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We present the first experimental observation of exact backscattering of light by an ensemble of particles in ambient air. Our experimental set-up operates in the far-field single scattering approximation, covers the exact backscattering direction with accuracy (θ = π ± ε with ε = 3.5 × 10−3 rad) and efficiently collects the particles backscattering radiation, while minimizing any stray light. Moreover, by using scattering matrix formalism, the observation of the particles UV-backscattering signal allowed to measure the particles depolarization of water droplets and salt particles in air, for the first time, in the exact backscattering direction. We believe this result may be useful for comparison with the existing numerical models and for remote sensing field applications in radiative transfer and climatology.
© 2013 Optical Society of America
1. Introduction
Light backscattering has become an efficient tool for providing information on the optical properties of condensed or gaseous matter, with numerous applications in various research fields, such as biomedicine [1], fundamental research [2] or environmental and climate research [3–6], to quote only a few. Besides its quite simple geometry and its handiness for in situ applications [7], the backscattering direction has raised great interest as it is one of the most sensitive directions to the size and the shape of the sample [8,9]. Moreover, in the single scattering approximation, the polarization property can be used to provide insights on the symmetry of particulate matter, since, for homogenous spherical particles backscattering, the light polarization is preserved [10]. Although more than a century has now elapsed since G. Mie presented this theory [11], even though measurements of water clouds do not contradict the Mie theory, it is surprising that its experimental proof has never been achieved in the exact backscattering direction for an ensemble of particles in air, such as spherical water droplets, while, in the literature, a considerable number of papers apply the Mie theory, as for environmental purposes, such as in remote sensing and radiative transfer applications.
Basically, polarization-resolved backscattering studies are performed in the framework of the Mueller matrix or scattering phase matrix. The observation of exact backscattering (i.e. at the exact backscattering angle θ = 180°) is difficult to achieve, as it requires a high angular resolution (to avoid blocking the incident light with the detector) and a high dynamical range (to discriminate the potentially low backscattering radiation from background stray light). In the literature, these two difficulties have been overcome in experiments related to condensed matter phases, such as solid GaAs crystals [2], solid biological tissues [12,13], liquid water [14], PSL spheres in liquid water [14] or, more recently, for liquid animal blood [1]. However, up to now, to our knowledge, no laboratory measurement exists that covers the exact backscattering angle for an ensemble of particles in air. O. Muñoz and J. Hovenier recently reviewed [15] the existing light scattering laboratory experiments measuring one or more elements of the scattering matrix. The closest value to the exact backscattering direction is θ = 179.6° [16]. To cover the exact backscattering direction, polynomial extrapolations or numerical algorithms have been proposed [17], but their inherent assumptions must be discussed and may lead to quite considerable errors, as recently discussed by M. Schnaiter et al. [18]. Hence, there is a need for laboratory scattering matrix measurements in the exact backscattering direction for an ensemble of particles in air, for at least two reasons. Firstly, it may help validating numerical simulations based on T-matrix or DDA numerical codes, which are never assumption-free, especially when the particles exhibit complex morphologies. Secondly, it may be also useful in passive or active lidar remote sensing field experiments, which operate in the backscattering geometry with narrow field of view and where T-matrix computations have been coupled with polarization lidar [5, 6]. In a recent paper [6], we also demonstrated that, to efficiently partition particles in a two/three-component ambient air mixture with a precise UV-VIS polarization lidar, it is necessary to specifically address the depolarization properties of each particle component, which is feasible if laboratory measurements exist for each particles ensemble in air, that cover the exact backscattering direction. Ideally, such laboratory measurements should operate in the far-field single scattering approximation, to ease the comparison with both the numerical simulations and the field experiments.
