General application of the genetic algorithm to the estimation of the Liquid Crystal Director in PA-LC devices

. The study of liquid crystal (LC) director distribution is an important area of research in materials science and technology. Parallel-aligned liquid crystal (PA-LC) devices have been extensively studied due to their applications in liquid crystal displays, optical devices, and sensors. Estimating the LC director distribution is a critical step in designing and optimising PA-LC devices. This work shows the results derived from applying novel optimisation techniques to estimate the liquid crystal (LC) director distribution in parallel-aligned liquid crystal (PA-LC) devices. Moreover, the genetic algorithm (GA) has been applied and compared with the minimisation of the Frank-Oseen free energy through the Euler-Lagrange equations. The GA is a stochastic optimisation technique that can e ff ectively explore the search space and find the global optimum. Overall, this study’s results demonstrate the GA’s e ff ectiveness in estimating the LC director distribution in PA-LC devices. This approach can improve the performance and design of liquid crystal displays, optical devices, and sensors. Furthermore, it can be extended to other fields where the optimisation of complex systems is required. Further research is needed to optimise the GA parameters and to explore its potential in other applications.


Introduction
Liquid crystal (LC)-based devices are widely used because they offer continuous modulation of the phase with high spatial resolutions.Spatial light modulators (SLMs) can modify the amplitude and phase of a light field by controlling the orientation of the LC inside the device [1][2][3].An electric field is applied to control the distribution of nematic liquid crystals, which triggers a change in the index of the material.Hence, the phase and polarisation state of light passing through.A cover glass confines the liquid crystal layer with a transparent conductive coating and another substrate layer with patterned electrodes.For liquid crystal on silicon (LCoS) devices, this substrate layer contains the electronics and logic needed to control the voltage in each pixel individually.In recent years, researchers have focused on analysing pixel crosstalk's effects in liquid crystal-based SLMs.This phenomenon is based on the non-ideal response across the pixel area, which depends on the state of its neighbours.Crosstalk between neighbouring pixels smooths out the phase at the border between two pixels, resulting in significant effects for patterns with strong variations or high spatial frequency modulation patterns, such as a "sawtooth" blazed grating.Therefore, accurately estimating the orientation of the liquid crystals, described by the director n, a unit vector pointing in the direction of the long molecule axis, is crucial for analysing the influence of these nonlinear behaviours in SLMs.One of the most conventional approaches for estimating n is to seek the director configuration that minimises the total free energy [1,4].Recently, the genetic algorithm has been applied to LC orientation by considering simplifications and assumptions specific to each structure under study [5].In this work, the GA method has been used without constraints to have a model able to analyse different types of LC-based devices.

Modeling the director distribution of the liquid crystal
The director configuration that minimises the total free energy is obtained from the Frank-Oseen free energy density defined as where K ii are the elastic constants, E the electric field, and ϵ ϵ ϵ the dielectric tensor defined by Eq. ( 3) in [1].The minimal energy solution for the director distribution fulfils the Euler-Lagrange equations fully defined in Eq. ( 5) in [1].This approach implies a complex derivation and an iterative process that must be reformulated if some term in f changes.This situation can happen if specific anchoring conditions are needed or if holographic-polymer dispersed liquid crystal (H-PDLC) elements are analysed following this procedure [6].

Genetic algorithm
The GA is a search procedure based on evolutionary theory capable of finding solutions to complex and nonanalytical problems.An initial random population is typically chosen, which is used to generate another population subject to the rules of DNA replication.The algorithm exits when the generation of a new population no longer results in an improvement in the fitness parameter of individuals [7,8].The GA is available on several platforms; in this case, we are considering its MATLAB implementation.

Results
To validate the approach presented here, we simulated an LC infinite cell.Specifically, we used a cell with a small pretilt and a parallel-aligned configuration, with the director lying in the xz-plane [1,9].The experimental setup is fully defined in Fig. 1 in [10].Fig. 1 shows in solid line the results obtained through the minimisation of the free energy through Euler-Lagrange equations [1].The discrete set of points represents the results obtained through GA.Fig. 2 shows a more complex scenario where two finite pixels with different voltages are considered.Both the director and voltage distribution is shown in Fig. 2(b), whereas in Fig. 2(a), the phase retardance for this setup is represented.The results are consistent with those shown in Fig. 6 in [1].

Conclusions
This work presents the results of applying GA to estimate the director distribution.We used a general formalism of the functional to minimise the free energy instead of a specific simplified functional for each cell type, as seen in other contributions like in [5].Our results were validated, demonstrating the accuracy and potential of this approach.For future work, the authors plan to improve the performance of the GA method.

Figure 2 .
Figure 2. Binary grating oriented along the x-direction.Blue arrows depict the director configuration, while the electric field is represented by the equipotential lines of the electric potential and electric field lines (red lines with arrowheads).