Abstract

Light scattering techniques are widely used in many fields of condensed and soft matter physics. Usually these methods are based on the study of the scattered light in the far field. Recently, a new family of near field detection schemes has been developed, mainly for the study of small angle light scattering. These techniques are based on the detection of the light intensity near to the sample, where light scattered at different directions overlaps but can be distinguished by Fourier transform analysis. Here we report for the first time data obtained with a dynamic near field scattering instrument, measuring both polarized and depolarized scattered light. Advantages of this procedure over the traditional far field detection include the immunity to stray light problems and the possibility to obtain a large number of statistical samples for many different wave vectors in a single instantaneous measurement. By using the proposed technique we have measured the translational and rotational diffusion coefficients of rod-like colloidal particles. The obtained data are in very good agreement with the data acquired with a traditional light scattering apparatus.

©2009 Optical Society of America

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  1. F. Scheffold and R. Cerbino, “New trends in light scattering,” Curr. Opin. Colloid Interface Sci. 12(1), 50–57 (2007).
    [Crossref]
  2. F. Ferri, “Use of a charge coupled device camera for low-angle elastic light scattering,” Rev. Sci. Instrum. 68, 2265–2274 (1997).
    [Crossref]
  3. L. Cipelletti and D. Weitz, “Ultralow-angle dynamic light scattering with a charge coupled device camera based multispeckle, multitau correlator,” Rev. Sci. Instrum. 70, 3214–3221 (1999).
    [Crossref]
  4. K. Schatzel, “Suppression of multiple-scattering by photon cross-correlation techniques,” J. Mod. Opt. 38(9), 1849–1865 (1991).
    [Crossref]
  5. P. N. Pusey, “Suppression of multiple scattering by photon cross-correlation techniques,” Curr. Opin. Colloid Interface Sci. 4(3), 177–185 (1999).
    [Crossref]
  6. J. C. Thomas and S. Tjin, “Fiber optics dynamic light-scattering (FODLS) from moderately concentrated suspensions,” J. Colloid Interface Sci. 129, 15–31 (1989).
    [Crossref]
  7. D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60(12), 1134–1137 (1988).
    [Crossref] [PubMed]
  8. S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Y. A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. express 15(26), 18,275–18,282 (2007).
  9. D. Brogioli, A. Vailati, and M. Giglio, “Heterodyne near-field scattering,” Appl. Phys. Lett. 81, 4109–4111 (2002).
    [Crossref]
  10. D. Brogioli, “Near Field Speckles,” Ph.D. thesis, Università degli Studi di Cagliari (2002). Available at the url: www.geocities.com/dbrogioli/nfs_phd.
  11. M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g(r),” Phys. Rev. Lett. 85, 1416–1419 (2000).
    [Crossref] [PubMed]
  12. M. Giglio, M. Carpineti, A. Vailati, and D. Brogioli, “Near-field intensity correlations of scattered light,” Appl. Opt. 40(24), 4036–4040 (2001).
    [Crossref]
  13. M. Wu, G. Ahlers, and D. Cannell, “Thermally induced fluctuations below the onset of Reyleight-Benard convection,” Phys. Rev. Lett. 75, 1743–1746 (1995).
    [Crossref] [PubMed]
  14. S. P. Trainoff and D. S. Cannell, “Physical optics treatment of the shadowgraph,” Phys. Fluids 14, 1340–1363 (2002).
    [Crossref]
  15. D. Brogioli, A. Vailati, and M. Giglio, “A schlieren method for ultra-low angle light scattering measurements,” Europhys. Lett. 63, 220–225 (2003).
    [Crossref]
  16. J. Oh, J. O. de Zárate, J. Sengers, and G. Ahlers, “Dynamics of fluctuations in a fluid below the onset of Rayleigh-Bénard convection,” Phys. Rev. E 69, 21,106-1-13 (2004).
    [Crossref]
  17. D. Brogioli, F. Croccolo, V. Cassina, D. Salerno, and F. Mantegazza, “Nano-particle characterization by using Exposure Time Dependent Spectrum and scattering in the near field methods: how to get fast dynamics with low-speed CCD camera.” Opt. Express 16(25), 20,272–20,282 (2008).
    [Crossref]
  18. R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76(9), 093,110–11 (2005).
    [Crossref]
  19. F. Croccolo, D. Brogioli, A. Vailati, M. Giglio, and D. Cannell, “Use of dynamic Schlieren to study fluctuations during free diffusion,” Appl. Opt. 45, 2166–2173 (2006).
    [Crossref] [PubMed]
  20. F. Croccolo, D. Brogioli, A. Vailati, M. Giglio, and D. Cannell, “Non-diffusive decay of gradient driven fluctuations in a free-diffusion process,” Phys. Rev. E 76, 41,112-1-9 (2007).
    [Crossref]
  21. D. Magatti, M. D. Alaimo, M. A. C. Potenza, and F. Ferri, “Dynamic heterodyne near field scattering,” Appl. Phys. Lett. 92, 241,101-1-3 (2008).
    [Crossref]
  22. R. Cerbino and V. Trappe, “Differential Dynamic Microscopy: Probing Wave Vector Dependent Dynamics with a Microscope,” Phys. Rev. Lett. 100, 188,102-1-4 (2008).
    [Crossref] [PubMed]
  23. G. H. Koenderink, H. Y. Zhang, D. G. A. L. Aarts, M. P. Lettinga, A. P. Philipse, and G. Nagele, “On the validity of Stokes-Einstein-Debye relations for rotational diffusion in colloidal suspensions,” Faraday Discuss. 123, 335–354 (2003).
    [Crossref] [PubMed]
  24. T. Bellini, V. Degiorgio, F. Mantegazza, F. Ajmone-Marsan, and C. Scarnecchia, “Electrokinetic properties of colloids of variable charge. 1. Electrophoretic and electrooptic characterization,” J. Chem. Phys. 103, 8228–8237 (1995).
    [Crossref]
  25. V. Degiorgio, R. Piazza, T. Bellini, and M. Visca, “Static and dynamic light scattering study of fluorinated polymer colloids with a crystalline internal structure,” Adv. Colloid Interface Sci. 48, 61–91 (1994).
    [Crossref]
  26. R. Piazza, J. Stavans, T. Bellini, and V. Degiorgio, “Light scattering study of crystalline latex particles,” Opt. Commun. 73(4), 263–267 (1989).
    [Crossref]
  27. T. Bellini, R. Piazza, C. Sozzi, and V. Degiorgio, “Electric Birefringence of a Dispersion of Electrically Charged Anisotropic Particles,” Europhys. Lett. 7(6), 561–565 (1988).
    [Crossref]
  28. B. Berne and R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics (Dover, New York, 2000).
  29. J. Goodman, Introduction to Fourier Optics (Roberts & Company, Englewood, 2005).
  30. T. Sugimoto and K. Sakata, “Preparation of monodisperse pseudocubic α - Fe2O3 particles from condensed ferric hydroxide gel,” J. Colloid Interface Sci. 152(2), 587–590 (1992).
    [Crossref]
  31. S. Sacanna, “Novel Routes to Model Colloids: ellipsoids, lattices and stable meso-emulsions,” Ph.D. thesis, Utrecht University (2007). ISBN 978-90-393-4598-6.

