Abstract

We present a method of using an unmodified differential interference contrast microscope to acquire quantitative information on scatter and absorption of thin tissue samples. A simple calibration process is discussed that uses a standard optical wedge. Subsequently, we present a phase-stepping procedure for acquiring phase gradient information exclusive of absorption effects. The procedure results in two-dimensional maps of the local angular (polar and azimuthal) ray deviation. We demonstrate the calibration process, discuss details of the phase-stepping algorithm, and present representative results for a porcine skin sample.

© 2011 Optical Society of America

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  1. H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, X. Li, J. D. Rogers, A. Heifetz, D. Kunte, H. K. Roy, A. Taflove, and V. Backman, “Optical methodology for detecting histologically unapparent nanoscale consequences of genetic alterations in biological cells,” Proc. Natl. Acad. Sci. USA 105, 20124–10129 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. D. G. Fischer, S. A. Prahl, and D. D. Duncan, “Monte Carlo modeling of spatial coherence: free-space diffraction,” J. Opt. Soc. Am. A 25, 2571–2581 (2008).
    [CrossRef]
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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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    [CrossRef] [PubMed]
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  14. S. B. Mehta and C. J. R. Sheppard, “Sample-less calibration of the differential interference contrast microscope,” Appl. Opt. 49, 2954–2968 (2010).
    [CrossRef] [PubMed]
  15. C. B. Müller, K. Weiß, W. Richtering, A. Loman, and J. Enderlein, “Calibrating differential interference contrast microscopy with dual-focus fluorescence correlation spectroscopy,” Opt. Express 16, 4322–4329 (2008).
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    [CrossRef]
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    [CrossRef]
  24. S. V. King, A. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, “Quantitative phase microscopy through differential interference imaging,” J. Biomed. Opt. 13, 024020 (2008).
    [CrossRef] [PubMed]
  25. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 4th ed., Course of Theoretical Physics Series (Reed Elsevier, 2000), Volume  2.
  26. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E.Wolf, ed. (Elsevier, 1988), Vol.  26, pp. 349–393.
    [CrossRef]
  27. J. Novák, P. Novák, and A. Mikš, “Multi-step phase-shifting algorithms insensitive to linear phase shift errors,” Opt. Commun. 281, 5302–5309 (2008).
    [CrossRef]
  28. W. J. Tropf, M. E. Thomas, and T. J. Harris, “Properties of crystals and glasses,” in Devices Measurements, & Properties, 2nd ed., M.Bass, E.W.Van Stryland, D.R.Williams, and W.L.Wolfe, eds., Handbook of Optics (McGraw-Hill, 1995), Vol.  II, pp. 33.3–33.101.
  29. S. B. Mehta and C. J. R. Sheppard, “Quantitative phase retrieval in the partially coherent differential interference contrast (DIC) microscope,” presented at Focus on Microscopy, Krakow, Poland, 5–8 April 2009.
  30. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, 1970).
  31. S. B. Mehta and C. J. R. Sheppard, “Partially coherent image formation in differential interference contrast (DIC) microscope,” Opt. Express 16, 19462–19479 (2008).
    [CrossRef] [PubMed]
  32. Q. Kemao, S. Fangjun, and W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. 11, 1220–1223 (2000).
    [CrossRef]
  33. B. Gutmann and H. Weber, “Phase-shifter calibration and error detection in phase-shifting applications: a new method,” Appl. Opt. 37, 7624–7631 (1998).
    [CrossRef]
  34. S. Schwartz, D. B. Murphy, K. R. Spring, and M. W. Davidson, “de Sénarmont bias retardation in DIC microscopy,” http://www.microscopyu.com/pdfs/DICMicroscopy.pdf (Nikon MicroscopyU, 2003).

