Annotation of the results obtained in 2023
In 2023, to complete the project, matrices of light scattering on randomly oriented single particles of polyhedral shape with sizes from tens to 1000 wavelengths of incident light were numerically calculated. The calculations used wavelengths typical for lidars: 0.355, 0.532, 1.064 microns and others. In order for the results obtained to be used for problems of laser sensing of the atmosphere, as well as data interpretation during polarimetric remote sensing of solar system objects, the refractive index was chosen for a) ice crystals in cirrus clouds; b) atmospheric and cosmic dust particles; and c) regolith. To numerically study coherent effects in the light scattering matrix in scattering directions close to the backscattering direction, which manifest themselves in the form of surges in the functions of the dependence of scattering matrix elements on the scattering angle, we used the physical optics method developed at the IAO SB RAS by the project executors. In its latest implementation, the numerical algorithm of the method allows one to calculate the optical characteristics of particles with both convex and non-convex shapes. The method of physical optics takes a leading position in solving the problem of light scattering on atmospheric ice particles for active remote sensing problems. In 2023, the project implementers explored the lower limit of applicability of the physical optics method using the example of several Platonic solids. To validate the physical optics method developed at the IOA SB RAS, various numerical methods were used: IITM, PGOM , ADDA. The calculation results showed good agreement between the methods. In addition, the project investigated the angular sizes and shapes of surges for all elements of the scattering matrix for a randomly oriented particle of irregular shape . The method of physical optics has a high computational complexity when solving the problem of light scattering in all scattering directions, especially for large particles. Therefore, for particles that are large compared to the wavelength of incident light, it was decided to combine the advantages of geometric and physical optics methods, namely, to “stitch” two solutions. In narrow regions of scattering directions near the forward and backward scattering directions, where the geometric optics approximation produces singularities and leads to significant errors, it has been proposed to use a solution obtained within the framework of physical optics. And in other scattering directions, where the elements of the scattering matrix do not have features, it was proposed to use the method of geometric optics. A numerical study of the angular sizes and shapes of surges for various shapes and sizes of particles for all elements of the scattering matrix was carried out in 2023 using the example of an ensemble of 50 large ice crystals of irregular shape. It is shown that, regardless of the shape of the particle, there is an increase in the intensity of the scattered light centered in the backscatter direction, which is called an intensity surge or coherent backscatter peak. The angular width of the coherent peak practically does not change with the particle shape. For an individual particle from the ensemble, both strong negative (negative) polarization and positive polarization were detected for element (21) of the scattering matrix. However, averaging over an ensemble of particles leads to negative polarization, the angular width of which approximately corresponds to the minimum of the coherent peak. It is also shown that the angular size of the negative polarization extremum can be used to determine the size of convex non-spherical particles . As part of the project, in 2023, a comparison was made of the elements of light scattering matrices calculated by the physical optics method for an ensemble of randomly oriented large ice crystals of irregular shape with the results obtained for an agglomerate of fragments. The comparison showed that both for the agglomerate of fragments and for the ensemble average of large crystals, positive polarization is observed at small phase angles. The main idea of the project was to explore the analogy in solving two problems: 1) the problem of light scattering by a single particle of irregular shape and 2) the problem of multiple light scattering in chaotic media. To do this, in 2023, the problem of light scattering on single randomly oriented polyhedra of irregular shape with dimensions much greater than the wavelength of incident light was solved. As a result of calculations of the light scattering matrix on large randomly oriented polyhedra of irregular shape, it was concluded that the degree of linear polarization at small phase angles can be both negative and positive. It is shown that the phenomenon of negative polarization is not universal. Within the framework of the project, it was shown for the first time that the degrees of linear polarization for randomly oriented particles are oscillating alternating functions of the phase angle, but the sign of these functions at small phase angles is an unstable quantity that depends on the shape of the trajectory of light beams inside the particle. Numerical calculations have shown that in the case of a randomly oriented polyhedral particle of a convex shape, the trajectories of light beams that make the main contribution to coherent backscattering are mainly reduced to sliding trajectories. In sliding trajectories, each reflection inside the particle occurs at an angle close to total internal reflection, and, therefore, the beam energy after multiple reflections is practically conserved. It is shown that the backscattering peak is formed due to the interference between direct and conjugate trajectories, and their degree of linear polarization can be either positive or negative near the backscattering direction. It can be confidently stated that it is the appearance of total internal reflections along the beam trajectory that is the reason for the change in the signs of polarization surges. In order to confirm this statement, we considered the mechanism of formation of negative polarization on a real particle of irregular shape. The same particle was calculated for three different refractive indices: 1.2, 1.3116 and 1.5. The results showed that at refractive indices of 1.2 and 1.3116 there is strong negative polarization. However, at a refractive index of 1.5, the polarization dip became significantly smaller. This indirectly confirms the influence of total internal reflection on negative polarization. For a more detailed analysis, the light scattering matrix was averaged over an ensemble of 50 arbitrary shapes of randomly oriented particles . which confirms the conclusions obtained for a single particle. A study of the angular dimensions and shape of surges for particles of non-convex shape was carried out in 2023 based on a model for a hollow hexagonal column. Such a particle is a hexagonal prism with two identical pyramidal depressions from the bases , which in some sense resembles the shape of an hourglass. In nature, most particles that cause light scattering have an irregular geometric shape. For example, these are aerosol particles in the atmosphere, particles of regolith covering the surface of celestial bodies, etc. When the particle size is smaller or slightly exceeds the wavelength of incident light, the problem of light scattering by particles of irregular shape is usually solved numerically, for example, by the ADDA discrete dipole method . But if the particle size significantly exceeds the wavelength, then all standard methods for solving Maxwell’s equations in this case become unacceptable, since they lead to exorbitant costs of computer resources. In this case, an alternative is the method of physical optics, which has been developed in recent years at the Institute of Optical Optics SB RAS. Calculations have shown that for large hollow columns the light intensity peak is destroyed as the height of the recess increases. In addition, in contrast to the results that were obtained for small hollow columns, the sign of the degree of linear polarization is unstable. This is explained by the significant contribution of grazing beams with total internal reflections.

