Abstract

We investigate the polarization characteristics of coherent enhanced backscattering (EBS) using the pseudo-spectral time domain method implemented on staggered grid and local Fourier basis (SLPSTD) [Opt. Express 18, 9236 (2010)]. The studies are focused on Mie scatterers with findings profound to the understanding of polarization evolution in the scattering process. For linear polarization studies, the low-order scattering component of EBS is azimuthally anisotropic. A relationship between the degree of anisotropy and the photon’s penetration depth is established to characterize the depolarization progress. For circular polarization, exact numerical solutions disclose the origin of polarization memory effect and the helicity-flipping phenomenon. The region responsible for helicity-flipping is identified. Our numerical technique can be potentially applied to subsurface imaging that explores polarization memory effect.

©2010 Optical Society of America

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References

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  1. M. Ding and K. Chen, “Staggered-grid PSTD on local Fourier basis and its applications to surface tissue modeling,” Opt. Express 18(9), 9236–9250 (2010).
    [Crossref] [PubMed]
  2. S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000).
    [Crossref] [PubMed]
  3. Y. Liu, Y. L. Kim, X. Li, and V. Backman, “Investigation of depth selectivity of polarization gating for tissue characterization,” Opt. Express 13(2), 601–611 (2005).
    [Crossref] [PubMed]
  4. Y. L. Kim, Y. Liu, V. M. Turzhitsky, R. K. Wali, H. K. Roy, and V. Backman, “Depth-resolved low-coherence enhanced backscattering,” Opt. Lett. 30(7), 741–743 (2005).
    [Crossref] [PubMed]
  5. Y. L. Kim, P. Pradhan, H. Subramanian, Y. Liu, M. H. Kim, and V. Backman, “Origin of low-coherence enhanced backscattering,” Opt. Lett. 31(10), 1459–1461 (2006).
    [Crossref] [PubMed]
  6. A. Lagendijk, M. B. van der Mark, and A. Lagendijk, “Observation of weak localization of light in a finite slab: Anisotropy effects and light path classification,” Phys. Rev. Lett. 58(4), 361–364 (1987).
    [Crossref] [PubMed]
  7. M. P. van Albada, M. B. van der Mark, and A. Lagendijk, “Polarisation effects in weak localization of light,” J. Phys. D Appl. Phys. 21(10S), 28–31 (1988).
    [Crossref]
  8. E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
    [Crossref] [PubMed]
  9. R. Lenke, R. Tweer, and G. Maret, “Coherent backscattering of turbid samples containing large Mie spheres,” J. Opt. A-Pure Appl. Op. 4(3), 293–298 (2002).
  10. H. Subramanian, P. Pradhan, Y. L. Kim, Y. Liu, X. Li, and V. Backman, “Modeling low-coherence enhanced backscattering using Monte Carlo simulation,” Appl. Opt. 45(24), 6292–6300 (2006).
    [Crossref] [PubMed]
  11. J. Sawicki, N. Kastor, and M. Xu, “Electric field Monte Carlo simulation of coherent backscattering of polarized light by a turbid medium containing Mie scatterers,” Opt. Express 16(8), 5728–5738 (2008).
    [Crossref] [PubMed]
  12. M. I. Mishchenko, J. M. Dlugach, and L. Liu, “Azimuthal asymmetry of the coherent backscattering cone: theoretical results,” Phys. Rev. A 80(5), 053824 (2009).
    [Crossref]
  13. A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Second Edition (Artech House, 2000).
  14. Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett. 15(3), 158–165 (1997).
    [Crossref]
  15. S. H. Tseng, J. H. Greene, A. Taflove, D. Maitland, V. Backman, and J. T. Walsh., “Exact solution of Maxwell’s equations for optical interactions with a macroscopic random medium,” Opt. Lett. 29(12), 1393–1395 (2004).
    [Crossref] [PubMed]
  16. K. M. Koo, Y. Takiguchi, and R. R. Alfano, “Weak localization of photons: contributions from the different scattering pathlengths,” IEEEPhoton. Technol. Lett. 58, 94–96 (1989).
  17. Y. L. Kim, P. Pradhan, M. H. Kim, and V. Backman, “Circular polarization memory effect in low-coherence enhanced backscattering of light,” Opt. Lett. 31(18), 2744–2746 (2006).
    [Crossref] [PubMed]
  18. S. A. Kartazayeva, X. Ni, and R. R. Alfano, “Backscattering target detection in a turbid medium by use of circularly and linearly polarized light,” Opt. Lett. 30(10), 1168–1170 (2005).
    [Crossref] [PubMed]
  19. R. E. Nothdurft and G. Yao, “Applying the polarization memory effect in polarization-gated subsurface imaging,” Opt. Express 14(11), 4656–4661 (2006).
    [Crossref] [PubMed]
  20. R. E. Nothdurft and G. Yao, “Effects of turbid media optical properties on object visibility in subsurface polarization imaging,” Appl. Opt. 45(22), 5532–5541 (2006).
    [Crossref] [PubMed]
  21. T. W. Lee and S. C. Hagness, “A compact wave source condition for the pseudospectral time-domain method,” IEEE Antennas Wirel. Propag. Lett. 3(14), 253–256 (2004).
    [Crossref]
  22. Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain(PSTD) algorithm,” IEEE Trans. Geosci. Rem. Sens. 37(2), 917–926 (1999).
    [Crossref]
  23. S. H. Tseng, Y. L. Kim, A. Taflove, D. Maitland, V. Backman, and J. T. Walsh., “Simulation of enhanced backscattering of light by numerically solving Maxwell’s equations without heuristic approximations,” Opt. Express 13(10), 3666–3672 (2005).
    [Crossref] [PubMed]
  24. F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B Condens. Matter 40(13), 9342–9345 (1989).
    [Crossref] [PubMed]

