To describe the scattered light measured by the instruments above, we use Stokes's formalism: 1Isca(θ)Qsca(θ)Usca(θ)Vsca(θ)=σsca⋅ΔV4πr2⋅P(θ)‾‾⋅IinQinUinVin. Here, the incident and scattered light are described using Stokes's parameters for intensity (I) and the polarization ellipse (Q, U, and V) (Hansen and Travis, 1974). r is the distance of the detector from the scattering event. When interpreting this equation, it is helpful to remember that the total scattering, i.e., integrated over all angles, should be equal to the product of the scattering coefficient (σsca), the volume of the scattering medium (ΔV), and the incident light intensity (Iin). Thus, it becomes clear that the aerosol scattering matrix, P(θ)‾‾, is (a) the only factor with an angular dependence and (b) normalized such that it will integrate over all angles to equal 4π. We can think of the aerosol scattering matrix as a function which evaluates the probability that incident light will be scattered in a given direction, while preserving information regarding its polarization. P(θ)‾‾, defined in Eq. (2), is a 4×4 matrix which due to symmetry consists of six unique elements for randomly oriented particles that do not possess intrinsic optical activity (Bohren and Huffman, 1983). 2P(θ)‾‾=P11(θ)P12(θ)00P12(θ)P22(θ)0000P33(θ)P34(θ)00-P34(θ)P44(θ) Using different approximation methods, each of these elements can be calculated for a particle of known size and composition. Under single-scatter conditions, the elements of an aerosol population are the scattering cross-section-weighted sum of the elements from individual particles. Mie theory is the most commonly used method for calculating the intensity and polarization state of light after scattering with spherical aerosols and thus is the foundation of aerosol microphysical retrievals (Dubovik and King, 2000; Mie, 1908). For the LiNeph, the incident light can be defined with respect to the orientation of the observing camera relative to the polarization of the linearly polarized lasers. For the perpendicular (“Perp”) camera shown in Fig. 1, the Stokes vector used to evaluate Eq. (3) is as follows: 3IinQinUinVinPerp=1-100. This is because the axis of the CCD (along the x axis) is approximately orthogonal to both the propagation (along the y axis) and the polarization of the lasers (along the z axis). In reality, a small offset in the z axis (∼3.6 mm) from the optical axis of the wide angle lens introduces a small angle which describes the scattering plane rotation angle, η (Dolgos, 2014). For now, we assume η is zero, although we revisit this assumption in Sect. 2.4 as it has important implications for the accuracy of the measurement. Solving Eq. (1) for this idealized case means that the measured parameter, Iscat,Perp(θ), contains information about two elements from the scattering matrix, P11(θ) and P12(θ), as shown in Eq. (4): 4Isca,Perp(θ)=σsca⋅ΔV2πr2⋅[P11(θ)-P12(θ)]. Similar treatment for the parallel (“Para”) camera shows that the combined measurements can be used to solve for both P11 (commonly referred to as the scattering phase function) and P12, as shown in Eqs. (5) and (6): 5IinQinUinVinPara=1100,6Isca,Para(θ)=σsca⋅ΔV2πr2⋅[P11(θ)+P12(θ)]. P12 is typically reported for convenience as the degree of linear polarization (DoLP), -P12/P11. Below we will discuss the capabilities and limitations of the new aircraft-deployable LiNeph, as well as present some initial data from the FIREX-AQ field campaign studying wildfire smoke onboard the NASA DC-8.

The LiNeph uses two continuous wave laser beams as the light sources to measure the light scattering of an aerosol sample at two different wavelengths. The emissions from the OBIS 660 nm LX 100 mW diode laser (Coherent, Santa Clara CA, USA) and the LuxX 405-120 diode laser (Omicron, Rodgau, Germany) are directed into the aerosol sample chamber using turning mirrors, shown in Fig. 1a. For all the work presented here, the lasers were operated at 15 % of full power, 15 and 18 mW for the 660 and 405 nm lasers, respectively. Before entering the aerosol sample chamber, the lasers pass through a Glan–Taylor polarizer (GT10-A, Thorlabs, Newton MA, USA) to ensure linear polarization and then an anti-reflective coated window (VPW42-A, Thorlabs, Newton MA, USA). We use a series of four black nylon 3D-printed apertures to reduce stray light entering the chamber. The stray light reflects off the interior of the black-painted sample cell and is imaged by the detectors, resulting in increased noise when there is low signal. The lasers have a diagonal offset which enables aerosol scattering from both beams to be imaged by both cameras, as shown in Fig. 1b.

