1. Introduction

Optical and microphysical properties of cirrus clouds are one of the main sources of uncertainties in the up-to-date numerical models of weather forecasting and the global climate change. Therefore, these properties have been studied for many years by various remote-sensing techniques including lidars operating from both the ground, e.g [1], and space [2]. Conventional lidars measure the backscattering coefficient which gives mainly the vertical profiles of the particle number density in the clouds. The microphysical properties, i.e. the size, the shape and the orientation of the ice crystals constituting cirrus clouds, are investigated in lidar sounding when the polarization properties of lidar signals are involved. Thus, Sassen [1] suggests that large values of the lidar depolarization ratio are attributed to crystal clouds contrary to water-drop ones giving negligible depolarization. Later Noel et al. [3] tried to associate the depolarization ratio magnitudes with shapes of the ice crystals. However, plate-like ice crystals, which often reveal their preferentially horizontal orientation, also result in negligible depolarization if a lidar is pointing vertically. For example, the space-borne lidar CALIPSO [2] was turned at 3° off nadir in purpose to avoid the specular scattering produced by the horizontally oriented crystals.

The problem of discrimination of the horizontally oriented crystals can be solved experimentally using a scanning or tilted lidar whose axis direction, i.e. line-of-sight, is variable. Platt et al. [4] were the first who used the polarization lidar scanning near zenith to distinguish between the horizontally and randomly oriented crystals. Later similar measurements were carried out by Noel and Sassen [5]. Also Del Guasta et al. [6] applied the polarimetric lidar tilted at 30° off zenith to distinguish the horizontally oriented plate-like ice crystals using two quantities: the linear and circular depolarization ratios.

In general, the polarization properties of the light backscattered by the ice crystals are described not only by the linear and circular depolarization ratios, they are also presented by 16 elements of the Mueller matrix [7]. A few lidar instruments measuring the full Mueller matrix have been fabricated until now. In particular, a vertically pointing lidar measuring the Mueller matrix operates in Russia, Tomsk [8]. Recently, similar measurements of the full Mueller matrix characterizing the Asian dust in Korea, Daejeon, were reported, too [9]. The same vertically pointing lidar is under construction in China, Hefei. Also, in the USA, Boulder, the full Mueller matrix of both ice crystals and rain drops in the atmosphere was measured by a lidar pointing not vertically but tilted at some fixed angles off zenith [10].

In [8], Kaul at al. suggested that the preferential orientation of ice crystals in cirrus clouds could be not only quasi-horizontal, but the crystals could be preferably oriented in some azimuth direction, too. These authors showed that the full Mueller matrix obtained with a vertically pointing lidar was capable to discriminate the azimuth orientation; their experimental data supported this suggestion. Recently, Hayman et al. [10] studying cirrus clouds with their tilted lidar failed to detect the azimuthally oriented crystals at the lidar tilts of 4° and 22°, and only the tilt of 32° allowed them to discover such particles. Thus, the question whether the azimuth orientation of the crystals can be detected with a vertically pointing polarization lidar has not been completely clarified yet.

All the above-mentioned works have the following shortcoming: they are mostly empiric. The reason for this is that the problem of light backscattering by cirrus clouds has not been solved satisfactorily until now because of severe demands to computer capacities [11]. Only recently, some progress in the theoretical studies has been achieved using the physical-optics approximation [12,13]. In this paper, the Mueller matrix for the ice crystals with both the zenith and azimuth preferential orientations has been calculated for the first time by means of the physical-optics approximation. In this paper, we consider only the case of a vertically pointing lidar. Also, the simplest shapes of the ice crystals such as the hexagonal plates and columns have been considered. Nevertheless, these theoretical data can be useful for, at least, qualitative interpretation of the empirical data obtained by vertically pointing polarization lidars [8,9,14].

2. Polarization characteristics

As known [7,15,16], the backscattering Mueller or phase matrix M describing all polarization properties of the backscatter becomes diagonal in the case of random crystal orientation

In the matrix of Eq. (1), the polarization elements are expressed through the quantity d called the depolarization parameter. Instead of this value, the linear ${\delta }_l$ and circular ${\delta }_c$ depolarization ratios are often used. They are related with each other by the following equations

We emphasize that the diagonal view of the Mueller matrix is valid not only for random crystal orientations but it is true when the scattering media are rotationally symmetric around the lidar axis. Thus, in both cases of the random and rotationally symmetric orientations of the crystals any polarization lidar measurements are reduced to only one of the quantities d, ${\delta }_l$, and ${\delta }_c$, so there is no necessity to measure the other elements of the Mueller matrix.

