Abstract

Some different image formation models have been proposed for Nomarski’s differential interference contrast (DIC) microscope. However, the nature of coherence of illumination in DIC, of key importance in image formation, remains to be elucidated. We present a partially coherent image formation model for DIC and demonstrate that DIC microscope images the coherent difference of shifted replicas of the specimen; but imaging of the each component is partially coherent. Partially coherent transfer functions are presented for various DIC configurations. Plots of these transfer functions and experimental images provide quantitative comparison among various DIC configurations and elucidate their imaging properties. Approximations for weak or slowly varying specimens are also given. These improved models should be of great value in designing phase retrieval algorithms for DIC.

© 2008 Optical Society of America

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References

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  1. F. Zernike, "How I discovered phase contrast," Science 121, 345-349 (1955).
    [CrossRef] [PubMed]
  2. M. Franc¸on and S. Mallick, Polarization Interferometers: Applications in Microscopy and Macroscopy (Wiley-Interscience, 1971).
  3. M. Pluta, Advanced Light Microscopy, vol. 2 Specialized Methods (PWN-Polish Scientific Publishers,Warszawa, 1989).
  4. R. D. Allen, G. B. David, and G. Nomarski, "The Zeiss-Nomarski differential interference equipment for transmitted-light microscopy." Z Wiss Mikrosk 69, 193-221 (1969).
    [PubMed]
  5. R. D. Allen, J. L. Travis, N. S. Allen, and H. Yilmaz, "Video-enhanced contrast polarization (AVEC-POL) microscopy: a new method applied to the detection of birefringence in the motile reticulopodial network of Allogromia laticollaris." Cell Motil 1, 275-289 (1981).
    [CrossRef] [PubMed]
  6. C. J. R. Sheppard and T. Wilson, "Fourier imaging of phase information in scanning and conventional optical microscopes," Phil. Trans. Roy. Soc. London, Series A 295, 513-536 (1980).
    [CrossRef]
  7. C. Cogswell and C. Sheppard, "Confocal differential interference contrast(DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging," J. Microsc. 165, 81-101 (1992).
    [CrossRef]
  8. 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]
  9. C. Preza, "Rotational-diversity phase estimation from differential-interference-contrast microscopy images," J. Opt. Soc. Am. A 17, 415-424 (2000).
    [CrossRef]
  10. M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. Microsc. 199, 79-84 (2000).
    [CrossRef] [PubMed]
  11. M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
    [CrossRef] [PubMed]
  12. 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]
  13. M. Shribak, J. LaFountain, D. Biggs, and S.  Inou’e, "Quantitative orientation-independent differential interference contrast (DIC) microscopy," Proceedings of SPIE 6441, 64411L (2007).
    [CrossRef]
  14. J. A. O’Sullivan and C. Preza, "Alternating minimization algorithm for quantitative differential-interference contrast (DIC) microscopy," Proceedings of SPIE 6814, 68140Y (2008).
    [CrossRef]
  15. Z. Kam, "Microscopic differential interference contrast image processing by line integration (LID) and deconvolution," Bioimaging 6, 166-176 (1998).
    [CrossRef]
  16. M. Franc¸on, Optical Interferometry (Academic Press, 1966).
  17. O. von Hofsten, M. Bertilson, and U. Vogt, "Theoretical development of a high-resolutiondifferentialinterference-contrast optic for x-raymicroscopy," Opt. Express 16, 1132-1141 (2008). http://www.opticsexpress.org/abstract.cfm?URI=oe-16-2-1132.
    [CrossRef] [PubMed]
  18. R. Danz, A. Vogelgsang, and R. Kathner, "PlasDIC - a useful modification of the differential interference contrast according to Smith/Nomarski in transmitted light arrangement," Photonik (2004). www.zeiss.com/C1256F8500454979/0/366354E1E8BA8703C1256F8E003BBCB9/$file/plasdic photonik 2004march e.pdf.
  19. R. Danz, P. Dietrich, A. Soell, C. Hoyer, and M. Wagener, "Arrangement and method for polarization-optical interference contrast," (2006). US Patent No. 7046436.
  20. M. Pluta, Advanced Light Microscopy, vol. 1 Principles and Basic Properties (PWN-Polish Scientific Publishers, Warszawa, 1988).
  21. H. H. Hopkins, "On the diffraction theory of optical images," Proc. R. Soc. Lond. A, Math. and Phys. Sci. 217, 408-432 (1953).
    [CrossRef]
  22. C. J. R. Sheppard and A. Choudhury, "Image formation in the scanning microscope," J. Mod. Opt. 24, 1051-1073 (1977).
  23. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, Cambridge, 1999).
    [PubMed]
  24. A. A. Lebedeff, "L’interf´erom`etre `a polarisation et ses applications," Rev. d’Opt 9, 385-413 (1930). ("Polarization interferometer and its applications").
  25. F. H. Smith, "Interference Microscope," (1952). US patent no. 2601175.
  26. G. Nomarski, "Microinterf’erom`etre diff’erentiel `a ondes polaris’ees," J. Phys. Radium 16, 9-13 (1955). ("Differential microinterferometer with polarized waves").
  27. M. Franc¸on, "Polarization interference microscopes," Appl. Opt. 3, 1033-1036 (1964).
    [CrossRef]
  28. P. Munro and P. T¨or¨ok, "Vectorial, high numerical aperture study of Nomarski’s differential interference contrast microscope," Opt. Express 13, 6833-6847 (2005). http://www.opticsexpress.org/abstract.cfm?URI=oe-13-18-6833.
    [CrossRef] [PubMed]
  29. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  30. T. Wilson and C. J. R. Sheppard, "Coded apertures and detectors for optical differentiation," in Int. Optical Computing Conference, vol. 232, pp. 203-209 (Washington DC, 1980).
  31. T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscope (Academic Press, London, 1984).
  32. B. M¨oller, "Imaging of a straight edge in partially coherent illumination in the presence of spherical aberrations," J. Mod. Opt. 15, 223-236 (1968).
  33. C. Preza, "Phase estimation using rotational diversity for differential interference contrast microscopy," Ph.D. thesis, Washington University (1998).