Sample flow is pulled through the instrument sample cell by an external diaphragm pump and controlled by a mass flow controller (MCR-50, Alicat, Tuscon AZ, USA). For the FIREX-AQ mission aboard the NASA DC-8, a sample flow rate of 15 L min-1 was used to maximize the sample exchange rate in the ∼3 L sample volume and thus improve the ability of the instrument to resolve spatial changes in aerosol concentration as the aircraft penetrated a smoke plume. For ground-based measurements, lower flow rates could be used if the aerosol composition is not expected to change rapidly. Since some particles may be hygroscopic, the instrument exhaust is characterized using a temperature and relative humidity probe (HMP110, Vaisala, Vantaa, Finland). The sample cell pressure is monitored using a precision pressure transducer (PPT0015AXN5VA, Honeywell, Charlotte SC, USA).

The two LiNeph CCD detector arrays (16-bit, 2750×2200 pixel, cooled to -40 ∘C, Trius-SX694, Starlight Xpress, Bracknell, UK) record the images from the orthogonally mounted wide-angle, f-theta type lenses (FE185C046HA-1; Fujifilm, Tokyo, Japan). These images show the light scattered by everything in the field of view of the wide-angle lenses, including the instrument optics and interior, gases with non-negligible scattering cross sections, and particles. Since the particles are the species of interest, a high-efficiency particulate arrestance (HEPA) filter was interposed upstream of the sample volume to remove particles approximately every 5 min for a 45 s duration; see Fig. 1a. The two-way valves (MDM-060DT, Hanbay Inc., Virginia Beach VA, USA) were automated and controlled using a custom Labview program (National Instruments, Austin TX, USA) that also handled the data acquisition.

The particle-free, background images with identical optical and detector conditions (laser power and CCD exposure time) are averaged from before and after a sample period, and the resulting image is subtracted from sample images. An example background-subtracted image is shown in Fig. 2. The two arcs are from particles illuminated by the 405 and 660 nm lasers on the top and bottom, respectively, distorted by the wide-angle lens. The reported units are bits, which shows the full scale of the 16-bit detector. Bits are converted into a differential scattering coefficient (σ∘), Mm-1 sr-1, which will be a function of the CCD exposure time, as described in Sect. 2.3. In this image, the lasers propagate from left to right, and thus lower (higher) pixel columns show forward (backward) scattering. In addition to light scattered directly by the particles, the CCD arrays also detect stray or multiply scattered light. An example of multiply scattered light is shown in Fig. 2. Columns 20–60 and rows 60–100 show the light scattered by the particles and then again by the other wide-angle lens. Our background subtraction cannot account for these secondary scattering events, but we minimize the effect by darkening the interior of the instrument where possible and by excluding the affected pixels from the analysis.

From the background-subtracted image, two Gaussian functions are fit to each pixel column, one for each laser, excluding parts of the image that do not overlap with the laser path to the extent that is possible. The area under these Gaussian fits proportional to the particle scattering matrix elements (P11+P12) and (P11-P12) for the cameras oriented parallel and perpendicular to laser polarization, respectively. The precision of this method of analysis depends on the temporal stability of both the detector and the subtracted elements, which in turn rely on the stability of the pressure of the sample and the power of the lasers. If any of these elements, or the detector, changed in sensitivity, then the background subtracted images would be biased. During FIREX-AQ, we accounted for varying sample pressure by taking filter samples before and after any changes in aircraft altitude. Since automated filter samples are collected every 5 min, barring operator deferment, we will show in Sect. 3.1 that no drift in detector response was detected for greater than 10 min. Further, filter periods at the same altitude during the research flights (>2 h) showed a similar response, indicating that the lasers and detectors are stable and that the filtered air images are valid representations of instrument background and scattered light from gaseous species.