However, if the crystals reveal their preferential orientation in the plane perpendicular to the lidar axis, which will be called the azimuth orientation, some new elements of the Mueller matrix become nonzero. On the one hand, the appearance of these elements proves that the azimuth orientation of the crystals takes place. On the other hand, these elements provide additional information about size, the shapes and the orientation of the crystals as compared to the information available in the case of the rotational symmetry of crystal orientations. If the plane of polarization coincides with the base direction of the preferential azimuth orientation of the crystals, the Mueller matrix has the view

In measurements, the base direction of the preferential orientation and the plane of polarization, generally speaking, do not coincide. Let us denote by $\phi$ the angle between these directions. Then the experimentally measured Mueller matrix is found by means of the rotation matrix R for the Stokes parameters as

It is worth noting that the element $m_{44}$ does not change by the transformation of Eq. (5) and it is simply related with the conventional circular depolarization ratio ${\delta }_c$ [7] as

3. Model of crystal orientations

The hexagonal ice plates and columns are determined by their size and orientation. Let us denote by D and h the diameter of the basal hexagonal face and height, respectively, as shown in Fig. 1(a). The origin of the Сartesian coordinate system x, y, and z is placed in the crystal center with the z-axis pointing to zenith. The crystal orientation is determined by three Euler angles α, β, and γ depicted in Fig. 1(b). Here the main axis passes through the centers of the hexagonal faces forming the normal N. For any crystal orientation, the direction of the main axis is determined by the point on the unit orientation sphere or by the azimuth α and zenith β angles counted from the x and z axis, respectively. The rotation about the main axis is determined by the third Euler angle γ shown in Fig. 1(b).

 

Fig. 1 The model of crystal orientations.

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At present, the knowledge about the distributions of the ice crystals over their orientations in cirrus clouds is quite poor. Moreover, no one considered theoretically the preferential azimuth orientation of the crystals until now. Therefore, in this paper, we assume the simplest model of crystal orientations. In spite of the fact that this model is far from the reality, it allows one to obtain the main qualitative regularities.

In particular, we assume the uniform distribution of the crystals over the rotation angle γ, i.e. the probability density function is equal to $p(\gamma )=1/2\pi$. Then let us proceed to the distributions of the normal N on the unit orientation sphere shown in Figs. 1(c) and 1(d). The hexagonal plates and columns are oriented horizontally if $\beta =0$ for the plates and $\beta =\pi /2$ for the columns. The probability density function $p(\alpha ,\beta )$ on the unit sphere is normalized as ${\displaystyle \underset{0}{\overset{2\pi }{\int }}d\alpha }{\displaystyle \underset{0}{\overset{\pi }{\int }}p(\alpha ,\beta )\sin \beta d\beta }=1$. The uniform distribution all over the unit sphere $p(\alpha ,\beta )=1/4\pi$ corresponds to the random orientation. Thus, all orientation distributions vary within the fixed orientation, on the one side, and the random orientation, on the other side.

Our orientation model assumes that the probability density function $p(\alpha ,\beta )$ is a constant within the shaded domains on the unit sphere in Figs. 1(b) and 1(c) and zero outside the above domains, i.e.

Since this orientation model can lead to criticism, let us indicate the following steps for its improvement in further works. Firstly, the problem of the ice crystal orientation in the atmosphere is a complicated problem of aerodynamics [17]. Small crystals $\sim 10\text{μm}$ in diameter should be randomly oriented because of Brownian motion (thermal noise). As known [17], drag is produced as vorticity generated at the crystal surface. For the small crystals, this aerodynamics effect is negligible. As the crystals grow bigger, the asymmetric drag appearing for an inclined plate-like crystal reorients the crystal to the horizontal orientation [18,19]. This reorientation is stabilized by viscosity. This is the reason for the horizontal orientation of the plate-like ice crystals observed in the atmosphere. Thus, a further orientation model should take into account the orientation forces depending on both size and shapes of the crystals.

