Spatial distribution of doubly scattered polarized laser radiation in the focal plane of a lidar...

Spatial distribution of doubly scattered polarized laserradiation in the focal plane of a lidar receiver

Vadim Griaznov,1 Igor Veselovskii,1,* Paolo Di Girolamo,2 Michail Korenskii,1 and Donato Summa2

1Physics Instrumentation Center, Troitsk, Moscow Region, 142190, Russia2DIFA, Università degli Studi della Basilicata, Potenza, Italy

*Corresponding author: [email protected]

Received 30 January 2007; revised 20 June 2007; accepted 14 August 2007;posted 14 August 2007 (Doc. ID 79555); published 19 September 2007

Depolarization lidars are widely used to study clouds and aerosols because of their ability to discriminatebetween spherical particles and particles of irregular shape. Depolarization of cloud backscatteredradiation can be caused also by multiple scattering events. One of the ways to gain information aboutparticle parameters in the presence of strong multiple scattering is the measurement of radial andazimuthal dependence of the polarization patterns in the focal plane of receiver. We present an algorithmfor the calculation of corresponding polarized patterns in the frame of double scattering approximation.Computations are performed for various receiver field of views, for different parameters of the scatteringgeometry, e.g., cloud base and sounding depth, as well as for different values of cloud particle size andrefractive index. As the spatial distribution of cross-polarized radiation is of cross shape and rotated at45° with respect to laser polarization, the use of a properly oriented cross-shaped mask in the receiverfocal plane allows the removal of a significant portion of the depolarized component of the backscatteredradiation produced by double scattering. This has been verified experimentally based on cloud depolar-ization measurements performed at different orientations of the cross-shaped mask. Results obtainedfrom measurements are in agreement with model predictions. © 2007 Optical Society of America

OCIS codes: 280.0280, 280.1100, 280.3640, 290.4210.

1. Introduction

Lidar signals from aerosol layers and clouds alwaysinclude the contribution from multiple scattering.This contribution is usually considered as an addi-tional factor that complicates lidar data analysis. Atthe same time, the angular distribution of multiplescattering returns provides information about parti-cle parameters, which was confirmed by successfuloperation of lidars with multiple field of view (MFOV)[1–6]. In conventional MFOV lidars the azimuthallyintegrated return is considered, but it is well knownthat the azimuthal dependence of scattered powercan produce characteristic patterns carrying addi-tional information about the scattering particles.

The azimuthal distribution of backscattered radi-ation from a polarized laser source was first observedby Carswell and Pal in experiments with controlled-environment and atmospheric clouds [7,8]. Depen-

dence of cross-polarized patterns on particle size,shape, and optical density was experimentally stud-ied by Roy et al. [9] in an aerosol chamber. Resultsachieved demonstrate that scattering patterns con-tain retrievable information on cloud parameters. Touse this information, an appropriate model should bedeveloped to calculate the power and polarizationstate of the lidar return for a given field of view andazimuth angle. The model should also take into ac-count the parameters of scattering geometry such ascloud base and sounding depth.

The experimentally observed azimuthal patternscan be well described in the frame of a model sug-gested by Rakovic and Kattawar [10], which consid-ers doubly scattered polarized radiation by sphericalparticles. Unfortunately this model cannot be usedin a straightforward way to simulate lidar perfor-mances because it considers the following:

(a) It considers height-integrated scattering; thusit does not relate the pattern parameters to the lidarscattering geometry.

0003-6935/07/276821-10$15.00/0© 2007 Optical Society of America

6821 APPLIED OPTICS � Vol. 46, No. 27 � 20 September 2007

(b) It considers the cornerlike sequence of two scat-tering processes, i.e., backscattered photons propagat-ing parallel to laser emitted photons.

In our paper the approach presented in Ref. [10] isadopted to the case of MFOV lidar sounding, andexpressions for radial (FOV) and azimuthal distribu-tion of irradiance in the focal plane of the lidar re-ceiver are derived. The obtained formulas are used toillustrate the dependence of copolarized and cross-polarized patterns on particle size and refractive in-dex. Simulated scattering patterns are in agreementwith cloud depolarization measurements performedin Potenza (Southern Italy) by the DIFA–Universityof Basilicata Raman lidar system (BASIL).