The calibration of the LiNeph requires the sampling of gases and aerosols of known size and composition. Figure S1 in the Supplement shows the lab set up for calibration of the LiNeph. For calibrations using a pure gas, either CO2 or He, the LiNeph is pumped down to 125 hPa using an IDP3 scroll pump (Agilent, Santa Clara CA, USA) and backfilled with the gas of choice to ambient pressure. To ensure complete flushing of the sample volume, this process is repeated 3 times before a “He-only” or “CO2-only” measurement is made. For aerosol measurements, the sample diaphragm pump is disconnected and a nebulizer is used to generate 2 L min-1 of positive-pressure flow containing particles of known size and refractive index through the instrument. To ensure a consistent sample throughout the instrument volume, we verify that the aerosol loading and RH have been constant for least 5 min before beginning to take a measurement for calibration.

Here, we also present some data from the 2019 FIREX-AQ aircraft campaign. Figure 3 shows a schematic of the aerosol sampling, pre-conditioning, and measurement components aboard the NASA DC-8. During this campaign the LiNeph was mounted adjacent to integrating nephelometers (TSI Model 3563, Shoreview MN, USA) from the NASA Langley Aerosol Research Group (LARGE). All the instruments discussed here sampled from the LARGE University of Hawaii aerosol isokinetic inlet, which has a geometric diameter upper cut size of 4–5 µm (McNaughton et al., 2007; Chen et al., 2011). The LARGE group also operated a Laser Aerosol Spectrometer (model 3340A, TSI, Shoreview MN, USA) from the aerosol inlet and measured the dry particle size distribution at a frequency of 1 Hz (Moore et al., 2021). The aerosol sampled by the LiNeph was also dried to less than 20 % RH using Nafion driers and passed through a cyclone with a calculated cut size of 1.5 µm aerodynamic diameter. The aircraft cabin was temperature controlled, resulting in average sampling temperatures of ∼26 ∘C, although temperatures could rise as high as 36 ∘C when sampling at low altitudes for an extended duration. A dry scroll pump (TriScroll 300, Agilent, Santa Clara CA, USA) provided the vacuum for both the integrating nephelometers and the LiNeph. The flow controller specific to the LiNeph controlled the flow to 15 L min-1 (volumetric liters per minute). We will compare the integrated scattering measured by the LiNeph with the scattering derived from the measurements from the NOAA Aerosol Optical Properties (AOP) instrument suite, which includes cavity ring-down spectrometers (CRDS) and photoacoustic aerosol spectrometers (PAS) at wavelengths of 405, 532, and 664 nm (Langridge et al., 2011; Lack et al., 2012). The AOP instrument package sampled from the same aircraft inlet as the LiNeph and LARGE nephelometers, but located less than 2 m away. These measurements of aerosol extinction and absorption can be used to calculate the integrated aerosol scattering at the wavelengths interrogated by the LiNeph.

There are two important differences between the aerosol measured by the LiNeph and the AOP instrument suite. Firstly, the AOP uses an impactor to remove dry aerosols with aerodynamic diameters >2.5 µm, while the LiNeph cyclone cut point is 1.5 µm. However, in smoke plumes the difference between the total light scattering of PM1.5 vs. PM2.5 is negligible due to the overwhelming abundance of submicron particles. This was confirmed using size distributions from the LAS, which showed few particles with diameters greater than 1 µm (Moore et al., 2021). Secondly, during high particle concentrations that were common in the sampled smoke plumes, the aerosol sampled by the CRDS needed to be diluted. Without the dilution system, the uncertainty of the CRDS extinction measurement for dry scattering coefficients >100 Mm-1 is ±5 %, but with the dilution system in line, the uncertainty is estimated to be ±30 %. This added uncertainty was characterized in the field and may be due to incomplete mixing of the filtered and unfiltered sample flows.

