Abstract

Scattering by infinite hexagonal ice prisms is calculated using Maxwell’s equations in the discrete dipole approximation for size parameters x = πD/λ up to x = 400 (D = prism diameter). Birefringence is included in the calculations. Applicability of the geometric optics approximation is investigated. Excellent agreement between wave optics and geometric optics is observed for large size parameter in the outer part of the 22 degree halo feature. For smaller ice crystals halo broadening is predicted, and there is appreciable “spillover” of the halo into shadow scattering angles < 22 degrees. Ways to retrieve ice crystal sizes are suggested based on the full width at half-maximum of the halo, the power at < 22deg, and the halo polarization.

© 2014 Optical Society of America

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References

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    [Crossref]
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2014 (1)

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Rad. Transfer 138, 17–35 (2014).
[Crossref]

2013 (2)

V. Shcherbakov, “Why the 46° halo is seen far less often than the 22° halo?” J. Quant. Spectrosc. Radiat. Transfer 124, 37–44 (2013).
[Crossref]

P. Yang, L. Bi, B. A. Baum, K.-N. Liou, G. W. Kattawar, M. I. Mishchenko, and B. Cole, “Spectrally consistent scattering, absorption, and polarization properties of atmospheric ice crystals at wavelengths from 0.2 to 100 μm,” J. Atmos. Sci. 70, 330–347 (2013).
[Crossref]

2012 (1)

2009 (1)

P. C. Chaumet and A. Rahmani, “Efficient iterative solution of the discrete dipole approximation for magnetodi-electric scatterers,” Opt. Letters 34, 917–919 (2009).
[Crossref]

2008 (1)

2007 (1)

A. G. Borovoi, A. V. Burnashov, and A. Y. S. Cheng, “Light scattering by horizontally oriented ice crystal plates,” Journal of Quantitative Spectroscopy and Radiative Transfer 106, 11–20 (2007).
[Crossref]

2004 (1)

J. Tang, Y. Shen, Y. Zheng, and D. Qiu, “An efficient and flexible computational model for solving the mild slope equation,” Coastal Engineering 51, 143– 154 (2004).
[Crossref]

2003 (3)

1999 (1)

1996 (2)

A. Macke, M. I. Mishchenko, and B. Cairns, “The influence of inclusions on light scattering by large ice particles,” J. Geophys. Res. 101, 23311–23316 (1996).
[Crossref]

A. Macke, J. Mueller, and E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[Crossref]

1995 (1)

K. F. Evans and G. L. Stephens, “Microwave radiative transfer through clouds composed of realistically shaped ice crystals. part i. single scattering properties,” J. Atmos. Sci. 52, 2041–2057 (1995).
[Crossref]

1994 (2)

1993 (1)

1991 (1)

1990 (1)

G. L. Stephens, S.-C. Tsay, P. W. Stackhouse, and P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climatic feedback,” J. Atmos. Sci. 47, 1742–1754 (1990).
[Crossref]

1989 (1)

Y. Takano and K.-N. Liou, “Solar radiative transfer in cirrus clouds. Part I: Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[Crossref]

1982 (1)

1979 (1)

1917 (1)

A. Ehringhaus, “Beiträge zur Kenntnis der dispersion der doppelbrechung einiger kristalle,” Neues Jahrbuch für Mineralogie, Geologie und Paläontolgie, Beilange Band, B 41, 342–419 (1917).

Adam, J. A.

J. A. Adam, Mathematics in Nature: Modeling Patterns in the Natural World (Princeton University, 2011).

Baran, A. J.

Baum, B. A.

P. Yang, L. Bi, B. A. Baum, K.-N. Liou, G. W. Kattawar, M. I. Mishchenko, and B. Cole, “Spectrally consistent scattering, absorption, and polarization properties of atmospheric ice crystals at wavelengths from 0.2 to 100 μm,” J. Atmos. Sci. 70, 330–347 (2013).
[Crossref]

Y.-K. Lee, P. Yang, M. I. Mishchenko, B. A. Baum, Y. X. Hu, H.-L. Huang, W. J. Wiscombe, and A. J. Baran, “Use of circular cylinders as surrogates for hexagonal pristine ice crystals in scattering calculations at infrared wavelengths,” Appl. Opt. 42, 2653–2664 (2003).
[Crossref] [PubMed]

Bi, L.

