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The article proposes the classification of lenses with different symmetrical beam angles and offers a scale as a spot-light’s palette. A collection of spotlight’s images was created and classified according to the proposed scale. The analysis of 788 pcs of existing lenses and reflectors with different LEDs and COBs carried out, and the dependence of the axial light intensity from beam angle was obtained. A transfer training of new deep convolutional neural network (CNN) based on the pre-trained GoogleNet was performed using this collection. GradCAM analysis showed that the trained network correctly identifies the features of objects. This work allows us to classify arbitrary spotlights with an accuracy of about 80 %. Thus, light designer can determine the class of spotlight and corresponding type of lens with its technical parameters using this new model based on CCN.

The paper discusses the possibility of remote detection of a continuous laser beam propagating in a scattering continental and coastal atmosphere, when it is recorded outside the axial zone. In the single scattering approximation, estimates of the radianceat the registration site are carried out, which are compared with the threshold characteristics of existing photodetectors in the visible and IR spectral regions. It is shown that the laser radiation (LR) of the beam is reliably recorded in the range of angles (0–180)° at metrological range of visibility equal (5–20) km at night conditions. At twilight, under the same conditions, detection capabilities are significantly reduced. A significant increase of the LR beam radiance contrast with a decrease in its divergence has been shown experimentally in the field observations. At twilight, a decrease in the beam’s radiance contrast is seen. A beam with a divergence equal to 2 ceases to be distinguishable at angles equal to (80–90)°, and a beam with a divergence of 4 – at angles (60–70)°.In this case, the contrast difference reaches up to 10 times.

The article describes the role of polarisation in calculation of multiple reflections. A mathematical model of multiple reflections based on the Stokes vector for beam description and Mueller matrices for description of surface properties is presented. On the basis of this model, the global illumination equation is generalised for the polarisation case and is resolved into volume integration. This allows us to obtain an expression for the Monte Carlo method local estimates and to use them for evaluation of light distribution in the scene with consideration of polarisation. The obtained mathematical model was implemented in the software environment using the example of a scene with its surfaces having both diffuse and regular components of reflection. The results presented in the article show that the calculation difference may reach 30 % when polarisation is taken into consideration as compared to standard modelling.

The wavelength is that natural scale that determines the applicability domains of the ray approximation and the wave approximation of light. If the change of the radiation power spatial density is significant at the wavelength scale, then we deal with the light diffraction phenomenon, which is a subject to the wave optics. Consider the diffraction phenomenon at the diaphragm. It is possible to distinguish the near zone with significant wave inhomogeneities (i.e. the Fresnel zone) and the far Fraunhofer diffraction zone, in which the wave becomes close to homogeneous (the so-called quasi-homogeneous) and the ray approximation is possible. The problem is that there is no explicit relationship between the radiance of the rays before and after diaphragm. Method for determining the boundary conditions for the radiance in the Fraunhofer zone through the radiance of the incident radiation is proposed in the paper. This approach for computing the radiance field in the Fraunhofer zone can be generalized to other problems of optics, thereby providing the possibility of using computationally efficient ray-approximation-based methods to determine the light fields.

The article is devoted to the creation of a surface radiance factor mathematical model. The basis of the model is the solution of the boundary value problem of the radiative transfer equation (RTE). The surface is considered as a structure consisting of several turbid layers, each of which is characterized by its optical parameters. The top of the structure is randomly rough, uncorrelated, Fresnel. The lower boundary reflects perfectly diffusely. The complexity of solving the RTE boundary value problem for real layers is due to the fact that the suspended particles in each layer are always much longer than the wavelength. This leads to a strong anisotropy of the radiance angular distribution according to Mie theory. The solution comes down to a system of equations by the discrete ordinates method that consists of several hundred of differential equations. Subtraction of the anisotropic part from the solution based on an approximate analytical solution of the RTE allows avoiding this problem. The approximation is based on a slight decrease in the anisotropic part of the angular spectrum. The matrix-operator method determines the general solution for a complex multilayer structure. The calculation speed can be increased without compromising the accuracy of the solution with the help of the synthetic iterations method. The method consists of two stages: the first one repeats the described one with a small number of ordinates; on the second one the iteration of it is performed. The model is realised in the Matlab software.

