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Quantifying the coherent backscatter enhancement of non-spherical particles with discrete dipole approximation

Chen Zhou, Xue Han, and Lei Bi

Open Access Open Access

Abstract

The prevailing backscattering peak associated with the scattering phase function of large non-absorptive particles can be interpreted with the coherent backscatter enhancement (CBE) theory, but has not been explicitly quantified with numerical simulations based on solving Maxwell’s equations. In this paper, representative numerical simulations performed with the discrete-dipole-approximation (DDA) model are used to quantify the effect of CBE on the single-scattering phase function. For each scattering case, the particle volume was divided into multiple thin slices parallel to the incident beam. The dipole polarizations in the j’th slice in response to the incident field of the i’th slice were computed, and then the corresponding contribution to the scattering phase function was calculated. Interference between conjugate terms representing reversible wave paths is constructive at the backscattering direction, which corresponds to the CBE. Subsequently, the contribution of CBE to the scattering phase function was quantified by comparing the electric fields calculated with and without the interference between conjugate terms. Results from these numerical simulations are consistent with conclusions obtained from the CBE theory. The simulations also quantitatively explain why it is difficult to identify a CBE-induced backscattering peak for the phase function of small particles and strong-absorptive particles.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

In lidar remote sensing of clouds and aerosols, prior assumptions on the phase function near the backscattering angle are required, but accurate observations and simulations on the near-backscattering properties of large non-spherical particles are still challenging [1,2]. Recent studies indicate that there is a backscattering peak associated with the single-scattering phase function of large non-absorptive spheres [3], cubic particles [4,5], smooth and roughened hexagonal particles [5–9], spheroids [10,11], super-spheroids [12], and complex concave particles [13]. The backscattering peak of certain particles can be partially explained by waves propagating in opposite directions near the particle perimeter [3,4].

More generally, the prevailing backscattering peak can be explained with the coherent backscatter enhancement (CBE) theory [10], which expands the single-scattering solution obtained from Maxwell’s equations in terms of iterative series in an order-of-scattering form, and the interference between conjugate terms representing reversible sequences of elementary scatterers (which can be considered as dipoles) is constructive at the backscattering direction, leading to a backscattering peak. Though the iterative series could not be used to directly calculate the phase function of large particles, it can be used to demonstrate why the backscatter is enhanced analytically. The mechanism of CBE in single scattering is similar to the coherent backscattering in multiple scattering [14–16], and is a form of weak localization. The theory is applicable to both faceted particles (e. g., smooth hexagonal particles) and non-faceted particles (e. g., spheroids), and explains why the angular width of the CBE-induced backscattering peak line for a specific particle habit is inversely proportional to the particle size parameter. However, the analytical results in the previous study [10] do not quantify the contribution of CBE to the scattering phase function.

The CBE of large faceted particles can also be explained using ray tracing technique, where electromagnetic waves are approximated as light rays or beams [5,17–20]. The propagation of these rays or beams is determined by Fresnel’s law. The analytical wave-based explanation of CBE in Ref. [10] and the ray-based explanation in Ref. [19] reflects the same interference processes from different perspectives. The ray tracing technique could be used to quantify CBE for large faceted particles, but it could not explain the CBE of non-faceted particles such as spheres, spheroids, and roughened ice cloud particles, due to the precondition of ray approximation. For example, the backscattering glory [3] associated with surface waves in the case of spheres cannot be explained by ray tracing.

In this study, we designed a set of numerical simulations with the discrete-dipole-approximation (DDA), which is based on solving Maxwell’s equations and can be applied to arbitrary particle shapes, to quantify the contribution of CBE on the single-scattering phase function.

2. Methods

2.1 ADDA model

In this paper, ADDA 1.4.0-alpha [21,22] was used to analyze the scattering properties of non-spherical particles.

For a specific scatterer, the ADDA model divides the volume of the scatter into small cubical subvolumes, and each cubical subvolume is approximated to be a coupled dipole in light scattering. When an incident plane wave interacts with the scatterer, each dipole responses to the incident electromagnetic field, and then emits electromagnetic radiation that contributes to the scattering electric field and interacts with other dipoles (Fig. 1(a)). The dipoles keep interacting with each other until the local field of each dipole is stable [23,24]. The ADDA model solves the electric dipole moment of each dipole, and then calculates the scattering electric field.

 figure: Fig. 1.

