Abstract

The dependence of the far-field coherence and radiant intensity of light scattered from a liquid crystal on the correlation length for the fluctuations of molecules in the crystal are analyzed theoretically. The results are useful for understanding data obtained in scattering experiments that study the hydrodynamic properties of liquid crystals excited into internal flow by a dc electric field or by application of sound vibrations. It is found that if the molecular correlation length is small relative to the dimensions of the illuminated volume, the scattered field behaves according to the van Cittert-Zernike theorem, and the motion of the molecules is better studied by measurement of the radiant intensity. If instead the molecular correlation length is much longer than the dimensions of the illuminated volume, the van Cittert-Zernike theorem does not apply, and the spatial correlation properties of the motion of the molecules can be studied by measurement of the far-field coherence.

© 1978 Optical Society of America

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  1. R. Bartolino, M. Bertolotti, F. Scudieri, and D. Sette, "Coherence Of Light Passing Through a Nematic Liquid Crystal Under a d.c. Electric Field," Appl. Opt. 12, 2917–2922 (1973).
  2. F. Scudieri, M. Bertolotti, and R. Bartolino, "Light Scattering by a Liquid Crystal: a New Quasi-Thermal Source," Appl. Opt. 13, 181–185 (1977).
  3. M. Bertolotti, F. Scudieri, and S. Verginelli, "Spatial Coherence of Light Scattered by Media with Large Correlation Length of Refractive Index Fluctuations," Appl. Opt. 15, 1842–1844 (1976).
  4. M. Bertolotti and F. Scudieri, "Spatial Coherence of Light Scattered by Liquid Crystals," Proceedings of the Fourth Rochester Conference on Coherence and Quantum Optics, L. Mandel and E. Wolf, eds. Plenum, New York, in press).
  5. W. H. Carter and E. Wolf, "Coherence and radiometry with quasihomogeneous planar sources," J. Opt. Soc. Am. 67, 785–796 (1977).
  6. H. P. Baltes and B. Steinle, "Information on Planar Sources from Far-Zone Spatial Coherence," Lett. Nuovo Cimento 18, 313–318 (1977).
  7. H. P. Baltes, B. Steinle, and G. Antes, "Radiometric and Correlation Properties of Bounded Planar Sources," Proceedings of the Fourth Rochester Conference on Coherence and Quantum Optics, L. Mandel and E. Wolf, eds. (Plenum, New York, in press).
  8. W. H. Carter, "Three-dimensional wave theory of optical image formation from scattered sound," J. Opt. Soc. Am. 60, 1366–1374 (1970), see Eq. (B10) in Appendix B. This equation assumes that the scatterer is sufficiently weak to be treated in the first Born approximation and that the light is scattered principly only into directions which make small angles with the +z axis as defined by Eq. (B8) in this Appendix.
  9. L. Mandel and E. Wolf, "Coherence Properties of Optical Fields," Rev. Mod. Phys. 37, 231–287 (1965), see Eq. (3.32).
  10. A. C. Schell, "The Multiple Plate Antenna," Doctoral dissertation (Massachusetts Institute of Technology, 1961) (unpublished), Sec. 7.5.
  11. R. A. Shore, "Partially Coherent Diffraction by a Circular Aperture," in Electromagnetic Theory and Antennas, E. C. Jordan, ed. (Pergammon, London, 1963), Part 2, pp. 787–795.
  12. A. K. Jaiswal, G. P. Agrawal, and C. L. Mehta, "Coherence Functions in the Far Field Diffraction Plane," Nuovo'Cimento B15, 295–307 (1973).
  13. E. Marchand and E. Wolf, "Angular correlation and the far-zone behavior of partially coherent fields," J. Opt. Soc. Am. 62, 379–385 (1972), Eq. (37).
  14. G. C. Sherman, J. J. Stamnes, and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760–776 (1976). See particularly Sect. 5A, where we assume case (a) obtains.
  15. This change of variables is discussed in more detail in Ref. 5, after Eq. (3.2).
  16. , R. Bracewell, The Fourier Transform and Its Applications, (McGraw-Hill, New York, 1965); see the table on p. 244 for the convolution theorem in two dimensions.
  17. E. W. Marchand and E. Wolf, "Generalized Radiometry for Radiation from Partially Coherent Sources," Opt. Commun. 6, 305–308 (1972); see Eq. (12) in which the (2Π)-2 constant should be deleted.
  18. P. G. de Gennes, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1974); see Chapter 4, and also see Ref. 4, Sect. 3.
  19. E. Wolf, and W. H. Carter, "Angular Distribution of Radiant Intensity from Sources of Different Degrees of Spatial Coherence," Opt. Comm. 13, 20.4–209 (1975).