We convert each LiNeph image into σ∘ (scattering intensity as a function of scattering angle, Mm-1 sr-1) by applying two calibrations. σ∘ measured by each CCD array (see Eqs. 4 and 6) can then be used to solve for the normalized scattering matrix elements of interest, P11 and P12.

First, we convert the pixel intensity to σ∘ (Mm-1 sr-1) by comparing the area under a Gaussian fit at a given scattering angle (i.e., pixel column) to the theoretical scattering of particle-free air or CO2, both of which are well described by Rayleigh scattering (Manfred et al., 2018; Dolgos and Martins, 2014). Second, we establish which pixel column corresponds to which scattering angle by identifying local maxima and minima observed in the measured phase function of NIST-traceable polystyrene latex spheres (PSLs). Since PSLs are well characterized with respect to size, dispersion, shape, and refractive index, we can calculate the expected scattering matrix elements with a high degree of confidence. Figure 4 shows the good agreement between the measured (red markers) and calculated (lines) P11 and DoLP. The two calculated values of P11 and DoLP represent two theoretical instrument geometries, described below.

Although the optical axis of each lens is parallel or perpendicular to the polarization of the laser, the beams themselves are offset from the optical axis by ∼3.6 mm. This is roughly defined by the geometry of the series of four concentric apertures through which the lasers are introduced into the sample cell. Due to the nature of the wide angle lens, this can result in a scattering plane rotation angle, η, shown in Fig. 1 and described in detail by Dolgos et al. (2014). In our data reduction, we use the simplifying assumption that η is zero, although the instrument geometry dictates that η is non-zero. This results in a small error in the calculated P11 and DoLP.

In Fig. 4, we explore the magnitude of the error by simulating σ∘ measurements for two instrumental geometries and then using the same data reduction to calculate P11 and DoLP for both cases. The first instrument geometry is an idealized case where η is equal to zero (gold line), as in the data reduction. In the second instrument geometry, we use estimated, non-zero values of η to simulate σ∘ measurements (teal line) but still use the assumption that η=0 to calculate the P11 and DoLP. We do not observe improved agreement between the observed and calculated values for P11 nor DoLP by using the estimated η, thus we treat our estimated η as upper limits. η varies with scattering angle and does not exceed 40∘; see Figs. S2–S5. For some scattering angles (e.g., near 30 and 135∘), there is stray light from the instrument background which introduces additional noise as extraneous features.

The conversion from pixel intensity (bits) to σ∘ (Mm-1 sr-1), referred to as the differential scattering calibration in this work, also accounts for two types of image distortion. The first is distortion by the wide-angle lens, where the image of the area illuminated by the laser beam appears wider (is projected onto more CCD pixels) close to a scattering angle of 90∘ (see Fig. 2). The second distortion is due to the varying scattering path length for a given scattering angle. That is, the length of the volume of air defined by the laser and the ∼0.5∘ pixel viewing angle is shortest close to 90∘. Thus, the differential scattering calibration can have a small effect on the angular calibration by slightly shifting the pixel location of the PSL phase function local minima and maxima. The feature shifts are small and become negligible with just one iteration of the pixel column-to-angle and differential scattering calibration analyses. The PSL angular calibration is shown in Fig. S6, where dots indicate the raw, initial fitting, and the open circles indicate the final calibration. Error bars on the calibrated data indicate the 95 % confidence interval of the linear regression (scattering angle as a function of pixel column) that is the angular calibration. For each calibration point, we calculated the 95 % confidence interval by the propagating the variance associated with the linear regression slope and intercept while accounting for covariance between the slope and intercept. The average 95 % confidence interval for the 14 points is 0.9±0.2∘.

We determine the differential scattering coefficient calibration by measuring σ∘ of CO2 using circularly polarized light for each laser and each detector for a single alignment geometry. Circularly polarized light is necessary for this portion of the calibration because for linearly polarized light scattered in the Rayleigh regime, P11(θ)+P12(θ) (the scattering intensity observed by the Para CCD assuming η=0∘) at 90∘ is extremely small and hard to measure accurately. For circularly polarized light, the light scattering at 90∘ is 1/2 of the scattering at 0∘, as shown in Fig. 5.

