## Abstract

A color and polarization separating backlight can be obtained by using a surface-relief grating made of birefringent material as an outcoupling structure on top of the lightguide. A rigorous finite element diffraction model was applied to study the polarization effect of such a grating. The diffraction of plane waves by the anisotropic grating was studied for general conical incidence.

©2007 Optical Society of America

## 1. Introduction

Liquid crystal displays (LCDs) have been adapted to a broad range of applications as convenient flat panel displays. But the light efficiency of the current LCD systems is below 10%. The low efficiency is mainly because of the use of the absorption-based polarizers and color filters. The absorption by the polarizers and color filter is approximately 50% and 70%, respectively. Therefore, alternative approaches for polarization- and color-separation have to be considered. Ideally one would like to use one device which can separate both color and polarization and which is not based on absorption. By applying a diffraction grating on the lightguide in the LCDs, colors can be separated[1][2]. If the grating is made of anisotropic material, one can achieve that light of one polarization is diffracted and light of the orthogonal polarization is not[3]. The configuration of such a backlight for LCDs is shown in Fig. 1. This structure was placed in a side-lit configuration with a cold-cathode fluorescence lamp (CCFL). The grooves of the grating are filled with a birefringent material having its optical axis parallel to the grooves. The ordinary refractive index of the birefringent material is matched to the grating material, whereas its extraordinary refractive index is significantly higher. As a result, light with polarization parallel to the grooves is diffracted, whereas for the relevant angles of incidence, the light with polarization perpendicular to the grooves is totally reflected back into the lightguide.

The principle and performance of both the non-polarized and polarized version of color-separating backlights have been discussed in a previous paper [3]. In a color-separating backlight the light distribution, which contains a multitude of angles and mainly three wavelengths, is diffracted into three slightly overlapping angular distributions for the main colors. We showed experimental results of the (polarized) color-separated angular distribution of the light extracted from a structure like that of Fig. 1. We also showed simulations using a commercially available computer program [4], based on rigorous coupled wave theory for non-birefringent materials. We were able to make a good comparison between measured and simulated data in the plane perpendicular to the grooves. However, it was not possible to do simulations for other (conical) directions. Note that in general a backlight contains light of several directions and all these should be taken into account to understand its efficiency.

The aim of this paper is to discuss a method for simulating diffraction by
birefringent gratings at conical angles, applied to the problem of polarized
color-separating backlights. To study the performance of the diffraction gratings
containing birefringent material, a numerical program is required. We applied a
numerical code based on finite element method (FEM) in the simulations [5]. With our program any arbitrary conical angle of incidence
can be calculated rigorously. Results of simulations are shown in this paper for
when the incident wave vectors are in the plane perpendicular to the grooves
(*normal incidence*) and for wave vectors with conical
incidence.

## 2. Configuration

The example and numerical values that we use are similar to those of Ref.[3]. The geometry and configuration of the grating in our study is depicted in Fig. 2. A surface-relief grating with a period of 400 nm and duty cycle of 0.5 is applied onto a lightguide made of polycarbonate. The grooves of the grating are filled with a birefringent material which is liquid crystal (LC) having its optical axis along the grooves. The wavelength dependent refractive indices are shown in Table.1. The grating is rectangular and the depth of the grooves is 140 nm.

## 3. Theory

#### 3.1. Anisotropic media that are translation invariant with respect to one direction

For a configuration that consists of isotropic materials and that is
translational invariant with respect to one coordinate, say the
*y*-coordinate, it is well-known that Maxwell’s
equations are equivalent to a system of two coupled second order partial
differential equations for only the components *E _{y}*
and

*H*[6]. This is also true when the sources and fields depend on the

_{y}*y*-coordinate by a factor

*e*, for some

^{ikyy}*k*. When

_{y}*k*= 0, the two partial differential equations for

_{y}*E*and

_{y}*H*are uncoupled and therefore one can separate s- and p-polarization. For s-polarization,

_{y}*H*vanishes whereas for p-polarization

_{y}*E*is zero. When

_{y}*k*≠ 0, the two partial differential equations are coupled and the polarizations mix.

_{y}These observation apply in particular to periodic diffraction gratings that
consist of isotropic materials. In this section we want to investigate to what
extent the separation in two polarizations is still possible in the case of
gratings made of anisotropic materials. We will fist show that in an anisotropic
medium of which the dielectric permittivity
*ε*͇ is independent of the
*y*-direction, all field components can be expressed in terms of
*E _{y}* and

*H*.

