Advances in Condensed Matter Physics

Volume 2019, Article ID 1539865, 11 pages

https://doi.org/10.1155/2019/1539865

## Waveguide Propagation of Light in Polymer Porous Films Filled with Nematic Liquid Crystals

^{1}Saint Petersburg National Research University of Information Technologies, Mechanics and Optics (ITMO University), Kronverkskiy Prospekt 49, 197101 Saint Petersburg, Russia^{2}MIREA - Russian Technological University, Vernadskogo Ave., 78, Moscow 119454, Russia

Correspondence should be addressed to D. V. Shmeliova; ur.liam@avoilemhs

Received 5 October 2018; Accepted 13 January 2019; Published 13 February 2019

Academic Editor: Charles Rosenblatt

Copyright © 2019 A. D. Kiselev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We theoretically analyze the waveguide regime of light propagation in a cylindrical pore of a polymer matrix filled with liquid crystals assuming that the effective radial optical anisotropy is biaxial. From numerical analysis of the dispersion relations, the waveguide modes are found to be sensitive to the field-induced changes of the anisotropy. The electrooptic properties of the polymer porous polyethylene terephthalate (PET) films filled with the nematic liquid crystal 5CB are studied experimentally and the experimental results are compared with the results of the theoretical investigation.

#### 1. Introduction

Liquid crystals (LCs) are known to be of primary importance in the modern display industry. A unique combination of physical properties of LCs provides a means to control light propagation through micron-sized LC layers using electric fields at low operating voltages and small energy consumption. The key feature of liquid crystal materials that underlies excellent electrooptical characteristics of LCs and makes them promising for photonic applications is the high sensitivity of LC orientational structures to both external fields and conditions at bounding surfaces. In particular, the use of LC in microstructured waveguides [1, 2] and waveguides based on photonic crystals [3, 4] is mainly determined by this feature providing an efficient tool to adjust optical characteristics of waveguide photonic devices by means of the external (electric) field.

Utilization of LCs in nondisplay applications such as photonic devices [5] and a variety of sensors [6, 7] faces a number of challenges. These include a slow response time of orientational transformations for the devices of THz photonics based on thick layers of LC [8], high operating voltages needed to tune propagation of light in photonics fibers filled with LC [9], and relatively low sensitivity of planar LC layers to impurities in biosensensing applications [7]. Some of these problems can be solved by usage of composite materials, like PDLC, LC emulsions, and porous films filled with LC.

In this paper the electrooptical properties of the porous polymer films filled with nematic liquid crystals will be our primary concern. Recently such type of composite material—the polymer polyethylene terephthalate (PET) film—with normally oriented open pores of submicron and micron sizes, filled with LC, was experimentally studied in [10, 11]. It was shown that such systems are very promising for applications in photonic waveguide and terahertz (THz) devices [11, 12].

In particular, we have observed pronounced changes of light intensity transmitted through the film induced by applying low frequency ac electric field. Similarly, modulation of the light intensity can also be caused by heating of the sample via absorption of blue light by the azo-dye layer adsorbed on the internal surfaces of the pores. The hypothesis put forward in [11] to explain the experimental results assumes that breaking of the waveguide regime of light propagation is responsible for the light modulation.

These results motivate theoretical considerations of this paper that aimed to examine the waveguide regime of light propagation in cylindrical LC pores. In our calculations, we shall treat the LC material as an effective medium with variable anisotropy embedded in an isotropic environment. Though similar approach was previously used in [2], it was not applied to analyze the waveguide regime. Rigorous analysis of the problem based on numerical solution of Maxwell’s equations [13] showed that waveguide propagation of light in a cylindrical cavity surrounded by isotropic media is crucially influenced by the orientational structure formed inside the pore [14–16]. On the other hand, the type of the orientational configuration is governed by a number of parameters such as the Frank elastic constants, the surface anchoring strengths and orientation of the easy axes, and the diameter of LC pores [17–21]. The paper is organized as follows.

