.15.018141

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Jigang Wu, Zahid Yaqoob, Xin Heng, Xiquan Cui, and Changhuei Yang, "Harmonically matched grating-based full-field quantitative high-resolution phase microscope for observing dynamics of transparent biological samples," Opt. Express 15, 18141-18155 (2007)

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Related Topics
Table of Contents Category
- Medical Optics and Biotechnology
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Diffraction gratings (050.1950)
- Phase measurement (120.5050)
- Medical and biological imaging (170.3880)

About this Article
History
- Original Manuscript: September 6, 2007
- Revised Manuscript: December 4, 2007
- Manuscript Accepted: December 14, 2007
- Published: December 19, 2007
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 3, Iss. 1

Accessible

Open Access

Abstract

We have developed a full-field high resolution quantitative phase imaging technique for observing dynamics of transparent biological samples. By using a harmonically matched diffraction grating pair (600 and 1200 lines/mm), we were able to obtain non-trivial phase difference (other than 0° or 180°) between the output ports of the gratings. Improving upon our previous design, our current system mitigates astigmatism artifacts and is capable of high resolution imaging. This system also employs an improved phase extraction algorithm. The system has a lateral resolution of 1.6 µm and a phase sensitivity of 62 mrad. We employed the system to acquire high resolution phase images of onion skin cells and a phase movie of amoeba proteus in motion.

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References

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Year
|
Author
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Publication

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–698 (1942).
[Crossref]
F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects Part II,” Physica 9, 974–986 (1942).
[Crossref]
R. D. Allen, G. B. David, and G. Nomarski, “The Zeiss-Nomarski differential interference equipment for transmitted light microscopy,” Z. wiss. Mikr. 69, 193–221 (1969).
[PubMed]
K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]
K. J. Chalut, W. J. Brown, and A. Wax, “Quantitative phase microscopy with asynchronous digital holography,” Opt. Lett. 15, 3047–3052 (2007).
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]
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]
B. Rappaz, P. Marquet, E. Cuche, Y. Emery, C. Depeursinge, and P. J. Magistretti, “Measurement of the integral refractive index and dynamic cell morphometry of living cells with digital holographic microscopy,” Opt. Express 13, 9361–9373 (2005).
[Crossref] [PubMed]
J. Kuhn, T. Colomb, F. Montfort, F. Charriere, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15, 7231–7242 (2007).
[Crossref] [PubMed]
T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. 30, 1165–1167 (2005).
[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]
M. V. Sarunic, S. Weinberg, and J. A. Izatt, “Full-field swept-source phase microscopy,” Opt. Lett. 31, 1462–1464 (2006).
[Crossref] [PubMed]
D. O. Hogenboom, C. A. DiMarzio, T. J. Gaudette, A. J. Devaney, and S. C. Lindberg, “Three-dimensional images generated by quadrature interferometry,” Opt. Lett. 23, 783–785 (1998).
[Crossref]
J. Wu, Z. Yaqoob, X. Heng, L. M. Lee, X. Cui, and C. Yang, “Full field phase imaging using a harmonically matched diffraction grating pair based homodyne quadrature interferometer,” Appl. Phys. Lett. 90, 151123 (2007).
[Crossref]
Z. Yaqoob, J. Wu, X. Cui, X. Heng, and C. Yang, “Harmonically-related diffraction gratings-based interferometer for quadrature phase measurements,” Opt. Express 14, 8127–8137 (2006).
[Crossref] [PubMed]
M. A. Choma, C. Yang, and J. A. Izatt, “Instantaneous quadrature low-coherence interferometry with 3x3 fiber-optic couplers,” Opt. Lett. 28, 2162–2164 (2003).
[Crossref] [PubMed]
Z. Yaqoob, J. Fingler, X. Heng, and C. Yang, “Homodyne en face optical coherence tomography,” Opt. Lett. 31, 1815–1817 (2006)
[Crossref] [PubMed]
M. Pilu, A. W. Fitzgibbon, and R. B. Fisher, “Ellipse-specific direct least-square fitting,” in Proceedings of IEEE International Conference on Image Processing (Lausanne, 1998), vol. 3, pp. 599–602.
D. G. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley & Sons, 1998), Section 4.5.
C. Yang, “Molecular Contrast Optical Coherence Tomography: A Review,” Photochemistry and Photobiology 81, 215–237 (2005)
[Crossref]

2007 (4)

K. J. Chalut, W. J. Brown, and A. Wax, “Quantitative phase microscopy with asynchronous digital holography,” Opt. Lett. 15, 3047–3052 (2007).