2. Model Description

The scattering geometry considered in the model isshown in Fig. 1. We assume cloud homogeneity withrespect to particle size and extinction. A nondiver-gent laser beam propagates along the Z axis. Extinc-tion of atmosphere below the cloud is neglected, andthe first scattering event occurs at point A inside thecloud, having base at height za. After a second scat-tering event at point B, photons return back to thereceiver at angle �. The total photon path is 2R � z� v � w. For typical lidars with a variable field ofview, the angle � is less than 0.005 rad [9], so in thederivation below we suggest that the electric vector ofa doubly scattered wave is parallel to the XOY plane.Besides we use the apparent approximations: sin �� and cos � � 1.

The approach to calculate the scattered radiation isbased on the results presented in Ref. [10]. The inci-dent wave can be expressed in irradiance unitsthrough the vector E0 � dP0�ds0 � E0E00, where E0 isthe incident irradiance and E00 is the normalizedStokes vector. The incident power is dP0 � E0ds0,and the power at point A is dP1 � dP0 exp��aza ��c�z � za��, where �a and �c are the extinction coef-ficients below the cloud base and inside the cloud.The power dP1 illuminates a volume dV1 � ds1dz,and its scattering properties are characterized by thephase matrix M��. The relation between scattered

and incident waves is

dP2

dV1d�1� �sM��

R��dP1

ds1, (1)

with dP2 representing the power scattered from thevolume dV1.

In this equation the matrix

R�� 1 0 0 00 cos 2� sin 2� 00 �sin 2� cos 2� 00 0 0 1

�is used to rotate the initial coordinate system XYZaround the OZ axis through the angle � (Ref. [11]);the solid angle d�1 is equal to ds2�v2, where ds2 is theinfinitesimal area illuminated by scattered light fromvolume dV1, and �s is the total scattering coefficient.From Eq. (1) the scattered power is

dP2 � �sM��R��dP0 exp��aza � �c�z � za��d�1dz.(2)

At point B the power is attenuated by a factor ofexp��cv�,

dP3 � dP2 exp��cv� � �s exp��aza � �c�z � za� � �cv� M��R��dP0d�1dz, (3)

and the light scattered from volume dV2 � ds2dv isthen collected by the receiver. The received power hasthe form

dP4

dV2d�2� �sM��

dP3

ds2exp��cw �

za

cos �� a

za

cos ��,

(4)

or using expression (3),

dP4 � �s2 exp��2��M��M��R��E0ds0

ds2

v2 d�2dzdv,

(5)

where

� � �aza1 � cos �

cos � � �c�R � za1 � cos �

cos � �� aza � �c�R � za�.

Let dV2� be a volume oriented along OB, and dV2�� dV2 � ds2�dw. Then

dP4 � �s2 exp��2��M��M��R��E0ds0

ds2�

v2 d�2dzdw. (6)Fig. 1. Scattering geometry considered in the model.

20 September 2007 � Vol. 46, No. 27 � APPLIED OPTICS 6822

The radiance radiated from ds2� and attenuated onthe way to the telescope is

dP4

ds2�d�2 cos �� s

2 exp��2��M��M��R��E0ds0

dzdw

v2 cos �, (7)

and it is equal to the radiance through ds3:

dP4

ds3d�2� cos �� s

2 exp��2��M��M��R��E0ds0

dzdw

v2 cos �, (8)

where d�2� is the solid angle subtended by ds2�.Our goal is the calculation of copolarized and cross-

polarized scattered components as a function of azi-muthal angle �; therefore the received signal shouldbe represented in the same coordinate systems as theincident beam. This dictates introduction of the sec-ond rotation matrix R��. Taking into account thatd�2� � sin �d�d�, we can express the irradiancethrough ds3 as

dE � R��dP4

ds3� �s

2 exp��2��R��M��M��R��

E0ds0

dzdw

v2 sin � cos �d�d�. (9)

The angle � can be expressed through � as � �� . As the angle � is small, we can expand thematrix M�� in the Taylor series around � � �:

M�� M�� M�� d�M��

d�.

