## Transformation of light backscattering phase matrices of crystal clouds depending on the zenith sensing angle |

Optics Express, Vol. 21, Issue 11, pp. 13408-13418 (2013)

http://dx.doi.org/10.1364/OE.21.013408

Acrobat PDF (1169 KB)

### Abstract

Problems encountered in the interpretation of results of laser sensing of crystal clouds are considered. The parameters characterizing the cloud particle orientation are determined through the backscattering phase matrix elements. It is demonstrated how these parameters are related to the probability density of particle distribution over the spatial orientation angles. Trends in the change of the backscattering phase matrices attendant to variations of the zenith sensing angle are shown on the example of a monodisperse ice particle ensemble.

© 2013 OSA

## 1. Introduction

1. 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. Transf. **85**(2), 183–193 (2004). [CrossRef]

2. C. M. R. Platt, “Lidar backscatter from horizontal ice crystal plates,” J. Appl. Meteorol. **17**(4), 482–488 (1978). [CrossRef]

6. Y. Balin, B. Kaul, G. Kokhanenko, and D. Winker, “Application of circularly polarized laser radiation for sensing of crystal clouds,” Opt. Express **17**(8), 6849–6859 (2009). [CrossRef] [PubMed]

7. 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. Ocean. Technol. **26**(7), 1214–1228 (2009). [CrossRef]

*λ/d*, where

*d*is the particle diameter (of the order of 50 μm). In this case, deviation angle 0.5–1° will be sufficient. However, in addition to the orientation effect of the aerodynamic momentum, particles are subject to the destructive effect of turbulence and flutter [8

8. J. D. Klett, “Orientation model for particles in turbulence,” J. Atmos. Sci. **52**(12), 2276–2285 (1995). [CrossRef]

4. V. Noel and K. Sassen, “Study of planar ice crystal orientation in ice clouds from scanning polarization lidar observations,” J. Appl. Meteorol. **44**(5), 653–664 (2005). [CrossRef]

*et al*. [10

10. M. Del Guasta, 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**(20), 4878–4887 (2006). [CrossRef] [PubMed]

11. A. Borovoi and N. Kustova, “Specular scattering by preferentially oriented ice crystals,” Appl. Opt. **48**(19), 3878–3885 (2009). [CrossRef] [PubMed]

12. A. Borovoi, A. Konoshonkin, N. Kustova, and H. Okamoto, “Backscattering Mueller matrix for quasi-horizontally oriented ice plates of cirrus clouds: application to CALIPSO signals,” Opt. Express **20**(27), 28222–28233 (2012). [CrossRef] [PubMed]

## 2. Main special features of the BSPM

**M**relating the Stokes vectors of radiation scattered in the direction toward the source

**S**with the Stokes vector

**S**of radiation incident on an ensemble of particles contained in an elementary volume Δ

_{0}*V*:

13. 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**(36), 6620–6628 (2004). [CrossRef] [PubMed]

14. M. Hayman, S. Spuler, B. Morley, and J. VanAndel, “Polarization lidar operation for measuring backscatter phase matrices of oriented scatterers,” Opt. Express **20**(28), 29553–29567 (2012). [CrossRef] [PubMed]

13. 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**(36), 6620–6628 (2004). [CrossRef] [PubMed]

**M**is reduced to a simpler form by the following transformation:where

_{exp}**е**

_{z}direction. In practice, the procedure of finding the angle φ

_{0}involves variation of the rotation angle φ to minimize the sum of the squared elements which in Eq. (2) are represented by zeros for the ideal case that would take place without experiment errors. This condition is satisfied when the angle φ

_{0}has been found at which the reference plane (

**е**

_{x}

**е**

_{z}) coincides with the symmetry plane of cloud particles. Or in other words, it coincides with the preferred orientation direction in the plane perpendicular to the wave vector. The possibility of reduction of the experimental BSPM to the form given by Eq. (2) is caused by the symmetry properties that are a consequence of the fact that any plane comprising the

