## 1. Introduction

Many applications, such as diffraction lenses, optical data storage, image capture devices and solar cell applications, amongst others, are related to holographic polymer dispersed liquid crystals (H-PDLCs). The combination of a light-sensitive monomer and liquid crystal (LC) mixture and their exposition to an interference pattern sets up H-PDLC devices. The highly irradiated zones produce a lack of monomer due to polymerisation. This effect provokes an increase of the monomer concentration in the illuminated areas, thereby producing a periodic pattern. The grating is based on a set of small LC droplets surrounded by a polymer. Many researchers have focused their efforts into improving the diffraction efficiency or the angular selectivity [

1,

2] with lower driving voltages [

3]. It is also worth noting novel applications of H-PDLC, e.g., H-PDLC with variable period [

4] and H-PDLC gratings used for autostereoscopic display [

5,

6], amongst others. In order to achieve these goals, many theoretical and numerical approaches have been developed, e.g., the application of Montemezzani’s [

7] coupled waves theory for gratings to shaped-droplets [

8], models of droplet axis reorientation [

9] and the effective medium theory [

10] applied to the analysis of reflective H-PDLC. Kubitskiy et al. [

11] have contributed significantly to the application of FDTD to H-PDLC. It is worth noting the analysis performed in [

12], wherein the LC director distribution was computed by means of the application of the Monte-Carlo method, and a two-dimensional FDTD scheme was used for obtaining the light propagation along the grating. Wang et al. [

13] and Gui et al. [

14] also contributed to the analysis of H-PDLC using FDTD method for one-dimensionally periodic structures [

13], and H-PDLC with embedded silver nanoparticles [

14], respectively. It is also interesting to mention contributions more focused on the analysis of the properties of spatial light modulators based on a nematic zero-twist liquid-crystal (NLC); e.g., [

15] computed the LC director by minimising the electric and elastic-free energies for one-dimensionally periodic structures, [

16] analysed crosstalk in liquid spatial light modulators following a rigorous study of the LC director distribution as well and [

17,

18] followed a similar approach, also considering three-dimensional simulations and FDTD numerical method. The authors applied a two-dimensional split-field (SF) FDTD method to the analysis of ellipsoidal droplets in H-PDLC gratings with random properties [

19]. In this contribution previously mentioned, the Monte-Carlo method was applied to random ellipsoidal droplets for obtaining the LC director distribution. Many parameters related to the physical packing of the droplets were considered, e.g., the density packing, grating period, upper and lower limits of the ellipsoid size and initial LC director state. The generation of a random particle and its packing is not a straightforward issue. Actually, granular packing studies are relevant to physicists, biologists and engineers [

20,

21]. The viscosities of particle suspensions are affected by the packing fraction [

22]. Collisional processes in astronomy are also analysed numerically by creating a set of fragments with different shapes (cubes, spheres, etc.) [

23]. Here, random ellipsoids were generated and allocated inside the polymerised region of the H-PDLC. It is worth noting that the packing limit for spheres has been well established at a density of

$\Omega \approx 0.64$ [

21,

24]. However, in this work lower limits are considered in order to be closer to non-homogeneous H-PDLC droplet configurations, thereby triggering light scattering.

In this work, a full tensorial three-dimensional numerical analysis was carried out using the SF-FDTD method. Maxwell’s curl equations govern the propagation of light waves. FDTD simulations solve Maxwell’s equations by approximating both time and spatial derivatives by means of central differences [

25,

26] in a discrete grid. The FDTD method is a powerful approach since it provides information of the electromagnetic waves as a function of time and space. SF-FDTD is a specified version of the standard FDTD method for the analysis of electromagnetic waves with an oblique angle of incidence along with periodic media. This method has been recently used in many applications wherein diffractive elements are analysed [

26,

27,

28]. The analysis of anisotropic media implies considering the tensorial behaviour of the permittivity tensor

$\u03f5$, and increasing the complexity of the method and the computational resources as well [

27,

28]. The finite difference method has a computational cost that grows exponentially with the grid size. Thus, considering anisotropic media and 3D full simulations can be demanding for volume gratings or periodic structures with dimensions larger than the work wavelength. Here, some acceleration strategies have been considered in order to increase the performance, thereby reducing the running time per simulations, making this approach competitive. SF-FDTD implementation was completely developed in C++ by the authors. This complete control on the software has permitted us to include new features such as parallel computing based on GPUs and CUDA. Interested readers can find more details related to the implementation of the method in [

26,

27,

28,

29,

30,

31] regarding the computational acceleration through parallel and GPU computing. It is worth noting a free implementation of the method (WOLFSIM:

https://sourceforge.net/projects/wolfsim/). The details of this implementation are provided in the outstanding works of Oh and Escuti [

26] and Miskiewicz et al. [

28].

