The linear depolarisation ratio (δ) of an ensemble of ice particles in the cloud chamber is measured with the SIMONE (Streulichtintensitätsmessungen zum optischen Nachweis von Eispartikeln – Scattering Intensity Measurements for the Optical Detection of Ice Particles) instrument . A schematic setup is shown in Fig. a.

The instrument measures the scattered intensity in the near forward direction (2°) and in the near-backscattering direction (178°) from a 488 nm continuous wave laser beam propagating through the cloud chamber. Well-mixed conditions and random particle orientations persist due to operating a mixing fan in the cloud chamber. The detection volume of approximately 7 cm3 is defined by the overlap between the laser beam and field of view of the detector. The laser beam is linearly polarised and the polarisation axis can be rotated with a liquid crystal rotator. For this study, a polarisation parallel to the scattering plane is chosen. In the 178° direction the scattered light is split with a polarising beam splitter and the resulting co- and cross-polarised states are analysed using two photomultiplier tubes (Perkin Elmer MP-1383). The response of the two photomultiplier tubes are calibrated for each campaign using a Spectralon target with a known linear depolarisation ratio that can be placed in the scattering center inside the cloud chamber . For the RICE03 campaign, data from a different instrument with the same operation principle (SIMONE-Junior), an emission wavelength of 552 nm and a detection volume of approximately 30 cm3 were used, which is described in .

During the calibration process, which is identical for SIMONE and SIMONE-Junior, the polarisation of the incident light is switched between parallel and perpendicular orientation with reference to the scattering plane. A calibration factor is multiplied to the channel, which records the intensity with parallel polarisation, in order to obtain the same linear depolarisation ratio for the linearly polarised incident light at both orientations. Figure b shows the calibration process exemplary for measurement campaign RICE03, where a calibration factor of 1.4908 was derived for SIMONE-Junior. The calibration factor takes into account the different gains of the detectors, different losses of the polarisation filtering and effects of possible differences in alignment of the detectors. The measurement uncertainty of δ is 2.1 %, derived as the standard deviation of the linear depolarisation ratio of the scattering target measured during the calibration process. Experiments with supercooled liquid droplets were conducted at the AIDA cloud chamber to validate the measurement uncertainty because the linear depolarisation ratio of liquid and thus spherical droplets vanishes . The mean linear depolarisation ratio of 12 experiments of supercooled liquid clouds at an initial gas temperature of -30°C is (0.7±0.3)%. The derivation from the theoretical value of 0 % is well below the measurement uncertainty of 2.1 % derived from the calibration with the scattering target. Additional information about the supercooled liquid droplet experiments is provided in Appendix .

The linear depolarisation ratio for incoming light with a polarisation parallel to the scattering plane δ is calculated as : 1δ=I⟂-I⟂,bgI∥-I∥,bg where I⟂-I⟂,bg is the background-subtracted light intensity with a polarisation perpendicular to the scattering plane and I∥-I∥,bg with a polarisation parallel to the scattering plane. The linear depolarisation ratio from the SIMONE instrument is averaged over time intervals of 10 s. Hereafter we refer to it as δ. Based on an estimation of the optical depth from the particle microphysics measurements, below 10 % of the intensity detected with SIMONE can be affected by multiple scattering effects for 89.9 % of the measurement data.

The particle size, shape and small-scale morphological complexity are optically characterised using the Particle Phase Discriminator 2 Karlsruhe edition (PPD-2K) and the Small Ice Detector 3 (SID-3) . Both instruments are optical particle counters that detect the scattered light of individual particles. A sample air flow passes the measurement volume that is defined by the overlap between the laser beam with a wavelength of 532 nm from a frequency-doubled Nd:YAG laser and the field of view of the trigger optics. The design of the trigger optics differs in both instruments. In PPD-2K, a beam splitter diverts 8 % of the forward scattering light to the trigger detector. SID-3 uses two nested trigger detectors with half angles of 9.25° at an angle of 50° to the forward scattering direction. In both instruments, the trigger intensity is detected with a photomultiplier tube and used to determine the size of the individual particles with a maximum count rate of 11 kHz. A special feature of SID-3 and PPD-2K is that they use an intensified Photek ICCD218 camera to record the spatial intensity distribution of the forward scattered light over an annulus between approximately 5° and 26° with a resolution of 780pixels×592pixels (SID-3) and between 7.4° and 25.6° with a resolution of 582pixels×592pixels (PPD-2K). Images are taken for a subset of the triggered particle events due to the camera's maximum imaging rate of 30 Hz. These diffraction patterns contain information on the particle habit and morphological complexity at scales of the wavelength of the used light. Both instruments were operated simultaneously. Here we use PPD-2K for getting the size information due to higher counting statistics compared to SID-3 and SID-3 for getting information on the crystal shape and degree of morphological complexity, similar to .