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]

J. Wu, Z. Yaqoob, X. Heng, L. M. Lee, X. Cui, and C. Yang, “Full field phase imaging using a harmonically matched diffraction grating pair based homodyne quadrature interferometer,” Appl. Phys. Lett. 90, 151123 (2007).
[Crossref]

J. Kuhn, T. Colomb, F. Montfort, F. Charriere, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15, 7231–7242 (2007).
[Crossref] [PubMed]

2006 (4)

M. V. Sarunic, S. Weinberg, and J. A. Izatt, “Full-field swept-source phase microscopy,” Opt. Lett. 31, 1462–1464 (2006).
[Crossref] [PubMed]

Z. Yaqoob, J. Fingler, X. Heng, and C. Yang, “Homodyne en face optical coherence tomography,” Opt. Lett. 31, 1815–1817 (2006)
[Crossref] [PubMed]

Z. Yaqoob, J. Wu, X. Cui, X. Heng, and C. Yang, “Harmonically-related diffraction gratings-based interferometer for quadrature phase measurements,” Opt. Express 14, 8127–8137 (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]

2005 (4)

C. Yang, “Molecular Contrast Optical Coherence Tomography: A Review,” Photochemistry and Photobiology 81, 215–237 (2005)
[Crossref]

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]

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. 30, 1165–1167 (2005).
[Crossref] [PubMed]

B. Rappaz, P. Marquet, E. Cuche, Y. Emery, C. Depeursinge, and P. J. Magistretti, “Measurement of the integral refractive index and dynamic cell morphometry of living cells with digital holographic microscopy,” Opt. Express 13, 9361–9373 (2005).
[Crossref] [PubMed]

2003 (1)

M. A. Choma, C. Yang, and J. A. Izatt, “Instantaneous quadrature low-coherence interferometry with 3x3 fiber-optic couplers,” Opt. Lett. 28, 2162–2164 (2003).
[Crossref] [PubMed]

1998 (1)

D. O. Hogenboom, C. A. DiMarzio, T. J. Gaudette, A. J. Devaney, and S. C. Lindberg, “Three-dimensional images generated by quadrature interferometry,” Opt. Lett. 23, 783–785 (1998).
[Crossref]

1988 (1)

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

1969 (1)

R. D. Allen, G. B. David, and G. Nomarski, “The Zeiss-Nomarski differential interference equipment for transmitted light microscopy,” Z. wiss. Mikr. 69, 193–221 (1969).
[PubMed]

1942 (2)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–698 (1942).
[Crossref]

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects Part II,” Physica 9, 974–986 (1942).
[Crossref]

Allen, R. D.

R. D. Allen, G. B. David, and G. Nomarski, “The Zeiss-Nomarski differential interference equipment for transmitted light microscopy,” Z. wiss. Mikr. 69, 193–221 (1969).
[PubMed]

Badizadegan, K.

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]

Best-Popescu, C. A.

Brown, W. J.

K. J. Chalut, W. J. Brown, and A. Wax, “Quantitative phase microscopy with asynchronous digital holography,” Opt. Lett. 15, 3047–3052 (2007).

Chalut, K. J.

K. J. Chalut, W. J. Brown, and A. Wax, “Quantitative phase microscopy with asynchronous digital holography,” Opt. Lett. 15, 3047–3052 (2007).

Charriere, F.

Choi, W.

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]

Choma, M. A.

M. A. Choma, C. Yang, and J. A. Izatt, “Instantaneous quadrature low-coherence interferometry with 3x3 fiber-optic couplers,” Opt. Lett. 28, 2162–2164 (2003).
[Crossref] [PubMed]

Colomb, T.

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

Cuche, E.

Cui, X.

Dasari, R. R.

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]

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. 30, 1165–1167 (2005).
[Crossref] [PubMed]

David, G. B.