2008 (3)

D. Brogioli, F. Croccolo, V. Cassina, D. Salerno, and F. Mantegazza, “Nano-particle characterization by using Exposure Time Dependent Spectrum and scattering in the near field methods: how to get fast dynamics with low-speed CCD camera.” Opt. Express 16(25), 20,272–20,282 (2008).
[Crossref]

D. Magatti, M. D. Alaimo, M. A. C. Potenza, and F. Ferri, “Dynamic heterodyne near field scattering,” Appl. Phys. Lett. 92, 241,101-1-3 (2008).
[Crossref]

R. Cerbino and V. Trappe, “Differential Dynamic Microscopy: Probing Wave Vector Dependent Dynamics with a Microscope,” Phys. Rev. Lett. 100, 188,102-1-4 (2008).
[Crossref] [PubMed]

2007 (3)

F. Croccolo, D. Brogioli, A. Vailati, M. Giglio, and D. Cannell, “Non-diffusive decay of gradient driven fluctuations in a free-diffusion process,” Phys. Rev. E 76, 41,112-1-9 (2007).
[Crossref]

F. Scheffold and R. Cerbino, “New trends in light scattering,” Curr. Opin. Colloid Interface Sci. 12(1), 50–57 (2007).
[Crossref]

S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Y. A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. express 15(26), 18,275–18,282 (2007).

2006 (1)

2005 (1)

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76(9), 093,110–11 (2005).
[Crossref]

2004 (1)

J. Oh, J. O. de Zárate, J. Sengers, and G. Ahlers, “Dynamics of fluctuations in a fluid below the onset of Rayleigh-Bénard convection,” Phys. Rev. E 69, 21,106-1-13 (2004).
[Crossref]

2003 (2)

D. Brogioli, A. Vailati, and M. Giglio, “A schlieren method for ultra-low angle light scattering measurements,” Europhys. Lett. 63, 220–225 (2003).
[Crossref]

G. H. Koenderink, H. Y. Zhang, D. G. A. L. Aarts, M. P. Lettinga, A. P. Philipse, and G. Nagele, “On the validity of Stokes-Einstein-Debye relations for rotational diffusion in colloidal suspensions,” Faraday Discuss. 123, 335–354 (2003).
[Crossref] [PubMed]

2002 (2)

S. P. Trainoff and D. S. Cannell, “Physical optics treatment of the shadowgraph,” Phys. Fluids 14, 1340–1363 (2002).
[Crossref]

D. Brogioli, A. Vailati, and M. Giglio, “Heterodyne near-field scattering,” Appl. Phys. Lett. 81, 4109–4111 (2002).
[Crossref]

2001 (1)

2000 (1)

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g(r),” Phys. Rev. Lett. 85, 1416–1419 (2000).
[Crossref] [PubMed]

1999 (2)

P. N. Pusey, “Suppression of multiple scattering by photon cross-correlation techniques,” Curr. Opin. Colloid Interface Sci. 4(3), 177–185 (1999).
[Crossref]

L. Cipelletti and D. Weitz, “Ultralow-angle dynamic light scattering with a charge coupled device camera based multispeckle, multitau correlator,” Rev. Sci. Instrum. 70, 3214–3221 (1999).
[Crossref]

1997 (1)

F. Ferri, “Use of a charge coupled device camera for low-angle elastic light scattering,” Rev. Sci. Instrum. 68, 2265–2274 (1997).
[Crossref]

1995 (2)

M. Wu, G. Ahlers, and D. Cannell, “Thermally induced fluctuations below the onset of Reyleight-Benard convection,” Phys. Rev. Lett. 75, 1743–1746 (1995).
[Crossref] [PubMed]

T. Bellini, V. Degiorgio, F. Mantegazza, F. Ajmone-Marsan, and C. Scarnecchia, “Electrokinetic properties of colloids of variable charge. 1. Electrophoretic and electrooptic characterization,” J. Chem. Phys. 103, 8228–8237 (1995).
[Crossref]

1994 (1)

V. Degiorgio, R. Piazza, T. Bellini, and M. Visca, “Static and dynamic light scattering study of fluorinated polymer colloids with a crystalline internal structure,” Adv. Colloid Interface Sci. 48, 61–91 (1994).
[Crossref]