PPD-2K measures the trigger intensity I of each particle. To relate I to the particle size, the spherical equivalent diameter d is used, which is the diameter of a sphere that scatters the same intensity in the direction of the PPD-2K trigger field of view at polar angles between 7.4° and 25.6° as the recorded particle. d is calculated from the trigger intensity I using the following equation : 2d=a⋅Ib where calibration coefficient a depends on the laser power and PMT gain and calibration coefficient b=0.522 depends on the trigger geometry and must be around 0.5 because the scattered intensity is proportional to the geometric cross section of the particle . For each measurement campaign, the factor a is calibrated with spherical droplets where the size is determined comparing the Mie fringes of the diffraction patterns to Mie theory (see for details). The ice crystal maximum dimension is estimated to be approximately 1.0–2.5 times larger than the spherical equivalent diameter depending on the crystal shape and complexity (see Appendix ).

Besides particle size, other microphysical features can be extracted from the diffraction patterns recorded by PPD-2K and SID-3, such as particle shape and degree of crystal complexity. The degree of crystal complexity can be derived from a speckle pattern texture analysis of the diffraction patterns, and is represented by the ice crystal normalised energy feature parameter ke . ke is a measure for particle complexity with scales around the laser wavelength of the incident light, 532 nm, including surface roughness, polycrystallinity and (stepped) hollowness . The parameter ranges between about 3.8 and 7.0 and a threshold of kethr=4.6 was defined by for the SID-3 measurements, where lower values correspond to pristine particles and higher values for morphologically complex particles. Similarly to , ke is only calculated for particles with trigger intensities between 10 counts and 25 counts in this work to reduce the effect of particle size biases on ke, which is caused by varying mean intensity of the diffraction pattern images. In this work, the probability of coincident particle sampling is below 1 % with a maximum detected particle concentration of 42.6 and 22.3 cm-3 measured by PPD-2K and SID-3 .

Furthermore, the particle shape is derived from the diffraction patterns. If there are clear maxima in the azimuthally integrated polar profile (Mie fringes) the particle is considered spherical and is classified as a droplet (see Fig. d). Otherwise, a discrete fast Fourier transform (FFT) of the polar integrated azimuth intensity profile is performed for the shape identification following the procedure of . Maximum Fourier coefficients of order 2 and 4 indicate a columnar shape (Fig. c) and maximum Fourier coefficients of order 3 and 6 indicate a hexagonal plate (Fig. a). If the maximum coefficient is of a non-symmetric order the particles are interpreted as irregulars (Fig. b). It needs to be noted that if a particle is classified as irregular it does not necessarily mean that it has an irregular shape. Irregular diffraction patterns may also result from sufficiently roughed columnar or hexagonal ice particles, which do not show their usual, distinct diffraction patterns. Furthermore, the fraction of columns and plates should be seen as a lower limit because the random particle orientations can lead to bent arcs in the scattering patterns of pristine hexagonal ice particles, which may occasionally be falsely interpreted as irregulars. Out of 100 random ice particles classified as irregulars by the Fourier method, 11 hexagonal ice particles with bent arcs were manually identified.

In this work, the measured single particle scattering information is converted into particle size distributions using 50 bins in a size range between about 7 and 70 µm depending on the campaign-specific size calibration. The particle size distributions are averaged over 10 s and a log-normal size distribution function is fitted to obtain the geometric mean diameter (GMD) and the standard deviation of the logarithm of GMD (σ). The log-normal particle size distribution is defined as : 3n(d)=n02π⋅d⋅log⁡σ⋅e-log⁡2d/GMD2log⁡2σ where n0 is the total concentration and d is the particle diameter. Figure e shows the retrieved log-normal parameters σ over GMD. Most σ are in a range between 1.1 and 1.5 with some outliers to higher values. σ has a mean value of 1.26±0.13. For all curve fits with GMD<10µm, σ has a mean value of 1.21±0.08.

