## Application of circularly polarized laser radiation for sensing of crystal clouds

Optics Express, Vol. 17, Issue 8, pp. 6849-6859 (2009)

http://dx.doi.org/10.1364/OE.17.006849

Acrobat PDF (375 KB)

### Abstract

The application of circularly polarized laser radiation and measurement of the fourth Stokes parameter of scattered radiation considerably reduce the probability of obtaining ambiguous results for radiation depolarization in laser sensing of crystal clouds. The uncertainty arises when cloud particles appear partially oriented by their large diameters along a certain azimuth direction. Approximately in 30% of all cases, the measured depolarization depends noticeably on the orientation of the lidar reference plane with respect to the particle orientation direction. In this case, the corridor of the most probable depolarization values is about 0.1–0.15, but in individual cases, it can be noticeably wider. The present article considers theoretical aspects of this phenomenon and configuration of a lidar capable of measuring the fourth Stokes parameter together with an algorithm of lidar signal processing in the presence of optically thin cloudiness when molecular scattering cannot be neglected. It is demonstrated that the element *a*_{44} of the normalized backscattering phase matrix (BSPM) can be measured. Results of measurements are independent of the presence or absence of azimuthal particle orientation. For sensing in the zenith or nadir, this element characterizes the degree of horizontal orientation of long particle diameters under the action of aerodynamic forces arising during free fall of particles.

© 2009 Optical Society of America

## 1. Introduction

3. C. M. R. Platt. Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals. J. Appl. Meteorol. **17**, 1220–1224 (1978). [CrossRef]

4. 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**, 1111–1114 (1981). [CrossRef]

7. 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]

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

7. 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]

*a*

_{44}of the normalized (BSPM) can be measured. Moreover, results of measurements are independent of the presence or absence of azimuthal particle orientation. For sensing in the zenith or nadir, this element characterizes the degree of horizontal orientation of long particle diameters under the action of aerodynamic forces arising during free fall of particles. This offers the possibilities for detecting layers of ice particles with high reflectivity, for example, the layers responsible for the optical phenomenon of the lower Sun observed from a board of airborne vehicles above crystal clouds due to light reflection from horizontally oriented crystal faces. Exactly the possibility of detecting such situations is the main advantage of application of circular polarization. In this case, the distribution of scattered radiation differs significantly from the distribution observed for random orientation of particles.

9. D. M. Winker, W. H. Hunt, and M. J. McGill, “Initial performance assessment of CALIOP," Geophys. Res. Lett. , **34**, L19803, doi:10.1029/2007GL030135 (2007). [CrossRef]

## 2. Backscattering phase matrix and orientation parameters.

**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*:

**M**should be considered as a matrix of the two-component medium:

**A**denotes the BSPM of the aerosol component, and

**∑**denotes the BSM of the molecular component.

*V*contains sufficiently large number of independently scattering ice particles, so that the microphysical volume parameters characterize the cloud as a whole. The BSPM of the ensemble of independently scattering particles is equal to the sum of BSPMs of individual particles. By virtue of the symmetry of the diagonal BSPM elements, the relation [10],

7. 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]

**A′**is obtained from the experimental matrix

**A**using the transformation

**R**(Φ) is the operator of rotation of the coordinate system around the wave vector of incident (and scattered) radiation through the angle Φ:

*A*

_{44}=

*A*′

_{44}.

**A′**, obtained using transformation in Eq. (6), the

*reduced*matrices. It should immediately be noted that the reduction operation makes sense for the BSPMs with nonzero nondiagonal elements. This is the case when a preferred azimuthal direction is present in the cloud. For example, long axes of columns are partly oriented in one preferable direction. The physical meaning of operation in Eq. (6) consists in virtual cloud rotation around the

*z*axis of the lidar coordinate system in which the Stokes parameters are determined. The coordinates are specified by three unit vectors

**e**

_{x}×

**e**

_{y}=

**e**

_{z}. The optical lidar axis coincides with the

*z*axis. The

*x0z*plane is taken for the plane of reference.

**43**, 6620–6628 (2004). [CrossRef]

_{0}of the operator

**R**at which the matrix is described by Eq. (5). The operation is ambiguous. If instead of Φ

_{0}we substitute Φ

_{0}±π/2, the matrix will be described by Eq. (5) again, but the signs of the elements

*A*

_{12},

*A*

_{21}and

*A*

_{34},

*A*

_{43}will change. In [7

**43**, 6620–6628 (2004). [CrossRef]

_{0}value with negative elements

*A*

_{12}and

*A*

_{21}was taken. Then the angle Φ

_{0}counted from the

*x*axis specifies the azimuthal direction perpendicular to which the long particle diameters are oriented.