*z*axis can be chosen for the reference backscattering plane. For any arbitrary BSPM, conditions are always satisfied. The last equality is a consequence of the reciprocity theorem [15]. A detailed analysis of the symmetry properties can be found in [16, 17

17. J. W. Hovenier and C. V. M. Van Der Mee, “Testing scattering matrices a compendium of recipes,” J. Quant. Spectrosc. Radiat. Transf. **55**(5), 649–661 (1996). [CrossRef]

18. M. I. Mishchenko and J. W. Hovenier, “Depolarization of light backscattered by randomly oriented nonspherical particles,” Opt. Lett. **20**(12), 1356–1358 (1995). [CrossRef] [PubMed]

19. C. J. Flynn, A. Mendoza, Y. Zheng, and S. Mathur, “Novel polarization-sensitive micropulse lidar measurement technique,” Opt. Express **15**(6), 2785–2790 (2007). [CrossRef] [PubMed]

*z*axis), the BSPM can be written in the following form:where

*d*is the parameter which determines the depolarized part of scattered radiation when the cloud is illuminated by linearly polarized light [20

20. G. G. Gimmestad, “Reexamination of depolarization in lidar measurements,” Appl. Opt. **47**(21), 3795–3802 (2008). [CrossRef] [PubMed]

**e**

_{x},

**e**

_{z}) coincides with this direction. In this case, unlike Eq. (5),

13. 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**(36), 6620–6628 (2004). [CrossRef] [PubMed]

**43**(36), 6620–6628 (2004). [CrossRef] [PubMed]

*M*

_{11},

*M*

_{14},

*M*

_{41}, and

*M*

_{44}do not change when the angle

*φ*is changed.

## 3. Modeling of the BSPM

*φ*so that its symmetry axis lie in the

*xz*reference plane where

*φ*= 0 and calculate its BSPM

**M**(0,

*θ*,

*γ*). Then this matrix should be averaged with equal probabilities over every possible angle γ taking into account the sixth-order symmetry:As a result of averaging, the BSPM

**M**(0,

*θ*) assumes the form of Eq. (2) with zero elements

*φ*, the rotational transformation

*n*identical particles in unit volume and probability density of angular distribution

*θ*and then stored in a tabular form. In this case, for each

*θ*value, averaging should be performed as in Eq. (7). Below we take advantage of one result of such calculations. If analogous calculations are available for particles of other sizes, BSPM models can be constructed for polydisperse clouds of hexagonal particles by summation of matrices taking into account the contribution of each particle type to the total concentration. Undoubtedly, this method is vary laborious and requires the creation of a vast data bank of crystal particle matrices.

## 4. Orientation of particles

23. H.-R. Cho, J. V. Iribarne, and W. G. Richards, “On the orientation of ice crystals in a cumulo-nimbus cloud,” J. Atmos. Sci. **38**(5), 1111–1114 (1981). [CrossRef]

8. J. D. Klett, “Orientation model for particles in turbulence,” J. Atmos. Sci. **52**(12), 2276–2285 (1995). [CrossRef]

8. J. D. Klett, “Orientation model for particles in turbulence,” J. Atmos. Sci. **52**(12), 2276–2285 (1995). [CrossRef]

*θ*suggested in [9]:where

*θ*is the deflection angle of the normal to the hexagonal plate base from the zenith direction or from the horizontal direction in the case of columns, I

_{0}(

*k*) is the zeroth order modified Bessel function of the first kind. The distribution parameter here iswhere Λ

*is the form factor for plates (*

_{p,c}*p*) or columns (

*c*),

*d*is the maximum particle diameter, ν is the kinematic viscosity of air,

*ε*is the energy dissipation rate, and

*A*and

_{p,c}*b*are empirical constants for the fall velocity

_{p,c}*u*specified by the formula

*k*.

## 5. Model BSPM of monodisperse cloud of ice plates versus the orientation parameter *k* and zenith sensing angle

**M**(0,

*θ*) for hexagonal plates with circumscribed circle diameter of 400 μm and thickness of 30.64 μm are presented. Figure 2 illustrates some of this data. In this case, small and frequent interference peaks of the original figure have been smoothed. We believe that this is justified, because the position of these peaks depends on the change of the crystal sizes by the value of the order of the wavelength.