It has been shown that the regular part of the solution (RPS) which remains after separating the anisotropic part of the solution (APS) in the small-angle modification of the spherical harmonics method (SHM) is a smooth quasi-isotropic function with individual peaks in the angular distribution. The smooth part of the RPS without peaks can be determined in the two-streaming or diffuse approximation. The first iteration of the angular distribution of the radiance significantly refines the solution and allows one to restore the abovementioned angular peaks. The quasi-diffusion approximation—separation of the APS on the basis of the SHM, the determination of the RPS in the diffusion approximation, and the refinement of the solution on the basis of the first iteration—does not depend on the symmetry of the problem and, therefore, can be generalized to the case of an arbitrary geometry of the medium.

Now days regulatory documents for non-special lighting systems, the illuminance and various parameters derived from it are normalized as a quantitative characteristic. In most cases, all calculations are carried out for illuminance on the floor of the room or on an imaginary working plane located at the height of the table. However, illuminance is an integral characteristic of incident light, while the human eye responds to light reflected from the surface. That is, if we take a completely black surface with a reflection coefficient equal to zero, then formally you can get the required illuminance on it, while visually we will not see anything, since nothing will be reflected from the surface. In terms of the human eye, luminance must be normalized instead of illuminance. Recently, the calculation and measurement of luminance was an extremely difficult task, so the it is understandable, that current regulatory documents describes almost illuminance normalization, but not luminance. This paper aims to modeling luminance spatial-angular distribution, which enables us to run the assessment of the lighting quality.

The article is devoted to the modelling of light reflection from real surfaces. Most of the existing models are often created to solve a certain task and therefore can not be used in other realms. To create a model useful for different purposes, we propose a reflective surface representation as a scattering layer bounded by the diffuse bottom and a randomly rough Fresnel upper boundary. Such an approach allows one to account for light polarization in the scattering layer and slope correlation of the randomly rough boundary which paves the way for the observation of some physical effects that take place in nature (for instance, statistical lens emergence). The first results of the light reflection modelling were taken at such initial parameters as to compare to those obtained in other research. Former occurred to be qualitatively of the same form as the latter. The model needs to undergo further validation in numerous experiments to prove its serviceability for different types of reflective surfaces.

The transmission of light is one of the key optical processes in the terrestrial environment (the atmosphere and underlying surfaces). The dependence of light transmittance on the illumination/observation conditions and optical properties of the atmosphere–underlying system can be studied using the integro-differential radiative transfer equation. However, for numerous applications a set of analytical equations is needed to describe the transmitted light intensity and flux. In this paper, we describe various analytical techniques to study light transmittance through light scattering and absorbing media. A physical significance and improved mathematical accuracy of approximations are provided using the analytical models for the diffusion exponent, average cosine of the light field, spherical and plane albedos. The accuracy of various approximations is studied using exact radiative transfer calculations with various scattering phase functions, single-scattering albedos, observational conditions, and optical depths.

An efficient approach to solving the equation of radiation transfer in a turbid medium with strongly anisotropic scattering is proposed based on the small-angle approximation of the theory of radiation transfer with allowance for the dispersion of the radiation propagation path length, as well as with allowance for polarization. The approach is applied to the solution of the problem of the field of a pointlike isotropic radiation source in a two-dimensional medium.

The paper suggests a new improved approach to calculating the anisotropic part of a diffuse light field in a turbid medium. We discuss some specific applications related to specific problems of radiative transfer theory, which demonstrate the accuracy and flexibility of this approach in practice.

The problem of the propagation of the short light pulse in the scattering medium is considered within the radiative transfer theory. The novel approach to the accounting for the propagation path length dispersion in the small scattering angle approximation limit is proposed. The new type of the approximate solution for the medium with the high anisotropy of the scattering is formulated. The results of the numerical simulations of the proposed transient solution are presented. The application of the new solution to the polarized radiation problems is discussed.

