Abstract

I describe an improved implementation of a previously reported interferometric device, the sampling field sensor (SFS) [Appl. Opt. 47, B32–B43 (2008)]. It provides X, Y, and XY shearing interferometric information simultaneously (space multiplexed) with amplitude and polarization information while using time-multiplexed phase shifting. Its simple common-path configuration makes it compact and vibration insensitive, as demonstrated by the λ/125 phase estimation repeatability that was below the coherent noise floor (estimated at λ/50). The SFS may be viewed as an efficient, robust and accurate full-field optical–digital interface, easy to integrate with traditional imaging systems. This is demonstrated by using the sensor as the focal plane array of a transmitted-light microscope in a straightforward setup using an illumination path polarization phase shifter. This work is focused on a qualitative demonstration and presents phase, amplitude, and polarization images of different types of human cheek cells and Caenorhabditis elegans larvae.

© 2008 Optical Society of America

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References

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  1. A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529-541 (1981).
    [CrossRef]
  2. J. Curlander and R. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, 1991).
  3. R. Tumbar, D. L. Marks, and D. J. Brady, “Robust, common-path, phase-shifting interferometer and optical profilometer,” Appl. Opt. 47, B32-B43 (2008).
    [CrossRef] [PubMed]
  4. D. Psaltis, “Coherent optical information systems,” Science 298, 1359-1363 (2002).
    [CrossRef] [PubMed]
  5. R. Tumbar and D. J. Brady, “Sampling field sensor with anisotropic fan-out,” Appl. Opt. 41, 6621-6636 (2002).
    [CrossRef] [PubMed]
  6. R. Tumbar and D. J. Brady, “Interferometric sensor and method to detect optical fields,” U.S. Patent 6,639,683 (2003).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  16. M. Laikin, Lens Design, 3rd ed. (Marcel Dekker, 2001).
  17. R. H. Dyck, “Multiple-frame CCD image sensor with overlying photosensitive layer,” U.S. Patent 5796433 (1998).
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    [PubMed]
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    [PubMed]
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    [CrossRef]
  26. W. J. Brown and M. G. Farquhar, “Accumulation of coated vesicles bearing mannose 6-phosphate receptors for lysosomal enzymes in the Golgi region of I-cell fibroblasts,” Proc. Natl. Acad. Sci. USA 81, 5135-5139 (1984).
    [CrossRef] [PubMed]
  27. F. Dubois, C. Yourassowsky, O. Monnom, J.-C. Legros, O. Debeir, P. Van Ham, R. Kiss, and C. Decaestecker, “Digital holographic microscopy for the three-dimensional dynamic analysis of in vitro cancer cell migration,” J Biomed. Opt. 11, 054032 (2006).
    [CrossRef] [PubMed]
  28. P. Ferraro, D. Alferi, S. De Nicola, L. De Petrocellis, A. Finizio, and G. Pierattini, “Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,” Opt. Lett. 31, 1405-1407 (2006).
    [CrossRef] [PubMed]

2008

2006

F. Dubois, C. Yourassowsky, O. Monnom, J.-C. Legros, O. Debeir, P. Van Ham, R. Kiss, and C. Decaestecker, “Digital holographic microscopy for the three-dimensional dynamic analysis of in vitro cancer cell migration,” J Biomed. Opt. 11, 054032 (2006).
[CrossRef] [PubMed]

P. Ferraro, D. Alferi, S. De Nicola, L. De Petrocellis, A. Finizio, and G. Pierattini, “Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,” Opt. Lett. 31, 1405-1407 (2006).
[CrossRef] [PubMed]

2005

H. Sun, M. A. Player, J. Watson, D. C. Hendry, R. G. Perkins, G. Gust, and D. M. Paterson, “The use of digital/electronic holography for biological applications,” J. Opt. A Pure Appl. Opt. 7, S399-S407 (2005).
[CrossRef]

2004

2003

2002

1999

D. Zicha, E. Genot, G. A. Dunn, and I. M. Kramer, “TGFbeta1 induces a cell-cycle-dependent increase in motility of epithelial cells,” J Cell Sci. 112, 447-454 (1999).
[PubMed]

