## Abstract

Abstract: **A multigrid** computational model has been developed to assess the performance of refractive liquid crystal lenses, which is up to 40 times faster than previous techniques. Using this model, the optimum geometries producing an ideal parabolic voltage distribution were deduced for refractive liquid crystal lenses with diameters from 1 to 9 mm. The ratio of insulation thickness to lens diameter was determined to be 1:2 for small diameter lenses, tending to 1:3 for larger lenses. The model is used to propose a new method of lens operation with lower operating voltages needed to induce specific optical powers. The operating voltages are calculated for the induction of optical powers between + 1.00 D and + 3.00 D in a 3 mm diameter lens, with the speed of the simulation facilitating the optimization of the refractive index profile. We demonstrate that the relationship between additional applied voltage and optical power is approximately linear for optical powers under + 3.00 D. The versatility of the computational simulation has also been demonstrated by modeling of in-plane electrode liquid crystal devices.

©2012 Optical Society of America

## 1. Introduction

Liquid crystal lenses have shown potential for a number of display and optical applications, with recent developments in electronic ophthalmic devices [1]. A unique property of liquid crystal lenses is that they are switchable, and application of a voltage activates the lens. This is perfect for presbyopia correction, non-mechanical optical zooms and holographic applications. There are many types of liquid crystal lens design, including solid lenses [2], refractive gradient index lenses [3–8] and diffractive lenses [1,9].

Refractive Liquid Crystal Lenses (RLCLs) utilize a parabolic voltage distribution across a liquid crystal (LC) layer to induce a gradient-index lens. Higher voltages are applied at the peripheries of the lens than the centre and consequently the extraordinary refractive index (*n _{e}*) is reduced at the edges of the optical area. This results in a refractive focus along one polarisation vector. A parabolic voltage distribution can be applied using various techniques, including a curved electrode system [3,4], modal control [5] or using an insulating dielectric layer [6,7,10].

A RLCL design by Ye et al. is shown in Fig. 1 [1,9], which uses two electrodes to induce a parabolic voltage distribution across a LC layer via a glass insulation layer placed between the LC and the electrodes. Calibration of the thickness of the insulating layer is vital for RLCL operation; insufficient insulation will result in a non-parabolic voltage distribution and excessive thickness results in high operating voltages. The insulation thickness required to produce a parabolic voltage distribution can be calculated by modeling the electric potential within the device, with a ratio of 2-3:1 between lens diameter and insulation thickness reported [6]. Further optimization of the RLCL dimensions using numerical modeling can be conducted to deduce the most effective insulation thickness within this range. An issue arises from simulation times and computational resources required for high resolution numerical simulations of the electric potential. Typical simulations of 3 mm diameter RLCLs require a simulation box of 300 by 150 data points, resulting in calculation times of 1-2 hours using a standard desktop computer, which is acceptable for testing existing lenses, but is generally considered too slow to use as a design tool.

The computer model reported in this paper improves upon previous approaches using numerical techniques by utilizing a multigrid algorithm [11]. This simulation technique has two significant advantages over standard numerical solutions. Firstly, simulations are up to 40 times faster due to increased computational efficiency, allowing hundreds of simulations to be conducted daily with a typical desktop computer. Secondly, the number of multigrid iterations can be changed depending on the geometry of the device. For complex geometries, a higher resolution starting grid can be used to minimize truncation errors and hence yield more accurate calculations. For large and simple geometries, a low resolution starting grid is appropriate to reduce calculation times. With the speed and flexibility of the model one can simulate the electric potential in a range of devices with only minor modifications to the program required.

## 2. Electric field and potential calculations

The process of solving the electric potential within a LC device requires a numerical solution to Maxwell’s equations. Assuming no enclosed charge in the system, the electric potential in a cylindrically symmetric system is given as

*ϕ*denotes the electric potential, ε is the permittivity of the material(s) and x and z are coordinates shown in Fig. 2 . In terms of computational modeling, standard numerical techniques are utilized using MATLAB (R2008A, MathsWorks, Natick, USA), including a successive over-relaxation (SOR) solution and red black ordering. The multigrid algorithm reduces simulation time by initially solving the equation numerically for a coarse grid, with further iterations on progressively finer grids until an adequately accurate solution is realized [11].

## 3. Simulation of RLCLs

A RLCL has relatively simple boundary conditions, as shown in Fig. 2. The system lacks free boundaries resulting in a closed simulation box. The model assumes an isotropic permittivity; hence the average electric permittivity of the LC is used for calculations. This is an approximation; in a LC material a field induced change in the director modifies the electric permittivity in the system. However, this simplification allows quick, approximate calculations of the potential profile across the system to give a good indication of device performance.

A discretized profile of the director across the LC layer is calculated using a simplified 1-dimensional model. The extraordinary refractive index (*n _{e}*) is calculated for each electric potential simulation point across the top of the LC layer, giving an approximation of the refractive profile. In the following simulations, the dielectric and refractive properties of 4-cyano-4-pentylbiphenyl (5CB) are used for modeling the LC layer. While this 1-dimensional model neglects non-linear director field interactions, the approximations are acceptable for providing quick estimates of operating voltages and geometries rather than in-depth calculations. A typical simulation of a 3 mm diameter RLCL is shown in Fig. 3
. The electric potential in the device can be calculated using the multigrid model and the electric potential across the LC layer is deduced. The thickness of the glass and the active and control voltages have been set to produce a parabolic voltage profile.