In order to simulate H-PDLC structures, some considerations must be clarified. H-PDLC devices are based on a set of droplets filled with LC material surrounded by a polymer. The polymer is considered isotropic, so the

${\u03f5}_{ii}$ components of the tensor are well known in this case. However, the refractive index perceived by input light in the different LC droplets depends on the external control voltage and the induced orientation of the director

$\mathbf{n}$ inside the LC droplet. Therefore, it is necessary to estimate the director’s orientation as a function of the geometry, the LC characteristics and the external control voltage. The minimisation of the total free energy [

16,

32] was the formalism chosen for solving this aspect. The minimisation of the free energy considers the contributions of the deformation, surface and electric field [

30]; i.e., the solution considers the influence of

${K}_{11},{K}_{22},{K}_{33},{K}_{24}$ elastic constants, anchoring strength and external field. The impact of the elastic constants has been widely analysed. The influence of the saddle splay constant

${K}_{24}$ is not usually considered for LC-based devices such as spatial modulators. However, it has been reported that the

${K}_{24}$ term has a direct influence on the droplet structure, even in the limit of the zero anchoring strength for H-PDLC devices [

33,

34] when the surface-to-volume ratio is high [

32]. On the other hand, the influence of the

${K}_{13}$ parameter has been neglected in this work since it has been demonstrated that its anchoring strength at the LC interface is negligible [

33]. Once the director’s orientation is obtained, the components of the dielectric tensor of the LC material are computed easily through the well-known dependence between the LC material and the dielectric tensor

$\u03f5$ via

$\mathbf{n}$ [

16].

The scheme here proposed was applied to an H-PDLC sample with random droplets. After the sample is defined, the input parameters regarding physical characteristics of the LC, polymer or external voltage are considered for the minimisation of the free energy. The estimated director distribution permits to obtain the different components of the permittivity tensor, which is considered for the numerical FDTD simulation that provides the electromagnetic field in three dimensions. The analysis carried out includes different anchoring strengths, and different values for the external field applied. The influences on the diffraction efficiency of these parameters is analysed as a function of the incidence angle of the input light.

The strategy considered differs significantly from our previous work [

19]. Here, three-dimensional simulations were carried out, which were done to analyse two-dimensionally periodic structures instead of only one-dimensional periodic structures. Moreover, a rigorous formalism based on the minimisation of the free energy was used for determining the director distribution of the LC droplets, instead of considering statistical approaches such as the one considered in [

19]. This solution brings about the opportunity of modelling accurately LC-based devices in a sub-micron regime, taking into account the physical parameters of the media considered, and the external control voltage or the LC constants, for instance. The model brings about the possibility of performing an inverse procedure in order to fit physical parameters while taking into account experimental data, or predicting the behaviour of an H-PDLC sample before its production. The results herein presented show the potential of the setup since the scheme provides the diffraction efficiency as a function of the external field amplitude for a different angle of incidence and anchoring conditions.

## 3. Results

Figure 3 shows the diffraction efficiency

$\eta $ as a function of the incidence angle

${\theta}_{0}$ for different external control voltages (

$\Phi $). It can be seen how the diffraction efficiency for ± 1st orders is dramatically reduced as the voltage is increased. For the highest voltage, the LC director is aligned towards the normal layer direction, thereby producing a matching between the LC ordinary refractive index and the one of the polymer. In this extreme situation (

$\Phi $ = 200 V), the grating almost disappears, and a homogeneous dielectric layer is perceived by input light. It is worth noting that the magnitude of the external voltage is usually very high in PDLC structures [

32]. Some differences between the ±1st-orders can be produced by the normalisation (

$\left|\mathbf{n}\right|$ = 1) of the LC director during the time relaxation procedure. This normalisation can produce a preferred orientation of the LC director, inducing a small asymmetry of the grating.