1992 (1)

T. Sugimoto and K. Sakata, “Preparation of monodisperse pseudocubic α - Fe2O3 particles from condensed ferric hydroxide gel,” J. Colloid Interface Sci. 152(2), 587–590 (1992).
[Crossref]

1991 (1)

K. Schatzel, “Suppression of multiple-scattering by photon cross-correlation techniques,” J. Mod. Opt. 38(9), 1849–1865 (1991).
[Crossref]

1989 (2)

J. C. Thomas and S. Tjin, “Fiber optics dynamic light-scattering (FODLS) from moderately concentrated suspensions,” J. Colloid Interface Sci. 129, 15–31 (1989).
[Crossref]

R. Piazza, J. Stavans, T. Bellini, and V. Degiorgio, “Light scattering study of crystalline latex particles,” Opt. Commun. 73(4), 263–267 (1989).
[Crossref]

1988 (2)

T. Bellini, R. Piazza, C. Sozzi, and V. Degiorgio, “Electric Birefringence of a Dispersion of Electrically Charged Anisotropic Particles,” Europhys. Lett. 7(6), 561–565 (1988).
[Crossref]

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60(12), 1134–1137 (1988).
[Crossref] [PubMed]

Aarts, D. G. A. L.

G. H. Koenderink, H. Y. Zhang, D. G. A. L. Aarts, M. P. Lettinga, A. P. Philipse, and G. Nagele, “On the validity of Stokes-Einstein-Debye relations for rotational diffusion in colloidal suspensions,” Faraday Discuss. 123, 335–354 (2003).
[Crossref] [PubMed]

Ahlers, G.

J. Oh, J. O. de Zárate, J. Sengers, and G. Ahlers, “Dynamics of fluctuations in a fluid below the onset of Rayleigh-Bénard convection,” Phys. Rev. E 69, 21,106-1-13 (2004).
[Crossref]

M. Wu, G. Ahlers, and D. Cannell, “Thermally induced fluctuations below the onset of Reyleight-Benard convection,” Phys. Rev. Lett. 75, 1743–1746 (1995).
[Crossref] [PubMed]

Ajmone-Marsan, F.

T. Bellini, V. Degiorgio, F. Mantegazza, F. Ajmone-Marsan, and C. Scarnecchia, “Electrokinetic properties of colloids of variable charge. 1. Electrophoretic and electrooptic characterization,” J. Chem. Phys. 103, 8228–8237 (1995).
[Crossref]

Alaimo, M. D.

D. Magatti, M. D. Alaimo, M. A. C. Potenza, and F. Ferri, “Dynamic heterodyne near field scattering,” Appl. Phys. Lett. 92, 241,101-1-3 (2008).
[Crossref]

Bandyopadhyay, R.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76(9), 093,110–11 (2005).
[Crossref]

Bellini, T.

T. Bellini, V. Degiorgio, F. Mantegazza, F. Ajmone-Marsan, and C. Scarnecchia, “Electrokinetic properties of colloids of variable charge. 1. Electrophoretic and electrooptic characterization,” J. Chem. Phys. 103, 8228–8237 (1995).
[Crossref]

V. Degiorgio, R. Piazza, T. Bellini, and M. Visca, “Static and dynamic light scattering study of fluorinated polymer colloids with a crystalline internal structure,” Adv. Colloid Interface Sci. 48, 61–91 (1994).
[Crossref]

R. Piazza, J. Stavans, T. Bellini, and V. Degiorgio, “Light scattering study of crystalline latex particles,” Opt. Commun. 73(4), 263–267 (1989).
[Crossref]

T. Bellini, R. Piazza, C. Sozzi, and V. Degiorgio, “Electric Birefringence of a Dispersion of Electrically Charged Anisotropic Particles,” Europhys. Lett. 7(6), 561–565 (1988).
[Crossref]

Berne, B.

B. Berne and R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics (Dover, New York, 2000).

Brogioli, D.

D. Brogioli, F. Croccolo, V. Cassina, D. Salerno, and F. Mantegazza, “Nano-particle characterization by using Exposure Time Dependent Spectrum and scattering in the near field methods: how to get fast dynamics with low-speed CCD camera.” Opt. Express 16(25), 20,272–20,282 (2008).
[Crossref]

F. Croccolo, D. Brogioli, A. Vailati, M. Giglio, and D. Cannell, “Non-diffusive decay of gradient driven fluctuations in a free-diffusion process,” Phys. Rev. E 76, 41,112-1-9 (2007).
[Crossref]

F. Croccolo, D. Brogioli, A. Vailati, M. Giglio, and D. Cannell, “Use of dynamic Schlieren to study fluctuations during free diffusion,” Appl. Opt. 45, 2166–2173 (2006).
[Crossref] [PubMed]

D. Brogioli, A. Vailati, and M. Giglio, “A schlieren method for ultra-low angle light scattering measurements,” Europhys. Lett. 63, 220–225 (2003).
[Crossref]

D. Brogioli, A. Vailati, and M. Giglio, “Heterodyne near-field scattering,” Appl. Phys. Lett. 81, 4109–4111 (2002).
[Crossref]

M. Giglio, M. Carpineti, A. Vailati, and D. Brogioli, “Near-field intensity correlations of scattered light,” Appl. Opt. 40(24), 4036–4040 (2001).
[Crossref]

D. Brogioli, “Near Field Speckles,” Ph.D. thesis, Università degli Studi di Cagliari (2002). Available at the url: www.geocities.com/dbrogioli/nfs_phd.

Cannell, D.

