On the relation between ice-crystal scattering phase function at 180° and particle size: implication to lidar-based remote sensing of cirrus clouds

Jiachen Ding; Ping Yang; Ping Yang; Ping Yang; Gorden Videen; Gorden Videen; Gorden Videen

doi:10.1364/OE.491395

1. Introduction

Evaluation of the impact of ice clouds on the global radiative energy budget and water cycle is largely dependent on the measurement of global ice-cloud microphysics including particle sizes. Measured ice-cloud microphysical properties can also be used to validate and improve cloud microphysics schemes used in climate models (e.g., [1–3]). Spaceborne-lidar-based remote sensing [4,5] plays a critical role in inferring global microphysical properties of ice clouds (e.g., [6,7]), because of the vast spatial coverage and high vertical resolution of spaceborne lidar observations.

Accurate quantification of the backscattering properties of ice crystals is vital to lidar remote sensing of ice clouds. A lidar measures backscattered light from ice clouds in the exact 180° scattering direction. To infer the microphysical properties of ice clouds from the backscattered signal, we must first know how the particle size and shape (e.g., [8–12]) affect the backscattered signal.

In particular, the backscattered intensity is proportional to the scattering phase function of scattering particles at the 180° scattering angle (${P_{11}}({180^\circ } )$). Using light-scattering simulations, we find that for some particles, the random-orientation-averaged ${P_{11}}({180^\circ } )$ has a linear or quadratic dependence on the particle size. The particles are weakly absorptive and have two adjacent facets that are perpendicular to each other. Borovoi et al. [13–15] used a physical-optics approximation to study backscattering of ice crystals with sizes much larger than the wavelength of incident light, and found that the differential backscattering cross section ${C_{\textrm{bsca}}}$ of the hexagonal column has a cubic relation to the particle size. Konoshonkin et al. [16] shows that ${C_{\textrm{bsca}}}$ follows power laws with respect to the particle size. ${C_{\textrm{bsca}}}$ is proportional to ${P_{11}}({180^\circ } )$ multiplied by the scattering cross section ${C_{\textrm{sca}}}$. If a particle is much larger than the wavelength of incident light, ${C_{\textrm{sca}}}$ is approximately equal to twice the particle projected area ${A_\textrm{p}}$, which is proportional to the particle size squared. Moreover, Konoshonkin et al. [16] reports that the (2, 2) element of the backscattering Mueller matrix can be expressed in a power-law function with respect to particle size, which varies for different particle shapes. The significant dependence of ${C_{\textrm{bsca}}}$ on particle size is attributed to the corner-reflection (CR) structure in an ice crystal [17].

In this study, we use a synergistic combination of the invariant imbedding T-matrix method (IITM) and physical geometric-optics method (PGOM) [18] to investigate the relations between ${P_{11}}({180^\circ } )$ and particle size (the ${P_{11}}({180^\circ } )- L$ relation) for different particle shapes. Then, we analyze and explain ${P_{11}}({180^\circ } )- L$ relations for particles with the CR structure using the PGOM formulation and Fraunhofer diffraction theory. A High Spectral Resolution Lidar (HSRL) [19], such as the Atmospheric Lidar to be onboard the planned EarthCARE satellite [5], can directly measure the lidar ratio of ice clouds. ${P_{11}}({180^\circ } )$ is proportional to the inverse of the lidar ratio [11] and thus can be derived from an HSRL measurement. The known ${P_{11}}({180^\circ } )- L$ relation can then be used to retrieve ice-cloud particle sizes. Alternatively, the particle size can be measured by other instruments such as radar. Because the ${P_{11}}({180^\circ } )- L$ relation depends on particle shapes, collocated ${P_{11}}({180^\circ } )$ and particle-size measurements provide particle-shape information. Therefore, an in-depth understanding of the ${P_{11}}({180^\circ } )- L$ relation will benefit robust retrievals of ice-cloud microphysics based on HSRL measurements. Section 2 introduces the IITM and PGOM used in the study. Section 3 discusses light-scattering simulations of ${P_{11}}({180^\circ } )- L$, and analyzes the ${P_{11}}({180^\circ } )- L$ relation. Section 4 concludes this study.

2. Light-scattering computational methods

This study focuses on the far-field scattering of an incident monochromatic plane wave by a dielectric particle. The electric vector of the incident wave is expressed as

(1)$${{\mathbf{E}}^{\textrm{inc}}}({\mathbf r} )= {\mathbf E}_0^{\textrm{inc}}\textrm{exp}({i{\mathbf k} \cdot {\mathbf r}} ), $$

where the superscript ‘inc’ denotes the incident wave, ${\mathbf k}$ is the wave vector and ${\mathbf r}$ is the position vector. $k = |{\mathbf k} |$ is the modified wavenumber defined as $2\pi /\lambda $, and $\lambda $ is the wavelength of the incident wave in the ambient medium assumed to be nonabsorptive in this study. The time-harmonic term $\textrm{exp}({ - i\omega t} )$ is omitted in Eq. (1) for brevity, where $\omega $ is angular frequency and t is time.

Both the incident and scattered electric fields can be decomposed into two orthogonal components. The orthogonal components of the electric fields are related by a scattering amplitude matrix S [20],

(2a)$$\left[ {\begin{array}{c} {E_\parallel^{\textrm{sca}}({\mathbf r} )}\\ {E_ \bot^{\textrm{sca}}({\mathbf r} )} \end{array}} \right] = \frac{{\textrm{exp}({ikr} )}}{{ - ikr}}{\mathbf S}({\hat{r},\hat{k}} )\left[ {\begin{array}{c} {E_{0,\parallel }^{\textrm{inc}}}\\ {E_{0, \bot }^{\textrm{inc}}} \end{array}} \right], $$

(2b)$${\mathbf S}({\hat{r},\hat{k}} )= \left[ {\begin{array}{cc} {{S_2}}&{{S_3}}\\ {{S_4}}&{{S_1}} \end{array}} \right], $$

where the superscript ‘sca’ denotes the scattered wave, r ($r \gg \lambda $) is the distance between the scattering particle and the observer, $\hat{k}$ is a unit vector along the direction of incidence, and $\hat{r} = {\mathbf r}/r$ is the scattering directional vector. Subscripts ‘${\parallel} $’ and ‘$\bot $’ denote the quantities whose vectors are parallel and perpendicular to the scattering plane formed by $\hat{k}$ and $\hat{r}$. Note that the scattering amplitude matrix is for a particle with a fixed orientation.

