1. Introduction

The problem of light scattering by nonspherical particles of irregular shape is of great interest [1]. The particles of irregular shape are ubiquitous in nature. For example, the aerosol and dust particles in the atmosphere have irregular shapes. Even in cirrus clouds consisting of ice crystals the irregular shapes like aggregates are often predominant [2]. In astrophysics, surfaces of the Moon and other planets are covered by regolith particles of irregular shapes. In biomedical optics, tissues are composed of complex constituents, and so on.

When size of the particles a is smaller than or comparable to light wavelengths λ, the problem of light scattering by such irregular particles can be solved using the “exact” numerical methods like the T-matrix method, discrete dipole approximation, finite difference time domain method [1], discontinuous Galerkin time domain method (DGTD) [3], and so on. However, in the case of large particles, where the size parameter $x = \pi a/\lambda$ is large, e.g., $x \ge 50$, these exact methods are not successful because of great demands to computer resources. Here the geometric-optics approximation (GOA) looks reasonable. However, geometric optics ignores wave phenomena that become essential at some scattering angles, especially in the forward and backward directions. Thus, the problem of light scattering by large irregular particles becomes a challenge in computational physics.

Because of the computational costliness, there are few papers where the problem of light scattering by particles of irregular shape at large size parameter $x > 100$ is solved using the exact numerical methods. Thus, Grynko et al. [3] applied their DGTD method to particles of the Gaussian random shape at $10 < x < 150$. Also there is a series of papers like [4,5] devoted to light scattering by large ice crystal particles of cirrus clouds. Here an ice hexagonal column at $x < 100$ was chosen as a basic shape. Then this shape was distorted in two ways. First, it was a shape distortion by changing the dihedral angles among the faces. Second, any face was assumed to be rough. To calculate the scattered light for such particles of irregular shapes, these authors used two exact methods: the pseudo-spectral time-domain method (PSTD) and invariant imbedding T-matrix method (II-TM). Then Ding et al. [2] presented their calculations of the backscattering peak obtained with the II-TM method for randomly oriented 8-column ice aggregate with roughened faces at $40 < x < 130$. It is worthwhile to note that physical interpretation of the numerical data obtained in [2–5] meets some difficulties and uncertainties.

In this paper, we calculate the light scattered by a randomly oriented large faceted particle of irregular shape at 60 < x < 240. The random orientation of a particle is assumed similarly to the rigorous definition discussed recently in [6]. We emphasize that only randomly oriented particles are considered in this paper. Otherwise, if a particle has a fixed orientation, intensity of the scattered light reveals a complicated speckle pattern [7]. After averaging over random orientation in our calculations, the speckle pattern disappears and our intensities of scattered light become the statistically averaged quantities.

The convex irregular shapes like those shown in Fig. 1 were randomly generated by a computer code. The scattered light is calculated using the physical-optics approximation (PhOA) developed by the authors [8–10]. Though the PhOA is not an exact method, it is a quite reasonable approximation, at least, for large particles where facet sizes are larger than incident wavelength. In comparison with the exact methods, the PhOA has two advantages. First, the PhOA does not demand extremely large computer resources. Second, the numerical results obtained are obviously interpreted from the physical point of view using the concept of plane-parallel beams created inside a faceted particle.

 

Fig. 1. Samples of irregular particle shapes.

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Note that, in literature, there is a discussion whether a backscattering peak appears in cirrus clouds [2,5] that is of importance for lidar studies. Earlier [8–10] we applied the PhOA to calculate light backscatter by cirrus clouds consisting of large ice crystals of pristine shapes like hexagonal columns and plates. We showed that the backscatter for such shapes was determined predominantly by 8 beams where the beams were formed by corner reflection from the dihedral angles of 90° between crystal facets. These 8 beams were predominant in the backscattering cone of [170°, 180°] forming a backscattering peak within the PhOA, while these beams formed a singularity at $\theta = 180^\circ$ in the geometric-optics approximation (GOA). It was called the corner-refection effect.

In the case of the irregular shapes shown in Fig. 1, dihedral angles of 90° don’t exist and there is a question whether the backscattering peak appears for such particle shapes. In this paper, we show that the light scattering by a faceted irregular particle at averaging over random particle orientations is split into the coherent and incoherent components. Such a split of scattered light is similar to the phenomena well known for multiple scattering media [11]. We obtain that a large particle of faceted shape can create both the coherent and incoherent backscattering peaks with different angular widths.

2. Physical-optics and geometric-optics approximations

We consider light scattering by a set of convex particles of different shapes shown in Fig. 1 where shapes were randomly generated by a computer code. The number of particle facets was chosen as about 20, the average size was about 20 µm, and the refractive index of the particles was assumed, for specificity, as 1.3116 + i0.