In this paper, we propose a new experimental set-up to observe the backscattering radiation for an ensemble of particles in ambient air, in the exact backscattering direction θ = 180°. For the first time to our knowledge, the requirement of a high angular resolution and a high dynamical range of the optical signal is fulfilled for an ensemble of particles in ambient air in the exact backscattering direction. This finding has been achieved by using a pulsed laser source (to discriminate the particles backscattering signal from the background stray light) and a polarizing beamsplitter cube (to retro-reflect the backscattering radiation towards a photo-detector), which provides the exact backscattering geometry after precise alignment. With this set-up, the exact backscattering direction is covered with a scattering angle range of 3.5 × 10−3 radian, namely θ = 180.0 ± 0.2 deg. The UV-light has been chosen to increase our sensitivity to nano-sized particles, in the range of a few cents of nanometers. Such nanoparticles are incriminated as one of the major uncertainties in global warming and climate research [19]. Moreover, by modulating the laser linear polarization, from the observation of the particles backscattering signal, we accurately measured the particles depolarization of an ensemble of water droplets in air, then of an ensemble of salt particles in air. The paper is organized as follows. In Section 2, we present the principle of our exact backscattering measurement and derive the theoretical expression of the particles backscattering signal for an ensemble of randomly-oriented particles in air, under the far-field single scattering approximation. The procedure for measuring the particles depolarization is also detailed. We then present our new experimental set-up in Section 3, by providing justifications for covering the exact backscattering direction and optimizing the collection of the particles backscattering radiation, while minimizing the background stray light. Our experimental observations of the particles backscattering signal are presented in Section 4 for water droplets in ambient air, then for salt particles in ambient air. Then, the particles depolarization is measured, for the first time in the exact backscattering direction. Spherical water droplets were used to test our experimental set-up, while salt particles were presented as an example of non spherical particles. We hence checked that our apparatus exhibits no depolarization for spherical particles: the obtained depolarization was dp = (0.04 ± 0.08) % corresponding to a lidar particles depolarization ratio δp = (0.02 ± 0.05) %. The statistical and systematical errors are then discussed. We believe that this study may be useful for environmental purposes, such as for remote sensing and radiative transfer applications, from both the field application and the numerical simulation points of view.
2. Backscattering of light by particles in ambient air
In this section, we present the principle of an exact backscattering measurement for an ensemble of randomly-oriented particles in ambient air. We then derive the expression of the particles backscattering signal by using Mueller matrices. The principle of the particles depolarization measurement is also described.
2.1 Framework for polarization-resolved backscattering measurements
Polarization-resolved backscattering is generally studied in the framework of the Mueller matrices, in which the Stokes vectors Stinc and St of the incident and backscattered radiations are related by the so-called Mueller matrix or particles scattering phase matrix Fp [20]. In the last two decades, M.I. Mishchenko et al. [21,22] theoretically described the case where particles are not static but randomly-oriented and placed in an unbounded host surrounding medium such as ambient air. In this case, in the single-scattering approximation, the scattering matrix Fp has only six non-vanishing elements, which can be expressed as a function of F11,p, F22,p and F14,p, as detailed in the appendix. The matrix elements F11,p and F22,p are used to measure the particles deviation from isotropy, through the particles depolarization dp = 1 − F22,p/F11,p, which is null for spherical particles. In polarization lidar atmospheric remote sensing applications [5, 6, 23], the propensity of the scattering medium to depolarize laser light is measured by the particles depolarization ratio δp = (F11,p − F22,p)/(F11,p + F22,p), which can be easily linked to dp as follows δp = dp/(2 – dp). The above formalism is generally applied in the far-field approximation [22, 24] where the scattering volume can be treated as a point source of scattered radiation. For the far-field approximation to apply, several conditions must be fulfilled [22]. Firstly, the distance D from the backscattering volume to the observation point must be large compared with the particles size a and with the laser wavelength λ (D >> a, λ). Secondly, the phases of the scattered partial wavelets only coincide in the far-field zone if D >> kaira2/2, where kair is the wavevector of the light in the air surrounding medium. Thirdly, when the scattering direction changes as little as π/(2kaira), changes in the scattering pattern may be resolved if π/(2kaira) >> dc/(2D), where dc is the diameter of the collecting lens. Finally, the randomly-oriented particles must be separated by a distance d greater than the particles size a, so that the position of a particle be not affected by the presence of its neighbors.
In the literature, several light scattering matrix experiments have been built and operated at high scattering angles (θ ≥ 168°), approaching the exact backscattering direction (θ = 180°). Table 1 presents their principal characteristics for an ensemble of particles in ambient air by giving the scattering angle range, the wavelength λ of the radiation, the nature (continuous/pulsed) of the laser source and the detector field of view (FOV). To our knowledge, none of the existing apparatus cover the exact backscattering direction and the UV spectral range has never been explored. Moreover, from a detailed reading of the corresponding papers, it seems difficult to know if the far-field single scattering conditions are fulfilled. When measuring a scattering signal at high scattering angles, close to the backward direction, two main difficulties arise, which have been identified by J.W. Hovenier et al. [25]. The first intricacy is relative to the finite size of the detector, which may block the incident radiation in the backscattering geometry. The insertion of a beamsplitter, as often performed for condensed matter phases [12–14, 26], usually limits the accuracy of the backscattering measurement because its specifications are generally imperfectly known and represent an important artifact, source of systematic error in the backscattering measurement [26]. In addition, inserting a beamsplitter plate may create some stray light, affecting the particles backscattering signal, as for solid biological tissues [12]. The second intricacy is relative to the intensity of the backscattering signal itself, which might be low for an ensemble of particles in air, so that any stray light might overcome the particles backscattering signal. To overcome this difficulty, for condensed matter phases, lock-in detection is usually applied on continuous incident radiation [12,13]. For a particles ensemble in air, the particles backscattering signal is so low that even this technique has not been successfully applied.