The scattering phase function ${P_{11}}$ is defined as

(3)$${P_{11}}({\hat{r},\hat{k}} )= \frac{{2\pi }}{{{k^2}{C_{\textrm{sca}}}}}\mathop \sum \nolimits_{j = 1}^4 {|{{S_j}} |^2}. $$

The backscattering case corresponds to $\hat{r} ={-} \hat{k}$. The random orientation-averaged ${P_{11}}$ can be computed by averaging the ${C_{\textrm{sca}}}$-weighted ${P_{11}}$ at all particle orientations [21]. The random orientation-averaged ${P_{11}}$ satisfies

(4)$$\frac{1}{2}\mathop \int \nolimits_0^\pi {P_{11}}(\mathrm{\Theta } )\mathrm{sin\Theta }d\mathrm{\Theta } = 1, $$

where $\mathrm{\Theta }$ is the scattering angle given by $\mathrm{\Theta } = \textrm{arccos}({\hat{r} \cdot \hat{k}} )$. The lidar ratio can be directly obtained from ${P_{11}}({180^\circ } )$ as $4\pi /[{{P_{11}}({180^\circ } )\varpi } ]$ [11], where $\varpi $ is the single-scattering albedo.

In this study, we use the IITM and PGOM to compute ${P_{11}}({180^\circ } )$ for various particle shapes and sizes. In the following discussion, ${P_{11}}$ is the random orientation-averaged quantity.

The IITM computes the T-matrix based on the electromagnetic volume integral [22–24]. The invariant imbedding technique [25] is used to solve the volume integral equation. IITM is assumed to be numerically exact, because its formulations can be directly derived from Maxwell’s equations without any physical approximations. The Legendre polynomial expansion coefficients of ${P_{11}}$ can be analytically calculated from the T-matrix elements [26]. ${P_{11}}({180^\circ } )$ is computed by

(5)$${P_{11}}({180^\circ } )= \mathop \sum \nolimits_{l = 0}^\infty {({ - 1} )^l}\alpha _1^l, $$

where $\alpha _1^l$ is the Legendre polynomial expansion coefficient of ${P_{11}}$ at expansion degree l. The term ${({ - 1} )^l}$ is the $l$th-degree Legendre polynomial value at $180^\circ $. Limited by computational capability, IITM can only be applied to particles with small to moderate size parameters. The size parameter is defined as k multiplied by a particle characteristic radius, such as half of the maximum dimension $.\; $ Yang et al. [18] shows the consistency between the IITM and PGOM for moderately sized particles. We use the PGOM to compute ${P_{11}}({180^\circ } )$ for particles with moderate and large size parameters.

The PGOM combines the geometric-optics ray-tracing technique with electromagnetic surface and volume integrals [27,28]. In a far-field approximation, the scattered electric field can be expressed in terms of a surface integral,

(6)$${\left. {{{\mathbf{E}}^{{\text{sca}}}}\left( {\mathbf{r}} \right)} \right|_{kr \to \infty }} = \frac{{{\text{exp}}\left( {ikr} \right)}}{{ - ikr}}\frac{{{k^2}}}{{4\pi }}\hat{r} \times \mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc} {\left\{ {\hat{n} \times {\mathbf{E}}\left( {{\mathbf{r'}}} \right) - \hat{r} \times \left[ {\hat{n} \times {\mathbf{H}}\left( {{\mathbf{r'}}} \right)} \right]} \right\}} {\text{exp}}\left( { - ik\hat{r} \cdot {\mathbf{r'}}} \right)dS$$

where the integral is over a closed surface surrounding the particle. $\hat{n}$ is the normal vector toward the outside of the closed surface. ${\mathbf E}({{\mathbf r^{\prime}}} )$ and ${\mathbf H}({{\mathbf r^{\prime}}} )$ are the electric and magnetic fields on the exterior side of the closed surface, which are computed by geometric-optics ray tracing in the PGOM. The ray-tracing technique is further improved by Bi et al. [29], Sun et al. [30] and Yang et al. [18], where a bundle of rays or a beam is traced. Beam tracing uses the beam-splitting technique to trace beams with polygon shapes and is much more efficient than ray tracing. Beam tracing is equivalent to tracing a bundle of rays within a beam.

The explicit expression of the scattering amplitude matrix can be derived from Eq. (6) by decomposing the electric and magnetic fields into two orthogonal components. In the PGOM framework, we have

(7)$${\mathbf S} = \mathop \sum \nolimits_{p = 0} \mathop \sum \nolimits_{n = 1} {{\mathbf S}_p}, $$

where the inner summation is for different beams and the outer summation is for beam-tracing orders. The dependence of ${{\mathbf S}_p}$ on the beam index n is omitted for brevity. p is the number of times a beam interacts with particle facets. $p = 0$ is for diffraction, $p = 1$ is for external reflection, $p = 2$ is for a beam refracting into the particle and refracting out without internal reflection, and $p \ge 3$ is for an outgoing beam with $p - 2$ times of internal reflection. The beam-tracing process starts from splitting the incident beam, which has the shape of the particle projection along the direction of incidence. Each split beam is uniquely incident on a particle facet. The incident beam is then reflected by the particle surface and refracts into the particle. The surface integrals of the incident and externally reflected fields contribute to the diffraction and external reflection components ${{\mathbf S}_0}$ and ${{\mathbf S}_1}$, respectively. The refracted beam inside the particle is split in the same manner, and each split beam is reflected and refracted on an internal surface of the particle. The fields in the beams refracted out of the particle are used in the surface integral and contribute to ${{\mathbf{S}}_p},p > 1$. At each reflection-refraction event, a beam is split and the beam tracing is continued for each of the split beams.