Fig. 1. Illustration of conjugate terms representing reversible wave paths in the ADDA sensitivity experiments. (a) Interactions between the incident waves (blue), dipoles (black dots) and the scattering field (red). The black arrows denote interactions between dipoles. (b) Illustration of conjugate reversible wave paths with the same scattering angle. The phase difference between the red and blue wave paths is zero when θ is zero, and is non-zero when θ>0. (c) Contribution of dipole polarizations in the i’th slice to the scattering field (Iii), when the incident field in the i’th slice is set to be same as the reference experiment and incident fields in other slices is set to be zero. Each black dot denotes a dipole. (d) Contribution of the dipoles in the j’th slice to the scattering electric field, when the incident field in the i’th slice is set to be same as the reference experiment and incident fields in other slices is set to be zero (Iij). (e) Contribution of dipole polarizations in the i’th slice in response to the incident field in the j’th slice (Iji). (f) Superposition of conjugate reversible wave paths, where conjugate reversible paths (blue and red paths) interfere constructively at the backscattering angle. The black arrows denote a wave path without conjugate reversible path.

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2.2 Equations for the coherent backscatter enhancement

When a particle interacts with an incident plane wave ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} ^{inc}}$, the electric field $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} $ at any location $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} $ can be expressed using the volume integral equation (Eq. (2.16) of [25])

$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} \left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} } \right) = {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} ^{inc}}\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} } \right) + \mathop \smallint \nolimits_{{V_{int}}}^{} d{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} \mathrm{^{\prime}}}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over G} \left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} ,{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }\mathrm{^{\prime}}}} \right)\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} \left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} \mathrm{^{\prime}}} \right)k_1^2\left( {{{\tilde{m}}^2}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }\mathrm{^{\prime}}}} \right) - 1} \right), $$
where $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over G} $ is the free space dyadic Green’s function, ${V_{int}}$ denotes the space inside the scatter, $k_1^{}$ denotes wave number, and $\tilde{m}$ is the refractive index of the interior relative to that of the exterior. Note that the Green's function is strongly singular and principal-volume definition of the integral is implied with separate delta-function term [26]. Then the scattering electric field at location ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _s}$ is
$${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} ^{sca}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right) = \mathop \smallint \nolimits_{{V_{int}}}^{} d{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} \mathrm{^{\prime}}}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over G} \left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s},{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }\mathrm{^{\prime}}}} \right)\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} \left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} \mathrm{^{\prime}}} \right)k_1^2\left( {{{\tilde{m}}^2}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }\mathrm{^{\prime}}}} \right) - 1} \right), $$

Inserting Eq. (1) into Eq. (2), we have ([10])

$$\begin{aligned} {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }^{sca}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right) &= \mathop \smallint \nolimits_{{V_{int}}}^{} d{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_1}k_1^2\left( {{{\tilde{m}}^2}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_1}} \right) - 1} \right)\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over G} \left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s},{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_1}} \right)[{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }^{inc}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_1}} \right)\\& + \mathop \smallint \nolimits_{{V_{int}}}^{} d{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_2}k_1^2\left( {{{\tilde{m}}^2}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_2}} \right) - 1} \right)\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over G} \left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_1},{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_2}} \right)\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} \left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_2}} \right)] \end{aligned}$$
we keep inserting Eq. (1) into Eq. (3) for infinite times, and the scattering term is expanded into iterative series in an order-of-scattering form,
$$\begin{aligned} &{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }^{sca}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right) = \mathop \smallint \nolimits_{{V_{int}}}^{} d{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_1}k_1^2\left( {{{\tilde{m}}^2}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_1}} \right) - 1} \right)\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over G} \left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s},{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_1}} \right){{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }^{inc}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_1}} \right) + \\ &\mathop \sum \nolimits_{n = 2}^\infty \mathop \smallint \nolimits_{{V_{int}}}^{} \ldots \mathop \smallint \nolimits_{{V_{int}}}^{} d{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_1} \ldots d{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_n}k_1^{2\textrm{n}}\mathop \prod \nolimits_{i = 1}^n \left( {{{\tilde{m}}^2}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_i}} \right) - 1} \right)\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over G} \left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s},{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_1}} \right) \ldots \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over G} \left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_{n - 1}},{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_n}} \right){{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }^{inc}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_n}} \right) \end{aligned}$$
where the first term denotes direct interactions between elementary scatters (or dipoles in ADDA) and incident wave, and the second term denotes contributions from a wave path representing an order-of-scattering. In the previous study [10], it is demonstrated analytically that the interference between conjugate terms (wave path ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _n}$, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _{n - 1}}$,…, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _1}$, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _s}$, and wave path ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _1}$, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _2}$,…, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _n}$, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _s}$) representing reversible sequences of elementary scatterers (or dipoles, Fig. 1(b)) is constructive at the backscattering direction and less constructive at off-backscattering directions, resulting in the CBE.