2010

2009

2008

H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, X. Li, J. D. Rogers, A. Heifetz, D. Kunte, H. K. Roy, A. Taflove, and V. Backman, “Optical methodology for detecting histologically unapparent nanoscale consequences of genetic alterations in biological cells,” Proc. Natl. Acad. Sci. USA 105, 20124–10129 (2008).
[CrossRef]

D. G. Fischer, S. A. Prahl, and D. D. Duncan, “Monte Carlo modeling of spatial coherence: free-space diffraction,” J. Opt. Soc. Am. A 25, 2571–2581 (2008).
[CrossRef]

C. B. Müller, K. Weiß, W. Richtering, A. Loman, and J. Enderlein, “Calibrating differential interference contrast microscopy with dual-focus fluorescence correlation spectroscopy,” Opt. Express 16, 4322–4329 (2008).
[CrossRef] [PubMed]

S. V. King, A. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, “Quantitative phase microscopy through differential interference imaging,” J. Biomed. Opt. 13, 024020 (2008).
[CrossRef] [PubMed]

J. Novák, P. Novák, and A. Mikš, “Multi-step phase-shifting algorithms insensitive to linear phase shift errors,” Opt. Commun. 281, 5302–5309 (2008).
[CrossRef]

S. B. Mehta and C. J. R. Sheppard, “Partially coherent image formation in differential interference contrast (DIC) microscope,” Opt. Express 16, 19462–19479 (2008).
[CrossRef] [PubMed]

2006

W. Gao, “Changes of polarization of light beams on propagation through tissue,” Opt. Commun. 260, 749–754 (2006).
[CrossRef]

C. Preza, S. V. King, and C. J. Cogswell, “Algorithms for extracting true phase from rotationally-diverse and phase-shifted DIC images,” Proc. SPIE 6090, 60900E (2006).
[CrossRef]

M. Shribak and S. Inoué, “Orientation-independent differential interference contrast microscopy,” Appl. Opt. 45, 460–469 (2006).
[CrossRef] [PubMed]

2004

2002

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[CrossRef]

N. N. Boustany, R. Drezek, and N. V. Thakor, “Calcium-induced alterations in mitochondrial morphology quantified in situ with optical scatter imaging,” Biophys. J. 83, 1691–1700 (2002).
[CrossRef] [PubMed]

2001

2000

Q. Kemao, S. Fangjun, and W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. 11, 1220–1223 (2000).
[CrossRef]

1999

1998

1997

1996

1983

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Backman, V.

H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, X. Li, J. D. Rogers, A. Heifetz, D. Kunte, H. K. Roy, A. Taflove, and V. Backman, “Optical methodology for detecting histologically unapparent nanoscale consequences of genetic alterations in biological cells,” Proc. Natl. Acad. Sci. USA 105, 20124–10129 (2008).
[CrossRef]

Barbastathis, G.

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, 1970).

Boustany, N. N.

Capoglu, I. R.

H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, X. Li, J. D. Rogers, A. Heifetz, D. Kunte, H. K. Roy, A. Taflove, and V. Backman, “Optical methodology for detecting histologically unapparent nanoscale consequences of genetic alterations in biological cells,” Proc. Natl. Acad. Sci. USA 105, 20124–10129 (2008).
[CrossRef]

Cogswell, C. J.

S. V. King, A. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, “Quantitative phase microscopy through differential interference imaging,” J. Biomed. Opt. 13, 024020 (2008).
[CrossRef] [PubMed]

C. Preza, S. V. King, and C. J. Cogswell, “Algorithms for extracting true phase from rotationally-diverse and phase-shifted DIC images,” Proc. SPIE 6090, 60900E (2006).
[CrossRef]

Conchello, J.-A.

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E.Wolf, ed. (Elsevier, 1988), Vol.  26, pp. 349–393.
[CrossRef]

Dana, K. J.

K. J. Dana, “Three dimensional reconstruction of the tectorial membrane: an image processing method using Nomarski differential interference contrast microscopy,” Master’s thesis (Massachusetts Institute of Technology, 1992).

Davidson, M. W.

S. Schwartz, D. B. Murphy, K. R. Spring, and M. W. Davidson, “de Sénarmont bias retardation in DIC microscopy,” http://www.microscopyu.com/pdfs/DICMicroscopy.pdf (Nikon MicroscopyU, 2003).

Dogariu, A.

Drezek, R.