1998

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Achromatic phase shifting by a rotating polarizer,” Opt. Commun. 154, 249-254 (1998).
[CrossRef]

1997

C. J. Cogswell, N. I. Smith, K. G. Larkin, and P. Hariharan, “Quantitative DIC microscopy using a geometric phase shifter,” Proc. SPIE 2984, 72-81 (1997).
[CrossRef]

1995

1987

1984

W. J. Brown and M. G. Farquhar, “Accumulation of coated vesicles bearing mannose 6-phosphate receptors for lysosomal enzymes in the Golgi region of I-cell fibroblasts,” Proc. Natl. Acad. Sci. USA 81, 5135-5139 (1984).
[CrossRef] [PubMed]

1983

1981

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529-541 (1981).
[CrossRef]

1977

R. Hard, R. Zeh, and R. D. Allen, “Phase-randomized laser illumination for microscopy,” J. Cell Sci. 23, 335-343 (1977).
[PubMed]

1955

F. Zernike, “How I discovered phase contrast,” Science 121, 345-349 (1955).
[CrossRef] [PubMed]

G. Nomarski and A. R. Weill, “Application to metallography of interference methods with two waves in polarized light--application a la metallographie des methodes interferentielles a deux ondes polarisees,” Rev. Metall. 52, 121-134 (1955).

Alferi, D.

Allen, R. D.

R. Hard, R. Zeh, and R. D. Allen, “Phase-randomized laser illumination for microscopy,” J. Cell Sci. 23, 335-343 (1977).
[PubMed]

Badizadegan, K.

Brady, D. J.

Brown, W. J.

W. J. Brown and M. G. Farquhar, “Accumulation of coated vesicles bearing mannose 6-phosphate receptors for lysosomal enzymes in the Golgi region of I-cell fibroblasts,” Proc. Natl. Acad. Sci. USA 81, 5135-5139 (1984).
[CrossRef] [PubMed]

Burow, R.

Cogswell, C. J.

C. J. Cogswell, N. I. Smith, K. G. Larkin, and P. Hariharan, “Quantitative DIC microscopy using a geometric phase shifter,” Proc. SPIE 2984, 72-81 (1997).
[CrossRef]

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1988), Vol. XXVI, p. 349.
[CrossRef]

Curlander, J.

J. Curlander and R. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, 1991).

Dasari, R. R.

De Nicola, S.

De Petrocellis, L.

Debeir, O.

F. Dubois, C. Yourassowsky, O. Monnom, J.-C. Legros, O. Debeir, P. Van Ham, R. Kiss, and C. Decaestecker, “Digital holographic microscopy for the three-dimensional dynamic analysis of in vitro cancer cell migration,” J Biomed. Opt. 11, 054032 (2006).
[CrossRef] [PubMed]

Decaestecker, C.

F. Dubois, C. Yourassowsky, O. Monnom, J.-C. Legros, O. Debeir, P. Van Ham, R. Kiss, and C. Decaestecker, “Digital holographic microscopy for the three-dimensional dynamic analysis of in vitro cancer cell migration,” J Biomed. Opt. 11, 054032 (2006).
[CrossRef] [PubMed]

Deflores, L. P.

Dubois, F.

F. Dubois, C. Yourassowsky, O. Monnom, J.-C. Legros, O. Debeir, P. Van Ham, R. Kiss, and C. Decaestecker, “Digital holographic microscopy for the three-dimensional dynamic analysis of in vitro cancer cell migration,” J Biomed. Opt. 11, 054032 (2006).
[CrossRef] [PubMed]

Dunn, G. A.

D. Zicha, E. Genot, G. A. Dunn, and I. M. Kramer, “TGFbeta1 induces a cell-cycle-dependent increase in motility of epithelial cells,” J Cell Sci. 112, 447-454 (1999).
[PubMed]

Dyck, R. H.