Table 1. Existing light scattering experiments for particles in air, close to the exact backscattering direction. The scattering angle θ, the wavelength of the radiation λ and the field of view FOV are given, together with the corresponding sample and the continuous / pulsed character of the chosen laser source. Our work provides laboratory measurements in the exact backscattering direction, with a high signal-to-noise ratio.
2.2 Principle of an exact backscattering measurement for nanoparticles in ambient air
Figure 1 is a 3D-scheme showing the principle of our particles backscattering measurement (see Section 3 for a detailed description of our experimental set-up). We overcome the first intricacy by inserting a polarizing beamsplitter cube (PBC) on the optical pathway from the laser source to the particles scattering medium. In this way, the particles backscattering radiation is detected after retro-reflection on a PBC, whose specifications and acceptance angle are accurately known, as detailed in Section 3. Moreover, the particles backscattering radiation is discriminated from the background stray light by measuring the time interval Δt = 2D/c taken by a laser pulse to reach the detector, located at a distance D from the particles backscattering medium, as in lidar applications.
Fig. 1 Principle of an exact backscattering measurement for an ensemble of nanoparticles in ambient air: the particles backscattering radiation is discriminated from the background stray light by measuring the time interval Δt = 2D/c taken by a laser pulse to reach the detector, after retro-reflection on a polarizing beamsplitter cube (PBC). The polarization (p, s) of the laser pulses are given, the experiment is performed in the laboratory ambient air and a quarter-wave plate (QWP) is used to modulate the incident laser linear polarization.
As underlined by M.I. Mishchenko [24], any measurement of particles scattering consists in a two-stage procedure: the scattering signal S is first measured in the absence of the particles (in which case, S = S0), then, in their presence. The particles backscattering signal Sp is then basically deduced by subtracting S0 from S:
The S0 signal is due to ambient air backscattering and to the partial reflection of the incident laser pulse on optical components such as a quarter-wave plate (QWP) which induces some stray light on our detector, as discussed in Section 4. If a laser pulse is emitted at a time ti, the particles backscattering signal Sp(t) is always null, except for t = ti + Δt, where Δt = 2D/c represents the time-of-flight of the laser pulse from the laser cavity to the detector, located at a distance D from the particles. When the laser pulse duration τ is taken into account, backscattering occurs along the z-propagation direction over a length λ = cτ/2, which defines the spatial extension of the backscattering volume in the z-direction. Accordingly, the particles backscattering signal Sp(t) extends over a time interval close to τ. In the framework of the Mueller matrices, suitable for polarization-resolved experiments, in the far-field single scattering approximation, the Stokes vector Stp of the particles backscattering radiation at time (ti + Δt) relates to the Stokes vector Stinc(ti) of a laser pulse emitted at time ti as follows:where D is the distance from the particles to the detector and α/D2 accounts for the collection solid angle (α also depends on the laser power density and on the detector efficiency). The Mp matrix is the Mueller matrix that accounts for the modification of the polarization state of the laser pulse during its propagation in the particles medium and in the air surrounding medium. The particles backscattering signal Sp corresponds to the first component of the Stokes vector Stp, as our detector only measures the total light intensity. It is then obtained by projecting the Stp-vector on the [1, 0, 0, 0]T vector, as detailed by M. Hayman et al. [28]. Using the Mueller matrix formalism allows to decompose the matrix Mp as the product of the successive Mueller matrices encountered by the laser pulse during its propagation from the laser cavity to the detector:The Mueller matrices ME and MR, to be detailed in the appendix, correspond to emitter and receiver optical devices, and include the retro-reflecting PBC and a QWP. In between ME and MR, the particles backscattering is taken into account through the particles scattering phase matrix Fp introduced in Section 2.1. The propagation of the incident laser pulse in the air surrounding medium is described by the matrices Tair,k and Tair,-k, where k = 2π/λ u is the incident laser wavevector represented in Fig. 1 (u is the unit vector in the z-direction). It is important to note that, by combining Eqs. (1)-(3), the particles ensemble scattering matrix Fp can be determined by varying the ME or / and MR Mueller matrices. To modify ME and MR, we inserted a quarter-wave plate (QWP) after the retro-reflecting PBC on the optical pathway from the laser source to the particles. By rotating the angle ψ between the horizontal (x,z)-plane and the fast axis of the QWP, we modulate the incident laser linear polarization and measure the corresponding Sp-signal as a function of ψ. The detailed calculation of Sp(ψ) corresponding to our {PBC + QWP} emitter / receiver optical configuration is given in the appendix, and we get:Hence, by performing accurate particles backscattering signals measurements for a set of different ψ-angles, it is possible to accurately evaluate the F11,p and F22,p-coefficients to retrieve the particles depolarization dp = 1 – F22,p/F11,p for an ensemble of nanoparticles in ambient air, and this, for the first time in the exact backscattering direction.The principle of this experiment is in analogy with a UV-polarization lidar, for which we recently measured particles depolarization as low as 0.6% in the troposphere [23]. Such a value is very close to the UV-molecular depolarization from which it has been partitioned, and corresponds to particles backscattering signals as low as 5 × 10−8 m−1.sr−1 in the UV spectral range [23]. In this paper, a typical particles number concentration Np = 105 part.cm−3 may be expected from the generator of spherical nanoparticles having a 100 nm-diameter. Using Mie theory, we computed the particles backscattering cross-section Cback = 2.5 × 10−13 cm2.part−1.sr−1 to get the corresponding particles backscattering coefficient 2.5 × 10−6 m−1.sr−1, well above the detection threshold measured in our free troposphere lidar experiment. We are hence confident that the particles backscattering signal can be detected in our laboratory for an ensemble of nanoparticles in ambient air, leading to accurate laboratory particles depolarization measurements, for the first time performed in the exact backscattering direction and under far-field scattering conditions.
