Laboratory evaluation of the scattering matrix elements of mineral dust particles from 176.0° up to 180.0°-exact backscattering angle

Accepted Manuscript

Laboratory evaluation of the scattering matrix elements of mineral dust particles from 176.0° up to 180.0°-exact backscattering angle Alain Miffre , Danael ¨ Cholleton , Patrick Rairoux PII: DOI: Reference:

S0022-4073(18)30558-2 https://doi.org/10.1016/j.jqsrt.2018.10.019 JQSRT 6256

To appear in:

Journal of Quantitative Spectroscopy & Radiative Transfer

Received date: Revised date: Accepted date:

31 July 2018 9 October 2018 10 October 2018

Please cite this article as: Alain Miffre , Danael ¨ Cholleton , Patrick Rairoux , Laboratory evaluation of the scattering matrix elements of mineral dust particles from 176.0° up to 180.0°exact backscattering angle, Journal of Quantitative Spectroscopy & Radiative Transfer (2018), doi: https://doi.org/10.1016/j.jqsrt.2018.10.019

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ACCEPTED MANUSCRIPT Highlights Scattering matrix of mineral dust is measured in laboratory from 176.0° to 180.0°. A laboratory polarimeter has hence been built and validated on spherical particles. The slopes of the scattering matrix elements tend towards zero at π-angle. Our laboratory findings are in good agreement with T-matrix numerical code.

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Laboratory evaluation of the scattering matrix elements of mineral dust particles from 176.0° up to 180.0°-exact backscattering angle Alain Miffre1, Danaël Cholleton1 and Patrick Rairoux1 University of Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, F-69622, Villeurbanne, France Corresponding author: A. Miffre ([email protected]) 1

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Abstract In this paper, the scattering matrix elements of an ensemble of mineral dust particles are for the first time evaluated in laboratory for scattering angles ranging from 176.0° to the πbackscattering angle of 180.0° with a high angular resolution of 0.4° and compared with the outputs of T-matrix numerical code. Elastic light scattering is addressed at near and exact backscattering angles with a newly-built laboratory polarimeter, validated on spherical ( )/ ( ) of the scattering particles following the Lorenz-Mie theory. The ratios ( ) matrix elements of mineral dust particles are then precisely evaluated in laboratory from 176.0° up to 180.0° with a 0.4° angular resolution (even 0.2° between 179.2° and 180.0°), which is new. When approaching the π-backscattering angle, the slopes of the scattering matrix elements are almost zero, as theoretically predicted by J.W. Hovenier (2014). Moreover, our laboratory findings are found in good agreement with the outputs of the Tmatrix numerical code, showing the ability of the spheroidal model to describe light-scattering by mineral dust also from near to exact backscattering. Atmospheric implications for polarization lidar retrievals are then discussed in terms of linear and circular depolarization ratios for mineral dust. These results, which complement other existing light scattering experiments, may be used to extrapolate light scattering by mineral dust particles up the πbackscattering angle, which is useful in radiative transfer and climatology, in which backscattering is involved.

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1. Introduction

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Keywords Scattering matrix; Backscattering; Mineral dust; Extrapolations; T-matrix.

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Mineral dust is a major constituent of the atmosphere, contributing to the global water cycle by acting as a freezing nucleus and to the global carbon cycle by fertilizing ecosystems such as the Amazonian rainforest [1]. It also contributes to atmospheric chemistry for example by inducing surface photo-catalytic processes and may lead to new particle formation in the atmosphere, as we published [2]. By scattering and absorption, mineral dust also contributes to the Earth‟s radiative budget and may have a cooling effect [3]. Despite the important role of mineral dust in the Earth‟s atmosphere, the light scattering properties of mineral dust are still subject to uncertainties, mainly due to the complexity of these particles which are highly inhomogeneous, present a wide range of sizes and exhibit a highly irregular shape, with sharp edges and potential surface roughness [4]. The literature on light scattering by mineral dust particles is abundant with field and laboratory experiments, completed with numerical simulations. Among all light scattering directions, the backward scattering direction has drawn attention for both practical and fundamental reasons. Practically speaking, light backscattering is involved in ground-based and satellite-based lidar remote sensing instruments which provide a major source of global data on mineral dust [5,6], which are needed for radiative and climate forcing assessments. Lidar remote sensing provides fast, reliable and unique vertical profiles of mineral dust particles backscattering, under in-situ atmospheric conditions of temperature and humidity. There, the highly irregular shape of mineral dust particles is accounted for by carefully analyzing the polarization state of the electromagnetic radiation, allowing to evaluate the linear depolarization ratio, after robust calibration of the polarization detector [7]. Indeed, as

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explained in textbooks on light scattering [8], the polarization state of the laser is not preserved when non-spherical particles are involved in the scattering event, leading to linear depolarization, an optical signature of particles deviation from isotropy. Hence, if a sufficiently high sensitivity is achieved on the polarization lidar detector, the complex vertical layering of mineral dust can be revealed in the free troposphere, even when mineral dust are involved in two [9-11] or three component particle external mixtures, as we published [12]. However, the downside of such field measurements is that mineral dust particles cannot be identified with a sole polarization lidar backscattering experiment, as the measured depolarization is not mineral dust particles specific but dedicated to particles mixtures [11]. In this context, controlled laboratory measurements are interesting as they allow studying the backscattering property of a determined ensemble of mineral dust particles, which may provide accurate inputs to better constrain lidar inversions [11, 12].

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Light backscattering is also important from a more fundamental point of view, as it may lead to the so-called coherent backscattering effect [8]. Also, as noted in [13], the backward scattering direction has been identified as one of the most sensitive directions to the particles heterogeneities and surface structure, including possible surface roughness. Hence, as discussed in [14] through numerical simulations, the most prominent effects of surface roughness are seen close to the exact backscattering direction and the diagonal scattering matrix elements are affected the most in the backscattering direction. We could then investigate the influence of surface roughness on lidar backscattering profiles [15]. Also, near the backward scattering direction, light scattering numerical simulations exhibit a narrow double-lobe feature when studying small-scale surface roughness [16]. Hence, studying near and exact backscattering by mineral dust appears very interesting. In that context, Hovenier and Guirado demonstrated in [17] that the slopes of the scattering matrix elements should tend towards zero when the direction of the scattered light tends towards the π-backscattering angle. This theoretical result was established by introducing an extended scattering matrix, based on symmetry arguments.