F. Croccolo, D. Brogioli, A. Vailati, M. Giglio, and D. Cannell, “Non-diffusive decay of gradient driven fluctuations in a free-diffusion process,” Phys. Rev. E 76, 41,112-1-9 (2007).
[Crossref]

F. Croccolo, D. Brogioli, A. Vailati, M. Giglio, and D. Cannell, “Use of dynamic Schlieren to study fluctuations during free diffusion,” Appl. Opt. 45, 2166–2173 (2006).
[Crossref] [PubMed]

M. Wu, G. Ahlers, and D. Cannell, “Thermally induced fluctuations below the onset of Reyleight-Benard convection,” Phys. Rev. Lett. 75, 1743–1746 (1995).
[Crossref] [PubMed]

Cannell, D. S.

S. P. Trainoff and D. S. Cannell, “Physical optics treatment of the shadowgraph,” Phys. Fluids 14, 1340–1363 (2002).
[Crossref]

Carpineti, M.

M. Giglio, M. Carpineti, A. Vailati, and D. Brogioli, “Near-field intensity correlations of scattered light,” Appl. Opt. 40(24), 4036–4040 (2001).
[Crossref]

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g(r),” Phys. Rev. Lett. 85, 1416–1419 (2000).
[Crossref] [PubMed]

Cassina, V.

D. Brogioli, F. Croccolo, V. Cassina, D. Salerno, and F. Mantegazza, “Nano-particle characterization by using Exposure Time Dependent Spectrum and scattering in the near field methods: how to get fast dynamics with low-speed CCD camera.” Opt. Express 16(25), 20,272–20,282 (2008).
[Crossref]

Cerbino, R.

R. Cerbino and V. Trappe, “Differential Dynamic Microscopy: Probing Wave Vector Dependent Dynamics with a Microscope,” Phys. Rev. Lett. 100, 188,102-1-4 (2008).
[Crossref] [PubMed]

F. Scheffold and R. Cerbino, “New trends in light scattering,” Curr. Opin. Colloid Interface Sci. 12(1), 50–57 (2007).
[Crossref]

Chaikin, P. M.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60(12), 1134–1137 (1988).
[Crossref] [PubMed]

Cipelletti, L.

L. Cipelletti and D. Weitz, “Ultralow-angle dynamic light scattering with a charge coupled device camera based multispeckle, multitau correlator,” Rev. Sci. Instrum. 70, 3214–3221 (1999).
[Crossref]

Croccolo, F.

D. Brogioli, F. Croccolo, V. Cassina, D. Salerno, and F. Mantegazza, “Nano-particle characterization by using Exposure Time Dependent Spectrum and scattering in the near field methods: how to get fast dynamics with low-speed CCD camera.” Opt. Express 16(25), 20,272–20,282 (2008).
[Crossref]

F. Croccolo, D. Brogioli, A. Vailati, M. Giglio, and D. Cannell, “Non-diffusive decay of gradient driven fluctuations in a free-diffusion process,” Phys. Rev. E 76, 41,112-1-9 (2007).
[Crossref]

F. Croccolo, D. Brogioli, A. Vailati, M. Giglio, and D. Cannell, “Use of dynamic Schlieren to study fluctuations during free diffusion,” Appl. Opt. 45, 2166–2173 (2006).
[Crossref] [PubMed]

de Zárate, J. O.

J. Oh, J. O. de Zárate, J. Sengers, and G. Ahlers, “Dynamics of fluctuations in a fluid below the onset of Rayleigh-Bénard convection,” Phys. Rev. E 69, 21,106-1-13 (2004).
[Crossref]

Degiorgio, V.

T. Bellini, V. Degiorgio, F. Mantegazza, F. Ajmone-Marsan, and C. Scarnecchia, “Electrokinetic properties of colloids of variable charge. 1. Electrophoretic and electrooptic characterization,” J. Chem. Phys. 103, 8228–8237 (1995).
[Crossref]

V. Degiorgio, R. Piazza, T. Bellini, and M. Visca, “Static and dynamic light scattering study of fluorinated polymer colloids with a crystalline internal structure,” Adv. Colloid Interface Sci. 48, 61–91 (1994).
[Crossref]

R. Piazza, J. Stavans, T. Bellini, and V. Degiorgio, “Light scattering study of crystalline latex particles,” Opt. Commun. 73(4), 263–267 (1989).
[Crossref]

T. Bellini, R. Piazza, C. Sozzi, and V. Degiorgio, “Electric Birefringence of a Dispersion of Electrically Charged Anisotropic Particles,” Europhys. Lett. 7(6), 561–565 (1988).
[Crossref]

Dixon, P. K.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76(9), 093,110–11 (2005).
[Crossref]

Durian, D. J.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76(9), 093,110–11 (2005).
[Crossref]

Ferri, F.

D. Magatti, M. D. Alaimo, M. A. C. Potenza, and F. Ferri, “Dynamic heterodyne near field scattering,” Appl. Phys. Lett. 92, 241,101-1-3 (2008).
[Crossref]

F. Ferri, “Use of a charge coupled device camera for low-angle elastic light scattering,” Rev. Sci. Instrum. 68, 2265–2274 (1997).
[Crossref]

Giglio, M.

F. Croccolo, D. Brogioli, A. Vailati, M. Giglio, and D. Cannell, “Non-diffusive decay of gradient driven fluctuations in a free-diffusion process,” Phys. Rev. E 76, 41,112-1-9 (2007).
[Crossref]

F. Croccolo, D. Brogioli, A. Vailati, M. Giglio, and D. Cannell, “Use of dynamic Schlieren to study fluctuations during free diffusion,” Appl. Opt. 45, 2166–2173 (2006).
[Crossref] [PubMed]

D. Brogioli, A. Vailati, and M. Giglio, “A schlieren method for ultra-low angle light scattering measurements,” Europhys. Lett. 63, 220–225 (2003).
[Crossref]

D. Brogioli, A. Vailati, and M. Giglio, “Heterodyne near-field scattering,” Appl. Phys. Lett. 81, 4109–4111 (2002).
[Crossref]

M. Giglio, M. Carpineti, A. Vailati, and D. Brogioli, “Near-field intensity correlations of scattered light,” Appl. Opt. 40(24), 4036–4040 (2001).
[Crossref]

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g(r),” Phys. Rev. Lett. 85, 1416–1419 (2000).
[Crossref] [PubMed]

Gittings, A. S.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76(9), 093,110–11 (2005).
[Crossref]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (Roberts & Company, Englewood, 2005).