(10)

After substituting Eq. (10) in Eq. (9), we have

dE � �s2R��M�� · M�� M��R��E0

exp��2��

v2 ds0dzdw sin � cos �d�d�. (11)

For further analysis we need to express the variablesz, v, and w through �, �, and R. The required rela-tionships for z and w and their derivatives can bederived from Fig. 1. These are

z R �

sin� � �

2

sin�

2 cos�

2

, z �

�R

2 cos�

2

·sin

�

2

sin2�

2

,

w R �

cos�

2

cos�

2 cos� � �

2

, w �

�

�R sin�

2

2 cos�

2 cos2� � �

2 .

(12)

The corresponding Jacobian is

z w � R � �

z �

z R

w �

w R��

R sin�

2

2 sin2�

2 cos�

2 cos2� � �

2 .

(13)

After inserting Eq. (13) into Eq. (11), the receivedirradiance has the form

dE � �s2R��M�� · M�� M��R��E0

exp��2��

R cos �ds0d�dRd�d�, (14a)

and after using appoximation cos � � 1,

dE � �s2R��M�� · M�� M��R��E0

exp��2��

R ds0d�dRd�d�. (14b)

Replacing infinitesimal quantities ds0, dR, d�, andd� with the corresponding finite differencies and de-noting m��, �� M�� · M�� M��,F��, �, �� R��m��, ��R��, Meff��, �, �, R� � �Mij�� exp��2��R��min

�max F��, �, ��d�, and C � �s2E0�s0

�R��, the integral value of irradiance can be re-written as

E � CMeff��, �, R�E00, (15)

where the lower and upper limits of � are foundfrom Fig. 1: �min � R��R � za� and �max � � � �za��R � za�.

We are interested in considering separately co-polarized and cross-polarized lidar returns. Thecorresponding polarizers are described through thematrices

T �12�

1 1 0 01 1 0 00 0 0 00 0 0 0

�, T� �12�

1 �1 0 0�1 1 0 00 0 0 00 0 0 0

�. (16)

The received signals are E � CT Meff��, �, R�E00 andE� � CT�Meff��, �, R�E00.

If the incident radiation is linearly polarized alongthe OX axis, copolarized and cross-polarized irradi-ances at the receiver entrance are


E �C2�

M11 � M21 � M12 � M22

M11 � M21 � M12 � M22

00

�,

E� �C2�

M11 � M21 � M12 � M22

M11 � M21 � M12 � M22

00

�. (17)

Measured irradiances are the first elements of thesevectors:

E � C�M11 � M21 � M12 � M22��2,

E� � C�M11 � M21 � M12 � M22��2, (18)

and the total irradiance is Etot � C�M11 � M12�.Let us suppose that the medium is invariant under

rotation and reflection; for example, it consists ofrandomly oriented isotropic cylinders, spheroids, oranisotropic spheres. Its scattering matrix is given by([11], p. 414)

Msca�� a b 0 0b c 0 00 0 d �e0 0 e f

�.After substituting it into expression for Meff, the ma-trix elements have the form

M11 �exp��2��

R ��min

�max

�a�� a�� a�� a��

� b�� b�� b�� b��d�, (19)

M12 �exp��2��

R cos 2��min

�max

�a�� b��

� �a�� b�� b�� c�� b�� c��d�, (20)

M21 �exp��2��

R cos 2��min

�max

�b�� a��

� �b�� a�� c�� b�� c�� b��d�, (21)

M22 �exp��2��

R �cos2 2��min

�max

�b�� b��

� �b�� b�� c�� c��

� �c�� c��d� � sin2 2��min

�max

�d�� d��

� �d�� d�� e�� e��

� �e�� e��d��. (22)

If only isotropic spheres are considered, then c � aand f � d, and the matrix elements become simpler.For the spheres the elements of phase matrixa, b, d, e are calculated through Mie theory as isshown in Ref. [11].

Expressions (18)–(22) show that measured copolar-ized and cross-polarized irradiances are periodicalfunctions of azimuthal angle �. When only azimuth-ally integrated power is of interest, expressions (19)–(22) can be integrated, and for spherical particlescopolarized and cross-polarized returns become

E �� Cexp��2��

2R ��min

�max

d��3��a�� a��

� b�� b�� a�� a�� b�� b�� d�� d�� e�� e�� d�� d�� e�� e��,

E�� Cexp��2��

2R ��min

�max

d��a�� a��

� b�� b�� a�� a�� b�� b�� d�� d�� e�� e�� d�� d�� e�� e��. (23)

All computations in this paper were performed onlyfor the spheres, but as it follows from the generalform of matrix Msca, the azimuthal periodicity mayoccur also for a medium that is invariant under ro-tation and reflection.