R. D. Allen, G. B. David, and G. Nomarski, “The Zeiss-Nomarski differential interference equipment for transmitted light microscopy,” Z. wiss. Mikr. 69, 193–221 (1969).
[PubMed]

Depeursinge, C.

Devaney, A. J.

D. O. Hogenboom, C. A. DiMarzio, T. J. Gaudette, A. J. Devaney, and S. C. Lindberg, “Three-dimensional images generated by quadrature interferometry,” Opt. Lett. 23, 783–785 (1998).
[Crossref]

DiMarzio, C. A.

D. O. Hogenboom, C. A. DiMarzio, T. J. Gaudette, A. J. Devaney, and S. C. Lindberg, “Three-dimensional images generated by quadrature interferometry,” Opt. Lett. 23, 783–785 (1998).
[Crossref]

Emery, Y.

Fang-Yen, C.

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]

Feld, M. S.

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]

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. 30, 1165–1167 (2005).
[Crossref] [PubMed]

Fingler, J.

Z. Yaqoob, J. Fingler, X. Heng, and C. Yang, “Homodyne en face optical coherence tomography,” Opt. Lett. 31, 1815–1817 (2006)
[Crossref] [PubMed]

Fisher, R. B.

M. Pilu, A. W. Fitzgibbon, and R. B. Fisher, “Ellipse-specific direct least-square fitting,” in Proceedings of IEEE International Conference on Image Processing (Lausanne, 1998), vol. 3, pp. 599–602.

Fitzgibbon, A. W.

Gaudette, T. J.

D. O. Hogenboom, C. A. DiMarzio, T. J. Gaudette, A. J. Devaney, and S. C. Lindberg, “Three-dimensional images generated by quadrature interferometry,” Opt. Lett. 23, 783–785 (1998).
[Crossref]

Ghiglia, D. G.

D. G. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley & Sons, 1998), Section 4.5.

Goda, K.

Kuhn, J.

Laposata, M.

Lee, L. M.

Lindberg, S. C.

D. O. Hogenboom, C. A. DiMarzio, T. J. Gaudette, A. J. Devaney, and S. C. Lindberg, “Three-dimensional images generated by quadrature interferometry,” Opt. Lett. 23, 783–785 (1998).
[Crossref]

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]

Magistretti, P. J.

Manley, S.

Marquet, P.

Montfort, F.

Nomarski, G.

R. D. Allen, G. B. David, and G. Nomarski, “The Zeiss-Nomarski differential interference equipment for transmitted light microscopy,” Z. wiss. Mikr. 69, 193–221 (1969).
[PubMed]

Oh, S.

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]

Pilu, M.

Popescu, G.

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. 30, 1165–1167 (2005).
[Crossref] [PubMed]

Pritt, M. D.

D. G. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley & Sons, 1998), Section 4.5.

Rappaz, B.

Sarunic, M. V.

M. V. Sarunic, S. Weinberg, and J. A. Izatt, “Full-field swept-source phase microscopy,” Opt. Lett. 31, 1462–1464 (2006).
[Crossref] [PubMed]

Wax, A.

K. J. Chalut, W. J. Brown, and A. Wax, “Quantitative phase microscopy with asynchronous digital holography,” Opt. Lett. 15, 3047–3052 (2007).

Weinberg, S.

M. V. Sarunic, S. Weinberg, and J. A. Izatt, “Full-field swept-source phase microscopy,” Opt. Lett. 31, 1462–1464 (2006).
[Crossref] [PubMed]

Wu, J.

Yang, C.

Z. Yaqoob, J. Fingler, X. Heng, and C. Yang, “Homodyne en face optical coherence tomography,” Opt. Lett. 31, 1815–1817 (2006)
[Crossref] [PubMed]

C. Yang, “Molecular Contrast Optical Coherence Tomography: A Review,” Photochemistry and Photobiology 81, 215–237 (2005)
[Crossref]

M. A. Choma, C. Yang, and J. A. Izatt, “Instantaneous quadrature low-coherence interferometry with 3x3 fiber-optic couplers,” Opt. Lett. 28, 2162–2164 (2003).
[Crossref] [PubMed]

Yaqoob, Z.