2010 (1)

2009 (1)

M. I. Mishchenko, J. M. Dlugach, and L. Liu, “Azimuthal asymmetry of the coherent backscattering cone: theoretical results,” Phys. Rev. A 80(5), 053824 (2009).
[Crossref]

2008 (1)

2006 (5)

2005 (4)

2004 (2)

S. H. Tseng, J. H. Greene, A. Taflove, D. Maitland, V. Backman, and J. T. Walsh., “Exact solution of Maxwell’s equations for optical interactions with a macroscopic random medium,” Opt. Lett. 29(12), 1393–1395 (2004).
[Crossref] [PubMed]

T. W. Lee and S. C. Hagness, “A compact wave source condition for the pseudospectral time-domain method,” IEEE Antennas Wirel. Propag. Lett. 3(14), 253–256 (2004).
[Crossref]

2000 (1)

S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000).
[Crossref] [PubMed]

1999 (1)

Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain(PSTD) algorithm,” IEEE Trans. Geosci. Rem. Sens. 37(2), 917–926 (1999).
[Crossref]

1997 (1)

Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett. 15(3), 158–165 (1997).
[Crossref]

1989 (2)

K. M. Koo, Y. Takiguchi, and R. R. Alfano, “Weak localization of photons: contributions from the different scattering pathlengths,” IEEEPhoton. Technol. Lett. 58, 94–96 (1989).

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B Condens. Matter 40(13), 9342–9345 (1989).
[Crossref] [PubMed]

1988 (1)

M. P. van Albada, M. B. van der Mark, and A. Lagendijk, “Polarisation effects in weak localization of light,” J. Phys. D Appl. Phys. 21(10S), 28–31 (1988).
[Crossref]

1987 (1)

A. Lagendijk, M. B. van der Mark, and A. Lagendijk, “Observation of weak localization of light in a finite slab: Anisotropy effects and light path classification,” Phys. Rev. Lett. 58(4), 361–364 (1987).
[Crossref] [PubMed]

1986 (1)

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[Crossref] [PubMed]

Akkermans, E.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[Crossref] [PubMed]

Alfano, R. R.

S. A. Kartazayeva, X. Ni, and R. R. Alfano, “Backscattering target detection in a turbid medium by use of circularly and linearly polarized light,” Opt. Lett. 30(10), 1168–1170 (2005).
[Crossref] [PubMed]

K. M. Koo, Y. Takiguchi, and R. R. Alfano, “Weak localization of photons: contributions from the different scattering pathlengths,” IEEEPhoton. Technol. Lett. 58, 94–96 (1989).

Backman, V.