3. Simulation

Expressions (18)–(22) allow us to calculate theirradiances: E ��, ��, E��, ��, and Etot��, �� of back-scattered light, which we will call copolarized, cross-polarized, and total scattering patterns. In thissection the derived formulas are used to illustrate themain properties of these patterns. The results arepresented for the angular spectra of irradiance at thereceiver entrance, and the power in the focal plane ofthe receiver can be easily recalculated from E ,��, ��.


The irradiance in all figures is given in arbitraryunits.

When analyzing the influence of scattering geom-etry, it is important to establish the dependence ofpatterns parameters on sounding depth �z � zc

� za, where zc is the height at which backscatteringoccurs. In single-scattering, vertically pointing lidarmeasurement, photons detected with a time delay Twith respect to laser firing time are backscatteredfrom a height cT�2, where c is the speed of light. InMFOV lidar measurements, photons detected with atime delay T can be backscattered from differentheights: zc varies from R to R�1 � �max�, where 2�max isthe maximum FOV of the receiver. For typicalground-based sounding conditions with R � 2000 mand �max � 5 mrad, the variation of zc is 10 m. In theforthcoming analysis this deviation is neglected, andwe assume that zc � R.

Simulations are performed for a log-normal parti-cle size distribution (PSD) with modal radius r0 anddispersion ln � � 0.1. We should point out here thatthe choice of the size distribution was not made tomatch to any real atmospheric situation, but ratherto establish the dependence of patterns parameterson particle size and to remove the effects related tomorphology-dependent resonances. Laser wave-length for all computations in this section is � �1064 nm.

Radial-azimuthal distribution of backscatteredpower depends on particle size. This is illustrated byFigs. 2 and 3 showing the contour plots of E ��, �� andEtot��, �� for r0 � 0.2, 1, 10, and 50 �m. The smallervalues of power correspond to the contours locatedfarther from the center of each plot. The irradiancesare plotted in �x, �y coordinates, which are related tothe radial and azimuthal angles as �x � � cos � and�y � � sin �. Computations are performed for a cloud

base za � 500 m and a sounding depth �z � zc � za

� 50 m. Azimuthal dependence of E ��, �� is quitecomplicated and is determined by the properties ofthe so-called Mie functions i1, i2, describing corre-spondingly the scattering cross section for waves withpolarization perpendicular and parallel to the scat-tering plane [11]. For small particles the patternlooks dipolelike, while for large particles the patternsize is similar in the OX and OY directions. For par-ticles with r0 exceeding 20 �m, the modulation depthof azimuthal variations is strongly decreased, and at50 �m [Fig. 2(d)] the pattern is close to circular.Based on the performed calculations, we can expectthat the maximum information content of azimuthalpower distribution (for a wavelength of 1.06 �m) isattributed to the radii interval 0.4–20 �m. The totalreturn Etot��, �� for small particles also has dipolelikeshape [Fig. 3(a)] because depolarization introducedby small spherical particles is small. With increasingsize, the pattern acquires a circular shape [Fig. 3(d)].

The pattern for the cross-polarized return is clo-verlike and rotated at 45° with respect to the laserpolarization direction, as is shown in Fig. 4. Thisshape is preserved for all particle sizes or refractiveindices: for any value of �, backscattered irradianceis a periodical function of azimuth angle only, i.e.,E��, �� 1 � cos 4��. Such periodical structure ofthe azimuthal dependence is inherent to double scat-tering processes, and higher scattering orders willonly cause a blur in the pattern [9].

