# Spectro-polarimetric backscattering of atmospheric particles

Killian Aleau, Romain Ceolato, Matthew Berg, Guillaume Huss, Olga Muñoz, Gorden Videen

► To cite this version:

Killian Aleau, Romain Ceolato, Matthew Berg, Guillaume Huss, Olga Muñoz, et al. Spectro-polarimetric backscattering of atmospheric particles. Journal of Quantitative Spectroscopy and Radiative Transfer, 2024, 327, pp.109132. 10.1016/j.jqsrt.2024.109132. hal-04963830

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# Spectro-polarimetric backscattering of atmospheric particles

Killian Aleau<sup>a</sup>, Romain Ceolato<sup>a,*</sup>, Matthew Berg<sup>b</sup>, Guillaume Huss<sup>c</sup>, Olga Muñoz<sup>d</sup>, Gorden Videen<sup>e,f</sup>

## ARTICLE INFO

### Keywords:

- Backscattering
- Spectral
- Depolarization
- Aerosol
- Particles
- Lidar

## ABSTRACT

This study presents a new experiment to measure the polarization-resolved two-dimensional light-scattering patterns, near and including the backscattering direction, of fixed particles representative of some atmospheric-aerosol types. The measurements are conducted across a broad spectrum, from 450 to 850 nm, and for different polarization states. A supercontinuum laser, achromatic optics, and polarizers are used to measure the scattering patterns. Important quantities for applications such as lidar are also measured including the backscattering phase-function and the linear depolarization ratio in the exact backscattering direction. Microscopic images of the particles are provided by digital in-line holography as well. Thus, in some cases, it is possible to interpret the spectro-polarimetric measurements with knowledge of the size and shape of the particles involved. Spherical particles of known properties are used to calibrate the method. Then, Arizona test dust, Gobi and Sahara Desert dust, and ash from the Eyjafjallajökull and La Palma volcanic eruptions are studied to illustrate the method's potential for lidar applications.

## 1. Introduction

Atmospheric aerosols are important in multiple environmental processes. They originate from diverse sources, including natural phenomena like dust storms, volcanic eruptions, and wildfires, as well as anthropogenic sources like emissions from factories. Studying the properties of atmospheric aerosols is important to understand their impact on climate, air quality, visibility, and human health. To do so, a variety of remote-sensing techniques may be used, such as lidar. Lidar is an active remote-sensing method using laser light to measure certain properties of aerosols [1]. Short pulses are emitted from a lidar unit into the atmosphere and are backscattered by aerosol particles. The particle properties are then inferred by detecting the backscattered light with sensitive detectors and analyzing the resulting signals. Specific quantities of interest to lidar include the backscattering phase function and linear depolarization ratio for different wavelengths [2], i.e., spectro-polarimetric backscattering (SPB) measurements. Advances in remote-sensing technology have led to the emergence of innovative instruments capable of characterizing aerosols with

enhanced precision [3,4]. Modern systems enable the measurement of aerosol properties at either specific scattering angles or across a broad range of angles [5,6] and wavelengths [7,8]. Our work aims to add to these capabilities through fundamental laboratory-based experiments.

Here, we measure the backscattered-light intensity of stationary particles. The particles span a range of sizes approximately 1 – 100 µm and in some cases are representative of atmospheric aerosols. The measurements cover scattering angles in two (angular) dimensions across the 450–850 nm spectrum. Based on work by Berg and Holler [9], we integrate a digital in-line holography (DIH) measurement to provide microscopic images revealing the particle size and shape. Thus, the combination allows us to interpret the SPB measurements with some degree of knowledge of the particle morphology. Calibration of the method is achieved with spherical particles of known size and refractive index. We then apply the method to different particles including, Arizona test dust (ATD), a common atmospheric mineral-dust proxy, Gobi and Saharan Desert dust [10], and Eyjafjallajökull and La Palma [10,11] volcanic ash particles to explore the potential of the method. Finally, we attempt to establish connections between the particle morphologies and

<footnote><sup>a</sup> ONERA, The French Aerospace Lab, 2 Avenue Edouard Belin, Toulouse FR 31055, France</footnote>

<footnote><sup>b</sup> Department of Physics, Kansas State University, 1228 N. Martin Luther King Jr. Dr., Manhattan, KS 66506-2601, USA</footnote>

<footnote><sup>c</sup> Leukos Laser, Limoges 87000, France</footnote>

<footnote><sup>d</sup> Instituto de Astrofísica de Andalucía, CSIC, Granada, Spain</footnote>

<footnote><sup>e</sup> Space Science Institute, 4750 Walnut Street, Boulder Suite 205, CO 80301, USA</footnote>

<footnote><sup>f</sup> Department of Atmospheric Sciences, Texas A&M University, College Station, TX, USA</footnote>

<footnote>* Corresponding author.
E-mail address: romain.ceolato@onera.fr (R. Ceolato).</footnote>
various features in the measured data potentially useful for lidar applications.

## 2. Experiment

Fig. 1 shows the layout of the experiment, where a supercontinuum laser (LEUKOS, Samba 450) emits pulsed light with a pulse length of approximately 1 ns at a repetition rate of 30 kHz and a spectrum of $\lambda = 450\text{nm} - 2400\text{nm}$. The laser achieves this by passing a monochromatic beam at $\lambda = 1064\text{ nm}$ through a fiber, producing the spectrum via non-linear effects [12]. The resulting beam is then passed through a tunable filter (Leukos, Bebop) consisting of two low-pass and high-pass filters that can select a single (quasi-monochromatic) wavelength between $\lambda = 450\text{nm} - 850\text{ nm}$, in 5 nm increments. Note that this means that our closest comparison to the popular lidar wavelength of $\lambda = 532\text{ nm}$ in the following will be $\lambda = 535\text{ nm}$. The beam passes through a beamsplitter cube (BS1), is reflected by a mirror (M), and then passes through a linear polarizer (LP: Thorlabs, WP25M-VIS) allowing the polarization direction to be set, which is chosen here to be vertical, i.e., normal to the optical table. Next, the beam passes through a non-polarizing plate beam-splitter (BS2: Thorlabs, BSW10R) and proceeds to the sample. The sample consists of various microparticles deposited on a slightly tilted, uncoated glass window (W). Here the coordinate system is defined such that the beam propagates along the positive z-axis as shown. The portion of light backscattered by the particles along the negative z-axis, Fig. 1 (a), is then reflected by BS2 toward an achromatic lens (L3: Thorlabs, AC254-030-AB-ML) of focal length $f = 30\text{ mm}$. The purpose of the lens is two-fold: it both increases the angular range ultimately resolved and filters stray light using an iris (IR). The maximum scattering angle, denoted $\theta_m$, is determined by the diameter of the lens's clear aperture $D_L$ and the distance between the particles and the lens, $s_{obj} + f$, as given by

$$ \theta_m = \tan^{-1} \left[ \frac{D_L}{2(s_{obj} + f)} \right], \qquad (1) $$

where $s_{obj}$ is the object distance in the thin-lens formula, i.e. the distance

between the particle and L3 via reflection from BS2 minus $f$. A rotatable linear polarizer (RLP: Thorlabs, WP25M-VIS), or analyzer, is positioned between L3 and IR to resolve the polarization state of the backscattered light. The analyzer is mounted in a motorized rotation stage, allowing its polarization axis to be oriented at an angle $\psi$ between 0 and 180° with respect to the optical table. That is, an angle of $\psi = 0°$ corresponds to horizontal polarization, meaning parallel to the table. Backscattered light is then received by the scattering sensor, which is a low-noise sCMOS sensor (Hammamatsu, 13440) with a resolution of 2048 × 2048 pixels with $c = 2.4\ \mu\text{m}$ pixel size, 40 fps framerate, global shutter, and 1 electron per pixel electronic-noise level, which significantly reduces the noise present in the scattering measurements.