2008

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. A. O’Sullivan and C. Preza, "Alternating minimization algorithm for quantitative differential-interference contrast (DIC) microscopy," Proceedings of SPIE 6814, 68140Y (2008).
[CrossRef]

O. von Hofsten, M. Bertilson, and U. Vogt, "Theoretical development of a high-resolutiondifferentialinterference-contrast optic for x-raymicroscopy," Opt. Express 16, 1132-1141 (2008). http://www.opticsexpress.org/abstract.cfm?URI=oe-16-2-1132.
[CrossRef] [PubMed]

2007

M. Shribak, J. LaFountain, D. Biggs, and S.  Inou’e, "Quantitative orientation-independent differential interference contrast (DIC) microscopy," Proceedings of SPIE 6441, 64411L (2007).
[CrossRef]

2005

2004

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

2000

C. Preza, "Rotational-diversity phase estimation from differential-interference-contrast microscopy images," J. Opt. Soc. Am. A 17, 415-424 (2000).
[CrossRef]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. Microsc. 199, 79-84 (2000).
[CrossRef] [PubMed]

1999

1998

Z. Kam, "Microscopic differential interference contrast image processing by line integration (LID) and deconvolution," Bioimaging 6, 166-176 (1998).
[CrossRef]

1992

C. Cogswell and C. Sheppard, "Confocal differential interference contrast(DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging," J. Microsc. 165, 81-101 (1992).
[CrossRef]