Though the theoretical mechanisms behind Eq. (4) are clear, and the summation of the series in Eq. (4) is finite, it is not practical to directly sum up such kind of series numerically [27]. Alternatively, we designed a set of idealized experiments with ADDA to quantify CBE. While the mechanisms behind DDA can be understanded in an order-of-scattering way [27], ADDA uses an iterative method to solve the scattering field, which have a much larger applicability domain than adding up the scattering-order series.

2.3 ADDA experiment design

For a specific scatter, we first divided the volume of the scatter into small dipoles. In the light scattering calculations, the incident wave was set to be a linearly polarized plane wave, and the scatter is randomly oriented. To calculate the phase matrices of the particle with random orientation, the particle was rotated with random Euler angles, and the scattering matrices were calculated using the ADDA model. To obtain the full 4 × 4 phase matrices, the electric field of the incident wave was initially set to be parallel to the scattering plane, and then set to be perpendicular to the scattering plane. 100 sets of different Euler angles were calculated and averaged to calculate the phase matrices of randomly oriented particles. These calculations were used as the reference experiments.

For each particle orientation, the volume of the particle was subsequently divided into a number of (N = 20 in this paper) thin slices parallel to the incident beam, where the thickness of each slice is much smaller than the wavelength, and then a same number of sensitivity experiments (N = 20) were performed. In each sensitivity experiment, the incident field in a specific slice was set to be same as the reference experiment, but the incident field in other slices were set to be zero. The scattering electric vector for the i’th sensitivity experiment is $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} _i^{sca}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right)$. The slicing is performed separately for each orientation, and there are altogether 2000 ADDA calculations for each particle. After the ADDA calculations accomplished, we saved the dipole polarizations, and calculated the contribution of the dipoles in the j’th slice to the scattering electric field (Fig. 1(c)-(e)) at a specific far-field location, $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} _{i,j}^{sca}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right)$. Then we have

$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} _{}^{sca}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right) = \mathop \sum \nolimits_{i = 1}^N \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} _i^{sca}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right) = \mathop \sum \nolimits_{i = 1}^N \mathop \sum \nolimits_{j = 1}^N \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} _{i,j}^{sca}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right). $$

Equation (5) is exact for the case of each orientation. When $i \ne j$, $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} _{i,j}^{sca}$ and $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} _{j,i}^{sca}$ represent the contributions of conjugate reversible wave paths (Figs. 1(d)-(e)) to the scattering electric field, and should interfere constructively at backscattering angles according to the CBE equations [10]. We rearranged the terms in Eq. (5), and then the intensity of the scattered light at a specific far-field location can be written as