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Rad. Transfer 138, 17–35 (2014).
[Crossref]

P. Yang, L. Bi, B. A. Baum, K.-N. Liou, G. W. Kattawar, M. I. Mishchenko, and B. Cole, “Spectrally consistent scattering, absorption, and polarization properties of atmospheric ice crystals at wavelengths from 0.2 to 100 μm,” J. Atmos. Sci. 70, 330–347 (2013).
[Crossref]

Borovoi, A. G.

A. G. Borovoi, A. V. Burnashov, and A. Y. S. Cheng, “Light scattering by horizontally oriented ice crystal plates,” Journal of Quantitative Spectroscopy and Radiative Transfer 106, 11–20 (2007).
[Crossref]

Bryan, G.

W. R. Cotton, G. Bryan, and S. C. Van den Heever, Storm and Cloud Dynamics (Academic Press, 2010).

Bryant, G. W.

P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003).
[Crossref]

Burnashov, A. V.

A. G. Borovoi, A. V. Burnashov, and A. Y. S. Cheng, “Light scattering by horizontally oriented ice crystal plates,” Journal of Quantitative Spectroscopy and Radiative Transfer 106, 11–20 (2007).
[Crossref]

Cai, Q.

Cairns, B.

A. Macke, M. I. Mishchenko, and B. Cairns, “The influence of inclusions on light scattering by large ice particles,” J. Geophys. Res. 101, 23311–23316 (1996).
[Crossref]

Chaumet, P. C.

P. C. Chaumet and A. Rahmani, “Efficient iterative solution of the discrete dipole approximation for magnetodi-electric scatterers,” Opt. Letters 34, 917–919 (2009).
[Crossref]

P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003).
[Crossref]

Cheng, A. Y. S.

A. G. Borovoi, A. V. Burnashov, and A. Y. S. Cheng, “Light scattering by horizontally oriented ice crystal plates,” Journal of Quantitative Spectroscopy and Radiative Transfer 106, 11–20 (2007).
[Crossref]

Cole, B.

P. Yang, L. Bi, B. A. Baum, K.-N. Liou, G. W. Kattawar, M. I. Mishchenko, and B. Cole, “Spectrally consistent scattering, absorption, and polarization properties of atmospheric ice crystals at wavelengths from 0.2 to 100 μm,” J. Atmos. Sci. 70, 330–347 (2013).
[Crossref]

Cotton, W. R.

W. R. Cotton, G. Bryan, and S. C. Van den Heever, Storm and Cloud Dynamics (Academic Press, 2010).

Draine, B. T.

Ehringhaus, A.

A. Ehringhaus, “Beiträge zur Kenntnis der dispersion der doppelbrechung einiger kristalle,” Neues Jahrbuch für Mineralogie, Geologie und Paläontolgie, Beilange Band, B 41, 342–419 (1917).

Evans, K. F.

K. F. Evans and G. L. Stephens, “Microwave radiative transfer through clouds composed of realistically shaped ice crystals. part i. single scattering properties,” J. Atmos. Sci. 52, 2041–2057 (1995).
[Crossref]

Flatau, P. J.

Greenler, R.

R. Greenler, Rainbows, Halos, and Glories (Cambridge University, 1980).

Hu, Y. X.

Huang, H.-L.

Kattawar, G. W.

P. Yang, L. Bi, B. A. Baum, K.-N. Liou, G. W. Kattawar, M. I. Mishchenko, and B. Cole, “Spectrally consistent scattering, absorption, and polarization properties of atmospheric ice crystals at wavelengths from 0.2 to 100 μm,” J. Atmos. Sci. 70, 330–347 (2013).
[Crossref]

Können, G.

Können, G. P.

Landi Degl’Innocenti, E.

E. Landi Degl’Innocenti, “Physics of Polarization,” in “Astrophysical Spectropolarimetry: Proceedings of the XII Canary Islands Winter School of Astrophysics, Puerto de la Cruz, Tenerife, SpainNovember 13–24, 2000”, J. Trujillo-Bueno, F. Moreno-Insertis, and F. Sánchez, eds. (Cambridge University, 2002), pp. 1–52.