R. H. Dyck, “Multiple-frame CCD image sensor with overlying photosensitive layer,” U.S. Patent 5796433 (1998).

Eiju, T.

Elssner, K. E.

Farquhar, M. G.

W. J. Brown and M. G. Farquhar, “Accumulation of coated vesicles bearing mannose 6-phosphate receptors for lysosomal enzymes in the Golgi region of I-cell fibroblasts,” Proc. Natl. Acad. Sci. USA 81, 5135-5139 (1984).
[CrossRef] [PubMed]

Feld, M. S.

Ferraro, P.

Finizio, A.

Genot, E.

D. Zicha, E. Genot, G. A. Dunn, and I. M. Kramer, “TGFbeta1 induces a cell-cycle-dependent increase in motility of epithelial cells,” J Cell Sci. 112, 447-454 (1999).
[PubMed]

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, 1968).

Grzanna, J.

Gust, G.

H. Sun, M. A. Player, J. Watson, D. C. Hendry, R. G. Perkins, G. Gust, and D. M. Paterson, “The use of digital/electronic holography for biological applications,” J. Opt. A Pure Appl. Opt. 7, S399-S407 (2005).
[CrossRef]

Hard, R.

R. Hard, R. Zeh, and R. D. Allen, “Phase-randomized laser illumination for microscopy,” J. Cell Sci. 23, 335-343 (1977).
[PubMed]

Hariharan, P.

C. J. Cogswell, N. I. Smith, K. G. Larkin, and P. Hariharan, “Quantitative DIC microscopy using a geometric phase shifter,” Proc. SPIE 2984, 72-81 (1997).
[CrossRef]

P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504-2506 (1987).
[CrossRef] [PubMed]

Helen, S. S.

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Achromatic phase shifting by a rotating polarizer,” Opt. Commun. 154, 249-254 (1998).
[CrossRef]

Hendry, D. C.

H. Sun, M. A. Player, J. Watson, D. C. Hendry, R. G. Perkins, G. Gust, and D. M. Paterson, “The use of digital/electronic holography for biological applications,” J. Opt. A Pure Appl. Opt. 7, S399-S407 (2005).
[CrossRef]

Hori, T.

T. Hori, “Method for making high-frame-rate CCD imaging devices from otherwise ordinary and inexpensive CCD devices,” U.S. Patent 6255134 (2001).

Iwai, H.

Kiss, R.

F. Dubois, C. Yourassowsky, O. Monnom, J.-C. Legros, O. Debeir, P. Van Ham, R. Kiss, and C. Decaestecker, “Digital holographic microscopy for the three-dimensional dynamic analysis of in vitro cancer cell migration,” J Biomed. Opt. 11, 054032 (2006).
[CrossRef] [PubMed]

Kothiyal, M. P.

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Achromatic phase shifting by a rotating polarizer,” Opt. Commun. 154, 249-254 (1998).
[CrossRef]

Kramer, I. M.

D. Zicha, E. Genot, G. A. Dunn, and I. M. Kramer, “TGFbeta1 induces a cell-cycle-dependent increase in motility of epithelial cells,” J Cell Sci. 112, 447-454 (1999).
[PubMed]

Laikin, M.

M. Laikin, Lens Design, 3rd ed. (Marcel Dekker, 2001).

Larkin, K. G.

C. J. Cogswell, N. I. Smith, K. G. Larkin, and P. Hariharan, “Quantitative DIC microscopy using a geometric phase shifter,” Proc. SPIE 2984, 72-81 (1997).
[CrossRef]

Legros, J.-C.

F. Dubois, C. Yourassowsky, O. Monnom, J.-C. Legros, O. Debeir, P. Van Ham, R. Kiss, and C. Decaestecker, “Digital holographic microscopy for the three-dimensional dynamic analysis of in vitro cancer cell migration,” J Biomed. Opt. 11, 054032 (2006).
[CrossRef] [PubMed]

Lim, J. S.

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529-541 (1981).
[CrossRef]

Marks, D. L.

McDonough, R.

J. Curlander and R. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, 1991).

Merkel, K.

Monnom, O.