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The above recent advances highlight the importance of studying near and exact backscattering of light by mineral dust in laboratory. Light scattering numerical simulations, though they are more and more accurate, rely on several assumptions that should be validated through accurate controlled-laboratory experiments. This is especially true for mineral dust particles which are complex-shaped particles for which there is no analytical solution to the macroscopic Maxwell equations [18]. The literature dedicated to light scattering measurements by mineral dust particles in laboratory is abundant with pioneering work by West et al. [19], then a major contribution by Volten et al. [20] who studied mineral dust scattering matrix elements at two wavelengths over a scattering angle range from 5° up to 173°, then extended from 3° up to 177° by Munoz et al. [21], as referenced in the widely used Amsterdam light scattering database [22]. Since that date, several authors also contributed to analyze light scattering by mineral dust particles in laboratory, such as Glen and Brooks [23] or Järvinen et al. [24]. Despite these major advances, to our knowledge, mineral dust light scattering experiments do not cover the exact backward scattering direction so that as underscored in [25], “the phase matrix can be obtained only at specific wavelengths in limited angular scattering regions, for example from 3° to 177°”, though depolarization has been measured for large-size ice crystals [26] and backscattering patterns have been recorded for single trapped volcanic ash particles [13]. Hence, light scattering measurements are lacking at near and exact backscattering for an ensemble of mineral dust particles. Recently, Videen et al. [27] discussed on the interpolation of light scattering responses from irregularly-shaped particles and noted that the greatest discrepancy between the experiment and the modelled data occurs in the backscatter region. Hence, measurements of the scattering matrix for very large scattering angles are coveted [28]. In the absence of such light scattering measurements at large scattering angle, extrapolations have been performed to obtain data over the entire scattering angle range from 0° to 180°, as required for accurate radiative transfer calculations. Examples of such extrapolations are given in the literature [29-32].

ACCEPTED MANUSCRIPT Though a synthetic scattering matrix has hence built with major benefits [32], the added data points are artificial and the assumptions inherent to these extrapolations should be precisely checked, as analyzed by Liu et al. [29] and more recently by Huang et al. [25]. This requires to increase the accessible range of laboratory light scattering experiments to cover the gap from 177° up the π-backscattering angle with a high angular resolution (better than 1°) for an ensemble of mineral dust particles.

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In this paper, we based our experimental work on the theoretical result established by Hovenier and Guirado [17]. A new light scattering experimental set-up has been designed and built to enlarge the accessible range of light scattering measurements from 176.0° up to 180.0° with a high angular resolution of 0.4°. This task is challenging as the angular resolution is high (even equal to 0.2° between 179.2° and 180.0°). Otherwise, the involved scattering cross-sections are rather moderate, due to the size of the generated dust samples, which lies in the range up to a few micrometres, to be representative of mineral dust after long-range transport. Also, backscattering from non-spherical particles is generally weaker than for spherical particles [33]. In [34], we reached a first step towards experimental verification of Hovenier‟s theorem by addressing exact light backscattering by mineral dust particles in laboratory. The present paper lies in complement to this work, by precisely evaluating the ( )/ ( ) of the scattering matrix elements of mineral dust particles in the ratios ( ) far-field single scattering approximation to compare our laboratory findings with numerical simulations.

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The paper is organized as follows. The principle of the experiment at large scattering angles (from 176.0° up to 180.0°) is presented in Section 2. The methodology to evaluate the ratios ( ) ( )/ ( ) of the scattering matrix elements in laboratory is detailed and the detected scattered light intensity is expressed as a function of ( ) in the framework of the scattering matrix formalism, applied at near and exact backscattering angles ( ). The experimental set-up is presented in Section 3 by detailing how to cover the scattering angle range from 176.0° up to 180.0° with a high angular resolution. To allow precise evaluation of ( ), special care has been taken to precisely analyse the modification of the polarization of the incident and scattered radiations by all the optical components, to reveal the modification of the polarization during the (back)scattering event. Our set-up is then validated in Section 4 on spherical particles, which follow the Lorenz-Mie theory [35]. The scattering matrix elements of Arizona Test Dust (ATD) particles are then evaluated from scattered light intensity laboratory measurements carried out between 176.0° and 180.0° with 0.4° angular resolution (0.2° angular resolution between 179.2° and 180.0°). Moreover, these laboratory findings are found in good agreement with T-matrix numerical simulation, showing the ability of the spheroidal model to describe light scattering by mineral dust also from near up to exact backscattering angle. We believe this result may be used for further interpreting light scattering experiments involving mineral dust particles.

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2. Light scattering at near and exact backscattering angle 2.1 Theoretical considerations We here consider elastic light scattering at wavelength λ by an ensemble of particles with complex refractive index , which are not static but move in an unbounded host medium such as ambient air. The scattering event is described in the framework of the scattering T matrix [ ( )], relating the incident and scattered Stokes vectors ( ) which describe the polarization state of the incident and scattered light :

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where is the distance from the particles ensemble to the detector and the and subscripts refer to the incident and scattered radiations. The first Stokes component corresponds to the light intensity, and to the linear polarization, while deals with circular polarization. The scattering geometry is defined by the scattering angle ( ) between the wave-vectors of the incident and scattered radiations. These wave-vectors define the scattering plane, used as a reference plane for defining the Stokes vectors of both incident and scattered waves. Assuming that particles are randomly-oriented, when particles present a plane of symmetry and / or when particles and their mirror particles are present in equal number [36], the scattering matrix [ ( )] simplifies to a block-diagonal matrix:

] ( ) ( )

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The dimensionless scattering matrix elements ( ) depend on the scattering angle , on the particles size, shape and chemical composition, and have no azimuthal dependence since the particles form a macroscopically isotropic and mirror-symmetric scattering medium. ( ) As detailed by Van del Hulst [37], in the backward scattering direction , ( ) while application of the backscattering theorem [37] leads ( ) ( ) ( ), as detailed in [36]. Hence, since ( ) ( ) to , the backscattering matrix [ ( )] only has two non-vanishing elements, namely ( ) and ( ) ( ) ( )

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At scattering angle , the scattering matrix elements can be normalized with respect to ( ) at the same scattering angle as follows: the scattering phase function

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As explained in [38], the scattering matrix ratios exact backscattering angle, we hence get: ( ) ( ) ( )

( )

( )

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Moreover, for spheres, ( ) [39] so that ( ) angle, ( )

( ) ( ) hence at most equal unity. At specific ( ( (

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( ) ( ) ( ) and ( ) whatever the scattering angle ( ). Hence, for spheres at backscattering and ( ) ( ) .

2.2 Methodology To evaluate the ratios ( ) of the scattering matrix elements at near ( ) and exact ( ) backscattering, we apply the methodology described in Fig. 1. The exact

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backscattering geometry ( ) is achieved as we previously published [34, 40] by precisely aligning (1 mm out of 10 meters) a retro-reflecting polarizing beamsplitter cube (PBC) along the z-optical axis from a laser source to the particles scattering volume. The methodology to ( ) ( )⁄ ( ) is described in our previous contributions [34, 40]. evaluate the ratio In a few words, in the exact backward scattering direction, stray scattered light from optical components can be significant as noticed by [13, 28] and to overcome this difficulty, the particles backscattering radiation is discriminated from background stray light by achieving time-resolved measurements synchronized with the laser pulse, to address the time-of( ) is flight taken by the laser pulse to reach the detector after the scattering event. precisely evaluated by measuring the backscattered light intensity for a complete rotation of a quarter-wave plate (QWP), to gain in accuracy, as detailed in Section 2.4 below.