${{\mathbf S}_p}$ is expressed as

(8)$${{\mathbf S}_p} = {D_p}{{\mathbf L}_p}{\mathbf U}_p^s, $$

where matrix ${{\mathbf L}_p}$ accounts for the transformation of the electric field polarization reference plane from the $p$th-order reflection-refraction plane to the incident scattering plane [18]. Matrix ${\mathbf U}_p^s$ describes the reflection and refraction processes, and related reference plane transformation [27]. ${D_p}$ is a surface integral,

(9)$${D_p} = \frac{{{k^2}{e^{i{\delta _p}}}}}{{4\pi }}\mathop \iint \nolimits_{{A_p}}^{} {\text{exp}}\left[ {ik\left( {{{\hat{k}}_p} - \hat{r}} \right) \cdot {\mathbf{r'}}} \right]{d^2}{\mathbf{r'}},$$

where the surface integral is over the intersection area ${A_p}$ of a beam and the particle surface, ${\delta _p}$ is the phase of the wave in the beam and ${\hat{k}_p}$ is the propagation direction of the beam. When ${\hat{k}_p} \ne \hat{r}$, ${D_p}$ can be evaluated analytically based on the Stokes theorem [29]. When ${\hat{k}_p} = \hat{r}$, the surface integral in Eq. (9) is equal to the area of ${A_p}$. Equation (9) explicitly shows that a beam propagating along direction ${\hat{k}_p}$ contributes to all scattering directions $\hat{r}$.

According to Eqs. (3) and (8), ${P_{11}}$ computed by PGOM can be expressed as

(10a)$${P_{11}} = \frac{{4\pi }}{{{k^2}{C_{\textrm{sca}}}}}({{F_a} + {F_b}} ), $$

(10b)$${F_a} \equiv \frac{1}{2}\mathop \sum \nolimits_{p = 0} \mathop \sum \nolimits_{n = 1} \mathop \sum \nolimits_{j = 1}^4 {|{S_{p,j}^n} |^2}, $$

(10c)$${F_b} \equiv \frac{1}{2}\mathop \sum \nolimits_{p = 0} \mathop \sum \nolimits_{q \ne p} \mathop \sum \nolimits_{n = 1} \mathop \sum \nolimits_{m = 1} \mathop \sum \nolimits_{j = 1}^4 ({S_{p,j}^nS{{_{q,j}^m}^\ast }} )+ \frac{1}{2}\mathop \sum \nolimits_{p = 0} \mathop \sum \nolimits_{n = 1} \mathop \sum \nolimits_{m \ne n}\mathop \sum \nolimits_{j = 1}^4 ({S_{p,j}^nS{{_{p,j}^m}^\ast }} ), $$

where ${F_a}$ is the contribution of individual beams without interference with other beams, and ${F_b}$ accounts for the interference among different beams. The superscripts m and n are the beam indices. The asterisk symbol indicates complex conjugate.

3. Simulation results and analysis

We consider six particle shapes shown in Fig. 1, namely, a regular hexagonal column (RHC, Fig. 1(a)), triangular prism (TP) (Fig. 1(b)), droxtal (DRX, Fig. 1(c)), irregular hexagonal column without perpendicular facets (IRHC1, Fig. 1(d)), irregular hexagonal column with two perpendicular facets (IRHC2, Fig. 1(e)) and cube (Fig. 1(f)). The definitions of characteristic sizes L for individual shapes are illustrated in Fig. 1. Hexagonal columns and droxtal shapes are found in atmospheric H₂O ice crystals [31]. The cube shape can be used to model CO₂ ice crystals, which are observed in the Martian atmosphere [32]. The TP shape is selected only for theoretical study.

Fig. 1. ${P_{11}}$ versus scattering angles for $kL$ equal to 500.0, 1000.0 and 2000.0, computed by PGOM. (a): a regular hexagonal column (RHC); (b): triangular prism (TP); (c): droxtal (DRX); (d): irregular hexagonal column without perpendicular facets (IRHC1); (e): irregular hexagonal column with 2 perpendicular facets (IRHC2); and (f): cube. The refractive index for the RHC, TP, IRHC1 and IRHC2 cases is $1.3117 + 1.48 \times {10^{ - 9}}i$, for the DRX case is $1.3243 + 2.00 \times {10^{ - 11}}i$, and for the cube case is $1.4180 + 7.28 \times {10^{ - 7}}i$. The inset plot in each panel shows scattering angles from 174° to 180°.

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Except for the IRHC1 and IRHC2, all shapes are regular and contain symmetries. In the IRHC2, the top facet is perpendicular to one of the side facets. The RHC, TP, IRHC2 and cube have so-called CR structures, where there are orthogonal facets. To generate the IRHC1, we first randomly and independently tilt each facet of an RHC. Then, we obtain the intersections of the eight tilted facets and construct the IRHC1. The IRHC2 is constructed in a similar manner, except that we keep the top facet and a side facet of an RHC fixed, and tilt other facets. The vertex coordinate data (x, y, z Cartesian coordinates) of the IRHC1 and IRHC2 with $kL = 1$ are given in Table 1. The geometric parameters of the droxtal shape are defined in [33].

Table 1. Vertex Coordinates of the Irregular Hexagonal Columns 1 and 2 with ${\boldsymbol kL} = 1$

View Table

In the present light-scattering computations, the refractive indices used are $1.3117 + 1.48 \times {10^{ - 9}}i$, $1.3243 + 2.00 \times {10^{ - 11}}i$, and $1.4180 + 7.28 \times {10^{ - 7}}i$ for the RHC, IRHC and TP, for the DRX, and for the OCT and cube, respectively. The three refractive-index values are for H₂O ice with a temperature of 210 K [34] at wavelengths 532 nm and 355 nm, and CO₂ ice in visible light [35], respectively.

Figure 1 shows the ${P_{11}}$ of the particles with the six shapes at all scattering angles and with size parameters $kL$ equal to 500.0, 1000.0 and 2000.0. As shown in the inset plots of Fig. 1, all particles have backscattering peaks. For RHC, TP, IRHC2 and cube, their backscattering peaks are stronger with a larger $kL$. For DRX and IRHC1, the similar dependence of backscattering peak magnitude on $kL$ is not manifested.