Each sensor pixel corresponds to a specific scattering direction, described by the polar and azimuthal angles ($\theta, \phi$). As depicted in Fig. 1 (c), the polar scattering angle $\theta$ for a given pixel depends on the pixel's position as,

$$ \theta(i, j) = \tan^{-1} \left[ \frac{(s_{\text{img}} + f) p(i, j)}{(s_{\text{obj}} + f) d} \right]. \qquad (2) $$

In Eq. (2), $p(i, j)$ represents a pixel's distance from the center of the sensor array enumerated by the indexes $i$ and $j$ along the sensor's x and y-axes, and $p(i, j) = c \sqrt{i^2 + j^2}$ with $c$ being the pixel size. The distance $s_{\text{img}}$ is the lens's image distance, and $d$ is the distance between the lens's image plane and the sensor plane. A second laser (Leukos, H LX-G) with wavelength $\lambda = 532\text{ nm}$ is used to record in-line holograms of the particles. This beam is expanded by lenses L1 and L2 to approximately 1 cm in diameter and is combined with the SPB beam by BS1. As shown in Fig. 1(b), a second sensor, called the holography sensor (FLIR, Blackfly USB3), records the intensity interference pattern, or hologram, produced by the particles on W.

### 2.1. Holographic imaging

A sense for the particle morphology and size of a sample is provided by contact-free imaging via digital in-line holography (DIH). This

<figure><img src="image_4.png" /><figcaption>Fig. 1. Optical arrangement for the SPB experiment. A supercontinuum laser provides pulses of wavelength-tunable light that illuminate stationary particles on a window (W). Light backscattered by the particles is imaged onto the scattering sensor providing the backscattered intensity pattern in polar and azimuthal angles. Another laser illuminates the particles with an expanded beam where scattered and unscattered light interfere to produce an in-line hologram across the holography sensor. Images of some of the particles within the footprint of the supercontinuum beam on the window can then be obtained via image reconstruction from the recorded digital hologram.</figcaption></figure>
involves recording the interference pattern due to the portion of the $\lambda = 532$ nm beam scattered by the particles around the forward direction, $\theta = 0^{\circ}$, and the unscattered portion of the same beam. The pattern is resolved by the holography sensor and constitutes a raw in-line hologram $I^{holo}$. A separate measurement without particles present on the window (W) in Fig. 1 is made, which is called the reference measurement, $I^{ref}$. Then, a contrast hologram $H^{con}$ is formed by the difference between these measurements, $H^{con} = I^{ref} - I^{holo}$. An image of the particles on W can then be reconstructed from $H^{con}$ by applying the Fresnel transform in a process called image reconstruction [13–15].

Image reconstruction involves simulating the process of illuminating the contrast hologram by the holography beam ($\lambda = 532$ nm) and calculating the resulting diffraction pattern from it via scalar-diffraction theory in the Fresnel approximation, i.e.,

the hologram passes through W before reaching the sensor, and thus, is refracted and slightly deflected; effects that are absent in the conventional approach [9] where the particles reside on the opposite side (sensor side) of W.

Although the DIH imaging could readily be replaced by a conventional microscope, and likely provide better resolution images, we employ holography to demonstrate that sufficient particle size and shape information can be provided to interpret the SPB measurements. The motivation for doing so relates to aerosols. If aspects of this experiment were eventually adapted to examine free-flowing aerosols, rather than stationary particles, then a microscope would become problematic. The limited depth of focus and working distance of a microscope prevents imaging a flowing stream of particles due to the random trajectories that particles take as they flow, e.g., from a nozzle.

$$E(x, y, z) = \frac{\exp\left(\frac{2\pi i z}{\lambda}\right)}{i\lambda z} E_0 \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} H^{\text{con}}(x', y') \exp\left\{\frac{i\pi}{\lambda z}[(x-x')^2 + (y-y')^2]\right\} dx' dy' \quad (3)$$

In Eq. (6), $E(x, y, z)$ is the diffracted wave-amplitude in a plane parallel to the holography-sensor array where $H^{con}$ is defined, and $E_0$ is a constant related to the intensity of the holography beam. A more useful form for the wave amplitude $E$ is the convolution [15]:

$$E(x, y, z) = E_0 [H^{\text{con}} * h](x, y, z), \quad (4)$$

where,

$$h(x, y, z) = \frac{\exp\left(\frac{2\pi i z}{\lambda}\right)}{i\lambda z} \exp\left[\frac{i\pi}{\lambda z}(x^2 + y^2)\right] \quad (5)$$

is the impulse response and * denotes the convolution operation. According to the convolution theorem, Eq. (4) can be computed using forward $\mathcal{F}$ and inverse $\mathcal{F}^{-1}$ Fourier transforms as,

$$E(x, y, z) = E_0 \mathcal{F}^{-1}[\mathcal{F}(H^{\text{con}}) \times \mathcal{F}(h)](x, y, z), \quad (6)$$

where $\times$ denotes element-wise multiplication. Eq. (9) can now be evaluated with Fast Fourier Transforms (FFT), which significantly reduces the computation time [13,16]. Provided $z$ in Eq. (6) is the distance between W and the sensor array, a focused image of the particles forms as $|E(x, y, z)|^2$. In practice, the correct value for $z$ here is determined by repeatedly evaluating Eq. (6) until the image is brought into focus.

Mathematica code to perform image reconstruction via Eq. (6) is provided by Berg [15]. The result depicts individual particles in gray-scale, with a theoretical image resolution of twice the sensor pixel-size, i. e., $\delta = 2c = 4.8\mu\text{m}$ here. As shown in Fig. 1, the hologram measurement is performed within the same optical arrangement as the SPB measurement without any movement of the sample window. This permits a direct comparison between the particle images and the SPB measurements, like the work of Berg and Holler [9], where forward scattering is considered (rather than the backscattering here). All optical elements in Fig. 1 remain in place during the holography and SPB measurements except for the holography sensor, which is removable so that specular back-reflection from that sensor's window does not affect the SPB measurements.

Because the holography beam is collimated and the Fresnel transform is used for reconstruction, each pixel in the reconstructed image is the same size as the holography-sensor pixels [15]. However, a degree of distortion may occur due to the particles being on the illuminated side of the window, recall Fig. 1(b). That is, the forward-scattered light forming

In such cases, digital holography is the most capable way to image moving particles.