_{y}The time-harmonic source-free Maxwell equations in non-magnetic anisotropic materials are,

where *ε*
_{0} and
*μ*
_{0} are the dielectric permittivity
and the magnetic permeability of vacuum respectively, and
*ε*͇ is the relative dielectric
permittivity of the anisotropic crystal which is a complex symmetric tensor of
rank 2 given by:

with respect to the cartesian coordinate system
(*x*, *y*, *z*). We write the
tensor *ε*͇ as a sum of its real and
imaginary parts,

The real and imaginary parts of *ε*͇
commute, which means that for any vector **v** we have,

This implies that the tensors *ε*′ and
*ε*″ have the same eigenvectors and
that therefore they can be diagonalized on the same (orthogonal) basis.

The configuration and the material properties are assumed to be invariant with
respect to translations in the *y*-direction, but may vary in a
cross-sectional plane *y* = constant. Hence
*ε*͇ may be a function of
(*x*, *z*):
*ε*͇(*x*,
*z*).

We consider an electromagnetic field which depends harmonically on
*y*, i.e.

for some real *k _{y}*. It is easy to see that by using
Maxwell’s equations, the

*x*- and

*z*-components of the electromagnetic field can be expressed in terms of the

*y*-components:

$$=-\frac{{\omega}^{2}{\epsilon}_{0}{\mu}_{0}}{D}\frac{{k}_{y}}{{\mathrm{\omega \mu}}_{0}}\mathcal{N}\left(\genfrac{}{}{0ex}{}{{\epsilon}_{\mathrm{xy}}}{{\epsilon}_{\mathrm{zy}}}\right){E}_{y}+\frac{i}{D}\frac{{k}_{y}}{\omega {\mu}_{0}}\mathcal{N}\frac{\partial}{\partial x}\left(\genfrac{}{}{0ex}{}{{k}_{y}{E}_{y}}{{\mathrm{\omega \mu}}_{0}{H}_{y}}\right)+\frac{i}{D}\frac{{k}_{y}}{\omega {\mu}_{0}}\mathcal{Q}\frac{\partial}{\partial z}\left(\genfrac{}{}{0ex}{}{{k}_{y}{E}_{y}}{\omega {\mu}_{0}{H}_{\mathrm{y}}}\right)+\frac{i}{\omega {\mu}_{0}}\left(\genfrac{}{}{0ex}{}{\frac{\partial}{\partial x}}{\frac{\partial}{\partial y}}\right){E}_{y},$$

where

$$={\left({\omega}^{2}{\epsilon}_{0}{\mu}_{0}\right)}^{2}\left({\epsilon}_{\mathrm{xx}}{\epsilon}_{\mathrm{zz}}-{\epsilon}_{\mathrm{xz}}{\epsilon}_{\mathrm{zx}}\right)-{\omega}^{2}{\epsilon}_{0}{\mu}_{0}{k}_{y}^{2}\left({\epsilon}_{\mathrm{xx}}+{\epsilon}_{\mathrm{zz}}\right)+{k}_{y}^{4}.$$

Note that *D* is the determinant of matrix Q;. It is
assumed here that *D* ≠ 0. The
*y*-components of Maxwell’s equations (1) and (2) imply:

where ∇͂ is defined by,

By substitution of Eqs. (8), (9) into the right-hand sides of Eqs. (13), (14) we get a coupled system for
*E _{y}* and

*H*,

_{y}$$+\frac{\omega {\epsilon}_{0}}{D}\left({\epsilon}_{\mathrm{yx}}{\epsilon}_{\mathrm{yz}}\right)\mathcal{Q}\frac{\partial}{\partial z}\left(\genfrac{}{}{0ex}{}{{k}_{y}{E}_{y}}{\omega {\mu}_{0}{H}_{y}}\right)+\frac{{k}_{y}}{\omega {\mu}_{0}}{\omega}^{2}{\epsilon}_{0}{\mu}_{0}\stackrel{\u0342}{\nabla}\bullet \left[\frac{1}{D}\mathcal{N}\left(\genfrac{}{}{0ex}{}{{\epsilon}_{\mathrm{xy}}}{{\epsilon}_{\mathrm{zy}}}\right){E}_{y}\right]$$