${P_{11}}({180^\circ } )- kL$ relations are shown in Fig. 2 for the particles with the six shapes and $kL$ ranging from 2 to $3 \times {10^4}$. Limited by computational resources, IITM is applied for $kL$ up to 200 for RHC and droxtal shapes, up to 150 for IRHC1 and cube, and up to 100 for TP and IRHC2. ${P_{11}}({180^\circ } )$ is a random-orientation-averaged quantity. The random-orientation averaging is carried out analytically in IITM computations [24]. PGOM computes single-scattering properties of a particle at many random orientations. Then, the single-scattering properties are averaged numerically to obtain the random-orientation-averaged quantities. We find that, in PGOM computations, the number of random orientations (${N_{\textrm{ro}}}$) must be large enough to obtain converged ${P_{11}}({180^\circ } )$ values with respect to ${N_{\textrm{ro}}}$. The required ${N_{\textrm{ro}}}$ generally increases with increased particle size. In PGOM computations of ${P_{11}}({180^\circ } )$, we assign ${N_{\textrm{ro}}}$ to be $2 \times {10^6}$, $4 \times {10^6}$, $8 \times {10^6}$, ${10^7}$, $4 \times {10^7}$ and $8 \times {10^7}$ for $100 \le kL < {10^3}$, ${10^3} \le kL < 2 \times {10^3}$, $2 \times {10^3} \le kL < 5 \times {10^3}$, $5 \times {10^3} \le kL < {10^4}$, ${10^4} \le kL < 2 \times {10^4}$, and $2 \times {10^4} \le kL < 3 \times {10^4}$, respectively. With the assigned ${N_{\textrm{ro}}}$, the ${P_{11}}({180^\circ } )$ values computed by PGOM almost do not change if ${N_{\textrm{ro}}}$ is increased, in the cases of the six considered shapes. In the moderate size parameter range ($kL$ from 100 to 300 or so), ${P_{11}}({180^\circ } )$ values computed by PGOM and IITM may not be identical, due to uncertainties from the geometric optics approximation and numerical implementation, but the ${P_{11}}({180^\circ } )- kL$ relations are similar in both IITM and PGOM results.

Fig. 2. ${P_{11}}({180^\circ } )$ versus particle size parameters $kL$ for (a): a regular hexagonal column (RHC); (b): triangular prism (TP); (c): droxtal (DRX); (d): irregular hexagonal column without perpendicular facets (IRHC1); (e): irregular hexagonal column with 2 perpendicular facets (IRHC2); and (f): cube. The refractive index for the RHC, TP, IRHC1 and IRHC2 cases is $1.3117 + 1.48 \times {10^{ - 9}}i$, for the DRX case is $1.3243 + 2.00 \times {10^{ - 11}}i$, and for the cube case is $1.4180 + 7.28 \times {10^{ - 7}}i$. The solid gray curves are the computed ${P_{11}}({180^\circ } )$ values with $kL$ ranging from 2 to 30,000 by the IITM and PGOM. The inset plot in each panel is for $kL$ from 2 to 300, where the IITM results are highlighted in cyan. For the RHC (panel a), TP (panel b) and IRHC2 (panel e) shapes, the ${P_{11}}({180^\circ } )- kL$ relations are fitted by a linear function $y = f(x )$ denoted in the corresponding panels. For the cube (panel f) shape, the ${P_{11}}({180^\circ } )- kL$ relation is fitted by a linear function $y = f({{x^2}} )$ denoted in the panel f. R is the correlation coefficient between ${P_{11}}({180^\circ } )$ and $kL$.

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Figures 2(a), 2(b) and 2(e) show that the ${P_{11}}({180^\circ } )- kL$ relations for the RHC, TP and IRHC2 are approximately linear in the considered size-parameter ranges. We fit the ${P_{11}}({180^\circ } )- kL$ relations with a linear function, and the correlation between ${P_{11}}({180^\circ } )$ and $kL$ is strong. The linear relation is obvious in both IITM and PGOM results for $kL$ larger than 30 in the case of RHC and TP, while the linear ${P_{11}}({180^\circ } )- kL$ relation appears when $kL$ is larger than 1,000 with IRHC2. In the IITM results of RHC and TP, the ${P_{11}}({180^\circ } )- kL$ relation has some oscillations, but generally ${P_{11}}({180^\circ } )$ increases linearly with respect to $kL$. For the DRX and IRHC1 cases shown in Figs. 2(c)–2(d), ${P_{11}}({180^\circ } )$ has small variations with respect to $kL$ when $kL$ is larger than 50, and tends to converge to a constant value with increasing $kL$. For the cube case shown in Fig. 2(f), ${P_{11}}({180^\circ } )$ is approximately proportional to ${({kL} )^2}$. The correlation coefficient between ${P_{11}}({180^\circ } )$ and ${({kL} )^2}$ is almost equal to one.

A real cloud usually contains ice crystals with different shapes. In the mixture of ice crystal habits, if a shape has a linear or quadratic ${P_{11}}({180^\circ } )- kL$ relation, the ${P_{11}}({180^\circ } )- kL$ relation of the ice crystal mixture would be also linear or quadratic. This is similar to adding nearly constant terms to a linear or quadratic function, and the resulting function is still linear or quadratic.

In cases where ${P_{11}}({180^\circ } )$ is highly correlated with $kL$, all particle shapes have the CR structure, which is consistent with the finding by Borovoi et al. [13]. With the CR structure, the incident field can be reflected exactly $180^\circ $ relative to the direction of incidence, which results in strong backscattering and is highly dependent on the particle size parameter. To examine the detailed mechanism controlling the ${P_{11}}({180^\circ } )- kL$ relation in a particle with the CR structure, we investigate the contributions of various beam-tracing components to backscattering in the PGOM.