N. N. Boustany, R. Drezek, and N. V. Thakor, “Calcium-induced alterations in mitochondrial morphology quantified in situ with optical scatter imaging,” Biophys. J. 83, 1691–1700 (2002).
[CrossRef] [PubMed]

Duncan, D. D.

Enderlein, J.

Fangjun, S.

Q. Kemao, S. Fangjun, and W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. 11, 1220–1223 (2000).
[CrossRef]

Fischer, D. G.

Gao, W.

W. Gao, “Changes of polarization of light beams on propagation through tissue,” Opt. Commun. 260, 749–754 (2006).
[CrossRef]

Gbur, G.

Gutmann, B.

Harris, T. J.

W. J. Tropf, M. E. Thomas, and T. J. Harris, “Properties of crystals and glasses,” in Devices Measurements, & Properties, 2nd ed., M.Bass, E.W.Van Stryland, D.R.Williams, and W.L.Wolfe, eds., Handbook of Optics (McGraw-Hill, 1995), Vol.  II, pp. 33.3–33.101.

Heifetz, A.

H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, X. Li, J. D. Rogers, A. Heifetz, D. Kunte, H. K. Roy, A. Taflove, and V. Backman, “Optical methodology for detecting histologically unapparent nanoscale consequences of genetic alterations in biological cells,” Proc. Natl. Acad. Sci. USA 105, 20124–10129 (2008).
[CrossRef]

Inoué, S.

Kemao, Q.

Q. Kemao, S. Fangjun, and W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. 11, 1220–1223 (2000).
[CrossRef]

King, S. V.

S. V. King, A. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, “Quantitative phase microscopy through differential interference imaging,” J. Biomed. Opt. 13, 024020 (2008).
[CrossRef] [PubMed]

C. Preza, S. V. King, and C. J. Cogswell, “Algorithms for extracting true phase from rotationally-diverse and phase-shifted DIC images,” Proc. SPIE 6090, 60900E (2006).
[CrossRef]

Knüttel, A.

Kou, S. S.

Kumar, G.

Kunte, D.

H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, X. Li, J. D. Rogers, A. Heifetz, D. Kunte, H. K. Roy, A. Taflove, and V. Backman, “Optical methodology for detecting histologically unapparent nanoscale consequences of genetic alterations in biological cells,” Proc. Natl. Acad. Sci. USA 105, 20124–10129 (2008).
[CrossRef]

Kuo, S. C.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 4th ed., Course of Theoretical Physics Series (Reed Elsevier, 2000), Volume  2.

Li, X.

H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, X. Li, J. D. Rogers, A. Heifetz, D. Kunte, H. K. Roy, A. Taflove, and V. Backman, “Optical methodology for detecting histologically unapparent nanoscale consequences of genetic alterations in biological cells,” Proc. Natl. Acad. Sci. USA 105, 20124–10129 (2008).
[CrossRef]

Libertun, A.

S. V. King, A. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, “Quantitative phase microscopy through differential interference imaging,” J. Biomed. Opt. 13, 024020 (2008).
[CrossRef] [PubMed]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 4th ed., Course of Theoretical Physics Series (Reed Elsevier, 2000), Volume  2.

Liu, Y.

H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, X. Li, J. D. Rogers, A. Heifetz, D. Kunte, H. K. Roy, A. Taflove, and V. Backman, “Optical methodology for detecting histologically unapparent nanoscale consequences of genetic alterations in biological cells,” Proc. Natl. Acad. Sci. USA 105, 20124–10129 (2008).
[CrossRef]

Loman, A.

Mehta, S. B.

Metaxas, D. N.

Mikš, A.

J. Novák, P. Novák, and A. Mikš, “Multi-step phase-shifting algorithms insensitive to linear phase shift errors,” Opt. Commun. 281, 5302–5309 (2008).
[CrossRef]

Mujat, C.

Müller, C. B.

Murphy, D.

D. Murphy, “Differential interference contrast (DIC) microscopy and modulation contrast microscopy,” in Fundamentals of Light Microscopy and Digital Imaging (Wiley-Liss, 2001), pp. 153–168.

Murphy, D. B.