Z. Yaqoob, J. Fingler, X. Heng, and C. Yang, “Homodyne en face optical coherence tomography,” Opt. Lett. 31, 1815–1817 (2006)
[Crossref] [PubMed]

Zernike, F.

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–698 (1942).
[Crossref]

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects Part II,” Physica 9, 974–986 (1942).
[Crossref]

Appl. Phys. Lett. (1)

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. Express (3)

Opt. Lett. (7)

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. 30, 1165–1167 (2005).
[Crossref] [PubMed]

K. J. Chalut, W. J. Brown, and A. Wax, “Quantitative phase microscopy with asynchronous digital holography,” Opt. Lett. 15, 3047–3052 (2007).

M. A. Choma, C. Yang, and J. A. Izatt, “Instantaneous quadrature low-coherence interferometry with 3x3 fiber-optic couplers,” Opt. Lett. 28, 2162–2164 (2003).
[Crossref] [PubMed]

Z. Yaqoob, J. Fingler, X. Heng, and C. Yang, “Homodyne en face optical coherence tomography,” Opt. Lett. 31, 1815–1817 (2006)
[Crossref] [PubMed]

M. V. Sarunic, S. Weinberg, and J. A. Izatt, “Full-field swept-source phase microscopy,” Opt. Lett. 31, 1462–1464 (2006).
[Crossref] [PubMed]

D. O. Hogenboom, C. A. DiMarzio, T. J. Gaudette, A. J. Devaney, and S. C. Lindberg, “Three-dimensional images generated by quadrature interferometry,” Opt. Lett. 23, 783–785 (1998).
[Crossref]

Photochemistry and Photobiology (1)

C. Yang, “Molecular Contrast Optical Coherence Tomography: A Review,” Photochemistry and Photobiology 81, 215–237 (2005)
[Crossref]

Phys. Rev. Lett. (1)

Physica (2)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–698 (1942).
[Crossref]

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects Part II,” Physica 9, 974–986 (1942).
[Crossref]

Prog. Opt. (1)

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

Z. wiss. Mikr. (1)

R. D. Allen, G. B. David, and G. Nomarski, “The Zeiss-Nomarski differential interference equipment for transmitted light microscopy,” Z. wiss. Mikr. 69, 193–221 (1969).
[PubMed]

Other (2)

D. G. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley & Sons, 1998), Section 4.5.

Supplementary Material (2)

» Media 1: AVI (360 KB)

» Media 2: AVI (1710 KB)

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Fig. 1. (a). Simple beamsplitter based interferometer: 180o phase shift between output ports; (b) Single shallow grating based interferometer: still 180o phase shift between output ports; (c) G1G2 grating based interferometer: non-trivial phase shift can be obtained.

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Fig. 2. Experimental setup for phase imaging. BS: beam splitter; O1 and O2: objective lenses 1 and 2; P: pinhole; L1-4: lens 1-4; S: sample; G1G2: the harmonically matched grating pair (G1G2 grating) on a holographic plate.

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Fig. 3. Geometric aberration induced by the grating. (a) Previous imaging setup; (b) Astigmatism of the focal spots of previous setup; (c) Current imaging setup; (d) No aberration in the focal spot of current setup; (e) Image of letter “C” acquired by the previous setup; (f) Image of letter “C” acquired by the current setup; (g) Aberration of illumination beam caused by the grating diffraction in current setup.

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Fig. 4. “Stretch” and “compression” distortion caused by the diffraction of the grating in the imaging system.

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Fig. 5. Compare unwrap algothms. (a) Wrapped image; (b) Unwrapped image by simple unwrap algorithm; (c) Unwrapped image by Flynn’s algorithm.

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Fig. 6. Measurement of the temporal phase stability. (a) Phase image of a cover glass, the two spots that are used to measure the phase stability are indicated; (b) Fluctuation of the phase of spot 2 with respect to spot 1 versus time, the standard deviation is 62 mrad.

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Fig. 7. Images of “CIT” logo by our iamging system. (a) Intensity image; (b) Phase image; (c) 3D reconstruction of the phase image; (d) Step-height measurement.

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Fig. 8. Images of onion skin cells. (a) Intensity image; (b) Phase image; (c) 3D reconstruction of the phase image.

