Abstract

We describe implementation and demonstration of a polarization technique adapted for lidar to measure all unique elements of the volume backscatter phase matrix. This capability allows for detection of preferential orientation within a scattering volume, and may improve scattering inversions on oriented ice crystals. The technique is enabled using a Mueller formalism commonly employed in polarimetry, which does not require the lidar instrument be polarization preserving. Instead, the accuracy of the polarization measurements are limited by the accuracy of the instrument characterization. A high spectral resolution lidar at the National Center for Atmospheric Research was modified to demonstrate this polarization technique. Two observations where the instrument is tilted off zenith are presented. In the first case, the lidar detects flattened large raindrops oriented along the same direction due to drag forces from falling. The second case is an ice cloud approximately 5 km above lidar base that contains preferentially oriented ice crystals in a narrow altitude band.

© 2012 OSA

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  1. T. A. Seliga and V. N. Bringi, “Potential use of radar differential reflectivity measurements at orthogonal polarizations for measuring precipitation,” J. Appl. Meteor.15, 69–76 (1976).
    [CrossRef]
  2. J. Vivekanandan, G. Zhang, and E. Brandes, “Polarimetric radar estimators based on a constrained gamma drop size distribution model,” J. Appl. Meteor.43, 217–230 (2004).
    [CrossRef]
  3. R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, 1980).
  4. Y. Tanko and K. Liou, “Solar radiative transfer in cirrus clouds. Part II: Theory and computation of multiple scattering in an anisotropic medium,” J. Atmos. Sci.46, 20–36 (1989).
    [CrossRef]
  5. A. Heymsfield and J. Iaquinta, “Cirrus crystal terminal velocity,” J. Atmos. Sci.57, 916–938 (2000).
    [CrossRef]
  6. C. D. Westbrook, “The fall speeds of sub-100μm ice crystals,” Q. J. R. Meteorol. Soc.134, 1243–1251 (2008).
    [CrossRef]
  7. V. Noel and K. Sassen, “Study of planar ice crystal orientation in ice clouds from scanning polarization lidar observations,” J. Appl. Meteor.44, 653–664 (2005).
    [CrossRef]
  8. V. Noel and H. Chepfer, “Study of ice crystal orientation in cirrus clouds based on satellite polarized radiance measurements,” J. Atmos. Sci.61, 2073–2081 (2005).
    [CrossRef]
  9. K. Masuda and H. Ishimoto, “Influence of particle orientation on retrieving cirrus cloud properties by use of total and polarized reflectances from satellite measurements,” J. Quant. Spectrosc. Radiat. Transfer85, 183–193 (2004).
    [CrossRef]
  10. M. Hayman and J. P. Thayer, “General description of polarization in lidar using Stokes vectors and polar decomposition of Mueller matrices,” J. Opt. Soc. Am. A29, 400–409 (2012).
    [CrossRef]
  11. Y. Balin, B. Kaul, G. Kokhanenko, and D. Winker, “Application of circularly polarized laser radiation for sensing of crystal clouds,” Opt. Express17, 6849–6859 (2009).
    [CrossRef] [PubMed]
  12. C. M. R. Platt, “Lidar backscatter from horizontal ice crystal plates,” J. Appl. Meteorol.17, 482–488 (1978).
    [CrossRef]
  13. L. Thomas, J. C. Cartwright, and D. P. Wareing, “Lidar observations of the horizontal orientation of ice crystals in cirrus clouds,” Tellus B42, 2011–2016 (1990).
    [CrossRef]
  14. S. A. Young, C. M. R. Platt, R. T. Austin, and G. R. Patterson, “Optical properties and phase of some midlatitude, midlevel clouds in ECLIPS,” J. Appl. Meteorol.39, 135–153 (2000).
    [CrossRef]
  15. Y. Hu, P. Yang, B. Lin, G. Gibson, and C. Hostetler, “Discriminating between spherical and non-spherical scatterers,” J. Quant. Spectrosc. Radiat. Transfer79–80, 757–764 (2004).
  16. C. Zhou, P. Yang, A. E. Dessler, Y. Hu, and B. A. Baum, “Study of horizontally oriented ice crystals with CALIPSO observations and comparision with monte carlo radiative transfer simulations,” J. Appl. Meteor. Climatol.51, 1426–1439 (2012).
    [CrossRef]
  17. W. H. Hunt, D. M. Winker, M. A. Vaughan, K. A. Powell, P. L. Lucker, and C. Weimer, “CALIPSO lidar description and performance assessment,” J. Atmos. Oceanic Technol.26, 1214–1228 (2009).
    [CrossRef]
  18. C. D. Westbrook, A. J. Illingworth, E. J. O’Connor, and R. J. Hogan, “Doppler lidar measurements of oriented planar ice crystals falling from supercooled and glaciated layer clouds,” Q. J. R. Meteor. Soc.136, 260–276 (2010).
    [CrossRef]
  19. F-M Bréon and B. Dubrulle, “Horizontally oriented plates in clouds,” J. Atmos. Sci.61, 2888–2898 (2004).
    [CrossRef]
  20. A. J. Heymsfield and M. Kajikawa, “An improived approach to calculating terminal velocities of plate-like crystals and graupel,” J. Atmos. Sci.44, 1088–1099 (1987).
    [CrossRef]
  21. M. D. Gusta, E. Vallar, O. Riviere, F. Castagnoli, V. Venturi, and M. Morandi, “Use of polarimetric lidar for the study of oriented ice plates in clouds,” Appl. Opt.45, 4878–4887 (2006).
    [CrossRef]
  22. M. D. Shupe, D. D. Turner, V. P. Walden, R. Bennartz, M. P. Cadeddu, B. B. Castellani, C. J. Cox, D. R. Hudak, M. S. Kulie, N. B. Miller, R. R. Neely, and W. D Neff, “High and dry: New observations of tropospheric and cloud properties above the Greenland Ice Sheet,” B. Am. Meteorol. Soc. doi:, In Press.
    [CrossRef]
  23. R. R. Neely, M. Hayman, R. Stillwell, J. P. Thayer, R. M. Hardesty, M. O’Neill, M. D. Shupe, and C. Alvarez, “Polarization lidar at Summit, Greenland for the detection of cloud phase and particle orientation,” J. Atmos. Oceanic Technol. In Review.
  24. B. V. Kaul, I. V. Samokhvalov, and S. N. Volkov, “Investigating particle orientation in cirrus clouds by measuring backscattering phase matrices with lidar,” Appl. Opt.43, 6620–6628 (2004).
    [CrossRef]
  25. J. S. Tyo, Z. Wang, S. J. Johnson, and B. G. Hoover, “Design and optimization of partial Mueller matrix polarimeters,” Appl. Opt.49, 2326–2332 (2010).
    [CrossRef] [PubMed]
  26. K. M. Twietmeyer and R. A. Chipman, “Optimization of Mueller matrix polarimeters in the presence of error sources,” Opt. Express16, 11589–11603 (2008).
    [CrossRef] [PubMed]
  27. M. H. Smith, “Optimization of a duel-rotating-retarder Mueller matrix polarimeter,” Appl. Opt.412488–2493 (2002).
    [CrossRef] [PubMed]
  28. E. Eloranta, Chapter 5: High Spectral Resolution Lidar in Lidar: Range-Resolved Optical Remote Sensing of the AtmosphereNew York, U.S.A. (Springer, 2005).
  29. P. Piironen and E. Eloranta, “Deomonstration of a high-spectral-resolution lidar based on an iodine absorption filter,” Opt. Lett.19, 234–236 (1994).
    [CrossRef] [PubMed]
  30. G. G. Matvienko, I. V. Samokhvalov, and B. V. Kaul, “Research of the cirrus structure with a polarization lidar: parameters of particle orientation in crystal clouds,” in Proc. SPIE5571, 393–400 (2004).
    [CrossRef]
  31. H. van de Hulst, Light Scattering by Small Particles (Wiley, 1981).
  32. M. Mishchenko and J. Hovenier, “Depolarization of light backscattered by randomly oriented nonspherical particles,” Opt. Lett.20, 1356–1358 (1995).
    [CrossRef] [PubMed]
  33. C. J. Flynn, A. Mendoza, Y. Zheng, and S. Mathur, “Novel polarization-sensitive micropulse lidar measurement technique,” Opt. Express15, 2785–2790 (2007).
    [CrossRef] [PubMed]
  34. G. Gimmestad, “Reexamination of depolarization in lidar measurements,” Appl. Opt.47, 3795–3802 (2008).
    [CrossRef] [PubMed]
  35. M. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles, (Academic, 2000).
  36. S. M. Kay, Fundamentals of Statistical Signal Processing, I: Estimation Theory, (Prentice Hall, 1993).
  37. M. Hayman and J. Thayer, “Lidar polarization measurements of PMCs,” J. Atmos. Sol. Terr. Phys.73, 2110–2117 (2011).
    [CrossRef]
  38. A. Zardecki and A. Deepak, “Forward multiple scattering corrections as a function of detector field of view,” Appl. Opt.22, 2970–2976 (1983).
    [CrossRef] [PubMed]
  39. X. Zhu, “Explicit Jones transformation matrix for a tilted birefringent plate with its optic axis parallel to the plate surface,” Appl. Opt.33, 3502–3506 (1994).
    [CrossRef] [PubMed]
  40. S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. I. Algorithms,” J. Opt. Soc. Am. A10, 2371–2382 (1993).
    [CrossRef]
  41. S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A10, 2383–2393 (1993).
    [CrossRef]