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In this paper, we complement our existing backscattering experimental set-up to allow ( ) ( ) ( ) ( ), ( ) at nearevaluating the different scattering matrix ratios backscattering angles ( ) by achieving light scattering measurements for a set of incident polarization states, namely ( ), as explained below in Section 2.5. The scattering angle is varied by modifying the wave-vector of the incident radiation, while the wavevector of the scattered radiation remains identical to that used in the exact backscattering experimental set-up, as schemed in Fig. 1. In this way, the same light detector is used for evaluating the scattering matrix ratios ( ) at both near ( ) and exact ( ) backscattering angles, which minimizes biases in the ( )-evaluation. As detailed in ( )/ ( ) of scattering matrix elements are evaluated at Section 2.5, the ratios ( ) ) by measuring the detected scattered light intensity for a complete near backscattering ( ( )-evaluation. The rotation of a quarter-wave plate (QWP), to gain in accuracy in the ( ) is provided from 176.0° up to 180.0° with 0.4° angular resolution by evaluation of precisely rotating a mirror (not represented in Fig. 1, see Section 3.1), allowing to modify the wave-vector of the incident radiation by 0.2° steps, which in turn varies the scattering angle ( ).

Fig. 1 Principle of the laboratory experimental set-up to evaluate the ratios ( ) ( )/ ( ) of the scattering matrix elements at near ( , in green) and exact ( , in blue) backscattering angles for particles in ambient air. ( ) is evaluated at near and exact backscattering angles with the same polarization detector, composed of a quarter-wave plate (QWP), a retro-reflecting polarizing beamsplitter cube (PBC) and a light detector (PM), by measuring the scattered light intensity ( ) for a set of incident polarizations ( ). The scattering angle ( ) is varied from 176.0° up to 180.0° with 0.4° angular resolution by precisely rotating a mirror (not represented in the figure, see Section 3.1 for details), thus varying the incident wavevector while the light detector is left at a fixed position, set to the exact backward scattering direction ( ), as previously published [40]. For the sake of clarity, we mention that is the angle between the fast axis of the QWP and the ( ) laser

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scattering plane, counted counter-clockwise for an observer looking from the PBC to the particles. The particles are located at a = 5 meters distance, to allow timeresolved light (back)scattering measurements. The ( ) polarization components are defined with respect to the ( ) laser scattering plane.

ACCEPTED MANUSCRIPT 2.3 Detected scattered light intensity )

at near and exact backscattering (

Basically, the Stokes vectors ( ) and ( ) of the incident and detected radiations are ( ) of the set-up and ( ) ( ) ( ). According to Fig. related with the Mueller matrix 1, the Mueller matrix of the set-up is equal to ( ) where ( ) is the scattering matrix given by Eq. (2) for and by Eq. (3) for . At scattering angle , the detected scattered intensity is then given by the first component of ( ): )

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Where is the electro-optics efficiency of the detector, is the incident laser power and ( ) = [1, 0, 0, 0] is a projection unitary raw vector. To gain in accuracy in the ( )-evaluation, the scattered light intensity is measured for different positions of the QWP, over a complete rotation of QWP ( -modulation angle). As detailed in Appendix A, the corresponding expression of ( ) can be obtained by developing the Mueller matrices of the PBC and the QWP [41]. After a few calculations detailed in Appendix A, we get for the detected scattered ) light intensity at near and exact backscattering angles ( (

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where , and determine the Stokes vector of the incident radiation ( ) (values of , and are given in Section 2.4 and 2.5). In Eq. (7), to ease the reading, the dependence of the scattering matrix elements with the scattering angle has been omitted and ( ) is the detected scattered phase function, as being proportional to .

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2.4 Scattering matrix elements at exact backscattering (

( )

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At exact backscattering angle ( ), due to the co-axial geometry, the incident Stokes vector is determined by the combination of the PBC and the QWP and expresses as ( ) = [1, ( ), ( ) ( )]T. As detailed in Appendix A, by combining Eq. (3) and (7), we get for the detected backscattered intensity:

where the coefficients ( )

( ) ( )

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depend on the backscattering matrix elements ( ))

(9-a) (9-b) (9-c)

( ) ( ) (Eq. 5-a), ( )-function is expected to be Since and the ( ) ( ) periodic. The periodicity of is then an experimental means to verify that ( ). Moreover, since ( ) ( ) (Eq. 5-b), ( ) ( ) ( ) and ( ) , so that following Eq. (8), the -ratio can be determined ( ) can be precisely from the ( )-variations independently on ( ). As a result, evaluated from the -ratio as published [34]: ( )

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2.5 Scattering matrix elements at near backscattering (

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Interestingly, the same formalism can be applied to near-backscattering angles ( ). Following Eq. (7), and as detailed in Appendix A, the detected scattered intensity can then be written as follows: ( )

(

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( ), ( ) and ( ) at As a result, to evaluate the diagonal scattering matrix elements scattering angle , three different laser polarizations must be considered: (or ) polarization (linear in the laser scattering plane, corresponding to ) to determine ; + (or -) polarization (linear at 45° from the scattering plane, corresponding to ) to determine ; (or )-polarization (right or left circular, corresponding to ) to determine . Table 1 provides the values of the to coefficients for each incident laser polarization ( ). Table 1 ( ) and ( ) can be determined either from ± or shows that -polarizations. )

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Tab. 1 Expression of the coefficients the to appearing in the expression of the scattered light intensity in Eq. (11) as a function of the scattering matrix elements ( ) ( ) ( ) for incident polarization states ( ). The dependence of with scattering angle has been omitted to ease the reading.

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Following Eq. (11), Fig. 2 displays the variations of ( ) for each incident laser polarization in two case studies: a first set of -values represented in full lines, corresponding to spherical particles with then a second set of values displayed in dashed-lines corresponding to non-spherical particles with . To better envision the modification of ( ) when , and are modified, in both case studies. According to Eq. (11), the periodicity of ( ) is determined by the coefficient and equals for and polarizations curves ( ) then for other incident polarizations ( ). Moreover, opposite -phase variations are obtained in Figure 2 when incident polarization states ( -, ) are considered instead of ( +, ), as expected from Table 1 where and coefficients are then changed to their opposite. We then concentrate on Fig. 2(a) to (c). The sensitivity of ( ) to particles deviation from sphericity ( ) is seen in Fig. 2(a) where the scattered intensity ( ) exhibits several minima equal to , to be related to particles deviation from isotropy or linear depolarization. The sensitivity of ( ) to a modification in and is seen in Fig. 2(b) by comparing the full ( ) and dashed-lines ( ) curves, which exhibit different secondary maxima and minima. The sensitivity of ( ) to is seen in Fig. 2(c) where the minima are equal to and related to circular depolarization. These latter minima are not null, even for spherical particles

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(full lines), since 2.1.

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Fig. 2 Numerical simulation of the particles near backscattering intensity ( ) as a function of the -modulation angle of the QWP, obtained by applying Eq. (11) for , + and laser incident polarization (to be seen from panels (a) to (c)), then and laser incident polarization (from panels (d) to (f)), leading to opposite -phase variations. Two case studies are considered: spherical particles (in full-lines), with , then non-spherical particles (in dashedlines), with . In both case studies .