S. Schwartz, D. B. Murphy, K. R. Spring, and M. W. Davidson, “de Sénarmont bias retardation in DIC microscopy,” http://www.microscopyu.com/pdfs/DICMicroscopy.pdf (Nikon MicroscopyU, 2003).

Novák, J.

J. Novák, P. Novák, and A. Mikš, “Multi-step phase-shifting algorithms insensitive to linear phase shift errors,” Opt. Commun. 281, 5302–5309 (2008).
[CrossRef]

Novák, P.

J. Novák, P. Novák, and A. Mikš, “Multi-step phase-shifting algorithms insensitive to linear phase shift errors,” Opt. Commun. 281, 5302–5309 (2008).
[CrossRef]

Pasternack, R. M.

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Piestun, R.

S. V. King, A. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, “Quantitative phase microscopy through differential interference imaging,” J. Biomed. Opt. 13, 024020 (2008).
[CrossRef] [PubMed]

Pradhan, P.

H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, X. Li, J. D. Rogers, A. Heifetz, D. Kunte, H. K. Roy, A. Taflove, and V. Backman, “Optical methodology for detecting histologically unapparent nanoscale consequences of genetic alterations in biological cells,” Proc. Natl. Acad. Sci. USA 105, 20124–10129 (2008).
[CrossRef]

Prahl, S. A.

Preza, C.

S. V. King, A. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, “Quantitative phase microscopy through differential interference imaging,” J. Biomed. Opt. 13, 024020 (2008).
[CrossRef] [PubMed]

C. Preza, S. V. King, and C. J. Cogswell, “Algorithms for extracting true phase from rotationally-diverse and phase-shifted DIC images,” Proc. SPIE 6090, 60900E (2006).
[CrossRef]

C. Preza, D. L. Snyder, and J.-A. Conchello, “Theoretical development and experimental evaluation of imaging models for differential-interference-contrast microscopy,” J. Opt. Soc. Am. A 16, 2185–2199 (1999).
[CrossRef]

Qian, Z.

Richtering, W.

Rogers, J. D.

H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, X. Li, J. D. Rogers, A. Heifetz, D. Kunte, H. K. Roy, A. Taflove, and V. Backman, “Optical methodology for detecting histologically unapparent nanoscale consequences of genetic alterations in biological cells,” Proc. Natl. Acad. Sci. USA 105, 20124–10129 (2008).
[CrossRef]

Roy, H. K.

H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, X. Li, J. D. Rogers, A. Heifetz, D. Kunte, H. K. Roy, A. Taflove, and V. Backman, “Optical methodology for detecting histologically unapparent nanoscale consequences of genetic alterations in biological cells,” Proc. Natl. Acad. Sci. USA 105, 20124–10129 (2008).
[CrossRef]

Schmitt, J. M.

Schwartz, S.

S. Schwartz, D. B. Murphy, K. R. Spring, and M. W. Davidson, “de Sénarmont bias retardation in DIC microscopy,” http://www.microscopyu.com/pdfs/DICMicroscopy.pdf (Nikon MicroscopyU, 2003).

Sheppard, C. J. R.

Shribak, M.

Snyder, D. L.

Spring, K. R.

S. Schwartz, D. B. Murphy, K. R. Spring, and M. W. Davidson, “de Sénarmont bias retardation in DIC microscopy,” http://www.microscopyu.com/pdfs/DICMicroscopy.pdf (Nikon MicroscopyU, 2003).

Subramanian, H.

H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, X. Li, J. D. Rogers, A. Heifetz, D. Kunte, H. K. Roy, A. Taflove, and V. Backman, “Optical methodology for detecting histologically unapparent nanoscale consequences of genetic alterations in biological cells,” Proc. Natl. Acad. Sci. USA 105, 20124–10129 (2008).
[CrossRef]

Taflove, A.

H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, X. Li, J. D. Rogers, A. Heifetz, D. Kunte, H. K. Roy, A. Taflove, and V. Backman, “Optical methodology for detecting histologically unapparent nanoscale consequences of genetic alterations in biological cells,” Proc. Natl. Acad. Sci. USA 105, 20124–10129 (2008).
[CrossRef]

Tatarski, V. I.