2012 (2)

M. Hayman and J. P. Thayer, “General description of polarization in lidar using Stokes vectors and polar decomposition of Mueller matrices,” J. Opt. Soc. Am. A29, 400–409 (2012).
[CrossRef]

C. Zhou, P. Yang, A. E. Dessler, Y. Hu, and B. A. Baum, “Study of horizontally oriented ice crystals with CALIPSO observations and comparision with monte carlo radiative transfer simulations,” J. Appl. Meteor. Climatol.51, 1426–1439 (2012).
[CrossRef]

2011 (1)

M. Hayman and J. Thayer, “Lidar polarization measurements of PMCs,” J. Atmos. Sol. Terr. Phys.73, 2110–2117 (2011).
[CrossRef]

2010 (2)

J. S. Tyo, Z. Wang, S. J. Johnson, and B. G. Hoover, “Design and optimization of partial Mueller matrix polarimeters,” Appl. Opt.49, 2326–2332 (2010).
[CrossRef] [PubMed]

C. D. Westbrook, A. J. Illingworth, E. J. O’Connor, and R. J. Hogan, “Doppler lidar measurements of oriented planar ice crystals falling from supercooled and glaciated layer clouds,” Q. J. R. Meteor. Soc.136, 260–276 (2010).
[CrossRef]

2009 (2)

W. H. Hunt, D. M. Winker, M. A. Vaughan, K. A. Powell, P. L. Lucker, and C. Weimer, “CALIPSO lidar description and performance assessment,” J. Atmos. Oceanic Technol.26, 1214–1228 (2009).
[CrossRef]

Y. Balin, B. Kaul, G. Kokhanenko, and D. Winker, “Application of circularly polarized laser radiation for sensing of crystal clouds,” Opt. Express17, 6849–6859 (2009).
[CrossRef] [PubMed]

2008 (3)

2007 (1)

2006 (1)

2005 (2)

V. Noel and K. Sassen, “Study of planar ice crystal orientation in ice clouds from scanning polarization lidar observations,” J. Appl. Meteor.44, 653–664 (2005).
[CrossRef]

V. Noel and H. Chepfer, “Study of ice crystal orientation in cirrus clouds based on satellite polarized radiance measurements,” J. Atmos. Sci.61, 2073–2081 (2005).
[CrossRef]

2004 (6)

K. Masuda and H. Ishimoto, “Influence of particle orientation on retrieving cirrus cloud properties by use of total and polarized reflectances from satellite measurements,” J. Quant. Spectrosc. Radiat. Transfer85, 183–193 (2004).
[CrossRef]

J. Vivekanandan, G. Zhang, and E. Brandes, “Polarimetric radar estimators based on a constrained gamma drop size distribution model,” J. Appl. Meteor.43, 217–230 (2004).
[CrossRef]

F-M Bréon and B. Dubrulle, “Horizontally oriented plates in clouds,” J. Atmos. Sci.61, 2888–2898 (2004).
[CrossRef]

B. V. Kaul, I. V. Samokhvalov, and S. N. Volkov, “Investigating particle orientation in cirrus clouds by measuring backscattering phase matrices with lidar,” Appl. Opt.43, 6620–6628 (2004).
[CrossRef]

G. G. Matvienko, I. V. Samokhvalov, and B. V. Kaul, “Research of the cirrus structure with a polarization lidar: parameters of particle orientation in crystal clouds,” in Proc. SPIE5571, 393–400 (2004).
[CrossRef]

Y. Hu, P. Yang, B. Lin, G. Gibson, and C. Hostetler, “Discriminating between spherical and non-spherical scatterers,” J. Quant. Spectrosc. Radiat. Transfer79–80, 757–764 (2004).