Circular polarization was achieved by placing a zero-order quarter-wave plate (WPQSM05-405/670, Thorlabs, Newton MA, USA) after the Glan–Taylor prism for each laser. To remove the background signal from stray light scattering off the interior of the instrument, we purged the sample volume and backfilled with helium, which has a negligible scattering cross section. After subtracting the illuminated helium image from the illuminated CO2 image, we applied the same fitting protocol used for measuring aerosol scattering. The area under each of these Gaussian fits is shown as red circles in Fig. 5. The red circles are the uncalibrated σ∘ for 935 hPa of CO2 illuminated by 660 nm light as viewed by the “Parallel” CCD (see Fig. 1). The black dotted line shows σ∘ calculated using Mie theory, the measured sample pressure, and the scattering cross section of CO2 from Penndorf (1957) which includes a molecular depolarization term. The ratio of the theoretical σ∘ (Mm-1 sr-1) to the raw σ∘ (bits) is the differential scattering coefficient calibration, shown as orange triangles. A sixth-order polynomial fit is used to smooth and extrapolate the calibration function between the smallest (7∘) and largest (171∘) measured scattering angles. This calibration is applied to all raw data to correct for the lens distortion and varying path length as described above, as well as correct for differences in CCD array sensitivity. This allows the measured σ∘ to be compared, and therefore the scattering matrix elements of interest are isolated.

The precision of the LiNeph depends on both the stability of the instrument (laser power and detector response) and also the homogeneity of light-scattering entities in the sampled air. We can evaluate the stability of the instrument response in the lab first by ensuring homogeneity of the sampled air – that is by using particle-free CO2. Over the duration of 10 min, we observed no statistically significant change in light scattered by pure CO2 as measured by our CCD arrays, suggesting that over that time period there is no significant drift in either the sensitivity CCD arrays or the laser output. Further, no correlation (R2=0.007) was observed when comparing the light scattered at different angles (see Fig. S7), indicating that the noise observed in the measurement was due to either the Gaussian peak fitting routine and/or electronic noise within the CCD array. If the variation were due to laser power fluctuations, then the observed light scattering intensity at different angles would have been correlated.

Another potential source of instability in the instrument is mechanical vibration induced by the aircraft. The instrument uses a rigid optical cage system (30 mm cage components, Thorlabs, Newton MA, USA) to minimize susceptibility to mechanical perturbations. The cage system, the laser platform, and the sample cell are all mounted to a modified aluminum U channel (1630T45, McMaster-Carr, Sante Fe Springs CA, USA) that provides excellent rigidity. No deviation in the alignment was observed during the FIREX-AQ campaign as measured by the pixel position of the Gaussian fit maximum value.

We accomplish two things by using the area under a Gaussian fit of the signals in the pixel rows of the CCD at a given angle (pixel column) as a measure of scattered light. Firstly, we account for lens distortion of the laser beam diameter. Secondly, we effectively average electronic noise observed in individual pixels; i.e., there is less noise in the measured Gaussian fits of a measurement than there would be measuring just the pixel intensity at the peak of the signal. For example, multiple measurements of a peak pixel at a given scattering angle (e.g., 104∘) may have a relative standard deviation of 54 %, but the area under the Gaussian fit for the same angle has a relative standard deviation of 22 %. One might expect similar accuracy improvement by summing the pixels containing the laser, as is the case with the PI-Neph (Dolgos and Martins, 2014). Theoretically, the difference between the two methods should be small. One benefit of the Gaussian fit technique is that it may be less sensitive to which pixel rows one designates as representing the light scattered by the aerosol. The Gaussian fit technique can also readily account for a changing baseline (e.g., if multiple scattering illuminates the inside of the instrument). However, by summing the signal, one is less sensitive to the inhomogeneous background of the instrument (see Fig. 2).

