## Abstract

The finite-difference time-domain (FDTD) method is a powerful numerical algorithm used to directly solve Maxwell’s equations. We introduce the idea of the FDTD method and the techniques required for optical simulation of cholesteric liquid crystal (Ch-LC) devices. Bragg reflection characteristics of Ch-LC cells are investigated using the FDTD method. Three approaches to broadening the bandwidth of Bragg reflection are demonstrated: (1) using a higher birefringence LC, (2) using a cell with a gradient pitch length, and (3) using a cell with a new multidimensional structure of a Ch-LC.

©2006 Optical Society of America

## 1. Introduction

To analyze the light wave propagation in a cholesteric liquid crystal medium, the 4×4 matrix method has been developed [1]. The 4×4 matrix method is fast but difficult to use for solving two-dimensional (2-D) and three-dimensional (3-D) problems, especially when the geometry of the cell is complex. The finite-difference time-domain (FDTD) method directly solves Maxwell’s equations and can be used to accommodate multidimensional inhomogeneity of the dielectric tensor. In the present FDTD algorithm a broadband pulse wave source and total field-scattered field (TF-SF) formulations are used, and the computational space terminations are provided by a combination of the perfectly matched layer (PML) and periodic boundary conditions (PBC). In this paper, the FDTD method is used to investigate the ways to broaden the bandwidth of the Bragg reflection of a Ch-LC device.

## 2. Finite-difference time-domain method

The FDTD method is a direct method for the solution of the Maxwell curl equations. The most commonly used FDTD formulations are based on the Yee grid [2]. Figure 1 shows the general layout of the computational space used in the FDTD method. In the discretized formulation of the partial differential Maxwell equations, the time derivations of the electric and magnetic fields on a grid node can be approximated with the central difference equations [3]

where *n* stands for the time points. Note that the electric fields are updated at the “integer” time points and the magnetic fields at the “half” time points. The discretization of the equation, which yields the time derivative of the electric and magnetic fields in the Maxwell’s equations can be expressed with Eqs. (3) and (4).

Since the medium (LC) is anisotropic, ε in Eqs. (3) and (4) needs to be modified to a 3×3 tensor [4] as

and

${\epsilon}_{11}={{n}_{o}}^{2}+\left({{n}_{e}}^{2}-{{n}_{o}}^{2}\right){\mathrm{sin}}^{2}{\theta}_{c}{\mathrm{cos}}^{2}{\varphi}_{c},$

${\epsilon}_{12}={\epsilon}_{21}=\left({{n}_{e}}^{2}-{{n}_{o}}^{2}\right){\mathrm{sin}}^{2}{\theta}_{c}\mathrm{sin}{\varphi}_{c}\mathrm{cos}{\varphi}_{c},$

${\epsilon}_{13}={\epsilon}_{31}=\left({{n}_{e}}^{2}-{{n}_{o}}^{2}\right)\mathrm{sin}{\theta}_{c}\mathrm{cos}{\theta}_{c}\mathrm{cos}{\varphi}_{c},$

${\epsilon}_{22}={{n}_{o}}^{2}+\left({{n}_{e}}^{2}-{{n}_{o}}^{2}\right){\mathrm{sin}}^{2}{\theta}_{c}{\mathrm{sin}}^{2}{\varphi}_{c},$

${\epsilon}_{23}={\epsilon}_{32}=\left({{n}_{e}}^{2}-{{n}_{o}}^{2}\right)\mathrm{sin}{\theta}_{c}\mathrm{cos}{\theta}_{c}\mathrm{sin}{\varphi}_{c},$

${\epsilon}_{33}={{n}_{o}}^{2}+\left({{n}_{e}}^{2}-{{n}_{o}}^{2}\right){\mathrm{cos}}^{2}{\theta}_{c},$

where n_{e} stands for the extraordinary refractive index, n_{o} stands for the ordinary refractive index, θ _{c} is the angle between *z* axis and the director of LC (director is defined as the commonly oriented axis of rod-like LCs), and φ _{c} is the angle between the *x* axis and the projection of the director of LC on *x-y* plane.

Applied with this time updating scheme from Maxwell’s equations, we have to know the local material parameters. Once the dielectric tensor relative to the director distribution of LC at every grid point is defined, we can obtain the sequences of light propagation in any structure of LC devices. The FDTD method can be generalized to compute wave propagation in materials of arbitrary conductivity and magnetic permeability, but here we restrict the problem to nonconductive and nonmagnetic media as the prior works [5–10].

To obtain the frequency responses efficiently, we replaced the conventional continuous wave with one broadband pulse *p(t)* as the wave source in our FDTD program, which is described in Eq. (5), where *α*=3.7351×10^{29}, t_{0}=3.2725×10^{-15}, and *ω*
_{c}=3.456×10^{15}. The spectrum of the broadband pulse *p(t)* covers the wavelengths of visible light (400nm–700nm).