In the case of RHC, we find that the $p = 4$ beam-tracing component, where the incident beam is internally reflected twice before refracting out of the particle, dominates over the components contributing to ${P_{11}}({180^\circ } )$, as shown in Fig. 3. Results shown in Fig. 3 consider three scenarios in the calculation of the S matrix at $180^\circ $: 1) All contributions from beam-tracing orders 0 to 17 are included; 2) we exclude the $p = 4$ contribution but include all other contributions with $p \le 17$; and 3) we consider only the $p = 4$ contribution. We find that the contributions of $p > 17$ can be neglected. Then, we compute ${P_{11}}({180^\circ } )$ with Eq. (3) using the S matrix results in the three scenarios, the ${C_{\textrm{sca}}}$ result in scenario 1, and a random-orientation averaging calculation. Figure 3 shows that the ${P_{11}}({180^\circ } )$ values computed in the three scenarios are all linearly correlated with $kL$. The ${P_{11}}({180^\circ } )- kL$ relation in scenario 3 is close to scenario 1, which suggests that the $p = 4$ component dominates the contribution to ${P_{11}}({180^\circ } )$. We also find that interference among the contributions to backscattering by different beams is negligible compared with the interference within a beam that propagates exactly toward the backscattering direction, which is also known as incoherent backscattering [36].

Fig. 3. ${P_{11}}({180^\circ } )$ of an RHC with size parameters from 100 to 30,000 computed by the PGOM. The solid gray curve (Scenario 1 defined in the text) includes all beam-tracing contributions for p ranging from 0 to 15. The dotted red curve (Scenario 2 defined in the text) has all orders except for $p = 4$. The blue dashed curve (Scenario 3 defined in the text) includes only $p = 4$.

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Figure 4 illustrates the beam-tracing path of a $p = 4$ beam in the case of an RHC. The particle coordinate is coincident with the laboratory coordinate OXYZ. The direction of incidence has zenith angle $\theta $ and azimuth angle $\phi $ relative to the Z and X directions, respectively. The beam refracts into the particle at the bottom facet, then is reflected twice by an interior side and top facets of the particle, and refracts out of the particle at the bottom facet. The intersections of the beam with particle facets are numbered from 1 to 4 as shown in Fig. 4(a). When $\phi = 30^\circ $ and $\theta = 30^\circ $, the path of a ray in the beam tracing shown in Fig. 4(a) is on a 2D plane as shown in Fig. 4(b). The outgoing direction is exactly opposite to the direction of incidence.

Fig. 4. An illustration of a $p = 4$ beam in the case of an RHC. (a): The polygons labeled as 1 to 4 are the intersections of the beam with particle facets. The black solid arrows are the path of a ray in the beam tracing. $\theta $ and $\phi $ are the angles between the direction of incidence with the Z and X axes, respectively. (b): The path of a ray in the beam shown in a 2D plane when $\phi = 30^\circ $ and $\theta = 30^\circ $. ${Q_1}$ to ${Q_4}$ are the points where the ray is reflected or refracted.

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In the following discussions, we consider a beam refracted out of the scattering particle at intersection 4 (denoted as B4 hereafter) as illustrated in Fig. 4(a). Specifically, the contribution of B4 to the backscattering ${\mathbf S}$ matrix is computed by Eq. (8), which is denoted as ${{\mathbf S}_{p = 4}}$ in the following discussion. The integral ${D_{p = 4}}$ (Eq. (9)) together with the corresponding ${{\mathbf L}_{p = 4}}$ and ${\mathbf U}_{p = 4}^s$ matrices of B4 are used in Eq. (8) to compute the ${{\mathbf S}_{p = 4}}$ matrix. We fix the angle $\theta $ to be $30^\circ $ and vary $\phi $ from $25^\circ $ to $30^\circ $. For the specific beam, we define a function $P({kL,\phi } )$ as

(11a)$$F({kL,\phi } )= \frac{1}{2}\mathop \sum \nolimits_{j = 1}^4 {|{{S_{p = 4,j}}} |^2}, $$

(11b)$$P({kL,\phi } )= F({kL,\phi } )/{({kL} )^2}, $$

where ${S_{p = 4,j}},\; j = 1,\; 2,\; 3,\; 4$ are the four elements of ${{\mathbf S}_{p = 4}}$. Similar to Eq. (10b), $F({kL,\phi } )$ accounts for the contribution by B4 without interference with other beams. Note that the ${{\mathbf S}_{p = 4}}$ matrix elements are functions of $\theta $, $\phi $ and $kL$. When $\phi = 30^\circ $, the beam refracted out of the particle propagates along the exact opposite direction of the incident beam, and the integral ${D_{p = 4}}$ is proportional to the area of the intersection. According to Eq. (11), $P({kL,\phi = 30^\circ } )$ is proportional to ${({kL} )^2}$.

The surface integral in Eq. (9) is indeed consistent with the Fraunhofer diffraction integral [37]. If evaluated analytically, the integral contains a sinc function term. Specifically, $P({kL,\phi } )$ can be approximated by the function

(12)$$Q({kL,\phi } )\equiv {c_0}{({kL} )^2}\textrm{sin}{\textrm{c}^2}[{{c_1}kL({\phi - 30^\circ } )} ], $$

where ${c_0}$ and ${c_1}$ are constants. Note that $\textrm{sinc}(x )$ is equal to $\textrm{sin}(x )/x$ when $x \ne 0$ and equal to 1 when $x = 0$. $Q({kL,\phi } )$ can fit the $P({kL,\phi } )$ values very well when $\phi - 30^\circ $ is close to zero. Figure 5 shows the $P({kL,\phi } )- \phi $ relations for six different $kL$ values. For illustration, Figs. 6(a)–6(c) show the $P({kL,\phi } )- kL$ relations for $\phi = 28^\circ $, $\phi = 29.9^\circ $ and $\phi = 30^\circ $. For a fixed $\phi $, such as $\phi = 28^\circ $, $Q({kL,\phi } )$ is more accurate for a smaller $kL$. For a fixed $kL$, $Q({kL,\phi } )$ is more accurate for larger $\phi $. In Figs. 5 and 6, constants ${c_0}$ and ${c_1}$ are $2.95 \times {10^{ - 5}}$ and $4.36 \times {10^{ - 3}}$, respectively.