To analyse the size of ice crystals that are smaller than the lower size limit of PPD-2K (7 µm) a Fourier transform infrared spectrometer (FTIR) was used to determine the particle size during the HALO06 campaign . The FTIR measures the spectral extinction of the ice particle ensemble at wave numbers between 6000 and 800 cm-1. At mid-infrared wavelengths, the extinction spectra only vary slightly with particle shape unless highly irregular habits are involved. For this study, similar to , the ice particle size distribution was assumed to be log-normal and retrieved from the measured extinction spectra using T-matrix calculations for a fixed aspect ratio of 0.7 (circular columns). The FTIR retrieval provides reliable results down to mean particle maximum dimensions of about 1 µm.

Furthermore, a formvar replicator was operated in the cloud chamber during measurement campaigns RICE01 and RICE03 to generate replicas of the ice crystals on a 35 mm transparent plastic film strip. Details can be found in . The formvar replicas are analysed using a Zeiss IM35 inverted microscope with a magnification of up to 500 and an Imaging Source DFK41AU02 camera with a resolution of 1280pixels×960pixels. The images of the replicas allow to obtain additional information about the particle shape that cannot be derived from the diffraction patterns, such as information about hollowness, but the statistics are poorer.

The expansion experiments were conducted in the Aerosol Interactions and Dynamics in the Atmosphere (AIDA) cloud chamber at the Karlsruhe Institute of Technology . Here, the data from different ice nucleation measurement campaigns at cirrus temperatures between -75 and -39°C are analysed (see Table ). The temporal evolution of a typical experiment is shown in Fig. . The expansion experiments are conducted with the following procedure discussed in :

Preparation: The cloud chamber is cleaned by evacuation and flushing cycles and then humidified to ice-saturated conditions by forming a thin ice coating on the inner chamber walls.

To correlate the linear depolarisation ratio to the particle size at cirrus temperatures, the following conditions are applied:

The gas temperature of the cloud chamber Tgas is below -39°C.

Transition matrix (T-matrix) and conventional geometric optics (CGOM) ray tracing Monte Carlo methods are used to simulate the near-backscattering depolarisation ratio that is measured with the SIMONE instrument. Here we use the T-matrix code from to perform the T-matrix simulations of spheroidal particles, which are constrained to small size parameters below approximately 50 due to computational limitations. The assumption of spheroidal particle shapes is a rough approximation of the hexagonal shape of ice particles . Therefore, as a comparison to the T-matrix approach, which is widely used in LIDAR applications e.g., light scattering simulations of hexagonal ice crystals with the Invariant-Imbedding T-matrix Method (IITM) for size parameters of up to 158 are additionally compared to the measurement data.

For size parameters larger than 90 we use CGOM simulations of hexagonal particles with a tilted facet method and internal scatterers to generate complex ice crystals. The size parameter is defined as x=2πaλ with characteristic particle length a and wavelength λ of the light in the surrounding medium. A wavelength of λ=488nm and a refractive index of n=1.31 are used.

All simulations calculate the scattering matrix elements that are needed to obtain the linear depolarisation ratio. For this, the assumption of randomly oriented ice particles is used. The linear depolarisation ratio for incident polarisation parallel to the scattering plane δ for an arbitrary scattering angle θ is defined as : 4δ(θ)=S11(θ)-S22(θ)S11(θ)+2S12(θ)+S22(θ) where S11(θ), S12(θ) and S22(θ) are the scattering matrix elements obtained from the CGOM and T-matrix simulations for the scattering angle θ.

The T-matrix method is applied to simulate δ for the light scattered by randomly oriented pristine spheroids with aspect ratios ranging between 1.5 and 2.0. This range is based on the replica images that are presented later. In order to be comparable to the measurements, the T-matrix scattering matrix elements are integrated over a log-normal size distribution similar to with 82 particle sizes varying from 0.01–10.0 µm and a fixed σ of 1.21. This is the mean σ found in the log-normal fits to the PPD-2K particle size distributions with GMD smaller than 10 µm. The maximum dimension of the spheroids is used for the comparison to the PPD-2K spherical equivalent diameter.