Grier, D. G.

S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Y. A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. express 15(26), 18,275–18,282 (2007).

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60(12), 1134–1137 (1988).
[Crossref] [PubMed]

Kim, S. H.

S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Y. A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. express 15(26), 18,275–18,282 (2007).

Koenderink, G. H.

G. H. Koenderink, H. Y. Zhang, D. G. A. L. Aarts, M. P. Lettinga, A. P. Philipse, and G. Nagele, “On the validity of Stokes-Einstein-Debye relations for rotational diffusion in colloidal suspensions,” Faraday Discuss. 123, 335–354 (2003).
[Crossref] [PubMed]

Lee, S. H.

S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Y. A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. express 15(26), 18,275–18,282 (2007).

Lettinga, M. P.

G. H. Koenderink, H. Y. Zhang, D. G. A. L. Aarts, M. P. Lettinga, A. P. Philipse, and G. Nagele, “On the validity of Stokes-Einstein-Debye relations for rotational diffusion in colloidal suspensions,” Faraday Discuss. 123, 335–354 (2003).
[Crossref] [PubMed]

Magatti, D.

D. Magatti, M. D. Alaimo, M. A. C. Potenza, and F. Ferri, “Dynamic heterodyne near field scattering,” Appl. Phys. Lett. 92, 241,101-1-3 (2008).
[Crossref]

Mantegazza, F.

D. Brogioli, F. Croccolo, V. Cassina, D. Salerno, and F. Mantegazza, “Nano-particle characterization by using Exposure Time Dependent Spectrum and scattering in the near field methods: how to get fast dynamics with low-speed CCD camera.” Opt. Express 16(25), 20,272–20,282 (2008).
[Crossref]

T. Bellini, V. Degiorgio, F. Mantegazza, F. Ajmone-Marsan, and C. Scarnecchia, “Electrokinetic properties of colloids of variable charge. 1. Electrophoretic and electrooptic characterization,” J. Chem. Phys. 103, 8228–8237 (1995).
[Crossref]

Nagele, G.

G. H. Koenderink, H. Y. Zhang, D. G. A. L. Aarts, M. P. Lettinga, A. P. Philipse, and G. Nagele, “On the validity of Stokes-Einstein-Debye relations for rotational diffusion in colloidal suspensions,” Faraday Discuss. 123, 335–354 (2003).
[Crossref] [PubMed]

Oh, J.

J. Oh, J. O. de Zárate, J. Sengers, and G. Ahlers, “Dynamics of fluctuations in a fluid below the onset of Rayleigh-Bénard convection,” Phys. Rev. E 69, 21,106-1-13 (2004).
[Crossref]

Pecora, R.

B. Berne and R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics (Dover, New York, 2000).

Philipse, A. P.

G. H. Koenderink, H. Y. Zhang, D. G. A. L. Aarts, M. P. Lettinga, A. P. Philipse, and G. Nagele, “On the validity of Stokes-Einstein-Debye relations for rotational diffusion in colloidal suspensions,” Faraday Discuss. 123, 335–354 (2003).
[Crossref] [PubMed]

Piazza, R.

V. Degiorgio, R. Piazza, T. Bellini, and M. Visca, “Static and dynamic light scattering study of fluorinated polymer colloids with a crystalline internal structure,” Adv. Colloid Interface Sci. 48, 61–91 (1994).
[Crossref]

R. Piazza, J. Stavans, T. Bellini, and V. Degiorgio, “Light scattering study of crystalline latex particles,” Opt. Commun. 73(4), 263–267 (1989).
[Crossref]

T. Bellini, R. Piazza, C. Sozzi, and V. Degiorgio, “Electric Birefringence of a Dispersion of Electrically Charged Anisotropic Particles,” Europhys. Lett. 7(6), 561–565 (1988).
[Crossref]

Pine, D. J.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60(12), 1134–1137 (1988).
[Crossref] [PubMed]

Potenza, M. A. C.

D. Magatti, M. D. Alaimo, M. A. C. Potenza, and F. Ferri, “Dynamic heterodyne near field scattering,” Appl. Phys. Lett. 92, 241,101-1-3 (2008).
[Crossref]

Pusey, P. N.

P. N. Pusey, “Suppression of multiple scattering by photon cross-correlation techniques,” Curr. Opin. Colloid Interface Sci. 4(3), 177–185 (1999).
[Crossref]

Roichman, Y.

S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Y. A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. express 15(26), 18,275–18,282 (2007).

Sacanna, S.

S. Sacanna, “Novel Routes to Model Colloids: ellipsoids, lattices and stable meso-emulsions,” Ph.D. thesis, Utrecht University (2007). ISBN 978-90-393-4598-6.

Sakata, K.

T. Sugimoto and K. Sakata, “Preparation of monodisperse pseudocubic α - Fe2O3 particles from condensed ferric hydroxide gel,” J. Colloid Interface Sci. 152(2), 587–590 (1992).
[Crossref]

Salerno, D.

D. Brogioli, F. Croccolo, V. Cassina, D. Salerno, and F. Mantegazza, “Nano-particle characterization by using Exposure Time Dependent Spectrum and scattering in the near field methods: how to get fast dynamics with low-speed CCD camera.” Opt. Express 16(25), 20,272–20,282 (2008).
[Crossref]

Scarnecchia, C.