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (The National Oceanic and Atmospheric Administration, U. S. Department of Commerce and The National Science Foundation, 1971).

Teague, M. R.

Thakor, N. V.

N. N. Boustany, R. Drezek, and N. V. Thakor, “Calcium-induced alterations in mitochondrial morphology quantified in situ with optical scatter imaging,” Biophys. J. 83, 1691–1700 (2002).
[CrossRef] [PubMed]

N. N. Boustany, S. C. Kuo, and N. V. Thakor, “Optical scatter imaging: subcellular morphometry in situ with Fourier filtering,” Opt. Lett. 26, 1063–1065 (2001).
[CrossRef]

Thomas, M. E.

W. J. Tropf, M. E. Thomas, and T. J. Harris, “Properties of crystals and glasses,” in Devices Measurements, & Properties, 2nd ed., M.Bass, E.W.Van Stryland, D.R.Williams, and W.L.Wolfe, eds., Handbook of Optics (McGraw-Hill, 1995), Vol.  II, pp. 33.3–33.101.

Tropf, W. J.

W. J. Tropf, M. E. Thomas, and T. J. Harris, “Properties of crystals and glasses,” in Devices Measurements, & Properties, 2nd ed., M.Bass, E.W.Van Stryland, D.R.Williams, and W.L.Wolfe, eds., Handbook of Optics (McGraw-Hill, 1995), Vol.  II, pp. 33.3–33.101.

Waller, L.

Weber, H.

Weiß, K.

Wolf, E.

Xiaoping, W.

Q. Kemao, S. Fangjun, and W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. 11, 1220–1223 (2000).
[CrossRef]

Zheng, J.-Y.

Appl. Opt.

Biophys. J.

N. N. Boustany, R. Drezek, and N. V. Thakor, “Calcium-induced alterations in mitochondrial morphology quantified in situ with optical scatter imaging,” Biophys. J. 83, 1691–1700 (2002).
[CrossRef] [PubMed]

J. Biomed. Opt.

S. V. King, A. Libertun, R. Piestun, C. J. Cogswell, and C. Preza, “Quantitative phase microscopy through differential interference imaging,” J. Biomed. Opt. 13, 024020 (2008).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Meas. Sci. Technol.

Q. Kemao, S. Fangjun, and W. Xiaoping, “Determination of the best phase step of the Carré algorithm in phase shifting interferometry,” Meas. Sci. Technol. 11, 1220–1223 (2000).
[CrossRef]

Opt. Commun.

J. Novák, P. Novák, and A. Mikš, “Multi-step phase-shifting algorithms insensitive to linear phase shift errors,” Opt. Commun. 281, 5302–5309 (2008).
[CrossRef]

W. Gao, “Changes of polarization of light beams on propagation through tissue,” Opt. Commun. 260, 749–754 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. Natl. Acad. Sci. USA

H. Subramanian, P. Pradhan, Y. Liu, I. R. Capoglu, X. Li, J. D. Rogers, A. Heifetz, D. Kunte, H. K. Roy, A. Taflove, and V. Backman, “Optical methodology for detecting histologically unapparent nanoscale consequences of genetic alterations in biological cells,” Proc. Natl. Acad. Sci. USA 105, 20124–10129 (2008).
[CrossRef]

Proc. SPIE

C. Preza, S. V. King, and C. J. Cogswell, “Algorithms for extracting true phase from rotationally-diverse and phase-shifted DIC images,” Proc. SPIE 6090, 60900E (2006).
[CrossRef]

Other

K. J. Dana, “Three dimensional reconstruction of the tectorial membrane: an image processing method using Nomarski differential interference contrast microscopy,” Master’s thesis (Massachusetts Institute of Technology, 1992).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (The National Oceanic and Atmospheric Administration, U. S. Department of Commerce and The National Science Foundation, 1971).