1977

1976

M. Bertolotti, F. Scudieri, and S. Verginelli, "Spatial Coherence of Light Scattered by Media with Large Correlation Length of Refractive Index Fluctuations," Appl. Opt. 15, 1842–1844 (1976).

G. C. Sherman, J. J. Stamnes, and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760–776 (1976). See particularly Sect. 5A, where we assume case (a) obtains.

1975

E. Wolf, and W. H. Carter, "Angular Distribution of Radiant Intensity from Sources of Different Degrees of Spatial Coherence," Opt. Comm. 13, 20.4–209 (1975).

1973

R. Bartolino, M. Bertolotti, F. Scudieri, and D. Sette, "Coherence Of Light Passing Through a Nematic Liquid Crystal Under a d.c. Electric Field," Appl. Opt. 12, 2917–2922 (1973).

A. K. Jaiswal, G. P. Agrawal, and C. L. Mehta, "Coherence Functions in the Far Field Diffraction Plane," Nuovo'Cimento B15, 295–307 (1973).

1972

E. Marchand and E. Wolf, "Angular correlation and the far-zone behavior of partially coherent fields," J. Opt. Soc. Am. 62, 379–385 (1972), Eq. (37).

E. W. Marchand and E. Wolf, "Generalized Radiometry for Radiation from Partially Coherent Sources," Opt. Commun. 6, 305–308 (1972); see Eq. (12) in which the (2Π)-2 constant should be deleted.

1970

1965

L. Mandel and E. Wolf, "Coherence Properties of Optical Fields," Rev. Mod. Phys. 37, 231–287 (1965), see Eq. (3.32).

Agrawal, G. P.

A. K. Jaiswal, G. P. Agrawal, and C. L. Mehta, "Coherence Functions in the Far Field Diffraction Plane," Nuovo'Cimento B15, 295–307 (1973).

Antes, G.

H. P. Baltes, B. Steinle, and G. Antes, "Radiometric and Correlation Properties of Bounded Planar Sources," Proceedings of the Fourth Rochester Conference on Coherence and Quantum Optics, L. Mandel and E. Wolf, eds. (Plenum, New York, in press).

Baltes, H. P.

H. P. Baltes and B. Steinle, "Information on Planar Sources from Far-Zone Spatial Coherence," Lett. Nuovo Cimento 18, 313–318 (1977).

H. P. Baltes, B. Steinle, and G. Antes, "Radiometric and Correlation Properties of Bounded Planar Sources," Proceedings of the Fourth Rochester Conference on Coherence and Quantum Optics, L. Mandel and E. Wolf, eds. (Plenum, New York, in press).

Bartolino, R.

Bertolotti, M.

Bracewell, R.

, R. Bracewell, The Fourier Transform and Its Applications, (McGraw-Hill, New York, 1965); see the table on p. 244 for the convolution theorem in two dimensions.

Carter, W. H.

de Gennes, P. G.

P. G. de Gennes, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1974); see Chapter 4, and also see Ref. 4, Sect. 3.

Jaiswal, A. K.

A. K. Jaiswal, G. P. Agrawal, and C. L. Mehta, "Coherence Functions in the Far Field Diffraction Plane," Nuovo'Cimento B15, 295–307 (1973).

Lalor, E.

G. C. Sherman, J. J. Stamnes, and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760–776 (1976). See particularly Sect. 5A, where we assume case (a) obtains.

Mandel, L.

L. Mandel and E. Wolf, "Coherence Properties of Optical Fields," Rev. Mod. Phys. 37, 231–287 (1965), see Eq. (3.32).

Marchand, E.

Marchand, E. W.

E. W. Marchand and E. Wolf, "Generalized Radiometry for Radiation from Partially Coherent Sources," Opt. Commun. 6, 305–308 (1972); see Eq. (12) in which the (2Π)-2 constant should be deleted.

Mehta, C. L.

A. K. Jaiswal, G. P. Agrawal, and C. L. Mehta, "Coherence Functions in the Far Field Diffraction Plane," Nuovo'Cimento B15, 295–307 (1973).

Schell, A. C.

A. C. Schell, "The Multiple Plate Antenna," Doctoral dissertation (Massachusetts Institute of Technology, 1961) (unpublished), Sec. 7.5.

Scudieri, F.

Sette, D.

Sherman, G. C.