## 2.2. Spectro-polarimetric backscattering (SPB)

The polarization state of the backscattered light is measured by means of two broadband linear polarizers, LP and RLP (the analyzer), following Swami et al. [17]. The intent is to measure the linear depolarization ratio, denoted $\delta_L$. This ratio is common in polarimetric elastic backscatter lidar systems. By concentrating on $\delta_L$, we simplify the interpretation of the depolarization characteristics of the particles considered, with the aim of improving data analysis for remote-sensing applications. The first polarizer, LP, controls the initial polarization state of the beam incident on the particles on W, while the second polarizer RLP is positioned down-stream from the achromatic lens L3, allowing only linear polarization oriented at the angle $\psi$ to pass. Recall that the angle $\psi$ is measured from an axis parallel to the optical table, which equates to passing horizontal polarization when $\psi = 0^{\circ}$. Four specific RLP angles are considered: $\psi = 0^{\circ}$, $45^{\circ}$, $-45^{\circ}$, and $90^{\circ}$. This set of measurements allows for the determination of $\delta_L$ as well as the partial Mueller-matrix $M$, the latter of which is also relevant to aerosol differentiation in lidar [18]. The measurements are expressed in terms of the Stokes vector, denoted $S$ and defined in Eq. (7) below. The vector is formed by four polarimetric intensities, or elements, $I, Q, U$, and $V$. The $I$ Stokes element is simply the intensity received by a pixel in the scattering sensor. The remaining elements are then the intensities received when RLP is rotated to specific angles $\psi$. Thus, we introduce the notation $I_{\psi}$ to represent the pixel-level measurement when the RLP is oriented at the angle $\psi$. The Stokes vector is given by:

$$S = \begin{pmatrix} I \\ Q \\ U \\ V \end{pmatrix} = \begin{pmatrix} I_{90} + I_0 \\ I_{90} - I_0 \\ I_{45} - I_{-45} \\ I_R - I_L \end{pmatrix}, \quad (7)$$

The $V$ parameter, or the terms $IR$ and $I_L$ in Eq. (7), relate to right and left circularly polarized light and are not considered in this work. In effect, the Stokes vector here is only three elements, and thus, requires only a partial Mueller matrix in the following. Eq. (8) below defines $\delta_L$ as the ratio of the difference between $I$ and $Q$ to the sum of $I$ and $Q$. It can also be expressed as the ratio of $I_0$ to $I_{90}$, which represent the vertical and horizontal polarization states with respect to the optical table
<figure><img src="image_5.png" /><figcaption>Fig. 2. Calibration of the holographic imaging using a 1951 USAF resolution test target. Panel (a) shows the recorded hologram of the target and (b) shows the image reconstructed from it. In (c) is a zoomed-in view of the smallest element region.</figcaption></figure>

respectively:

$$ \delta_L = \frac{I - Q}{I + Q} = \frac{I_0}{I_{90}} \qquad (8) $$

## 3. Calibration

### 3.1. Holography calibration

A 1951 USAF resolution test target (Thorlabs, R1DS1P) is used to calibrate length scales in the holographic imaging. To do so, the sample window in Fig. 1 is temporarily replaced with the target and a hologram is recorded, see Fig. 2(a). An image of the target is then reconstructed from the hologram, Fig. 2(b). The combination of digits visible in the image correspond to groups of six line elements that relate to a specific image resolution in line pairs per millimeter (lp/mm). We first consider group-2 element-2 corresponding to 4 lp/mm to verify that the size of a pixel in the reconstructed image is the same as the sensor pixel size. This is done by counting the number of pixels across the element in the DIH image and equating that to the true element-size divided by the pixel size. Then, we consider the smallest group resolved in the reconstruction, which is group-7 element-4. This group is shown in Fig. 2(c) and corresponds to an image resolution of 2.7 µm. As such, the resolution of the holographic imaging, $\delta_{img}$, is taken as approximately $\delta_{img} \sim 3$ µm, which is slightly better than the theoretical estimate based on the Nyquist-Shannon sampling theorem, i.e., $\delta_{img} \sim 2c = 4.8$ µm [15]. Thus, $\delta_{img} \sim 3$ µm constitutes the uncertainty in any length-scale measurements presented later when determined from DIH images, e.g., the particle size.

### 3.2. Spectral and angular calibration

As light passes through the optical elements in Fig. 1, the polarization state can be affected by the elements because reflection and transmission characteristics are polarization dependent. As such, the Stokes vector derived from the scattering-sensor measurements does not represent the true Stokes vector for the particles on W. To account for this effect, measurements are performed to determine the partial Mueller matrix of the optical elements. The supercontinuum beam is temporarily reoriented to propagate from the location of the particles (at W) to the scattering sensor, so that the beam passes through the same optical elements that backscattered light from the particles will pass through in later measurements. This is done for different polarization states by selecting specific LP and RLP orientations abbreviated as V for vertical when $\psi = 90^{\circ}$, H for horizontal when $\psi = 0^{\circ}$, and P when $\psi = 45^{\circ}$. The Stokes vector $S_{opt}$ of the light reaching the sensor is determined by multiplying the partial Mueller matrix of the combined optical elements, $M_{opt}$, by the Stokes vector of the input light, $S_{inc}$, as

$$ S_{opt} = M_{opt} \cdot S_{inc} = \begin{pmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{pmatrix} \cdot \begin{pmatrix} I_{inc} \\ Q_{inc} \\ U_{inc} \end{pmatrix} = \begin{pmatrix} I_{opt} \\ Q_{opt} \\ U_{opt} \end{pmatrix} \qquad (9) $$

To obtain the Mueller matrix elements in Eq. (9), nine intensity measurements are conducted using the following LP-RLP combinations: HH, HV, VH, VV, VP, PV, HP, PH, PP. Each intensity is defined in terms of the partial matrix, where, as an example, the first four measurements are given in Eq. (10) below:
$$HH = \frac{1}{2}(1 \ 1 \ 0) \cdot \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{bmatrix} \cdot \frac{1}{2} \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} = \frac{1}{4}(m_{11} + m_{12} + m_{21} + m_{22}), \qquad (10a)$$

$$HV = \frac{1}{2}(1 \ -1 \ 0) \cdot \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{bmatrix} \cdot \frac{1}{2} \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} = \frac{1}{4}(m_{11} + m_{12} - m_{21} - m_{22}), \qquad (10b)$$

$$VH = \frac{1}{2}(1 \ 1 \ 0) \cdot \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{bmatrix} \cdot \frac{1}{2} \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix} = \frac{1}{4}(m_{11} - m_{12} + m_{21} - m_{22}), \qquad (10c)$$

$$VV = \frac{1}{2}(1 \ -1 \ 0) \cdot \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{bmatrix} \cdot \frac{1}{2} \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix} = \frac{1}{4}(m_{11} - m_{12} - m_{21} + m_{22}). \qquad (10d)$$

The matrix elements $M_{opt}$ are then

<table><tr><td>$m_{11}$ = HH + HV + VH +<br>VV</td><td>$m_{12}$ = HH + HV - VH -<br>VV</td><td>$m_{13}$ = 2PH + 2PV - $m_{11}$</td></tr><tr><td>$m_{21}$ = HH - HV + VH - VV</td><td>$m_{22}$ = HH - HV - VH +<br>VV</td><td>$m_{23}$ = 2PH - 2PV - $m_{21}$</td></tr><tr><td>$m_{31}$ = 2HP + 2VP - $m_{11}$</td><td>$m_{32}$ = 2HP - 2VP - $m_{12}$</td><td>$m_{33}$ = 4PP - 2PH - 2PV -<br>$m_{31}$</td></tr></table>

The measurement of $M_{opt}$ is repeated for each wavelength in the spectrum used in later measurements, i.e., $M_{opt} = M_{opt}(\lambda)$.