Light scattering by such large faceted particle can be readily calculated in the physical-optics approximation (PhOA) [8–10]. The PhOA means that light inside a particle is described by geometric optics. As a result, the electromagnetic field inside any faceted particle and on its surface becomes a superposition of plane-parallel beams that are formed by multiple reflection of light by particle facets. The main characteristic of a beam is its trajectory, i.e. a succession of facets which reflects the beam inside the particle. Also our PhOA code calculates shape, size, propagation direction, amplitude and polarization of the electric field for any beam on the exit facet of the particle (Fig. 2). Then, in the far zone (at distance $R > > {a^2}/\lambda$), every beam is transformed into an outgoing spherical wave according to the Maxwell equations that is equivalent to the Fraunhofer diffraction of the plane-parallel beam leaving an exit facet. When the diffraction procedure is omitted in our PhOA code, this code turns out immediately into the GOA code.

 

Fig. 2. Typical backscattering beams.

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Angular distribution of scattered light is usually described by the phase function

For randomly oriented particles, the quantities $I(\theta ,\varphi )$ and ${\sigma _{sca}}$ in Eq. (1) should be averaged over orientations resulting in the following phase function

In the GOA, the phase function is formed by only reflection/refraction events. Consequently, $p(\theta )$ in the GOA does not depend on particle size, only shape essential. In the PhOA, on the contrary, the phase function depends on both shape and size.

3. Phase functions in the geometric-optics approximation

Figure 3 shows the phase functions for four shapes of Fig. 1 calculated within the GOA. We see some complicated spiky functions. Here at $\theta \to 180^\circ$ the phase function can be increasing, flat or decreasing that will be classified as the peaks, flatlands and gaps, respectively. The same values for 300 particle shapes are summarized in Fig. 4.

 

Fig. 3. Phase functions in the GOA for four particle shapes shown in Fig. 1.

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Fig. 4. Same as in Fig. 3 for 300 particle shapes for all scattering angles, the yellow line is the average.

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In Fig. 4, we obtain that in spite of the complicated behavior of the phase functions, there is a remarkable feature for the average of the phase functions over shapes. Namely, the averaged function marked yellow in Fig. 4 proves to be almost constant in the backscattering cone of [170°, 180°], i.e. the phase function calculated in the GOA for the ensemble of particles with the shapes like those of Fig. 1 has no backscattering peak after averaging over both particle orientations and particle shapes.

It is worthwhile noting that, unlike the case of pristine crystals [8–10], every curve in Figs. 3 and 4 in the cone of [170°, 180°] is a sum of contributions from thousands beams. This fact is illustrated in Fig. 5. Indeed, in Fig. 5, the solid colored curves correspond to several beams with predominant magnitudes of their contributions. We see that the beam contributions within the GOA are some spiky functions with different location and width of the spikes. The dotted lines in Fig. 5 are the sums for the various numbers N of the additive beams. We obtain that the sum of the spiky functions is smoothed at large N and the number N should be up to 1000. Also we were convinced that all the beams had the similar trajectories like those depicted in Fig. 2.

 

Fig. 5. Contributions to the phase function for particle shape 3 of Fig. 1 at different numbers of the beams added.

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4. Phase functions in the physical-optics approximation

Now let us go to the PhOA where diffraction and interference of the beams will be taken into account. Note that every plane-parallel beam leaving a particle facet like those shown in Fig. 2 is transformed by the Fraunhofer diffraction into an outgoing spherical wave in the wave zone of the particle. For brevity, we call these spherical waves also as the beams though it is not true geometrically.

For a given particle shape and orientation, our PhOA code calculates the 2×2 amplitude (Jones) matrix for every beam ${{\textbf j}_i}(\theta ,\varphi )$ on the scattering direction sphere $(\theta ,\varphi )$ that are summarized

The scattered intensity I for unpolarized incident radiation is the first element ${M_{11}}$ of the Mueller matrix. Consequently, we get

Interference between two beams depends on the phase shift δ between them, and δ is usually large $\delta > > 2\pi$ for a large particle. Besides, at random particle orientations, δ is a quickly varying value. As a result, the interference term should be mainly negligible ${p_{int}} \approx 0$ because of the sign-changing oscillations. However, sometimes the phase shift is, on the contrary, small $\delta < \pi$ resulting in a steady interference pattern. This statement is true under two conditions. First, it takes place for pairs of the beams where light passes only the same particle facets in both direct and inverse orders. Second, the scattering direction should be near backward, i.e. $\theta \approx 180^\circ$.

