Angular structure of radiation scattered by monolayer of polydisperse droplets of nematic liquid crystal

We considered light scattering by a polydisperse ensemble of droplets of a nematic liquid crystal. To model light scattering by a monolayer of polymer-dispersed spherical droplets of a nematic liquid crystal with cylindrical symmetry of its internal structure, we proposed a semianalytical modeling method. The method is based on interference approximation of the theory of multiple wave scattering, anomalous diffraction approximation, and effective-medium approximation. The method takes into account cooperative optical effects in concentrated, partially ordered layers and can be used to analyze the small-angle structure of the intensity of scattered radiation in relation to the concentration, size, polydispersity of liquid crystal droplets, orientation of their optical axes, and refractive indices of the liquid crystal and polymer. The obtained relations can be applied to solving direct and inverse problems of light scattering in composite liquid crystal materials using data of polarization measurements. We present graphical results of solving the direct problem for components of the polarization vector of scattered wave. These results illustrate the formation of an angular structure for monolayers with a high concentration of polydisperse droplets of the liquid crystal in the range of small scattering angles (0 < θ _s ≤ 8°).     

Light scattering by a nematic liquid crystal droplet: Wentzel–Kramers–Brillouin approximation

Light scattering by an optically anisotropic liquid crystal (LC) droplet of a nematic in an isotropic polymer matrix is considered in the Wentzel–Kramers–Brillouin (WKB) approximation. General relations are obtained for elements of the amplitude matrix of light scattering by a droplet of arbitrary shape and for the structure of the director field. Analytic expressions for the amplitude matrices are derived for spherical LC droplets with a uniformly oriented structure of local optical axes for strictly forward and strictly backward scattering. The efficiency factors of extinction and backward scattering for a spherical nonabsorbing LC droplet depending on the LC optical anisotropy, refractive index of the polymer, illumination conditions, and orientation of the optical axis of the droplet are analyzed. Verification of the obtained solutions has been performed.     

Study on the algorithm of computational ghost imaging based on discrete fourier transform measurement matrix

We have developed a model and realized an algorithm for the calculation of the coefficient of coherent (direct) transmission of light through a layer of liquid crystal (LC) droplets in a polymer matrix. The model is based on the Hulst anomalous diffraction approximation for describing the scattering by an individual particle and the Foldy-Twersky approximation for a coherent field. It allows one to investigate polymer dispersed LC (PDLC) materials with homogeneous and inhomogeneous interphase surface anchoring on the droplet surface. In order to calculate the configuration of the field of the local director in the droplet, the relaxation method of solving the problem of minimization of the free energy volume density has been used. We have verified the model by comparison with experiment under the inverse regime of the ionic modification of the LC-polymer interphase boundary. The model makes it possible to solve problems of optimization of the optical response of PDLC films in relation to their thickness and optical characteristics of the polymer matrix, sizes, polydispersity, concentration, and anisometry parameters of droplets. Based on this model, we have proposed a technique for estimating the size of LC droplets from the data on the dependence of the transmission coefficient on the applied voltage.     

A method for calculating the light extinction coefficients of a polymer layer with small nematic liquid-crystal droplets

A method is proposed for analyzing radiation extinction coefficients of polymer films containing small nematic liquid-crystal droplets with cylindrical symmetry. This method is based on the Rayleigh-Gans approximation and the effective scattering matrix. The orientational ordering of both liquid-crystal molecules in droplets and the droplets in a layer is described by multilevel order parameters. The extinction of radiation in polymer films is analyzed for the cases of oblique and normal incidence of light. Simple and convenient expressions are derived for calculating and estimating the extinction coefficients for the case of normal incidence of light on a layer containing spherical droplets. The error of the method is estimated.     

