Localized Conical Edge Modes of Higher Orders in Photonic Liquid Crystals

Vladimir A Belyakov,Sergei V Semenov

doi:10.3390/cryst9100542

Abstract

Most studies of the localized edge (EM) and defect (DM) modes in cholesteric liquid crystals (CLC) are related to the localized modes in a collinear geometry, i.e., for the case of light propagation along the spiral axis. It is due to the fact that all photonic effects in CLC are most pronounced just for a collinear geometry, and also partially due to the fact that a simple exact analytic solution of the Maxwell equations is known for a collinear geometry, whereas for a non-collinear geometry, there is no exact analytic solution of the Maxwell equations and a theoretical description of the experimental data becomes more complicated. It is why in papers related to the localized modes in CLC for a non-collinear geometry and observing phenomena similar to the case of a collinear geometry, their interpretation is not so clear. Recently, an analytical theory of the conical modes (CEM) related to a first order of light diffraction was developed in the framework of the two-wave dynamic diffraction theory approximation ensuring the results accuracy of order of δ, the CLC dielectric anisotropy. The corresponding experimental results are reasonably well described by this theory, however, some numerical problems related to the CEM polarization properties remain. In the present paper, an analytical theory of a second order diffraction CEM is presented with results that are qualitatively similar to the results for a first order diffraction order CEM and have the accuracy of order of δ2, i.e., practically exact. In particular, second order diffraction CEM polarization properties are related to the linear σ and π polarizations. The known experimental results on the CEM are discussed and optimal conditions for the second order diffraction CEM observations are formulated.

Highlights

There have been rather intense experimental and theoretical activities in the field of optical localized conical edge modes (CEMs) in chiral liquid crystals (CLC)
There is a small parameter for CLC, namely, the local dielectric anisotropy δ that allows to apply an approximate analytic approach and the two-wave dynamical diffraction theory, for the theoretical description of CEMs [6]
The above analytic description of the second order CEM is approximate, for many cases, it may be regarded as exact from a practical point of view, because the accuracy of results is determined by a small parameter δ2

Summary

Introduction

There have been rather intense experimental and theoretical activities in the field of optical localized conical edge modes (CEMs) in chiral liquid crystals (CLC). There is a small parameter for CLC, namely, the local dielectric anisotropy δ that allows to apply an approximate analytic approach and the two-wave dynamical diffraction theory, for the theoretical description of CEMs [6]. Such an approach was recently applied to the first order CEM in CLC [7]. The corresponding EMs and DMs in chiral liquid crystals, and more general, in spiral media, are very similar to the corresponding modes in one-dimensional scalar periodic structures They reveal abnormal reflection and transmission outside (for EMs) and inside (for DMs) the forbidden band gap [1,2] and distributed feedback (DFB) lasing at a low lasing threshold [3]. Direction and the frequency of the polarization vectors ni and nk and other parameters in (1) have to be found from the solution of the equation

CLC Eigen Waves at Conditions of the Second Order Diffraction

Absorbing CLC

Amplifying LC

Conclusions