$$\begin{aligned} &I\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right) = C{\left|{{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }^{sca}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right)} \right|^2} = C{\left|{\mathop \sum \nolimits_{i = 1}^N \mathop \sum \nolimits_{j = 1}^N \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E}_{i,j}^{sca}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right)} \right|^2}\\ &= C{\left|{\mathop \sum \nolimits_{i = 1}^N \mathop \sum \nolimits_{j = i + 1}^N \left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E}_{i,j}^{sca}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right) + \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E}_{j,i}^{sca}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right)} \right) + \mathop \sum \nolimits_{i = 1}^N \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E}_{i,i}^{sca}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right)} \right|^2} \end{aligned}$$
where C is a constant, and would be reduced later when the scattering phase function is normalized. The scattering field for individual components was computed for each orientation independently, then the intensity was averaged over different orientations, and thus the decomposition of scattering intensity into individual components remains valid after orientation averaging. To quantify the amplification due to the CBE, we calculated the contribution of individual components to the intensity of the scattering field,
$${I_{ij}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right) = C{\left|{\mathop \sum \nolimits_{i = 1}^N \mathop \sum \nolimits_{j = i + 1}^N \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E}_{i,j}^{sca}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right)} \right|^2}, $$
$${I_{ji}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right) = C{\left|{\mathop \sum \nolimits_{i = 1}^N \mathop \sum \nolimits_{j = i + 1}^N \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E}_{j,i}^{sca}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right)} \right|^2}, $$
$${I_{ii}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right) = C{\left|{\mathop \sum \nolimits_{i = 1}^N \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E}_{i,i}^{sca}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right)} \right|^2}, $$
$${I_{ij + ji}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right) = C{\left|{\mathop \sum \nolimits_{i = 1}^N \mathop \sum \nolimits_{j = i + 1}^N \left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E}_{i,j}^{sca}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right) + \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E}_{j,i}^{sca}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right)} \right)} \right|^2}, $$
where Iii denotes the irreversible term (a term without conjugate reversible term) (Fig. 1(c)), Iij and Iji denote conjugate terms containing conjugate reversible wave paths (Fig. 1(d)-(e)), and Iij + ji denotes a superposition of all conjugate terms. It is worth noting that conjugate wave paths also contribute to the irreversible term Iii, but the slices are parallel to incident beam, so the phase difference of each corresponding conjugate wave path is close to zero for near-backscattering angles, and the magnitude of CBE is approximately constant at near-backscattering angles according to Eq. (30) of [10], thus CBE does not affect the near-backscattering line shape of Iii. Then the CBE amplification factor induced by interferences between conjugate terms Iij and Iji is
$${\zeta _{ij}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right) = \frac{{{I_{ij + ji}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right)}}{{{I_{ij}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right) + {I_{ji}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right)}}, $$
and the total CBE amplification factor induced by interferences between all terms Iij, Iji and Iii is
$${\zeta _{all}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right) = \frac{{I\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right)}}{{{I_{ij}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right) + {I_{ji}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right) + {I_{ii}}\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_s}} \right)}}. $$

For the scattering component parallel to the scattering plane when the electric field of the incident beam is set to be parallel to the scattering plane, the amplification factor ${\zeta _{ij,\parallel }}$ is 2 at backscattering angle according to Eq. (30) of [10], and we will check whether ${\zeta _{ij,\parallel }}$ calculated from ADDA experiments is 2 at backscattering angle to test the theory of [10]. For the scattering component perpendicular to the reference plane, ${\zeta _{ij, \bot }}$ can be less than 2 at the backscattering angle according to [10]. ${\zeta _{all}}$ is usually smaller than 2, depending on the magnitude and phase difference between irreversible and reversible conjugate terms (theoretically the maximum value of ${\zeta _{all}}$ is up to 3).

3. Results

We first calculated the scattering properties of spheroids, which are widely used to model non-spherical scatters such as aerosols. The aspect ratio of the spheroids was set to be 3/4, the refractive index was set to be 1.33 + 0.0i, and six different size parameters were considered in these simulations. For particles with size parameters $\pi D/\lambda \ge 10$ (D is the maximum dimension, λ is wavelength), the dipoles were generated by the ADDA code [22] for each case (note that the input of ADDA option “-size” is not the size parameter $\pi D/\lambda $ used in this study), and the value of dipoles per lambda is 13.3. For size parameters smaller than 10, the dipoles are same as that for the particle with size parameter of 10, and the value of dipoles per lambda is greater than 13.3 correspondingly. To verify the accuracy of ADDA, we compared the results of ADDA with that calculated using the T-matrix [28], and the results show that ADDA agrees quite well with T-matrix (Fig. 2).