Fig. 5. $P({kL,\phi } )$ defined in Eq. (11) versus $\phi $ at six $kL$ values. The overplotting gray curves are the fitting functions $Q({kL,\phi } )$ defined in Eq. (12). (a): $kL$ equal to 500, 700 and 1,000; and (b): $kL$ equal to 20,000, 25,000 and 30,000.

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Fig. 6. $P({kL,\phi } )$ (defined in Eq. (11)), $\bar{P}({kL} )$ (defined in Eq. (13a)), $Q({kL,\phi } )$ (defined in Eq. (12)) and $\tilde{Q}({kL} )$ (defined in Eq. (14)) versus $kL$. The solid curves show computations using $P({kL,\phi } )$ and $\bar{P}({kL} )$. The dashed curves are fit using $Q({kL,\phi } )$ and $\tilde{Q}({kL} )$. (a-c): $P({kL,\phi } )$ and $Q({kL,\phi } )$ for $\phi $ = $28^\circ $, $29.9^\circ $ and $30^\circ $; (d): $\bar{P}({kL} )$ and $\tilde{Q}({kL} )$.

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We now compute the averaged $P({kL,\phi } )$ and $Q({kL,\phi } )$ over $\phi $ from $25^\circ $ to $30^\circ $,

(13a)$$\bar{P}({kL} )= \frac{1}{{{{({kL} )}^2}}}\mathop \int \nolimits_{25^\circ }^{30^\circ } F({kL,\phi } )d\phi , $$

(13b)$$\bar{Q}({kL} )= \frac{1}{{{{({kL} )}^2}}}\mathop \int \nolimits_{25^\circ }^{30^\circ } {({kL} )^2}Q({kL,\phi } )d\phi , $$

where $\bar{P}({kL} )$ is evaluated numerically and $\bar{Q}({kL} )$ has an approximate analytical expression:

(14)$$\bar{Q}({kL} )= {c_0}{({kL} )^2}\mathop \int \nolimits_{25^\circ }^{30^\circ } \textrm{sin}{\textrm{c}^2}[{{c_1}kL({\phi - 30^\circ } )} ]d\phi \approx \frac{{{c_0}\pi kL}}{{2{c_1}}} = \tilde{Q}({kL} ), $$

where the definite integral $\mathop \smallint \nolimits_{ - \infty }^\infty \textrm{sin}{\textrm{c}^2}(x )dx = \pi $ is used and the term $\textrm{sin}{\textrm{c}^2}[{{c_1}kL({\phi - 30^\circ } )} ]$ is close to zero for $\phi > 25^\circ $ with the considered $kL$ ranges. $\tilde{Q}({kL} )$ is a linear function of $kL$. Figure 6(d) compares the $\tilde{Q}({kL} )$ and $\bar{P}({kL} )$. $\bar{P}({kL} )$ is consistent with $\tilde{Q}({kL} )$ for $kL$ smaller than 15,000. For $kL > 15,000$, there is a bias between $\bar{P}({kL} )$ and $\tilde{Q}({kL} )$. Nevertheless, $\bar{P}({kL} )$ is approximately linearly proportional to $kL$.

According to Eq. (10), B4 can have interference with other beams in the contribution to ${P_{11}}({180^\circ } )$. By the principle of reciprocity, another beam that propagates along the exact opposite path of B4 exists (denoted as B4’), and the contributions of B4 and B4’ are coherent [38]. It can be shown straightforwardly from Eq. (10c) and the above discussion that the interference term of B4 and B4’ in computing ${P_{11}}({180^\circ } )$ also has an approximate linear relation to $kL$. The interference terms of B4 and other beams may not have simple relations to $kL$.

The above analysis of the far-field contribution by a single beam can be applied to explain the ${P_{11}}({180^\circ } )- kL$ relations for various particle shapes. Except for averaging the ${C_{\textrm{sca}}}$-weighted ${P_{11}}$ at all particle orientations, the random-orientation-averaged ${P_{11}}({180^\circ } )$ can be computed by

(15)$${P_{11}}({180^\circ } )= \frac{{\mathop \int \nolimits_0^{2\pi } d\phi \mathop \int \nolimits_0^\pi {C_{\textrm{sca}}}({\theta ,\phi } ){P_{11}}({180^\circ ,\theta ,\phi } )\textrm{sin}\theta d\theta }}{{\mathop \int \nolimits_0^{2\pi } d\phi \mathop \int \nolimits_0^\pi {C_{\textrm{sca}}}({\theta ,\phi } )\textrm{sin}\theta d\theta }}, $$

where the particle orientation is fixed, and the directions of incidence vary randomly. For RHC, TP and IRHC2, outgoing beams propagate along the exact backscattering direction only in certain ranges of the $\theta $ and $\phi $ angles. According to Eqs. (11)–(15), ${P_{11}}({180^\circ } )$ can have an approximate linear relation with respect to $kL$. In the case of a cube, any two neighboring facets are perpendicular to each other, and there are outgoing beams in the backscattering direction at most $\theta $ and $\phi $ angles. The contribution of each of these backscattering beams to ${P_{11}}({180^\circ ,\theta ,\phi } )$ is proportional to ${({kL} )^2}$. Therefore, ${P_{11}}({180^\circ } )$ is still proportional to ${({kL} )^2}$ according to Eq. (15). For particles without the CR structure, the interference among different beams (also known as the coherent contribution [36]) is significant compared to the incoherent contribution. The coherent contribution to ${P_{11}}({180^\circ } )$ does not have a simple relation to $kL$. ${P_{11}}({180^\circ } )$ tends to be a constant as $kL$ increases, as shown in Figs. 2(c)-(d) for DRX and IRHC1, which is also true in the case of a roughened hexagonal column aggregate [11].