) ) ) ) )

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Moreover, the ( )-variations can be adjusted with Eq. (11) to evaluate the coefficients to at scattering angle . According to Eq. (11), ( ) is proportional to ( ) so that the to coefficients are determined with the scattered phase function ( ) as a pre-factor. As a consequence, the scattering matrix elements ( ) can be evaluated at near backscattering angle from Eqs. (11) and (12) if expressions involving only the ratios of to coefficients are used. As a result, the ratios of scattering matrix elements at are given by: ) ) ) ) )

(

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(13-a) (13-b) (13-c) (13-d) (13-e)

) respectively refer to the value of the coefficients where subscripts ( to when considering ( ) incident polarization states. To gain in accuracy in the ( ) evaluation, the -polarization, which exhibits largest scattered intensities in Fig. 2, has been ( ) and ( ). Also, analogous expressions of ( ) can be obtained if used to evaluate ( ) polarization states are considered for the incident electromagnetic radiation.

3. Near and exact backscattering laboratory experiment

ACCEPTED MANUSCRIPT 3.1 Experimental set-up: the + -polarimeter

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Light scattering at near and exact backscattering angles ( ) is carried out with an home-built + -polarimeter shown in Fig. 3, which complements the home-built -polarimeter built in our previous contributions [34,40]. The + -polarimeter is composed of a pulsed laser source (to discriminate particles scattering from ambient air scattering) and a polarization sensitive light detector, composed of a QWP, a retro-reflecting PBC and a light detector. The laser source is a doubled Nd:YAG laser source, emitting laser pulses of = 7 ns duration at a 1 kHz repetition rate, at wavelength 532 nm, as in lidar remote sensing experiments [42,43]. The backscattering geometry is achieved by precisely inserting the retro-reflecting PBC on the optical pathway from the laser source to the particles scattering medium with a high accuracy (0.1 mm.m-1). The exact backscattering angle ( ) is then covered with accuracy, the value of 0.2° corresponding to the field of view ( ) of the detector [34,40]. To allow light scattering experiments at near backscattering angles, a rotating mirror ( ) is used to vary the scattering angle from 176.0° up to 180.0°. Indeed, as shown in Fig. 3, precise rotation of this mirror by an angle modifies the scattering angle . The downside of such methodology is that the reflectivity of the rotating mirror ( ) depends on the angle of incidence, so does the laser power incident on the ( ) ( ) ( ) cannot be particles beam. As a result, the angular dependence of ( )-variations easily plotted since cannot be easily distinguished from variations in the incident laser power when rotating the ( ) mirror. The near and exact backscattering experiments are indeed not performed simultaneously, otherwise both time-of-flights could not be distinguished as the same light detector is used in both geometries. Hence, plotting ( ) will demand modifications of the set-up, as explained in the outlook sectionAs shown in Fig. 3(a), ( ) is mounted on a high-precision translation stage (in light grey) along the -axis, allowing to set ( ) at the position ( ) corresponding to light scattering at angle . The translation stage slides on a very rigid bench (in dark grey) and special care has been taken to precisely position this bench at 90.0° from the z-axis. As a result, the scattering angle can be set with 0.1° accuracy, higher than the detector of Hence, the ( )evaluation has been carried out with 0.4° angular steps, corresponding to .

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After retro-reflection on PBC, the -polarized component of the detected scattered light intensity is measured at near and exact backscattering angle ( ) for each polarization ) of the incident electromagnetic radiation, precisely set with state ( , ( )-evaluation, the and as explained in Section 3.2. To gain in accuracy in the scattered light intensity is measured for a complete rotation of the QWP, leading to observed variations similar to that displayed in Fig. 2. Polarization cross-talks are fully negligible since a secondary PBC is inserted in the detector, while a well-specified interference filter at λ = 532 nm is used to specifically address elastic light scattering [7]. Likewise, special care has been taken to preserve the polarization state of the radiation before and after the scattering event by carrying out the light scattering experiment in ambient air, thus avoiding the use of chamber walls and windows, with inherent AR-coatings, that may provoke a strong scattering signal or/and modify its polarization. The particles are not static but move and are injected with a nozzle in the scattering volume (2 mm wide diameter, 2 mm large) after their generation with a commercial atomizer for spherical water droplets and a commercial dust generator for non-spherical particles. As a case study for non-spherical particles, Arizona Test Dust (ATD) particles are used as a proxy for mineral dust. The corresponding size distributions are measured from 10 nm up to 10 micrometres with a particle sizer, as schemed in Fig. 3(b). The particles sizer is composed of a Nano-Scan SMPS (TSI3910) and an optical particle counter (TSI, OPS3330). As in every light scattering experiment, particles scattering is evaluated by subtracting the detected scattered intensity in the presence of the particles from its background value in their absence. At a given scattering angle , the light intensity is recorded as a function of the

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). To gain in QWP -modulation angle for each incident polarization state ( accuracy, a complete -rotation is performed and the time duration of such a rotation is about 3 min (the angular resolution is 5° and each acquisition point represents an average over 2 500 lasers shots). For precise time-of-flight experiments with ns-pulsed laser, the light intensity ( ) is recorded as a function of time with a 12-bits oscilloscope, presenting a 2.5 GHz sampling rate. Finally, to account for the amount of light scattered during the whole laser pulse duration, the light intensity ( ) is integrated over the time duration of the ns-laser pulse. This experiment is then repeated over a complete scattering angle range from to with a 0.4° angular step. Moreover, between 179.2° and 180.0°, the angular resolution is increased to a 0.2° to provide more experimental data points in the vicinity of the -backscattering angle.

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( ) ( ) ( ) of the Fig. 3 Experimental set-up for evaluating the ratios scattering matrix elements at near and exact backscattering angles from 176.0° up 180.0° with 0.4° angular resolution. The laser beam and the detector are identical to that used in our previous publications dedicated to exact backscattering [40]. The scattering angle is modified by rotating the mirror ( ) by with a precise rotation stage (0.1° accuracy). The reflectivity of the mirror ( ) being higher for the polarization state, the incident radiation on mirror ( ) has for polarization state. , and respectively stand for polarizing beamsplitter cube, halfwavelength plate and quarter-wave plate. The position of the neutral lines of and have been precisely determined before the light scattering experiment.