Lee, Y.-K.

Liou, K.-N.

P. Yang, L. Bi, B. A. Baum, K.-N. Liou, G. W. Kattawar, M. I. Mishchenko, and B. Cole, “Spectrally consistent scattering, absorption, and polarization properties of atmospheric ice crystals at wavelengths from 0.2 to 100 μm,” J. Atmos. Sci. 70, 330–347 (2013).
[Crossref]

Y. Takano and K.-N. Liou, “Solar radiative transfer in cirrus clouds. Part I: Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[Crossref]

Q. Cai and K.-N. Liou, “Polarized light scattering by hexagonal ice crystals: theory,” Appl. Opt. 21, 3569–3580 (1982).
[Crossref] [PubMed]

Lynch, D. K.

D. K. Lynch, K. Sassen, D. O. Starr, and G. Stephens, Cirrus (Oxford University, 2001).

Macke, A.

M. I. Mishchenko and A. Macke, “How big should hexagonal ice crystals be to produce halos?” Appl. Opt. 38, 1626–1629 (1999).
[Crossref]

A. Macke, M. I. Mishchenko, and B. Cairns, “The influence of inclusions on light scattering by large ice particles,” J. Geophys. Res. 101, 23311–23316 (1996).
[Crossref]

A. Macke, J. Mueller, and E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[Crossref]

A. Macke, “Scattering of light by polyhedral ice crystals,” Appl. Opt. 32, 2780–2788 (1993).
[Crossref] [PubMed]

Mariotte, E.

E. Mariotte, Quatrieme essay. De la nature des couleurs (Michallett, E., Paris, 1681).

Mishchenko, M. I.

P. Yang, L. Bi, B. A. Baum, K.-N. Liou, G. W. Kattawar, M. I. Mishchenko, and B. Cole, “Spectrally consistent scattering, absorption, and polarization properties of atmospheric ice crystals at wavelengths from 0.2 to 100 μm,” J. Atmos. Sci. 70, 330–347 (2013).
[Crossref]

Y.-K. Lee, P. Yang, M. I. Mishchenko, B. A. Baum, Y. X. Hu, H.-L. Huang, W. J. Wiscombe, and A. J. Baran, “Use of circular cylinders as surrogates for hexagonal pristine ice crystals in scattering calculations at infrared wavelengths,” Appl. Opt. 42, 2653–2664 (2003).
[Crossref] [PubMed]

M. I. Mishchenko and A. Macke, “How big should hexagonal ice crystals be to produce halos?” Appl. Opt. 38, 1626–1629 (1999).
[Crossref]

A. Macke, M. I. Mishchenko, and B. Cairns, “The influence of inclusions on light scattering by large ice particles,” J. Geophys. Res. 101, 23311–23316 (1996).
[Crossref]

Moilanen, J.

W. Tape and J. Moilanen, Atmospheric Halos and the Search for Angle X, vol. 58 (American Geophysical Union, 2006).
[Crossref]

Mueller, J.

A. Macke, J. Mueller, and E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[Crossref]

Muller, S. H.

Qiu, D.

J. Tang, Y. Shen, Y. Zheng, and D. Qiu, “An efficient and flexible computational model for solving the mild slope equation,” Coastal Engineering 51, 143– 154 (2004).
[Crossref]

Rahmani, A.

P. C. Chaumet and A. Rahmani, “Efficient iterative solution of the discrete dipole approximation for magnetodi-electric scatterers,” Opt. Letters 34, 917–919 (2009).
[Crossref]

P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003).
[Crossref]

Raschke, E.

A. Macke, J. Mueller, and E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[Crossref]

Sassen, K.

D. K. Lynch, K. Sassen, D. O. Starr, and G. Stephens, Cirrus (Oxford University, 2001).

Shcherbakov, V.

V. Shcherbakov, “Why the 46° halo is seen far less often than the 22° halo?” J. Quant. Spectrosc. Radiat. Transfer 124, 37–44 (2013).
[Crossref]

Shen, Y.