4. Concluding remark

The IITM and PGOM computations show that the ${P_{11}}({180^\circ } )- L$ relation can be linear or quadratic for particles with CR structures, which is consistent with previous studies [13–16]. For particles without a CR structure, ${P_{11}}({180^\circ } )$ tends to converge to a constant with increasing L. The ${P_{11}}({180^\circ } )- L$ relation is explained under the PGOM beam-tracing framework. A CR structure can cause an incident beam to be reflected back along the exact backscattering direction. The contribution of the outgoing beam propagating around the backscattering direction to ${P_{11}}({180^\circ } )$ at a single incidence direction can be approximated by a sinc-function-squared term multiplied by an ${L^2}$ term. The integration of the sinc-function-squared term with respect to the incidence angle is approximately proportional to $1/L$. The resultant contribution, averaged over random orientations, is approximately proportional to L. The orientation-averaged ${P_{11}}({180^\circ } )$ can be computed by averaging the ${P_{11}}({180^\circ } )$ results for all incidence directions. If a particle has CR structures, the ${P_{11}}({180^\circ } )- L$ relation can be linear or quadratic. The dependence of the ${P_{11}}({180^\circ } )- L$ relation on particle shapes can be used to estimate ice-crystal shapes from lidar observation of ${P_{11}}({180^\circ } )$ and particle size measurement from other remote-sensing instruments.

Funding

Internal funding provided by Texas A&M University (137500-20001), and the endowment funds associated with the David Bullock Harris Chair in Geosciences (02-512231-10000).

Acknowledgments

The computations were conducted at the Texas A&M University High Performance Research Computing facilities. The authors thank Steven R. Schroeder for helping edit the grammar. The authors thank the four anonymous reviewers for their constructive suggestions toward improving the manuscript.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. U. Lohmann and E. Roeckner, “Design and performance of a new cloud microphysics scheme developed for the ECHAM general circulation model,” Clim. Dyn. 12(8), 557–572 (1996). [CrossRef]

2. T. Hashino, M. Satoh, Y. Hagihara, T. Kubota, T. Matsui, T. Nasuno, and H. Okamoto, “Evaluating cloud microphysics from NICAM against CloudSat and CALIPSO,” J. Geophys. Res. Atmos. 118(13), 7273–7292 (2013). [CrossRef]

3. Z. Guo, M. Wang, Y. Peng, and Y. Luo, “Evaluation on the vertical distribution of liquid and ice phase cloud fraction in Community Atmosphere Model Version 5.3 using spaceborne lidar observations,” Earth Sp. Sci.7(3), (2020).

4. D. M. Winker, J. Pelon, J. A. Coakley, S. A. Ackerman, R. J. Charlson, P. R. Colarco, P. Flamant, Q. Fu, R. M. Hoff, C. Kittaka, T. L. Kubar, H. Le Treut, M. P. Mccormick, G. Mégie, L. Poole, K. Powell, C. Trepte, M. A. Vaughan, and B. A. Wielicki, “The CALIPSO mission: A global 3D view of aerosols and clouds,” Bull. Am. Meteorol. Soc. 91(9), 1211–1230 (2010). [CrossRef]

5. A. J. Illingworth, H. W. Barker, A. Beljaars, et al., “The EarthCARE satellite: The next step forward in global measurements of clouds, aerosols, precipitation, and radiation,” Bull. Am. Meteorol. Soc. 96(8), 1311–1332 (2015). [CrossRef]

6. K. Sato and H. Okamoto, “Refinement of global ice microphysics using spaceborne active sensors,” J. Geophys. Res. 116(D20), D20202 (2011). [CrossRef]

7. F. Ewald, S. Groß, M. Wirth, J. Delanoë, S. Fox, and B. Mayer, “Why we need radar, lidar, and solar radiance observations to constrain ice cloud microphysics,” Atmos. Meas. Tech. 14(7), 5029–5047 (2021). [CrossRef]

8. S. Iwasaki and H. Okamoto, “Analysis of the enhancement of backscattering by nonspherical particles with flat surfaces,” Appl. Opt. 40(33), 6121–6129 (2001). [CrossRef]

9. L. Bi, P. Yang, G. W. Kattawar, B. A. Baum, Y. X. Hu, D. M. Winker, R. S. Brock, and J. Q. Lu, “Simulation of the color ratio associated with the backscattering of radiation by ice particles at the wavelengths of 0.532 and 1.064 µm,” J. Geophys. Res. 114, D00H08 (2009). [CrossRef]

10. A. Borovoi, A. Konoshonkin, N. Kustova, and H. Okamoto, “Backscattering Mueller matrix for quasi-horizontally oriented ice plates of cirrus clouds: Application to CALIPSO signals,” Opt. Express 20(27), 28222–28233 (2012). [CrossRef]

11. J. Ding, P. Yang, R. E. Holz, S. Platnick, K. G. Meyer, M. A. Vaughan, Y. Hu, and M. D. King, “Ice cloud backscatter study and comparison with CALIPSO and MODIS satellite data,” Opt. Express 24(1), 620–636 (2016). [CrossRef]

12. H. Okamoto, K. Sato, A. Borovoi, H. Ishimoto, K. Masuda, A. Konoshonkin, and N. Kustova, “Wavelength dependence of ice cloud backscatter properties for space-borne polarization lidar applications,” Opt. Express 28(20), 29178–29191 (2020). [CrossRef]

13. A. Borovoi, A. Konoshonkin, and N. Kustova, “Backscattering by hexagonal ice crystals of cirrus clouds,” Opt. Lett. 38(15), 2881–2884 (2013). [CrossRef]

14. A. Borovoi, A. Konoshonkin, and N. Kustova, “The physical-optics approximation and its application to light backscattering by hexagonal ice crystals,” J. Quant. Spectrosc. Radiat. Transf. 146, 181–189 (2014). [CrossRef]

15. A. G. Borovoi, A. V. Konoshonkin, N. V. Kustova, and I. A. Veselovskii, “Contribution of corner reflections from oriented ice crystals to backscattering and depolarization characteristics for off-zenith lidar profiling,” J. Quant. Spectrosc. Radiat. Transfer 212, 88–96 (2018). [CrossRef]

16. A. Konoshonkin, A. Borovoi, N. Kustova, and J. Reichardt, “Power laws for backscattering by ice crystals of cirrus clouds,” Opt. Express 25(19), 22341–22346 (2017). [CrossRef]

17. A. Borovoi, I. Grishin, E. Naats, and U. Oppel, “Backscattering peak of hexagonal ice columns and plates,” Opt. Lett. 25(18), 1388–1390 (2000). [CrossRef]

18. P. Yang, J. Ding, R. L. Panetta, K. Liou, G. W. Kattawar, and M. I. Mishchenko, “On the convergence of numerical computations for both exact and approximate solutions for electromagnetic scattering by nonspherical dielectric particles (invited review),” Prog. Electromagn. Res. 164, 27–61 (2019). [CrossRef]

19. E. E. Eloranta, “High Spectral Resolution Lidar,” in Lidar: Range–Resolved Optical Remote Sensing of the Atmosphere, C. Weitkamp, ed. (Springer, 2005), pp. 143–164.