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Fig. 9. (a). (360 KB) MOVIE: phase movie of movement of amoeba proteus. The nucleus and contractile vacuole can be clearly seen. In the movie, the food vacuoles are moving inside the amoeba. The size of one frame is 147 µm (width)×123 µm (height) (372×312 pixels) [Media 1] (b) (1.66 MB) Movie: intensity movie of movement of amoeba proteus [Media 2].

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Fig.B1. 1. Schematic of the phase noise assessment.

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Equations (43)

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

Port 1 : P_{1} = P_{r} ⁄ 2 + P_{s} ⁄ 2 + \sqrt{P_{r} P_{s}} \cos (ψ_{s} - ψ_{r} + θ)

Port 2 : P_{2} = P_{r} ⁄ 2 + P_{s} ⁄ 2 - \sqrt{P_{r} P_{s}} \cos (ψ_{s} - ψ_{r} + θ)

ϕ (x_{0}) = ∣ m ∣ [sgn (m) \frac{2 π x_{0}}{Λ} + \frac{π}{2}]

Port 1 : P_{1} = P_{r 1} + P_{s 1} + 2 \sqrt{P_{r 1} P_{s 1}} \cos (\frac{2 π x_{0}}{Λ} + \frac{π}{2} + θ)

Port 2 : P_{2} = P_{r 2} + P_{s 2} + 2 \sqrt{P_{r 2} P_{s 2}} \cos (\frac{2 π x_{0}}{Λ} - \frac{π}{2} + θ)

ϕ_{R, 1} = ϕ_{0, G 1 G 2} = 0, ϕ_{S, 1} = ϕ_{+ 1, G 1} = \frac{2 π x_{1}}{Λ_{1}} + \frac{π}{2},

ϕ_{R, 2} = ϕ_{- 1, G 2} = - \frac{2 π x_{2}}{Λ_{2}} + \frac{π}{2}, ϕ_{S, 2} = ϕ_{- 1, G 1} = - \frac{2 π x_{1}}{Λ_{1}} + \frac{π}{2}

Δ ϕ_{1} = ϕ_{S, 1} - ϕ_{R, 1} = \frac{2 π x_{1}}{Λ_{1}} + \frac{π}{2}, Δ ϕ_{2} = ϕ_{S, 2} - ϕ_{R, 2} = - \frac{2 π x_{1}}{Λ_{1}} + \frac{2 π x_{2}}{Λ_{2}}

Δ ϕ = Δ ϕ_{2} - Δ ϕ_{1} = \frac{4 π (x_{2} - x_{1})}{Λ_{1}} - \frac{π}{2}

x_{2} = f_{2} \tan [\sin^{- 1} (\frac{λ}{d} - \sin (\tan^{- 1} \frac{x_{1}}{f_{1}})) - \sin^{- 1} \frac{λ}{d}]

x_{2} = \frac{- f_{2} ⁄ f_{1}}{\sqrt{1 - λ^{2} ⁄ d^{2}}} x_{1}

P_{1} (i, j) = P_{r 1} (i, j) + P_{s 1} (i, j) + A \sqrt{P_{r 1} (i, j) P_{s 1} (i, j)} \cos (Δ ψ (i, j))

P_{2} (i, j) = P_{r 2} (i, j) + η (i, j) P_{s 1} (i, j)

+ A \sqrt{P_{r 2} (i, j) η (i, j) P_{s 1} (i, j)} \cos (Δ ψ (i, j) + Δ ϕ)

[\frac{1}{P_{r 1}} + \frac{η}{P_{r 2}} - \frac{2 \cos (Δ ϕ) \sqrt{η}}{\sqrt{P_{r 1} P_{r 2}}}] {P_{s 1}}^{2} - [2 \frac{P_{1} - P_{r 1}}{P_{r 1}} + 2 \frac{P_{2} - P_{r 2}}{P_{r 2}} -

2 \cos (Δ ϕ) \frac{(P_{1} - P_{r 1}) η + (P_{2} - P_{r 2})}{\sqrt{η P_{r 1} P_{r 2}}} + A^{2} \sin^{2} (Δ ϕ)] P_{s 1}