 

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Annotation of the results obtained in 2022
In 2022, for the implementation of the project, the matrix of light scattering on convex polyhedron was numerically calculated, the dimensions of which reached a thousand wavelengths of incident light. The shape of the polyhedron was random, it was generated by a computer. The incident radiation was light with wavelengths typical for lidars of 0.355, 0.532, 1.064, and other microns. The refractive index was chosen as the refractive index for ice at the corresponding wavelengths, so that the calculation results can be applied to the work on sounding cirrus clouds with lidars of the IAO SB RAS. The particle sizes varied from tens of microns to 1000 microns. In particular, for a wavelength of 0.355 microns, the ratio [(particle size)/(wavelength)] reached 3000, which is a record value for numerical solutions of such problems of wave scattering on particles. To calculate the matrices of light scattering on such randomly oriented ice crystals, the method of physical optics was used. The algorithm of the method of physical optics was developed by the authors of the project, and published on the website of the IAO SB RAS, in previous years. The results of the calculations are presented in 2022 in two articles in the journal Atmosphere MDPI and in one article Proceedings of SPIE. In these papers, the calculated values of the scattering matrices as a function of scattering angles are presented, including in the direction of scattering strictly backward θ = 180°, and in the vicinity of this direction θ = 180°, which is important for problems of laser sounding of the Earth's atmosphere. The method of physical optics used in the calculations has a number of advantages over standard methods for numerically solving problems of light scattering by large particles. Namely, the calculation of the scattered field can be carried out simultaneously in three modes: a) the mode of geometric optics, b) the mode of physical optics, taking into account only the diffraction of scattered light beams, and c) the mode of physical optics, where both the diffraction of beams and the interference between them are taken into account. A new result, obtained in 2022, is that we have shown that for large convex particles, the dominant contribution to the coherent backscattering peak comes from certain beam trajectories, which we call grazing trajectories. We call the reflection of a beam of light from a face of a crystal grazing if the angles of incidence and reflection differ slightly from the angle of 90°. We call the beam trajectory inside the particle gliding if most of the reflections in the trajectory were gliding. For the problems of lidar sensing, to which we apply the results of calculations, only scattering directions strictly backward θ = 180° or near the direction θ = 180° are important. Here, for coherent backscattering, one non-slip reflection of the light beam from the particle faces is sufficient. The rest of the reflections are gliding. Thus, we have shown for the first time that the coherent backscattering peak is formed for convex polyhedron mainly by grazing beam trajectories. In the calculations of the problem of light scattering by a large chaotically oriented particle, carried out in 2022, we calculated all 16 elements of the scattering matrix in all scattering directions. The advantage of our method of physical optics is that the beam interference, which is the desired coherent effect, can be calculated separately from the total field. Using this advantage of ours, we were able to answer the question: where do coherent effects manifest themselves, i.e. wave interference, in a scattered field? As it turned out, coherent effects manifest themselves in the form of surges in the distributions of scattering matrix elements over scattering angles. We numerically calculated these surges and showed that they appear in the same scattering angle ranges as the well-known coherent backscattering peak for light intensity. Moreover, it turned out that the angular sizes of both the coherent backscattering peak and the surges for other elements of the scattering matrix were approximately the same. Particular attention in the project was paid to the surge for element (12) of the scattering matrix. The fact is that in astrophysics already about 100 years ago it was noticed that the polarization of solar radiation reflected from planets and other celestial bodies near the backward scattering direction θ = 180° exhibits specific properties. Namely, the degree of linear polarization becomes negative, and has a minimum at a certain scattering angle θ0 , where θ0 < 180°. This phenomenon has been called the negative polarization effect. To date, over a dozen different theories have been published trying to explain this phenomenon, but all of them are found to be insufficiently satisfactory. Based on the results obtained in the course of the project, we came to the conclusion that the so-called negative polarization effect is precisely the burst of element (12) of the scattering matrix. A more detailed study of this issue is planned for the next year of the implementation of this project. The scientific novelty of the project was to test the idea that the solution of two seemingly different problems, i.e. the problems of radiation scattering by one large chaotically oriented particle of an irregular shape and the problems of multiple radiation scattering in chaotic media should exhibit the same qualitative regularities. We drew attention to a series of theoretical and experimental works on multiple scattering of narrow beams of light in various scattering media, where a specific azimuthal symmetry of the intensity of multiple scattered light was found, for example, see Rakovic´ et al, Applied Optics 38(15), 3399–3408 (1999). Based on the analogy between multiple wave scattering and the problem of wave scattering by a single large particle, we assumed that the solution of the problem of wave scattering by a single particle should manifest the same azimuthal symmetry as in the problem of multiple scattering. As shown by our calculations, the desired symmetry did not manifest itself. Thus, we have seen that there is no analogy with respect to azimuthal symmetry in the above two problems. Nevertheless, our calculations of the scattered field for particles with sizes up to 1000 microns gave another important result. We have shown that such a quantity, which is often used in lidar measurements, as the depolarization ratio, can be considered in cirrus clouds to be independent of the size of ice crystals. This fact can be effectively used to reconstruct the shape and size of ice crystals in lidar sounding of cirrus clouds.

 

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