3.2 Accuracy on the evaluation of

( )

( )

( )

Statistical errors are not a main concern in the + -polarimeter where each acquisition is an average over several thousand laser shots and statistical noise is mainly due to photon noise (shot and electronic noise are fully negligible). Systematic errors in the + -polarimeter are similar to that encountered in a polarization lidar experiment [7]. Such systematic errors originate from an imperfect definition of the polarization state of the incident radiation and potential misalignment between the emitter and the detector polarization axes. Also, potential

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(a) Polarization state of the incident radiation

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According to Eq. (12), the coefficients to as given in Table 1 are modified when the polarization state of the incident radiation differs from ( , s, +, -, ). Slight deviations may occur in these coefficients in our light scattering experiment since, according to Fresnel‟s formula, the reflectivity of the rotating mirror ( ) depends on the polarization state of the incoming radiation as well as on the angle of incidence. This systematic effect is hence more pronounced at 176.0° since ( ) has its highest reflectivity at 45° incidence corresponding to the -angle. To be quantitative, we analyzed the modifications induced on ( ) by potential deviations from polarization states ( , s, +, -, ). From textbook knowledge on polarization [41], the incident Stokes vector ( ) can be represented by a ( ) point on the Poincare sphere described by its longitude and latitude , corresponding to ellipticity. If and quantify the deviation from polarization states ( , s, +, -, ), the modified expressions of the coefficients to are gathered in Table 2, at first order in and . Table 2 is hence the analogous of Table 1 when and are not null. )

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Tab. 2 Same as Table 1 except that and , corresponding to slight corrections in the incident Stokes vectors are not null (at first order in and ). and are ( ) respectively the longitude and the latitude on the Poincare sphere and corresponds to ellipticity. As an example, for -polarization, the incident Stokes vector T ( ) ( ) ( ) ( ) ( ) is [ , , )] . For 45+ polarization state, it is (the position of neutral lines of and , which have been precisely determined, does not introduce any imperfection).

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Interestingly, , , and are independent on and : they were then chosen in Eq. (13) to provide the expression of ( ). If other coefficients are affected by and , a secondary after , was added before setting the polarization to ( ) with or/and , which substantially reduces and . For scattering angles very close to -angle (above 179.2°) however, due to the lack of space, a secondary could not be added after , which results in larger error bars in the angular scans ( ) presented in Section 4. To be quantitative, the longitude and the latitude were determined from Table 2 by noting that ( ) ( ) while ( ). Hence, though incident ( ) ) allow ( )-evaluation by following Eq. (13), incident polarization states ( ) are useful for determining polarization states ( and . Moreover, following Eq. ( ) is at most equal to (13) and Table 2, we may conclude that the error bar on (in ( ) ( ) ( ) is respectively equal fact , but ). Likewise, the error bar on to . Finally, the periodicity of ( ) for -polarization can be used as an experimental criterion to discuss on the nullity of and : if either or were not null, would no longer vanish as is the case in Table 1, and the periodicity of ( ) would then no longer be equal to but equal to .

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As we discussed in [7], to calibrate polarization lidar remote sensing experiments, any mismatch between the polarization axes of the incident and detected radiations may lead to quite considerable errors when interpreting the polarization of the detected backscattered radiation. Similarly, accurate determination of ( ) can only be obtained if the -polarization axes of and are perfectly aligned. The offset angle which a priori exists between these two -axes can however be accurately determined by rotating in the absence of . As detailed in Appendix B, in this configuration, the detected scattered intensity is given by: (14)

where is a modulation angle describing the rotation of . As a result, can be ( ), accurately determined by adjusting the variations of as explained in Appendix B where an example of -retrieval is given. Moreover, by rotating by , we can compensate for the offset angle to ensure that the -polarization axes of the incident and scattered radiation be perfectly aligned.

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To account for potential fluctuations in the particles generator, a light scattering detector has been added to our experiment, located at scattering angle ( apart from -optical axis). This light detector being polarization insensitive, the corresponding detected ( ) ( ) is proportional to light intensity is equal to where ( ) and hence to the particles number. As a result, the detected light intensity can be used to normalize the scattered intensity when is null, which is the case in our experiment since is very close to zero (see Section 4), and, otherwise, for ( )polarizations. Moreover, this normalization is very precise since and are recorded for each -modulation angle of QWP.

4. Results and discussion

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The methodology described in Section 2 is applied to the experimental set-up presented in Section 3 to evaluate the ratios of scattering matrix elements of spherical particles (to validate our experimental set-up using Lorentz-Mie theory), then non-spherical Arizona Test Dust (ATD)-particles, from 176.0° up to 180.0° scattering angle with 0.4° angular resolution (and even 0.2° between 179.2° and 180.0°).

4.1 Validation of the experimental set-up on spherical particles

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Figure 4 presents the recorded evolution of the scattered intensity ( ) for spherical particles at scattering angles = 176.4° and = 178.4° for incident polarization states ( ). Each polarization curve ( ) contains 72 experimental data points corresponding to the complete rotation of QWP with a 5° angular resolution for -modulation. The curves in Fig. 4 are similar to that obtained in Fig. 2 (in full lines), however shifted by , corresponding to the position of the QWP fast axis on its rotating mount. As expected, the periodicity of ( ) in Fig. 4(a), related to –polarization, is equal to at both scattering angles showing that and are very close to zero. Indeed, we evaluated (resp. 0.013) and (resp. 0.014) at 176.0° (resp. at 178.0°). As expected, Fig. 4(a) exhibits ( ) ( ) null minima, equal to , proving that ( ) at both scattering angles. The curves in Fig. 4(a) are hence almost superposed since and then only depend on which is very close to zero. Differences occur in the scattered intensity at scattering angles = 176.4° and = 178.4° when considering the and –polarization

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curves (see Fig. 4(b,c)): over a 2°-variation of the scattering angle, large variations in the detected scattered intensity ( ) are observed due to variations in and coefficients, showing the sensitivity of our experimental set-up. In Fig. 4(b), we can visualize the modification of ( ) and ( ) between = 176.4° and = 178.4°. Likewise, the modification of ( ) and ( ) between = 176.4° and = 178.4° is visualized in Fig. 4(c), related to circular –polarization. The minima of the –polarization ( ), tend towards zero when approaching the curves, which are equal to backscattering angle. These minima allow to experimentally study how the scattering matrix ( ) ( ) element ( ) evolves when the scattering angle approaches the backscattering angle. Besides, when approaching the -backscattering angle, ( ) becomes closer to zero (to reach zero at -angle), which in turn modifies the curvature of ( ) for ( ) is then modified.

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Fig. 4 Measured light scattered intensity ( ) (normalized with ) for spherical particles at scattering angles (in blue) and (in green), for successive incident polarization state ( ), seen from panels (a) to (c). The scattered intensity, recorded for a complete rotation of QWP corresponding to 72 experimental data points, is adjusted with Eq. (11) to determine the coefficients to with a high precision (R² = 99 %), allowing a precise evaluation of ( ) by applying Eq. (13).