Y. L. Kim, P. Pradhan, H. Subramanian, Y. Liu, M. H. Kim, and V. Backman, “Origin of low-coherence enhanced backscattering,” Opt. Lett. 31(10), 1459–1461 (2006).
[Crossref] [PubMed]

H. Subramanian, P. Pradhan, Y. L. Kim, Y. Liu, X. Li, and V. Backman, “Modeling low-coherence enhanced backscattering using Monte Carlo simulation,” Appl. Opt. 45(24), 6292–6300 (2006).
[Crossref] [PubMed]

Y. L. Kim, P. Pradhan, M. H. Kim, and V. Backman, “Circular polarization memory effect in low-coherence enhanced backscattering of light,” Opt. Lett. 31(18), 2744–2746 (2006).
[Crossref] [PubMed]

S. H. Tseng, Y. L. Kim, A. Taflove, D. Maitland, V. Backman, and J. T. Walsh., “Simulation of enhanced backscattering of light by numerically solving Maxwell’s equations without heuristic approximations,” Opt. Express 13(10), 3666–3672 (2005).
[Crossref] [PubMed]

Y. Liu, Y. L. Kim, X. Li, and V. Backman, “Investigation of depth selectivity of polarization gating for tissue characterization,” Opt. Express 13(2), 601–611 (2005).
[Crossref] [PubMed]

Y. L. Kim, Y. Liu, V. M. Turzhitsky, R. K. Wali, H. K. Roy, and V. Backman, “Depth-resolved low-coherence enhanced backscattering,” Opt. Lett. 30(7), 741–743 (2005).
[Crossref] [PubMed]

S. H. Tseng, J. H. Greene, A. Taflove, D. Maitland, V. Backman, and J. T. Walsh., “Exact solution of Maxwell’s equations for optical interactions with a macroscopic random medium,” Opt. Lett. 29(12), 1393–1395 (2004).
[Crossref] [PubMed]

Chen, K.

Ding, M.

Dlugach, J. M.

M. I. Mishchenko, J. M. Dlugach, and L. Liu, “Azimuthal asymmetry of the coherent backscattering cone: theoretical results,” Phys. Rev. A 80(5), 053824 (2009).
[Crossref]

Greene, J. H.

Hagness, S. C.

T. W. Lee and S. C. Hagness, “A compact wave source condition for the pseudospectral time-domain method,” IEEE Antennas Wirel. Propag. Lett. 3(14), 253–256 (2004).
[Crossref]

Jacques, S. L.

S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000).
[Crossref] [PubMed]

Kartazayeva, S. A.

Kastor, N.

Kim, M. H.

Kim, Y. L.

Koo, K. M.

K. M. Koo, Y. Takiguchi, and R. R. Alfano, “Weak localization of photons: contributions from the different scattering pathlengths,” IEEEPhoton. Technol. Lett. 58, 94–96 (1989).

Lagendijk, A.

M. P. van Albada, M. B. van der Mark, and A. Lagendijk, “Polarisation effects in weak localization of light,” J. Phys. D Appl. Phys. 21(10S), 28–31 (1988).
[Crossref]

A. Lagendijk, M. B. van der Mark, and A. Lagendijk, “Observation of weak localization of light in a finite slab: Anisotropy effects and light path classification,” Phys. Rev. Lett. 58(4), 361–364 (1987).
[Crossref] [PubMed]

A. Lagendijk, M. B. van der Mark, and A. Lagendijk, “Observation of weak localization of light in a finite slab: Anisotropy effects and light path classification,” Phys. Rev. Lett. 58(4), 361–364 (1987).
[Crossref] [PubMed]

Lee, K.

S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000).
[Crossref] [PubMed]

Lee, T. W.

T. W. Lee and S. C. Hagness, “A compact wave source condition for the pseudospectral time-domain method,” IEEE Antennas Wirel. Propag. Lett. 3(14), 253–256 (2004).
[Crossref]

Li, X.

Liu, L.

M. I. Mishchenko, J. M. Dlugach, and L. Liu, “Azimuthal asymmetry of the coherent backscattering cone: theoretical results,” Phys. Rev. A 80(5), 053824 (2009).
[Crossref]

Liu, Q. H.

Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain(PSTD) algorithm,” IEEE Trans. Geosci. Rem. Sens. 37(2), 917–926 (1999).
[Crossref]

Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett. 15(3), 158–165 (1997).
[Crossref]

Liu, Y.

MacKintosh, F. C.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B Condens. Matter 40(13), 9342–9345 (1989).
[Crossref] [PubMed]

Maitland, D.