1981

R. D. Allen, J. L. Travis, N. S. Allen, and H. Yilmaz, "Video-enhanced contrast polarization (AVEC-POL) microscopy: a new method applied to the detection of birefringence in the motile reticulopodial network of Allogromia laticollaris." Cell Motil 1, 275-289 (1981).
[CrossRef] [PubMed]

1980

C. J. R. Sheppard and T. Wilson, "Fourier imaging of phase information in scanning and conventional optical microscopes," Phil. Trans. Roy. Soc. London, Series A 295, 513-536 (1980).
[CrossRef]

1977

C. J. R. Sheppard and A. Choudhury, "Image formation in the scanning microscope," J. Mod. Opt. 24, 1051-1073 (1977).

1969

R. D. Allen, G. B. David, and G. Nomarski, "The Zeiss-Nomarski differential interference equipment for transmitted-light microscopy." Z Wiss Mikrosk 69, 193-221 (1969).
[PubMed]

1968

B. M¨oller, "Imaging of a straight edge in partially coherent illumination in the presence of spherical aberrations," J. Mod. Opt. 15, 223-236 (1968).

1964

1955

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

G. Nomarski, "Microinterf’erom`etre diff’erentiel `a ondes polaris’ees," J. Phys. Radium 16, 9-13 (1955). ("Differential microinterferometer with polarized waves").

1953

H. H. Hopkins, "On the diffraction theory of optical images," Proc. R. Soc. Lond. A, Math. and Phys. Sci. 217, 408-432 (1953).
[CrossRef]

Allen, N. S.

R. D. Allen, J. L. Travis, N. S. Allen, and H. Yilmaz, "Video-enhanced contrast polarization (AVEC-POL) microscopy: a new method applied to the detection of birefringence in the motile reticulopodial network of Allogromia laticollaris." Cell Motil 1, 275-289 (1981).
[CrossRef] [PubMed]

Allen, R. D.

R. D. Allen, J. L. Travis, N. S. Allen, and H. Yilmaz, "Video-enhanced contrast polarization (AVEC-POL) microscopy: a new method applied to the detection of birefringence in the motile reticulopodial network of Allogromia laticollaris." Cell Motil 1, 275-289 (1981).
[CrossRef] [PubMed]

R. D. Allen, G. B. David, and G. Nomarski, "The Zeiss-Nomarski differential interference equipment for transmitted-light microscopy." Z Wiss Mikrosk 69, 193-221 (1969).
[PubMed]

Arnison, M. R.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. Microsc. 199, 79-84 (2000).
[CrossRef] [PubMed]

Bertilson, M.

Biggs, D.

M. Shribak, J. LaFountain, D. Biggs, and S.  Inou’e, "Quantitative orientation-independent differential interference contrast (DIC) microscopy," Proceedings of SPIE 6441, 64411L (2007).
[CrossRef]

Choudhury, A.

C. J. R. Sheppard and A. Choudhury, "Image formation in the scanning microscope," J. Mod. Opt. 24, 1051-1073 (1977).

Cogswell, C.

C. Cogswell and C. Sheppard, "Confocal differential interference contrast(DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging," J. Microsc. 165, 81-101 (1992).
[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]

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. Microsc. 199, 79-84 (2000).
[CrossRef] [PubMed]

Conchello, J.-A.

David, G. B.

R. D. Allen, G. B. David, and G. Nomarski, "The Zeiss-Nomarski differential interference equipment for transmitted-light microscopy." Z Wiss Mikrosk 69, 193-221 (1969).
[PubMed]

Fekete, P. W.

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. Microsc. 199, 79-84 (2000).
[CrossRef] [PubMed]

Franc¸on, M.

Hopkins, H. H.

H. H. Hopkins, "On the diffraction theory of optical images," Proc. R. Soc. Lond. A, Math. and Phys. Sci. 217, 408-432 (1953).
[CrossRef]

Inou’e, S.