In contrast to E��, ��, azimuthal distribution ofthe copolarized return (as illustrated by Fig. 2) de-pends on � and r0. Figure 5 shows the azimuthaldistribution of E ��, �� for r0 � 1 �m (� � 10 and2 mrad) and 10 �m (� � 1 and 0.2 mrad). The dash-dotted curve in this same picture shows the azi-muthal distribution of E��, ��. For convenience of

Fig. 2. Copolarized scattering patterns for PSDs with modal ra-dius (a) 0.2 �m, (b) 1 �m, (c) 10 �m, and (d) 50 �m. Calculationsare performed for za � 500 m, �z � 50 m, and m � 1.33. Laserpolarization is oriented horizontally (along OX).

Fig. 3. Total scattering patterns for PSDs with modal radius (a)0.2 �m, (b) 1 �m, (c) 10 �m, and (d) 50 �m. Calculations are per-formed for za � 500 m, �z � 50 m, and m � 1.33. Laser polariza-tion is oriented horizontally (along OX).


comparison, the distributions are normalized to keepthe maximum value of irradiance equal to 1. Withincreasing particle size, the azimuthal variability ofE (modulation depth) becomes lower. Thus, as wealready mentioned, for large particles the azimuthaldistribution of E does not provide new information tocompare with radial distribution. We can concludealso that the main information contained in theazimuthal distribution of E can probably be ob-tained by measuring at three azimuthal angles � �0, ��4, and ��2.

One of the questions arising when the azimuthalpatterns are considered is whether their measure-ments allow us to estimate particle refractive index.Figure 6 shows the copolarized scattering patternscalculated for m � 1.5 and 1.6, with r0 � 10 �m andscattering geometry being the same as in Fig. 2. The

contribution of the wave with polarization per-pendicular to the scattering plane increases withm rising, and the pattern evolves from verticallyoriented (at small �) for m � 1.33 [Fig. 2(c)] to hor-izontally oriented [Fig. 6(b)]. The azimuthal distribu-tions of E ��, �� together with depolarization ratiosE��, ��Etot��, �� calculated for r0 � 10 �m and thevalues of refractive indices m � 1.33, 1.4, 1.5, and 1.6are shown in Fig. 7. Again, for convenience of com-parison, the distributions E ��, �� are normalized tokeep its maximum value equal to 1. With increasingm the maxima of E ��, �� at � � 90° rise with respectto the maxima at � � 0. The depolarization reachesits maximum value at � � 45° with a periodicity of��2. The maxima of the depolarization ratio dependon m, and Fig. 7 illustrates that combined use ofcopolarized and cross-polarized patterns may in prin-ciple allow the estimation of the particle refractiveindex. However, to conclude if the simultaneous re-trieval of particle size and index is possible and toestimate the uncertainty of such retrieval, the infor-mation content analysis of azimuthal patterns should

Fig. 4. Cross-polarized pattern for r0 � 10 �m and m � 1.33.Scattering geometry is the same as in Fig. 2.

Fig. 5. Azimuthal distribution of copolarized return for r0 �

1 �m (solid curve, � � 2, 10 mrad) and 10 �m (dotted curve, � �

0.2, 1 mrad). Computations are performed for m � 1.33, za

� 500 m, and �z � 50 m. Dash-dotted curve corresponds to azi-muthal distribution of cross-polarized return.

Fig. 6. Copolarized patterns for refractive index (a) 1.5, (b) 1.6,and r0 � 10 �m. Scattering geometry is the same as in Fig. 2.


be performed [12]. Such analysis is beyond the scopeof the current paper.

The scattering geometry parameters influence theradial size of the pattern. Numerous theoretical stud-ies were performed in the approximation of doublescattering for the case of conventional MFOV lidars(exploiting azimuthally integrated return) [6,13]. Re-sults of these studies demonstrate that FOV distri-bution of scattered power P�� depends mainly on theratio � � �z�za, where �z is the sounding depth. As� decreases, the scattered power concentrates tosmaller values of �. The analysis performed for theazimuthal patterns leads to the same conclusion.The radial distributions of E ��, � � 0� and E��, �� 45°� are shown in Fig. 8. Calculations are per-formed for za � 1000 m and �z � 25, 50, and 100 m.With decreasing �z (and so �), the distributions shiftto the smaller �. The patterns calculated for differentvalues of the cloud base za, but for the same values of�, coincide after normalization. Thus, scattering ge-ometry influences the pattern size but does not alterits shape. We should mention that the radial depen-dence of azimuthally integrated power calculated

through Eq. (23) is similar to the results obtained inRef. [6], thus providing an independent verification ofthe code.