The broadband pulse plane wave *p(t)* is generated and introduced into the computational domain of interest based on the total field-scattered field (TF-SF) formulation. According to the TF-SF formulation, the computational domain is divided into two regions, and the known incident plane wave is introduced on the interface between the two regions. The TF-SF formulation can easily generate a time-varying wave form of arbitrary propagation direction, polarization, and duration. Compared with the hard source formulation and the conventional scattered field method, the TF-SF formulation can prevent any nonphysical reflection toward the material of interest from the position of wave source, and offer the scattered field directly.

The PML is also used to prevent outgoing waves such as scattered waves from reflecting back to the main computational space. A PML is an artificial medium having fictitious exponential conductivities that vary from the normal to the tangential electric components. Therefore any arbitrary wave leaving the main domain of interest and entering a PML is absorbed exponentially without any reflection [11,12].

As for the PBC, we should discuss it in two aspects: normal and oblique incidence. It is well understood that field patterns from period to period must replicate themselves, and this phase shift vanishes for normal incidence of illumination. Periodicity in the field patterns is then exact, and implementing the FDTD method under PBC is straightforward. The complication arises in the case of oblique incidence of illumination where the phase shift from period to period is nonzero and must be taken into account. We choose the multiple unit method to deal with such cases [3]. The multiple unit method requires more memories and computational duration, but it allows artificial wave sources, which makes the analysis of frequency responses more efficient and is more straightforward for programming.

By collecting the near-field results and transforming them to far-field, we can obtain the far-field amplitude or intensity distribution.

## 3. Simulation results and discussion

#### 3.1. One-dimensional Ch-LC

### 3.1.1. Verification of the FDTD method

To validate the accuracy of our FDTD model, we compare the resulting simulated reflective spectra of a Ch-LC, obtained through the use of the FDTD method, with that obtained by use of the 4×4 Berreman matrix method (simulated using software DIMOS) in Fig. 2, wherein n_{e}=1.616, n_{o}=1.494, P_{o}=338 nm, and cell gap d=12P_{o}, where P_{o} is the pitch length of the Ch-LC. It can be seen clearly that the simulated results of using these two methods agree well with each other. Figure 3 shows the comparison of the FDTD simulated results and the measured reflective spectra of Ref. [13], wherein the physical parameters are n_{g}=1.533, n_{e}=1.616, no=1.494, Po=338nm, thickness of glass substrates dg=0.1 mm, and cell gap d=5 µm, where ng is the refractive index of glass substrates. The incident angles in the simulation and measurement are 0° and 22°, respectively.

In our simulation, the main peaks of the reflection are seen to coincide almost exactly with the measured spectra. However, it is found that the shape of fringes of the simulated spectra without considering the effect of glass substrates is slightly different from the measurement. The formation of the side band (fringes) of the reflective spectra is dependent on the cell gap, the refractive index of LC, and the substrates. Consideration of the influence of glass substrates may take more computation time and memory, but it can significantly improve accuracy, especially for the fringes of the reflective spectra.

### 3.1.2. Central wavelength of Bragg reflection

At normal incidence the central wavelength of Bragg reflection is *λ*_{o}*=nP*_{o}
cosΘ_{clc}, where Θ_{clc} is the angle between the helical axis and the cell normal, and *n* is the average refractive index defined by *n*=(*n*_{e}
+*n*_{o}
)/2 at normal incidence, and otherwise *n*=[(${n}_{e}^{\mathit{2}}$
+2${n}_{o}^{2}$)/3]^{1/2} at oblique incidence, i.e. when the helical axis is not parallel to the cell normal [13]. The reflective spectra of planar Ch-LC with different pitch lengths are shown in Fig. 4. Note that all of the incident lights and detections in the remaining simulations of this paper are unpolarized.

For an oblique wave incident at an angle of θ_{in} into a Ch-LC cell, the central wavelength of the Bragg reflection is cos *λ*_{o}*=nP*_{o}
cos Θ, where $\Theta ={\mathrm{sin}}^{-1}\left(\frac{1}{n}\mathrm{sin}{\theta}_{\mathit{in}}\right)$, which can be obtained using Snell’s law. Figure 5 describes the blue shift of Bragg reflection resulting from the different Θ_{clc}, and Fig. 6 shows the variations of the spectra with the incident angle.

#### 3.2. Broadband Bragg reflection

The broadband Bragg reflection is an important feature required for display application, especially in bistable black and white electronic books. In this section, applied with the FDTD method, we demonstrate three methods for broadening the bandwidth of Bragg reflection of a Ch-LC: (1) a higher birefringence LC, (2) a cell with gradient pitch length, and (3) a cell with multidimensional structure of a Ch-LC.

### 3.2.1. High birefringence

The bandwidth of the Bragg reflection from a Ch-LC cell is given by Δλ_{0}=ΔnP_{0}, where Δn is the birefringence of the LC [13]. Apparently, increasing the birefringence can broaden the bandwidth of the Bragg reflection. Figure 7 shows the spectra of three Ch-LC materials with n_{o}=1.5231 and n_{e}=(a) 1.6231, (b) 1.7231, and (c) 1.9231, respectively. The cell gap d=12P_{0}, and P_{0}=340 nm.