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 4th ed., Course of Theoretical Physics Series (Reed Elsevier, 2000), Volume  2.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E.Wolf, ed. (Elsevier, 1988), Vol.  26, pp. 349–393.
[CrossRef]

W. J. Tropf, M. E. Thomas, and T. J. Harris, “Properties of crystals and glasses,” in Devices Measurements, & Properties, 2nd ed., M.Bass, E.W.Van Stryland, D.R.Williams, and W.L.Wolfe, eds., Handbook of Optics (McGraw-Hill, 1995), Vol.  II, pp. 33.3–33.101.

S. B. Mehta and C. J. R. Sheppard, “Quantitative phase retrieval in the partially coherent differential interference contrast (DIC) microscope,” presented at Focus on Microscopy, Krakow, Poland, 5–8 April 2009.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, 1970).

D. Murphy, “Differential interference contrast (DIC) microscopy and modulation contrast microscopy,” in Fundamentals of Light Microscopy and Digital Imaging (Wiley-Liss, 2001), pp. 153–168.

S. Schwartz, D. B. Murphy, K. R. Spring, and M. W. Davidson, “de Sénarmont bias retardation in DIC microscopy,” http://www.microscopyu.com/pdfs/DICMicroscopy.pdf (Nikon MicroscopyU, 2003).

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Figures (14)

Fig. 1
Fig. 1

Illustration of the components of a Nomarski DIC microscope.

Fig. 2
Fig. 2

Phases of rays deflected by an optical wedge.

Fig. 3
Fig. 3

Illustration of the effect of wedge orientation with respect to shear direction.

Fig. 4
Fig. 4

Global fit of data to Eq. (16) for red color channel. Numbers, N, refer to the number of turns of the bias screw on the upper Nomarski prism; the angle γ γ s is the angular difference between the wedge orientation and shear direction.

Fig. 5
Fig. 5

Step-size estimates from the Carré algorithm with mean, β , plus or minus one standard deviation.

Fig. 6
Fig. 6

Carré-derived phases and fit to model. Fit model is Φ = Φ i + m cos ( γ γ s ) , where the slope and intercept values are respectively m = 120 ° and Φ i = 129 ° .

Fig. 7
Fig. 7

Montage of images illustrating processing of DIC data. (a) Original DIC image of porcine skin sample prepared with BrDU stain (red color channel); (b) grayscale encoded map of scatter angle, θ x ( x , y ) , along the axis of shear (range of angles displayed is ( 2 ° , 2 ° ) ); (c) mean image, a ( x , y ) [see Eq. (5)]; (d) modulation image, b ( x , y ) [see Eq. (5)]; (e) grayscale-encoded map of polar scatter angle, η ( x , y ) (range of angles displayed is ( 0 , 2 ° ) ); (f) grayscale-encoded map of azimuthal scatter angle, ξ ( x , y ) (range of angles displayed is ( 180 ° , 180 ° ) ).

Fig. 8
Fig. 8

Quadratic least-squares fit to valid estimates of phase step, β ( x , y ) ; units are degrees.

Fig. 9
Fig. 9

PDFs of valid phase-step angles and the corresponding fit shown in Fig. 8.

Fig. 10
Fig. 10

PDF of the modulation visibility for porcine skin sample.

Fig. 11
Fig. 11

PDFs of polar (top) and azimuthal (bottom) scatter angles for tissue sample. Also shown for the polar scatter angle is the best approximate HG phase function. Note that the HG phase function has been multiplied by sin η , thus giving the product the formal definition of a PDF. For the azimuthal angle, cos ξ = 0.148 , as opposed to cos ξ = 0 , for a completely uniform distribution.

Fig. 12
Fig. 12

Relationship between polar scatter angle, η ( x , y ) , and grayscale values of the modulation image, b ( x , y ) . Line is least-squares fit; correlation is computed point by point.

Fig. 13
Fig. 13

False color, log 10 , encoding of polar scatter angle PSD. DC is in center; axis limits are ± 1 / 2 p , where p is the pixel size ( 0.678 μm ).

Fig. 14
Fig. 14

Azimuthally integrated polar scatter angle PSD.