Optical equivalence of isotropic ensembles of ellipsoidal particles in the Rayleigh-Gans-Debye and anomalous diffraction approximations and its consequences

Light scattering by isotropic ensembles of ellipsoidal particles is considered in the Rayleigh-Gans-Debye approximation. It is proved that randomly oriented ellipsoidal particles are optically equivalent to polydisperse randomly oriented spheroidal particles and polydisperse spherical particles. Density functions of the shape and size distributions for equivalent ensembles of spheroidal and spherical particles are presented. In the anomalous diffraction approximation, equivalent ensembles of particles are shown to also have equal extinction, scattering, and absorption coefficients. Consequences of optical equivalence are considered. The results are illustrated by numerical calculations of the angular dependence of the scattering phase function using the T-matrix method and the Mie theory.     

Small-angle light scattering from polymer-dispersed liquid-crystal films

A method is developed for modeling and computing the angular distribution of light scattered forward from a single-layer polymer-dispersed liquid-crystal (PDLC) film. The method is based on effective-medium approximation, anomalous diffraction approximation, and far-field single-scattering approximation. The angular distribution of forward-scattered light is analyzed for PDLC films with droplet size larger than the optical wavelength. The method can be used to study field-and temperature-induced phase transitions in LC droplets with cylindrical symmetry by measuring polarized scattered light intensity.     

Coherent backscattering in nematic liquid crystals in an external magnetic field

We consider multiple light scattering in a nematic liquid crystal. Using the Monte Carlo method, we calculate for the first time the effect of a magnetic field on the shape of the peak of coherent backscattering taking into account the long-range action of fluctuations of the orientational order and anisotropy of the scattering length. For a small number of initial and final scattering events, we take into account the ordinary mode of light, which is weakly scattered in a nematic liquid crystal (NLC), whereas a strongly scattered extraordinary mode is taken into account for all scattering events. For simplicity, we use a single-constant approximation of the NLC elastic moduli. We show that the angular shape of the peak of coherent backscattering remains nearly unchanged, whereas the magnetic field and the scattering phase function vary by several orders of magnitude.     

Method for Describing the Angular Distribution of Optical Radiation Scattered by a Monolayer of Ordered Spherical Particles (Normal Illumination)

We have developed a method for describing the angular distribution of intensity of radiation scattered by a monolayer of homogeneous spatially ordered monodisperse spherical particles normally illuminated by a plane circularly polarized electromagnetic wave. The method is based on the quasicrystalline approximation (QCA) of the theory of multiple scattering of waves (TMSW) using the multipole expansion of fields and the tensor Green function in vectorial spherical wavefunctions. The method is applied for analyzing the characteristics of radiation scattered by a partially ordered monolayer and a monolayer with a nonideal lattice. The results of calculations are compared with the available experimental data on the position of the first-order diffraction peak on the angular and spectral dependences of the intensity of radiation scattered by a closely packed monolayer with a nonideal triangular lattice of SiO₂ particles. Good conformity of the results has been established.     

Phase transition in ellipsoidal droplets of nematic liquid crystals

We developed a simple method for the calculation of the director field distribution in the droplets of nematic liquid crystals (LCs) of any shape, allowing for the interaction of LCs with the droplet surface, as well as the influence of constant electric field. In contrast to different approaches, the approach that is developed in the present paper does not require any simplifying suppositions about the structure of the LC director field. The elastic-continuum theory is used, complemented with the possibility of consideration of point and linear defects. Calculations are performed using the Monte Carlo method on a simple grid. The triangulation technique is used to take the boundary conditions of droplets of a complex shape into account. The developed approach can be used for investigation of the properties of polymer-dispersed liquid crystals (PDLCs). The topological phase transitions in the nematic LC 4-cyano-4′-pentyl-biphenyl (5CB) in spherical and ellipsoidal droplets are investigated in this paper.     