Figure 4 shows the amplification factors calculated using Eqs. (8)–(9). For the parallel component of the phase function, the amplification factor ζij is exactly 2 at the backscattering angle for all cases, and is smaller than 2 at off-backscattering angles, consistent to the analytical results of the CBE theory (Eq. (31) of [10]). For the perpendicular component, the value of ζij is slightly smaller than 2 at the backscattering angle, which is also consistent with the discussions of [10]. The phase difference between reversible and non-reversible terms (Iij and Iii) is random, so the value of ζall is variable at the backscattering angle. The width of the backscattering peak is narrower for larger particles, consistent to [10]. For small spheroids (πD/λ=1 and 2), the amplification factor ζij of the parallel component is still 2 at the backscattering angle, but the amplification factor is quasi-uniform near the backscattering angle (Fig. 4(a)-(b)), so there is no apparent backscattering peak. These results indicate that the CBE theory is applicable to particles with maximum dimensions comparable to the wavelength, and explain quantitatively why a CBE-induced backscattering peak is difficult to be identified in the scattering phase function of small particles.

 figure: Fig. 2.

Fig. 2. Comparison of scattering phase function of randomly oriented spheroid calculated with T-matrix (a-c) and ADDA (d-f). The left column is for P11 component of the normalized phase matrix, the middle column is the phase function for the parallel component of scattering light (${P_\parallel }$) when the incident beam is linearly polarized, and the right column is the phase function for the perpendicular component (${P_ \bot }$). The aspect ratio of spheroids is set to be a/b = 3/4, and the refractive index is set to be 1.33 + 0.0i. Different colors denote particles with different size parameters. The x-axes are scattering angles, and the y-axes are the phase functions.

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 figure: Fig. 3.

Fig. 3. Decomposition of the parallel component of the scattering phase function (Fig. 2(e)). In each panel, the black line is the normalized phase functions for the parallel component (${P_\parallel }$, Fig. 2(e)), blue line is the contributions of Iij to the phase function, green line is the contributions of Iij + ji, cyan line is the contributions of Iii, and the red line denotes the summation of Iij, Iji and Iii. Each panel denotes results for a specific size parameter, and the size parameter is labeled on the title of each panel.

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 figure: Fig. 4.

Fig. 4. Amplification factor of spheroids with different size parameters as a function of scattering angle. (a) ζij for the parallel component. (b) ζall for the parallel component. (c) ζij for the perpendicular component. (b) ζall for the perpendicular component. Different colors denote particles with different size parameters.

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This method is applicable to particles with arbitrary habits, and we then applied these experiments to hexagonal particle (faceted shape, where the value of dipoles per lambda is set by default). The results of hexagons with different refractive index are shown in Figs. 5 and 6. The amplification factors for weak absorptive hexagons are similar to that of non-absorptive spheroids. For strong absorptive hexagons, the amplification factor ζij (parallel component) is still 2 at the backscattering angle, but the magnitude of Iii is much greater than that of Iij, and the value of ζall is smaller at the backscattering angle. Besides, the angular width of the CBE-induced backscattering peak is also wider for strong-absorptive particles. These results explain quantitatively why the effect of CBE is weaker for strong absorptive particles.

 figure: Fig. 5.

Fig. 5. Same as Fig. 3, but for hexagons with different refractive indexes. The aspect ratio of these hexagons is a/L = 0.3, and the size parameter is πL/λ=11.

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 figure: Fig. 6.

Fig. 6. Same as Fig. 4, but for the amplification factors of hexagons with different refractive indexes.

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4. Conclusions and discussions

In this paper, representative numerical simulations performed with the ADDA method are presented to quantify the effect of coherent backscatter enhancement on the scattering phase function. We first divided the volume of the scatter into small cubical subvolumes containing dipoles, and calculated the scattering phase function of the scatter located in a plane wave with linear polarization with the ADDA. Then we divided the volume of the scatter into multiple thin slices parallel to the incident beam, and performed a series of sensitivity experiments. In each sensitivity experiment, the incident electric field in a specific slice (the i’th slice) was set to be same as the reference experiment, and the incident field in other slices was set to be zero. We calculated the contribution of electric field in each slice to the scattering field, and then decomposed the total scattering field into contributions of the electric fields in the j’th slice in response to the incident field in the i’th slice (Iij). Interferences between conjugate terms Iij and Iji representing reversible wave paths were quantified, and the results show that the interferences are constructive at the backscattering direction, consistent with the analytical conclusions of the coherent backscatter enhancement (CBE) theory. The CBE of small particles is quasi-uniform at near-backscattering angle, so a backscattering peak could not be found. Nonreversible term (Iii) dominates in the scattering phase function of strong-absorptive particles, and conjugate reversible terms (Iij and Iji) dominates for large weak-absorptive particles, so the CBE-induced backscattering peak is more apparent for large weak-absorptive particles.