3. Experimental set-up for particles backscattering in ambient air
Our experimental set-up is detailed in Fig. 2. The laser pulses are generated by a tripled-Nd:YAG laser source, delivering 25 mJ energy at a 10 Hz repetition rate. The laser wavelength (λ = 355 nm) has been chosen to increase our sensitivity to nano-sized particles, in the range of a few cents of nanometers. The laser pulse duration τ = 5 ns is below the time-of-flight Δt = 2D/c = 40 ns, and, as in lidar applications, the laser cavity trigger (rise time of 100 ps) is used to fix the time origin. At the exit of the laser cavity, a half-wavelength plate HWPE (HWP-355-100-2, MG) and a polarizing beamsplitter cube PBCE (PBSO-355-050, MG) are used to set the p-polarization of the incident laser pulse, corresponding to the Stokes vector Stinc = [1, 1, 0, 0]T. In Fig. 2, the emitter optics system is composed of the PBC and the QWP, while the three components QWP, PBC and PBCD figure the receiver optics system. Their respective Mueller matrices ME and MR are detailed in the appendix. The p-polarization of the laser pulse is transmitted through the PBC (PBSO-355-100, MG), also used to retro-reflect, towards the detector, the s-polarization of the backscattering radiation. This air-spaced PBC has an extinction ratio Tp/Ts greater than 250:1 at λ = 355 nm and is 355 nm-AR-coated. A precise alignment procedure has been followed to achieve the backscattering geometry described in Fig. 1. During this procedure, the 355 nm-laser was also used as an alignment laser, which entered the detector by its exit, so as to precisely position the retro-reflecting PBC on the detector x-optical axis, in exact perpendicular direction to the z-optical axis, materialized by the 355 nm-laser pulses. The use of a diffuser and the observation of diffraction rings from several irises along the optical paths allowed defining the detector x-axis perpendicular to the z-optical axis, with a maximum deviation of 1 mm.10 m−1, corresponding to 0.1 mrad.
Fig. 2 Experimental set-up for measuring the exact backscattering of light pulses by an ensemble of particles in ambient air. The nanosecond time-resolved particles backscattering radiation is collected and detected after retro-reflection on a PBC. The nanoparticles were generated by atomization from a liquid water solution, then dried. An air-cooled 355 nm beam-dump (EKSMA optics) was placed a large distance from the particles to block the laser propagation.
The laser specifications determine the backscattering volume, which is defined, along the z-axis, by the length λ = cτ/2 = 0.75 meters, and in the (x, y) transverse plane, by the waist w(z) of the laser (beam-profile measurements led to w = 9 mm for z = D = 5 m). The particles flux enters this volume through a ¼”-injection nozzle, chosen for injecting all the particles in the backscattering volume, to increase the signal-to-noise ratio. The nanoparticles, generated with a commercial atomizer (TSI, model 3079), are not static but move in a 4 L.min−1 particles flow rate. The use of a commercial compressed-clean air nebulizer prevented from particles coagulation. After the diffusion drier (used to remove the water liquid phase), the moving particles enter the scattering volume before leaving the experiment through an exhaust pump. We hence generated water droplets in ambient air, or, alternatively, solid salt particles in ambient air. According to the manufacturer, their size was in the range of a few cents of nanometers. We controlled their number concentrations with a TSI particle counter, as discussed in Section 4.2. The role of the QWP, used to modulate the laser linear polarization, is also discussed in Section 4.2. Our AR-coated QWP (QWPM-355-10-4, MG) has been tilted from normal incidence by θi = 2.5° to compensate for some of its imperfections [29]. This also helped to minimize the partial reflections on the QWP. The influence of the tilt angle on the retrieved particles depolarization is discussed in Section 4.2 and in the appendix.