M. Shribak, J. LaFountain, D. Biggs, and S.  Inou’e, "Quantitative orientation-independent differential interference contrast (DIC) microscopy," Proceedings of SPIE 6441, 64411L (2007).
[CrossRef]

Kam, Z.

Z. Kam, "Microscopic differential interference contrast image processing by line integration (LID) and deconvolution," Bioimaging 6, 166-176 (1998).
[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]

LaFountain, J.

M. Shribak, J. LaFountain, D. Biggs, and S.  Inou’e, "Quantitative orientation-independent differential interference contrast (DIC) microscopy," Proceedings of SPIE 6441, 64411L (2007).
[CrossRef]

Larkin, K. G.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. Microsc. 199, 79-84 (2000).
[CrossRef] [PubMed]

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]

M¨oller, B.

B. M¨oller, "Imaging of a straight edge in partially coherent illumination in the presence of spherical aberrations," J. Mod. Opt. 15, 223-236 (1968).

Munro, P.

Nomarski, G.

R. D. Allen, G. B. David, and G. Nomarski, "The Zeiss-Nomarski differential interference equipment for transmitted-light microscopy." Z Wiss Mikrosk 69, 193-221 (1969).
[PubMed]

G. Nomarski, "Microinterf’erom`etre diff’erentiel `a ondes polaris’ees," J. Phys. Radium 16, 9-13 (1955). ("Differential microinterferometer with polarized waves").

O’Sullivan, J. A.

J. A. O’Sullivan and C. Preza, "Alternating minimization algorithm for quantitative differential-interference contrast (DIC) microscopy," Proceedings of SPIE 6814, 68140Y (2008).
[CrossRef]

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]

Preza, C.

J. A. O’Sullivan and C. Preza, "Alternating minimization algorithm for quantitative differential-interference contrast (DIC) microscopy," Proceedings of SPIE 6814, 68140Y (2008).
[CrossRef]

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, "Rotational-diversity phase estimation from differential-interference-contrast microscopy images," J. Opt. Soc. Am. A 17, 415-424 (2000).
[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]

Sheppard, C.

C. Cogswell and C. Sheppard, "Confocal differential interference contrast(DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging," J. Microsc. 165, 81-101 (1992).
[CrossRef]

Sheppard, C. J. R.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

C. J. R. Sheppard and T. Wilson, "Fourier imaging of phase information in scanning and conventional optical microscopes," Phil. Trans. Roy. Soc. London, Series A 295, 513-536 (1980).
[CrossRef]

C. J. R. Sheppard and A. Choudhury, "Image formation in the scanning microscope," J. Mod. Opt. 24, 1051-1073 (1977).

Shribak, M.

M. Shribak, J. LaFountain, D. Biggs, and S.  Inou’e, "Quantitative orientation-independent differential interference contrast (DIC) microscopy," Proceedings of SPIE 6441, 64411L (2007).
[CrossRef]

Smith, N. I.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. Microsc. 199, 79-84 (2000).
[CrossRef] [PubMed]

Snyder, D. L.

T¨or¨ok, P.

Travis, J. L.

R. D. Allen, J. L. Travis, N. S. Allen, and H. Yilmaz, "Video-enhanced contrast polarization (AVEC-POL) microscopy: a new method applied to the detection of birefringence in the motile reticulopodial network of Allogromia laticollaris." Cell Motil 1, 275-289 (1981).
[CrossRef] [PubMed]

Vogt, U.

von Hofsten, O.

Wilson, T.

C. J. R. Sheppard and T. Wilson, "Fourier imaging of phase information in scanning and conventional optical microscopes," Phil. Trans. Roy. Soc. London, Series A 295, 513-536 (1980).
[CrossRef]

Yilmaz, H.

R. D. Allen, J. L. Travis, N. S. Allen, and H. Yilmaz, "Video-enhanced contrast polarization (AVEC-POL) microscopy: a new method applied to the detection of birefringence in the motile reticulopodial network of Allogromia laticollaris." Cell Motil 1, 275-289 (1981).
[CrossRef] [PubMed]

Zernike, F.