The method in this paper is applicable to arbitrary particle shapes, but the computational cost of this method is high for large particles, so physical optics methods [17–20] might be preferred for the case of large faceted particles. The particles were divided to 20 thin slices parallel to the incident beam and perpendicular to the scattering plane in this study, and the results would be almost the same if they were divided to more slices instead; alternatively, the particles could also be divided to thin columns perpendicular to the scattering plane using the same technique, but the corresponding computational cost would be much greater. On the other hand, all the qualitative results in this paper are consistent with the CBE theory, so this paper fortifies the CBE theory in single scattering.

Funding

National Natural Science Foundation of China (42075127).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results can be obtained from the authors upon request.

References

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Data availability

Data underlying the results can be obtained from the authors upon request.

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Figures (6)
Fig. 1.
Fig. 1. Illustration of conjugate terms representing reversible wave paths in the ADDA sensitivity experiments. (a) Interactions between the incident waves (blue), dipoles (black dots) and the scattering field (red). The black arrows denote interactions between dipoles. (b) Illustration of conjugate reversible wave paths with the same scattering angle. The phase difference between the red and blue wave paths is zero when θ is zero, and is non-zero when θ>0. (c) Contribution of dipole polarizations in the i’th slice to the scattering field (Iii), when the incident field in the i’th slice is set to be same as the reference experiment and incident fields in other slices is set to be zero. Each black dot denotes a dipole. (d) Contribution of the dipoles in the j’th slice to the scattering electric field, when the incident field in the i’th slice is set to be same as the reference experiment and incident fields in other slices is set to be zero (Iij). (e) Contribution of dipole polarizations in the i’th slice in response to the incident field in the j’th slice (Iji). (f) Superposition of conjugate reversible wave paths, where conjugate reversible paths (blue and red paths) interfere constructively at the backscattering angle. The black arrows denote a wave path without conjugate reversible path.

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Fig. 2.
Fig. 2. Comparison of scattering phase function of randomly oriented spheroid calculated with T-matrix (a-c) and ADDA (d-f). The left column is for P11 component of the normalized phase matrix, the middle column is the phase function for the parallel component of scattering light ( ${P_\parallel }$ ) when the incident beam is linearly polarized, and the right column is the phase function for the perpendicular component ( ${P_ \bot }$ ). The aspect ratio of spheroids is set to be a/b = 3/4, and the refractive index is set to be 1.33 + 0.0i. Different colors denote particles with different size parameters. The x-axes are scattering angles, and the y-axes are the phase functions.

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Fig. 3.
Fig. 3. Decomposition of the parallel component of the scattering phase function (Fig. 2(e)). In each panel, the black line is the normalized phase functions for the parallel component ( ${P_\parallel }$ , Fig. 2(e)), blue line is the contributions of Iij to the phase function, green line is the contributions of Iij + ji, cyan line is the contributions of Iii, and the red line denotes the summation of Iij, Iji and Iii. Each panel denotes results for a specific size parameter, and the size parameter is labeled on the title of each panel.

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Fig. 4.
Fig. 4. Amplification factor of spheroids with different size parameters as a function of scattering angle. (a) ζij for the parallel component. (b) ζall for the parallel component. (c) ζij for the perpendicular component. (b) ζall for the perpendicular component. Different colors denote particles with different size parameters.

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Fig. 5.
Fig. 5. Same as Fig. 3, but for hexagons with different refractive indexes. The aspect ratio of these hexagons is a/L = 0.3, and the size parameter is πL/λ=11.

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Fig. 6.
Fig. 6. Same as Fig. 4, but for the amplification factors of hexagons with different refractive indexes.

