This review discusses the physical properties of nematic, cholesteric, and smectic liquid crystals. Molecular theories of the liquid crystal phases are discussed and the molecular field theories of the phase transitions between the various liquid crystal phases are presented. The elastic theory and hydrodynamics of liquid crystals is developed. A wide variety of phenomena in liquid crystals, including elastic distortions, disclinations, flow properties, fluctuations, light scattering, wave propagation, nuclear magnetic resonance, effects of magnetic and electric fields, electrohydrodynamics, and optical properties, is discussed.

Physics of liquid crystals

The theory and recent applications of waves in periodic structures are reviewed. Both the Floquet and coupled waves approach are analyzed in some detail. The theoretical part of the paper includes wave propagation in unbounded and bounded active or passive periodic media, wave scatterring from periodic boundaries, source radiation (dipole, Cerenkov, transition, and Smith-Purcell) in-periodic media, and pulse transmission through a periodic slab. The applications part covers the recent development in a variety of fields: distributed feedback oscillators, filters, mode convertors, couplers, second-harmonic generators, deflectors, modulators and transducers in the fields of integrated optics and integrated surface acoustics. We also review the work on insect compound eyes, mehanical structures ocean waves, pulse compressions, temperature waves, and cholestric liquid crystals. Particles interaction with crystals is briefly reviewed, especially in the case of zeolite crystals and supelattices. Recent advances in fabrication techniques for very fine gratings me also covered. Finally, speculations about future problems and development in the field of waves in periodic structures are given.

/pdf/waves-in-active-and-passive-periodic-structures-a-review-1rmm6yq97w.pdf

Waves in active and passive periodic structures: A review

By controlling the interaction of biological building blocks at the nanoscale, natural photonic nanostructures have been optimized to produce intense coloration. Inspired by such biological nanostructures, the possibility to design the visual appearance of a material by guiding the hierarchical self-assembly of its constituent components, ideally using natural materials, is an attractive route for rationally designed, sustainable manufacturing. Within the large variety of biological building blocks, cellulose nanocrystals are one of the most promising biosourced materials, primarily for their abundance, biocompatibility, and ability to readily organize into photonic structures. Here, the mechanisms underlying the formation of iridescent, vividly colored materials from colloidal liquid crystal suspensions of cellulose nanocrystals are reviewed and recent advances in structural control over the hierarchical assembly process are reported as a toolbox for the design of sophisticated optical materials.

/pdf/the-self-assembly-of-cellulose-nanocrystals-hierarchical-1zgc21j3er.pdf

The Self-Assembly of Cellulose Nanocrystals: Hierarchical Design of Visual Appearance.

A review is presented of the theory of optical properties of cholesteric liquid crystals (CLC). The diffraction nature of the unusual optical properties of CLC is exhibited in the simplest and most easily visualizable kinematic approximation of diffraction theory. A quantitative description of the optical properties is given based on the exact solution of Maxwell equations for light propagated along the optical axis of CLC and within the framework of the dynamic diffraction theory for an arbitrary direction of propagation. Considerable attention is devoted to the manifestation of the connection between the structural and optical properties of the CLC observed when light is propagated at an angle to the optical axis of the CLC and when the CLC structure is distorted by an external field. Investigations of the optical properties of absorbing CLC reveal that suppression of light absorption in the selective reflection band is analogous in its nature to the well-known Borrmann effect in x-ray diffraction. Different approaches to the optics of imperfect CLC which frequently most nearly correspond to the conditions of actual experiments are described. A theory of the optical properties of chiral smectic crystals is presented. This theory, on the whole, is analogous to that for CLC, but it predicts a number of qualitative differences in the optics of chiral smectic crystals. The theory and special features of the Vavilov-Cherenkov radiation in CLC is briefly presented. A comparison is made of the theoretical results with the experimental data and the most promising directions for theoretical and experimental research are indicated.

http://iopscience.iop.org/article/10.1070/PU1979v022n02ABEH005417/pdf

Optics of cholesteric liquid crystals

When a single complex variable χ is used to specify the ellipse of polarization of light passing through an anisotropic medium, a first-order, second-degree, ordinary differential equation (Riccati’s equation) governs the evolution of χ with distance along the direction of propagation. In this differential equation the properties of the medium are represented by the elements nij of its N-matrix, which was first introduced by Jones. Its solution χ(z,χ0) represents a trajectory in the complex plane which is traversed as the distance z is increased, starting from an initial polarization χ0 at z = 0. A stereographic projection onto a tangent sphere produces the corresponding trajectory in the more-familiar Poincare-sphere representation. The function χ(z,χ0) has been determined for propagation along (i) an arbitrary direction in a homogeneous anisotropic medium, and (ii) the helical axis of a cholesteric liquid crystal. The solution in the first case provides a unified law that leads to all the rules for the use of the Poincare sphere. For axial propagation in a cholesteric liquid crystal, it is found that two orthogonal polarizations are privileged in that the axes of their ellipses are forced to remain in alignment with the principal axes of birefringence of the molecular planes. The general solution (that satisfies the conditions of propagation) shows that the ellipse of polarization never repeats itself. As to the two parameters of the ellipse, the ellipticity is shown to be periodic with periodicity shorter than the pitch of the helical structure and the azimuth is aperiodic.

Simplified Approach to the Propagation of Polarized Light in Anisotropic Media—Application to Liquid Crystals*

Equations are derived for the propagation of electromagnetic waves along the axis of a cholesteric material, using constitutive equations based on a theoretical model for the structure of this type of material The resulting time-independent equations are a pair of coupled equations for left- and right-hand circularly polarized waves Solutions are obtained for these equations and are shown to provide good agreement with experimental observations on the circular dichroism and reflection of circularly polarized light for cholesteric materials

Gary H. Conners

Papers

Electromagnetic Wave Propagation in Cholesteric Materials