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

Appl. Opt.

Bioimaging

Z. Kam, "Microscopic differential interference contrast image processing by line integration (LID) and deconvolution," Bioimaging 6, 166-176 (1998).
[CrossRef]

Cell Motil

R. D. Allen, J. L. Travis, N. S. Allen, and H. Yilmaz, "Video-enhanced contrast polarization (AVEC-POL) microscopy: a new method applied to the detection of birefringence in the motile reticulopodial network of Allogromia laticollaris." Cell Motil 1, 275-289 (1981).
[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. Microsc.

C. Cogswell and C. Sheppard, "Confocal differential interference contrast(DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging," J. Microsc. 165, 81-101 (1992).
[CrossRef]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. Microsc. 199, 79-84 (2000).
[CrossRef] [PubMed]

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

J. Mod. Opt.

C. J. R. Sheppard and A. Choudhury, "Image formation in the scanning microscope," J. Mod. Opt. 24, 1051-1073 (1977).

B. M¨oller, "Imaging of a straight edge in partially coherent illumination in the presence of spherical aberrations," J. Mod. Opt. 15, 223-236 (1968).

J. Opt. Soc. Am. A

J. Phys. Radium

G. Nomarski, "Microinterf’erom`etre diff’erentiel `a ondes polaris’ees," J. Phys. Radium 16, 9-13 (1955). ("Differential microinterferometer with polarized waves").

Math. and Phys. Sci.

H. H. Hopkins, "On the diffraction theory of optical images," Proc. R. Soc. Lond. A, Math. and Phys. Sci. 217, 408-432 (1953).
[CrossRef]

Opt. Express

Proceedings of SPIE

M. Shribak, J. LaFountain, D. Biggs, and S.  Inou’e, "Quantitative orientation-independent differential interference contrast (DIC) microscopy," Proceedings of SPIE 6441, 64411L (2007).
[CrossRef]

J. A. O’Sullivan and C. Preza, "Alternating minimization algorithm for quantitative differential-interference contrast (DIC) microscopy," Proceedings of SPIE 6814, 68140Y (2008).
[CrossRef]

Science

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

Series A

C. J. R. Sheppard and T. Wilson, "Fourier imaging of phase information in scanning and conventional optical microscopes," Phil. Trans. Roy. Soc. London, Series A 295, 513-536 (1980).
[CrossRef]

Z Wiss Mikrosk

R. D. Allen, G. B. David, and G. Nomarski, "The Zeiss-Nomarski differential interference equipment for transmitted-light microscopy." Z Wiss Mikrosk 69, 193-221 (1969).
[PubMed]

Other

C. Preza, "Phase estimation using rotational diversity for differential interference contrast microscopy," Ph.D. thesis, Washington University (1998).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

T. Wilson and C. J. R. Sheppard, "Coded apertures and detectors for optical differentiation," in Int. Optical Computing Conference, vol. 232, pp. 203-209 (Washington DC, 1980).

T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscope (Academic Press, London, 1984).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, Cambridge, 1999).
[PubMed]

A. A. Lebedeff, "L’interf´erom`etre `a polarisation et ses applications," Rev. d’Opt 9, 385-413 (1930). ("Polarization interferometer and its applications").

F. H. Smith, "Interference Microscope," (1952). US patent no. 2601175.

M. Franc¸on and S. Mallick, Polarization Interferometers: Applications in Microscopy and Macroscopy (Wiley-Interscience, 1971).

M. Pluta, Advanced Light Microscopy, vol. 2 Specialized Methods (PWN-Polish Scientific Publishers,Warszawa, 1989).

M. Franc¸on, Optical Interferometry (Academic Press, 1966).

R. Danz, A. Vogelgsang, and R. Kathner, "PlasDIC - a useful modification of the differential interference contrast according to Smith/Nomarski in transmitted light arrangement," Photonik (2004). www.zeiss.com/C1256F8500454979/0/366354E1E8BA8703C1256F8E003BBCB9/$file/plasdic photonik 2004march e.pdf.