G. C. Sherman, J. J. Stamnes, and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760–776 (1976). See particularly Sect. 5A, where we assume case (a) obtains.

Shore, R. A.

R. A. Shore, "Partially Coherent Diffraction by a Circular Aperture," in Electromagnetic Theory and Antennas, E. C. Jordan, ed. (Pergammon, London, 1963), Part 2, pp. 787–795.

Stamnes, J. J.

G. C. Sherman, J. J. Stamnes, and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760–776 (1976). See particularly Sect. 5A, where we assume case (a) obtains.

Steinle, B.

H. P. Baltes and B. Steinle, "Information on Planar Sources from Far-Zone Spatial Coherence," Lett. Nuovo Cimento 18, 313–318 (1977).

H. P. Baltes, B. Steinle, and G. Antes, "Radiometric and Correlation Properties of Bounded Planar Sources," Proceedings of the Fourth Rochester Conference on Coherence and Quantum Optics, L. Mandel and E. Wolf, eds. (Plenum, New York, in press).

Verginelli, S.

Wolf, E.

W. H. Carter and E. Wolf, "Coherence and radiometry with quasihomogeneous planar sources," J. Opt. Soc. Am. 67, 785–796 (1977).

E. Wolf, and W. H. Carter, "Angular Distribution of Radiant Intensity from Sources of Different Degrees of Spatial Coherence," Opt. Comm. 13, 20.4–209 (1975).

E. W. Marchand and E. Wolf, "Generalized Radiometry for Radiation from Partially Coherent Sources," Opt. Commun. 6, 305–308 (1972); see Eq. (12) in which the (2Π)-2 constant should be deleted.

E. Marchand and E. Wolf, "Angular correlation and the far-zone behavior of partially coherent fields," J. Opt. Soc. Am. 62, 379–385 (1972), Eq. (37).

L. Mandel and E. Wolf, "Coherence Properties of Optical Fields," Rev. Mod. Phys. 37, 231–287 (1965), see Eq. (3.32).

Appl. Opt.

J. Math. Phys.

G. C. Sherman, J. J. Stamnes, and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760–776 (1976). See particularly Sect. 5A, where we assume case (a) obtains.

J. Opt. Soc. Am.

Lett. Nuovo Cimento

H. P. Baltes and B. Steinle, "Information on Planar Sources from Far-Zone Spatial Coherence," Lett. Nuovo Cimento 18, 313–318 (1977).

Nuovo’Cimento

A. K. Jaiswal, G. P. Agrawal, and C. L. Mehta, "Coherence Functions in the Far Field Diffraction Plane," Nuovo'Cimento B15, 295–307 (1973).

Opt. Comm.

E. Wolf, and W. H. Carter, "Angular Distribution of Radiant Intensity from Sources of Different Degrees of Spatial Coherence," Opt. Comm. 13, 20.4–209 (1975).

Opt. Commun.

E. W. Marchand and E. Wolf, "Generalized Radiometry for Radiation from Partially Coherent Sources," Opt. Commun. 6, 305–308 (1972); see Eq. (12) in which the (2Π)-2 constant should be deleted.

Rev. Mod. Phys.

L. Mandel and E. Wolf, "Coherence Properties of Optical Fields," Rev. Mod. Phys. 37, 231–287 (1965), see Eq. (3.32).

Other

A. C. Schell, "The Multiple Plate Antenna," Doctoral dissertation (Massachusetts Institute of Technology, 1961) (unpublished), Sec. 7.5.

R. A. Shore, "Partially Coherent Diffraction by a Circular Aperture," in Electromagnetic Theory and Antennas, E. C. Jordan, ed. (Pergammon, London, 1963), Part 2, pp. 787–795.

M. Bertolotti and F. Scudieri, "Spatial Coherence of Light Scattered by Liquid Crystals," Proceedings of the Fourth Rochester Conference on Coherence and Quantum Optics, L. Mandel and E. Wolf, eds. Plenum, New York, in press).

P. G. de Gennes, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1974); see Chapter 4, and also see Ref. 4, Sect. 3.

This change of variables is discussed in more detail in Ref. 5, after Eq. (3.2).

, R. Bracewell, The Fourier Transform and Its Applications, (McGraw-Hill, New York, 1965); see the table on p. 244 for the convolution theorem in two dimensions.

H. P. Baltes, B. Steinle, and G. Antes, "Radiometric and Correlation Properties of Bounded Planar Sources," Proceedings of the Fourth Rochester Conference on Coherence and Quantum Optics, L. Mandel and E. Wolf, eds. (Plenum, New York, in press).

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