Maynard, R.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[Crossref] [PubMed]

Mishchenko, M. I.

M. I. Mishchenko, J. M. Dlugach, and L. Liu, “Azimuthal asymmetry of the coherent backscattering cone: theoretical results,” Phys. Rev. A 80(5), 053824 (2009).
[Crossref]

Ni, X.

Nothdurft, R. E.

Pine, D. J.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B Condens. Matter 40(13), 9342–9345 (1989).
[Crossref] [PubMed]

Pradhan, P.

Roman, J. R.

S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000).
[Crossref] [PubMed]

Roy, H. K.

Sawicki, J.

Subramanian, H.

Taflove, A.

Takiguchi, Y.

K. M. Koo, Y. Takiguchi, and R. R. Alfano, “Weak localization of photons: contributions from the different scattering pathlengths,” IEEEPhoton. Technol. Lett. 58, 94–96 (1989).

Tseng, S. H.

Turzhitsky, V. M.

van Albada, M. P.

M. P. van Albada, M. B. van der Mark, and A. Lagendijk, “Polarisation effects in weak localization of light,” J. Phys. D Appl. Phys. 21(10S), 28–31 (1988).
[Crossref]

van der Mark, M. B.

M. P. van Albada, M. B. van der Mark, and A. Lagendijk, “Polarisation effects in weak localization of light,” J. Phys. D Appl. Phys. 21(10S), 28–31 (1988).
[Crossref]

A. Lagendijk, M. B. van der Mark, and A. Lagendijk, “Observation of weak localization of light in a finite slab: Anisotropy effects and light path classification,” Phys. Rev. Lett. 58(4), 361–364 (1987).
[Crossref] [PubMed]

Wali, R. K.

Walsh, J. T.

Weitz, D. A.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B Condens. Matter 40(13), 9342–9345 (1989).
[Crossref] [PubMed]

Wolf, P. E.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[Crossref] [PubMed]

Xu, M.

Yao, G.

Zhu, J. X.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B Condens. Matter 40(13), 9342–9345 (1989).
[Crossref] [PubMed]

Appl. Opt. (2)

IEEE Antennas Wirel. Propag. Lett. (1)

T. W. Lee and S. C. Hagness, “A compact wave source condition for the pseudospectral time-domain method,” IEEE Antennas Wirel. Propag. Lett. 3(14), 253–256 (2004).
[Crossref]

IEEE Trans. Geosci. Rem. Sens. (1)

Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain(PSTD) algorithm,” IEEE Trans. Geosci. Rem. Sens. 37(2), 917–926 (1999).
[Crossref]

IEEEPhoton. Technol. Lett. (1)

K. M. Koo, Y. Takiguchi, and R. R. Alfano, “Weak localization of photons: contributions from the different scattering pathlengths,” IEEEPhoton. Technol. Lett. 58, 94–96 (1989).

J. Phys. D Appl. Phys. (1)

M. P. van Albada, M. B. van der Mark, and A. Lagendijk, “Polarisation effects in weak localization of light,” J. Phys. D Appl. Phys. 21(10S), 28–31 (1988).
[Crossref]

Lasers Surg. Med. (1)

S. L. Jacques, J. R. Roman, and K. Lee, “Imaging superficial tissues with polarized light,” Lasers Surg. Med. 26(2), 119–129 (2000).
[Crossref] [PubMed]

Microw. Opt. Technol. Lett. (1)

Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett. 15(3), 158–165 (1997).
[Crossref]

Opt. Express (5)

Opt. Lett. (5)

Phys. Rev. A (1)

M. I. Mishchenko, J. M. Dlugach, and L. Liu, “Azimuthal asymmetry of the coherent backscattering cone: theoretical results,” Phys. Rev. A 80(5), 053824 (2009).
[Crossref]

Phys. Rev. B Condens. Matter (1)

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B Condens. Matter 40(13), 9342–9345 (1989).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

A. Lagendijk, M. B. van der Mark, and A. Lagendijk, “Observation of weak localization of light in a finite slab: Anisotropy effects and light path classification,” Phys. Rev. Lett. 58(4), 361–364 (1987).
[Crossref] [PubMed]

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: Analysis of the peak line shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[Crossref] [PubMed]

Other (2)

R. Lenke, R. Tweer, and G. Maret, “Coherent backscattering of turbid samples containing large Mie spheres,” J. Opt. A-Pure Appl. Op. 4(3), 293–298 (2002).