20. H. C. van de Hulst, Light Scattering by Small Particles (John Wiley and Sons, 1957).

21. M. I. Mishchenko and M. A. Yurkin, “On the concept of random orientation in far-field electromagnetic scattering by nonspherical particles,” Opt. Lett. 42(3), 494–497 (2017). [CrossRef]

22. B. R. Johnson, “Invariant imbedding T matrix approach to electromagnetic scattering,” Appl. Opt. 27(23), 4861–4873 (1988). [CrossRef]

23. L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 138, 17–35 (2014). [CrossRef]

24. B. Sun, L. Bi, P. Yang, M. Kahnert, and G. W. Kattawar, Invariant Imbedding T-Matrix Method for Light Scattering by Nonspherical and Inhomogeneous Particles, 1st ed. (Elsevier, 2020).

25. R. E. Bellman and G. M. Wing, An Introduction to Invariant Imbedding (John Wiley & Sons, 1975).

26. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University, 2002).

27. P. Yang and K. N. Liou, “Geometric-optics–integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35(33), 6568–6584 (1996). [CrossRef]

28. P. Yang and K. N. Liou, “Light scattering by hexagonal ice crystals: Solutions by a ray-by-ray integration algorithm,” J. Opt. Soc. Am. A 14(9), 2278–2289 (1997). [CrossRef]

29. L. Bi, P. Yang, G. W. Kattawar, Y. Hu, and B. A. Baum, “Scattering and absorption of light by ice particles: Solution by a new physical-geometric optics hybrid method,” J. Quant. Spectrosc. Radiat. Transfer 112(9), 1492–1508 (2011). [CrossRef]

30. B. Sun, P. Yang, G. W. Kattawar, and X. Zhang, “Physical-geometric optics method for large size faceted particles,” Opt. Express 25(20), 24044–24060 (2017). [CrossRef]

31. M. P. Bailey and J. Hallett, “A comprehensive habit diagram for atmospheric ice crystals: Confirmation from the laboratory, AIRS II, and other field studies,” J. Atmos. Sci. 66(9), 2888–2899 (2009). [CrossRef]

32. J. Ding, P. Yang, M. T. Lemmon, and Y. Zhang, “Simulations of halos produced by carbon dioxide ice crystals in the Martian atmosphere,” Geophys. Res. Lett. 50(8), e2023GL103457 (2023). [CrossRef]

33. P. Yang, L. Bi, B. A. Baum, K.-N. Liou, G. W. Kattawar, M. I. Mishchenko, and B. Cole, “Spectrally consistent scattering, absorption, and polarization properties of atmospheric ice crystals at wavelengths from 0.2 to 100 µ m,” J. Atmos. Sci. 70(1), 330–347 (2013). [CrossRef]

34. H. Iwabuchi and P. Yang, “Temperature dependence of ice optical constants: Implications for simulating the single-scattering properties of cold ice clouds,” J. Quant. Spectrosc. Radiat. Transfer 112(15), 2520–2525 (2011). [CrossRef]

35. S. G. Warren, “Optical constants of carbon dioxide ice,” Appl. Opt. 25(16), 2650–2674 (1986). [CrossRef]

36. V. Shishko, A. Konoshonkin, N. Kustova, D. Timofeev, and A. Borovoi, “Coherent and incoherent backscattering by a single large particle of irregular shape,” Opt. Express 27(23), 32984–32993 (2019). [CrossRef]

37. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

38. C. Zhou, “Coherent backscatter enhancement in single scattering,” Opt. Express 26(10), A508–A519 (2018). [CrossRef]

Irregular hexagonal column 1 (IRHC1)			Irregular hexagonal column 2 (IRHC2)
x	y	z	x	y	z
2.387139E-01	-1.236971E-01	2.582004E-01	3.245042E-01	-3.106455E-02	3.065691E-01
1.548793E-01	1.638560E-01	4.528270E-01	1.590269E-01	2.555505E-01	3.065691E-01
-1.500719E-03	1.231995E-01	4.914416E-01	-1.129401E-01	1.822362E-01	3.065691E-01
-1.048167E-01	-5.421677E-02	4.323061E-01	-2.461802E-01	-2.377703E-02	3.065691E-01
-1.040327E-01	-1.660373E-01	3.691289E-01	-7.249460E-02	-3.350597E-01	3.065691E-01
1.774577E-01	-2.710864E-01	1.994133E-01	1.025827E-01	-3.024082E-01	3.065691E-01
9.928173E-02	-3.187343E-01	-3.438840E-01	1.875823E-01	-1.692725E-01	-3.350001E-01
-2.795336E-01	-2.100393E-01	-3.666166E-01	-1.882701E-01	-2.621690E-01	-2.102735E-01
-3.377441E-01	1.560237E-01	-3.847918E-01	-3.524634E-01	3.259497E-02	-2.083783E-01
-2.957011E-01	2.514775E-01	-3.867872E-01	-2.304032E-01	2.513595E-01	-2.784614E-01
1.995722E-01	3.496799E-01	-3.671654E-01	9.890457E-02	3.596854E-01	-3.919662E-01
2.534240E-01	9.957453E-02	-3.540723E-01	3.136709E-01	-1.230082E-02	-3.965730E-01

On the relation between ice-crystal scattering phase function at 180° and particle size: implication to lidar-based remote sensing of cirrus clouds

Abstract

1. Introduction

2. Light-scattering computational methods

3. Simulation results and analysis

4. Concluding remark

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Tables (1)

Equations (20)

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