2002 (1)

2000 (2)

S. A. Young, C. M. R. Platt, R. T. Austin, and G. R. Patterson, “Optical properties and phase of some midlatitude, midlevel clouds in ECLIPS,” J. Appl. Meteorol.39, 135–153 (2000).
[CrossRef]

A. Heymsfield and J. Iaquinta, “Cirrus crystal terminal velocity,” J. Atmos. Sci.57, 916–938 (2000).
[CrossRef]

1995 (1)

1994 (2)

1993 (2)

1990 (1)

L. Thomas, J. C. Cartwright, and D. P. Wareing, “Lidar observations of the horizontal orientation of ice crystals in cirrus clouds,” Tellus B42, 2011–2016 (1990).
[CrossRef]

1989 (1)

Y. Tanko and K. Liou, “Solar radiative transfer in cirrus clouds. Part II: Theory and computation of multiple scattering in an anisotropic medium,” J. Atmos. Sci.46, 20–36 (1989).
[CrossRef]

1987 (1)

A. J. Heymsfield and M. Kajikawa, “An improived approach to calculating terminal velocities of plate-like crystals and graupel,” J. Atmos. Sci.44, 1088–1099 (1987).
[CrossRef]

1983 (1)

1978 (1)

C. M. R. Platt, “Lidar backscatter from horizontal ice crystal plates,” J. Appl. Meteorol.17, 482–488 (1978).
[CrossRef]

1976 (1)

T. A. Seliga and V. N. Bringi, “Potential use of radar differential reflectivity measurements at orthogonal polarizations for measuring precipitation,” J. Appl. Meteor.15, 69–76 (1976).
[CrossRef]

Alvarez, C.

R. R. Neely, M. Hayman, R. Stillwell, J. P. Thayer, R. M. Hardesty, M. O’Neill, M. D. Shupe, and C. Alvarez, “Polarization lidar at Summit, Greenland for the detection of cloud phase and particle orientation,” J. Atmos. Oceanic Technol. In Review.

Austin, R. T.

S. A. Young, C. M. R. Platt, R. T. Austin, and G. R. Patterson, “Optical properties and phase of some midlatitude, midlevel clouds in ECLIPS,” J. Appl. Meteorol.39, 135–153 (2000).
[CrossRef]

Balin, Y.

Baum, B. A.

C. Zhou, P. Yang, A. E. Dessler, Y. Hu, and B. A. Baum, “Study of horizontally oriented ice crystals with CALIPSO observations and comparision with monte carlo radiative transfer simulations,” J. Appl. Meteor. Climatol.51, 1426–1439 (2012).
[CrossRef]

Bennartz, R.

M. D. Shupe, D. D. Turner, V. P. Walden, R. Bennartz, M. P. Cadeddu, B. B. Castellani, C. J. Cox, D. R. Hudak, M. S. Kulie, N. B. Miller, R. R. Neely, and W. D Neff, “High and dry: New observations of tropospheric and cloud properties above the Greenland Ice Sheet,” B. Am. Meteorol. Soc. doi:, In Press.
[CrossRef]

Brandes, E.

J. Vivekanandan, G. Zhang, and E. Brandes, “Polarimetric radar estimators based on a constrained gamma drop size distribution model,” J. Appl. Meteor.43, 217–230 (2004).
[CrossRef]

Bréon, F-M

F-M Bréon and B. Dubrulle, “Horizontally oriented plates in clouds,” J. Atmos. Sci.61, 2888–2898 (2004).
[CrossRef]

Bringi, V. N.

T. A. Seliga and V. N. Bringi, “Potential use of radar differential reflectivity measurements at orthogonal polarizations for measuring precipitation,” J. Appl. Meteor.15, 69–76 (1976).
[CrossRef]

Cadeddu, M. P.

M. D. Shupe, D. D. Turner, V. P. Walden, R. Bennartz, M. P. Cadeddu, B. B. Castellani, C. J. Cox, D. R. Hudak, M. S. Kulie, N. B. Miller, R. R. Neely, and W. D Neff, “High and dry: New observations of tropospheric and cloud properties above the Greenland Ice Sheet,” B. Am. Meteorol. Soc. doi:, In Press.
[CrossRef]

Cartwright, J. C.

L. Thomas, J. C. Cartwright, and D. P. Wareing, “Lidar observations of the horizontal orientation of ice crystals in cirrus clouds,” Tellus B42, 2011–2016 (1990).
[CrossRef]

Castagnoli, F.

Castellani, B. B.

M. D. Shupe, D. D. Turner, V. P. Walden, R. Bennartz, M. P. Cadeddu, B. B. Castellani, C. J. Cox, D. R. Hudak, M. S. Kulie, N. B. Miller, R. R. Neely, and W. D Neff, “High and dry: New observations of tropospheric and cloud properties above the Greenland Ice Sheet,” B. Am. Meteorol. Soc. doi:, In Press.
[CrossRef]

Chepfer, H.

V. Noel and H. Chepfer, “Study of ice crystal orientation in cirrus clouds based on satellite polarized radiance measurements,” J. Atmos. Sci.61, 2073–2081 (2005).
[CrossRef]

Chipman, R. A.

Cox, C. J.

M. D. Shupe, D. D. Turner, V. P. Walden, R. Bennartz, M. P. Cadeddu, B. B. Castellani, C. J. Cox, D. R. Hudak, M. S. Kulie, N. B. Miller, R. R. Neely, and W. D Neff, “High and dry: New observations of tropospheric and cloud properties above the Greenland Ice Sheet,” B. Am. Meteorol. Soc. doi:, In Press.
[CrossRef]

Deepak, A.

Dessler, A. E.