The cross-polarized component is maximum at� � 45° for any particle size. This implies that theregions corresponding to the maximum depolariza-tion ratio in the focal plane of the lidar receivershould also be localized along these directions. Thisis illustrated by the azimuthal distribution of thedepolarization ratio shown in Fig. 9 for a PSD withr0 � 10 �m. It follows that if a cross-shaped mask isinserted in the focal plane of the receiver, the rotationof this mask should influence the depolarization mea-sured by a conventional lidar. Lidar measurementsperformed with variable FOV and a variable orien-

Fig. 7. Azimuthal distributions of (a) E ��, �� and (b) depolariza-tion ratio E��, ��Etot��, �� for refractive indices m � 1.33, 1.4, 1.5,and 1.6. Calculations are performed for r0 � 10 �m and � � 1mrad. Scattering geometry is the same as in Fig. 2.

Fig. 8. Radial dependence of E ��, � � 0� (solid) and E��, �

� 45°� (dash-dotted) on FOV. Computations are performed forza � 1000 m, �z � 25, 50, and 100 m; r0 � 10 �m.

Fig. 9. Azimuthal pattern of depolarization E��, ��Etot��, �� forr0 � 10 �m. Calculations are performed for m � 1.33, �max

� 5 mrad, za � 500 m, and �z � 50 m. The shades of gray corre-spond to variation of depolarization in the 0–0.8 interval.


tation cross-shaped mask are reported and discussedin the next section.

4. Experimental Results

Lidar measurements reported in this section wereperformed by the DIFA–University of BasilicataRaman lidar system (BASIL). Using Raman�elasticscattering from atmospheric molecules and particles,BASIL provides high temporal and spatial resolutionmeasurements of particle backscatter at 355 nm and532 nm, particle extinction at 355 nm, particle depo-larization at 355 nm, and atmospheric temperatureand water vapor mixing ratio both in daytime andnighttime. BASIL makes use of an Nd:YAG lasersource equipped with second and third harmonic gen-eration crystals and is capable of emitting pulses at355 nm and 532 nm, with an average power of 5 Wand 6 W, respectively. The receiver is built around atelescope in Newtonian configuration (40 cm diame-ter). Signal detection is accomplished by means ofphotomultipliers operated in photon counting mode.The receiver of BASIL was opportunely modified forthe purposes of this research effort in order to allowvariable-FOV measurements and measurements ofthe azimuthal dependence of lidar signals. All mea-surements illustrated in this section were performedat 355 nm.

When comparing experimental and simulationresults, we suggest that the laser beam divergence�las is small; thus the single scattered radiation iscontained inside the smallest field of view 2�min.It should be noted that this constraint is fulfilledonly for small receiver apertures D such that12��las � �D�zc�� min. For the lidar with D � 400mm, �las � 0.5 mrad, and 2�min � 1.5 mrad, this con-dition is fulfilled for zc � 400 m.

The particle depolarization ratio was determinedusing the expression

�par�z� ��

par�z��

par�z��

R��z� � 1R �z� � 1 �mol�z�, (24)

where

R �z� ��

par�z� � � mol�z�

� mol�z�

,

R��z� ��

par�z� � ��mol�z�

��mol�z�

are the measured scattering ratios for the paralleland perpendicular components, respectively, �

par,��

par, � mol, and ��

mol are particles and moleculesbackscattering coefficients, and �mol � ��

mol�� mol is

the molecular depolarization ratio, which is taken tobe 0.005 [14]. Measurements at different FOVs wereperformed in quick succession, varying FOV between0.55 and 8.8 mrad (considering an integration time of3 minutes per FOV). We recall at this point that FOV

corresponds to 2�. Different FOVs are selected byusing the field stops of different size located in thefocus of the telescope. Since measurements at differ-ent FOVs could not be performed simultaneously, wetried to select cloudy conditions characterized by thepresence of an almost stable and uniform cloud deckin order to minimize signal variability associatedwith variable cloud properties. Figure 10 showsthe depolarization profiles at different FOVs. Themeasurement was performed on 17 December 2005(00:05–00:24 GMT), and the cloud layer was found inthe range 0.7–1.3 km (above station level). Surfacetemperature at the lidar station was approximately8 °C, which suggests a freezing level well above thecloud layer. This figure clearly reveals a strong de-pendence of particle depolarization on the FOV.