Secondly, the distributions of size and the shapes of the crystals in cirrus are not well known yet. Depending on the environmental temperature and humidity, these distributions can strongly vary from bottom to top of the clouds (e.g., [20]). In reality, cirrus clouds are mixtures of ice crystals with different size and shapes that are distributed according to some probability densities. In this case, calculations of the backscattering Mueller matrix are computationally costly. Therefore, in this paper, the size for the hexagonal ice columns and plates is assumed to be fixed obeying the empirical equations for the aspect ratio suggested by Mitchel and Arnott [21].

4. Polarization Mueller matrixes versus the orientation distributions

In general, the Mueller matrix obtained by a polarization lidar has a view of Eq. (7). Then this matrix can be reduced to the matrix of Eq. (3) by the azimuth rotation according to Eq. (5). These are the elements of the reduced matrix (3) that are informative about the microphysical properties of the crystals, i.e. their size, shapes and orientation distributions. To clarify the relation between these microphysical properties and the elements of the Mueller matrix, the direct scattering problem is modeled in this section. Namely, we calculate the reduced Mueller matrix for hexagonal columns and plates of some typical sizes as functions of the widths of the orientation distributions over both zenith and azimuth angles. Our code of the physical-optics approximation is used [22]. The incident wavelength is 0.532 µm and the refractive index is assumed as 1.3116. Since the element $m_{44}$ does not depend on the rotations about the azimuth angle, we obtain the function $m_{44}(B)$ depending only on the zenith tilt while the other four functions $m_{12}(A,B)$, $m_{34}(A,B)$, $m_{22}(A,B)$, and $m_{33}(A,B)$ depend also on the azimuth tilt. It is worth noting that the functions $m_{22}(A,B)$ and $m_{33}(A,B)$ are not independent since they obey the condition

The results obtained for $m_{44}(B)$ are presented in Fig. 2. A distinct jump of $m_{44}(B)$ can be seen at the zenith tilt of 30° for both the hexagonal plates and columns. This result leads to the following important conclusion. The element $m_{44}(B)$ of the Mueller matrix and, consequently, the circular depolarization ratio at the low zenith tilts $B<30°$ essentially differ from the ones at the high zenith $B>30°$ tilts. Namely, we obtain $m_{44}<-0.5$ for $B<30°$ and $m_{44}>-0.5$ for $B>30°$. Consequently, the magnitude of the element $m_{44}$ allows us to divide the crystal orientations into two parts: the low zenith tilts $B<30°$ and high zenith tilts $B>30°$.

 

Fig. 2 The element $m_{44}$ versus the zenith tilts for the hexagonal ice plates (solid) and columns (dashed).

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This phenomenon has the following physical explanation. We remind that within the physical-optics treatment [23] the scattered light is formed by the geometric-optics ray trajectories inside the crystals. The ray trajectories are combined into the plane-parallel beams leaving the crystals. At the far distance from the particles, the outgoing beams are transformed into the Fraunhofer diffraction patterns about the geometric-optics propagation direction. It is shown that after averaging over crystal orientations the diffraction weakly distorts the polarization. Therefore the polarization elements of the Mueller matrix depend mainly on the kind of the ray trajectories inside the crystals which predominantly contribute to the scattered light. Size and shapes of the crystals are less important for the polarization elements.

Let us consider the typical ray trajectories shown in Fig. 3. Figure 3(a) shows the trivial specular reflection. Here the horizontally oriented hexagonal column in Fig. 3(a) rotating about the Euler angle γ summarizes the diffraction pattern formed by the specular-reflection beams. The main contribution to the integral is made by the reflection at the normal incidence. As known, the Mueller matrix of the normal reflection is diagonal $diag(1,1,-1,-1)$. The same is also true for the hexagonal plates at the low zenith tilts. Therefore, Fig. 2 demonstrates $m_{44}(B)\approx -1$ at $B<30°$ with good accuracy for the hexagonal plates. It means that the specular reflection is predominant for the plates at the low zenith tilts.