Now it is useful to notice a profound similarity between the backscatter by a large randomly oriented particle and the effect of coherent backscattering well-known for multiple scattering media [11]. Indeed, light passing several scatterers in the theory of multiple scattering is similar to our beams passing several facets inside a particle. In [11], the pairs with the inverse passage of the same scatterers create the famous backscattering coherent peak and the rest components of the field produce the incoherent term. Using such analogy, it is reasonable to rename the diffraction and interference terms of Eq. (6) as the incoherent and coherent phase functions

Figure 6 shows the total phase functions ${p_{total}} = {p_{coh}} + {p_{incoh}}$ calculated within the PhOA (solid lines) where the incoherent parts are presented by the dashed lines. An advantage of the PhOA is that its algorithm can calculate simultaneously the total and incoherent phase functions where the incoherent part is calculated by the same but simplified algorithm.

 

Fig. 6. Total phase functions (solid lines) and their incoherent parts (dashed lines) calculated with PhOA: a) for particle shapes 1, 2, 3, and 4 of Fig. 1 with the same size parameter x = 120; b) for particle shape 3 at different size parameters (x = 60, 120, and 240).

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Thus, Fig. 6(a) demonstrates appearance of the backscattering coherent peaks for four particle shapes which phase functions at $\theta \to 180^\circ$ in the GOA are quite different (see Fig. 3). The angular width of the coherent peaks is approximately the same and it is equal to

In Eq. (8), the value a can be treated as the diameter of the sphere circumscribing the particle. This result is explained as following. For the irregular particle shapes assumed, the pairs of beams, which travel inside a particle along the direct and inverse trajectories forming the coherent backscatter, leave the particle mostly near the opposite particle edges. In other words, such beam pairs travel predominantly near the particle surface because of multiple reflections at grazing incidence like the trajectory depicted in Fig. 2. On the contrary, intensity of such beam pairs leaving the particle near its projection center is lower because of the small reflection coefficients at normal light incidence.

It is interesting to note that in the theory of multiple scattering a magnitude of the coherent backscattering peak is characterized by the enhancement factor [11]. In our case of backscatter by a randomly oriented irregular particle, the enhancement factor corresponds to the ratio $\zeta = {p_{total}}(180^\circ )/{p_{incoh}}(180^\circ )$. Figure 6 demonstrates that our enhancement factor obeys the same inequality $\zeta \le 2$ as in the theory of multiple scattering.

The incoherent phase functions are shown in Fig. 7 for the particle shape 3 at different size parameters. Here the solid line with a lot of peaks and gaps is the phase function calculated in the GOA which does not depend on particle size. It is obvious that the ${p_{incoh}}(\theta )$ calculated with the PhOA is formed by the same plane-parallel beams leaving the particle. However, every beam contributes to the ${p_{incoh}}(\theta )$ by its diffraction pattern with the angular width of $\Delta \theta \approx \lambda /b,$ where b is the transversal size of the beam. Consequently, the smoothness of the ${p_{incoh}}(\theta )$ is larger for smaller particle size as seen in Fig. 7. Moreover, some part of energy abandons the backscattering cone of [170°, 180°] because of diffraction. Such energy loss increases for smaller particles where the beam size b is smaller. Consequently, the ${p_{incoh}}(\theta )$ increases for larger particle size and it should approach to the geometric-optics phase function at $x \to \infty$.

 

Fig. 7. Incoherent phase functions for the shape 3 at different size parameters.

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5. Coherent and incoherent backscattering peaks

In the previous section, we have calculated the phase functions within the PhOA at $60 < x < 240$ for a rather narrow model of particle shapes shown in Fig. 1. We obtained that the small peaks and gaps in Fig. 3 calculated within the GAO at $\theta \to 180^\circ$ have been smoothed by diffraction in the PhOA. As a result, the incoherent phase functions in the PhOA become factually some constants at $\theta \to 180^\circ$ (Figs. 6 and 7). Then, on the background of these constants, we obtain the backscattering peak associated with the coherent phase function.

However, such regularities are not universal. Indeed, the backscattering peaks that can be obtained within the GOA are not always smoothed by diffraction in the PhOA. In this case, the incoherent phase function also creates the backscattering peak in parallel to the coherent backscattering. Examples of such particles creating both the coherent and incoherent backscattering peaks are the conventional hexagonal ice columns. Here the dihedral angle of 90° between crystal faces produces the incoherent backscattering peak due to the corner-reflection effect [8–10].

Figure 8 presents the total and incoherent phase functions calculated within the PhOA for the randomly oriented hexagonal ice column with the height of 15.85 µm, hexagon diameter of 11.09 µm, and wavelength of 0.532 µm. Figures 6 and 8 demonstrate strong difference in the phase functions for such regular and irregular particle shapes.

