# Novel PCA-Based Lower-Dimensional Remapping of the Solution Space for a Genetic Algorithm Optimization: Estimating the Director Distribution in LC-Based SLM Devices

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*Appl. Sci.*

**2024**,

*14*(21), 9950; https://doi.org/10.3390/app14219950 (registering DOI)

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Physical Model of the SLM

#### 2.2. Lower-Dimensional Remapping: PCA Analysis

#### 2.3. Genetic Algorithm

## 3. Results

^{®}Core™ i7 13700, 16 GB RAM DDR4-3200 MHz, and 512 GB PCIe NVMe. It is worth noting that the GA algorithm proposed here is currently slower than the Euler–Lagrange method. The GA-PCA-based method is in a seminal stage, whereas the Euler–Lagrange solution takes advantage of MATLAB capabilities in handling matrix operations. Under these conditions, the GA-PCA needs 146.5 s/iteration, achieving good accuracy (see Table 1) after ten iterations. The Euler–Lagrange method takes 34.7 s on average in its running time simulations for the considered convergence criteria (variations in the output smaller than ${10}^{-4}$). Even so, it is worth mentioning that the application of GA without PCA would be unfeasible due to memory and resource constraints. However, the proposed scheme, despite its current limitations, enables simulations that were previously impossible to handle, demonstrating the practical benefits of this research.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

GA | Genetic Algorithm |

PCA | Principal Component Analysis |

LC | Liquid Crystal |

LCoS | Liquid Crystal on Silicon |

SLM | Spatial Light Modulator |

TN | Twisted Nematic |

PA | Parallel Aligned |

FO | Frank–Oseen |

TN-LC | Twisted nematic Liquid Crystal |

PALCoS | Parallel Aligned Liquid Crystal on Silicon |

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**Figure 1.**(

**a**) Section example of a 3D Random Smooth Scalar Field with periodic boundary conditions in the x and y axis. Arbitrary units. Moran’s I = 0.5702. (

**b**) Cumulative Explained Variance of 5184 Principal Components of a 10,000 (20 × 24 × 12 voxels) 3D Random Smooth Scalar Field Samples set. The 99.5% of the variance is represented by dashed lines with the number of principal components required.

**Figure 3.**Numerical comparison of Genetic Algorithm for different LC-based simple systems. (

**a**) Twist angle distribution $\theta \left(z\right)$ at various applied voltages for a PA-LCoS. (

**b**) Twist angle distribution $\varphi \left(z\right)$ at various applied voltages for a TN-LCoS.

**Figure 4.**Numerical calculations for a 2-pixel period binary grating of 4 $\mathsf{\mu}$m. The external voltage applied to the electrode is 2.8 V on the top ($z={l}_{\mathrm{LC}}$) and the ground electrode is in the bottom ($z=0$). (

**a**,

**b**) represent the phase response computed by Equation (12) in [8] for horizontal (grating in x-direction) and vertical (grating in y-direction) gratings, respectively. (

**c**–

**f**) represents the visualisation of the distribution of the voltage $\mathsf{\Phi}$ (colour map) and the director orientation $\mathbf{n}$ (blue lines): (

**c**) $\widehat{K}=\widehat{x}$, y = 2 $\mathsf{\mu}$m and $xz$-plane. (

**d**) $\widehat{K}=\widehat{y}$, x = 2 $\mathsf{\mu}$m and $yz$-plane. (

**e**) $\widehat{K}=\widehat{x}$, x = 2 $\mathsf{\mu}$m and $yz$-plane. (

**f**) $\widehat{K}=\widehat{y}$, x = 2 $\mathsf{\mu}$m and $xz$-plane.

**Table 1.**Mean error between the tilt ($\theta $) and twist ($\varphi $) angle between the genetic algorithm (GA) and traditional method for minimising the Frank–Oseen elastic free energy (FO) detailed in [8].

Error Table | ||||
---|---|---|---|---|

V | Horizontal Grating | Vertical Grating | ||

$|{\mathit{\theta}}_{\mathrm{GA}}-{\mathit{\theta}}_{\mathrm{FO}}|$($\xb0$) | $|{\mathit{\varphi}}_{\mathrm{GA}}-{\mathit{\varphi}}_{\mathrm{FO}}|$($\xb0$) | $|{\mathit{\theta}}_{\mathrm{GA}}-{\mathit{\theta}}_{\mathrm{FO}}|$($\xb0$) | $|{\mathit{\varphi}}_{\mathrm{GA}}-{\mathit{\varphi}}_{\mathrm{FO}}|$($\xb0$) | |

0.0 V | 0.1596 | 0.0097 | 0.1991 | 0.0673 |

0.7 V | 0.1927 | 0.0109 | 0.3208 | 0.1249 |

1.4 V | 3.457 | 0.0876 | 2.819 | 0.6951 |

2.1 V | 4.6468 | 0.2756 | 2.2709 | 1.3073 |

2.8 V | 4.5691 | 0.4854 | 2.6296 | 1.5494 |

3.5 V | 4.2467 | 1.2012 | 3.7983 | 2.1426 |

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**MDPI and ACS Style**

Colomina-Martínez, J.; Sirvent-Verdú, J.J.; Bernabeu, A.P.; Lloret, T.; Nieto-Rodríguez, B.; Neipp, C.; Beléndez, A.; Francés, J.
Novel PCA-Based Lower-Dimensional Remapping of the Solution Space for a Genetic Algorithm Optimization: Estimating the Director Distribution in LC-Based SLM Devices. *Appl. Sci.* **2024**, *14*, 9950.
https://doi.org/10.3390/app14219950

**AMA Style**

Colomina-Martínez J, Sirvent-Verdú JJ, Bernabeu AP, Lloret T, Nieto-Rodríguez B, Neipp C, Beléndez A, Francés J.
Novel PCA-Based Lower-Dimensional Remapping of the Solution Space for a Genetic Algorithm Optimization: Estimating the Director Distribution in LC-Based SLM Devices. *Applied Sciences*. 2024; 14(21):9950.
https://doi.org/10.3390/app14219950

**Chicago/Turabian Style**

Colomina-Martínez, Jaume, Joan Josep Sirvent-Verdú, Andrés P. Bernabeu, Tomás Lloret, Belén Nieto-Rodríguez, Cristian Neipp, Augusto Beléndez, and Jorge Francés.
2024. "Novel PCA-Based Lower-Dimensional Remapping of the Solution Space for a Genetic Algorithm Optimization: Estimating the Director Distribution in LC-Based SLM Devices" *Applied Sciences* 14, no. 21: 9950.
https://doi.org/10.3390/app14219950