**M**(0,

*θ*), elements of the BSPM

**M**(

*n*) can be calculated from Eq. (7) for different values of the parameter

*k*in the distribution given by Eq. (8), and the matrix obtained corresponds to sensing in the zenith (or nadir) direction. In this case, there is no preferred direction for the azimuth angle

*φ*. The distribution over the orientation angles is written in the following form:where

*М*

_{11}) and the behavior of the diagonal BSPM elements normalized by

*М*

_{11}with increase in the orientation parameter

*k*are displayed. It can be seen that they are close to asymptotic values already at

*k*= 10.

*β*. In this case, the rotational symmetry is broken. The preferred plane

*z*0

*β*arises that comprises the zenith and sensing directions. For clarity, we now assume that all plates are oriented horizontally. Then projections of horizontally oriented plates onto the plane perpendicular to the sensing direction differ for directions lying in the

*z*0

*β*plane and transverse to it. For example, the projection of a round plate is an ellipse. All major axes of ellipses are oriented in the same azimuth direction transverse to the

*z*0

*β*plane. As a result, the rotational symmetry about the direction

*z*inherent in the particle ensemble is not retained for rotation about the direction

*β*.

*θ*substituted by

*θ′*and

*φ*substituted by

*φ′*, where

*θ′*is counted from the direction

*β*, and

*φ′*is counted from the points of intersection of planes

*θ′*= const with the

*z*0

*β*plane:For the strict zenith orientation of the plate normals, calculations by Eq. (12) would be equivalent to calculations by Eq. (8), but for the distribution over the angles φ, θ of the following form:This implies that the normals of all plates are deflected from the zenith direction by the angle

*β*in the azimuth direction

*φ*= π. Generally, the distribution function written in the coordinates

*φ′*,

*θ′*is transformed into a function of coordinates

*φ*,

*θ*. For this purpose, we now write down the following transformation:

*z*axis about the

*y*axis through the angle

*β*and the vector-column of coordinates

*x*,

*y*,

*z*of an arbitrary unit vector. On the right side, there is the vector-column of coordinates

*x′*,

*y′*,

*z′*of the same vector in the rotated system of coordinates. We further find the direction of this vector in the spherical system of coordinates with the

*z′*axis coinciding with the direction

*β*: The dependences

*β*. These functions show the weight of the normals of plates with orientation

*k*and several zenith sensing angles

*β*are presented in Table 3. For the limiting case of almost complete orientation (

*k*= 500), the dependences of the same parameters on the zenith sensing angle are shown in Fig. 4.

## 6. Discussion

*k*by more than two orders of magnitude (for

*k*= 500) in comparison with the intensity of reflection from the ensemble of randomly oriented particles shown in Fig. 3. The normalized diagonal BSPM elements change even faster. From Fig. 3 it can be seen that with increasing

*k*, they monotonically approach their asymptotic values

*k*= 10. From Table 2 it follows that

*k*= 10 implies not too strict orientation: the standard deviation is about 9°. It is obvious that the polarization is influenced to a greater degree by particles that at the present moment are close to the horizontal position and bring the main contribution to the intensity.

*k*< 3), the zenith sensing angles up to 6° have practically no effect on the polarization characteristics of scattered radiation. In particular, for sensing by polarized radiation, the BSPM is described by the matrix Eq. (5) with

*m*

_{14}=

*m*

_{41}= 0. The situation changes when it is impossible to neglect the nonzero value of the element

*m*

_{12}. Then Eq. (5) loses its meaning, since

*d*and the depolarization ratio

20. G. G. Gimmestad, “Reexamination of depolarization in lidar measurements,” Appl. Opt. **47**(21), 3795–3802 (2008). [CrossRef] [PubMed]

*d*is taken to mean the depolarization for sensing by linearly polarized radiation. Indeed, for sensing by radiation with unit power and the Stokes vector

_{l}*q*and depolarization

*d*, we can writeIt can be easily seen that Eq. (17) follows from Eq. (18), (19).