Furthermore, we use a dataset of numerically exact IITM simulations of hexagonal ice crystals published in . It simulates particle complexity as surface roughness. The degree of this surface roughness is defined by the variance (σ2) of a two-dimensional Gaussian distribution of local planar surface slopes. The scattering matrix elements are integrated over a log-normal size distribution with 31 particle sizes between 0.1 and 24.6 µm, using a fixed σ of 1.21 for particle sizes of up to 10 µm and a fixed σ of 1.26 for larger particle sizes. The dataset was computed using the refractive index of ice at 532 nm (n=1.3116-1.49×10-9i), which differs slightly from that at 488 nm. This difference has a marginal effect on light scattering in the backward direction. The aspect ratio ranges from 1.0–2.0, with decreasing maximum particle size for increasing aspect ratio due to increasing computational effort.

The CGOM simulations use hexagonal ice particles and average over 2 million particle orientations. The calculated scattering matrix elements are integrated over a log-normal size distribution with 42 particle sizes between 8.4 and 37.3 µm and a fixed σ of 1.26. This is the mean σ of all fits to PPD-2K cirrus particle size distributions. Ice crystal complexity is represented with the tilted facet approach where a random tilt angle is applied to the ice crystal surface. A distortion parameter describes the maximum possible angle of a random tilt that is applied to the crystal surface. With a distortion of 0 no tilt is applied and a distortion of 0.5 is equivalent to a random tilt of up to 45°. Additional ice crystal complexity can be simulated with the mean free path by adding internal scatterers to the simulation that randomly change the direction of the internal rays . In this work, the internal scattering is non-absorbing with a single scattering albedo of 0.999, representative of air bubbles. The mean free path is the mean distance that a simulated ray propagates through the crystal in the Monte Carlo simulation before changing direction due to the simulated internal scatterer. We have no direct information about concentration, size and type of internal scatterers inside the cloud chamber grown ice particles. Therefore, the mean free path is varied between 150 µm for high internal scattering and 104µm for negligible internal scattering to find the best overlap with the measurement data. The column length is used for the comparison to the PPD-2K spherical equivalent diameter.

The conversion between the particle sizes from the numerical simulations and the spherical equivalent diameter measured with PPD-2K is non-trivial and therefore an approximation is needed. In this work, we use the maximum dimension of the spheroidal particles and the column length of the hexagonal particles, which is common approach when analysing the optical properties of ice particles . To estimate possible errors, the spherical equivalent diameter measured with PPD-2K is calculated for different complex shaped ice particles in Fig. A1. The maximum particle dimension can be up to 40 % larger than the spherical equivalent diameter for rough columnar and bullet rosette shaped ice particles. However, the column length of columnar particles can also be smaller than the maximum particle dimension, depending on its aspect ratio . For example, a columnar particle with an aspect ratio of 1.5 has a maximum dimension, which is about 30 % larger than its length. These uncertainties in size conversion are shaded in the figures where the results of the numerical simulations are compared to the measurements with PPD-2K.

PPD-2K measured particle spherical equivalent diameters of up to 70.5 µm for ice crystals nucleated and grown in the AIDA chamber at gas temperatures below -39°C. Figure  shows histograms of the GMD of ice crystals grown between AIDA initial gas temperatures between -45 and -39°C and between -75 and -45°C, respectively. At initial gas temperatures between -45 and -39°C, the median GMD was 17.4 µm (Interquartile range (IQR) = 6.7 µm). In comparison, ice crystals grown at lower cirrus temperatures between -75 and -45°C had a median GMD of 10.6 µm (IQR=3.6µm). The GMD of ice crystals grown in the lower cirrus temperature range is 39 % smaller than of ice crystals grown in the higher cirrus temperature range. It is expected that ice crystals at higher temperatures grow to larger sizes, since at the same relative humidity the growth rate increases for increasing temperatures due to more available condensible water vapour . Ice crystals grown in the AIDA cloud chamber were limited to maximum sizes of about 70 µm due to sedimentation losses.