In Section 2 we present the details of our theoretical analysis and systematically study the effects of effective LC anisotropy on the waveguide regime of light propagation. In Section 3, we describe the experimental results obtained for the porous PET films with micron and submicron pores filled with LC 5CB. In our discussion of these results we, in particular, demonstrate a qualitative agreement between the experimental data and the theoretical predictions. Concluding remarks are given in Section 4.

#### 2. Theory

##### 2.1. Basic Equations

We begin with the Maxwell equations for a monochromatic electromagnetic wavewhere is the free space wavenumber and is the dielectric tensor, which propagate along the axis of a cylindrical waveguide (the axis). Assuming that the electromagnetic field can be written in the factorized form , we shall use the cylindrical coordinate system and separate out the longitudinal and transverse parts of both the electromagnetic field and the gradient operator as follows:where and . With the help of the relation , we obtain the transverse part of Maxwell’s equations (1) in the following form:In what follows, we, following the results of the paper [2], restrict ourselves to the case of effective radial biaxial anisotropy characterized by the constitutive relations of the following form:Note that the case of radial anisotropy was also considered in the theory of light scattering by optically anisotropic cylindrical particles developed in [22, 23].

From equations (4) it is not difficult to express the transverse components of the electromagnetic field , in terms of the longitudinal components , as follows:For convenience, we shall introduce the dimensionless parametersand write down the expressions for the transverse components in the explicit form:

Equations for the longitudinal components can be obtained using the divergence relations: (these follow from Maxwell’s equations (1)), which can be conveniently recast into the following form:After eliminating the transverse components from the relations (6)–(7) with the help of Eq. (13), we havewhere the operators are given byFor the Fourier harmonics of the longitudinal componentswhere is the azimuthal number enumerating the harmonics, we derive the following system of equations:where , and .

##### 2.2. Boundary Conditions and Dispersion Relations

For the special case of a zero mode with , the solution of the system (20) can be readily found and is given byIn general case, we introduce the matrix notationsand will search for the solutions of the system (20) that are regular at (on the waveguide axis) in the form of a power series:Substitution of the expansion (23) into Eq. (20) gives the equation for the coefficient and the recurrence relations for :The values of the parameter can be found from the singularity condition for the matrix :giving two positive values of the parameter that define two solutions which are regular at . Note that, similar to the case of Bessel functions of the first kind , we can use the expression for the determinant of the matrix (at ):to prove convergence of the power series (23), representing an entire function.

The obtained solution can be used not only for analyzing the waveguide regime, but also for describing the light scattering by radially anisotropic scatterers without recourse to the simplifying assumptions such as used in the literature [22, 23].

Now, following [2], we concentrate on the important special case with . In this case we havewhere .

Outside the waveguide at , the solution that exponentially decreases at infinity, , is given bywhere is the modified Bessel function [24] and is the refractive index of the ambient medium.

From the continuity conditions for the tangential components of the electromagnetic field at the waveguide boundarywe havewhere , and the matrix is given byDispersion relations for the harmonics of the electromagnetic field, linking and , can be deduced from the singularity condition for the matrix (32):

##### 2.3. Numerical Results

We begin our analysis with the case of waveguide modes with the zero azimuthal number: . In this case, the dispersion relations are simplified and reduced to the relations for two types of the waves: waves with and waves with The roots of Eqs. (35) and (36) are located between the corresponding zeros of the Bessel function and will be numbered in ascending order by the index . In our subsequent calculations, we shall use the estimates taken from the paper [11]: the pores of radius that ranged from nm to nm are filled with the nematic liquid crystal 5CB characterized by the refractive indices and (), the effective refractive index of the polymer matrix is (), and the wavelength of the laser radiation is nm. Clearly, in this case, we have .

Figure 1 shows how liquid crystal anisotropy (the value of the longitudinal refractive index ) affects the dispersion curves for the waveguide modes and . It can be seen that the curves for the modes, which are computed for the planar director configuration with the effective refractive index , are independent of (see Eq. (35)), whereas the difference between the dispersion curves for the and modes becomes more pronounced as the difference between the refractive indices and increases. The radial distributions of nonvanishing components of the electromagnetic fields for the waveguide modes and are presented in Figures 2 and 3, respectively.