F. Dubois, C. Yourassowsky, O. Monnom, J.-C. Legros, O. Debeir, P. Van Ham, R. Kiss, and C. Decaestecker, “Digital holographic microscopy for the three-dimensional dynamic analysis of in vitro cancer cell migration,” J Biomed. Opt. 11, 054032 (2006).
[CrossRef] [PubMed]

Nomarski, G.

G. Nomarski and A. R. Weill, “Application to metallography of interference methods with two waves in polarized light--application a la metallographie des methodes interferentielles a deux ondes polarisees,” Rev. Metall. 52, 121-134 (1955).

Oldenbourg, R.

Oppenheim, A. V.

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529-541 (1981).
[CrossRef]

Oreb, B. F.

Paterson, D. M.

H. Sun, M. A. Player, J. Watson, D. C. Hendry, R. G. Perkins, G. Gust, and D. M. Paterson, “The use of digital/electronic holography for biological applications,” J. Opt. A Pure Appl. Opt. 7, S399-S407 (2005).
[CrossRef]

Perkins, R. G.

H. Sun, M. A. Player, J. Watson, D. C. Hendry, R. G. Perkins, G. Gust, and D. M. Paterson, “The use of digital/electronic holography for biological applications,” J. Opt. A Pure Appl. Opt. 7, S399-S407 (2005).
[CrossRef]

Pierattini, G.

Player, M. A.

H. Sun, M. A. Player, J. Watson, D. C. Hendry, R. G. Perkins, G. Gust, and D. M. Paterson, “The use of digital/electronic holography for biological applications,” J. Opt. A Pure Appl. Opt. 7, S399-S407 (2005).
[CrossRef]

Popescu, G.

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Psaltis, D.

D. Psaltis, “Coherent optical information systems,” Science 298, 1359-1363 (2002).
[CrossRef] [PubMed]

Rathjen, C.

Schwider, J.

Shribak, M.

Sirohi, R. S.

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Achromatic phase shifting by a rotating polarizer,” Opt. Commun. 154, 249-254 (1998).
[CrossRef]

Smith, N. I.

C. J. Cogswell, N. I. Smith, K. G. Larkin, and P. Hariharan, “Quantitative DIC microscopy using a geometric phase shifter,” Proc. SPIE 2984, 72-81 (1997).
[CrossRef]

Spolaczyk, R.

Sun, H.

H. Sun, M. A. Player, J. Watson, D. C. Hendry, R. G. Perkins, G. Gust, and D. M. Paterson, “The use of digital/electronic holography for biological applications,” J. Opt. A Pure Appl. Opt. 7, S399-S407 (2005).
[CrossRef]

Tumbar, R.

Van Ham, P.

F. Dubois, C. Yourassowsky, O. Monnom, J.-C. Legros, O. Debeir, P. Van Ham, R. Kiss, and C. Decaestecker, “Digital holographic microscopy for the three-dimensional dynamic analysis of in vitro cancer cell migration,” J Biomed. Opt. 11, 054032 (2006).
[CrossRef] [PubMed]

Vaughan, J. C.

Watson, J.

H. Sun, M. A. Player, J. Watson, D. C. Hendry, R. G. Perkins, G. Gust, and D. M. Paterson, “The use of digital/electronic holography for biological applications,” J. Opt. A Pure Appl. Opt. 7, S399-S407 (2005).
[CrossRef]

Weill, A. R.

G. Nomarski and A. R. Weill, “Application to metallography of interference methods with two waves in polarized light--application a la metallographie des methodes interferentielles a deux ondes polarisees,” Rev. Metall. 52, 121-134 (1955).

Yourassowsky, C.

F. Dubois, C. Yourassowsky, O. Monnom, J.-C. Legros, O. Debeir, P. Van Ham, R. Kiss, and C. Decaestecker, “Digital holographic microscopy for the three-dimensional dynamic analysis of in vitro cancer cell migration,” J Biomed. Opt. 11, 054032 (2006).
[CrossRef] [PubMed]

Zeh, R.