A visual analysis of the ice crystal shapes was performed using microscope images of the formvar replicas during regrowth phases when the relative humidity was kept constant. This is done for a subsample of the cloud chamber experiments (see Table ). The aim is to identify differences in the growth characteristics on the formvar replicas at different growth conditions. The relative humidity with respect to ice (RHice) in the regrowth phases is varied between 105 % and 120 %. The number of columnar ice crystals and the number of hollow columnar ice crystals were manually counted per microscopic frame, which usually contained some tens of ice crystals. Out of the 11 096 imaged ice crystals on 324 microscope frames. 29 % were classified having columnar and 71 % of ice crystals were classified having other, mainly irregular shapes (see exemplary replicator images in Fig. a–e). The irregular shapes were predominantly compact crystals that sometimes showed several c-axes radiating from a center point, resembling budding rosettes (e.g. Fig. d). The columnar growth regime at temperatures below -40°C is consistent with previous laboratory studies, e.g. by . They reported that for relative humidities below RHice=125% mainly single columns grow, while for higher relative humidity (budding) rosettes are expected. In Table  the fraction of replicas classified as columnar is shown for different relative humidity with respect to ice between 105 % and 120 % and for different gas temperatures. At -40°C the column fraction ranges between 12 % and 25 % with no clear dependence on the relative humidity with respect to ice. At -50°C the fraction of replicas classified as columnar is larger than at -40°C, ranging from 23 % (RHice=105%) to 59 % (RHice=120%). At this temperature range the columnar fraction increases with increasing supersaturation in the regrowth phase. The higher column fraction with a mean of 41 % at -50°C in comparison to a mean of 19 % at -40°C agrees well with previous results of laboratory-grown ice crystals by , where at -50°C more single crystal columns are expected in comparison to the transition region between columnar and plate-like growth regimes at -40°C. The observed transition towards columnar growth with increasing relative humidity between -40 and -50°C for RHice<130% has also been reported. Example microscope images of formvar replicas from experiment 29, 32, 38 and 40 of the RICE04 campaign are shown in Fig. e, d, c, and a. Due to the small sample number, the fraction of columnar particles and hollow columnar particles seen in Fig.  is not representative. Nonetheless, the general occurrence of hollowness on the prism faces at both cirrus temperatures and the smaller particles sizes at -50°C in comparison to -40°C can be observed on the microscope images.

Statistically more robust information about ice crystal shapes can be derived from the SID-3 diffraction patterns using the Fourier analysis method. Of all particle images at cirrus temperatures of the investigated campaigns 0.4 % show diffraction patterns with features of spherical particles, 3.8 % show diffraction patterns with features of plates, 35 % show diffraction patterns with features of columns and 61 % show diffraction patterns with features of irregular ice particles. This is in good agreement with the fractions of columnar and irregular particles identified on the microscope images of the replicas taken during the regrowth phases of a subsample of the experiments (see Table ). In Fig. a the fractions of the ice crystal shapes are shown for the different campaigns and temperature groups. It can be noted that the fraction of columns increases from 19 % (RICE01) and 18 % (RICE03) for initial gas temperatures between -45 and -39°C to 44 % (RICE01), 36 % (RICE02) and 38 % (RICE03) between -75 and -45°C. This characteristic was also seen in the replica analysis.

Furthermore, different types of morphological complexities can be studied with the ice crystal replicas. The most common type of complexity is hollowness of the basal facets (Fig. a) with occasional air inclusions (Fig. e). Table  shows the fraction of hollow columns to all columns. This fraction varied between 33 % (RHice=105%) and 73 % (RHice=107%), with a mean of 46 % for experiments initiated at -40°C. At initial gas temperatures of -50°C the mean fraction of columns that are hollow per experiment ranges between 4 % (RHice=109%) and 70 % (RHice=112%), with a mean of 40 %. Hollow fractions below 30 % are only observed for experiments at initial gas temperatures of -50°C. Thus, for both investigated temperatures a significant mean fraction of columns (more than 40 %) show hollowness. This is in contrast to a laboratory study by , who estimated a critical supersaturation of 20 % for columns to develop hollowness at the basal facets at temperatures below -40°C. It is not always possible to visually identify hollowness on the microscope images of the formvar replicas. Therefore, the observed fraction of columnar ice particles and hollow columnar ice particles should be considered as a lower limit. In atmospheric cirrus observations found all bullets but only few columns to be hollow in microscope pictures of precipitation at cirrus temperatures at South Pole station between -73 and -35°C. observed hollow ends in 52 %–80 % of all bullet-rosette and columnar particles of formvar replicas from balloon-borne mid-latitude cirrus observations between -46 and -33°C. The increasing trend in the fraction of columns with hollow ends with temperature is in agreement with our cloud chamber observations. Another type of crystal complexity that was observed on the replicas are dislocations (Fig. b). They are thought to form from thermal stress in the changing growth conditions that can be present during the experiments .