To determine the Stokes vector of backscattered light from particles, $S_{SPB}$, from that measured by the scattering sensor, $S_m$, the partial Mueller matrix is inverted, i.e.,

$$S_{SPB} = M_{opt}^{-1} \cdot S_m \qquad (11)$$

The effectiveness of this calibration process is verified with a polarimetric camera (Thorcam, CS505MUP1) and two Lambertian Spectralon targets: (Labsphere, SRS-02) with a 2% surface reflectance and (Labsphere, SRS-99) with a 99% surface reflectance; either are placed at the location of the sample window W. The camera determines the polarization state using a linear-polarizer array across its sensor. Each pixel in the polarimetric camera corresponds to a linear polarizer that measures one of four polarization directions: $\psi = 0^{\circ}$, $45^{\circ}$, $90^{\circ}$, and –

45°. The beam illuminates the Spectralon target at normal incidence, and the camera is positioned to capture backscattered light at a polar scattering angle of $\theta = 175^{\circ}$. Four images are obtained for the four linear-polarization combinations: HH, HV, VH, and VV. For each image, the intensities are averaged under the assumption that the cross-polarization ratios $CP_+$ and $CP_-$ given by

$$CP_+ = \frac{I_{ref}}{Q_{ref}}, \quad CP_- = \frac{I_{ref}}{U_{ref}}, \qquad (12)$$

remain constant with respect to the small variation of $\theta$ across the camera pixels. In Eq. (12), $I_{ref}$ represents the intensity for polarization state P. While the polarization-correction procedure of Eq. (11) is effective, it slightly overestimates the depolarization compared to that from the polarization camera. The correction may not address all effects, such as stray light, and as a result, we perform an additional correction based on the Spectralon measurements. This involves considering the ratios of Eq. (12) as the target values to be obtained using the experimental arrangement with the scattering sensor, i.e., instead of the polarization camera. To do so, two coefficients, $a$ and $b$, are calculated as a function of wavelength as

$$a = \frac{I}{Q CP_{+}}, \quad b = \frac{I}{U CP_{-}} \qquad (13)$$

and are then added to $S_{SPB}$, thus further correcting $Q_{SPB}$ and $U_{SPB}$:

$$Q_{SPB,corr} = Q_{SPB} a, \quad U_{SPB,corr} = U_{SPB} b \qquad (14)$$

The $a$ and $b$ factors apply to all subsequent measurements.

The reference curves obtained with the polarization camera are plotted in Fig. 3 with the measured datapoints from the scattering sensor. Notably, the reference curve and the measured data for the SRS-02 target closely follow each other. However, the measurements for the SRS-99 target consistently display an approximately 4% discrepancy across all wavelengths. The discrepancy will be factored into later error calculations. Note that the larger (smaller) depolarization behavior observed for SRS-99 (SRS-02) is also seen in several other publications, see [19,20].

Overall, the polarization calibration process yields good results. An intriguing observation is that it appears that the error tends to increase in tandem with depolarization. This behavior may stem from insufficientities in the calibration, leading to more discrepancy when the light incident on BS2 is significantly depolarized. Reflection from the beamsplitter is polarization dependent, and thus, if the light is highly polarized in a specific direction, the error might be decreased compared to when the light is completely depolarized. This point underscores the

<figure><img src="image_6.png" /><figcaption>Fig. 3. Depolarization ratio as a function of wavelength for a reference Spectralon target measured with the polarization camera (solid line) and that measured with the scattering sensor (data points with error bars). The red line represents the 99% reflectance Spectralon target (SRS-99), while the blue line corresponds to the 2% reflectance target (SRS-02). The depolarization ratios at $\lambda = 535$nm are $\delta_L = 91\%$ and $\delta_L = 4.8\%$, respectively, for SRS-99 and SRS-02.</figcaption></figure>
importance of accurate calibration, particularly in scenarios involving depolarized light, where deviations can significantly impact the measurement accuracy. Also, note that while the Spectralon targets are reflectance standards, there are no polarimetric standards for more diffuse scattering objects like the particles investigated below, and thus, we view the outcome of this calibration procedure to be as favorable as possible.

Error estimation for the depolarization measurements involves two types of error. The first arises from the signal-to-noise ratio of the scattering sensor. We measure the random readout noise in the images by performing measurements without particles and with the laser off, i. e., a dark measurement. We do so as

$$ \Delta_1 = \delta_L \sqrt{\left(\frac{\sigma_n}{S_0}\right)^2 + \left(\frac{\sigma_n}{S_1}\right)^2}, \qquad (15) $$

where $\Delta_1$ is the absolute error due to readout noise, $\sigma_n$ represents the standard deviation between pixel intensity in the dark measurement, and $S_0$ and $S_1$ correspond to the intensity signals of different polarization states. The second error is the average deviation of the depolarization compared to the reference measurements from the Spectralon targets. We refer to Fig. 3 where an error of approximately 4% at $\delta_L \sim 80\%$ is observed. The error associated with this deviation, which depends on $\delta_L$ is

$$ \Delta_k = \sqrt{\frac{1}{N} \sum_{i=0}^{N} (\delta_{k,i} - \overline{\delta_{k,ref}})}. \qquad (16) $$

In Eq. (16), $k$ is the value of the Spectralon-target reflectance, $\delta_{k,i}$ denotes the measured depolarization for the $i^{th}$ wavelength $\lambda$ or scattering direction $(\theta, \phi)$, $\overline{\delta_{k,ref}}$ corresponds to the reference depolarization measured with the polarization camera, and $N$ represents the number of measurement points for either $\lambda$ or $(\theta, \phi)$. Since the error depends on the Spectralon value, and consequently, the depolarization, we establish a linear relationship between the errors at different depolarization levels as

$$ \Delta_2(\delta_L) = \delta_L \left( \frac{\Delta_{99} - \Delta_{02}}{\delta_{99} - \delta_{02}} \right) + \left[ \Delta_{02} - \left( \frac{\Delta_{99} - \Delta_{02}}{\delta_{99} - \delta_{02}} \right) \delta_{02} \right] \qquad (17) $$

Finally, assuming these errors are independent, the total error is

$$ \Delta(\delta_L) = \sqrt{\Delta_1^2 + \Delta_2^2}. \qquad (18) $$

Next, we use spherical microparticles to calibrate the angular aspects of the SPB measurements. These particles are quasi-monodisperse soda-lime glass spheres (Thermo Scientific Duke Standards, 9030) with diameters of 32 µm. The particles are not perfectly spherical, showing a sphericity of approximately 90% as determined from the holographic images. Specifically, an ellipse is fit to the holographic particle image, and $\eta$ is defined as the ratio of the major-to-minor semi-axes. The particles are in dry-powder form and a small quantity is dispersed across the sample window W by tapping a small spatula containing a minute amount of the powder over the glass. Compressed air is then used to remove most of the particles from W, and the process is repeated, if necessary, until only one particle is visible on W in the region where the supercontinuum beam will illuminate the glass.