*A*

_{33}= −

*A*

_{22}is satisfied in addition to relation in Eq. (3) and, as experimentally demonstrated in [7

**43**, 6620–6628 (2004). [CrossRef]

*A*

_{14}and

*A*

_{41}is sufficiently low.

*A*

_{22}=

*A*

_{11}−

*d*from Eq. (3) and using the equality

*A*

_{33}= −

*A*

_{22}, we write down

*A*

_{33}=

*d*−

*A*

_{11}and

*A*

_{44}= 2

*d*−

*A*

_{11}and then normalize by

*A*

_{11}. The meaning of the designation is obvious. The element

*a*

_{22}of the normalized matrix is numerically equal to the degree of polarization of radiation by a medium upon exposure to radiation linearly polarized in the reference plane or perpendicular to it. In this case,

*d*is the commonly used definition of this quantity.

*A*

_{14}and

*A*

_{41}mean that the particles forming the ensemble possess a mirror symmetry or that the ensemble comprises equal numbers of particles being mirror reflections of one another. As already mentioned above and shown in Fig. 1, zero values of elements

*A*

_{14}and

*A*

_{41}are most probable. Then a diagonal matrix is obtained the form of which was suggested in [12

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

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

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

**43**, 6620–6628 (2004). [CrossRef]

*A*

_{12}=

*A*

_{21}have the minus signs if the

*x0z*reference plane coincides with the lidar tilt plane. This occurs because of breaking of the rotational symmetry. Projections of particles onto the optical lidar axis are less than those on the direction perpendicular to the lidar tilt plane. For example, a round plate is represented by an ellipse extended in this direction. The effect arises of apparent orientation of the long particle diameters perpendicular to the reference plane. Negative values of the above-indicated matrix elements are confirmed by the BSPM calculated in [17], for hexagonal plates and columns. Exactly this effect was experimentally observed in [8

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

18. Massimo Del Guasta, Edgar Vallar, Olivier Riviere, Francesco Castagnoli, Valerio Venturi, and Marco Morandi “Use of polarimetric lidar for the study of oriented ice plates in clouds,” Appl. Opt. **45**, 4878–4887 (2006). [CrossRef] [PubMed]

*A*

_{11}:

*a*

_{14}and

*a*

_{44}are independent of the azimuth of the lidar reference plane and hence of the presence or absence of azimuthal orientation of cloud particles. The importance of this BSPM property is caused by the fact that the element

*a*

_{44}in the case of zenith or nadir sensing acts as a parameter describing the horizontal orientation of long particle diameters [7

**43**, 6620–6628 (2004). [CrossRef]

*χ*defined by the following combination of elements of the reduced BSPM:

*χ*changes from 0 (for random orientation of particles) to 1 (for strict orientation of particles in a given direction). The modal value of the experimental

*χ*distributions is 0.1. The presence of azimuthal orientation is additionally indicated by the nonzero value of the element

*a*′

_{12}.

*a*′

_{44},

*a*′

_{12}, and

*a*′

_{14}of the reduced normalized BSPM obtained in [7

**43**, 6620–6628 (2004). [CrossRef]

*a*′

_{44}< −0.4) is observed almost in 20% of all cases. The average value of

*a*′

_{12}is 〈

*a*′

_{12}〉 = −0.22, and the distribution mode is −0.15. The probability that the

*a*′

_{12}value lies in the interval [−0.4, −0.6] is approximately equal to 0.15. Values −0.6 >

*a*′

_{12}> −1 are observed in several fractions of percent.

*a*′

_{12}of the reduced BSPM corresponds to such mutual orientation of the lidar and cloud when the

*x0z*lidar reference plane is parallel to the direction perpendicular to which the long particle diameters are oriented. If the lidar is rotated around the z optical axis through the angle φ, the element

*a*

_{12}of the nonreduced BSPM, that is, of the experimental BSPM, will be

*a*′

_{12}to −

*a*′

_{12}. As demonstrated below, this leads to ambiguity of the depolarization measured in lidar experiments with linearly polarized laser radiation.

## 3. Depolarization of backscattered radiation

*P*is the polarization degree and

*q*,

*u*, and

*v*are the Stokes parameters normalized by the radiation intensity. Let us consider two methods of depolarization measurements in lidar research. The first and the most widespread method assumes that laser radiation is linearly polarized, and two components of the scattered radiation intensity are measured, one of which (

*I*

_{∥}) is polarized in the plane of laser radiation polarization, and another component (

*I*

_{⊥}) is cross-polarized. The second method assumes that laser radiation is circularly polarized, and a λ/4 phase plate is placed in the receiver in front of the analyzer of linear polarization. The fast axis of the plate is rotated about the

*x*axis counterclockwise through an angle of 45°.