The limit of quantification is defined for each scattering angle measured. We conservatively define the minimum signal required for quantification of aerosol scattering to be a Gaussian fit with an amplitude that is at least 10 times greater than the noise (1 standard deviation) measured in the background subtraction sample. Noise may result from either electronic noise in the CCD array or from light reflecting off the interior of the instrument body. This means that the noise varies spatially in each image; e.g., more noise is observed in forward scattering directions due to window glow where the lasers enter the sample cavity. Figure S8 shows that the noise in the Gaussian fit area observed at each scattering angle is ∼2 % of the signal or 200 bits, whichever is larger, for a 0.5 s exposure time. This means that increasing the CCD exposure time can allow measurements of the phase function when total scattering is low (e.g., σscat is ∼14 Mm-1 for the differential scattering calibration in Fig. 5). Varying the exposure duration can also be useful for phase functions that are strongly forward scattering and thus require a broad dynamic range. If the aerosol population is unchanging, two sets of differential scattering functions can be measured and then combined: one with a short exposure (to capture intense forward scattering without saturating the CCD) and one with a long exposure (to increase the signal-to-noise ratio for less intense backscattering angles). It is also conceivable that very high aerosol concentrations could attenuate the propagating laser, thereby biasing the observed scattering to the forward scattering angles. However, even with an extinction coefficient of 10 000 Mm-1, the laser would only be attenuated at most by 0.7 %. The maximum scattering coefficient observed during FIREX-AQ, within intense smoke plumes, was 8000 Mm-1 and thus we consider this to be a minor source of error.

High aerosol concentrations can also affect the accuracy of σ∘ measurements because of multiple scattering. In this instance a photon is scattered by a particle in a direction consistent with σ∘, but is scattered by a second particle before being detected. Gogoi et al. (2009) showed that there was a small but measurable multiple scattering effect (reduction in radiance measured) when the optical depth was greater than 0.01. The maximum distance from scattering entity to the detector in the LiNeph is ∼36 cm (for particles scattering in the forward direction), resulting in an optical depth of 0.028 at the highest observed integrated scattering coefficient. This suggests that for the higher concentrations (scattering coefficient greater than ∼3333 Mm-1), there may be a small negative bias. Monte Carlo radiative transfer simulations by Ge et al. (2011) show that for a field-of-view of 1∘ and with particle diameters of 500 nm, the negative bias will be less than 3 %.

Having addressed the inherent instrument uncertainties, we will now analyze the uncertainties associated with specific measurement environments and samples. There is a concern that, due to the large volume of the sample chamber of the LiNeph, there might not be a homogeneous sample illuminated by the lasers. It is important that each observed solid scattering angle contains a representative distribution of the aerosol. One instance where this would not be the case is if rare but highly scattering particles transit through the sample cell but only transect the laser at a few angles. This would result in spikes observed in the recorded phase function. This was not observed during FIREX-AQ because the aerosol distribution was dominated by very high concentrations of small particles, and because a PM1.5 cyclone removed larger dust and ash particles that may have been present. This was verified by the size distribution measurements with impactors with an even larger cut size, PM2.5.

Another potential source of error during ambient measurements is a rapidly changing βscat, e.g., when there is an increase in the aerosol number concentration. If the sampled βscat increases rapidly, the βscat gradient within the sample cell will be observed in the scattering phase function. For example, if the βscat at the instrument inlet increases during a measurement, there may be more scattering in the section of the sample cell corresponding to forward scattering angles than in the portion of the cell corresponding to backscattering. To minimize this effect for the sampling of wildfire plumes, the LiNeph was designed with a minimal internal volume, albeit at the expense of increased background noise due to stray light. During FIREX-AQ, the LiNeph was operated at 15 L min-1, which means an aerosol exchange rate of less than 12 s for the approximately 3 L sample cell if perfect mixing is assumed. Figure S9 shows the change in total measured CCD signal (no Gaussian fits or image processing) while measuring well-mixed smoke and interposing a HEPA filter. Imposing the filter at t=0 s results in the removal of smoke particles and leaves only the light-scattering gases. An exponential fit shows a 2.6 s time constant, which suggests that the sample cell should not be characterized as a well-mixed reactor. Plug or laminar flow through the center of the instrument may result in a functionally faster aerosol exchange rate. This allows for transition periods from background air to smoke plume air to be minimized. Additionally, we report angularly resolved radiance and polarimetric measurements only when the prior measurement of the integrated scattering is within 15 % of the current measurement, usually about 2.5 s later, indicating that we are not likely in a transition period that would skew the phase function shape. This criteria is equally important when merging long and short exposure radiance measurements to capture strong forward scattering and weak back or side scattering, as described Sect. 3.2.