R. Hard, R. Zeh, and R. D. Allen, “Phase-randomized laser illumination for microscopy,” J. Cell Sci. 23, 335-343 (1977).
[PubMed]

Zernike, F.

F. Zernike, “How I discovered phase contrast,” Science 121, 345-349 (1955).
[CrossRef] [PubMed]

Zicha, D.

D. Zicha, E. Genot, G. A. Dunn, and I. M. Kramer, “TGFbeta1 induces a cell-cycle-dependent increase in motility of epithelial cells,” J Cell Sci. 112, 447-454 (1999).
[PubMed]

Appl. Opt.

J Biomed. Opt.

F. Dubois, C. Yourassowsky, O. Monnom, J.-C. Legros, O. Debeir, P. Van Ham, R. Kiss, and C. Decaestecker, “Digital holographic microscopy for the three-dimensional dynamic analysis of in vitro cancer cell migration,” J Biomed. Opt. 11, 054032 (2006).
[CrossRef] [PubMed]

J Cell Sci.

D. Zicha, E. Genot, G. A. Dunn, and I. M. Kramer, “TGFbeta1 induces a cell-cycle-dependent increase in motility of epithelial cells,” J Cell Sci. 112, 447-454 (1999).
[PubMed]

J. Cell Sci.

R. Hard, R. Zeh, and R. D. Allen, “Phase-randomized laser illumination for microscopy,” J. Cell Sci. 23, 335-343 (1977).
[PubMed]

J. Opt. A Pure Appl. Opt.

H. Sun, M. A. Player, J. Watson, D. C. Hendry, R. G. Perkins, G. Gust, and D. M. Paterson, “The use of digital/electronic holography for biological applications,” J. Opt. A Pure Appl. Opt. 7, S399-S407 (2005).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

S. S. Helen, M. P. Kothiyal, and R. S. Sirohi, “Achromatic phase shifting by a rotating polarizer,” Opt. Commun. 154, 249-254 (1998).
[CrossRef]

Opt. Lett.

Proc. IEEE

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529-541 (1981).
[CrossRef]

Proc. Natl. Acad. Sci. USA

W. J. Brown and M. G. Farquhar, “Accumulation of coated vesicles bearing mannose 6-phosphate receptors for lysosomal enzymes in the Golgi region of I-cell fibroblasts,” Proc. Natl. Acad. Sci. USA 81, 5135-5139 (1984).
[CrossRef] [PubMed]

Proc. SPIE

C. J. Cogswell, N. I. Smith, K. G. Larkin, and P. Hariharan, “Quantitative DIC microscopy using a geometric phase shifter,” Proc. SPIE 2984, 72-81 (1997).
[CrossRef]

Rev. Metall.

G. Nomarski and A. R. Weill, “Application to metallography of interference methods with two waves in polarized light--application a la metallographie des methodes interferentielles a deux ondes polarisees,” Rev. Metall. 52, 121-134 (1955).

Science

F. Zernike, “How I discovered phase contrast,” Science 121, 345-349 (1955).
[CrossRef] [PubMed]

D. Psaltis, “Coherent optical information systems,” Science 298, 1359-1363 (2002).
[CrossRef] [PubMed]

Other

R. Tumbar and D. J. Brady, “Interferometric sensor and method to detect optical fields,” U.S. Patent 6,639,683 (2003).

Handbook of Optics, M. Bass, ed. (McGraw-Hill, 1995), vol. 2.

J. Curlander and R. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, 1991).

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1988), Vol. XXVI, p. 349.
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, 1968).

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

M. Laikin, Lens Design, 3rd ed. (Marcel Dekker, 2001).

R. H. Dyck, “Multiple-frame CCD image sensor with overlying photosensitive layer,” U.S. Patent 5796433 (1998).

T. Hori, “Method for making high-frame-rate CCD imaging devices from otherwise ordinary and inexpensive CCD devices,” U.S. Patent 6255134 (2001).