T. Bellini, V. Degiorgio, F. Mantegazza, F. Ajmone-Marsan, and C. Scarnecchia, “Electrokinetic properties of colloids of variable charge. 1. Electrophoretic and electrooptic characterization,” J. Chem. Phys. 103, 8228–8237 (1995).
[Crossref]

Schatzel, K.

K. Schatzel, “Suppression of multiple-scattering by photon cross-correlation techniques,” J. Mod. Opt. 38(9), 1849–1865 (1991).
[Crossref]

Scheffold, F.

F. Scheffold and R. Cerbino, “New trends in light scattering,” Curr. Opin. Colloid Interface Sci. 12(1), 50–57 (2007).
[Crossref]

Sengers, J.

J. Oh, J. O. de Zárate, J. Sengers, and G. Ahlers, “Dynamics of fluctuations in a fluid below the onset of Rayleigh-Bénard convection,” Phys. Rev. E 69, 21,106-1-13 (2004).
[Crossref]

Sozzi, C.

T. Bellini, R. Piazza, C. Sozzi, and V. Degiorgio, “Electric Birefringence of a Dispersion of Electrically Charged Anisotropic Particles,” Europhys. Lett. 7(6), 561–565 (1988).
[Crossref]

Stavans, J.

R. Piazza, J. Stavans, T. Bellini, and V. Degiorgio, “Light scattering study of crystalline latex particles,” Opt. Commun. 73(4), 263–267 (1989).
[Crossref]

Sugimoto, T.

T. Sugimoto and K. Sakata, “Preparation of monodisperse pseudocubic α - Fe2O3 particles from condensed ferric hydroxide gel,” J. Colloid Interface Sci. 152(2), 587–590 (1992).
[Crossref]

Suh, S. S.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76(9), 093,110–11 (2005).
[Crossref]

Thomas, J. C.

J. C. Thomas and S. Tjin, “Fiber optics dynamic light-scattering (FODLS) from moderately concentrated suspensions,” J. Colloid Interface Sci. 129, 15–31 (1989).
[Crossref]

Tjin, S.

J. C. Thomas and S. Tjin, “Fiber optics dynamic light-scattering (FODLS) from moderately concentrated suspensions,” J. Colloid Interface Sci. 129, 15–31 (1989).
[Crossref]

Trainoff, S. P.

S. P. Trainoff and D. S. Cannell, “Physical optics treatment of the shadowgraph,” Phys. Fluids 14, 1340–1363 (2002).
[Crossref]

Trappe, V.

R. Cerbino and V. Trappe, “Differential Dynamic Microscopy: Probing Wave Vector Dependent Dynamics with a Microscope,” Phys. Rev. Lett. 100, 188,102-1-4 (2008).
[Crossref] [PubMed]

Vailati, A.

F. Croccolo, D. Brogioli, A. Vailati, M. Giglio, and D. Cannell, “Non-diffusive decay of gradient driven fluctuations in a free-diffusion process,” Phys. Rev. E 76, 41,112-1-9 (2007).
[Crossref]

F. Croccolo, D. Brogioli, A. Vailati, M. Giglio, and D. Cannell, “Use of dynamic Schlieren to study fluctuations during free diffusion,” Appl. Opt. 45, 2166–2173 (2006).
[Crossref] [PubMed]

D. Brogioli, A. Vailati, and M. Giglio, “A schlieren method for ultra-low angle light scattering measurements,” Europhys. Lett. 63, 220–225 (2003).
[Crossref]

D. Brogioli, A. Vailati, and M. Giglio, “Heterodyne near-field scattering,” Appl. Phys. Lett. 81, 4109–4111 (2002).
[Crossref]

M. Giglio, M. Carpineti, A. Vailati, and D. Brogioli, “Near-field intensity correlations of scattered light,” Appl. Opt. 40(24), 4036–4040 (2001).
[Crossref]

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g(r),” Phys. Rev. Lett. 85, 1416–1419 (2000).
[Crossref] [PubMed]

van Blaaderen, S. M. Y. A.

S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Y. A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. express 15(26), 18,275–18,282 (2007).

van Oostrum, P.

S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Y. A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. express 15(26), 18,275–18,282 (2007).

Visca, M.

V. Degiorgio, R. Piazza, T. Bellini, and M. Visca, “Static and dynamic light scattering study of fluorinated polymer colloids with a crystalline internal structure,” Adv. Colloid Interface Sci. 48, 61–91 (1994).
[Crossref]

Weitz, D.

L. Cipelletti and D. Weitz, “Ultralow-angle dynamic light scattering with a charge coupled device camera based multispeckle, multitau correlator,” Rev. Sci. Instrum. 70, 3214–3221 (1999).
[Crossref]

Weitz, D. A.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60(12), 1134–1137 (1988).
[Crossref] [PubMed]

Wu, M.

M. Wu, G. Ahlers, and D. Cannell, “Thermally induced fluctuations below the onset of Reyleight-Benard convection,” Phys. Rev. Lett. 75, 1743–1746 (1995).
[Crossref] [PubMed]

Yi, G. R.

S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Y. A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. express 15(26), 18,275–18,282 (2007).

Zhang, H. Y.