 

Fig. 8. Total (solid) and incoherent (dashed) phase functions for the hexagonal ice column.

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Analogously to our previous paper [10], we can interpret Fig. 8 using two parameters: the averaged size of one backscattering beam b and the maximal distance a between two typical backscattering beams (see Fig. 2). Then the effective angular widths for the coherent and incoherent backscattering peaks should be $\Delta {\theta _{coh}} \approx \lambda /a$ and $\Delta {\theta _{incoh}} \approx \lambda /b$. Applying these equations to Fig. 8 we find that the parameter a is equal approximately to the column diameter while $b \approx a/3$ that is consistent with our previous results. Let us remind, that in the previous section, it was the diameter of the circumscribing sphere a that determined the angular width of the coherent backscattering peak. In this section, the value a is the diameter of the hexagonal face. This inconsistence illustrates that characteristics of particle size can be manifold for different shape models and they should be clearly defined every time.

In Fig. 8, the magnitude of the incoherent backscattering is predominant. Here the coherent phase function ${p_{coh}}(\theta ) = {p_{total}}(\theta ) - {p_{incoh}}(\theta )$ looks as small oscillations on the background of the incoherent phase function. Also we see that the coherent phase function at $\theta = 180^\circ$ is negative since we get the destructive interference in the backward scattering direction that was discussed earlier [10].

We note that the enhancement factor in Fig. 8 obeys, as before, the inequality $\zeta \le 2$. Such an inequality is valid for an ensemble of waves with random phases where every wave is not dominant as it takes place for particles of irregular shapes. In the case of the hexagonal column shown in Fig. 8, on the contrary, there are dominant beams associated with the corner-reflection effect. These beams create the incoherent backscattering peak because of mainly diffraction but not interference.

Let us emphasize that we have split the scattered light into the coherent and incoherent components explicitly because of discretization of the scattered light as a series summarizing the multiple-reflected beams. In the exact methods like the II-TM, PSTD [2,4,5], and DGTD [3] solving the problem of light scattering by large randomly oriented particles of complicated shapes, the incoherent component cannot be separated explicitly. Nevertheless, the results obtained in this paper could be qualitatively expanded to results of other calculations as well. Indeed, the backscattered surge obtained in [3] can be associated with the coherent backscattering peak. Here the discretization of the scattered light might be caused by the faceted shapes of the particles that were eventually used in [3]. Backscattering peak for roughened ice crystals of cirrus clouds was obtained numerically in [2,4,5] but its physical mechanism was not understood. In [5], Zhou and Yang supposed without any justification that this peak should be analogous to the coherent backscattering peak in multiple scattering media [11]. The physical sources of the waves travelling along the same path in the direct and inverse directions, which are needed for such interpretation, were not defined. We suggest that a roughened surface can be discretized where any element of the roughened surface redistributes scattered light inside a particle and then this element creates an outgoing spherical wave with random phase in the wave zone of the particle. In this case, appearance of the coherent backscattering peak can be explained exactly in such a way as it was done in this paper. However, it is not clear whether the incoherent backscattering peak should appear for the roughened hexagonal column considered in [5] and this problem needs further numerical studies.

6. Conclusions

In this paper, we have calculated light scattering by large (at size parameters $x > 100$) randomly oriented particles of faceted shapes shown in Fig. 1 using both the geometric-optics and physical-optics approximations. In the GOA, we show that the phase functions for the large particles of irregular shapes are some spiky functions with a lot of peaks and gaps of various location and widths. However, after averaging over particle shapes, these peaks and gaps are smoothed at $\theta \to 180^\circ$ and the averaged phase function in the GOA has no backscattering peaks. Though the number of particle facets in our model is not large (about 20) we believe that this number is enough to avoid possible numerical artefacts. Otherwise, increase of the facet number would essentially increase the computation time in the PhOA. Moreover, it could violate the condition of validity for the PhOA where edges of polyhedron facets were larger than wavelength.

In the PhOA, the light scattered by a single large particle of complicated shape is split, for the first time to our knowledge, into two qualitatively different parts: the coherent and incoherent components. Such a split was used earlier only for multiple scattering media. For the model of irregular particles shown in Fig. 1, the coherent part creates the coherent backscattering peak with the angular width of wavelength/(particle size) while the incoherent part has no backscattering peaks. Here the incoherent part approaches to the geometric-optics solution at large size parameter $x \to \infty$.

However, if a faceted shape of a particle contains the dihedral angles of 90° among any facets, the incoherent part creates the incoherent backscattering peak as well due to the corner-reflection effect. The angular width of the incoherent backscattering peak is equal to wavelength/(transversal size of backscattering plane-parallel beams).