Due to the optical limitation of the microscope and the replication technique, it is not possible to make conclusions about sub-micron scale complexity, such as surface roughness, from the microscope replica analysis. For this, we use the crystal complexity information derived from the SID-3 diffraction patterns. Figure b shows a statistical analysis of the optical complexity parameter ke for the different campaigns and temperature groups. For the ice crystals grown in the higher temperature range, the median ke is with 5.22 clearly higher than the threshold of 4.6 defined for morphologically complex ice crystals. In the lower cirrus temperature range median ke is with 4.54 below the threshold of complex ice crystals. One reason can be that higher growth rates at higher temperatures promote crystal complexity, as discussed in . In addition, small differences can be seen in the fraction of columnar ice crystals and ke for the different measurement campaigns in the same temperature groups. For example, RICE03 has a higher median ke in comparison to RICE02 and RICE01 in both temperature groups. These differences can be explained by different RHice during the experiments of the different campaigns. In the next section, we analyse the depolarisation properties for each of the two temperature ranges separately.

In Fig.  the measured δ is shown as a function of GMD derived from the PPD-2K particle size distributions for the previously defined temperature ranges. Figure a shows the data in the higher temperature range between -45 and -39°C, where δ has a minimum of 0.08 for a GMD around 12 µm and increases for both larger and smaller GMD up to 0.3. In the lower temperature range between -75 and -45°C during RICE01 and RICE02, δ has a constant low value between 0.05 and 0.10 for sizes between 8 and 25 µm (Fig. b). For smaller sizes below 8 µm observed during the HALO06 campaign, δ increases sharply and reaches values up to 0.5. These are two distinct temperature-dependent depolarisation ratio–size modes. However, the data from the RICE03 campaign show a similar behaviour as for the higher temperature range in Fig. a. This difference is likely caused by more complex crystals during the lower cirrus temperatures of RICE03 with a median higher than the threshold value of kethr=4.6 (see Fig. ). A detailed comparison to previous atmospheric observations is given in Sect. . In the following section, we investigate the link between δ and the optical crystal complexity.

In Fig.  the measured δ is shown as a function of the optical crystal complexity metric derived from SID-3 measurements for all available cirrus temperatures between -75°C<T<-39°C. Pearson product-moment correlation coefficients R are determined and a linear regression is added for R>0.5. This is done separately for each campaign where SID-3 was operated. The results are separated into size groups of (5±1)µm, (10±1)µm, (15±1)µm and (20±1)µm to limit the effect of a possible size dependence of ke.

For the smaller three particle size ranges of (5±1)µm, (10±1)µm and (15±1)µm |R|≤0.13 indicates no or weak correlation between ke and δ. Only for the largest size range of (20±1)µm a moderate correlation of R=0.54 is observed. The scarcity of particles in this largest size group during the RICE02 measurement campaign can be attributed to the lower initial gas temperature of -50°C, compared to RICE01 and RICE03, which included experiments at -50 and -40°C. Higher growth temperatures promote the formation of larger ice particles through increased growth rates at the same relative humidity .

The weak to moderate correlation between the optical complexity parameter ke and the depolarisation ratio δ contrasts with previous studies, which have reported increasing δ with increasing surface roughness in the 5–20 µm size range . Using numerical simulations, showed that δ increases with increasing surface roughness up to a threshold of σ2≈0.1, beyond which additional roughening has little to no further effect on δ. Most of the crystals in our experiments fall within the range ke>4.6, which could indicate that their surface roughness was already above this threshold – potentially explaining the weak dependence of δ on optical complexity. found that ke>4.6 is indicative of moderately to strongly roughened crystals, supporting this interpretation. Growing more pristine crystals with σ2<0.1 would require very low supersaturations and long growth times, conditions that are difficult to maintain in the AIDA cloud chamber due to sedimentation losses. Alternatively, other morphological features not captured by ke, such as crystal habits, may have varied across experiments and influenced δ, further weakening the observed correlation. This highlights the complexity of the relationship between δ and ice crystal morphology.