G. H. Koenderink, H. Y. Zhang, D. G. A. L. Aarts, M. P. Lettinga, A. P. Philipse, and G. Nagele, “On the validity of Stokes-Einstein-Debye relations for rotational diffusion in colloidal suspensions,” Faraday Discuss. 123, 335–354 (2003).
[Crossref] [PubMed]

Adv. Colloid Interface Sci. (1)

V. Degiorgio, R. Piazza, T. Bellini, and M. Visca, “Static and dynamic light scattering study of fluorinated polymer colloids with a crystalline internal structure,” Adv. Colloid Interface Sci. 48, 61–91 (1994).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

D. Brogioli, A. Vailati, and M. Giglio, “Heterodyne near-field scattering,” Appl. Phys. Lett. 81, 4109–4111 (2002).
[Crossref]

D. Magatti, M. D. Alaimo, M. A. C. Potenza, and F. Ferri, “Dynamic heterodyne near field scattering,” Appl. Phys. Lett. 92, 241,101-1-3 (2008).
[Crossref]

Curr. Opin. Colloid Interface Sci. (2)

F. Scheffold and R. Cerbino, “New trends in light scattering,” Curr. Opin. Colloid Interface Sci. 12(1), 50–57 (2007).
[Crossref]

P. N. Pusey, “Suppression of multiple scattering by photon cross-correlation techniques,” Curr. Opin. Colloid Interface Sci. 4(3), 177–185 (1999).
[Crossref]

Europhys. Lett. (2)

D. Brogioli, A. Vailati, and M. Giglio, “A schlieren method for ultra-low angle light scattering measurements,” Europhys. Lett. 63, 220–225 (2003).
[Crossref]

T. Bellini, R. Piazza, C. Sozzi, and V. Degiorgio, “Electric Birefringence of a Dispersion of Electrically Charged Anisotropic Particles,” Europhys. Lett. 7(6), 561–565 (1988).
[Crossref]

Faraday Discuss. (1)

G. H. Koenderink, H. Y. Zhang, D. G. A. L. Aarts, M. P. Lettinga, A. P. Philipse, and G. Nagele, “On the validity of Stokes-Einstein-Debye relations for rotational diffusion in colloidal suspensions,” Faraday Discuss. 123, 335–354 (2003).
[Crossref] [PubMed]

J. Chem. Phys. (1)

T. Bellini, V. Degiorgio, F. Mantegazza, F. Ajmone-Marsan, and C. Scarnecchia, “Electrokinetic properties of colloids of variable charge. 1. Electrophoretic and electrooptic characterization,” J. Chem. Phys. 103, 8228–8237 (1995).
[Crossref]

J. Colloid Interface Sci. (2)

T. Sugimoto and K. Sakata, “Preparation of monodisperse pseudocubic α - Fe2O3 particles from condensed ferric hydroxide gel,” J. Colloid Interface Sci. 152(2), 587–590 (1992).
[Crossref]

J. C. Thomas and S. Tjin, “Fiber optics dynamic light-scattering (FODLS) from moderately concentrated suspensions,” J. Colloid Interface Sci. 129, 15–31 (1989).
[Crossref]

J. Mod. Opt. (1)

K. Schatzel, “Suppression of multiple-scattering by photon cross-correlation techniques,” J. Mod. Opt. 38(9), 1849–1865 (1991).
[Crossref]

Opt. Commun. (1)

R. Piazza, J. Stavans, T. Bellini, and V. Degiorgio, “Light scattering study of crystalline latex particles,” Opt. Commun. 73(4), 263–267 (1989).
[Crossref]

Opt. express (1)

S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Y. A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. express 15(26), 18,275–18,282 (2007).

D. Brogioli, F. Croccolo, V. Cassina, D. Salerno, and F. Mantegazza, “Nano-particle characterization by using Exposure Time Dependent Spectrum and scattering in the near field methods: how to get fast dynamics with low-speed CCD camera.” Opt. Express 16(25), 20,272–20,282 (2008).
[Crossref]

Phys. Fluids (1)

S. P. Trainoff and D. S. Cannell, “Physical optics treatment of the shadowgraph,” Phys. Fluids 14, 1340–1363 (2002).
[Crossref]

Phys. Rev. E (2)

J. Oh, J. O. de Zárate, J. Sengers, and G. Ahlers, “Dynamics of fluctuations in a fluid below the onset of Rayleigh-Bénard convection,” Phys. Rev. E 69, 21,106-1-13 (2004).
[Crossref]

F. Croccolo, D. Brogioli, A. Vailati, M. Giglio, and D. Cannell, “Non-diffusive decay of gradient driven fluctuations in a free-diffusion process,” Phys. Rev. E 76, 41,112-1-9 (2007).
[Crossref]

Phys. Rev. Lett. (4)

R. Cerbino and V. Trappe, “Differential Dynamic Microscopy: Probing Wave Vector Dependent Dynamics with a Microscope,” Phys. Rev. Lett. 100, 188,102-1-4 (2008).
[Crossref] [PubMed]

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g(r),” Phys. Rev. Lett. 85, 1416–1419 (2000).
[Crossref] [PubMed]

M. Wu, G. Ahlers, and D. Cannell, “Thermally induced fluctuations below the onset of Reyleight-Benard convection,” Phys. Rev. Lett. 75, 1743–1746 (1995).
[Crossref] [PubMed]

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60(12), 1134–1137 (1988).
[Crossref] [PubMed]

Rev. Sci. Instrum. (3)

F. Ferri, “Use of a charge coupled device camera for low-angle elastic light scattering,” Rev. Sci. Instrum. 68, 2265–2274 (1997).
[Crossref]

L. Cipelletti and D. Weitz, “Ultralow-angle dynamic light scattering with a charge coupled device camera based multispeckle, multitau correlator,” Rev. Sci. Instrum. 70, 3214–3221 (1999).
[Crossref]

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: A tool to study time-varying dynamics,” Rev. Sci. Instrum. 76(9), 093,110–11 (2005).
[Crossref]

Other (4)

D. Brogioli, “Near Field Speckles,” Ph.D. thesis, Università degli Studi di Cagliari (2002). Available at the url: www.geocities.com/dbrogioli/nfs_phd.

