Solution of Maxwell’s equations to the problem of single scattering can be expanded into iterative series in an order-of-scattering form, where the interference between conjugate terms representing reversible sequences of elementary scatterers is constructive at the backscattering direction, resulting in a coherent backscatter enhancement (CBE). The backscattering phase function of randomly oriented particles is amplified by CBE with an amplification factor between 1 and 2 depending on particle habit and refractive index. The angular width of the CBE-induced backscattering peak line for a specific particle habit is inversely proportional to the particle size parameter. The CBE-induced backscattering peak has been identified in the scattering phase function of a wide range of randomly oriented particles, including non-absorptive spheres, spheroids, and hexagonal particles.
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Backscattering properties are essential in active remote sensing of clouds and aerosols. Previous studies have identified a backscattering peak associated with the scattering phase function of various large transparent particles including spheres [1–3], smooth/roughened hexagonal particles [4–9], and complex concave particles .
Previous studies indicate that the narrow backscattering peak associated with the single scattering phase function of these particles may be induced by constructive interferences between waves propagating conjugate reversible paths [3,7,8,11], a process similar to the so-called coherent backscattering for disordered media [12–19]. Despite the backscattering properties in single scattering have been found to be similar to coherent backscattering [20-21], the prevailing coherent backscattering theory applies only to multiple scattering, and new efforts are needed to explain the processes that lead to the backscatter enhancement in single scattering.
In this study, we transform the closed-form solution of Maxwell’s equations for single scattering to an order-of-scattering form, identify pairs of conjugate reversible sequences of elementary scatterers in the corresponding order-of-scattering expansions, and demonstrate that the interferences between the conjugate pairs of sequences are constructive at the exact backscattering direction. We define the corresponding enhancement of backscatter as the coherent backscatter enhancement (CBE) (Section 2). Subsequently we identify CBE in case studies (Section 3).
2. Deriving the coherent backscatter enhancement from Maxwell’s Equations
We consider a stable finite particle with refractive index , located in an infinite, homogeneous, isotropic and non-absorbing medium with refractive index of . There is a monochromatic plane incident light wave in the medium, and the wavelength is λ. The monochromatic Maxwell curl equations for this problem can be written as (Eqs. (2.1) and (2.2) of )
The solution of the inhomogeneous differential equation can be divided into two parts. The first part is a solution of the homogeneous equation with the right-hand side to be zero, and the second part is a particular solution of the inhomogeneous equation . The first part of the solution to Eq. (4) is:
We use the free space dyadic Green’s function to solve the particular solution, i. e., the scattered field. The Green’s function satisfiesEq. (4), we getEq. (4) is 22])22]Eq. (10), the scattering electric field in the far field is Eq. (10) into Eq. (12), we haveEq. (10) into Eq. (13) for infinite times, then we haveFig. 1) would interfere constructively, in a similar way as the case of coherent backscattering for disordered medium [18-19]. To demonstrate this, we decompose Eq. (14) into three termsEq. (16c) we have relabeled the integration variables by replacing the subscripts 1, 2, …, n with n, n-1, …, 2, 1. Equations (16b) and (16c) contain conjugate terms representing reversible sequences of elementary scatters.22,23]
The electric field for a linearly polarized incident plane wave can be written asEqs. (16b) and (16c) can be written asEqs. (17b), (17c), (21) and (24), and getEq. (26) would be
At the exact backscattering angle, , so according to Eq. (29) we have
For very small particles, , so at all scattering angles, and thus the amplification factor is also close to 2 at off-backscattering angles. As a result, the interferences between these conjugate terms is same for all scattering angles, and would not affect the scattering phase function of Rayleigh scattering significantly.
However, for large particles, the value of is not constantly zero, and the amplification factor in Eq. (25) ranges from 0 to 2 at large off-backscattering angles. As a result, the phase and magnitude of and is not exactly the same, and we haveEqs. (31) and (32), the interference between these conjugate terms is most constructive at the exact backscattering angle.
At near backscattering angles, the phase difference between conjugate terms is distributed between 0 and . The maximum dimension of the particle is D, and the off-backscatter angle α is close to 0, then we haveEq. (30)]. If 2παD/λ<π/2, then the amplification factor is always larger than 1, so the interference between and is constructive. When 2παD/λ>π/2, the amplification factor ranges from 0 to 2, and the interference between and is not necessarily constructive. The critical off-backscatter angle to reach 2παD/λ = π/2 is αc = λ/4D. Therefore, the characteristic angular width of a CBE-induced backscattering peak line would be inversely proportional to particle size parameter (πD/λ). This property can be used to identify CBE in case studies.