C. Zhou, P. Yang, A. E. Dessler, Y. Hu, and B. A. Baum, “Study of horizontally oriented ice crystals with CALIPSO observations and comparision with monte carlo radiative transfer simulations,” J. Appl. Meteor. Climatol.51, 1426–1439 (2012).
[CrossRef]

Dubrulle, B.

F-M Bréon and B. Dubrulle, “Horizontally oriented plates in clouds,” J. Atmos. Sci.61, 2888–2898 (2004).
[CrossRef]

Eloranta, E.

P. Piironen and E. Eloranta, “Deomonstration of a high-spectral-resolution lidar based on an iodine absorption filter,” Opt. Lett.19, 234–236 (1994).
[CrossRef] [PubMed]

E. Eloranta, Chapter 5: High Spectral Resolution Lidar in Lidar: Range-Resolved Optical Remote Sensing of the AtmosphereNew York, U.S.A. (Springer, 2005).

Flynn, C. J.

Gibson, G.

Y. Hu, P. Yang, B. Lin, G. Gibson, and C. Hostetler, “Discriminating between spherical and non-spherical scatterers,” J. Quant. Spectrosc. Radiat. Transfer79–80, 757–764 (2004).

Gimmestad, G.

Greenler, R.

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

Gusta, M. D.

Hardesty, R. M.

R. R. Neely, M. Hayman, R. Stillwell, J. P. Thayer, R. M. Hardesty, M. O’Neill, M. D. Shupe, and C. Alvarez, “Polarization lidar at Summit, Greenland for the detection of cloud phase and particle orientation,” J. Atmos. Oceanic Technol. In Review.

Hayman, M.

M. Hayman and J. P. Thayer, “General description of polarization in lidar using Stokes vectors and polar decomposition of Mueller matrices,” J. Opt. Soc. Am. A29, 400–409 (2012).
[CrossRef]

M. Hayman and J. Thayer, “Lidar polarization measurements of PMCs,” J. Atmos. Sol. Terr. Phys.73, 2110–2117 (2011).
[CrossRef]

R. R. Neely, M. Hayman, R. Stillwell, J. P. Thayer, R. M. Hardesty, M. O’Neill, M. D. Shupe, and C. Alvarez, “Polarization lidar at Summit, Greenland for the detection of cloud phase and particle orientation,” J. Atmos. Oceanic Technol. In Review.

Heymsfield, A.

A. Heymsfield and J. Iaquinta, “Cirrus crystal terminal velocity,” J. Atmos. Sci.57, 916–938 (2000).
[CrossRef]

Heymsfield, A. J.

A. J. Heymsfield and M. Kajikawa, “An improived approach to calculating terminal velocities of plate-like crystals and graupel,” J. Atmos. Sci.44, 1088–1099 (1987).
[CrossRef]

Hillman, L. W.

Hogan, R. J.

C. D. Westbrook, A. J. Illingworth, E. J. O’Connor, and R. J. Hogan, “Doppler lidar measurements of oriented planar ice crystals falling from supercooled and glaciated layer clouds,” Q. J. R. Meteor. Soc.136, 260–276 (2010).
[CrossRef]

Hoover, B. G.

Hostetler, C.

Y. Hu, P. Yang, B. Lin, G. Gibson, and C. Hostetler, “Discriminating between spherical and non-spherical scatterers,” J. Quant. Spectrosc. Radiat. Transfer79–80, 757–764 (2004).

Hovenier, J.

Hovenier, J. W.

M. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles, (Academic, 2000).

Hu, Y.

C. Zhou, P. Yang, A. E. Dessler, Y. Hu, and B. A. Baum, “Study of horizontally oriented ice crystals with CALIPSO observations and comparision with monte carlo radiative transfer simulations,” J. Appl. Meteor. Climatol.51, 1426–1439 (2012).
[CrossRef]

Y. Hu, P. Yang, B. Lin, G. Gibson, and C. Hostetler, “Discriminating between spherical and non-spherical scatterers,” J. Quant. Spectrosc. Radiat. Transfer79–80, 757–764 (2004).

Hudak, D. R.

M. D. Shupe, D. D. Turner, V. P. Walden, R. Bennartz, M. P. Cadeddu, B. B. Castellani, C. J. Cox, D. R. Hudak, M. S. Kulie, N. B. Miller, R. R. Neely, and W. D Neff, “High and dry: New observations of tropospheric and cloud properties above the Greenland Ice Sheet,” B. Am. Meteorol. Soc. doi:, In Press.
[CrossRef]

Hunt, W. H.

W. H. Hunt, D. M. Winker, M. A. Vaughan, K. A. Powell, P. L. Lucker, and C. Weimer, “CALIPSO lidar description and performance assessment,” J. Atmos. Oceanic Technol.26, 1214–1228 (2009).
[CrossRef]

Iaquinta, J.

A. Heymsfield and J. Iaquinta, “Cirrus crystal terminal velocity,” J. Atmos. Sci.57, 916–938 (2000).
[CrossRef]

Illingworth, A. J.

C. D. Westbrook, A. J. Illingworth, E. J. O’Connor, and R. J. Hogan, “Doppler lidar measurements of oriented planar ice crystals falling from supercooled and glaciated layer clouds,” Q. J. R. Meteor. Soc.136, 260–276 (2010).
[CrossRef]

Ishimoto, H.

K. Masuda and H. Ishimoto, “Influence of particle orientation on retrieving cirrus cloud properties by use of total and polarized reflectances from satellite measurements,” J. Quant. Spectrosc. Radiat. Transfer85, 183–193 (2004).
[CrossRef]

Johnson, S. J.

Kajikawa, M.

A. J. Heymsfield and M. Kajikawa, “An improived approach to calculating terminal velocities of plate-like crystals and graupel,” J. Atmos. Sci.44, 1088–1099 (1987).
[CrossRef]

Kaul, B.

Kaul, B. V.

G. G. Matvienko, I. V. Samokhvalov, and B. V. Kaul, “Research of the cirrus structure with a polarization lidar: parameters of particle orientation in crystal clouds,” in Proc. SPIE5571, 393–400 (2004).
[CrossRef]

B. V. Kaul, I. V. Samokhvalov, and S. N. Volkov, “Investigating particle orientation in cirrus clouds by measuring backscattering phase matrices with lidar,” Appl. Opt.43, 6620–6628 (2004).
[CrossRef]

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing, I: Estimation Theory, (Prentice Hall, 1993).