Scattering and Transmission by a Monolayer of Spheres: A Study on the Monolayer Structure

The angular distribution of scattered light and the transmission of radiation through a monolayer of monodisperse spherical particles at variable particle concentration are studied. The scattering of light by a single particle is calculated with the classical Lorentz‐Mie theory. For a monolayer of mono‐dispersed spherical particles, if the monolayer density is less than 0.5 and the particle size parameter is larger than 5, effects from multiple scattering and dependent scattering can be excluded so that only steric interactions are considered. It is found that the scattering pattern, especially in the forward and backward directions, and the transmission are strongly dependent on the monolayer density.     

Interference effects in backscattering of light by a layer of a discrete random medium

Multiple backscattering of light by a layer of a discrete random medium is considered. A brief derivation of equations for describing the coherent and incoherent components of scattered light is presented. These equations are solved numerically in the approximation of doubled scattering of light by a semi-infinite medium of spherical scatterers having a size comparable with the wavelength in order to study the effect of the properties of particles on the angular dependence of interference effects. Calculations show that the half-width of the interference peak decreases upon an increase in lateral scattering by particles and that the degree of polarization has a complex angular dependence on the properties of the particles. For an optically thin layer of the medium, the relations defining the interference peak half-width and the scattering angle upon extreme linear polarization as functions of the effective refractive index are given.     

The effect of multiple scattering in disperse media on polarization characteristics of scattered light

The effect of scattering of different multiplicity on polarization characteristics of scattered light is studied by the Monte Carlo computer simulation technique. The scattering multiplicity distribution versus the direction of scattering and dimensions of the scattering system is obtained for monodisperse systems of spherical particles of different size. The angular dependences of the elements of the light-scattering matrix (LSM) are calculated. It is shown that in a system of spherical particles, specific features of the LSM structure associated with multiple scattering have much in common with similar features of the LSM in systems of nonspherical particles under conditions of single scattering. The angular dependences of the degree of depolarization of the scattered light are studied.     

Anisotropic weak localization of light

We have observed angular anisotropy in weak localization of light from highly scattering, orientationally ordered, nematic liquid crystals. This demonstration of angular anisotropy in a multiple-scattering interference phenomenon was facilitated by a light scattering instrument with extraordinary angular resolution. The measured anisotropies were consistent with a simple model of coherent backscattering generalized for propagation-direction dependent mean free paths.     

Strain-induced matrix and droplets anisotropic deformation in liquid crystalline cellulose dispersed liquid crystal films

The present work deals with the preparation of new liquid crystalline cellulose dispersed liquid crystal films (100 μm) using a shearing casting technique. The matrix of the films presents the so-called band texture perpendicular to the shear direction. The nematic low molecular weight liquid crystal is encapsulated in micron and submicron size ellipsoids. The ratio between the lengths of the main axis and the short axis is around 1.23. The main axis is oriented, on average, 28° away from the shear direction. The evolution of the band texture and of the ellipsoidal liquid crystal droplets is investigated by polarizing optical microscopy and light scattering techniques as a function of the strain imposed along and perpendicular to the shear direction. Stretch along shear with strain equal to 0.8 seems to have no effect on the banded structure of the matrix, while the nematic liquid crystal ellipsoids slightly orient the main axis to the stress direction and their shape anisotropy increases by a factor of 2. Deformation in the direction transverse to the shear direction induces a deep change of the polymeric matrix and, at the end, a fibrillar structure is found. The nematic ellipsoids rotate their main axis and align along the stretch direction. Their shape anisotropy evolves from a value of 1.23 to 1 for intermediate deformations with strain equal to 0.5, to a final value of 5 with strain equal to 0.8, in the perpendicular direction.     

Light attenuation by disperse layers with a high concentration of oriented anisotropic spherical particles

An analysis is made of the applicability of Beer’s law to the calculation of coefficients of directed (coherent) transmittance of light-scattering media formed by oriented anisotropic spherical liquid-crystal droplets. It is shown on the basis of comparison with experimental data that the application of Beer’s law to a layer with a high concentration of anisotropic particles leads to large errors in the calculation of transmittance. The interference approximation, taking into account the interference of waves scattered by separate particles, leads to better agreement with the experiment than Beer’s law, which assumes a linear relation between the attenuation factor of a disperse medium and the particle concentration.     