_{l}*m*

_{44}. In [29,30

30. Y. S. Balin, B. V. Kaul, G. P. Kokhanenko, and I. E. Penner, “Observations of specular reflective particles and layers in crystal clouds,” Opt. Express **19**(7), 6209–6214 (2011). [CrossRef] [PubMed]

*m*

_{44}becomes positive in the case of chaotic particle distribution for large values of the depolarization parameter

*d*> 0.5, and that the direction of rotation of the polarization plane changes. In our calculations, this is also manifested for random particle orientation (see the first line of Table 3). This effect can also be manifested for large zenith sensing angles and strong orientation of plates in the horizontal direction. For example, this can be seen in the last line of Table 3. In Fig. 4 it is also seen that for

_{44}is positive.

*k*= 500), scanning of the sensing direction is in fact equivalent to measurement of the scattering matrix

**M**(θ) for a fixed particle position. The situation illustrated by Fig. 4 is close to that expected for distribution Eq. (13). The behavior of the elements

*m*(

_{ij}*β*) in this case reproduces almost completely the behavior of the matrix

**M**(0,

*θ*)/

*M*

_{11}(see Eq. (7) and Fig. 3).

## Summary

**M**(0,

*θ*) calculated with sufficiently small step in the polar angle

*θ*for zero azimuth angle. If there is a databank of such matrices for particles of various shapes and sizes, the BSPM transformation with changes of the zenith sensing angle can be calculated for composite polydisperse ensembles of particles. The distribution over the polar orientation angles used in this article is not commonly accepted, but it is not a matter of principle. The method presented above is applicable to any arbitrary distribution. The assumption on the random azimuth orientation is not obligatory as well. In the article, it is shown that the proximity of the diagonal matrix elements to their asymptotic values

## Acknowledgments

## References and links

1. | 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. Transf. |

2. | C. M. R. Platt, “Lidar backscatter from horizontal ice crystal plates,” J. Appl. Meteorol. |

3. | L. Thomas, J. C. Cartwright, and D. P. Wareing, “Lidar observations of the horizontal orientation of ice crystals in cirrus clouds,” Tellus B Chem. Phys. Meterol. |

4. | V. Noel and K. Sassen, “Study of planar ice crystal orientation in ice clouds from scanning polarization lidar observations,” J. Appl. Meteorol. |

5. | V. Noel and H. Chepfer, “Study of ice crystal orientation in cirrus clouds based on satellite polarized radiance measurements,” J. Atmos. Sci. |

6. | Y. Balin, B. Kaul, G. Kokhanenko, and D. Winker, “Application of circularly polarized laser radiation for sensing of crystal clouds,” Opt. Express |

7. | 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. Ocean. Technol. |

8. | J. D. Klett, “Orientation model for particles in turbulence,” J. Atmos. Sci. |

9. | B. V. Kaul and I. V. Samokhvalov, “Orientation of particles in |

10. | M. Del Guasta, 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. |

11. | A. Borovoi and N. Kustova, “Specular scattering by preferentially oriented ice crystals,” Appl. Opt. |

12. | A. Borovoi, A. Konoshonkin, N. Kustova, and H. Okamoto, “Backscattering Mueller matrix for quasi-horizontally oriented ice plates of cirrus clouds: application to CALIPSO signals,” Opt. Express |

13. | 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. |

14. | M. Hayman, S. Spuler, B. Morley, and J. VanAndel, “Polarization lidar operation for measuring backscatter phase matrices of oriented scatterers,” Opt. Express |

15. | H. C. Van de Hulst, |

16. | C. R. Hu, G. W. Kattawar, M. E. Parkin, and P. Herb, “Symmetry theorems on the forward and backward scattering Mueller matrices for light scattering from a non-spherical dielectric scatter,” Appl. Opt. |

17. | J. W. Hovenier and C. V. M. Van Der Mee, “Testing scattering matrices a compendium of recipes,” J. Quant. Spectrosc. Radiat. Transf. |

18. | M. I. Mishchenko and J. W. Hovenier, “Depolarization of light backscattered by randomly oriented nonspherical particles,” Opt. Lett. |