Kokhanenko, G.

Kulie, M. S.

M. D. Shupe, D. D. Turner, V. P. Walden, R. Bennartz, M. P. Cadeddu, B. B. Castellani, C. J. Cox, D. R. Hudak, M. S. Kulie, N. B. Miller, R. R. Neely, and W. D Neff, “High and dry: New observations of tropospheric and cloud properties above the Greenland Ice Sheet,” B. Am. Meteorol. Soc. doi:, In Press.
[CrossRef]

Lin, B.

Y. Hu, P. Yang, B. Lin, G. Gibson, and C. Hostetler, “Discriminating between spherical and non-spherical scatterers,” J. Quant. Spectrosc. Radiat. Transfer79–80, 757–764 (2004).

Liou, K.

Y. Tanko and K. Liou, “Solar radiative transfer in cirrus clouds. Part II: Theory and computation of multiple scattering in an anisotropic medium,” J. Atmos. Sci.46, 20–36 (1989).
[CrossRef]

Lucker, P. L.

W. H. Hunt, D. M. Winker, M. A. Vaughan, K. A. Powell, P. L. Lucker, and C. Weimer, “CALIPSO lidar description and performance assessment,” J. Atmos. Oceanic Technol.26, 1214–1228 (2009).
[CrossRef]

Masuda, K.

K. Masuda and H. Ishimoto, “Influence of particle orientation on retrieving cirrus cloud properties by use of total and polarized reflectances from satellite measurements,” J. Quant. Spectrosc. Radiat. Transfer85, 183–193 (2004).
[CrossRef]

Mathur, S.

Matvienko, G. G.

G. G. Matvienko, I. V. Samokhvalov, and B. V. Kaul, “Research of the cirrus structure with a polarization lidar: parameters of particle orientation in crystal clouds,” in Proc. SPIE5571, 393–400 (2004).
[CrossRef]

McClain, S. C.

Mendoza, A.

Miller, N. B.

M. D. Shupe, D. D. Turner, V. P. Walden, R. Bennartz, M. P. Cadeddu, B. B. Castellani, C. J. Cox, D. R. Hudak, M. S. Kulie, N. B. Miller, R. R. Neely, and W. D Neff, “High and dry: New observations of tropospheric and cloud properties above the Greenland Ice Sheet,” B. Am. Meteorol. Soc. doi:, In Press.
[CrossRef]

Mishchenko, M.

Morandi, M.

Neely, R. R.

M. D. Shupe, D. D. Turner, V. P. Walden, R. Bennartz, M. P. Cadeddu, B. B. Castellani, C. J. Cox, D. R. Hudak, M. S. Kulie, N. B. Miller, R. R. Neely, and W. D Neff, “High and dry: New observations of tropospheric and cloud properties above the Greenland Ice Sheet,” B. Am. Meteorol. Soc. doi:, In Press.
[CrossRef]

R. R. Neely, M. Hayman, R. Stillwell, J. P. Thayer, R. M. Hardesty, M. O’Neill, M. D. Shupe, and C. Alvarez, “Polarization lidar at Summit, Greenland for the detection of cloud phase and particle orientation,” J. Atmos. Oceanic Technol. In Review.

Neff, W. D

M. D. Shupe, D. D. Turner, V. P. Walden, R. Bennartz, M. P. Cadeddu, B. B. Castellani, C. J. Cox, D. R. Hudak, M. S. Kulie, N. B. Miller, R. R. Neely, and W. D Neff, “High and dry: New observations of tropospheric and cloud properties above the Greenland Ice Sheet,” B. Am. Meteorol. Soc. doi:, In Press.
[CrossRef]

Noel, V.

V. Noel and K. Sassen, “Study of planar ice crystal orientation in ice clouds from scanning polarization lidar observations,” J. Appl. Meteor.44, 653–664 (2005).
[CrossRef]

V. Noel and H. Chepfer, “Study of ice crystal orientation in cirrus clouds based on satellite polarized radiance measurements,” J. Atmos. Sci.61, 2073–2081 (2005).
[CrossRef]

O’Connor, E. J.

C. D. Westbrook, A. J. Illingworth, E. J. O’Connor, and R. J. Hogan, “Doppler lidar measurements of oriented planar ice crystals falling from supercooled and glaciated layer clouds,” Q. J. R. Meteor. Soc.136, 260–276 (2010).
[CrossRef]

O’Neill, M.

R. R. Neely, M. Hayman, R. Stillwell, J. P. Thayer, R. M. Hardesty, M. O’Neill, M. D. Shupe, and C. Alvarez, “Polarization lidar at Summit, Greenland for the detection of cloud phase and particle orientation,” J. Atmos. Oceanic Technol. In Review.

Patterson, G. R.

S. A. Young, C. M. R. Platt, R. T. Austin, and G. R. Patterson, “Optical properties and phase of some midlatitude, midlevel clouds in ECLIPS,” J. Appl. Meteorol.39, 135–153 (2000).
[CrossRef]

Piironen, P.

Platt, C. M. R.

S. A. Young, C. M. R. Platt, R. T. Austin, and G. R. Patterson, “Optical properties and phase of some midlatitude, midlevel clouds in ECLIPS,” J. Appl. Meteorol.39, 135–153 (2000).
[CrossRef]

C. M. R. Platt, “Lidar backscatter from horizontal ice crystal plates,” J. Appl. Meteorol.17, 482–488 (1978).
[CrossRef]

Powell, K. A.

W. H. Hunt, D. M. Winker, M. A. Vaughan, K. A. Powell, P. L. Lucker, and C. Weimer, “CALIPSO lidar description and performance assessment,” J. Atmos. Oceanic Technol.26, 1214–1228 (2009).
[CrossRef]

Riviere, O.

Samokhvalov, I. V.

B. V. Kaul, I. V. Samokhvalov, and S. N. Volkov, “Investigating particle orientation in cirrus clouds by measuring backscattering phase matrices with lidar,” Appl. Opt.43, 6620–6628 (2004).
[CrossRef]

G. G. Matvienko, I. V. Samokhvalov, and B. V. Kaul, “Research of the cirrus structure with a polarization lidar: parameters of particle orientation in crystal clouds,” in Proc. SPIE5571, 393–400 (2004).
[CrossRef]

Sassen, K.