Fig. 4 presents the SPB patterns measured for a single spherical 32 µm diameter particle for $\lambda = 535$ nm. A notable feature is the distortion of the symmetry of the scattering pattern. As described by Berg et al. [21], a perfect sphere illuminated with either vertically or horizontally polarized light will have zero depolarization and an angular scattering pattern with mirror symmetry about the vertical and horizontal scattering planes, i.e., the y-z and x-z planes, or in Fig. 4 about the $\phi = 90^\circ$ or $270^\circ$ and $\phi = 0^\circ$ or $180^\circ$ planes, respectively. The symmetry will be displayed for all polar scattering angles $\theta$ as well. While the patterns in Fig. 4 are close to showing this symmetry, some degree of aberration is evident, which can be attributed to the slight non-sphericity of the particle or aberrations from the optical elements, or both. Under the assumption that this particle is suitably spherical, we introduce an angular power-law based on the formula given by Drap and Lefèvre [22] to correct for the radial image distortion caused by lens L3 in Fig. 1. The radial coordinate in the image corresponds to the polar scattering angle $\theta$, and thus, we use the polynomial:

$$ \theta \rightarrow g\theta + q\theta^2. \qquad (19) $$

In Eq. (19), the parameters $g$ and $q$ are adjusted to bring the measured backscattering patterns in Fig. 4 into agreement with those simulated via Mie theory for a 32 µm glass particle with a refractive index of $m(\lambda = 535nm) = 1.52 + 10^{-6}i$. Power-law corrections of this form are also applied to small-angle light-scattering measurements in other work, see [5,23,16].

The intensity pattern in Fig. 4(a) exhibits a distinctive nested-ring form, which is expected for a spherical particle. Meanwhile, the depolarization pattern $\delta_L(\theta, \phi)$ in Fig. 4(b) reveals significant values centered on the azimuthal scattering angles $\phi = 45^\circ, 135^\circ, 225^\circ$ and $315^\circ$ in a cloverleaf pattern. Although the degree of total polarization of the light

<figure><img src="image_7.png" /><figcaption>Fig. 4. Measured (a) two-dimensional angular backscattered total intensity and (b) depolarization patterns for a single 32 µm diameter spherical glass particle on W in Fig. 1. In (a) is shown the first element of the Stokes vector $I(0, \phi)$ for the sum of the vertical and horizontal polarization states. In (b) is shown the linear depolarization ratio $\delta_L(0, \phi)$ corresponding to (a). The wavelength is $\lambda = 535$nm.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Total backscattered intensity *I* as a function of polar scattering angle θ (horizontal axis) and wavelength λ (vertical axis) for a single 32 µm diameter glass spherical particle. The two curves in the plots represent cuts through *I*(θ, λ) at either θ = 180° or λ = 535 nm. The plot in (a) shows the measurements while (b) displays the results of simulation using Mie theory. An image of the particle producing this scattering data is obtained from DIH and shown inset in (a). The particle diameter measured from this image is *D* = 32 ± 3µm.</figcaption></figure>

<figure><img src="image_9.png" /></figure>

scattered by a perfect sphere is 100% in all scattering directions when it
is illuminated by linearly polarized light [21], the degree of linear po-
larization is dependent on (θ, φ). Such strong linear depolarization can
be attributed to refraction within the particle and the resulting
non-uniform spatial character of the particle’s internal electric field as
explained by Berg et al. [21].

To ease later analyses, the scattering patterns are averaged over the azimuthal angle φ, resulting in a curve depicting scattering and depolarization as a function of either θ or λ as shown in Fig. 5. Specifically, the values of each scattering-sensor pixel associated with the same θ, but different φ, are averaged to obtain a single value. The averaging is done using the power-law correction of Eq. (19). Fig. 5(a) shows the azimuthal average of the pattern in Fig. 4(a), while Fig. 5(b) shows the results from simulations with Mie theory for a 32 µm diameter glass particle. One can see a favorable degree of agreement between the measurement and theory across the full range of θ and λ, which supports the efficacy of the angle correction of Eq. (19). Fig. 5(a) also shows the DIH image of the particle. An estimate for its diameter from the image is D = 32 ± 3µm, which is in excellent agreement with the known size from the manufacturer.

4. Results

We now consider SPB measurements and DIH imaging of a variety of
particle types. First, glass microspheres are considered again to further
test the SPB method against Mie theory, but for a different particle size
and number of particles. Then, Gobi and Saharan-Desert mineral dust
and volcanic ash are studied as proxies for important categories of at-
mospheric aerosol particles in lidar contexts.

4.1. Glass microspheres

To investigate the effect of particle size on the measurements, Fig. 6
presents the depolarization ratio δ<sub>L</sub>(θ,λ) for a collection of 10 µm diam-
eter quasi-spherical glass particles (Thermo Scientific, Duke Standards,
9010). The particles are deposited across W in the manner described in
Section 3.2, except here, multiple particles are allowed to remain on the
glass. Of particular interest is δ<sub>L</sub> for λ = 535nm at the exact backscat-
tering direction θ = 180°, a quantity we abbreviate δ<sub>L</sub>(180°,535nm).

<figure><img src="image_10.png" /><figcaption>**Fig. 6.** Linear depolarization ratio δ<sub>L</sub>(0,535nm) for a collection of 10 µm diameter glass particles. An image of several of the many particles producing this data is obtained from DIH and shown inset (bottom left) in the plot.</figcaption></figure>

According to Mie theory, this value should equal zero for any λ for a
single spherical particle. However, the data shows that δ<sub>L</sub>(180°,535nm)
= 4.3 ± 1.4%. Given the field-symmetry analysis of Berg et al. [21], we
attribute this behavior to the slight non-sphericity of the particles. Even
for these smaller 10 µm diameter particles, which have a size parameter
kR = 2πR/λ between 30 and 60 (due to the range of λ), refraction inside
the particle is significant and the non-sphericity readily breaks the
mirror symmetry of the internal field that otherwise would yield
δ<sub>L</sub>(180°,λ) = 0 for a perfect sphere. Note that similar behavior for
comparable values of kR is seen in Bi et al. [24] as well. Fig. 6 also shows
that other scattering angles exhibit yet larger δ<sub>L</sub>(0,535nm). A portion of
the full holographic image of the particles illuminated by the super-
continuum beam is shown inset in Fig. 6. This provides a sense for the
monodisperse size of these particles. The image also permits measuring
the size *D* and axis-ratio η of the particles, which is *D* = 10 ± 3µm and η
= 1.1, respectively.