19. | C. J. Flynn, A. Mendoza, Y. Zheng, and S. Mathur, “Novel polarization-sensitive micropulse lidar measurement technique,” Opt. Express |

20. | G. G. Gimmestad, “Reexamination of depolarization in lidar measurements,” Appl. Opt. |

21. | B. V. Kaul and I. V. Samokhvalov, “Orientation of particles in Ci crystal clouds. Part 2. Azimuth orientation,” Atmos. Oceanic Opt. |

22. | B. V. Kaul, “Effect of electric field on orientation of ice cloud particles,” Atmos. Oceanic Opt. |

23. | H.-R. Cho, J. V. Iribarne, and W. G. Richards, “On the orientation of ice crystals in a cumulo-nimbus cloud,” J. Atmos. Sci. |

24. | V. V. Kuznetsov, N. K. Nikiforova, and L. N. Pavlova, “On measuring the microstructure of crystal fogs by an Aspekt-10 television aerosol spectrometer,” Trudy Inst. Eksper. Meteorol. |

25. | M. Kajikawa, “Laboratory measurement of falling velocity of individual ice crystals,” J. Meteor. Soc. Japan |

26. | K. Sassen, “Remote sensing of planar ice crystals fall attitudes,” J. Meteorol. Soc. Jpn. |

27. | O. A. Volkovitskii, L. N. Pavlova, and A. G. Petrushin, |

28. | D. N. Romashov, “Backscattering phase matrix of monodisperse ensembles of hexagonal water ice crystals,” Atmos. Oceanic Opt. |

29. | Yu. S. Balin, B. V. Kaul, and G. P. Kokhanenko, “Observation of specularly reflective particles and layers in crystal clouds,” Atmos. Oceanic Opt. |

30. | Y. S. Balin, B. V. Kaul, G. P. Kokhanenko, and I. E. Penner, “Observations of specular reflective particles and layers in crystal clouds,” Opt. Express |

**OCIS Codes**

(280.3640) Remote sensing and sensors : Lidar

(290.1090) Scattering : Aerosol and cloud effects

(290.1350) Scattering : Backscattering

(290.5855) Scattering : Scattering, polarization

**ToC Category:**

Remote Sensing

**History**

Original Manuscript: April 18, 2013

Revised Manuscript: May 16, 2013

Manuscript Accepted: May 16, 2013

Published: May 28, 2013

**Virtual Issues**

Vol. 8, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Yury Balin, Bruno Kaul, Grigorii Kokhanenko, and David Winker, "Transformation of light backscattering phase matrices of crystal clouds depending on the zenith sensing angle," Opt. Express **21**, 13408-13418 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-11-13408