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The Fig.4 laboratory-measured data points nicely adjust with Eq. (11) to allow precise evaluation of the coefficients to , providing the ratios of scattering matrix elements ( ) ( ) ( ) by applying Eq. (13). By repeating this light scattering experiment from 176.0° up to 180.0° with 0.4° angular resolution (0.2° between 179.2° and 180.0°), we retrieve the values of ( ) displayed in Fig. 5(a) to (e) with corresponding error bars. Fig. 5 shows the ability of our experimental set-up to precisely evaluate variations in the ratios ( ) of scattering matrix elements. In particular, large variations in ( ), ( ) and ( ) are observed for the generated size distribution of spherical particles. Within experimental error bars, Fig. 5 exhibits the expected behaviour for the ratios ( ) as a function of the scattering angle for spherical particles. Indeed, in agreement with Eq. (5-a) and our previous contribution ( ) [34], we precisely evaluate . Moreover, within our experimental error ( ) bars, we may conclude that and ( ) ( ), as expected for spherical particles. Besides, since our experimental set-up operates in the far-field single scattering approximation [34,44], our laboratory findings can be compared with the analytical LorenzMie theory [35]. For that however, the refractive index and the size distribution of the generated particles must be precisely known. In our case study, the generated spherical particles are spherical water droplets mixed with NIST-calibrated spherical polystyrene latex spheres (PSL) and the generated size distribution (SD) is shown in Fig. 5(f). The benefit of using NIST-calibrated spheres mixed with spherical water droplets is that the backscattered intensity is enhanced, compared to an ensemble of dry PSL-sphere: within our experimental set-up, removal of water with a diffusion dryer led to undetectable light scattering intensities

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due to particle losses in the diffusion dryer. Adding more dry-PSL spheres would have led to increasing surfactants effects, which are undesirable, as explained below. The sphericity of the particles is nevertheless the important parameter to validate our laboratory polarimeter. The corresponding output of Lorenz-Mie numerical simulation is shown in dashed lines for each scattering matrix element ( ) in Fig. 5(a) to (e). The agreement is excellent for ( ) and rather good in other panels, except for ( ) and ( ) around 179.5°. Several arguments may however explain the observed discrepancies. Firstly, we should keep in mind that, due to the FOV of the detector, an uncertainty in the scattering angle exists and is equal to . Otherwise, the size distribution displayed in Fig. 5(f) is not error-free. A better characterization could be performed with an aerodynamic particle sizer but none was available in our laboratory. The optical particle sizer (OPS) being calibrated on PSL-spheres, the accuracy on the size measurement for the mixture of water droplets with PSL has to be discussed. The refractive index of water being smaller than that of PSL, the optical diameter is less than the volume equivalent diameter (Chien et al., 2016). These authors then conclude that “the magnitude of erroneous sizing is prominent for larger particles (above 4 microns) with irregular morphology” while we here deal with below 1 micrometer spherical water droplets (see first two modes in Fig. 5(f)). Moreover, to quantitatively retrieve the size of the generated water droplets, the refractive index of water was entered in the algorithm provided by the OPS-constructor for the first two modes of the size distribution plotted in Fig. 5(f) with particles diameter below 1 micrometer, corresponding to spherical water droplets.Moreover, in the literature, several values are given for the refractive index of PSL in suspension at 532 nm wavelength, as gathered in Miles et al. [45]. Otherwise, inconsistencies with Mie theory were observed by Singh et al. [46] when comparing the measured extinction cross-sections by CRDS on PSL with Mie theory, that were attributed to coating of surfactants. Indeed, PSL beads are embedded in a surfactant whose refractive index is unknown. The PSL beads may then agglomerate and the modification of the scattering matrix elements due to the possible presence of such agglomerates is not taken into account in our numerical simulation. For more precise comparison with Mie theory, we should then draw a core-shell surfactant-PSL particles model, which is however out of the scope of this work.

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( ) ( ) ( ) of the scattering matrix elements for scattering Fig. 5 Ratio angles from 176.0° up 180.0°, for spherical particles (Polystyrene latex spheres mixed with water droplets) from panels (a) to (e). The error bar on the scattering angle is (not represented to ease the reading), limited by the field of view of the detector (though the scattering angle is set with 0.1° precision, see Section 3.1). The error bars ( ) are retrieved as explained in Section 3.2; The uncertainties in the on coefficients to originating from the adjustment of ( ) over 72 points per polarization curve are negligible (R² = 99 %). Error bars above 179.2° are larger, as explained in Section 3.2. In panel (f) is also shown the size distribution of the generated spherical particles. The experimental points are adjusted (see dashed lines) with Lorenz-Mie theory, applicable to homogeneous spherical particles.

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Interestingly, when approaching the exact backward scattering angle, ( ) and ( ) tend towards their value at exact backscattering, equal to zero. Within our experimental error bars, Fig. 5 shows that the slope of these coefficients tends towards zero when approaching the exact backscattering angle. In other words, the ( ) curves tend to become parallel to the -axis when tends towards . This is especially true for ( ) which clearly reveals this behaviour and obviously true for ( ) whose slope is zero whatever the scattering angle. To our knowledge, this is the first time that this theoretical consideration set by J.W. Hovenier [17] is experimentally checked. For and however, within our experimental error bars, this behaviour is less pronounced and to draw a similar conclusion on these coefficients, more measurements should then be performed by adding a secondary PBC to reduce our experimental error bars.

4.2 Light scattering by mineral dust particles from 176.0° up to 180.0°

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Addressing light scattering by mineral dust particles near backscattering is much more difficult since backscattering from non-spherical particles is generally weaker than for spherical particles [33]. Another difficulty lies in the generated mineral dust samples which should be highly reproducible in number and size all during the light scattering experiment. For that, a commercial dust generator (SAG410) has been used to provide a constant dosing of desert dust particles by continuously loading a moving toothed belt, presenting defined spaces between the teeth to ensure a reproducible supply of powder to a disperser. As explained below, we could then generate mineral dust samples presenting a reproducible size distribution, using Arizona Test Dust (ATD) as a proxy for mineral dust. More precisely, ISO 12103-1 A1 ultrafine Arizona Test Dust particles are used, provided by Power Technology with size specification (http://www.powdertechnologyinc.com). The chemical composition of the dust samples was provided by the manufacturer and was confirmed with chemical colleagues, as we published [2].

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Figure 6 displays the variations of the light intensity ( ) scattered by the generated ATDparticles at scattering angles = 176.0° and = 178.0° for incident polarization states ( ). As for spherical particles, 72 experimental data points are acquired per polarization curve corresponding to a complete rotation of the QWP to gain in accuracy in the ( ) are -evaluation. We are hence confident that the observed variations of representative of a determined size and shape distribution of the generated mineral dust particles. Indeed, if the size of the sample was varying, the variations of ( ) would not exhibit constant maxima while rotating the QWP. Moreover, by setting the -modulation angle to one of these maxima, we could check that the scattered light intensity remained stable in time for a few hours (as long as the generator was fed with dust powder). Likewise, the shape of the generated samples is not varying, otherwise in Fig. 6(a), ( ) would exhibit varying minima (these minima are related to particles non-sphericity, as being equal to ). It is hence clear from Fig. 6(a) that mineral dust are non-spherical particles since the minima of ( ) are not null, at both scattering angles. As a result, ( ) differs from unity. As for spherical particles however, at both scattering angles, the periodicity of ( ) is equal to for the -polarization state, as expected since and are very close to zero. Evaluation of 2 and leads to 2 and 0.026 at 176.0° (resp. 0.013 and 0.014 at 178.0°). The detected scattered light intensity is higher at = 176.0° in the three panels since scattered light is reduced when approaching the backscattering angle. In contrary to what was observed in Fig. 4(b), the scattered light intensity exhibits a similar behaviour at both scattering angles in Fig. 6(b), showing that and are almost identical at these two angles. A similar conclusion can be drawn on in Fig. 6(c). Also, in Fig. 6(c), the minima, equal to , are far from zero, as a clear signature of circular depolarization of the generated mineral dust particles.