To illustrate the dependence of the depolarizationratio on the scattering geometry and particle size,Fig. 11 shows the ratio P��P �� calculated forr0 � 0.05, 1, 4, and 10 �m using expressions (23) for� � 355 nm. The depolarization ratio rises withan increasing of particle size from 0.07 (at r0� 0.05 �m) to 0.3 (at r0 � 4 �m). For particles with aradius above 4 �m, depolarization goes down. Exper-imental value of depolarization for 8.8 mrad FOV inFig. 10 is about 0.2, which is rather reasonable as-suming a typical size of cloud particles to be of tenmicrometers order. We should keep in mind thatthese simulations are performed for a uniform distri-bution of particles extinction and size throughout the

Fig. 10. Multi-FOV measurements of particle depolarization,with FOV varied in the range 0.55–8.8 mrad with log-equidistantincrements.


cloud, and the consideration of particle parametersvariability with height would make these dependen-cies more complicated.

For the purpose of studying the azimuthal depen-dence of the polarization pattern a cross-shapedmask was located in the focus of the telescope. This

mask has a diameter that corresponds to a FOV of8.8 mrad, with a circular blocking in the central partof the field stop ��1.5 mrad� to remove single scat-tering. The cross-shaped mask has arms, the angularapertures (approx. 64°) of which are optimized inorder to allow the removal of a significant portion ofdepolarized component produced by double scatter-ing.

Lidar measurements with the mask oriented at 0°and 45° with respect to laser polarization were per-formed in sequence (considering an integration timeof 3 minute per orientation). Figure 12 shows corre-sponding depolarization profiles, together with theparticle backscattering profile with the mask ori-ented at 45°. Cloud optical depth in the height inter-val 800–1200 m was calculated from the nitrogenRaman signals, and is 0.6. Estimations performed byusing the model of Eloranta [15] show that the scat-tering at considered FOV is mainly due to a doublescattering process. The figure clearly highlights thevariability of particle depolarization in dependence ofmask orientation. Particle depolarization is approxi-mately 3 times larger with the mask at 0° (peakdepolarization is 0.152) than with the mask at 45°(peak depolarization is 0.061). The strong discrimi-nation between the two mask orientations is in agree-ment with simulation results.

5. Conclusion

We derived the expressions to calculate radial andazimuthal distribution of lidar returns in the focalplane of a receiver in the double scattering approxi-mation. These expressions were used to study thedependence of the scattering patterns on scatteringgeometry and particle parameters. The influence ofthe scattering geometry is determined through theratio � � �z�za; its changes scale the pattern withoutchanging of the shape. At the same time, the shape ofthe copolarized pattern is sensitive to particle sizeand refractive index. Thus, we can expect that the useof azimuthal dependence should bring the additionalinformation to compare with the use of azimuthallyintegrated values [3,4]. Mathematically this meansthat the inverse problem is transformed to the systemof integral equations written for the set of azimuthalangles �. To estimate the optimal number of theseangles and expected advantages in terms of the ac-curacy of parameters retrieval, in the future we planto perform the analysis of information content of thedata, as was done in Ref. [6].

The determined azimuthal patterns are similarto the results presented by Rakovic and Kattawar[10]. This is not surprising, as in the first approxi-mation the azimuthal and radial dependence can beestimated from cornerlike geometry suggesting � ��zc, where � is the radial distance from the laser-beam axis. We should recall that, in contrast to theRakovic and Kattawar approach [10], we consider amodel that is particularly suited for the lidar sound-ing: scattering occurs from a certain altitude range,and � dependence of the backward phase function (inlinear approximation) is considered. In our model we

Fig. 12. Particle depolarization profiles measured with the cross-shaped mask oriented at 0° and 45° together with the particlebackscattering coefficient.