S. Sacanna, “Novel Routes to Model Colloids: ellipsoids, lattices and stable meso-emulsions,” Ph.D. thesis, Utrecht University (2007). ISBN 978-90-393-4598-6.

B. Berne and R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics (Dover, New York, 2000).

J. Goodman, Introduction to Fourier Optics (Roberts & Company, Englewood, 2005).

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Figures (7)

Fig. 1.
Fig. 1. Scheme of the optical set-up. The He-Ne laser, containing the polarizer, generates a collimated laser beam, which is attenuated by a neutral filter and bent upwards by a mirror. The beam is expanded by means of a negative focal length lens, making it diverge slightly before going through the sample cell. Scattered light is acquired in the near field together with transmitted light, through a 40x microscope objective, which conjugates a plane close to the sample onto the CCD sensor. Between the microscope objective and the CCD sensor, a rotating polarizer, which can be oriented at any angle, is inserted as analyzer.

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Fig. 2.
Fig. 2. Images collected by the CCD using the SINF technique (panels a and c) and their power spectra S(Q) (panels b and d). Data taken at ϑ = 0 for two different exposure times: Δt = 48ms for panels a and b; Δt = 497ms for panels c and d. Data for PTFE dispersion, 0.25% volume fraction in water.

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Fig. 3.
Fig. 3. Exposure time dependent spectra S ϑ(Q,Δt), measured as functions of the scattering vector Q, at two different angles ϑ between polarizer and analyzer. Left panel: data taken at different exposure time Δt with the analyzer at ϑ = 0 (VV, polarized component). Right panel: same as left panel, but with analyzer at ϑ= 82.0° (VV + VH, sum of polarized and depolarized components). The keys provide the different values of the exposure time Δt. Data taken for PTFE as in Fig. 2.

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Fig. 4.
Fig. 4. Exposure-time dependent spectra S(Q, Δt), measured as functions of the exposure time Δt, at two different angles ϑ between polarizer and analyzer. Squares: Q = 1.23 · 106m-1; dots: Q = 4.42·106m-1. Filled symbols: ϑ = 0; open symbols: ϑ = 82.0°. The lines represent the fitting curves obtained according to Eq. (17), see text for details. Fitting parameters at Q = 1.23 · 106m-1, τVV = 0.5s and τVH = 6.2ms; at Q = 4.42 · 106m-1, τVV = 36ms and τVH = 5.5ms. Data taken for PTFE as in Fig. 2.

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Fig. 5.
Fig. 5. Decay times τVV and τVH , obtained by the described fitting procedure, and plotted as functions of the scattering wave vector Q. Lines represent a fitting on τVV and τVH with Eq. (3) and Eq. (4). Values obtained by the fit DT = 1.4·10-12m2/s and Θ = 23.8s-1. Data taken for PTFE as in Fig. 2.

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Fig. 6.
Fig. 6. Decay times and fitting lines as for Fig. 5. Values obtained by the fit: DT = 2.3 · 10-12m2/s; Θ = 28.2s-1, α = 0.477. Data taken for the DR1 sample in water, at 1% volume fraction.

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Fig. 7.
Fig. 7. Left panel: depolarized VH intensity autocorrelation function CVH I (t) measured as a function of delay time t, by using traditional SIFF apparatus, at Q = 2.5 · 106m-1. The lines represent a fit with a stretched exponential or a single exponential decay. Fitting parameters: Θ = 27.8s-1; α = 0.523. Right panel: depolarized VH ETDS measured as a function of exposure time Δt, by using SINF apparatus, at Q = 2.1 · 106m-1. The lines represent a fit with fα (x) from Eq. (20) (which is related to a stretched exponential), or a fit with f(x) from Eq. (18) (which is related to a single exponential). Fitting parameters: Θ = 28.2s-1, α = 0.477. Data taken for DR1 sample as in Fig. 6.

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Equations (21)

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

C E V V ( Q , t ) ∝ e − t / τ V V
C E V H ( Q , t ) ∝ e − t / τ V H ,
τ V V = 1 D T Q 2
τ V H = 1 D T Q 2 + 6 Θ
C I V V ( Q , t ) ∝ β e − 2 t / τ V V + 1
C I V H ( Q , t ) ∝ β e − 2 t / τ V H + 1 ,
C I V V ( Q , t ) ∝ β e − 2 ( t / τ V H ) α + 1 ,
Q ( q ) = 2 K 1 − 1 − ( q K ) 2
S ( Q ) = T ( Q ) I ( Q ) + B ( Q )
E = E 0 + E V V + E V H
E = E 0 cos ϑ + E V V cos ϑ + E V H sin ϑ
I = ∣ E ∣ 2 ∝ ∣ E 0 ∣ 2 + ℜ [ E V V E 0 * ] + ℜ [ E V H E 0 * ] tan ϑ
S ϑ ( Q , Δ t ) = S V V ( Q , Δ t ) + S V H ( Q , Δ t ) tan 2 ϑ
S V V ( Q , Δ t ) = S ϑ = 0 ( Q , Δ t )
S V H ( Q , Δ t ) ∝ S ϑ = 82 ( Q , Δ t ) − S ϑ = 0 ( Q , Δ t )
S ( Q , Δ t ) ∝ 2 Δ t 2 ∫ 0 Δ t d s ( Δ t − s ) C E ( Q , s )
S ( Q , Δ t ) ∝ f ( Δ t τ )
f ( x ) = 2 x 2 ( e − x − 1 + x )
S V H ( Q , Δ t ) ∝ f α ( Δ t τ V H )
f α ( x ) = 2 α x [ γ ( 1 α , x α ) − 1 x γ ( 2 α , x α ) ] ,
γ ( a , x ) = ∫ 0 x e − t t a − 1 d t

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