The optimum operational geometries of RLCLs were calculated for various lens diameters. To select the optimum geometry of a RLCL, a series of comparisons were made between the potential profiles across the liquid crystal layer and an ideal parabola for a range of insulation thicknesses. To allow comparison of shape, the voltage was normalized with the minimum potential across the LC layer defined as zero and the maximum potential across the LC layer defined as one. A comparison of normalized potential curves with an ideal parabola is shown graphically in Fig. 4 . High quality parabolic voltage distributions, which correspond to superior optical quality, are shown to be clearly dependent upon the device geometry.

The relationship between the RLCL diameter and the insulation thickness for high quality parabolic voltage distributions is shown in Fig. 5(a) . Previously, the ratio of RLCL diameter to insulation thickness was found to be broadly in the range from 2:1 to 3:1 [6]. Our model allows the exact relationship of this ratio with respect to lens diameter to be specified, thus providing valuable insight into the optimization of the lens geometry required for parabolic voltage profiles. Many different simulations are needed to identify the optimum insulation thickness, so the speed of the multigrid solution is vital. We have discovered using our model that the optimum ratio of insulation thickness to lens diameter is strongly dependent on the lens geometry, taking values close to 2:1 for smaller lenses (~1-2 mm in diameter) and increasing towards 3:1 as the lens diameter increases.

Combining the electric potential simulation with a 1-dimesional director model allows us to assess and minimize the driving voltages needed to induce different positive focal powers. Our model compares the resulting refractive profile to a parabolic profile to assess the optimum driving voltages. In order to reduce the voltages, we propose a new method of operation whereby the refractive profile is controlled by both the active and control electrodes to give deviations in refractive index from an initial value of *n _{e}* in the centre of the lens, rather than varying from

*n*in the lens peripheries as used previously. Figure 5(b) shows the calculated operating voltages for a 3 mm diameter RLCL with a 50 µm 5CB LC layer giving optical powers between + 0.50 D and + 3.00 D. This new method of driving the lens reduces the operating voltages for low optical powers. Furthermore, the voltages can be tuned to give the most parabolic refractive profile, hence optimizing the optical quality. The calculation speed is again vital in optimizing the refractive profile due to the number of calculations required. In these lenses, an approximately linear relationship between the variable voltage and the additional optical power was observed, with each additional + 0.50 D corresponding to an increase of approximately 4 V. It was noted that at the peripheries of the lens the potential profile deviated from a parabola as the lens became more powerful, as shown in Fig. 6 . This is expected when a large change in

_{o}*n*is required for lenses of high optical powers, as the relationship between

_{e}*n*and potential is increasingly non-linear at high voltages.

_{e}## 4. Further applications

Due to the efficient nature of the multigrid computer model, the system was adapted to simulate in-plane electrode systems which are vital in many LC display applications such as in-plane switching devices and proposed blue phase displays. The system contains free boundaries above and below the electrodes requiring boundary conditions approximating infinity, an incredibly time consuming constraint for standard numerical simulations. Analysis was undertaken using the computer model described here to assess the smoothness of the electric field within these devices for different LC materials. Low ratios of LC to glass electric permittivities (ζ) resulted in smoother electric fields as shown in Fig. 7 , with distortions in the electric field at the interface minimized if the permittivity of the glass and LC are matched.

## 5. Conclusions

An efficient multigrid algorithm has been developed to model the electric potential in various LC devices with vastly reduced calculation times. This is an important tool for RLCL design; optimum geometries and operating voltages for RLCLs of various optical powers and sizes can be quickly screened using the model. The simulation gives a greater understanding to the previously calculated ratio of diameter to insulation thickness of between 2:1 and 3:1 for RLCLs, with additional insight into how this ratio changes with respect to RLCL diameter. The model predicts a ratio of 2:1 for small lenses of 1-2 mm in diameter, tending towards 3:1 as the lens diameter increases. Using the model we propose a new method of lens operation which minimizes operating voltages by varying the refractive profile from *n _{e}* in the centre, rather than from

*n*at the lens periphery. The speed of the multigrid simulation is vital in the optimization of the refractive profile, which maximizes the parabolicity of the refractive index profile. The model shows an approximately linear relationship between active electrode operating voltage and the optical power of the lens. Details of the limitations of RLCL performance are also shown using the model, with director profiles of high powered lenses exhibiting deviation from the parabolic distribution at the peripheries due to the increasingly non-linear relationship between

_{o}*n*and applied voltage at higher voltages. Finally we illustrate the versatility of our model with the simulation of in-plane electrodes systems.

_{e}## Acknowledgments

The authors acknowledge a Collaborative Award in Science and Industry (CASE) from the Medical Research Council (MRC) and Ultravision CLPL.

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