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Figures (8)

Fig. 1
Fig. 1

(a) SFS ( 250 × 250 ), (b) optical layout, (c) sampling mask geometry, (d) fan-out pattern, (e) fan-out overlap from adjacent sampling points (shown displaced for clarity).

Fig. 2
Fig. 2

LC phase-shifter test: (a) layout, (b) phase-shift signal, (c) calibration curve.

Fig. 3
Fig. 3

Digital phase microscope (laser beam path outlined): 1, laser; 2, attenuator; 3, polarization control optics; 4, LC; 5, spatial filter; 6, beam collimation; 7, turning mirror; 8, microscope body; 9, digital camera for intensity-only images; 10, turning mirror and lens coupler assembly ( 6 × ); 11, 250 × 250 SFS.

Fig. 4
Fig. 4

Typical SFS data: (a) birefringence angle (radians); (b), (c), (d)  X Y , X, and Y phase gradients (radians), respectively; (e) SFS intensity image (arbitrary units); (f) phase estimation repeatability (radians) over the target. Axes in SFS pixels, object sampling 0.92 μm .

Fig. 5
Fig. 5

Landmark image: SFS (a) X and (b) Y images [ cos ( clean  grad ) ] corresponding to DIC image; (c) SFS gradient magnitude; (d) as in (a) and (b), different shear direction; (e) raw intensity in the non-SFS channel; (f) raw intensity (x shear). Arrows, Fresnel fringes.

Fig. 6
Fig. 6

Reconstructed Landmark1 at 7 × EM : (a) lighted surface (height in radians and dimensions in micrometers), (b) weight map ( 0 = black ), (c) two-dimensional map, (d) reconstruction error (radians).

Fig. 7
Fig. 7

Typical human oral epithelial cell ( 7 × EM ). Baseline- corrected reconstruction: (a) surface view, (b) with added color-coded height, and (c) color coded (phase) two-dimensional image. (d) Uncompensated reconstruction. (e)  | ϕ | . (f) Weight map. Bar, 20 μm .

Fig. 8
Fig. 8

C. elegans 7 × EM .3: (a) lighted surface, (b)  Δ ϕ , (c)  Δ I , (d)  | ϕ | , (e) birefringence map. Axes, micrometers. Bar, 20 μm .

Equations (7)

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

σ ϕ = 6 N frames N pix I 0 ,
I X ( θ ) = k 5 I A + k 4 I B + 2 k 4 k 5 I A I B cos ( θ + ϕ A ϕ B + ϕ 4 5 ) for     θ = 0 to 2 π , I Y ( θ ) = k 2 I A + k 7 I C + 2 k 2 k 7 I A I C cos ( θ + ϕ A ϕ C + ϕ 2 7 ) for     θ = 0 to 2 π , I X Y ( θ ) = k 6 I A + k 3 I D + 2 k 6 k 3 I A I D cos ( θ + ϕ A ϕ D + ϕ 6 3 ) for     θ = 0 to 2 π , I Pol ( θ ) = k 1 I E + k 8 I O + 2 k 1 k 8 I E I O cos ( θ + ϕ E ϕ O + ϕ 1 8 ) for     θ = 0 to 2 π ,
ϕ A ϕ B = tan 1 { 2 [ I X ( 3 π / 2 ) I X ( π / 2 ) ] I X ( 0 ) 2 I X ( π ) + I X ( 2 π ) } ϕ 4 5 .
I A = I E + I O = 0.25 × [ I Pol ( 0 ) + I Pol ( π / 2 ) + I Pol ( π ) + I Pol ( 3 π / 2 ) ] .
ϕ E ϕ O = tan 1 ( 2 × ( I Pol ( 3 π / 2 ) I Pol ( π / 2 ) ) I Pol ( 0 ) 2 I Pol ( π ) + I Pol ( 2 π ) ) ϕ 1 8 , I E / I O = 1 2 M + ( 1 2 M ) 1 / 2 2 M ,
M = 1 4 [ ( I Pol ( 0 ) I A 1 ) 2 + ( I Pol ( 3 π / 2 ) I A 1 ) 2 ] 1 .
( G x , G y )

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