R. Danz, P. Dietrich, A. Soell, C. Hoyer, and M. Wagener, "Arrangement and method for polarization-optical interference contrast," (2006). US Patent No. 7046436.

M. Pluta, Advanced Light Microscopy, vol. 1 Principles and Basic Properties (PWN-Polish Scientific Publishers, Warszawa, 1988).

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Figures (8)

Fig. 1.
Fig. 1.

Comparison of light paths in the direction of shear for three DIC configurations. Imaging in the orthogonal direction is conventional. For simplicity, Wollaston prisms are shown instead of modified Wollaston prisms employed in DIC. The direction of polarization of light in the plane normal to the optical axis is color coded as shown in the legend at top-left. Black line indicates un-polarized light. For all configurations, the light path has been segmented in three key components - the illumination path consisting of the condenser and optical elements in its FFP, the degree of coherence in specimen plane, and the imaging path consisting of the objective and optical elements beyond the objective including the tube-lens and the eyepiece (which are not shown here). Horizontal dimensions have been greatly exaggerated for depicting sub-resolution shear with clarity. Phase bias between two wavefronts coming out of Wo is assumed to be 2ϕ.

Fig. 2.
Fig. 2.

Objective and condenser pupils used for computing transfer functions with S=0.7, shear 2Δ=1/4m0 and bias 2ϕ=π/2.

Fig. 3.
Fig. 3.

Slices through the four dimensional partially coherent transfer function computed for a Nomarski-DIC system with S=0,7, shear 2Δ=1/4m0 , and bias 2ϕ=π/2. Shear azimuth is assumed to be in X direction. (m; p) denotes frequency variable pairs in the X direction, whereas (n;q) denotes frequency variable pairs in the Y-direction. (a) An excerpt at n=q=0 from the image sequence (Media 1) showing 2D slices of the 4D transfer function along m and p with varying n and q. The slice at n=q=0 is a sufficient description for imaging of a specimen varying only in the X direction. (b) An excerpt at m=p=0 from the image sequence (Media 2) showing 2D slices of the 4D transfer function along n and q with varying m and p. The slice at m=p=0 is a sufficient description for imaging of a specimen varying only in the Y direction.

Fig. 4.
Fig. 4.

An excerpt from the image sequence (Media 3) comparing partially coherent transfer functions for S=1,…,0.1 for three configurations in the direction of shear. This is a snapshot at S=0,7. The shear is assumed to be 2Δ=1/4m0 and bias 2ϕ=π/2. Transfer functions are shown for Nomarski-DIC (top row), Köhler-DIC (middle row) and PlasDIC (bottom row). The transfer functions (left column) for all three configurations are separated into their even parts (middle column) and odd parts (right column). To allow clear visualization of frequency support and shape of the transfer function, the color look up table of each plot is stretched to the minimum and maximum values being displayed. Colorbar next to each plot allows comparison of relative strengths of the transfer functions at different values of S (for a given configuration).

Fig. 5.
Fig. 5.

Even part of weak object transfer function (left column), odd part of weak object transfer function (middle column) and phase gradient transfer functions (right column) for three DIC configurations.

Fig. 6.
Fig. 6.

Two excerpts from an image sequence (Media 4) showing variation in the phase gradient contrast with S in images of mouse intestine section (Invitrogen Fluocells prepared slide # 4) imaged with S=0.3,…,1.2. 20X 0.75 NA objective and 0.9NA condenser were used. Shear direction is horizontal. Quasi-monochromatic source of light was created by passing light from a Halogen lamp through a 550nm interference filter. Size of image is 18.3µm×18.3µm. To be able to compare contrast in images, all images were taken with exposures such that they fill the camera’s dynamic range almost equally. Further, all images have been converted to 32-bit floating point format and then pixel values were normalized to a maximum value of 1 before displaying.