For the scattering waves that are perpendicular to the reference plane, the conjugate terms representing reversible sequences of elementary scatterers are still in phase at the exact backscattering angle, but the magnitude of conjugate terms are not necessarily the same [Eq. (21)]. Therefore, the interference between the conjugate terms is still constructive at backscattering angle for the perpendicular component, but the amplification factor would be less than 2. In this study, we could not strictly derive the backscatter enhancement at the perpendicular plane, but we will show the properties of enhancement in the perpendicular component in later case studies.
In general, the constructive interferences between conjugate terms representing reversible sequences of elementary scatterers may enhance the scattering wave intensity at backscattering angle, and we define this enhancement as the coherent backscatter enhancement (CBE).
Although the CBE equations apply to all particles, the phase function of a specific oriented particle is usually full of pronounced oscillations, and one may need to perform ensemble averaging to smooth out the speckle pattern and thereby better expose CBE. This is similar to the case of coherent backscattering of sparse random multi-particle groups, where ensemble averaging serves to expose the diffuse radiative-transfer regime and coherent backscattering in the form of a narrow backscattering peak [18-19]. The importance of ensemble averaging in coherent backscattering has been demonstrated by direct computer solutions of the Maxwell equations , and explains the opposition optical phenomena exhibited by particulate surfaces of many solar-system objects . It is worth noting that the ensemble average of single scattering phase function, which is widely used to describe single scattering properties in remote sensing , does not contain information from multiple scattering.
3. Case studies
To verify the CBE theory for single scattering, we study and identify the CBE-induced backscattering peak in specific cases.
Our case analyses begin with the backscattering properties of roughened hexagonal particles [6,7,11], which has been investigated in . Figure 2(a) shows the phase function of roughened hexagonal ice crystals with 2πa/λ = πL/λ = 50, where 2a is the cross section diameter of the crystal, and L is the height of the hexagon. The phase function for roughened hexagonal particles are calculated from the pseudo-spectral time domain method (PSTD) [6,27], which solves Maxwell’s equations numerically. The phase function is generally flat at off backscattering angles (150°<θ<175°), with a peak near backscattering direction [Fig. 2(b)]. The angular width of the backscattering peak is inversely proportional to the particle size parameter [Fig. 2(c)] , consistent with the prediction of our theory. Figures 3(a) and 3(b) show the parallel and perpendicular components of light scattered by these roughened hexagonal particles when the incident beam is linearly polarized with a Stokes vector of I0[1 1 0 0]T. The intensity of the parallel component is larger than the perpendicular component, but the amplification factor and the angular width of the backscattering peak are comparable for the parallel and perpendicular components.
Figure 2(d) shows the backscattering phase function of a smooth regular hexagon, which is calculated from the imbedded T-matrix method (II-TM [11,28],). There is a general increasing trend of phase function with small oscillations between 170° and 180°, which is primarily caused by the corner-reflection effect [4,5]. In addition to the increasing trend, there is a narrow peak in addition to the previously mentioned increasing trend (177.5-180° for blue line), which is induced by interferences from coherent addition of the conjugate corner-reflection rays in the view of geometric optics . The angular width of the narrow peak is inversely proportional to the particle size [Fig. 2(f)], consistent with our CBE theory. Figures 3(c) and 3(d) show that the parallel component dominates over the perpendicular component when the incident beam is linearly polarized, but the angular width of the narrow backscattering peak is comparable for the parallel and perpendicular components. It is worth noting that when the surfaces of the regular hexagon are slightly tilted, the increasing trend diminishes, but the narrow peak still exists [5,8], implying that the coherent backscatter enhancement theory also applies to irregular smooth hexagonal particles.
Figure 2(g) shows the phase function of spheres, calculated from Mie theory . The backscattering phase function of sphere consists of a wide peak between 175 and 180, plus a narrow backscattering peak near 180° [Fig. 2(h)]. The wide backscatter peak is primarily contributed from spherical surface waves [1,2], and the narrow peak is induced by the interference between conjugate “long” and “short” surface wave paths  in the view of geometric optics. At the exact backscattering angle, the phase shift between “long” and “short” surface waves is zero according to the symmetry of sphere, so the conjugate surface waves interfere constructively, with an amplification factor of 2. When α is small, the phase difference between the conjugate “short” and “long” surface wave paths is 2πDα/λ1. The angular width of the narrow peak for spheres is also inversely proportional to the particle size parameter [Fig. 2(i)]. Results for spheres are not shown in Fig. 3, because the perpendicular component is always zero in response to the linearly polarized incident beam.