V. Noel and K. Sassen, “Study of planar ice crystal orientation in ice clouds from scanning polarization lidar observations,” J. Appl. Meteor.44, 653–664 (2005).
[CrossRef]

Seliga, T. A.

T. A. Seliga and V. N. Bringi, “Potential use of radar differential reflectivity measurements at orthogonal polarizations for measuring precipitation,” J. Appl. Meteor.15, 69–76 (1976).
[CrossRef]

Shupe, M. D.

M. D. Shupe, D. D. Turner, V. P. Walden, R. Bennartz, M. P. Cadeddu, B. B. Castellani, C. J. Cox, D. R. Hudak, M. S. Kulie, N. B. Miller, R. R. Neely, and W. D Neff, “High and dry: New observations of tropospheric and cloud properties above the Greenland Ice Sheet,” B. Am. Meteorol. Soc. doi:, In Press.
[CrossRef]

R. R. Neely, M. Hayman, R. Stillwell, J. P. Thayer, R. M. Hardesty, M. O’Neill, M. D. Shupe, and C. Alvarez, “Polarization lidar at Summit, Greenland for the detection of cloud phase and particle orientation,” J. Atmos. Oceanic Technol. In Review.

Smith, M. H.

Stillwell, R.

R. R. Neely, M. Hayman, R. Stillwell, J. P. Thayer, R. M. Hardesty, M. O’Neill, M. D. Shupe, and C. Alvarez, “Polarization lidar at Summit, Greenland for the detection of cloud phase and particle orientation,” J. Atmos. Oceanic Technol. In Review.

Tanko, Y.

Y. Tanko and K. Liou, “Solar radiative transfer in cirrus clouds. Part II: Theory and computation of multiple scattering in an anisotropic medium,” J. Atmos. Sci.46, 20–36 (1989).
[CrossRef]

Thayer, J.

M. Hayman and J. Thayer, “Lidar polarization measurements of PMCs,” J. Atmos. Sol. Terr. Phys.73, 2110–2117 (2011).
[CrossRef]

Thayer, J. P.

M. Hayman and J. P. Thayer, “General description of polarization in lidar using Stokes vectors and polar decomposition of Mueller matrices,” J. Opt. Soc. Am. A29, 400–409 (2012).
[CrossRef]

R. R. Neely, M. Hayman, R. Stillwell, J. P. Thayer, R. M. Hardesty, M. O’Neill, M. D. Shupe, and C. Alvarez, “Polarization lidar at Summit, Greenland for the detection of cloud phase and particle orientation,” J. Atmos. Oceanic Technol. In Review.

Thomas, L.

L. Thomas, J. C. Cartwright, and D. P. Wareing, “Lidar observations of the horizontal orientation of ice crystals in cirrus clouds,” Tellus B42, 2011–2016 (1990).
[CrossRef]

Travis, L. D.

M. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles, (Academic, 2000).

Turner, D. D.

M. D. Shupe, D. D. Turner, V. P. Walden, R. Bennartz, M. P. Cadeddu, B. B. Castellani, C. J. Cox, D. R. Hudak, M. S. Kulie, N. B. Miller, R. R. Neely, and W. D Neff, “High and dry: New observations of tropospheric and cloud properties above the Greenland Ice Sheet,” B. Am. Meteorol. Soc. doi:, In Press.
[CrossRef]

Twietmeyer, K. M.

Tyo, J. S.

Vallar, E.

van de Hulst, H.

H. van de Hulst, Light Scattering by Small Particles (Wiley, 1981).

Vaughan, M. A.

W. H. Hunt, D. M. Winker, M. A. Vaughan, K. A. Powell, P. L. Lucker, and C. Weimer, “CALIPSO lidar description and performance assessment,” J. Atmos. Oceanic Technol.26, 1214–1228 (2009).
[CrossRef]

Venturi, V.

Vivekanandan, J.

J. Vivekanandan, G. Zhang, and E. Brandes, “Polarimetric radar estimators based on a constrained gamma drop size distribution model,” J. Appl. Meteor.43, 217–230 (2004).
[CrossRef]

Volkov, S. N.

Walden, V. P.

M. D. Shupe, D. D. Turner, V. P. Walden, R. Bennartz, M. P. Cadeddu, B. B. Castellani, C. J. Cox, D. R. Hudak, M. S. Kulie, N. B. Miller, R. R. Neely, and W. D Neff, “High and dry: New observations of tropospheric and cloud properties above the Greenland Ice Sheet,” B. Am. Meteorol. Soc. doi:, In Press.
[CrossRef]

Wang, Z.

Wareing, D. P.

L. Thomas, J. C. Cartwright, and D. P. Wareing, “Lidar observations of the horizontal orientation of ice crystals in cirrus clouds,” Tellus B42, 2011–2016 (1990).
[CrossRef]

Weimer, C.

W. H. Hunt, D. M. Winker, M. A. Vaughan, K. A. Powell, P. L. Lucker, and C. Weimer, “CALIPSO lidar description and performance assessment,” J. Atmos. Oceanic Technol.26, 1214–1228 (2009).
[CrossRef]

Westbrook, C. D.

C. D. Westbrook, A. J. Illingworth, E. J. O’Connor, and R. J. Hogan, “Doppler lidar measurements of oriented planar ice crystals falling from supercooled and glaciated layer clouds,” Q. J. R. Meteor. Soc.136, 260–276 (2010).
[CrossRef]

C. D. Westbrook, “The fall speeds of sub-100μm ice crystals,” Q. J. R. Meteorol. Soc.134, 1243–1251 (2008).
[CrossRef]

Winker, D.

Winker, D. M.

W. H. Hunt, D. M. Winker, M. A. Vaughan, K. A. Powell, P. L. Lucker, and C. Weimer, “CALIPSO lidar description and performance assessment,” J. Atmos. Oceanic Technol.26, 1214–1228 (2009).
[CrossRef]

Yang, P.