The whole detector is housed in a secured dark box to minimize stray light. It is composed of a second PBC (PBCD), a collecting lens (LC), two supplementary lenses (L1) and (L2), an interference filter (IF) and a photomultiplier tube (PMT). The secondary PBC is used to minimize the polarization cross-talk Rp × Ts equal to a few 10−5, which is fully negligible. Accordingly, only the s-polarization of the backscattering radiation is detected. The 355 nm-IF is used to remove the non-elastic contribution from the collected backscattering signal. As shown by David et al. [2012], it also helps to minimize the molecular contribution to the collected backscattering signal, which improves the signal-to-noise ratio of the particles backscattering signal Sp by minimizing S0. At the exit of the Licel PMT (DC-350 MHz electrical bandwidth), the photo-electrons are sampled with a 12 bits digital oscilloscope (Le Croy HDO4054, 300 MHz, 2.5 GS/s), necessary for a time-resolved precise measurement of the low particles backscattering signal Sp. Special care has been taken to efficiently collect the particles backscattering signal Sp, which might be low for particles in air, while minimizing the stray light contribution S0. The iris I (diameter Ø) is used for that purpose by limiting the amount of collected stray light. The converging lens (L1) is used to parallelize the backscattering radiation before entering PBCD whose acceptance angle is equal to 2°. The diverging lens (L2) is used to focus the backscattered photons on the 6 mm-diameter photocathode of a Licel photomultiplier tube, used as a photo-detector. Let us consider a small scattering volume element materialized by a point P(x, y, z) of the particles scattering medium. The backscattering radiation induced by P is collected by the lens (LC) under the solid angle ΔΩ = 2π × (1 – cos α), where 2α is the apex angle of the cone defined by the clear aperture of lens (LC) and the distance D = OP0 from the collecting lens (LC) to the point P0 = P(0, 0, z = D). Only a fraction η of this collected light reaches the PMT. To optimize the collection of the Sp-signal, we have built a numerical program, based on matrix geometric optics, to analyze our multi-component optical system. It evaluates the product ΔΩ × η as a function of the position of the point P in the scattering volume element. For the set of input parameters given in Table 2, η reaches 100% for D > 4 meters. When the scattering point P deviates from the z-optical axis by Δx = ± 5 mm, a 100% η-collection efficiency is obtained for Ø = 1 mm if D > 5 meters. We then set the distance D to 5 meters, to ensure the largest possible ΔΩ × η-value. Note that this D-distance is sufficiently high for the retro-reflecting PBC to operate well-below its acceptance angle of 2°. We then deduced the distances DØ and D1 from geometrical optics. In this way, our experimental set-up operates at the exact backscattering angle θ = 180.0°, with a scattering angle range ε equal to (Δx + dc/2)/D = 0.2 deg, if dc is the clear aperture of the collecting lens (LC). Hence, for the first time to our knowledge, our experimental set-up covers the exact backscattering direction with accuracy, namely θ = π ± ε with ε = 3.5 × 10−3 rad. In addition, our set-up fulfills the far-field scattering conditions settled in Section 2.1, corresponding to Eqs. 8, 9, 10, 51, 71, 73 from Mishchenko et al. [22], for the following set of numerical values: a = 1 μm, D = 5 m, λ = 355 nm, kair = 1.77 × 107 m−1, dc = 25.4 mm and d = 1.87 × 10−6 m. Finally, the particles can be assumed as randomly-oriented. Though not easy to check, this assumption can be addressed from the Kolmogorov scale, which determines the smallest scale for observing a vortex in a viscous air flow [30]. In our experiment, the Kolmogorov scale, fixed by the Reynolds number of our flow and by our geometry, is equal to 6 × 10−5 meter, which is very low and well below the millimeter characteristic dimension of our scattering medium.
Table 2. Characteristics of our optical set-up for collecting the particles backscattering radiation. The matrix optics numerical program computes the distances D, DØ and D1 for the following set of input values: dc = d1 = d2 = 25.4 mm, Ø = 1 mm, using a 100 mm distance between (L1) and (L2).
4. Results and discussion
Using our experimental set-up, we here report on the first experimental observation of exact backscattering of light by an ensemble of particles in ambient air. To test our experimental set-up, we have considered two particles case studies: water droplets in air, as an example of spherical particles, then salt particles in air, as an example of non-spherical particles. We first present the raw data corresponding to the salt particles backscattering signal induced by UV-laser pulses. The particles UV-depolarization is then precisely evaluated for water droplets and salt crystals in ambient air with accuracy (the error is in the ‰-range).