In this section, the measurement data for the two temperature ranges are compared to two types of T-matrix simulations and to conventional geometric optics numerical (CGOM) simulations at the near-backscattering direction of 178°. Figure  shows δ from T-matrix numerical simulations of spheroidal particles and CGOM simulations of columns with hollow basal facets, together with the measurement data from Fig. . The simulation parameters are detailed in Sect. . With the assumption of spheroidal particle shapes – a rough approximation of the real, much more complex ice particle shapes – the T-matrix simulations reproduce the size-dependence of the measurement data well in the size range of about 2–9 µm. Overall, the T-matrix results of spheroidal particles are only meaningful in the lower temperature range (Fig. b), as for the warmer temperature range the ice crystals are too large to compute the light scattering properties with the method of .

The CGOM simulations of δ use hollow columns where each basal facet is hollowed to a depth of 33.3 % of the length of the column. The schematics are shown in Fig. c. This type and length of hollowness is consistent with the ice crystal replicas that were found on the microscope images of the formvar slides (e.g. see Fig. a) and with the hollowness parametrisation of . The consideration of hollowness lowers the linear depolarisation ratios to the range of the measurement data, likely due to the presence of additional planar surfaces inside the crystal . Using a mean free path around 2000 µm, the δ values of the higher cirrus temperature range between -45 and -39°C can be reproduced. Using a mean free path of about 4000 µm, the CGOM simulations reproduce the δ values of the lower cirrus temperature range between -75 and -45°C. The implementation of ice crystal complexity with hollowness, surface roughness and internal scatterers in the CGOM ray tracing methods allows to reproduce the measured δ. CGOM simulations of solid columns overestimate δ, as shown in Appendix , where ice crystal complexity is varied with distortion parameters from 0 (pristine) to 0.5 (highly complex). The CGOM simulation results are almost identical for aspect ratios of 1.75 and 2.0, and for the wavelength of λ=552nm, which was used during the RICE03 campaign (see Appendix  and ). CGOM simulations of hollow columns using mean free path from 104µm, indicating negligible internal scattering, to 500 µm, indicating high internal scattering are shown in Appendix .

CGOM simulations use the geometric optics approximation and therefore only yield reliable results for size parameters larger than about 102. Recent numerically exact T-matrix simulations have the capability to fill the size range between the smallest particle sizes and ray tracing simulations in the geometric optics approximation for complex shaped hexagonal particles . Figure  shows δ from IITM simulations with aspect ratios between 1.0 and 2.0. There is a sharp increase in δ with increasing particle size from δ≈0 at particle sizes<1µm to 0.2<δ<0.5 at particle sizes around 3 µm. For particle sizes larger than about 3 µm, δ reaches a plateau with smaller variation with size. The value of δ on the plateau increases with increasing surface roughness σ2. The simulations with aspect ratios of 1.5 and 2.0 show slightly higher results for δ in comparison to simulations with an aspect ratio of 1.0. While the simulated δ is lower than the CGOM simulations of solid hexagonal particles (see Appendix ), it overlaps with the CGOM simulations of hollow columns at low mean free paths (see Appendix ). Yet, the δ from the IITM simulations still generally overestimates the measurement data. Only for the pristine case (σ2=0) with aspect ratio 1.0, the simulations partly overlap with the measurement data. However, based on the microphysical data at least some degree of complexity on the ice crystal replica images and the optical small-scale complexity parameter is expected.

Similarities and differences in the evolution of δ with particle size from the IITM simulations are found in comparison to the T-matrix simulations. The sharp increase in δ for the smallest particle sizes in the IITM simulations is also seen in the T-matrix simulations. However, the peak of δ for the T-matrix simulations is at a lower size of about 2 µm in comparison to the IITM simulations. Furthermore, the subsequent plateau of δ is observed in both simulations, but at lower sizes and for lower δ in the T-matrix simulations in comparison to the IITM simulations. Further refining the ice crystal complexity model of the IITM simulations, for example, by incorporating hollowness, could enhance its agreement with the measurement data.