J. Tang, Y. Shen, Y. Zheng, and D. Qiu, “An efficient and flexible computational model for solving the mild slope equation,” Coastal Engineering 51, 143– 154 (2004).
[Crossref]

Stackhouse, P. W.

G. L. Stephens, S.-C. Tsay, P. W. Stackhouse, and P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climatic feedback,” J. Atmos. Sci. 47, 1742–1754 (1990).
[Crossref]

Starr, D. O.

D. K. Lynch, K. Sassen, D. O. Starr, and G. Stephens, Cirrus (Oxford University, 2001).

Stephens, G.

D. K. Lynch, K. Sassen, D. O. Starr, and G. Stephens, Cirrus (Oxford University, 2001).

Stephens, G. L.

K. F. Evans and G. L. Stephens, “Microwave radiative transfer through clouds composed of realistically shaped ice crystals. part i. single scattering properties,” J. Atmos. Sci. 52, 2041–2057 (1995).
[Crossref]

G. L. Stephens, S.-C. Tsay, P. W. Stackhouse, and P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climatic feedback,” J. Atmos. Sci. 47, 1742–1754 (1990).
[Crossref]

Takano, Y.

Y. Takano and K.-N. Liou, “Solar radiative transfer in cirrus clouds. Part I: Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[Crossref]

Tang, J.

J. Tang, Y. Shen, Y. Zheng, and D. Qiu, “An efficient and flexible computational model for solving the mild slope equation,” Coastal Engineering 51, 143– 154 (2004).
[Crossref]

Tape, W.

W. Tape, Atmospheric Halos (American Geophysical Union, 1994).
[Crossref]

W. Tape and J. Moilanen, Atmospheric Halos and the Search for Angle X, vol. 58 (American Geophysical Union, 2006).
[Crossref]

Tinbergen, J.

Tsay, S.-C.

G. L. Stephens, S.-C. Tsay, P. W. Stackhouse, and P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climatic feedback,” J. Atmos. Sci. 47, 1742–1754 (1990).
[Crossref]

Van den Heever, S. C.

W. R. Cotton, G. Bryan, and S. C. Van den Heever, Storm and Cloud Dynamics (Academic Press, 2010).

Weickmann, H. K.

Wendling, P.

Wendling, R.

Wessels, H. R. A.

Wiscombe, W. J.

Yang, P.

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Rad. Transfer 138, 17–35 (2014).
[Crossref]

P. Yang, L. Bi, B. A. Baum, K.-N. Liou, G. W. Kattawar, M. I. Mishchenko, and B. Cole, “Spectrally consistent scattering, absorption, and polarization properties of atmospheric ice crystals at wavelengths from 0.2 to 100 μm,” J. Atmos. Sci. 70, 330–347 (2013).
[Crossref]

Y.-K. Lee, P. Yang, M. I. Mishchenko, B. A. Baum, Y. X. Hu, H.-L. Huang, W. J. Wiscombe, and A. J. Baran, “Use of circular cylinders as surrogates for hexagonal pristine ice crystals in scattering calculations at infrared wavelengths,” Appl. Opt. 42, 2653–2664 (2003).
[Crossref] [PubMed]

Zheng, Y.

J. Tang, Y. Shen, Y. Zheng, and D. Qiu, “An efficient and flexible computational model for solving the mild slope equation,” Coastal Engineering 51, 143– 154 (2004).
[Crossref]

Appl. Opt. (8)

Coastal Engineering (1)

J. Tang, Y. Shen, Y. Zheng, and D. Qiu, “An efficient and flexible computational model for solving the mild slope equation,” Coastal Engineering 51, 143– 154 (2004).
[Crossref]

J. Atmos. Sci. (5)

G. L. Stephens, S.-C. Tsay, P. W. Stackhouse, and P. J. Flatau, “The relevance of the microphysical and radiative properties of cirrus clouds to climate and climatic feedback,” J. Atmos. Sci. 47, 1742–1754 (1990).
[Crossref]

K. F. Evans and G. L. Stephens, “Microwave radiative transfer through clouds composed of realistically shaped ice crystals. part i. single scattering properties,” J. Atmos. Sci. 52, 2041–2057 (1995).
[Crossref]