+ [\frac{{(P_{1} - P_{r 1})}^{2}}{P_{r 1}} + \frac{{(P_{2} - P_{r 2})}^{2}}{P_{r 2} η} - 2 \cos (Δ ϕ) \frac{(P_{1} - P_{r 1}) (P_{2} - P_{r 2})}{\sqrt{η P_{r 1} P_{r 2}}}] = 0

\frac{P_{s 1}}{P_{r 1}} + \frac{P_{s 2}}{P_{r} 2} - 2 \sqrt{\frac{P_{s 1}}{P_{r 1}} \frac{P_{s 2}}{P_{r 2}}} \cos (Δ ϕ) < \frac{A^{2}}{4} \sin^{2} (Δ ϕ)

\frac{P_{s 1}}{P_{r 1}} + \frac{P_{s 2}}{P_{r 2}} < 1

P_{s 1} = \frac{- b \pm \sqrt{Δ}}{2 a}, where Δ = b^{2} - 4 ac

δ P_{s 1} = \frac{\partial P_{s 1}}{\partial P_{1}} δ P_{1} + \frac{\partial P_{s 1}}{\partial P_{2}} δ P_{2}

= \frac{- \partial b ⁄ \partial P_{1} \pm (\partial Δ ⁄ \partial P_{1}) ⁄ 2 \sqrt{Δ}}{2 a} δ P_{1} + \frac{- \partial b ⁄ \partial P_{2} \pm (\partial Δ ⁄ \partial P_{2}) ⁄ 2 \sqrt{Δ}}{2 a} δ P_{2}

s_{0} + s_{1} = \frac{2 \frac{P_{1} - P_{r 1}}{P_{r 1}} + 2 \frac{P_{2} - P_{r 2}}{P_{r 2}} - 2 \cos (Δ ϕ) \frac{(P_{1} - P_{r 1}) η + (P_{2} - P_{r 2})}{\sqrt{η P_{r 1} P_{r 2}}} + A^{2} \sin^{2} (Δ ϕ)}{\frac{1}{P_{r 1}} + \frac{η}{P_{r 2}} - \frac{2 \cos (Δ ϕ) \sqrt{η}}{\sqrt{P_{r 1} P_{r 2}}}}

\Rightarrow 2 s_{0} < \frac{2 \frac{P_{1} - P_{r 1}}{P_{r 1}} + 2 \frac{P_{2} - P_{r 2}}{P_{r 2}} - 2 \cos (Δ ϕ) \frac{(P_{1} - P_{r 1}) η + (P_{2} - P_{r 2})}{\sqrt{η P_{r 1} P_{r 2}}} + A^{2} \sin^{2} (Δ ϕ)}{\frac{1}{P_{r 1}} + \frac{η}{P_{r 2}} - \frac{2 \cos (Δ ϕ) \sqrt{η}}{\sqrt{P_{r 1} P_{r 2}}}}

s_{0} (\frac{1}{P_{r 1}} + \frac{η}{P_{r 2}} - \frac{2 \cos (Δ ϕ) \sqrt{η}}{\sqrt{P_{r 1} P_{r 2}}})

< \frac{P_{1} - P_{r 1}}{P_{r 1}} + \frac{P_{2} - P_{r 2}}{P_{r 2}} - \cos (Δ ϕ) \frac{(P_{1} - P_{r 1}) η + (P_{2} - P_{r 2})}{\sqrt{η P_{r 1} P_{r 2}}} + \frac{A^{2}}{2} \sin^{2} (Δ ϕ)

= \frac{s_{0} + A \sqrt{P_{r 1} s_{0}} \cos (Δ ψ)}{P_{r 1}} + \frac{η s_{0} + A \sqrt{P_{r 2} η s_{0}} \cos (Δ ψ + Δ ϕ)}{P_{r 2}}

- \cos Δ ϕ \frac{[s_{0} + A \sqrt{P_{r 1} s_{0}} \cos (Δ ψ)] η + [η s_{0} + A \sqrt{P_{r 2} η s_{0}} \cos (Δ ψ + Δ ϕ)]}{\sqrt{η P_{r 1} P_{r 2}}} + \frac{A^{2}}{2} \sin^{2} (Δ ϕ)