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Fig. 6 Same as Fig. 4 for mineral dust (ATD), used as a proxy for non-spherical particles.

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By adjusting these experimental data points with Eq. (11), the coefficients to can be retrieved for each scattering angle to precisely evaluate the ratios ( ) of scattering matrix elements displayed in Fig. 7(a) to 7(e), by applying Eq. (13). Within our experimental error ( ) ( ) differs from bars, we can clearly conclude that and ( ) at all scattering angles as optical signatures of linear and circular depolarization from mineral dust particles. At ( ) exact backscattering angle, we evaluate and Fig. 7(c) shows that ( ) ( ) tends towards , as expected when approaching the exact backward ( ) scattering angle. Likewise, ( ) tends towards , as expected. Despite the sensitivity of our apparatus which was demonstrated on spherical particles (on for example), with the generated size distribution of mineral dust to be seen in Fig. 7(f), the scattering matrix elements slightly vary in the scattering angle range between 176.0° and 180.0°. It follows that the slopes of the ratios of scattering matrix elements are rather constant within our error bars. However, the tendency predicted by J. Hovenier [17] is visible of ( ) which varies from -0.23 at 176.0° up to -0.14° at 180.0° with a slope that is varying around 176.0° to become equal to zero close to the -angle. This suggests that, to experimentally check J.W. Hovenier‟s theoretical consideration with more precision, a larger scattering angle range should be used. This is however beyond the scope of this contribution.

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Though the highly irregular shape of mineral dust is very difficult to account for in mathematical models [15], we tested the applicability of the widely used spheroidal model, for the first time to our knowledge in the scattering angle range between 176.0° and 180.0°. The aspect ratios were then distributed either as equiprobable ( ( ) with ) or with a power-law shape distribution, in an attempt to better account for polarization effects [49]. Using for the refractive index of ATD at nm [34], we then applied Mishchenko‟s T-matrix numerical code [50] to retrieve the ratios of scattering matrix elements after size-integration using Fig. 7(f). To derive the size distribution for non-spherical ATD-particles, we took benefit from recent publication by Chien et al. (2016), allowing to correlate optical diameters to aerodynamic diameters. More precisely, Equations (5) and (9) from Chien et al. (2016) were applied, using 1.5 for the shape factor for non-spherical ATDparticles, to convert the mobility diameters (from our SMPS) and the optical diameter (from our particle counter) to volume equivalent diameters, as plotted in Fig. 7(f). Our numerical results are presented in Fig. 7(a) to (e) in dashed lines for the shape distribution and in dotted lines for the shape distribution. Interestingly, within our experimental error bars, the output of T-matrix numerical simulation agrees with our laboratory findings ( ) and ( ). The agreement is for ( ), ( ) and also for a majority of points for better when using the shape distribution rather than the equiprobable shape distribution of spheroids, in agreement with [34], where we observed a similar behaviour at specific backscattering angle. This conclusion cannot however be supported by electron microscopic images, since as underscored by Kahnert et al. [33]: ““single spheroids do not share the single-scattering properties of non-spheroidal particles with the same aspect ratios, so one

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should not think that when using spheroids to mimic scattering by more complex particles, best results would be achieved using aspect ratios of the target particles for the spheroids.” A ( ) even with the slight discrepancy of 0.06 is however observed for shape distribution. As explained in Section 4.1, we should however keep in mind that the uncertainty on the scattering angle is and that the size distribution displayed in Fig. 7(f) is not errorfree. Also, the refractive index of ATD is itself not error-free: its calculation in [34] is based on the Aspens formula which requires prior knowledge of the refractive indices of each oxide present in ATD as well as their volume fractions. To go further in the analysis, it would be interesting to compare T-matrix outputs with laboratory evaluations of the ratios of scattering matrix elements over a larger scattering angle range, starting for example from 165.0° up to 180.0°. This was however not feasible with our apparatus due to the lack of space, but a future experiment may eventually be planned, which is however far beyond the scope of this contribution. Interestingly, our laboratory evaluation of the scattering matrix elements agrees with that computed in the well-known paper by Dubovik et al. [51]: in the near and exact ⁄ backscattering region, his computed phase matrix elements were indeed equal to: ⁄ ⁄ ⁄ ⁄ , , , and . Care should however be taken since scattering matrix elements depend on the particles size distribution and refractive index (from Saudi Arabia in [51]).

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Fig. 7 Same as Fig. 5 for mineral dust ATD-particles. The data point at 179.8° for 45+ polarization (panel (b)) is unfortunately absent as could not be acquired which prevents from normalizing ( ). In panel (f), the size distribution of the generated ATD samples is expressed as a function of the volume equivalent diameter, using [47, 48].

4.3 Atmospheric lidar implications

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As explained in the introduction, light backscattering by mineral dust particles is often addressed in the atmosphere through polarization lidar remote sensing instruments [7-12]. The downside of such measurements is that the measured depolarization ratio is nevertheless that of a mixture which may differ from the depolarization specific to mineral dust particles, as we explained [11]. In a few words, it is only in the absence of detectable spherical particles, that the depolarization of the particles mixture equals that of nonspherical particles [11]. The above laboratory measurements, which are specific to mineral dust, may be used to retrieve the dust particles depolarization by including lidar instruments operating close to the specific -angle (off-axis), which is desirable to better constrain lidar inversions involving mineral dust. Following Mishchenko and Hovenier [36], the linear ( ) and circular ( ) depolarization ratios relate to the laboratory measurements at scattering angle as follows: (15)

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where the positive (negative) sign corresponds to incident p (s) polarization state for and to incident RC (LC) polarization state for . Using Fig. 7(a,c,d), the linear and circular lidar depolarization can then be evaluated from our laboratory measurements. In the exact ) while, using Equation (5-b), we get backscattering geometry ( ), ( ) ( ( ) ( ) . When approaching the -angle, care should be taken as ( ) may ( ) may vary. In our case study however, not be null and is almost constant when the scattering angle varies from 176.0 up to 180.0° and equals to ( ) %, on average over the scattering angle. This should not be however considered as a general established fact since other size distributions and other dust samples may lead potential differences. The circular depolarization ratio exhibits more pronounced variations, due to variations in ( ), increasing from 62.6 % at 176.0° scattering angle up to 75.4 % at 180.0°. Hence, for the first time to our knowledge, within our experimental error bars, we may conclude that the relationship ( ) ( ) ( ( )) only applies at exact backscattering angle, as theoretically set by Mishchenko and Hovenier [36]. Retrieving the lidar ratio would also be beneficial for lidar inversions based on Klett‟s inversion algorithm. Since ( ), such a retrieval would however require prior knowledge of both the extinction cross-section and the scattering phase function ( ), which is beyond the scope of this contribution.