Fig. 7.
Fig. 7.

Comparing partially coherent transfer functions C(m; p) at S=0.4 with images of an optical fiber in the direction of shear for (a) Nomarski-DIC and (b) Köhler-DIC for different values of bias. The optical fiber (core R.I.=1.581, clad R.I.=1.487, core diameter=10 µm, clad diameter=50 µm) was aligned with its axis perpendicular to the direction of shear. Images were taken with a 40X 0.9 NA lens and 0.9NA condenser lens stopped down to S=0.4. The fiber was immersed in water under a 0.17 mm glass coverslip, sealed with nailpolish, and imaged with quasi-monochromatic illumination as in Fig. 6.

Fig. 8.
Fig. 8.

Same as Fig. 7 but with S=1.

Equations (19)

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h K ( x , y ) = R [ h BF ( x + Δ , y ) exp ( i ϕ ) ] ( 1 R ) [ h BF ( x Δ , y ) exp ( i ϕ ) ] ,
h K ( x , y ) = 0.5 [ h BF ( x + Δ , y ) exp ( i ϕ ) h BF ( x Δ , y ) exp ( i ϕ ) ] .
P K ( ξ , η ) = 0.5 { P BF ( ξ , η ) exp [ i ( 2 π ξ Δ ϕ ) ] P BF ( ξ , η ) exp [ i ( 2 π ξ Δ ϕ ) ] }
= i P BF ( ξ , η ) sin ( 2 π ξ Δ ϕ ) .
C K ( m , n ; p , q ) = P cond ( ξ , η ) 2 P K ( ξ m , η n ) P K * ( ξ p , η q ) d ξ d η .
m 0 = N A obj λ ,
I K ( x , y ) = T ( m , n ) T * ( p , q ) C K ( m , n ; p , q ) exp { 2 π i [ ( m p ) x + ( n q ) y ] } d m d n d p d q .
t N ( x , y ) = t ( x + Δ , y ) exp ( i ϕ ) t ( x Δ , t ) exp ( i ϕ ) .
T N ( m , n ) = T ( m , n ) { exp [ i ( 2 π m Δ ϕ ) ] exp [ i ( 2 π m Δ ϕ ) ] }
= 2 i T ( m , n ) sin ( 2 π m Δ ϕ ) .
I ( x , y ) = T N ( m , n ) T N * ( p , q ) C BF ( m , n ; p , q ) exp { 2 π i [ ( m p ) x + ( n q ) y ] } d m d n d p d q
C BF ( m , n ; p , q ) = P cond ( ξ , η ) 2 P BF ( ξ m , η n ) P BF * ( ξ p , η q ) d ξ d η
C N ( m , n ; p , q ) = 4 C BF ( m , n ; p , q ) sin ( 2 π m Δ ϕ ) sin ( 2 π p Δ ϕ ) .
M K ( m , n ; p , q ) = P K ( m , n ) P K * ( p , q ) = 4 M BF sin ( 2 π m Δ ϕ ) sin ( 2 π p Δ ϕ )
C K coh ( m , n ) = P K ( m , n ) = i P BF ( m , n ) sin ( 2 π m Δ ϕ )
C N coh ( m , n ) = i P BF ( m , n ) sin ( 2 π m Δ ϕ )
C N ( m ; p ) = 4 C BF ( m ; p ) sin ( 2 π m Δ ϕ ) sin ( 2 π p Δ ϕ ) ,
C BF ( m ; p ) = P cond ( ξ , η ) 2 P BF ( ξ m , η ) P BF * ( ξ p , η ) d ξ d η ,
C K ( m ; p ) = P cond ( ξ , η ) 2 P K ( ξ m , η ) P K * ( ξ p , η ) d ξ d η ,

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