Figure 2(j) shows the phase function of spheroids calculated from T-matrix method , where the spheroid aspect ratio is set to 0.75. The narrow backscattering peak persists for these spheroids, and its angular width is similar as spheres and hexagons. Figures 3(e) and 3(f) show that the parallel component dominates over the perpendicular component when the incident beam is linearly polarized, and the angular width of the narrow backscattering peak is comparable for the parallel and perpendicular components. Therefore, it is indicated that the coherent backscatter enhancement also applies to these spheroids.
5. Conclusions and discussions
In conclusion, conjugate terms in the order-of-scattering form solution of Maxwell’s equations interfere constructively at the backscattering direction, which may induce a coherent backscatter enhancement. CBE at the direction parallel to incident electric vector has been demonstrated and verified by case studies, while the CBE of perpendicular component is shown in case studies without strict mathematical demonstration. The angular width of the CBE-induced backscattering peak is inversely proportional to the particle size parameter. The coherent backscatter enhancement applies to a wide range of particles, including non-absorptive spheres, spheroids, smooth hexagonal particles, and roughened hexagonal particles, which are widely used to simulate the single scattering properties of atmospheric clouds and aerosols.
Though the CBE theory is applicable to all kinds of single large particles, the narrow backscattering peak may not exist under certain circumstances. When the backscatter is primarily contributed from non-reversible terms [the first term in Eq. (15)], the coherent amplification factor would be close to 1, and it would be impossible to find a backscattering peak. When there is a significant peak or trough induced by other processes (e. g., the corner effect of regular smooth hexagonal particles), it would be less easy to identify the relatively small peak induced by CBE.
The processes that lead to CBE in single scattering is very similar to the coherent backscattering in multiple scattering, and could be regarded as a form of weak localization. Actually, equations derived for CBE also apply to disordered medium containing numerous small particles. This can be done by considering the medium as a large particle, where for spaces between sub-particles. For a massive disordered medium with given number and spatial distribution of small particles, the average distance between adjacent particles is proportional to the medium size, so the average path length l* within the medium is proportional to the disordered medium size D. According to the CBE theory in this study, the angular width of the coherent backscatter enhancement for the disordered medium would be proportional to the wavelength divided by the average path length l*, consistent with results from the coherent backscattering theory .
In the reality, CBE from both single scattering and multiple scattering may contribute to the backscatter signals received by instruments. For very sparse medium with large non-absorptive particles (e. g., clouds), the coherent backscattering peak induced by multiple scattering may be difficult to detect due to its extremely narrow angular width. For example, the coherent backscattering induced by multiple scattering between cloud particles has a characteristic angular width of ~10−10 rad at the wavelength of 0.5μm , the characteristic angular width for the backscattering peak of ice cloud crystals is ~10−3 rad (estimated based on Fig. 2), and the off-backscattering view angle of lidar on board a low-earth-orbit satellite is ~10−5 rad (calculated based on the speed of light and satellite). Therefore, the coherent backscattering from multiple scattering between cloud particles could not be observed by space-borne lidar, but the lidar returns are affected by CBE from single scattering. On the contrary, the coherent enhancement from multiple scattering dominates when light waves interact with condense media containing small particles, because the angular width of coherent backscattering peak line is relatively large due to short average wave path length , and the CBE-induced peak cannot be found in the phase function of extremely small particles.
This study provides a general perspective to CBE, and different approaches may be needed to quantify the effect of CBE in detailed case studies. In particular, we expect that the Debye’s series would be useful in future case studies, due to its ability to identify conjugate surface wave paths of spheres  and its applicability to non-spherical particles [32,33].
National R & D Program of China (2016YFC0202000: Task 2); Nanjing University (NJU).
The author thanks Dr. P. Yang and Dr. M. Mishchenko for scientific suggestions, Dr. L. Bi and Dr. C. Liu for PSTD/II-TM simulations and technical suggestions, and Dr. M. Zelinka and Dr. C. Terai for writing suggestions.
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