$$-i\frac{{k}_{y}}{\omega {\mu}_{0}}\stackrel{\u0342}{\nabla}\bullet \left[\frac{1}{D}\mathcal{N}\frac{\partial}{\partial x}\left(\genfrac{}{}{0ex}{}{{k}_{y}{E}_{y}}{\omega {\mu}_{0}{H}_{y}}\right)\right]-i\frac{{k}_{y}}{\omega {\mu}_{0}}\stackrel{\u0342}{\nabla}\bullet \left[\frac{1}{D}\mathcal{Q}\frac{\partial}{\partial z}\left(\genfrac{}{}{0ex}{}{{k}_{y}{E}_{y}}{\omega {\mu}_{0}{H}_{y}}\right)\right]\phantom{\rule{.2em}{0ex}}-\frac{i}{\omega {\mu}_{0}}\Delta {E}_{y},$$

where

Note that M = DN^{-1}. Even when
*k _{y}* = 0, both Eqs. (16) and (17) contain

*E*and

_{y}*H*. Consider the case of a grating made of anisotropic materials which is periodic with respect to the

_{y}*x*and invariant with respect to translations in the

*y*-direction. If the material above the grating is isotropic and the incident wave is in the (

*x*,

*z*)-plane perpendicular to the grooves, i.e. when

*k*= 0, then when the incident field is s- or p-polarized, the total field will be a mixture of both polarizations. In this respect the anisotropic and isotropic gratings totally differ.

_{y}Now we consider an anisotropic medium for which the *y*-axis (i.e.
the axis of translational invariance) is principal axis of
*ε*͇ for all points
(*x*, *y*, *z*). Then the
dielectric tensor is of the form,

and the coupled system Eqs. (16) and (17) becomes:

Furthermore Eqs. (8) and (9) become

Now assume again that *k _{y}* = 0, then Eq. (20) contains only

*H*and Eq. (21) contains only

_{y}*E*:

_{y}Then the system is uncoupled for *E _{y}* and

*H*. Eqs. (24) and (25) imply that when

_{y}*k*= 0 we can distinguish two types of polarization, namely:

_{y}**S-polarization**
*H _{y}* = 0,

*H*≠ 0,

_{x}*H*≠ 0,

_{z}*E*=

_{x}*E*= 0,

_{z}*E*≠ 0;

_{y}**P-polarization**
*E _{y}* = 0,

*E*≠ 0,

_{x}*E*≠ 0,

_{z}*H*=

_{x}*H*= 0,

_{z}*H*≠ 0.

_{y}Hence when the *y*-axis is the principal axis, the splitting in
uncoupled s- and p-polarization for *k _{y}* = 0 is
analogous to the isotropic case. But,

*E*and

_{x}*E*have other values than in the isotropic case because Eq. (22) depends on off-diagonal elements of the tensor.

_{z}In our specific case, the LC is uniaxial material. For such uniaxial materials,
of which the *y*-axis coincides with the optical axis, the
dielectric permittivity tensor is diagonal:

with *ε _{xx}* =

*ε*=

_{zz}*n*

^{2}

_{o}and

*ε*=

_{yy}*n*

^{2}

_{e}. When

*k*= 0 the s- and p-polarization can be separated. We can choose

_{y}*n*=

*n*

_{o}for p-polarization and

*n*=

*n*for s-polarization for analysis.

_{e}In general when *k _{y}* ≠ 0, the system is
coupled as can be easily seen from Eqs. (16) and (17).

#### 3.2. Numerical Method

In a general three-dimensional anisotropic grating problem all the electric and
magnetic field components are coupled when *k _{y}*
≠ 0. Several authors have studied the scattering of an incident plane
wave by an anisotropic grating. The Coupled Wave Method for isotropic gratings
has been extended to anisotropic gratings by Rokushima et al. [7], Glytsis et al. [8] and Mori et al. [9]. The Chandezon method (or C-method) has been applied to
anisotropic gratings by Harris et al. [10]. Li [11] has extended his Fourier modal method (FMM) to
anisotropic gratings. The FMM is a modification of the coupled wave method to
improve convergence for the case of TM polarization. In Li’s paper
conical incident angles are allowed.