\Rightarrow \frac{A \sqrt{s_{0}}}{\sqrt{P_{r 1}}} \sin (Δ ψ) \sin (Δ ψ + Δ ϕ) - \frac{A \sqrt{η s_{0}}}{\sqrt{P_{r 2}}} \sin Δ ϕ \sin (Δ ψ) + \frac{A^{2}}{2} \sin^{2} (Δ ϕ) > 0

\frac{A \sqrt{s_{0}}}{\sqrt{P_{r 1}}} \sin (Δ ψ + Δ ϕ) - \frac{A \sqrt{η s_{0}}}{\sqrt{P_{r 2}}} \sin (Δ ψ) + \frac{A^{2}}{2} \sin (Δ ϕ) > 0

\Rightarrow [\frac{A \sqrt{s_{0}}}{\sqrt{P_{r 1}}} \cos (Δ ϕ) - \frac{A \sqrt{η s_{0}}}{\sqrt{P_{r 2}}}] \sin (Δ ψ) + \frac{A \sqrt{s_{0}}}{\sqrt{P_{r 1}}} \sin (Δ ϕ) \cos (Δ ψ) + \frac{A^{2}}{2} \sin (Δ ϕ) > 0

- \sqrt{{[\frac{A \sqrt{s_{0}}}{\sqrt{P_{r 1}}} \cos (Δ ϕ) - \frac{A \sqrt{η s_{0}}}{\sqrt{P_{r 2}}}]}^{2} + {[\frac{A \sqrt{s_{0}}}{\sqrt{P_{r 1}}} \sin (Δ ϕ)]}^{2}} + \frac{A^{2}}{2} \sin (Δ ϕ) > 0

\Rightarrow \frac{P_{s 1}}{P_{r 1}} + \frac{P_{s 2}}{P_{r 2}} - 2 \sqrt{\frac{P_{s 1}}{P_{r 1}} \frac{P_{s 2}}{P_{r 2}}} \cos (Δ ϕ) < \frac{A^{2}}{4} \sin^{2} (Δ ϕ)

P_{1} = P_{r} + P_{s} + 2 \sqrt{P_{r} P_{s}} \cos Δ ψ

P_{2} = P_{r} + P_{s} + 2 \sqrt{P_{r} P_{s}} \sin Δ ψ

2 \sqrt{P_{r} P_{s}} \cos Δ ψ = P_{1} - P_{r} - P_{s}

2 \sqrt{P_{r} P_{s}} \sin Δ ψ = P_{2} - P_{r} - P_{s}

σ_{x_{1}} = \sqrt{\frac{h ν}{η τ} P_{1}}, σ_{x_{2}} = \sqrt{\frac{h ν}{η τ} P_{2}}

δ ψ \approx \frac{\sqrt{{σ_{x_{1}}}^{2} + {σ_{x_{2}}}^{2}}}{2 \sqrt{P_{r} P_{s}}} = \frac{1}{2} \sqrt{\frac{h ν}{η τ} \frac{P_{1} + P_{2}}{P_{r} P_{s}}} = \frac{\sqrt{2}}{2} \sqrt{\frac{h ν}{η τ} \frac{P_{r} + P_{s} + \sqrt{P_{r} P_{s}} (\cos Δ ψ + \sin Δ ψ)}{P_{r} P_{s}}}

\leq \frac{\sqrt{2}}{2} \sqrt{\frac{h ν}{η τ} \frac{P_{r} + P_{s} + \sqrt{2 P_{r} P_{s}}}{P_{r} P_{s}}}

δ ψ \approx \frac{\sqrt{{σ_{x_{1}}}^{2} + {σ_{x_{2}}}^{2}}}{2 \sqrt{P_{r} P_{s}}} = 0.015 \sqrt{\frac{{P^{2}}_{1} + {P^{2}}_{2}}{P_{r} P_{s}}}

= 0.015 \sqrt{\frac{2 {(P_{r} + P_{s})}^{2} + 4 P_{r} P + 4 (P_{r} + P_{s}) \sqrt{P_{r} P_{s}} (\cos Δ ψ + \sin Δ ψ)}{P_{r} P_{s}}}

\leq 0.015 \sqrt{\frac{2 {(P_{r} + P_{s})}^{2} + 4 P_{r} P + 4 (P_{r} + P_{s}) \sqrt{2 P_{r} P_{s}}}{P_{r} P_{s}}}

Abstract

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