4.1 Observation of exact backscattering for particles in ambient air
Figure 3(a) presents the backscattering signal S observed on the oscilloscope as a function of the time elapsed from the laser pulse emission, for two values of the angle ψ between the horizontal plane and the fast axis of the QWP. For each ψ-angle, the background signal S0 (dashed lines of the upper panel) has a time evolution showing the partial reflection of the laser pulse on the tilted QWP (to be seen around t = 10 ns) and the air particles backscattering signal, along the optical pathway from the laser source to the detector. As shown in Fig. 3(a), the background signal S0 depends on the angle ψ and remained constant between the S and S0-acquisitions. In the presence of the particles (full lines), S differs from S0 and this difference is the particles backscattering signal Sp as shown by Eq. (1). As plotted in Fig. 3(b), the signal Sp, which results from the difference of two signals, exhibits no systematic bias as a function of time, which means that the stray light has been efficiently removed from Sp. Moreover, the Sp-amplitude varies from zero to a few milli-Volts, so that the PMT, connected to a 50 Ω load resistance, is used in linear regime. Additionally, the linearity of Sp with the laser intensity was confirmed by varying the laser intensity over two decades, by using the HWPE and PBCE polarization components, to preserve the exact backscattering geometry. We are hence confident that the Sp-signal corresponds to the detection of the particles backscattering. Due to the laser pulse duration (τ = 5 ns), the particles backscattering signal extends over a few nanoseconds.
Fig. 3 Observation of exact backscattering of light by an ensemble of particles in air. Case study of salt particles. (a) Backscattering signal S as a function of time, for two ψ-angles of the QWP, in the presence (full-lines) and in the absence (dashed-lines, S = S0) of the particles. For ψ = 80.5° (black curve), at times lower than 20 ns, the S and S0 black curves merge on a unique line. (b) Particles backscattering signal Sp as a function of time obtained by applying Eq. (1). The sign of the PMT raw data have been changed to obtain a positive voltage and the signals result from an average over 150 laser shots. The time dependence of the signal S has been recorded at each time to ensure that the PMT remained in its linear regime (output voltage below 50 mV).
4.2 Particles depolarization in the exact backscattering direction
As shown in Fig. 3(b), the particles backscattering signal Sp is modified when the angle ψ of the QWP is varied. Following the methodology presented in Section 2.2, we have measured the particles backscattering signal Sp for a set of 12 values of ψ, by measuring the corresponding signals S and S0. To account for the amount of light backscattered during the whole laser pulse duration, the particles backscattering signal Sp = S – S0 has been integrated over the time t for each laser shot. Then, to reduce the statistical error, we averaged this time integral over 150 laser shots and plotted the retrieved signal Spτ in Fig. 4 as a function of ψ. The variations of Spτ have been adjusted with the cos(4ψ)-curve corresponding to Eq. (4). We hence evaluated the particles depolarization dp = 1 – F22,p /F11,p. Water droplets have been used as an example of spherical particles. In addition, we have used salt particles in ambient air as an example of nonspherical particles. From the fitting procedure, the water droplets depolarization was found equal to dp = (0.04 ± 0.08) %, while we got dp = (8.40 ± 0.30) % for the generated salt particles. These dp-values correspond to the following lidar particles depolarization ratios: δp = (0.02 ± 0.05) % for water droplets and δp = (4.38 ± 0.16) % for the generated salt particles, in the exact backscattering direction. The given statistical error bars are very low, as drastically reduced by laser shot averaging, by the 12-bits acquisition and by the 12 ψ-values fitting procedure. We have analyzed the statistical error bars affecting the retrieved particles depolarization on the basis of our recent research for improving the accuracy of lidar measurements [23,31]. The statistical noise is mainly due to photon noise, and shot noise and electronic noise are not a major concern. Hence, the achieved accuracy on dp indeed results from the 12-bits acquisition and from the 12 ψ-values fitting procedure, performed on particles backscattering signals averaged over 150 laser shots. Our particles depolarization measurement is also affected by systematic errors. The laser intensity and particles number fluctuations are however very low: over the duration of the experiment, we measured a mean laser energy of (24.0 ± 0.4) mJ.pulse−1 and a mean particles number concentration of Np = (5.23 ± 0.15) × 106 part.cm−3. In addition, the acquisition has been performed by choosing the 12 ψ-values in an almost random order, to minimize the systematic error on dp, due to possible experimental fluctuations, including Np-fluctuations. Hence, the obtained residue plot exhibits no clear structure. As shown in Fig. 4 in dashed-lines, normalization of the Spτ-signal by the particles number concentration does not noticeably affect the retrieved particles depolarization. In addition, pulse-to-pulse laser mode fluctuations were also minimized by using a Gaussian mirror in the laser cavity, to favor the TEM00 Gaussian mode. The exact position of the QWP along the Gaussian beam of the laser is also not a main concern [29]. However, the QWP is tilted from normal incidence by an angle θi = 2.5° while Eq. (4) assumes that the QWP operates at normal incidence. We however show in the appendix that for θi = 2.5° and for ψ-values between 84° and 96°, the Mueller matrix of a tilted QWP equals that of a QWP at normal incidence with a very low relative error, equal to 10−5.