A. Macke, J. Mueller, and E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[Crossref]

Y. Takano and K.-N. Liou, “Solar radiative transfer in cirrus clouds. Part I: Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[Crossref]

P. Yang, L. Bi, B. A. Baum, K.-N. Liou, G. W. Kattawar, M. I. Mishchenko, and B. Cole, “Spectrally consistent scattering, absorption, and polarization properties of atmospheric ice crystals at wavelengths from 0.2 to 100 μm,” J. Atmos. Sci. 70, 330–347 (2013).
[Crossref]

J. Geophys. Res. (1)

A. Macke, M. I. Mishchenko, and B. Cairns, “The influence of inclusions on light scattering by large ice particles,” J. Geophys. Res. 101, 23311–23316 (1996).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Quant. Spectrosc. Rad. Transfer (1)

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Rad. Transfer 138, 17–35 (2014).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (1)

V. Shcherbakov, “Why the 46° halo is seen far less often than the 22° halo?” J. Quant. Spectrosc. Radiat. Transfer 124, 37–44 (2013).
[Crossref]

Journal of Quantitative Spectroscopy and Radiative Transfer (1)

A. G. Borovoi, A. V. Burnashov, and A. Y. S. Cheng, “Light scattering by horizontally oriented ice crystal plates,” Journal of Quantitative Spectroscopy and Radiative Transfer 106, 11–20 (2007).
[Crossref]

Neues Jahrbuch für Mineralogie, Geologie und Paläontolgie, Beilange Band, B (1)

A. Ehringhaus, “Beiträge zur Kenntnis der dispersion der doppelbrechung einiger kristalle,” Neues Jahrbuch für Mineralogie, Geologie und Paläontolgie, Beilange Band, B 41, 342–419 (1917).

Opt. Express (1)

Opt. Letters (1)

P. C. Chaumet and A. Rahmani, “Efficient iterative solution of the discrete dipole approximation for magnetodi-electric scatterers,” Opt. Letters 34, 917–919 (2009).
[Crossref]

Phys. Rev. B (1)

P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003).
[Crossref]

Other (9)

E. Mariotte, Quatrieme essay. De la nature des couleurs (Michallett, E., Paris, 1681).

J. A. Adam, Mathematics in Nature: Modeling Patterns in the Natural World (Princeton University, 2011).

D. K. Lynch, K. Sassen, D. O. Starr, and G. Stephens, Cirrus (Oxford University, 2001).

W. R. Cotton, G. Bryan, and S. C. Van den Heever, Storm and Cloud Dynamics (Academic Press, 2010).

R. Greenler, Rainbows, Halos, and Glories (Cambridge University, 1980).

W. Tape, Atmospheric Halos (American Geophysical Union, 1994).
[Crossref]

W. Tape and J. Moilanen, Atmospheric Halos and the Search for Angle X, vol. 58 (American Geophysical Union, 2006).
[Crossref]

G. P. Können, Polarized Light in Nature (University of Cambridge, 1985).

E. Landi Degl’Innocenti, “Physics of Polarization,” in “Astrophysical Spectropolarimetry: Proceedings of the XII Canary Islands Winter School of Astrophysics, Puerto de la Cruz, Tenerife, SpainNovember 13–24, 2000”, J. Trujillo-Bueno, F. Moreno-Insertis, and F. Sánchez, eds. (Cambridge University, 2002), pp. 1–52.

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Figures (6)