 

Fig. 3 Typical ray trajectories: specular reflection (a), grazing-ray trajectory (b), and corner reflection (c).

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There is also another kind of ray trajectories which is essential only for the hexagonal columns at the low zenith tilts $B<30°$. The trajectory is depicted in Fig. 2(b). These trajectories are formed by multiple grazing reflections inside the crystals. Therefore, they are called the grazing trajectories. A set of such parallel rays is combined into narrow plane-parallel beams leaving the crystals. These grazing beams cannot leave the columns at the exact backward direction but their propagation directions are close to the backward one. The light intensity inside these narrow beams is high because of the full internal reflections saving the light energy. Note that their propagation directions do not depend on the column diameters according to geometric optics. However, the width of the narrow grazing beams increases with increasing column diameters. Consequently, their contribution by means of the Fraunhofer diffraction decreases with increasing diameters when the column rotates about the angle γ. The polarization element $m_{44}$ of the grazing beams differs essentially from the quantity (–1) inherent to the specular reflection. As a result, Fig. 2 demonstrates the deviation of $m_{44}$ from (–1) for the columns at $B<30°$ because of the contribution from the grazing beams. This contribution increases for smaller column diameters because of the diffraction. According to the data presented in Fig. 2, we suggest that the grazing beams are essential when the column diameters are less than 50 µm.

At the high zenith tilts $B>30°$, on the contrary, neither the specular reflection nor the grazing beams are essential. Here the main contribution to the backscatter is obtained from the corner-reflection trajectories shown in Fig. 3(c). Predominance of the corner-reflection trajectories at the high zenith tilts $B>30°$ was discussed earlier [12,13]. As seen in Fig. 2, the condition $m_{44}>-0.5$ becomes the indicator of the high zenith tilts $B>30°$ for both the hexagonal columns and plates.

The first element of the Mueller matrix ${\sigma }_{\pi }$ is not directly responsible for polarization; it is the product of the crystal number density by the backscattering cross section for unpolarized incident light. Nevertheless, for completing the physical picture of the backscatter, the backscattering cross section has been also presented in Fig. 4. We see that at the low zenith tilts $B<10°$ the backscatter from the plates essentially exceeds the backscatter from the columns of the same size. This fact is easily explained by the predominant contribution from the specular trajectories shown in Fig. 2(a). At the high zenith tilts $B>60°$, on the contrary, the backscatter from the columns is essentially larger compared with the plates. In this case, the corner-reflector trajectories of Fig. 3(c) are predominant.

 

Fig. 4 The backscattering cross section for the hexagonal ice plates (solid) and columns (dashed). The crystal size is marked by the same color as in Fig. 2.

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When the crystals reveal either the rotationally symmetric or random orientations, the Mueller matrix reduces to Eq. (1) and $m_{44}$ becomes a single element which completely determines any polarization measurements. The existence of the preferential azimuth orientation leads to the appearance of two off-diagonal elements $m_{12}$ and $m_{34}$ in Eqs. (3) and (7). Besides, the diagonal elements $m_{22}$ and $m_{33}$ are already not related by the equation $m_{22}=-m_{33}$, taking place for the rotationally symmetric orientations. Due to Eq. (11) we have four independent quantities. Two of them are $m_{12}$ and $m_{34}$while the other two are ensured by the diagonal elements.

Let us answer the question: what elements of the reduced Mueller matrix of Eq. (3) are most sensitive to the azimuth orientation of the crystals? In other words, what elements of the Mueller matrix are the reliable indicators of the azimuth orientation? The answer follows from our numerical data presented in Figs. 5-8 for the reduced matrix $M_1$.

 

Fig. 5 The element $m_{12}$ versus the azimuth A and zenith B tilts: (a) plate, D = 31.6 µm, h = 9.53 µm; (b) plate, D = 100 µm, h = 16 µm; (c) column, D = 22.1 µm, h = 31.6 µm; (d) column, D = 123.8 µm, h = 316 µm.

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Fig. 6 The same as in Fig. 5 for the element $m_{34}$.

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Fig. 7 The same as in Fig. 5 for the element $m_{22}$.

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Fig. 8 The same as in Fig. 5 for the element $m_{33}$.

