Optical measurements of the twist constant and angle in nematic liquid crystal cells

We present a reliable optical method for measuring the twist elastic constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {K}_2$$\end{document}K2 and for assessing the total twist angle in a standard nematic twist cell. The method relies on the use of a non-standard configuration of crossed polarisers and a twist cell, which allows us to measure accurately the twist-cell parameters by reducing the degeneracy between them. Grid patching and an efficient beam propagation method are utilised in the numerical models used for fitting the experimental data. The modelling shows that the polarisation dynamics in a twist cell is non-trivial and much more complex than in a planar cell. The twist elastic constant of three commonly used liquid crystals (5CB, 6CHBT and E7) was successfully extracted from cross-polarised intensity measurements.


Alignment equations
We use two spherical coordinate systems to solve the liquid crystal (LC) alignment equations in order to avoid coordinate singularities and preserve the norm of the director.In the case of a planar cell, a single spherical coordinate system can be chosen such that coordinate singularities are avoided.However, this is not possible for a twist cell, so two non-coaxial spherical coordinate systems are needed, a technique called grid patching [1][2][3] .
The choice of Cartesian coordinates is such that the LC cell surface is in the xy-plane, while the voltage is applied along the z-axis, and light propagates through the cell along the z-axis as well.Using this Cartesian coordinate system, the two spherical coordinate systems are defined as specified in Fig. S1.Here, θ 1 and φ 1 , θ 2 and φ 2 are the polar and the azimuthal angles for spherical coordinate system 1 and 2, respectively.In this way, the director can always be defined uniquely in at least one spherical coordinate system.In Cartesian coordinates the arbitrary orientation of the polarisers and the twist cell (from Fig. 1) is defined by the following angles in the xy-plane measured from the x-axis: ζ L for the first (left) polariser, ϕ L for the director at the input (left) boundary, ϕ R for the director at the output (right) boundary and ζ R for the analyser (right polariser).The angles ϕ L and ϕ R are the left and right φ 1 azimuthal angles at the LC boundaries when no voltage is applied.These angles determine not only the angular position, but also the twist of the cell, e.g., if ϕ L = 0 • and ϕ R = 90 • , then the cell undergoes a perfect ϕ = ϕ R − ϕ L = 90 • twist.The form of the director in the first coordinate system, Fig. S1a, is n n n = [sin θ 1 cos φ 1 , sin θ 1 sin φ 1 , cos θ 1 ] and the corresponding non-dimensionalised alignment equations are (S4) Coordinate system 1 is considered to be the default system as the alignment equations (S3) and (S4) for coordinate system 2 are more complicated than the alignment equations (S1) and (S2) for coordinate system 1.Coordinate system 2 is only used when the director in coordinate system 1 is too close to the coordinate singularity.This transition is specified in the Solver description section.

Oldano's method
Oldano 5 uses Berreman's formalism 6 to describe light propagation in a uniaxial stratified medium by solving Maxwell's equations in matrix form for the case of a plane monochromatic wave with wavenumber k 0 and frequency ω incident in the xz-plane 7 dψ ψ ψ dz = ik 0 Dψ ψ ψ, (S5) where ψ ψ ψ is the Berreman vector and D is the Berreman matrix, which are defined as 6 In these expressions E x , E y , H x and H y are the x and y components of the complex electric and magnetic field respectively, ε is the permittivity tensor and m = n i sin θ i , where n i is the refractive index of the incident medium and θ i is the angle of incidence.
Two different representations are used for the electromagnetic waves: the Berreman vector ψ ψ ψ defined in equation (S6), and another four-dimensional complex vector φ φ φ defined as 7 where a 1 and a 2 are the amplitudes of the two forward propagating waves, while a −1 and a −2 are the amplitudes of the two backward propagating waves whose polarisation does not change as they propagate through a linear homogeneous medium 5 .These four waves are the eigenvectors of the Berreman matrix D and form a basis.In a continuous stratified medium the relationship between ψ ψ ψ and φ φ φ is given by where T is a 4 × 4 transformation matrix composed of the eigenvectors of the Berreman matrix D. Using the φ φ φ notation Maxwell's equations from equation (S5) become the propagation equation where D 0 is the diagonal matrix defined by D = T D 0 T −1 .Solving the propagation equation (S10) for φ φ φ in the basis of the propagation modes is simpler than using Maxwell's equations (S5) for ψ ψ ψ as deriving the boundary conditions for φ φ φ is much easier than for ψ ψ ψ.

Solver description
In this section the methods and techniques discussed so far are combined into a cross-polarised intensity (CPI) model for a twist LC cell.The transmitted intensity of a twist cell placed between polarisers is computed as a function of the applied sinusoidal voltage.The model was implemented in MATLAB® for the purpose of fitting experimentally measured CPI traces normalised between 0 and 1.
In order to compute the twist-cell CPI the following list of parameters is needed: the extraordinary refractive index n e and the ordinary refractive index n o at the laser wavelength λ , the dielectric coefficients ε ∥ and ε ⊥ , the splay, twist and bend elastic constants K 1 , K 2 and K 3 , the cell thickness d and the pretilt θ 0 .The left and right φ 1 azimuthal angles at the LC cell boundaries z = ±1 when no voltage is applied, ϕ L and ϕ R , are also required, as well as the angles corresponding to the axes of the polarisers, ζ L and ζ R .When an experimental CPI trace is fitted, the input (previously known) LC cell parameters are: n e , n o , θ θ θ 0 0 0  S1.Twist angle dependence of the twist-cell CPI fit of an in-house E7 cell.The values of the twist elastic constant and the pretilt are extracted from the fits in Fig. 7b.

Figure S1 .
Figure S1.Azimuthal and polar angle definition for the two spherical coordinate systems used for the computation of the director field time derivatives.