C. Zhou, P. Yang, A. E. Dessler, Y. Hu, and B. A. Baum, “Study of horizontally oriented ice crystals with CALIPSO observations and comparision with monte carlo radiative transfer simulations,” J. Appl. Meteor. Climatol.51, 1426–1439 (2012).
[CrossRef]

Y. Hu, P. Yang, B. Lin, G. Gibson, and C. Hostetler, “Discriminating between spherical and non-spherical scatterers,” J. Quant. Spectrosc. Radiat. Transfer79–80, 757–764 (2004).

Young, S. A.

S. A. Young, C. M. R. Platt, R. T. Austin, and G. R. Patterson, “Optical properties and phase of some midlatitude, midlevel clouds in ECLIPS,” J. Appl. Meteorol.39, 135–153 (2000).
[CrossRef]

Zardecki, A.

Zhang, G.

J. Vivekanandan, G. Zhang, and E. Brandes, “Polarimetric radar estimators based on a constrained gamma drop size distribution model,” J. Appl. Meteor.43, 217–230 (2004).
[CrossRef]

Zheng, Y.

Zhou, C.

C. Zhou, P. Yang, A. E. Dessler, Y. Hu, and B. A. Baum, “Study of horizontally oriented ice crystals with CALIPSO observations and comparision with monte carlo radiative transfer simulations,” J. Appl. Meteor. Climatol.51, 1426–1439 (2012).
[CrossRef]

Zhu, X.

Appl. Opt. (7)

J. Appl. Meteor. (3)

T. A. Seliga and V. N. Bringi, “Potential use of radar differential reflectivity measurements at orthogonal polarizations for measuring precipitation,” J. Appl. Meteor.15, 69–76 (1976).
[CrossRef]

J. Vivekanandan, G. Zhang, and E. Brandes, “Polarimetric radar estimators based on a constrained gamma drop size distribution model,” J. Appl. Meteor.43, 217–230 (2004).
[CrossRef]

V. Noel and K. Sassen, “Study of planar ice crystal orientation in ice clouds from scanning polarization lidar observations,” J. Appl. Meteor.44, 653–664 (2005).
[CrossRef]

J. Appl. Meteor. Climatol. (1)

C. Zhou, P. Yang, A. E. Dessler, Y. Hu, and B. A. Baum, “Study of horizontally oriented ice crystals with CALIPSO observations and comparision with monte carlo radiative transfer simulations,” J. Appl. Meteor. Climatol.51, 1426–1439 (2012).
[CrossRef]

J. Appl. Meteorol. (2)

C. M. R. Platt, “Lidar backscatter from horizontal ice crystal plates,” J. Appl. Meteorol.17, 482–488 (1978).
[CrossRef]

S. A. Young, C. M. R. Platt, R. T. Austin, and G. R. Patterson, “Optical properties and phase of some midlatitude, midlevel clouds in ECLIPS,” J. Appl. Meteorol.39, 135–153 (2000).
[CrossRef]

J. Atmos. Oceanic Technol. (1)

W. H. Hunt, D. M. Winker, M. A. Vaughan, K. A. Powell, P. L. Lucker, and C. Weimer, “CALIPSO lidar description and performance assessment,” J. Atmos. Oceanic Technol.26, 1214–1228 (2009).
[CrossRef]

J. Atmos. Sci. (5)

F-M Bréon and B. Dubrulle, “Horizontally oriented plates in clouds,” J. Atmos. Sci.61, 2888–2898 (2004).
[CrossRef]

A. J. Heymsfield and M. Kajikawa, “An improived approach to calculating terminal velocities of plate-like crystals and graupel,” J. Atmos. Sci.44, 1088–1099 (1987).
[CrossRef]

V. Noel and H. Chepfer, “Study of ice crystal orientation in cirrus clouds based on satellite polarized radiance measurements,” J. Atmos. Sci.61, 2073–2081 (2005).
[CrossRef]

Y. Tanko and K. Liou, “Solar radiative transfer in cirrus clouds. Part II: Theory and computation of multiple scattering in an anisotropic medium,” J. Atmos. Sci.46, 20–36 (1989).
[CrossRef]

A. Heymsfield and J. Iaquinta, “Cirrus crystal terminal velocity,” J. Atmos. Sci.57, 916–938 (2000).
[CrossRef]

J. Atmos. Sol. Terr. Phys. (1)

M. Hayman and J. Thayer, “Lidar polarization measurements of PMCs,” J. Atmos. Sol. Terr. Phys.73, 2110–2117 (2011).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Quant. Spectrosc. Radiat. Transfer (2)

K. Masuda and H. Ishimoto, “Influence of particle orientation on retrieving cirrus cloud properties by use of total and polarized reflectances from satellite measurements,” J. Quant. Spectrosc. Radiat. Transfer85, 183–193 (2004).
[CrossRef]

Y. Hu, P. Yang, B. Lin, G. Gibson, and C. Hostetler, “Discriminating between spherical and non-spherical scatterers,” J. Quant. Spectrosc. Radiat. Transfer79–80, 757–764 (2004).

Opt. Express (3)

Opt. Lett. (2)

Proc. SPIE (1)

G. G. Matvienko, I. V. Samokhvalov, and B. V. Kaul, “Research of the cirrus structure with a polarization lidar: parameters of particle orientation in crystal clouds,” in Proc. SPIE5571, 393–400 (2004).
[CrossRef]

Q. J. R. Meteor. Soc. (1)

C. D. Westbrook, A. J. Illingworth, E. J. O’Connor, and R. J. Hogan, “Doppler lidar measurements of oriented planar ice crystals falling from supercooled and glaciated layer clouds,” Q. J. R. Meteor. Soc.136, 260–276 (2010).
[CrossRef]

Q. J. R. Meteorol. Soc. (1)

C. D. Westbrook, “The fall speeds of sub-100μm ice crystals,” Q. J. R. Meteorol. Soc.134, 1243–1251 (2008).
[CrossRef]

Tellus B (1)

L. Thomas, J. C. Cartwright, and D. P. Wareing, “Lidar observations of the horizontal orientation of ice crystals in cirrus clouds,” Tellus B42, 2011–2016 (1990).
[CrossRef]

Other (7)

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

H. van de Hulst, Light Scattering by Small Particles (Wiley, 1981).

E. Eloranta, Chapter 5: High Spectral Resolution Lidar in Lidar: Range-Resolved Optical Remote Sensing of the AtmosphereNew York, U.S.A. (Springer, 2005).