Unlike the measurement in Figs. 4 and 5, where only a single 32 µm
diameter microsphere is deposited on the sample window W, many 10
µm diameter microspheres are present in Fig. 6. Because the particles are
randomly distributed on W, some may be positioned close enough to a
neighbor to suggest that multiple scattering is significant. If so, that
could account for the nonzero δ<sub>L</sub> values observed at θ = 180°. The ho-
lographic image inset in Fig. 6 shows only a selection of the many 10 µm
particles on W that are illuminated by the supercontinuum beam. An
analysis of the full holographic image showing all the particles within
this beam is conducted to estimate the average separation ⟨r⟩ between
neighboring particles, which is ⟨r⟩ ~ 115 µm. The portion of the full
holographic image chosen for Fig. 6 does not depict this degree of inter-
particle separation due to the desire to adequately present the size and
shape of several individual particles in a small inset image in the figure.
Nevertheless, using an average separation of ⟨r⟩ ~ 115 µm suggests
that, in fact, the particles overall are sufficiently separated from each
other to suppress the effects of multiple scattering [25,26]. That is,
multiple scattering is expected to be negligible if k⟨r⟩ > 60, which is
satisfied in this case. Note, however, that the 10 µm diameter particles
are larger than those used in [25,26], and so, we cannot definitively
claim that multiple scattering does not contribute to some degree.

Another possible explanation for the nonzero values of δ<sub>L</sub> at θ = 180° in Fig. 6 relates to multiple scattering between the microspheres and the glass window W supporting the particles. We neglect this possibility for several reasons. First, if W were perfectly normal to the supercontinuum beam, the symmetry of the window-particle scattering arrangement would prohibit nonzero values for δ<sub>L</sub> at θ = 180° according to Berg et al. [21]: although W is slightly tilted with respect to this beam, in violation of the symmetry arguments, the tilt angle is small, < 1°, and we do not expect it to cause a significant departure from the expected behavior. Second, and perhaps most relevant, is that the effect of W without particles is accounted for in the polarization calibration described in Section 3.2. For these reasons, we conclude that the likely origin for the nonzero values of δ<sub>L</sub> at θ = 180° in Figs. 4–6 is primarily the slight non-sphericity of the particles.

## 4.2. Desert mineral dust

The term "dust" refers to solid particles with sizes ranging from approximately one to a few hundred microns. The particles are examples of a general class of coarse-mode aerosols, i.e., large particles that broadly include mineral particles originating from deserts and agricultural areas, plant fragments, volcanic ash, and particles generated in various workplace environments. Arizona test dust (ATD) is a commercially available sample that serves as a representative and heterogeneous mixture of different mineral particles. It consists of irregularly shaped particles like those commonly found in the atmosphere and is extensively studied in the literature where its light-scattering behavior can, perhaps, be regarded as well-known. Here, ATD particles are deposited on W in the same manner as above and SPB and DIH imaging is conducted. The results are presented in Fig. 7. The density of particles on W is adjusted, via the application of compressed air until sufficiently disperse that the single-scattering approximation should hold [25]. The particles shown in the inset image in Fig. 7, obtained from DIH, are approximately 25 ± 3µm in size. The average axis-ratio is ⟨η⟩ = 1.4, also determined from the DIH image. Generally, the ATD is highly polydisperse and it is difficult to quantify a size distribution for particles with such irregular shapes and size variability.

The focus here is to study the linear depolarization ratio and compare
it with existing lidar measurements where a value of δ<sub>L</sub> ~ 35% is ex-
pected [27–29]. The ATD particles, like other non-spherical particles,

<figure><img src="image_11.png" /><figcaption>**Fig. 7.** Linear depolarization ratio δ<sub>L</sub>(0,λ) for Arizona test dust (ATD). The two curves in each plot represent cuts through δ<sub>L</sub>(0,λ) at either θ = 180° (left panel) or λ = 535 nm (right panel). An image (bottom left) of several of the many particles producing this data is obtained from DIH and shown inset in the plot.</figcaption></figure>

exhibit a non-uniform backscattering pattern (not shown). Fig. 7 reveals
that as λ increases from λ = 500 nm to λ ~ 600 nm, so does δ<sub>L</sub>(180°,λ),
whereas δ<sub>L</sub>(180°,λ) then decreases and fluctuates for λ > 600 nm. There
is also a decrease in δ<sub>L</sub>(0,535nm) as θ increases toward the exact back-
scattering direction. We find that δ<sub>L</sub>(180°, 535 nm) ~ 29 ± 3.0%,
which is comparable to the literature values. Note, however, that the
size of the particles in Fig. 7, i.e., ~ 25 µm, is larger than most dust
aerosols measured by lidar [30–32]. The difference in size may account
for the slightly smaller values of δ<sub>L</sub>(180°,535nm) found here in com-
parison to the literature values.

Next, we consider Gobi and Saharan-Desert dust, and the results are
presented in Fig. 8. Both samples are collected at sites far from their
sources, and thus, are formerly aerosolized atmospheric particles. The
Saharan dust sample is collected from the 2004 Sahara dust intrusion
event at the Observatory of Sierra Nevada (OSN) in Granada, Spain.
Located 2896 m above sea level, the OSN is at least 1500 km from the
dust source. The Gobi sample was collected in Beijing during a heavy
dust storm. Further information on the samples is provided by Gómez
Martín et al. [10].

From the DIH images of these samples (Fig. 8 insets), an approxi-
mate, average particle-size of 22 ± 3µm and 23 ± 3µm, for Gobi and
Saharan dust, respectively is found. An approximate axis-ratio of ⟨η⟩ =
1.4(Gobi) and ⟨η⟩ = 1.5 (Saharan) is also found. Fig. 8 shows a more
erratic depolarization behavior than the ATD, Fig. 7, both as a function
of θ and λ. In particular, δ<sub>L</sub>(180°, 535 nm) ~ 36 ± 2.0% for Gobi
dust, Fig. 8(a), and δ<sub>L</sub>(180°, 535 nm) ~ 33 ± 8.0% for Saharan dust,
Fig. 8(b). Values like those obtained at λ = 532nm for ATD are seen [33].
Note that the δ<sub>L</sub>(0,λ) measurements in Fig. 8(a) and(b) are within a
standard deviation of each other, and so, it is difficult to claim a
dust-type differentiation capability based on these results. Compared to
the ATD, these particles show an overall smaller magnitude variation in
δ<sub>L</sub>. There are more particles present on W for the desert dust samples
than the ATD sample, which may accentuate an averaging effect for δ<sub>L</sub>
and account for the decreased variation.
<figure><img src="image_12.png" /><figcaption>Fig. 8. Same as Fig. 7 except for Gobi Desert dust (a) and Saharan Desert dust (b).</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 9. Same as Figs. 7 and 8 except for ash from the Eyjafjallajökull volcano (a) and the La Palma volcano (b).</figcaption></figure>

4.3. Volcanic ash

Volcanic ash clouds generated during explosive eruptions have profound impacts on climate, weather, and health. Among the many types of ash particles, those with size less than ~ 60 µm are important due to their prolonged residence in the atmosphere. Furthermore, the scattering of light by volcanic ash significantly affects air quality and visibility, and ingestion of the particles by aircraft engines poses substantial safety concerns. The irregular shape and wide size-distributions of ash particles complicate a determination of their scattering behavior. Variations in composition, surface roughness, and internal structure pose significant uncertainties for remote-sensing measurements with respect to accurately quantifying the concentration, composition, and other properties of the aerosols. Notably, the measurement of linear depolarization exhibits a high degree of variability depending on the specific ash type, and extensive scattering data exists for volcanic ash [34,35].