In this paper we apply the Finite Element Method (FEM) [12], [5] to model anisotropic gratings. The FEM is a general
numerical method for solving boundary value problems in mathematical physics.
When applied to Maxwell’s equations for 2D gratings, a 2D
computational domain Ω (Fig.2) in a plane perpendicular to the grooves is defined
which has width in the *x*-direction of one period and which
contains all nonplanar and anisotropic materials in its interior. The domain is
truncated in the vertical direction by a so-called Perfect Matched Layer (PML)[13]. On this computational domain, a boundary value problem
for the vector Helmholtz equation for either the electric or the magnetic field
is formulated. (Note that we do not solve the coupled system for the
*E _{y}*- and

*H*-components that was derived in the previous section). The computational domain is meshed using triangles or quadrilaterals and the electric of magnetic field is approximated by edge elements.

_{y}The FEM is relatively difficult to implement, especially in 3D. However, it is very flexible because both periodic and non-periodic configurations and all kinds of materials can be modeled. In particular, anisotropic and inhomogeneous materials with arbitrary principal axes can be treated without problems. In this respect the FEM differs substantially from the Couple Wave Method and the FMM for which the extension to anisotropic materials is highly non-trivial.

To ensure that the FEM code gives accurate results when applied to the anisotropic gratings considered in this paper, we have compared the results obtained for a flat multi-layer consisting of anisotropic media with an analytic model based on the expansion in terms of plane waves. Such a multi-layer is a very good test case for the FEM code because, in contrast with for example the FMM, the FEM does not simplify when applied to an anisotropic multi-layer. We found that when the edge elements of lowest order are used whith 50 mesh points per wavelength, the computed field is quite accurate for all conical angles of incidence.

#### 3.3. Birefringent gratings

The periodic grating which is applied onto the lightguide, was designed in such a
way that for all angles of incidence of interest, only the -1^{st} order is transmitted and propagates. If we consider a plane wave incident
from the lightguide on to the grating with wave vector
**k ^{i}** and incident angles

*θ*and ϕ

^{i}^{i}, the tangential components of the wave vector are given by,

where *k*
_{0} =
2*π*/*λ* is the wave
number in vacuum and *n ^{i}* is the refractive index in
the medium of the incident wave. Polar angle

*θ*and azimuthal angle ϕ are defined and depicted in Fig. 2. The wave vector of a diffracted transmitted wave can be expressed as,

where *n ^{d}* is the refractive index of the medium into
which the light is partially transmitted and ϕ

^{d}and

*θ*are diffraction angles. The components of the wave vector of the

^{d}*m*transmitted order and the components of the incident wave vector are related by:

^{th}where *p* is the period or pitch of the grating. The angular
directions of the *m ^{th}* diffracted order can be
calculated for varying incident angles

*θ*and ϕ

^{i}^{i}from the following equations:

In our grating the pitch *p* = 400 nm and therefore for most
wavelengths and for the angles of incidence 48° ≤
*θ ^{i}* ≤ 90°
and 0° ≤ ϕ

^{i}≤ 90° only the - 1

^{st}order is propagating. For blue light with short wavelength the -2

^{nd}transmitted order is also propagating, but only for very large incident angles.

## 4. Simulations and discussions

#### 4.1. Normal incidence

As we discussed in Section 3.1, when the principal axis of the anisotropic
material is along the grooves and the plane of incidence is perpendicular to the
grooves, we can distinguish s- and p-polarized waves. In this case, the
polarization is conserved. When the incident field is s-polarized, then one
transmitted diffracted order propagates which is s-polarized as well. According
to our definition of angles, ϕ^{d} = ϕ^{i} = 0 in this case. We will call this case *normal
incidence*.

The dependence of the intensity of the -1^{st} diffracted transmitted order as function of incident angle
*θ ^{i}* is shown in Fig. 3 for both s- and p-polarization. The transmission
of s- polarization is much larger than that of p-polarization. The small nonzero
intensity of the transmitted order for p-polarization is due to the small
mismatch between the refractive index of the polycarbonate and the ordinary
refractive index of the LC. It can be seen that the first transmitted order
appears above a certain incident angle. The 0

^{th}transmitted orders become evanescent above 40°, and the intensity of the -1

^{st}transmitted order then changes abruptly. The field increases as the angle becomes larger. For a side-lit lightguide system, the main rays have incident angles

*θ*; larger than 48°. For the angles of interest, the transmission of s-polarization is high comparing to that of p-polarization. But the absolute transmission for s-polarization is still fairly small. This is not a problem in our application. Because the lightguide is very long and therefore the light that reflected back into the lightguide is recycled many times.