Fig. 11. Depolarization ratio P��P �� of azimuthally inte-grated power for r0 � 0.05, 1, 4, and 10 �m as a function of receiverFOV. Calculations are performed for m � 1.33, � � 0.355 �m,za � 500 m, and �z � 50 m.


suggested that clouds are uniform in particle size anddensity. This approach can be further developed byincluding corresponding altitude dependences. All re-sults of this paper were obtained for a double scat-tering geometry, so these are strictly applicable onlyto small sounding depths in clouds and to thin aerosollayers.

Strong depolarization induced by multiple scatter-ing limits the potential of conventional depolarizationlidars measuring the azimuthally integrated return.Still discrimination between spherical particles andparticles of irregular shape is possible if azimuthaldependence of return is considered. The azimuthalpattern for cross-polarized return from spherical par-ticles has a specific cloverlike shape, which does notdepend on particle size. So if the cross-shape mask isinserted in the receiver focal plane, the depolariza-tion ratios obtained for two orientations of the mask(along the laser polarization and at 45°) differ, whileparticles of irregular shape do not provide such azi-muthal dependence [9]. The presented experimentalmeasurements should be considered as preliminary,and for the final conclusion about applicability of thistechnique the sounding of different types of theclouds should be performed.

References1. R. F. Cahalan, M. McGill, J. Kolasinski, T. Varnai, and K.

Yetzer, “THOR—cloud thickness from offbeams lidar returns,”J. Atmos. Ocean Technol. 22, 605–627 (2005).

2. L. R. Bissonnette and D. L. Hutt, “Multiply scattered aerosollidar returns: inversion method and comparison with in situmeasurements,” Appl. Opt. 34, 6959–6975 (1995).

3. G. Roy, L. C. Bissonnette, C. Bastille, and G. Vallee, “Estima-tion of cloud droplet size density distribution from multiple-field-of-view lidar returns,” Opt. Eng. 36, 3404–3415 (1997).

4. G. Roy, L. Bissonnette, C. Bastille, and G. Vallee, “Retrieval of

droplet-size density distribution from multiple-field-of-viewcross-polarized lidar signals: theory and experimental valida-tion,” Appl. Opt. 38, 5202–5211 (1999).

5. L. R. Bissonnette, G. Roy, and N. Roy, “Multiple-scattering-based lidar retrieval: method and results of cloud probing,”Appl. Opt. 44, 5565–5581 (2005).

6. I. Veselovskii, M. Korenskii, V. Griaznov, D. N. Whiteman, M.McGill, G. Roy, and L. Bissonnette, “Information content ofdata measured with a multiple-field-of-view lidar,” Appl. Opt.45, 6839–6848 (2006).

7. A. I. Carswell and S. R. Pal, “Polarization anisotropy in lidarmultiple scattering from clouds,” Appl. Opt. 19, 4123–4126(1980).

8. S. R. Pal and A. I. Carswell, “Polarization anisotropy in lidarmultiple scattering from atmospheric clouds,” Appl. Opt. 24,3464–3471 (1985).

9. N. Roy, G. Roy, L. R. Bissonnette and J.-R. Simard, “Measure-ment of the azimuthal dependence of cross-polarized lidarreturns and its relation to optical depth,” Appl. Opt. 43, 2777–2785 (2004).

10. M. J. Rakovic and G. W. Kattawar, “Theoretical analysis ofpolarization patterns from incoherent backscattering of light,”Appl. Opt. 37, 3333–3338 (1998).

11. F. B. Bohren and D. R. Huffman, Absorption and Scattering ofLight by Small Particles (Wiley, 1983).

12. I. Veselovskii, A. Kolgotin, D. Müller, and D. N. Whiteman,“Information content of multiwavelength lidar data with re-spect to microphysical particle properties derived from eigen-value analysis,” Appl. Opt. 44, 5292–5303 (2005).

13. I. Polonskii, E. Zege, and I. L. Katsev, “Lidar sounding of warmclouds and determination of their microstructure parameters,”Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 37, 624–632(2001).

14. A. Behrendt and T. Nakamura, “Calculation of the calibrationconstant of polarization lidar and its dependency on atmo-spheric temperature,” Opt. Express 10, 805–817 (2002).

15. E. W. Eloranta, “Practical model for the calculation of multiplyscattered lidar returns,” Appl. Opt. 37, 2464–2472 (1998).


Spatial distribution of doubly scattered polarized laser radiation in the focal plane of a lidar...

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