In Fig. 9, we consider two samples of ash obtained from the Icelandic volcano Eyjafjallajökull and the Canary Island of La Palma. The Eyjafjallajökull ash sample was collected following the April 2010 eruption at 5 km from the source [10,11]. The La Palma ash sample consists of pyroclastic material collected from a surface deposit during the 2021 eruption of the Cumbre Vieja volcanic ridge at La Palma Island, Spain.

Like the dust particles, the Icelandic ash displays significant variation in the depolarization ratio $\delta_L$, see Fig. 9(a). The average depolarization is $\delta_L(180°, 535 nm) \sim 25 \pm 1.4\%$. In terms of its spectral dependence, $\delta_L$ varies less with $\lambda$ but exhibits a peak at $\lambda = 650$ nm. The Icelandic ash particles exhibit an average size of 21µm and are separated by approximately 100 µm on the window as determined from the DIH image. The average particle axis-ratio is approximately $\langle\eta\rangle = 1.6$, which is larger than that of desert dust. One factor that may account for the different behavior of $\delta_L(\theta, \lambda)$ here as compared to the desert dust is the greater absorption, or imaginary part of the refractive index, that the ash particles have. That is, the visual appearance of the ash is black
<figure><img src="image_14.png" /><figcaption>**Fig. 10.** Linear depolarization ratio $\delta_L(\lambda)$ around the backscatter direction for the particle types considered in Figs. 7–9. The depolarization $\delta_L(\theta, \lambda)$ is integrated over the scattering-angle range $179^{\circ} \le \theta \le 180^{\circ}$ to yield the $\delta_L(\lambda)$ shown.</figcaption></figure>

compared to the more yellow-white appearance of the dust, a hallmark of significant broadband absorption in the visible spectrum.

For the La Palma ash, Fig. 9(b), the depolarization is significantly more uniform in scattering angle $\theta$ as compared to the Icelandic ash and shows an overall increase with $\lambda$, with a peak around $\lambda = 700$ nm. The average depolarization is $\delta_L(180^{\circ}, 535$ nm) $\sim 17 \pm 2.2\%$. These $\delta_L$ values are smaller than those reported in existing lidar observations, which typically show $\delta_L \sim 35\%$ [36]. However, as mentioned, the particles here are significantly larger than typical atmospheric aerosols, which may influence the comparative behavior. Section 5 proposes an analysis based on particle morphology, suggesting that shape differences have a significant impact on $\delta_L$. The particles exhibit an average size of 23µm and are separated by approximately 79 µm on the window as determined from the DIH image. The average particle axis-ratio is approximately $\langle\eta\rangle = 1.4$, which is larger than that of desert dust. We suspect that the particle size distribution is also broader than the Icelandic ash, which could indicate a greater proportion of large particles. The increased variability of particle size may account for the more uniform $\delta_L$ behavior seen in Fig. 9(b). Unlike the dust results in Fig. 8, the results for both ash samples do imply an ability to differentiate particle types based on depolarization characteristics and suggest an association between $\delta_L$ and particle non-sphericity and absorption.

### 4.4. Summary of results

Here we further analyze the depolarization $\delta_L(\theta, \lambda)$ for the various particle types. The behavior of $\delta_L$ as a function of wavelength is especially significant for lidar [37]. The intent is to consolidate the information presented above in a graph, Fig. 10, that illustrates $\delta_L(\theta, \lambda)$ integrated between $\theta = 179^{\circ}$ and $\theta = 180^{\circ}$, i.e., $\delta_L(\lambda)$ for short. The integration is meant to recognize the finite angular range resolved by lidar systems. One can see in Fig. 10 a relatively small depolarization for volcanic ash, around $\delta_L(\lambda) \sim 25\%$, while dust from both the Gobi and the Sahara Deserts shows nearly equivalent depolarization around $\delta_L(\lambda) \sim 30\%$. Arizona test dust shows the largest values, around $\delta_L(\lambda) \sim 40\%$. Lidar data from Haarig et al. [37] reveals a depolarization peak at $\lambda = 532$ nm and a decrease for larger wavelengths, which somewhat coincides with our observations for some particles, although ATD shows an increase in depolarization toward $\lambda = 800$ nm.

It should be repeated that, although the depolarization observed for ATD may appear large compared to some atmospheric lidar measurements, such comparisons should be approached with caution. The particles here are much larger than those generally observed in lidar studies. It is also relevant that the hygroscopicity of aerosols can have a significant impact on their radiative properties [38–40], especially the depolarization ratio, and the experiment does not consider such effects. To summarize the salient results of the work, Table 1 shows the primary

<caption>**Table 1**</caption>

Summary of lidar-relevant quantities measured for the different particle types in Figs. 6–9. The average particle size and axis ratio $\langle\eta\rangle$ are obtained from the DIH images as described in Section 3.2. The remaining quantities are obtained from the SPB measurements. The linear depolarization ratio for $\theta = 180^{\circ}$ and $\lambda = 535$ nm is given by $\delta_L(180^{\circ}, 535$ nm) and is one quantity commonly used in lidar. The quantity $\langle\delta_L(535$ nm)\rangle_{\theta}$ is the depolarization ratio for $\lambda = 535$ nm averaged over all $\theta$, while $\langle\delta_L(180^{\circ})\rangle_{\lambda}$ is the ratio for $\theta = 180^{\circ}$ but averaged over all $\lambda$.

<table><thead><tr><th></th><th>size~[µm]</th><th>⟨η⟩</th><th>δ<sub>L</sub>(180°,535nm)~[%]</th><th>⟨δ<sub>L</sub>(535nm)⟩<sub>θ</sub>~[%]</th><th>⟨δ<sub>L</sub>(180°)⟩<sub>λ</sub>~[%]</th></tr></thead><tbody><tr><td>Spheres</td><td>10 ± 3</td><td>1.1</td><td>4.3 ± 1.4</td><td>35 ± 2.1</td><td>5.2 ± 0.6</td></tr><tr><td>ATD</td><td>25 ± 3</td><td>1.4</td><td>29 ± 3.0</td><td>32 ± 2.4</td><td>38 ± 3.0</td></tr><tr><td>Gobi dust</td><td>22 ± 3</td><td>1.4</td><td>36 ± 2.0</td><td>42 ± 2.5</td><td>36 ± 2.0</td></tr><tr><td>Sahara dust</td><td>23 ± 3</td><td>1.5</td><td>33 ± 8.0</td><td>32 ± 2.0</td><td>33 ± 1.8</td></tr><tr><td>Icelandic ash</td><td>21 ± 3</td><td>1.6</td><td>25 ± 1.4</td><td>23 ± 1.7</td><td>25 ± 1.4</td></tr><tr><td>La Palma ash</td><td>23 ± 3</td><td>1.4</td><td>17 ± 2.2</td><td>15 ± 1.4</td><td>22 ± 1.4</td></tr></tbody></table>
lidar-relevant quantities resulting from the SPB and DIH measurements above. In addition to the particle size, $\langle\eta\rangle$, and $\delta_L(180^{\circ},535nm)$, the depolarization ratios averaged over all $\theta$ measured, $\langle\delta_L(535nm)\rangle_{\theta}$, or over all $\lambda$ measured, $\langle\delta_L(180^{\circ})\rangle_{\lambda}$, are presented to give a more general sense for the behavior for different particle types.