**S**

^{L}

_{0}= (1 1 0 0)

^{T}for radiation linearly polarized in the

*x0z*reference plane,

**S**

^{c}

_{0}= (1 0 0 −1) for circularly polarized radiation.

**L**

^{∥}and

**L**

_{⊥}of linear polarizers that transmit radiation in the reference plane and in the perpendicular direction:

**G**here are the first lines of the matrices

**G**

^{∥}=

**L**

^{∥}

**P**and

**G**

^{⊥}=

**L**

^{⊥}

**P**, where

**P**is the Müller matrix of the λ/4 plate placed at an angle of 45°, as described above.

**a**assuming that

*A*

_{11}= 1 and that laser radiation has unit intensity. The intensities of the parallel and cross-polarized components in the first method are described by the formulas

*a*

_{11}≡ 1,

*a*

_{12}=

*a*

_{21}, and

*k*is the proportionality factor unimportant for the examined case. The quantity

*u*=

*v*= 0. Then from Eqs. (13), (16), and (17) we obtain the depolarization value

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

*a*′

_{12}∣ ≡ 0. The ambiguity is caused by the fact that for sensing of an ensemble of particles oriented in an azimuthal direction, the parameters

*u*and

*v*can differ from zero and hence must be substituted into Eq. (13); however, they are not determined in the experiment. For vivid presentation, we now estimate the uncertainty interval for the modal value

*a*′

_{12}=−0.15 mentioned in Section 2. Taking

*a*

_{22}= 0.6 as the most probable value, we obtain that according to Eq. (12), the depolarization can change within the limits of 0.35–0.47 depending on the lidar orientation relative to the cloud. However, much more uncertain situations can be realized. Probably, the reader engaged in lidar polarization research of clouds will recall cases in which the intensity of the cross-polarized component was equal or even exceeded that of the parallel component. According to Eq. (18), the equality

*I*

_{⊥}=

*I*

_{∥}implies the complete depolarization, and for

*I*

_{⊥}>

*I*

_{∥}, definition (18) loses its meaning at all. The similar situation is possible when the lidar reference plane is tilted at an angle close to 45° to the direction of preferable azimuthal orientation of particles. Scattered radiation can have high enough degree of polarization, but small value of the second and large value of the third Stokes parameters. One of such BSPM presented in [7

**43**, 6620–6628 (2004). [CrossRef]

*u*=

*v*= 0 is satisfied, but ∣

*a*′

_{12}∣ depends on the tilt angle of the optical lidar axis. Therefore, the ambiguity of lidar depolarization is retained. Exactly for this reason we suggest to use the circular polarization which excludes the risk of obtaining ambiguous estimates of horizontal particle orientation and gives values of the

*a*′

_{44}BSPM element independent of the azimuthal orientation of particles.

## 4. A lidar for measurements with circularly polarized radiation

*x0z*reference plane. To change the polarization from linear to circular, a λ/4 quartz plate is inserted into the beam path. The fast axis of the plate is at an angle of 45° to the reference plane. The angle is counted counterclockwise as viewed counter to the laser beam.

*D*= 0.2 m and

*f*= 2 m). The field stop forms a field-of-view angle of 1 mrad. The Wollason prism (WP), forming two beams with mutually orthogonal polarization states, is inserted into the scattered beam path. The plane in which the ordinary and extraordinary beams lie is perpendicular to the

*x0z*plane. The

*x*axes of the transmitter and receiver coordinate systems coincide, and their

*y*and

*z*axes have opposite directions. In front of the Wollaston prism, a λ/4 quartz plate is placed. Its fast axis is at an angle of 45° to the reference plane. The angle is counted from the

*x*axis counterclockwise as viewed counter to incident radiation. Two FÉU-84 photomultipliers were used as receivers. The error in calibration of the relative sensitivity of photodetectors did not exceed 3%. This lidar configuration allows the column-vector

**S**

^{c}

_{0}considered above and row-vectors

**G**

_{∥}and

**G**

_{⊥}determined by Eq. (15) to be obtained.

## 5. Calculation of the element *a*_{44} normalized by the element *a*_{11} for the BSPM of the aerosol component from lidar signals

*a*

_{11}defines the backscattering coefficient of natural light. Let us write down the equation of laser sensing in the following form:

*P*(

*h*) is lidar return signal power and

**s**(

*h*) is the Stokes vector of scattered radiation normalized by the intensity;

*c*is the velocity of light;

*W*

_{0}is the laser pulse energy;

*D*is the receiving antenna area;

*h*=

*ct*/2 is the distance to the scattering volume at the moment of time

*t*/2;

**M**(

*h*) is the BSPM of the ensemble of particles occupying this volume;

*T*

^{2}= exp[−2∫

^{h}

_{0}(

*h*′,φ,θ)

*dh*′]; and ε(

*h*′,φ,θ) is the extinction coefficient.