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### References

- 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. Transf.85(2), 183–193 (2004). [CrossRef]
- C. M. R. Platt, “Lidar backscatter from horizontal ice crystal plates,” J. Appl. Meteorol.17(4), 482–488 (1978). [CrossRef]
- L. Thomas, J. C. Cartwright, and D. P. Wareing, “Lidar observations of the horizontal orientation of ice crystals in cirrus clouds,” Tellus B Chem. Phys. Meterol.42, 2011–2016 (1990).
- V. Noel and K. Sassen, “Study of planar ice crystal orientation in ice clouds from scanning polarization lidar observations,” J. Appl. Meteorol.44(5), 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(16), 2073–2081 (2004). [CrossRef]
- Y. Balin, B. Kaul, G. Kokhanenko, and D. Winker, “Application of circularly polarized laser radiation for sensing of crystal clouds,” Opt. Express17(8), 6849–6859 (2009). [CrossRef] [PubMed]
- 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. Ocean. Technol.26(7), 1214–1228 (2009). [CrossRef]
- J. D. Klett, “Orientation model for particles in turbulence,” J. Atmos. Sci.52(12), 2276–2285 (1995). [CrossRef]
- B. V. Kaul and I. V. Samokhvalov, “Orientation of particles in Ci crystal clouds. Part 1. Orientation at gravitational sedimentation,” J. Atmos. Oceanic Opt.16, 866–870 (2005).
- M. Del Guasta, 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(20), 4878–4887 (2006). [CrossRef] [PubMed]
- A. Borovoi and N. Kustova, “Specular scattering by preferentially oriented ice crystals,” Appl. Opt.48(19), 3878–3885 (2009). [CrossRef] [PubMed]
- A. Borovoi, A. Konoshonkin, N. Kustova, and H. Okamoto, “Backscattering Mueller matrix for quasi-horizontally oriented ice plates of cirrus clouds: application to CALIPSO signals,” Opt. Express20(27), 28222–28233 (2012). [CrossRef] [PubMed]
- 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(36), 6620–6628 (2004). [CrossRef] [PubMed]
- M. Hayman, S. Spuler, B. Morley, and J. VanAndel, “Polarization lidar operation for measuring backscatter phase matrices of oriented scatterers,” Opt. Express20(28), 29553–29567 (2012). [CrossRef] [PubMed]
- H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957, Dover, New York, 1981).
- C. R. Hu, G. W. Kattawar, M. E. Parkin, and P. Herb, “Symmetry theorems on the forward and backward scattering Mueller matrices for light scattering from a non-spherical dielectric scatter,” Appl. Opt.26, 4159–4173 (1987).
- J. W. Hovenier and C. V. M. Van Der Mee, “Testing scattering matrices a compendium of recipes,” J. Quant. Spectrosc. Radiat. Transf.55(5), 649–661 (1996). [CrossRef]
- M. I. Mishchenko and J. W. Hovenier, “Depolarization of light backscattered by randomly oriented nonspherical particles,” Opt. Lett.20(12), 1356–1358 (1995). [CrossRef] [PubMed]
- C. J. Flynn, A. Mendoza, Y. Zheng, and S. Mathur, “Novel polarization-sensitive micropulse lidar measurement technique,” Opt. Express15(6), 2785–2790 (2007). [CrossRef] [PubMed]
- G. G. Gimmestad, “Reexamination of depolarization in lidar measurements,” Appl. Opt.47(21), 3795–3802 (2008). [CrossRef] [PubMed]
- B. V. Kaul and I. V. Samokhvalov, “Orientation of particles in Ci crystal clouds. Part 2. Azimuth orientation,” Atmos. Oceanic Opt.19, 38–42 (2006).
- B. V. Kaul, “Effect of electric field on orientation of ice cloud particles,” Atmos. Oceanic Opt.19, 751–754 (2006).
- H.-R. Cho, J. V. Iribarne, and W. G. Richards, “On the orientation of ice crystals in a cumulo-nimbus cloud,” J. Atmos. Sci.38(5), 1111–1114 (1981). [CrossRef]
- V. V. Kuznetsov, N. K. Nikiforova, and L. N. Pavlova, “On measuring the microstructure of crystal fogs by an Aspekt-10 television aerosol spectrometer,” Trudy Inst. Eksper. Meteorol.7(112), 101–106 (1983).
- M. Kajikawa, “Laboratory measurement of falling velocity of individual ice crystals,” J. Meteor. Soc. Japan51, 263–272 (1972).
- K. Sassen, “Remote sensing of planar ice crystals fall attitudes,” J. Meteorol. Soc. Jpn.58, 422–429 (1980).
- O. A. Volkovitskii, L. N. Pavlova, and A. G. Petrushin, Optical Properties of Crystal Clouds (Gidrometeoizdat, Leningrad, 1984).
- D. N. Romashov, “Backscattering phase matrix of monodisperse ensembles of hexagonal water ice crystals,” Atmos. Oceanic Opt.12, 376–384 (1999).
- Yu. S. Balin, B. V. Kaul, and G. P. Kokhanenko, “Observation of specularly reflective particles and layers in crystal clouds,” Atmos. Oceanic Opt.24, 293–299 (2011).
- Y. S. Balin, B. V. Kaul, G. P. Kokhanenko, and I. E. Penner, “Observations of specular reflective particles and layers in crystal clouds,” Opt. Express19(7), 6209–6214 (2011). [CrossRef] [PubMed]

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