#### 4.2. General incidence

In a more general case, when the plane of incidence is not perpendicular to the
grooves, the diffracted transmitted field of the s- and p-polarized incident
fields couple. Hence the transmitted field does not have the same polarization
as the incident field anymore. We did simulations for both s- and p-polarized
incident fields. The diffracted transmitted field is decomposed into two linear
polarized components. One is along the direction of the grooves, which we call
*s*-*polarization*. The other is perpendicular
to the grooves, which is called *p-polarization*. Please notice
that the definition of the transmitted s- and p-polarizations differs from that
of the incident polarizations. The definition of the transmitted polarization
agrees with what is customary [3].

In a backlight configuration as that of Fig. 1 light is present with a multitude of angles
(Lambertian). Typically, the polar angles
*θ ^{i}* range from 48° to
90°. The distribution in diffracted angles

*θ*has a typical width of 20°. The maximum of the distribution for diffracted green light is close to

^{d}*θ*= 67°. With fixed polar angle

^{i}*θ*= 67°, the azimuthal angle ϕ

^{i}^{i}was varied from 0° to 90°. The calculated intensities of the -1

^{st}transmitted order are shown in Fig. 4 for the three wavelengths 450, 535 and 632 nm.

From those figures we can see that for *θ ^{i}*
= 67° the diffracted order becomes evanescent when the incident angle ϕ

^{i}is larger than approximately 40°. The transmitted field strongly depends on the angle. The transmission of s-polarization is relatively high at small incident angle ϕ

^{i}, whereas the transmission of p-polarization is low.

The angular distribution of the wave components of the diffracted transmitted
field was calculated according to the Eqs. (31) and (32), and is illustrated in Fig. 5. To give insight in the angular distribution of
the diffracted light, the relative intensity of the -1^{st} diffracted transmitted order from Fig. 4 is plotted in Fig. 6 as function of the diffraction angle
*θ ^{d}* (instead of angle ϕ

^{i}as is done in Fig. 4).

The transmission of s-polarization is much higher than that of p-polarization.
For s-polarized incident light, especially at small diffraction angle
*θ ^{d}*, the large intensity
difference between transmitted s-component and p-component results in a high
contrast ratio, as is shown in Fig. 7(a). The contrast ratio for unpolarized light is
somewhat lower, as shown in Fig. 7(b), due to the diffracted orders of
p-polarization. By using materials with better matching refractive indices for
the p-polarization, a higher contrast ratio can be achieved. Our measurements of
the angular distribution of extracted s- and p-polarized light [3], repeated in Fig.8, are in agreement with these simulated results. In
practice the contrast ratio is not higher than 23, presumably due to non-perfect
alignment of the liquid crystal in the grating.

## 5. Conclusions

Diffractive outcoupling can be used to make efficient backlights. A diffraction grating containing birefringent material was studied with a rigorous model. The simulation results showed that the diffracted transmitted intensity strongly depends on incident angles and has a high contrast ratio for all angles of interest. The simulation results are in good agreement with experiments. In the case presented here, the efficiencies are not very high (around 2%). Note that these efficiencies refer to one interaction with the grating. For a long lightguide the total extraction efficiency will be much higher. Depending on the application, with a long or short lightguide, a higher extraction efficiency may be desired. This can be achieved by the use of a higher index contrast or by adapting the grating depth[3]. If not all light is extracted before the other end of the lightguide is reached, it is desirable to use a diffusing mirror at that side to recycle the light. To avoid mixing of different colors, we proposed recently[14] a new concept using a grating with a smaller period resulting in the blue light diffracted along the normal and green and red around it. If combined with a suitable pixel lay-out and a proper lens array that directs the light to the desired pixels, an efficient display system can be achieved.

The rigorous model can be used as a tool to optimize gratings for certain applications. In general it is suitable for simulations of subwavelength structure consisting of anisotropic materials for variant applications, such as, switchable holographic grating made of liquid-crystal films

## Acknowledgments

The research is part of STW - Dutch Technology Foundation project DOE.6823.

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