Fig. 4 Time integral over the pulse duration of the particles backscattering signal Sp averaged over 150 laser shots as a function of the angle ψ of the QWP used to modulate the incident laser linear polarization. Case study of water droplets particles. The error bar on the reading of the ψ -angle is equal to 0.5°. The plotted error bar on Spτ is too low to be visible (it is equal to 1σ and calculated from the statistical error obtained by averaging the time integral of Sp(t) over 150 laser shots). The full-line black curve (dashed-line blue curve) corresponds to the adjustment of the data by using Eq. (4) (after particles number normalization). In both cases, no systematic bias is visible on the residue plot plotted in the lower panel.
Within our error bars, the observed particles depolarization dp = (0.04 ± 0.08) % is compatible with the expected zero-depolarization value for homogeneous spherical particles, which in turn favors our experimental set-up. In addition, the single-scattering approximation, assumed from the very beginning, now appears to be realistic, since multiple scattering would have led to particles depolarization, as shown by M.I. Mishchenko et al. [32]. Indeed, the right-hand panel of Fig. 5 in [32] provides specific quantitative evidence that at particle packing densities typically encountered with this laboratory setup, the assumption of the single-scattering regime is quite safe. Moreover, our experimental set-up has the ability to measure non-zero particles depolarization, corresponding to nonspherical particles, as are the generated salt particles. The retrieved salt particles depolarization δp = (4.38 ± 0.16) % is in the range of what is usually observed in the atmosphere by using a polarization lidar close to the sea-salt particles source region [33]. However, sea-salt particles may differ from our generated salt particles. In addition, even close to the source region, comparison of our laboratory measurement with field measurements remains difficult, as salt particles are present in the atmosphere in the form of what is called externally-mixed aerosols [6]. Comparing the obtained value with a laboratory reference literature is difficult as no apparatus exactly operates in the exact backscattering direction. For the closest value to the exact backscattering direction [16], the observed particles depolarization for salt particles is larger than ours. However, extrapolation up to 180.0° may lead to quite important errors [18], and in our experiment, the relative humidity is probably too high for the particles to depolarize as in Sakaï et al.’s experiment [16], despite the use of a diffusion dryer. Comparison with numerical simulations applying T-matrix or DDA-numerical codes [5, 9, 24] is highly desirable, though they are not assumption-free. In this context, numerical calculation of δp should be performed as a function of both the particles size and the ambient air relative humidity. This is clearly beyond the scope of this study as it would require complementary measurements in the surrounding medium of the particles.
5. Conclusion and outlooks
In this paper, a new experimental set-up has been developed to precisely measure the exact backscattering of light by an ensemble of nanoparticles in ambient air. As detailed in Section 2, a pulsed laser source and a polarizing beamsplitter cube (PBC) have been used to fulfill the needed requirements of a high angular resolution and a high dynamical range to cover the exact backscattering direction, for the first time for particles in ambient air. The exact backscattering direction is covered with accuracy: θ = π ± ε radians with ε = 3.5 × 10−3 radian. In Section 3 where our new experimental set-up is detailed, special care has been taken to optimize the collection of the particles backscattering signal Sp, while minimizing the background stray light signal S0, and this in the UV spectral range, to increase our sensitivity to nano-sized particles. In addition, for the first time to our knowledge, our light-scattering experimental set-up fulfills the far-field single-scattering approximation, which is generally applied in both numerical simulations and field experiments. We then reported in Section 4 on the first experimental observation of backscattering of light by water droplets in ambient air, as an example of spherical particles, then by salt particles in ambient air, as an example of non-spherical particles. The particles backscattering signal Sp has been retrieved from the detected signal S by subtracting the background signal S0 obtained in the absence of the particles, the latter being constant for several minutes. Moreover, by modulating the incident laser polarization, we measured the particles depolarization with accuracy. For polarization lidar remote sensing field applications, this particles depolarization dp can be converted into the so-called particles lidar depolarization ratio δp. Several outlooks have then been proposed in Section 4.2 where the retrieved particles depolarization has been discussed. To conclude with, this first experimental achievement of a polarimetric measurement in the exact backscattering direction opens new outlooks in coupling laboratory light scattering matrix measurements with both numerical simulations and field observations.
6. Appendix
Expression of the particles backscattering signal Sp
The aim of this appendix is to derive the expression of the detected particles backscattering signal Sp for our emitter/receiver optics configuration, composed a polarizing beamsplitter cube (PBC) and a quarter-wave plate (QWP).