## 5. Discussion

We now investigate a correlation between the SPB measurements and particle shape. Such a correlation could be useful to better characterize aerosol particles in lidar studies, e.g., to link particle size or type to lidar measurements. The SPB measurements of Figs. 7–9 are repeated multiple times, eight times for the ATD, three times for the Gobi and Saraha dust, eight times for the Icelandic ash, and twice for the La Palma ash. By analyzing the DIH images in each trial, an average particle axis-ratio $\langle\eta\rangle$ is found and the corresponding SPB measurements provide $\delta_L(\theta,\lambda)$ as before. Because the particle distribution for a given particle type is different in each trial, the values of $\langle\eta\rangle$ are different and will not necessarily correspond to the single-trial values associated with Figs. 7–9 and reported Table 1 above. Furthermore, the measured depolarization ratio $\delta_L$ is integrated over a narrow angular range in the backscattering direction, $179^{\circ} \le \theta \le 180^{\circ}$ as in Fig. 10, and averaged over all wavelengths, from 450nm $\le \lambda \le 850nm$, or $\langle\delta_L\rangle_{\lambda}$ for short. The results are then plotted as a function of $\langle\eta\rangle$ in Fig. 11. The error bars on each datapoint in the plot express the spectral variability, i.e., they show the standard deviation of the angle-integrated $\delta_L$ values in the average over $\lambda$.

Fig. 11 reveals an approximate empirical relationship between $\langle\delta_L\rangle_{\lambda}$ and $\langle\eta\rangle$. Let $\langle\delta_L^{\text{dust}}\rangle_{\lambda}$ represent the wavelength-averaged depolarization for the dust samples (ATD, Gobi, and Sahara) and $\langle\delta_L^{\text{ash}}\rangle_{\lambda}$ represent the ash samples (Icelandic, La Palma), then the relationship in Fig. 11 is:

$$ \langle\delta_L^{\text{dust}}\rangle_{\lambda} = 0.60\langle\eta\rangle - 0.48, \qquad (20) $$

and

$$ \langle\delta_L^{\text{dust}}\rangle_{\lambda} = 0.28\langle\eta\rangle - 0.17. \qquad (21) $$

The data shows that dust particles, ATD in particular, have the strongest dependence of $\langle\delta_L\rangle_{\lambda}$ on $\langle\eta\rangle$. The ash particles show a weaker

dependence and show a smaller wavelength variability (the error bars). All of the dust types show a similar behavior, indicated by the blue region in the plot, while the ash types also show a similar behavior, shown by the red region. This observation reinforces the idea that backscatter depolarization is strongly influenced by the chemical and morphological properties of the particles, and thus, is helpful to differentiate between particle types in remote sensing. Eqs. (20) and (21), may also be useful as a reference for future studies investigating depolarization, particularly for lidar.

We note that the average axis ratios $\langle\eta\rangle$ found for the dust and ash in Fig. 11 do not exceed $\langle\eta\rangle = 1.8$, which is important when comparing our results to existing studies in the literature. For example, the simulations using the T-matrix method [41] show a depolarization behavior largely in agreement with Fig. 11, although that work goes well beyond the $\langle\eta\rangle = 1.8$ limit we consider.

Comparing the depolarization values in this work to lidar observations of desert dust in the literature at $\lambda = 523nm$ reveals some agreement [37,42]. However, our values appear slightly larger, by around 5%. Such discrepancies may arise from differences in particle size distributions, particle surface roughness, and other factors. Notably, dust plumes are a mix of fine and coarse particles, while our measurements focus on coarse particles. Despite the discrepancies, the work here involves a significantly larger number of measurement points in both $\theta$ and $\lambda$, providing a more comprehensive characterization of the scattering and depolarization properties.

For volcanic ash, a comparison with lidar observations at $\lambda = 523nm$ [34,35] shows slightly smaller depolarization values than what is commonly reported. Like dust, ash plumes consist of a combination of fine and coarse particles, with the fine particles more likely to be aged and the coarse fresh. Recent polarimetric lidar observations around the Cumbre Vieja volcanic area (La Palma) have reported significant variations in linear depolarization ratios [43].

Finally, we summarize some limitations in our study. The use of linear polarizers precludes the measurement of circular polarization, an important quantity in remote-sensing aerosol characterization. The use of the glass window to support stationary particles introduces at least some degree of interference with the way that true aerosol particles would scatter light. Future measurements could partly address these limitations by examining and ensemble of free-flowing aerosols, or even

<figure><img src="image_15.png" /><figcaption>Fig. 11. Linear depolarization ratio $\delta_L$ integrated over the angular range $179^{\circ} \le \theta \le 180^{\circ}$ and averaged over all wavelength $450nm \le \lambda \le 850nm$, labeled as $\langle\delta_L\rangle_{\lambda}$, and plotted as a function of the average particle axis ratio $\langle\eta\rangle$ for the dust and ash particles.</figcaption></figure>
the examination of single aerosol particles using a triggered laser [9] and with a holographic component to obtain particle images [44]. Further improvements could involve the use of liquid crystal quarter-wave plates operating at multiple wavelengths, allowing for the generation and measurement of circular polarization [45]. An intriguing measurement would be the backscattering Angström exponent (BAE), which could provide important insight into the variation of scattering characteristics with wavelength [46–48]. While traditionally measured with monochromatic laser sources, extending these measurements to dual wavelengths [49], could also enhance the characterization of aerosol properties via remote sensing.

## 6. Conclusion

Our work provides a measurement method to acquire broadband two-dimensional angular backscattering and linear depolarization characteristics of stationary particles representative of certain atmospheric aerosols. Integration of digital in-line holography yields microscopic images of the particles producing the measurements, providing a sense for the particle size, shape, and abundance. The method is calibrated with respect to polarization and angular sensitivity using size-standard spherical microparticles. Then, a variety of samples, including desert dust and volcanic ash, are studied with particular emphasis on the backscatter linear depolarization ratio $\delta_L$. Some key results are $\delta_L = 36 \pm 2.0$ and $\delta_L = 33 \pm 8.0$ for Gobi and Sahara Desert dust, and $\delta_L = 25 \pm 1.4$ and $\delta_L = 17 \pm 2.2$ for the Icelandic and La Palma volcanic ash, respectively, at $\theta = 180^\circ$ and for $\lambda = 535$nm. We discuss various limitations of the measurements and assess the results in the context of existing lidar studies. In all, the work offers a new and capable method to characterize the broadband light scattering and polarimetry of particle samples and may have use to advance lidar studies of aerosols in many applications.

## CRediT authorship contribution statement

**Killian Aleau:** Writing – original draft, Visualization, Software, Data curation. **Romain Ceolato:** Writing – review & editing, Validation, Resources, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization. **Matthew Berg:** Visualization, Writing – review & editing, Software, Methodology. **Guillaume Huss:** Project administration, Funding acquisition. **Olga Muñoz:** Writing – review & editing, Formal analysis. **Gorden Videen:** Writing – review & editing, Formal analysis.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Data availability

Data will be made available on request.

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