Polarization of light transmitted through a polymer film with nanosized droplets of liquid crystal

A method is developed for analyzing the state of polarization of a plane wave transmitted through a polymer-dispersed liquid-crystal (PDLC) film with nanosized liquid-crystal (LC) droplets. This method is based on the anisotropic-dipole approximation for describing scattering by a separate droplet and on the Foldy-Twersky approximation for describing propagation of light in a film. Equations are obtained that relate the ellipsometric parameters of coherent (direct) light transmitted through a PDLC film to the order parameters that characterize the morphological and structural properties of the film. Elliptic and circular polarizations and the rotation of the plane of polarization of a wave transmitted through a film are investigated under the normal illumination of the PDLC film by a linearly polarized plane wave. The order parameters of the PDLC film are determined as a function of a control field under the transition from a partially ordered structure of optical axes of LC droplets to a homeotropic structure.     

Coherent transmission and reflection of a two-dimensional planar photonic crystal

A method for modeling the radial distribution function for particles of a two-dimensional planar photonic crystal in the form of a monolayer of spatially ordered monodisperse spherical particles is proposed. The coherent transmission and reflection coefficients for layers under normal illumination are calculated in the quasi-crystalline approximation of the multiple wave scattering theory. The dependence of the coherent transmission and reflection of the layer on the degree of ordering of the spherical particles is investigated. The influence of the long-range order on the coherent transmission and reflection coefficients for layers with triangular, square, and hexagonal lattices is estimated. Monolayers of weakly absorbing dielectric and strongly absorbing metallic particles are considered.     

Coherent transmittance of a polymer dispersed liquid crystal film in a strong field: Effect of correlation and polydispersity of droplets

A method for computing the coherent transmittance of a film of polymer dispersed nematic liquid crystals aligned by an external field in the case of an oblique incidence of light is developed, and a theoretical analysis of this quantity is performed. The effects of close packing are considered in terms of the interference approximation of the theory of multiple wave scattering. The correlation arising in the spatial distribution of polydisperse droplets is taken into account by introducing partial distribution functions. It is shown that an increase in the concentration of liquid crystal droplets is accompanied by an increase in the coherent transmittance of the film.     

Propagation of light through a monolayer of particles: Analysis of phase and coherent transmittance

Phase variations of a wave transmitted through a monolayer of spherical scatterers are studied as functions of size, optical constants, and particle concentration for light incident normally to the surface of the layer. The analysis is performed in the quasi-crystal approximation of the theory of multiple scattering of waves and in the single scattering approximation. The results obtained allow one to estimate the limits of applicability of the single scattering approximation to layers with partial ordering of scatterers in analysis of the transmitted wave phase. The variations of the phase of the wave in the range of the parameters where the coherent component of the transmitted beam exhibits quenching are studied. It is shown that small variations in the refractive index of the particles may give rise to strong variations of the phase. This effect can be used for phase modulation of light beams, e.g., in liquid crystal films controlled by an electric (or magnetic) field.     

Solar Radiation Scattering by Aerosol Axially Symmetric Objects

We have obtained analytical expressions for the scattering coefficient and the angular distribution of radiation for the aerosol train (tail) modeled by a homogeneous cylinder with given scattering coefficient and phase function of the elementary volume. The analytical formulas for the angular distribution of scattered radiation have been obtained in the single-scattering approximation. Using the Monte Carlo method, we have calculated the directional scattering coefficients of a homogeneous monodisperse tail consisting of spherical water particles with a mean radius of 1 μm for various diametrical optical thicknesses of the tail. The relations obtained can be used for passive remote sensing of aerosol air pollutions. __________ Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 72, No. 5, pp. 604–610, September–October, 2005.