The development of the small angle approximation of the radiative transfer equation (RTE) solution for the slab taking into account the photon path distribution function in the forward hemisphere of sighting directions is offered. The solution is based on the Taylor series expansion of the sighting angle cosine up to the 2nd order without the RTE conversion to the diffusion approximation. The obtained solution practically coincides with the exact solution in the forward hemisphere of directions up to the deep regime.

Several numerical and analytical solutions of the radiative transfer equation (RTE) were compared for plane albedo in a problem of solar light reflection by sea water. The study incorporated the simplest case—a semi-infinite one-dimensional plane—parallel absorbing and scattering homogeneous layer illuminated by a monodirectional light beam. Inelastic processes (such as Raman scattering and fluorescence), polarization and air–water surface refraction–reflection effects, were not considered. Algorithms were based on the invariant imbedding method and two different variants of the discrete ordinate method (DOM). Calculations were performed using parameters across all possible ranges (single-scattering albedo ω 0 and refracted solar zenith angle θ 1 ), but with a special emphasis on natural waters. All computations were made for two scattering phase functions, which included an almost isotropic Rayleigh phase function and strongly anisotropic double-peaked Fournier–Forand–Mobley phase function. Models were validated using quasi-single-scattering (QSSA) and exponential approximations, which represent the extreme cases of ω 0 →0 and ω 0 →1, respectively. All methods yielded relative differences within 1.8% for modeled natural waters. An analysis of plane albedo behavior resulted in the development of a new extended QSSA approximation, which when applied in conjunction with the extended Hapke approximation developed earlier, resulted in a maximum relative error of 2.7%. The study results demonstrated that for practical applications, the estimation of inherent optical properties from observed reflectance can best be achieved using an extended Hapke approximation.

The paper deals with the polarized radiative transfer within a slab irradiated by a collimated infinitely wide beam of arbitrary polarized light. The efficiency of the proposed analytical solution lies in the assumption that the complete vectorial radiative transfer solution is the superposition of the most anisotropic and smooth parts, computed separately. The vectorial small-angle modification of the spherical harmonics method is used to evaluate the anisotropic part, and the vectorial discrete ordinates method is used to obtain the smooth one. The azimuthal expansion is used in order to describe the light field spatial distribution for the case of abnormal irradiance and to obtain some known neutral points in the sky especially useful for polarized remote sensing of the atmosphere.

In this paper we propose the fast, but the accurate algorithm for numerical modeling of light fields in the turbid media slab. For the numerical solution of the radiative transfer equation (RTE) it is required its discretization based on the elimination of the solution anisotropic part and the replacement of the scattering integral by a finite sum. The solution regular part is determined numerically. A good choice of the method of the solution anisotropic part elimination determines the high convergence of the algorithm in the mean square metric. The method of synthetic iterations can be used to improve the convergence in the uniform metric. A significant increase in the solution accuracy with the use of synthetic iterations allows applying the two-stream approximation for the regular part determination. This approach permits to generalize the proposed method in the case of an arbitrary 3D geometry of the medium.

We present an analysis of a number of different approximations for the plane transmittance Tp and diffuse attenuation coefficient Kd of a semi-infinite, unbounded, plane-parallel, and optically homogeneous layer. The maximally wide optical conditions (from the full absorption to the full scattering and from the fully forward to the fully backward scattering) were considered. The approximations were analyzed from the point of view of their physical limitations and closeness to the numerical solution of the radiative transfer equation for the plane transmittance. The main criterion for inclusion of the models for analysis was the possibility of practical use, i.e., approximations were well parameterized and included only easily measured or estimated parameters. A detailed analysis of errors for different Tp and Kd models showed that the two-stream radiative transfer Ben-David model yields the best results over all optical conditions and depths. However, the quasi-single-scattering and polynomial Gordon’s approximations proved to be the best for the depths close to zero.

The propagation of the short electromagnetic pulse in the turbid medium is studied on the basis of equivalence of the radiative transfer theory and coherence functions approach. It is shown that traditional Markovian approximation suffers from the systematic error due to neglected propagation path length distribution. The corrected solution of the problem is derived and analyzed.