### 3.2.2. Gradient pitch length

Using a Ch-LC with a gradient pitch length can also broaden the bandwidth of the Bragg reflection. In a planar texture, the twist angle of a Ch-LC with a uniform pitch length is defined by

and the twist angle of a Ch-LC with a gradient pitch length is

where φ _{o} is the angle between the front LC director and the x axis (see Fig. 1), *d* is the cell gap, P_{0} and P_{1} are the pitch lengths of LCs near the entrance substrate and the emerging substrate, respectively. Figure 8 shows the reflective spectra of a Ch-LC with uniform pitch length P_{0}=280 nm and gradient pitch lengths with P_{0}=280 nm and P_{1}=380 nm. It can be easily seen that the effect of gradient pitch length on the bandwidth is significant.

### 3.2.3. Two-dimensional structure of a Ch-LC

It is well known that we can broaden the bandwidth of the Bragg reflection by widening the distribution range of Θ_{clc}, the angle between the helical axis of a Ch-LC, and the cell normal through surface treatment on the substrate or doping polymer in the LC [14–17]. However, none of these methods is very effective in controlling the distribution. In this paper, we can well control the distribution of Θ_{clc} with a multidimensional Ch-LC structure design.

Figure 9 shows a general layout of a 2-D Ch-LC structure. *W* is the width of one unit of the periodic structure, d is the cell gap, and θ a is the angle between the tangential plane of the substrates and the *x-y* plane, which is a function of (*x, y, z*). Assume that the Ch-LC in this structure is in the planar state and the helical axes uniformly distribute along y axis. The parameters in our simulation are: n_{e}=1.75, n_{o}=1.5231, P_{0}=350 nm, *W*=41.4 µm, d=2.8 µm, and θ _{a,max}=40°. We will discuss its optical characteristics in the next subsections 3.2.3-1 to 3.2.3–4.

*Normal incidence* For normal incidence, the simulated far-field distribution of reflectance (λ=550 nm) is shown in Fig. 10. Since the Ch-LC in the cell is in the planar state and the helical axes uniformly distribute along y axis, the reflective light is distributed to multiple directions, which makes the reflectance at each viewing angle relatively low. To observe the real efficiency of reflection, we collect all the reflective signals and calculate the total reflectance. The total reflectance for λ=550 nm is 0.2694, which means 26.94% of the incident power (λ=550 nm) is reflected and distributed to multiple directions. Figure 11 shows the wavelength dependence of the total reflectance and Fig. 12 shows the reflection spectra in different viewing angle directions (θ_{view}=0°, 20°, 45°, and 60°) at normal incidence.

*Oblique incidence* For oblique incidence (θ _{in}=30°), the far-field distribution of reflectance (λ=550 nm) is shown in Fig. 13, and the total reflectance for λ=550 nm is 0.5366. The wavelength dependence of total reflectance for λ=550 nm and the reflection spectra in different viewing angle directions (θ _{view}=0°, 15°, 30°, and 45°) are shown in Figs. 14 and 15, respectively. It was found that when the angle of incidence increases, the effect of Bragg reflection is more pronounced and the bandwidth of reflection decreases. This phenomenon can be explained briefly by referring to Fig. 16. For normal incidence, the light beams “see” a uniform distribution of the helix axes of a Ch-LC, which makes the effect of Bragg reflection not so apparent. On the contrary, the obliquely incident lights can mainly “see” the middle section of the slope, wherein the angle of incidence at each point is relatively small and approximately equal, therefore the effect of Bragg reflection is conspicuous.

*Imitative ambient lighting conditions* Assuming there are 13 rays of broadband pulse plane waves [expressed in Eq. (5)] propagating into the cell with θ _{in}=0°, ±5°, ±10°, …, and ±30°, we can superpose the reflection at various incident angles to simulate the reflection spectra under imitative ambient lighting conditions at different viewing angles, as depicted in Fig. 17. Notably, the value of the reflectance is relative to the normal incident spectrum. It is lower than the conventional 1-D Ch-LC due to the wide range distribution of reflection. However, increasing the number of incident rays with different incident angles will raise the reflectance as well. Moreover, the superposed reflection spectra are more broadband than the conventional 1-D Ch-LC, which is a desired feature for wide viewing angle black and white reflective LC displays.

*Chromaticity variation* Assuming the light source is CIE Illuminant D65 (the black-body radiator at 6500K), we can calculate the chromaticity values of 1-D and 2-D Ch-LCs at various viewing angles, as depicted in Fig. 18. It is found that the reflective light of the 2-D Ch-LC is very close to white light, and its chromatic aberration at different viewing angles is much smaller compared with that of the conventional 1-D Ch-LC.

## 4. Conclusion

We have implemented the powerful optical simulation tool, the FDTD method, to onedimensional and multidimensional inhomogeneous structures of liquid crystal devices. In this paper, we investigated the properties of Ch-LC cells and demonstrated three different ways to broaden the bandwidth of Bragg reflection such as using a higher birefringence LC, a cell with a gradient pitch length, and a cell with a new multidimensional structure of a Ch-LC. The multidimensional structure of a Ch-LC can achieve broadband reflection and a wide viewing angle, which are desired features for black and white reflective LC displays.

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