M. D. Shupe, D. D. Turner, V. P. Walden, R. Bennartz, M. P. Cadeddu, B. B. Castellani, C. J. Cox, D. R. Hudak, M. S. Kulie, N. B. Miller, R. R. Neely, and W. D Neff, “High and dry: New observations of tropospheric and cloud properties above the Greenland Ice Sheet,” B. Am. Meteorol. Soc. doi:, In Press.
[CrossRef]

R. R. Neely, M. Hayman, R. Stillwell, J. P. Thayer, R. M. Hardesty, M. O’Neill, M. D. Shupe, and C. Alvarez, “Polarization lidar at Summit, Greenland for the detection of cloud phase and particle orientation,” J. Atmos. Oceanic Technol. In Review.

M. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles, (Academic, 2000).

S. M. Kay, Fundamentals of Statistical Signal Processing, I: Estimation Theory, (Prentice Hall, 1993).

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Figures (7)

Fig. 1
Fig. 1

Layout of HSRL transceiver. The laser is initially vertically polarized and in its original polarization operation, the use of a QWP allows the instrument to measure circular depolarization ratios. By rotating the QWP, the transmitted and detected polarizations actively change, and the rank of the measurement matrix A is increased. Additional optics to the left of the QWP such as folding mirrors and polarizers have been omitted. The high backscatter receiver channel has also been omitted.

Fig. 2
Fig. 2

Poincaré Sphere representation of output polarizations for a horizontally polarized input resulting from an ideal QWP (blue) and the titled QWP (red) used in the HSRL and described in Eq. (23). Tilting the QWP tilts the output polarization trajectory and the first order term results in a path that is not quite periodic for 180° rotation.

Fig. 3
Fig. 3

Measured backscatter, F11(R) on parallel (blue), uncorrected cross polarized (red solid line) and corrected cross polarized (red dashed line) channels. Multiplying the received photon counts on the cross polarized channel by 1/K(R) results in retrieval of nearly identical backscatter measurements on the two channels.

Fig. 4
Fig. 4

Backscatter ratio (top), depolarization (middle) and diattenuation (bottom) of a rain storm containing oriented rain drops observed on July 16, 2012. The areas of large diattenuation suggest the presence of large raindrops.

Fig. 5
Fig. 5

Observed normalized backscatter phase (Mueller) matrix of heavy rainfall with a mixed phase cloud above it. The profile is time integrated over the highly diattenuating rain in Fig. 4. Diagonal matrix terms are dashed and off diagonals are solid. The cloud above 2.5 km is strictly depolarizing (randomly oriented) with off diagonal terms all zero. The rain below 2.5 km consists of oriented oblate spheroids which have a common preferred orientation.

Fig. 6
Fig. 6

Backscatter ratio (top), depolarization (middle) and diattenuation (bottom) of an ice cloud containing oriented ice crystals observed on July 2, 2012. The presence of the oriented ice crystals is not clear from the backscatter and depolarization profiles alone, but non-zero diattenuation indicates preferential orientation occurring mostly in an altitude band near 5 km..

Fig. 7
Fig. 7

Observed normalized backscatter phase (Mueller) matrix of ice cloud with a thin layer of oriented ice crystals. Diagonal matrix terms are dashed and off diagonal are solid. Most of the profile is strictly depolarizing, but there is a narrow (approximately 0.5 km) altitude range starting at 5 km that contains non-zero off diagonal scattering terms, indicating the presence of oriented ice crystals.

Tables (1)

Tables Icon

Table 1 NCAR HSRL Specifications

Equations (28)

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

N ( R ) = o T M R X [ ( G ( R ) A R 2 Δ R ) T atm ( k s , R ) F ( k i , k s , R ) T atm ( k i , R ) M T X S T X + S B ] ,
o T = [ η 0 0 0 ] ,
F ( π ) = β [ 1 0 0 f 14 0 1 d 0 0 0 0 d 1 0 f 14 0 0 2 d 1 ] ,
F ( 0 ) ( k i , k i ) = [ F 11 ( 0 ) F 12 ( 0 ) 0 F 14 ( 0 ) F 12 ( 0 ) F 22 ( 0 ) 0 0 0 0 F 33 ( 0 ) F 34 ( 0 ) F 14 ( 0 ) 0 F 34 ( 0 ) F 44 ( 0 ) ] ,
F 11 F 22 + F 33 F 44 = 0 .
F ( k i , k i ) = R ( φ ) F ( 0 ) ( k i , k i ) R ( φ ) = [ F 11 F 12 F 13 F 14 F 12 F 22 F 23 F 24 F 13 F 23 F 33 F 34 F 14 F 24 F 34 F 44 ] ,
N = D T F ( k i , k i ) S
S = T atm ( k i , R ) M T X S T X ,
D T = o T M R X T atm ( k s , R ) .
f i + 4 ( j 1 ) = F i j
N = a T f
a i + 4 ( j 1 ) = D i S j
a = [ D 1 S 1 D 1 S 2 + D 2 S 1 D 1 S 3 D 3 S 1 D 1 S 4 + D 4 S 1 D 2 S 2 D 2 S 3 D 3 S 2 D 2 S 4 + D 4 S 2 D 3 S 3 D 3 S 4 D 4 S 3 D 4 S 4 ]
f = [ F 11 F 12 F 13 F 14 F 22 F 23 F 24 F 33 F 34 F 44 ] T .
A = [ a 1 T a 2 T ] ,
N = A f .
f = A 1 N .
Σ f 2 = A 1 Σ N 2 ( A 1 ) T ,
Σ N 2 = Σ shot 2 + Σ Atm 2 ,
Σ Atm 2 = A Σ F 2 A T .
Σ f 2 = A 1 Σ shot 2 ( A 1 ) T + Σ F 2 .
a ( θ ) = θ θ + Δ θ a i ( ϑ ) d ϑ ,
Γ ( θ ) = Γ 0 + Γ 1 cos ( θ θ 1 ) + Γ 2 cos ( 2 θ ) ,
f ( R ) = [ A | | A ] 1 [ N | | ( R ) 1 K ( R ) N ( R ) ] ,
K ( R ) = η G ( R ) η | | G | | ( R ) ,
f | | ( R ) = A | | 1 N | | ( R ) ,
f ( R ) = A 1 N ( R ) .
K ( R ) = F 11 ( R ) F 11 | | ( R ) ,

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