For the FIREX-AQ mission, we sampled smoke plumes aboard the NASA DC-8 traveling 159±6 m s-1 while sampling smoke. This means that aerosol composition would change rapidly as we entered and exited the ∼44 km wide smoke plume. We will show that the LiNeph had sufficient temporal resolution to capture the larger features of smoke plumes by comparing the integrated scattering measured by the LiNeph at 0.24 Hz with the integrated scattering calculated from 1 Hz measurements of extinction and absorption by the AOP instrument suite (Langridge et al., 2011). If the sample exchange rate was insufficient, the integrated scattering measured by the LiNeph will appear as a moving average of the AOP-derived scattering coefficient. Panel (a) of Fig. 6 shows that there is sufficient aerosol exchange to capture the major features in a large smoke plume, although the finer details are lost. The calculated Pearson correlation coefficient between the two measurements is 0.96.

Panel (b) of Fig. 6 expands this analysis by showing the integrated scattering for 6 (5) FIREX-AQ research flights, at 405 (660) nm. The nearest neighbors approach was used to account for truncated angles. Flights were excluded that were missing data from the AOP measurements. A linear fit to the remaining data points show 21 % (2 %) more scattering measured by the LiNeph than the AOP-derived scattering at 405 (660) nm, with an R2=0.99 (0.97). While this is within the specified accuracy of the AOP-derived scattering measurement for diluted samples, an analysis of undiluted measurements (Fig. S10) shows less variance in the measurements, R2=0.99 (0.98), and shows a consistently 30 % (24 %) higher scattering measured by the LiNeph vs. the AOP-derived scattering measurements, for 405 (660) nm. If we take the AOP-derived scattering as a truth measurement, i.e., without its own error, we can say that the LiNeph is precise within <2 %, although with a positive bias of ∼30 %, likely due to calibration error. This is consistent with the reported accuracy of similar techniques, i.e., the PI-Neph reports agreement with commercial integrating nephelometers to within 5 % (Espinosa et al., 2017).

Having established that the LiNeph was likely measuring a homogeneous sample of smoke, we can now investigate the directionality and polarization of light scattered by wildfire smoke. Figure 7 shows the angularly resolved radiance and polarimetry measurements at 405 nm of smoke during two transects of the Williams Flats fire plume on 7 August 2019. This fire was initiated by a lightning strike and consumed over 44 000 acres (178 km2) of fuel including timber, short grass, light slash from logging, and a coniferous overstory over 25 d (InciWeb, 2021). The fire emitted an intense smoke plume extending downwind over 104 km and up to 44 km wide. The traces in Fig. 7 show the mean plus two standard deviations of two sets of measurements. Each set of measurements is from a single transect perpendicular to the axis of the smoke plume. While spacing of plume transects during FIREX-AQ were intended to produce pseudo-Lagrangian data, in fact the aircraft frequently traveled downwind at a rate faster than the plume advection (ratio of smoke age to elapsed time during all of FIREX-AQ was 0.8–6.4 as reported in Fig. S3 of Wiggins et al., 2020). For the 7 August flight, this ratio was about 3. Smoke age was estimated using wind speed and distance of the measurement from the fire. Smoke in Transect 1 was emitted approximately 1 h prior to being sampled, and smoke in the second transect considered here, Transect 10, was emitted approximately 4.4 h prior to being sampled. Panel (a) of Fig. 7 shows that there is a significant difference in the directional scattering of 405 nm light by smoke, although panel (b) shows that the change in linear polarization as a function of scattering angle appears to be consistent between the two plumes. The change in directional scattering was likely due in part to the change in mean particle size between the two transects. Figure S11 shows the average normalized number-weighted size distributions for both transects as measured by the LAS with an applied ammonium sulfate calibration (Moore et al., 2021). The mode diameter of Transect 1 was 174 nm while Transect 10 was 225 nm, only 51 nm larger.