In order to demonstrate the potential of this approach, the electric field for the three spatial components is represented in

Figure 4 in the Bragg’s angle (18.8

${}^{\circ}$). In all cases the input light is linear vertical polarisation (

${E}_{y}$) with an oblique angle of incidence. Since for the input voltage

$\Phi $ = 0 V the grating is present, and the Bragg condition is achieved, the mixture of the 0th and 1st order can be identified in transmission (see

Figure 4b). Due to the inner anisotropy of the LC some light is transferred into the

$\widehat{x}$ and

$\widehat{z}$ components, detailed in

Figure 4a,c.

Figure 4d–f represents the same set of fields, but in this case for the maximum control voltage

$\Phi $=200 V. Here, the spatial variation of the refractive index is very small; thus, the input light remains almost intact. Note that the amplitudes of the

$\widehat{x}$ (

Figure 4d) and

$\widehat{z}$ (

Figure 4f) light contributions are up to 10 times lower compared to the amplitudes shown in

Figure 4a,c, respectively. It is interesting to clarify that all FDTD simulations were configured in order to ensure the steady-state performing more than 15,000 time-steps. This setup provides the behaviour of the light after being inside the grating for more than 30 ps. This time interval can be reduced slightly for an oblique angle of incidence due to the Courant condition that ensures FDTD stability. However, the authors have corroborated empirically that the steady-state has been reached in all situations.

Figure 5 shows the variation of the -1st-order (

Figure 5a,d,g), 0th-order (

Figure 5b,e,h) and 1st-order (

Figure 5c,f,i). The diffraction efficiency is represented as a function of the external voltage for normal incidence and Bragg angles ± 18.8

${}^{\circ}$. The influence of the anchoring strength was also analysed by varying this parameter for each row of graphs. Actually, the first row of graphs shows the results for a weak anchoring situation (

${W}_{0}$ = 10

${}^{-5}$ J/m

${}^{2}$). The anchoring is increased in each row, from up to bottom in

Figure 5. It can be seen how the impact of the anchoring strength becomes evident for

${W}_{0}\approx {10}^{4}$ J/m

${}^{2}$ approximately. The higher the anchoring strength is, the lower the effect of higher control voltages is on the grating, thereby maintaining higher diffraction efficiencies even for the highest control voltage considered. These results demonstrate the necessity of applying huge voltages in H-PDLC devices, since it is well known that the surface and anchoring effects on the droplets are quite high [

32]. For providing more evidence of this behaviour, some LC distributions as a function of the anchoring and the external control voltages are summarised in

Figure 6. More specifically, the LC director for different anchoring strengths is shown in column order, whereas the voltage is increased for each row of graphs. The LC distribution for

Figure 6a,d,g,j, with

${W}_{0}$ = 10

${}^{-5}$ J/m

${}^{2}$, shows clearly how the LC director of the droplets gets oriented parallel to the

$\widehat{z}$-axis reasonably well even for

$\Phi $ = 132.2 V. Some differences can be identified in the surface area in the bottom droplets shown in

Figure 6g that are totally aligned in

Figure 6j. It is interesting to address that the LC director is almost not voltage influenced in the lower-mid voltage range since the parts of

Figure 6a,d are very similar.

Figure 6b,e,h,k covers the LC director distribution for

${W}_{0}$ = 10

${}^{4}$ J/m

${}^{2}$. For mid-range voltages (2nd and 3rd rows of graphs) the differences are noticeable, since the anchoring was increased, and the LC director, on boundaries, tended to be more static, even for high voltages. However,

Figure 6h,k shows that the increase of the control voltage is enough for forcing the alignment of the LC director. Finally,

Figure 6c,f,i,l covers the strong anchoring case (

${W}_{0}$ = 10

${}^{6}$ J/m

${}^{2}$). If

Figure 6c,f is compared with those related to weaker anchoring, it can be seen how the surface boundaries influence in a great manner the arrangement of the LC director in the low–mid range of the external voltage.

Figure 6l covers the case for both the highest control voltage and strongest anchoring. Here, some differences can be seen in the droplet allocated in the centre of this graph and some in the bottom. The LC director allocated inside these droplets shows a good vertical alignment induced by the influence of external voltage. However, the strong anchoring forces boundaries to remain almost intact compared to

Figure 6c,f,i.