*κ*

_{∥}and

*κ*

_{⊥}of the detectors in the corresponding channels. As a result, we obtain a pair of equations for the detector currents

_{⊥}/ κ

_{∥}.

**a**(

*h*) =

**A**(

*h*)/

*A*

_{11}(

*h*) is the normalized BSPM of the aerosol component,

**a**

_{1}is the row-vector, representing the first row of the matrix

**a**(the quantity

**A**

_{11}(

*h*)

**a**

_{1}

**S**

^{c}

_{0}is equal to the aerosol backscattering coefficient β

_{a}for circularly polarized radiation); σ is the BSPM of molecular scattering normalized by the element ∑

_{11}(σ

_{11}= 1, σ

_{22}= 0.97, σ

_{33}= σ

_{44}= −0.97, σ

_{ij}= 0);

*R*(

*h*) is the backscattering ratio. This quantity is determined by the well-known methods of reconstructing the optical parameters from lidar signals in the two-component molecular-aerosol atmosphere. For example, lidar signal calibration against a model profile of the molecular scattering described in [19

19. P. B. Russell, J. Y. Swissler, and P. M. McCormick, “Methodology of error analysis and simulation of lidar aerosol measurements,” Appl. Opt. **18**, 3783–3790 (1979). [PubMed]

*K*(

*h*) = (

*C*(

*h*) − 1)/ α(

*C*(

*h*) + 1)

*γ*(

*h*) → 0, and

*a*

_{14}= 0 are satisfied,

*a*

_{44}=

*C*(

*h*). That is, the required parameter is simply equal to the value determined directly from lidar return signals. The terms containing

*γ*(

*h*) are corrections for the molecular scattering contribution. For

*R*(

*h*) → 1, the error in determining

*R*(

*h*) can lead to inadmissibly large errors in determining the element

*a*

_{44}. The acceptable accuracy can be obtained for

*R*(

*h*) > 3.

*a*

_{14}. It is natural to take its average value ∣

*a*

_{14}∣ = 0 obtained in [7

**43**, 6620–6628 (2004). [CrossRef]

*a*

_{14}∣ > 0.1 does not exceed 0.05. The risk of an error is much lower than in lidar experiments with linearly polarized laser radiation when by default one takes

*a*

_{12}= 0. If the risk is excessive, the complete experiment should be carried out to determine elements

*a*

_{14}and

*a*

_{44}. To this end, additional measurements analogous to the above-described but with circular laser radiation polarization of opposite sign

**S**

^{c}

_{0}= (1 0 0 1) should be carried out. As a result, the second equation similar to Eq. (24) will be obtained. From the system of two equations, elements

*a*

_{14}and

*a*

_{44}can correctly be determined.

*F*

_{∥}(in units of 12-bit ADC code) is shown at the top of the figure. A 2D distribution of the

*a*

_{44}matrix element is shown at the bottom of the figure. It can be seen that the matrix element changes from

*a*

_{44}= −0.95 at altitudes beyond the cloud layer (between 6–7 and 8–9 km), which is typical of molecular scattering, to

*a*

_{44}= 0 within dense cloud layers. Significant time variations can be traced at fixed altitudes and significant altitude variations can be traced at fixed times. In this case, the fine structure of the

*a*

_{44}element not always coincides with that of the backscattering coefficient, which demonstrates the temporal and spatial variability of the orientation of crystal particles.

## 6. Conclusion

*a*

_{44}of the normalized BSPM and thereby elucidation of a debatable question on the role of particle orientation in modeling the radiative transfer in the atmosphere. Nowadays, a lidar whose characteristics have allowed us to start measurements coordinated with the schedule of the CALIPSO program is operated at the IAO SB RAS.