As detailed in Eq. (2), the incident radiation Stokes vector Stinc and the particles backscattering radiation Stokes vector Stp are related by a Mueller matrix Mp, such as Stp = Mp × Stinc, where Mp results from the successive Mueller matrices encountered on the light pathway from the laser source to the backscattering detector:
- • Emitting optics Mueller matrix ME
- • Propagation of the incident laser pulse in the air surrounding medium (Mueller matrices Tair, k and Tair, -k)
- • Backscattering in the particles medium (Mueller matrix Fp)
- • Receiver optics Mueller matrix MR
We here detail these successive Mueller matrices to retrieve the expression of the particles backscattering signal Sp.
A.1 Emitter and receiver Mueller matrices ME and MR
The Mueller matrix ME of the emitter optics device, composed of a PBC and a QWP, can be derived from any textbook on light polarization [33]:
where ψ is the angle between the fast axis of the QWP and the horizontal scattering plane (ψ is counted counterclockwise when looking in the z-propagation optical axis). The imperfections of the PBC are described by its s-transmission coefficient Ts << 1 (a perfect PBC would have Ts = 0 and Tp = 1 and would transmit the incident laser linear polarization corresponding to the Stokes vector [1, 1, 0, 0]T without any modification. The Mueller matrix of the QWP written in Eq. (5) corresponds to normal incidence [34]. The Jones matrix of a QWP tilted from normal incidence by an angle θi is given in [35]. The main correction factor is equal to 1 – (sin θi/nO)2 where nO is the ordinary refractive index. We have expressed the Mueller matrix of a tilted QWP as a function of ψ and θi. For our ψ-values (between 84° and 96°, see Fig. 4) and an incidence angle θi of 2.5°, we calculated that the Mueller matrix elements of the tilted QWP and found a fully negligible relative error of 10−5 when compared to the Mueller matrix elements of a QWP at normal incidence.In the backscattering geometry, the same optical components {PBC + QWP} are crossed in the opposite direction (−k) after particles backscattering. As a consequence, the receiver optics Mueller matrix MR can be derived from Eq. (6) by changing the angle ψ to its opposite, while using the PBC as a retro-reflector (we hence replace Tp with Rs and the imperfections of the retro-reflecting PBC are now addressed by its p-reflectance coefficient Rp << 1):
The use of a second PBC in the detector (see Section 3) enables to minimize the polarization cross-talk, then equal to Rp × Ts, in the range of 10−5. It follows that only the s-polarization of the backscattering radiation is measured on our detector.A.2 Particles backscattering Mueller matrix Fp
The scattering phase matrix of the particles scattering medium can be written as follows for particles in random orientation, in the single-scattering approximation [21]:
The F11,p and F22,p coefficients define the particles depolarization dp = 1 – F22,p/F11,p, while, for particles in random orientation F41,p = F14,p.A.3 Expression of the particles backscattering signal
As shown by K. Sassen [36], extinction is not sensitive to the polarization state of the light. Hence, the Mueller matrices Tair, k and Tair, -k associated to the air surrounding medium do not modify the polarization state of the incident laser pulse and be considered as transmission matrices. As a consequence, backscattering from the air surrounding medium is not a main concern and as detailed in Section 4, it can be subtracted to the backscattering signal to extract the particles backscattering signals Sp. We may hence write the Mp-Mueller matrix as follows Mp = MR × Fp × ME. By neglecting any polarization cross-talk which we minimized (i.e. assuming Rp = Ts = 0) and combining Eqs (1), (5) and (6), we get:
As detailed in Section 2.2, the particles backscattering signal Sp is then obtained by projecting the Stokes vector Stp = α/D2 × Mp × Stinc of the particles backscattering radiation on the vector [1, 0, 0, 0]T, since our photo-detector only measures the total light intensity, which corresponds to the first Stokes parameter. We have calculated Sp for a linearly polarized laser pulse, corresponding to the Stokes vector Stinc = [1, 1, 0, 0]T. In the detailed calculation of Sp, F14 and F41 only appear in the form of their difference F14 – F41, which is null for randomly-oriented particles [21]. After a few calculations, we get the following expression for Sp(ψ), which hence only depends on F11,p and F22,p:By measuring the Sp-signal for a set of different ψ-values, then fitting the curve Sp(ψ) as performed in Section 4.2, the F11,p and F22,p -coefficients can be accurately determined, which leads to an absolute measurement of the particles depolarization dp = 1 – F22,p/F11,p.Acknowledgments
The authors thank Olga Muñoz and Richard Perkins for fruitful discussions, Région Rhône-Alpes and CNRS for financial funding of part of this work. La thèse effectuée par Grégory David est soutenue par la Région Rhône-Alpes à hauteur de 32 116 euros.
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