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5. Conclusion and outlooks

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In this paper, a controlled laboratory experiment has been built and implemented to measure elastic light scattering by mineral dust particles from 176.0° up to the backscattering angle with 0.4° angular resolution (0.2° resolution between 179.2° and 180.0°). This + -laboratory polarimeter provides evaluations of the scattering matrix ratios ( ) at near and exact backscattering angles ( ).The scattering angle is varied by modifying the wave-vector of the incident radiation, while the wave-vector of the scattered radiation remains identical to that used in the exact backscattering experimental set-up, allowing the same light detector to be used for evaluating the scattering matrix ratios ( ) at near and exact backscattering angles, which minimizes biases in the ( )-evaluation. As explained in Section 2, our experiment relies on the robust scattering matrix formalism applied at near and exact backscattering angles. We could then relate the scattered light ( ) ( ) ( ) ( ), ( ) of the scattering matrix elements, for intensity to the ratios a set of incident polarization states, namely ( ), corresponding to different Stokes vectors for the incident radiation. The experimental set-up, the acquisition procedure and the systematic biases affecting the ( )-evaluation have been extensively discussed in Section 3. The sensitivity and the accuracy of the newly-built + -laboratory polarimeter have then

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been experimentally studied in Section 4 where the ratios of scattering matrix elements have been evaluated for spherical particles, to validate our set-up which then follows the Lorenz( ) ( )). We here experimentally showed that the slope of the Mie theory ( ( ) ratios of scattering matrix elements tend towards zero when approaching the -angle, as theoretically predicted by Hovenier [17]. Moreover, using Arizona Test Dust (ATD)-particles as a proxy, the ratios of scattering elements of mineral dust particles were evaluated in laboratory from 176.0° up to 180.0° backscattering angle with 0.4° angular resolution (0.2° between 179.2° and 180.0°). With the generated size distribution, we found out that was clearly below 1 (equal to 0.58 on average), that was almost constant and equal to -0.5 ( ) while ( ) started from --0.23 at 176.0° to tend to at the -angle, with a slope progressively tending towards zero when approaching the -angle, as predicted by ( ) and J.W. Hovenier [17]. Likewise, ( ) were almost null over the studied scattering angle range. Moreover, these laboratory findings could be compared with the output of the Tmatrix numerical code. A rather good agreement was found for most scattering matrix ratios ( ), especially when the shape distribution of spheroids was used, which demonstrates the ability of the spheroidal model to describe light scattering by mineral dust also from near to exact backscattering angle. If, despite the sensitivity of our experimental setup, the scattering matrix elements of generated mineral dust were found rather constant over the scattering angle range from 176.0° to 180.0°, this may not be a general established fact: other size distributions of other mineral dust particles may eventually lead to substantial variations. Actually, there is no light scattering database for aerosols in the scattering angle range from 176.0 to 180.0°. In that context, we hope that this work may contribute to enlarge the applicability of existing aerosols database [22], by complementing the existing experimental works and their outcomes. An outlook of this work is to further enlarge the accessible angular range below 176.0° to explore larger variations of scattering matrix ratios ( ) and then more precisely check J.W. Hovenier „s theoretical consideration [17] and numerical simulations. Another outlook would be to address light scattering from 176.0° to 180.0° at two different wavelengths, which may help to better constrain numerical simulations. Also, the experimental set-up should be modified to plot the shape of the scattering phase ( ) ( ), which would then for the first time be defined with respect the function ( ) exact -backscattering angle. Such a quantity would be of interest for dust particles and to interpret the shape of the backscattering peak for asteroids, satellites, and rings, where coherent backscattering is important. Moreover, absolute retrievals of the phase function ( ) of dust would be even more interesting. For that, our light detector should be calibrated to evaluate ( ). Such a robust calibration has been achieved in polarization lidar experiments by introducing controlled amounts of polarization cross-talks on both polarization lidar detector channels as we published [7]. In our laboratory experiment, only the s-component of the backscattering intensity can be measured (the incident radiation on the PBC, which is p-polarized, will saturate the light detector hence preventing from measuring the p-polarization of the backscattered radiation). Modification of the set-up are hence necessary, but such important modifications are beyond the scope of this contribution.”

Appendix A Detected scattered light intensity at near and exact backscattering The goal of this appendix is to present the detailed calculus of the detected scattered light intensity ( ) at near ( ) and exact ( ) backscattering angles. For the sake of clarity, we recall that is the modulation angle describing the rotation of quarter-wave plate QWP, as introduced in Fig. 1. As a starting point, we recall the Mueller matrices of and from [41]. If the imperfections of the PBC are characterized by and that for by , we have :

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where the dependence of the scattering matrix elements with scattering angle has been omitted to ease the reading. The expression of the scattered light intensity at near backscattering, as given in Eq. (7), can then be retrieved by considering the definition of the ( ), after a few incident Stokes vector ( ) . Then, if we linearize calculations, we get the expression of the coefficients to given in Eq. (12) appearing in the expression of the scattered light intensity given in Eq. (11). This equation can be used to derive the backscattered light intensity. In this specific case, the co-axial geometry imposes that the Stokes vector of the incident radiation is ( ) = [1, ( ), ( ) ( )]T, which provides, after a few calculations, the expression of the backscattered intensity given in Eq. (8), with coefficients given in Eq. (9).

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Appendix B

Determination of the offset angle

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The goal of this appendix is to detail the procedure allowing accurate determination of the offset angle which a priori exists between the -polarization axes of (emitter) and (detector), as discussed in Section 3.2(b). To precisely evaluate the offset angle between the -polarization axes of and (see Fig. 3), is removed from the + -polarimeter while is rotated (modulation angle instead of ). The raw vector ( ) ( ) and then defined in Appendix A is then modified as follows: ( ) becomes equal to ( , ). Besides, the incident Stokes vector now corresponds to the combination of and , i.e. ( ) = [1, ( ), ( ) )]T which allows retrieving Eq. (14). As shown in Fig. B, can then be accurately determined from a complete rotation of , since the position of neutral lines of was carefully determined before this light scattering experiment.

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Fig. B Precise determination of the offset angle between the -polarization axes of and (see Fig. 3) with the + -polarimeter. In this example, the scattering angle is equal to 178.0°. Precise adjustment of experimental data points with Eq. (14) provides (178.0°) = -7.59 ± 0.10°. When varying the scattering angle, the position of and are modified so that the offset angle is slightly different, but is accurately determined by following the same procedure.

Acknowledgments

The authors are grateful to M.I. Mishchenko for making his T-matrix numerical code available. CNRS is acknowledged for financial support.

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References

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[2] Dupart, Y., S. King, B. Nekat, A. Novak, A. Wiedensohler, H. Hermann, G. David, B. Thomas, A. Miffre, P. Rairoux, B. D‟Anna and C. George. Mineral dust photochemistry induces nucleation events in the presence of SO2. Proc. Natl. Acad. Sc. 2012; 109; 51: 20842–47, doi: 10.1073/pnas.1212297109.

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