## References and links

1. | K. Sassen and D.K. Lynch, “What are cirrus clouds?” in |

2. | Yu. F. Arshinov, B. V. Kaul, and I. V. Samokhvalov, “Study of crystal clouds by measuring the backscattering phase matrices with polarization lidar: Particle orientation in cirrus.” in |

3. | C. M. R. Platt. Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals. J. Appl. Meteorol. |

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

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

6. | B. V. Kaul and I. V. Samokhvalov, “Orientation of particles in Ci crystal clouds. Part 1. Orientation at gravitational sedimentation,” J. Atmos. Oceanic Opt. |

7. | 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. |

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

9. | D. M. Winker, W. H. Hunt, and M. J. McGill, “Initial performance assessment of CALIOP," Geophys. Res. Lett. , |

10. | H. C. van de Hulst, |

11. | B. V. Kaul, “Symmetry of light backscattering matrices of nonspherical aerosol particles,” J. Atmos. Oceanic Opt. |

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

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

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

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

16. | B. V. Kaul, “Influence of electric field on ice cloud orientation,” J. Atmos. Oceanic Opt. |

17. | D. N. Romashov, “Backscattering matrix for monodisperse ensembles of hexagonal ice crystals,” J. Atmos. Oceanic Opt. |

18. | Massimo Del Guasta, Edgar Vallar, Olivier Riviere, Francesco Castagnoli, Valerio Venturi, and Marco Morandi “Use of polarimetric lidar for the study of oriented ice plates in clouds,” Appl. Opt. |

19. | P. B. Russell, J. Y. Swissler, and P. M. McCormick, “Methodology of error analysis and simulation of lidar aerosol measurements,” Appl. Opt. |

**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 and Sensors

**History**

Original Manuscript: November 6, 2008

Revised Manuscript: January 22, 2009

Manuscript Accepted: March 14, 2009

Published: April 10, 2009

**Citation**

Yurii Balin, Bruno Kaul, Grigorii Kokhanenko, and David Winker, "Application of circularly polarized laser radiation for sensing of crystal clouds," Opt. Express **17**, 6849-6859 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-8-6849

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### References

- K. Sassen and D. K. Lynch, "What are cirrus clouds?" in Cirrus, OSA Technical Digest (Opt. Soc. Am., Washington DC, 1998), pp. 2-3.
- Yu. F. Arshinov, B. V. Kaul, and I. V. Samokhvalov, "Study of crystal clouds by measuring the backscattering phase matrices with polarization lidar: Particle orientation in cirrus," in Cirrus, OSA Technical Digest (Opt. Soc. Am., Washington DC, 1998), pp. 131-134.
- C. M. R. Platt, Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals," J. Appl. Meteorol. 17, 1220-1224 (1978). [CrossRef]
- 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, 1111 - 1114 (1981). [CrossRef]
- J. D. Klett, "Orientation model for particles in turbulence," J. Atmos. Sci. 52, 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. 18, 866- 870 (2005).
- 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]
- V. Noel and K. Sassen, "Study of planar ice crystal orientations in ice clouds from scanning polarization lidar observations," J. Appl. Meteor. 44, 653-664 (2005). [CrossRef]
- D. M. Winker, W. H. Hunt, and M. J. McGill, "Initial performance assessment of CALIOP," Geophys. Res. Lett. 34, L19803, doi:10.1029/2007GL030135 (2007). [CrossRef]
- H. C. van de Hulst, Light Scattering by Small Particles (John Wiley and Sons, Inc. New York; Chapman and Hall, Ltd. London, 1957).
- B. V. Kaul, "Symmetry of light backscattering matrices of nonspherical aerosol particles," J. Atmos. Oceanic Opt. 13, 829-833 (2000).
- M. I. Mishchenko and J. W. Hovenier, "Depolarization of light backscattered by randomly oriented nonspherical particles," Opt. Lett. 20, 1356-1358 (1995). [CrossRef] [PubMed]
- G. G. Gimmestad, "Reexamination of depolarization in lidar measurements," Appl. Opt. 47, 3795-3802 (2008). [CrossRef] [PubMed]
- C. J. Flynn, A. Mendoza, Yu. Zheng, and S. Mathur, "Novel polarization-sensitive micropulse lidar measurement technique," Opt. Express 15, 2785-2790 (2007). [CrossRef] [PubMed]
- B. V. Kaul and I. V. Samokhvalov, "Orientation of particles in Ci crystal clouds. Part 2. Azimuth orientation," J. Atmos. Oceanic Opt. 19, 38- 42 (2006).
- B. V. Kaul, "Influence of electric field on ice cloud orientation," J. Atmos. Oceanic Opt. 19, 835- 840 (2006).
- D. N. Romashov, "Backscattering matrix for monodisperse ensembles of hexagonal ice crystals," J. Atmos. Oceanic Opt. 12, 376-384 (1999).
- 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, 4878-4887 (2006). [CrossRef] [PubMed]
- P. B. Russell, J. Y. Swissler, and P. M. McCormick, "Methodology of error analysis and simulation of lidar aerosol measurements," Appl. Opt. 18, 3783-3790 (1979). [PubMed]

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