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To begin with, let us consider the case of the low zenith tilts $B<30°$. Excluding the case of the column of the diameter 22.1 µm, in Figs. 5-8 we see that the magnitudes of all four elements at any azimuth tilts A practically coincide with the diagonal matrix $diag(1,1,-1,-1)$ inherent to the normal specular reflection. Thus, our calculations have proven that the specular reflection is predominant for these plates and columns at the low zenith tilts $B<30°$. The above diagonal matrix does not transform with the rotation of Eq. (5). As a result, Eq. (9) is true for any azimuthal distribution. In other words, any vertically pointing lidar is not capable to distinguish the azimuth orientation of the crystals at their low zenith tilts $B<30°$.

However, there is an exception shown in Fig. 2 and Figs. 5(c)-8(c). It concerns the columns of diameters less than 50 µm. These columns contribute to the backscatter by the grazing beams together with the specular reflection. The grazing beams increase the element $m_{44}$ in Fig. 2. Their contributions to the other elements are shown in Figs. 5(c)-8(c). We see that the contribution of the grazing beams is negligible for the off-diagonal elements $m_{12}$ and $m_{34}$, but it is noticeable for the diagonal element $m_{22}$ or, equivalently, $m_{33}$. Thus, we conclude that if some measurements provide $m_{44}<-0.5$, the crystals have the low zenith tilts $B<30°$. Then, if the conditions of Eq. (9) are violated, the azimuth tilts should be estimated by measuring the diagonal element $m_{22}$ or, equivalently, $m_{33}$. Here measurements of the off-diagonal elements are not informative.

Then turn to the case of the high zenith tilts $B>30°$, where $m_{44}>-0.5$. Here we see that all elements of the Mueller matrix reveal strong variations with the variables A and B in the domains with the high zenith $B>30°$ and low azimuth $A<30°$ tilts. However, this situation when the zenith tilts are large but the azimuth tilts are small is hardly possible to observe in the reality. Therefore we omit these domains. For the moderate values of both the zenith and azimuth tilts we see that it is the element $m_{34}$ that is most variable with the azimuth A tilts. Therefore we conclude that if some measurements provide $m_{44}>-0.5$, the crystals have high zenith tilts $B>30°$. Then, if the Mueller matrix is not diagonal, the off-diagonal element $m_{34}$ should be measured to find the azimuth tilt A of the crystals. Here the diagonal element $m_{22}$ or, equivalently, $m_{33}$ are less informative about the azimuth orientation. For example, if one finds the zenith tilt B from Fig. 2 for a given crystal size and shape, then the azimuth tilt A can be easily found from Fig. 6 plotted for the same size and shape of the crystal.

5. Conclusions

While the preferentially horizontal orientations of the ice crystals in cirrus clouds are well known and they have been widely studied with the help of lidars, the preferentially azimuthal orientations of the ice crystals in the atmosphere has not been clarified yet. Thus, Kaul et al. [8] observed the preferentially azimuthal orientations measuring the Mueller matrix by a vertically pointing lidar. However, Hayman et al. [10] failed to detect this kind of orientations at the lidar tilts 4° and 22° off zenith. At the same time, the lidar tilted at 32° managed to observe such orientations.

The purpose of this paper was to clarify the possibilities of the vertically pointing lidars to retrieve the preferentially azimuth orientations of the crystals. We have calculated the backscatter Mueller matrix for hexagonal ice columns and plates by our code of the physical-optics approximation. The simplest model for orientation distributions of the crystals characterized by the azimuth A and zenith B tilts of the crystals was used. We suggest that a vertically pointing lidar can discriminate between the crystals with the low $B<30°$ and high $B>30°$ zenith tilts by measurements of the element $m_{44}$ or, equivalently, the circular depolarization ratio. At the low zenith tilts $B<30°$, a vertically pointing lidar is not capable to retrieve the azimuth tilts for the plate-like crystals and large hexagonal columns. However, measurements of the element $m_{33}$ can detect the azimuth tilts of the quasi-horizontally oriented hexagonal columns whose diameters are less than 50 µm. In the case of high zenith tilts $B>30°$ of the crystals, a vertically pointing lidar can retrieve the azimuth tilts A for both the hexagonal columns and plates by measuring the element $m_{34}$.