In this work we suggest a precise calculation method for GOSAT program devoid of shortcomings of existing methods. The algorithm was tested on the plain unidirectional source which already has an exact solution to compare with. In that way the radiuses of mesh cylinders were a much larger than their height. Then we tested an algorithm on finite cylinders mesh and compared the results with Monte-Carlo simulation to be sure of its appropriateness. As a results of our work we got a fast and precise method for the calculation angular radiance with consideration of polarization.

The singularity subtraction on the vectorial modification of spherical harmonics method (VMSH) of the solution of the vectorial radiative transfer equation boundary problem is applied to the problem of influence of atmosphere parameters on the polarimetric system signal. We assume in this model different phase matrices (Mie, Rayleigh, and Henyey‐Greenstein), reflecting bottom and particle size distribution. The authors describe the main features of the model and some results of its implementation.

In this paper the mathematical model of the visibility range determination of a laser beacon in a navigational complex is offered. The model includes model of the optical properties of a coastal atmospheric haze, algorithm of the light field calculation in a turbid medium and the statistical model of the image perception by the given level of detection probability. Calculation of laser signal atmospheric attenuation is conducted with use small angle approximation method of spherical harmonics.

The paper deals with the polarized radiative transfer within a slab irradiated by a collimated infinity-wide beam of arbitrary polarized light. The efficiency of the proposed analytical solution lies in the assumption that the complete vectorial radiative transfer solution is the superposition of the most anisotropic and a smooth parts, computed separately. The vectorial small angle modification of the spherical harmonics method is used to evaluate the anisotropic part and the vectorial discrete ordinates method is used to obtain the smooth one. The azimuthal expansion is used in order to describe the light field spatial distribution for the case of abnormal irradiance and to obtain some known neutral points in the sky especially useful for polarized remote sensing of the atmosphere.

Scattering of terahertz (THz) radiation is considered to be a significant problem for many applications, since it has often multiple character and is followed by interference effects. We present the computational results of THz propagation through scattering layers, obtained by using Monte Carlo simulation scheme of radiation transfer. Dense package of particles is considered for enhancing the influence of structure factor. Interference effects are shown to play an essential role for scattering phase function and scattering angular distribution of THz radiance.

We offer the generalization of the vectorial small angle modification of the spherical harmonics method (VMSH) for an arbitrary angle and polarization sate of irradiance of a slab. Non diagonal elements of an aerosol scattering matrix were admitted. The smooth addition part for the VMSH is given. Thus we obtain a complete and accelerated solution of the vectorial radiative transfer equation.

In this paper the problem of the light reflection from a turbid medium slab is considered. The method of the single scattering separation on a scattering leading to the radiative propagation direction veering concerning a surface normal, and on a scattering not leading to that is offered. The reflectance factor is represented as a series on the scattering multiplicities with a single change of the direction. For each multiplicity the precise linear integro-differential equation with homogeneous boundary conditions is arrived. The application of the discrete ordinates method brings to the linear matrix equations. The solution of these equations without using the small angle approximation in the form of the matrix exponential curves is obtained. The application range of the quasi-single scattering approximation is uniquely determined.

The analytical expression for solving the radiative transfer problem of the arbitrary polarized light in scattering and absorption media in small angle approximation for aerosol scattering matrix is offered. The comparison of the results obtained by using this analytical formula and by the direct matrix computation is realized.

The problem of the propagation of narrow radiation beams in a scattering medium is considered. The previously formulated small-angle approximation solution accounting for the path length spread is further developed. The numerical scheme for practical calculations is implemented, and the simulation results are presented and discussed. Applicability of the new solution to certain problems of optical communications and data transfer techniques is shown.

The problem of narrow-beam propagation in a medium with highly anisotropic scattering is considered within radiative transfer theory. A novel solution of the radiative transfer equation within small-angle approximation, accounting for the path length spread, is presented. A new stable numerical algorithm for the simulation of a narrow beam is developed and applied to a narrow beam in a two-dimensional scattering medium. A truly three-dimensional version of the proposed approach is formulated.