Fig. 1
Fig. 1 Hexagonal ice crystal with vertex-to-vertex diameter D. There are six side faces and two end (basal) faces. The c-axis is parallel to the sides and perpendicular to the end faces. Prisms are rotated randomly around the c-axis. In the GO approximation, halos discussed in this paper peak at the minimum deviation angle ζmin for which once-refracted light is propagating parallel to a side face before the second refraction.
Fig. 2
Fig. 2 Discrete dipole approximation results for incidence angle Θ = 90° and x = 200, averaged over rotations around the c-axis (sampled at Δβ = 3° intervals), for 2 values of d (see Table 1). Vertical dashed lines show the geometric optics halo angles θs,min for ordinary and extraordinary polarizations. (a) Muller matrix element S 11 ( 1 d ) (total halo intensity I S 11 ( 1 d ) (b) Fractional polarization P. The short dashed line indicates the polarization (P = 0.037) of the 22° halo peak in the GO limit.
Fig. 3
Fig. 3 (1/D)d2Csca/d cosθsdL as a function of scattering angle θs for hexagonal prism with light incoming perpendicular to c-axis, averaged over prism rotations around the c-axis. (a) Full range of θs; (b) zoom on 15° < θs < 30°. DDA results are shown for selected values of xπD/λ, as well as GO results (see text). Vertical dashed lines show the expected position θs,min of the inner edge of the 22° halo for ordinary and extraordinary polarizations in the ray-tracing limit.
Fig. 4
Fig. 4 Similar to Fig. 3, but for oblique incidence Θ = 60°. (a) Full range of allowed scattering angles θs (0 – 120°); (b) zoom on 18° < θs < 33°. For Θ = 60° the GO cusps are at 24.896° and 25.017° for ordinary and extraordinary rays. Note the good agreement between GO and the x = 400 results for θs > 26°.
Fig. 5
Fig. 5 (a) The “power spillover index” Ψ as a function of x for Θ = 90°. Ψ is a measure of scattered power interior to 21.9° (see Eq. 10). Line connecting calculated points is only to guide the eye. (b) Full width at half-maximum (FWHM) for the scattering peak near 22°, as a function of 1/x = λ/πD. Line connecting calculated points is only to guide the eye.
Fig. 6
Fig. 6 Polarization P = S21/S11 as a function of scattering angle θs for hexagonal prism with unpolarized light incident perpendicular to the c-axis (Θ = 90°), for x = 100, 200, 300, 400. (a) 19° < θs < 25°; (b) zoom into 21° < θs < 24°. Also shown (labelled x = ∞) are GO results, including birefringence. Vertical dashed lines show location of halo edge for ordinary and extraordinary polarizations. Horizontal dashed line shows Ph = 0.037 expected for GO neglecting birefringence (see Eq. 13). See text for discussion.

Tables (1)

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Table 1 Parameters for DDA calculations

Equations (13)

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ζ min = 2 arcsin [ ( n 2 cos 2 Θ ) 1 / 2 2 sin Θ ] 60 °
θ s , min = arccos [ 1 + sin 2 Θ ( cos ζ min 1 ) ] ,
S 11 ( 1 d ) = k 0 sin Θ d 2 C sca d ζ d L ,
cos θ s = 1 sin 2 Θ ( 1 cos ζ ) ;
d 2 C sca d θ s d L = sin θ s k 0 sin Θ sin ζ 1 [ S 11 ( 1 d ) ( ζ 1 ) + S 11 ( 1 d ) ( ζ 2 ) ] = sin θ s sin Θ [ S 11 ( 1 d ) ( ζ 1 ) + S 11 ( 1 d ) ( ζ 2 ) ] k 0 ( 1 cos θ s ) ( 2 sin 2 Θ + cos θ s 1 ) .
d 2 C sca d θ s d L ( θ s = 0 ) = 2 k 0 S 11 ( 1 d ) ( ζ = 0 ) ,
d 2 C sca d θ s d L ( θ s = θ s , max ) = 2 k 0 S 11 ( 1 d ) ( ζ = π ) .
m o ( E c ^ ) = 1.3112 + 10 5 i
m e ( E c ^ ) = 1.3126 + 10 5 i .
Ψ = 19 ° 21.9 ° ( d C sca / d θ s ) d θ s 19 ° 25 ° ( d C sca / d θ s ) d θ s .
( Δ ζ ) FWHM = 0.886 ( λ / w ) = 2.348 ( λ / D ) .
( Δ ζ ) FWHM 3.2 ( λ / D ) .
P h = ( n 3 + 4 n 2 ) 4 ( 3 + n 4 n 2 ) 4 ( n 3 4 n 2 ) 4 + ( 3 + n 4 n 2 ) 4 = 0.037 for n = 1.312 .

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