This modest growth could have been caused by changing emissions and/or atmospheric processing. The asymmetry parameter, described below, increases from 0.568 to 0.620, showing an increase in forward scattering that is consistent with increasing particle size. However, it is important to note that changes in particle composition, and hence refractive index, have also been observed as a consequence of photochemical aging in biomass burning aerosols. The degree of linear polarization provides additional information that may be useful in determining the degree to which changing refractive index and size account for the changing phase function. Analysis of the smoke optical properties and their changes with plume will be the subject of future study and is beyond the scope of this work.

One important application of phase function measurements is the calculation of the asymmetry parameter. The asymmetry parameter, g, is the intensity-weighted cosine average of the scattering angle (Andrews et al., 2006). It is calculated following Eq. (7): 7g=12∫0πcos⁡(θ)P11(θ)sin⁡(θ)dθ. The asymmetry parameter is used as a computationally efficient way to approximate the fraction of light that is scattered into the upper hemisphere, or up-scatter fraction, in radiative transfer models (Wiscombe and Grams, 1976). Despite its importance in understanding the direct radiative effect of aerosols in models, the asymmetry parameter is rarely measured directly. Instead, it is commonly estimated from hemispheric backscatter measurements by integrating nephelometers or else calculated from Mie theory (Andrews et al., 2006; Moosmüller and Ogren, 2017). Unfortunately, Marshall et al. (1995) used Mie theory to show that the typical method of estimating using hemispheric backscatter measurements will overestimate the asymmetry parameter for accumulation mode aerosols. Further, whether or not Mie theory is appropriate for predicting biomass burning aerosol phase functions is an area of active research (Manfred et al., 2018; Liu and Mishchenko, 2018).

Future work will explore the relationship between the asymmetry parameter and the hemispheric backscatter fraction, both of which can be derived from the phase function directly. It will also be of interest, although beyond the scope of this work, to evaluate whether Mie theory can be used, along with the particle size distribution measurements and assumed refractive indices, to predict the hemispheric backscattering measured by the integrated nephelometers and the LiNeph.

To calculate g, we first used a nearest-neighbors method to account for truncation, i.e., inability to measure scattering at the extreme forward and backwards (θ<7∘ or θ>171∘) angles. We investigated the effect of truncation on the asymmetry parameter using simulated phase functions calculated from measured particle size distributions during FIREX-AQ. We found that truncation affected the asymmetry parameter by less than 1 % due to the small particle size. Truncation will likely have a larger effect for particle size distributions with supermicron particles and therefore strong forward scattering.

The phase function measurements (e.g., Fig. 7) allow for precise measurements of the asymmetry parameter with a relative standard deviation of less than 3 %. But, as discussed in Sect. 2.4, the geometry of the LiNeph requires that the lasers be offset from the optical axis of the wide angle lenses, introducing a non-zero η. Based on measurements of PSLs (Figs. S2–S5), we can set an upper bound of this effect on the measured σ∘. We used the same method to investigate the effect of η on the measured asymmetry parameter. Figure S12 shows that for polydisperse lognormal aerosol size distributions with a mode around 200 nm, with varying refractive indices, the effect of a non-zero η on g is a bias of less than 2 %. For the largest modeled aerosol population, with a mode at 400 nm, the bias was 5 %. This effect is small in part because the geometry of the instrument results in offsetting biases when σ∘“Perp” and σ∘“Para” are combined to calculate P11. For the four P11 measurements of PSL calculated from the data in Figs. S2–S5, the average ratio of measured to Mie-calculated g was 1.01±0.09.