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Equations (12)

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E ⇀ ( r ⇀ ) = E ⇀ i n c ( r ⇀ ) + ∫ V i n t ⁡ d r ⇀ ′ G ↔ ( r ⇀ , r ⇀ ′ ) E ⇀ ( r ⇀ ′ ) k 1 2 ( m ~ 2 ( r ⇀ ′ ) − 1 ) ,
E ⇀ s c a ( r ⇀ s ) = ∫ V i n t ⁡ d r ⇀ ′ G ↔ ( r ⇀ s , r ⇀ ′ ) E ⇀ ( r ⇀ ′ ) k 1 2 ( m ~ 2 ( r ⇀ ′ ) − 1 ) ,
E ⇀ s c a ( r ⇀ s ) = ∫ V i n t ⁡ d r ⇀ 1 k 1 2 ( m ~ 2 ( r ⇀ 1 ) − 1 ) G ↔ ( r ⇀ s , r ⇀ 1 ) [ E ⇀ i n c ( r ⇀ 1 ) + ∫ V i n t ⁡ d r ⇀ 2 k 1 2 ( m ~ 2 ( r ⇀ 2 ) − 1 ) G ↔ ( r ⇀ 1 , r ⇀ 2 ) E ⇀ ( r ⇀ 2 ) ]
E ⇀ s c a ( r ⇀ s ) = ∫ V i n t ⁡ d r ⇀ 1 k 1 2 ( m ~ 2 ( r ⇀ 1 ) − 1 ) G ↔ ( r ⇀ s , r ⇀ 1 ) E ⇀ i n c ( r ⇀ 1 ) + ∑ n = 2 ∞ ⁡ ∫ V i n t ⁡ … ∫ V i n t ⁡ d r ⇀ 1 … d r ⇀ n k 1 2 n ∏ i = 1 n ⁡ ( m ~ 2 ( r ⇀ i ) − 1 ) G ↔ ( r ⇀ s , r ⇀ 1 ) … G ↔ ( r ⇀ n − 1 , r ⇀ n ) E ⇀ i n c ( r ⇀ n )
E ⇀ s c a ( r ⇀ s ) = ∑ i = 1 N ⁡ E ⇀ i s c a ( r ⇀ s ) = ∑ i = 1 N ⁡ ∑ j = 1 N ⁡ E ⇀ i , j s c a ( r ⇀ s ) .
I ( r ⇀ s ) = C | E ⇀ s c a ( r ⇀ s ) | 2 = C | ∑ i = 1 N ⁡ ∑ j = 1 N ⁡ E ⇀ i , j s c a ( r ⇀ s ) | 2 = C | ∑ i = 1 N ⁡ ∑ j = i + 1 N ⁡ ( E ⇀ i , j s c a ( r ⇀ s ) + E ⇀ j , i s c a ( r ⇀ s ) ) + ∑ i = 1 N ⁡ E ⇀ i , i s c a ( r ⇀ s ) | 2
I i j ( r ⇀ s ) = C | ∑ i = 1 N ⁡ ∑ j = i + 1 N ⁡ E ⇀ i , j s c a ( r ⇀ s ) | 2 ,
I j i ( r ⇀ s ) = C | ∑ i = 1 N ⁡ ∑ j = i + 1 N ⁡ E ⇀ j , i s c a ( r ⇀ s ) | 2 ,
I i i ( r ⇀ s ) = C | ∑ i = 1 N ⁡ E ⇀ i , i s c a ( r ⇀ s ) | 2 ,
I i j + j i ( r ⇀ s ) = C | ∑ i = 1 N ⁡ ∑ j = i + 1 N ⁡ ( E ⇀ i , j s c a ( r ⇀ s ) + E ⇀ j , i s c a ( r ⇀ s ) ) | 2 ,
ζ i j ( r ⇀ s ) = I i j + j i ( r ⇀ s ) I i j ( r ⇀ s ) + I j i ( r ⇀ s ) ,
ζ a l l ( r ⇀ s ) = I ( r ⇀ s ) I i j ( r ⇀ s ) + I j i ( r ⇀ s ) + I i i ( r ⇀ s ) .
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