A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Second Edition (Artech House, 2000).

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the system setup. To avoid the specular reflection, the incident light is slightly tilted from the normal of the medium surface by a 5 ° angle, so the coordinate system x ′ y z ′ is a rotation of the coordinate system x y z along the y axis by 5 ° . The backscattering angle θ and the azimuthal angle φ are defined in the x ′ y z ′ coordinate system, where the z ′ axis is the exact backward direction and the center of the EBS cone. The incidence polarization lies in the x ′ z ′ plane, whereas the polarization of the backscattered light is decomposed into a component parallel to x ′ z ′ plane ( I ∥ ) and its orthogonal component ( I ⊥ ).

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Fig. 2
Fig. 2

(a) Co-polarized and (b) cross-polarized EBS from random medium consisting of Mie scatterers under linearly polarized incidence. Each color map represents a view of the EBS intensity along the exact backward direction. In both (a) and (b), from left to right, the upper row corresponds to medium of optical thickness 1, 2 and 3, and the lower row 4, 5 and 6, respectively. The backscattering angle θ ranges from 0 ° at the center to 8 ° at the border. The azimuth angleφis defined from the direction of the incident polarization ( φ = 0 ° ). Azimuthal anisotropy appears as intensity fluctuation within each color map. The patterns are the most prominent for τ = 1 in both (a) and (b), but progressively become blurred as τ increases to 6.

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Fig. 3
Fig. 3

Polar diagrams of the integrated intensities (Eqs. (1)-(2), blue line) and their empirical fitting (Eqs. (3)-(4), red line): (a) τ = 1 , co-polarized; (b) τ = 6 , co-polarized; (c) τ = 1 , cross-polarized; and (d) τ = 6 , cross-polarized. Because only the curve’s shape is relevant to the degree of anisotropy, each individual curve is normalized to its own intensity at φ = 0 ° .

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Fig. 4
Fig. 4

Parameters characterizing the degree of anisotropy: (a) ε for the co-polarized EBS, and (b) γ for the cross-polarized EBS.

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Fig. 5
Fig. 5

Difference signal between co-polarized and cross-polarized backscatterings for various media thickness. The region θ < 0.25 ° is excluded to remove the specular reflection. The curves are found to be equidistant when τ > 2 , indicating the two helicity contents cannot be equalized as the photons propagate deeper and the incident circular polarization state is remembered. On the other hand, the signal is negative when τ < 3 , which means the overall backscattering from this region is helicity-flipped.

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Fig. 6
Fig. 6

Integrated intensity vs. optical thicknesses, (a) the 0.6 < τ < 2 portion is a third order polynomial fit to emphasize the helicity flipping; (b) the 2 < τ < 6 portion is a linear fit to reflect the polarization memory effect. In (a) the minimum is determined at τ = 0.95 . The descending part ( τ < 0.95 ) indicates the first single large-angle scattering is responsible for helicity flipping; the curve is still negative for 0.95 < τ < 2. 4 , indicating the overall effect still favors helicity flipping.

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Equations (8)

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

I ∥ s i m ( φ ) ≡ ∫ 0 ° 8 ° I ∥ s i m ( θ , φ ) sin θ d θ ,
I ⊥ s i m ( φ ) ≡ ∫ 0 ° 8 ° I ⊥ s i m ( θ , φ ) sin θ d θ ,
I ∥ f i t ( φ ) ≡ 1 / cos 2 φ a 2 + sin 2 φ a 2 ε 2 ,
I ⊥ f i t ( φ ) ≡ b [ 1 − γ cos ( 4 φ ) ] ,
χ ∥ 2 ≡ 1 M ∑ i = 1 M [ I ∥ f i t ( φ i ) − I ∥ s i m ( φ i ) ] 2 ,
χ ⊥ 2 ≡ 1 M ∑ i = 1 M [ I ⊥ f i t ( φ i ) − I ⊥ s i m ( φ i ) ] 2 ,
I d i f f ( θ ) ≡ 1 2 π ∫ 0 2 π [ I ⊙ ( θ , φ ) − I ⊗ ( θ , φ ) ] d φ .
I d i f f int ≡ ∫ 0.25 ° 8 °   I d i f f ( θ ) sin θ   d θ .

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