Tables (2)

Tables Icon

Table 1 Results of Global Fits to Data

Tables Icon

Table 2 Parameter Values from Carré Analyses

Equations (32)

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I ( x , y ) = A ( x , y ) { 1 + cos [ ϕ ( x + s , y ) ϕ ( x , y ) + Ψ ] } ,
I ( x , y ) = A ( x , y ) { 1 + cos [ Φ ( x , y ) + Ψ ] } ,
Φ ( x , y ) = s ϕ ( x , y ) x .
sin θ x = [ Φ ( x , y ) k s ] ,
tan η = tan 2 θ x + tan 2 θ y , tan ξ = tan θ y / tan θ x .
I j ( x , y ) = a ( x , y ) + b ( x , y ) cos [ Φ ( x , y ) + ( 2 j 3 2 ) β ( x , y ) ] ,
tan [ β ( x , y ) 2 ] = 3 [ I 1 ( x , y ) I 2 ( x , y ) ] [ I 0 ( x , y ) I 3 ( x , y ) ] [ I 1 ( x , y ) I 2 ( x , y ) ] + [ I 0 ( x , y ) I 3 ( x , y ) ] .
tan Φ ( x , y ) = tan [ β ^ ( x , y ) 2 ] [ I 0 ( x , y ) I 3 ( x , y ) ] + [ I 1 ( x , y ) I 2 ( x , y ) ] [ I 1 ( x , y ) + I 2 ( x , y ) ] [ I 0 ( x , y ) + I 3 ( x , y ) ] .
[ I 0 I 1 I 2 I 3 ] = [ 1 C 0 1 C 1 1 C 2 1 C 3 ] [ a b ] ; C j = cos [ Φ ( x , y ) + ( 2 j 3 2 ) β ^ ( x , y ) ] ,
[ I j C j I j ] = [ 4 C j C j C j 2 ] [ a b ] .
a ^ ( x , y ) = I j C j 2 C j I j C j 4 C j 2 ( C j ) 2 , b ^ ( x , y ) = 4 C j I j C j I j 4 C j 2 ( C j ) 2 .
ϕ ( x 2 , y ) ϕ ( x 1 , y ) = ϕ ( x + s , y ) ϕ ( x , y ) = k ( h 2 h 1 ) ( n 1 cos θ x ) ,
n sin ε = sin ( ε + θ x ) .
ϕ ( x + s , y ) ϕ ( x , y ) = k ( h 2 h 1 ) ( n 1 ) .
h 2 = h 1 + s cos ( γ γ s ) tan ε .
ϕ ( x + s , y ) ϕ ( x , y ) = k s ( n 1 ) tan ε cos ( γ γ s ) .
I = A { 1 + cos [ Φ 0 + m cos ( γ γ s ) + N ( β / 2 ) ] } ,
m = k s ( n 1 ) tan ε ,
s λ = m 2 π ( n 1 ) tan ε = 2.0 ,
Φ = Φ i + m cos ( γ γ s ) .
m = 120 ° , Φ i = 129 ° .
I j ( x , y ) = a ( x , y ) + b ( x , y ) cos [ Φ ( x , y ) ( 3 2 ) β ( x , y ) + j β ( x , y ) ] .
Φ i = 129 ° = Φ 0 + 3 β / 2 Φ 0 = 63 ° ,
resolution λ 0.77 NA = 2.57 .
θ x = sin 1 [ Φ Φ i 2 π ( s / λ ) ] ,
P HG ( η ) = 1 2 1 g 2 ( 1 2 g cos η + g 2 ) 3 / 2 .
S ( cos η ) = e μ s d 1 2 π δ ( 1 cos η ) + ( 1 e μ s d ) P ( cos η ) ,
4 π P ( cos η ) d Ω 1 and 4 π P ( cos η ) cos η d Ω g ,
g = 4 π S ( cos η ) cos η d Ω ,
g = e μ s d 4 π 1 2 π δ ( 1 cos η ) cos η d Ω + ( 1 e μ s d ) 4 π P ( cos η ) cos η d Ω ,
g = e μ s d + ( 1 e μ s d ) g .
g = 0.95 + 0.05 ( 0.9 ) = 0.995 .

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