# Optical characterization of dyed liquid crystal cells

Obeng Appiagyei Addai1, Ruilin Xiao2, Xiaoyu Zheng1 , Peter Palffy-Muhoray1,2
_1 Department of Mathematical Sciences, Kent State University, OH, USA _
_2 Advanced Materials and Liquid Crystal Institute, Kent State University, OH,
USA_ Email<EMAIL_ADDRESS>ORCID 0000-0003-0024-7989 Email<EMAIL_ADDRESS>ORCID 0000-0001-9001-098X Corresponding author, Email<EMAIL_ADDRESS>ORCID
0000-0002-3787-7741 Email<EMAIL_ADDRESS>ORCID 0000-0002-9685-5489

###### Abstract

The guest-host liquid crystal display, first proposed in 1968, relies on
controlling the orientation of dichroic dyes dissolved in a nematic liquid
crystal host. Controlling the orientation of the liquid crystal and of the
dissolved dye with an electric field allows control of the transmittance of
the cell. Knowing the dielectric properties at optical frequencies of the dye
and liquid crystal mixtures is crucial for the optimal design of guest-host
liquid crystal devices. In this work, the dielectric functions of various
layers in liquid crystal cells are described by models obeying the Kramers-
Kronig relations: the Sellmeier equation for transparent layers and causal
Gaussian oscillator model for absorbing layers. We propose a systematic way to
accurately model the dielectric response of each layer by minimizing the sum
of squared differences between the measured transmittance spectrum of a guest-
host cell in the near-UV/vis range and the prediction of the transmittance of
the modeled multilayer structure. By measuring the transmittance for incident
light polarized parallel and perpendicular to the nematic director allows us
to separately characterize the two principal dielectric functions of the
uniaxial sample. Our results show that the causal Gaussian oscillator model
can accurately characterize the dielectric functions of dyes in liquid
crystals.

keywords: dielectric functions, dichroic dyes, guest-host liquid crystals,
causal Gaussian oscillator model

##  1 Introduction

Dyes are colored since their absorption is wavelength dependent. They have a
wide range of applications, including information displays, such as LCDs. The
first guest-host liquid crystal display was proposed by Heilmeier and Zanoni
in 1968 [1], where dichroic dyes are dissolved in nematic liquid crystals
(NLCs). The transmittance of the NLC cell is controlled by an electric field
which can orient the LC molecules, which in turn orient the dichroic dye
molecules. The optical response depends on polarizabilities of the
constituents, hence their dielectric tensors at optical frequencies. Many
guest-host liquid crystal systems have been subsequently proposed and
realized; various aspects of these systems have been studied and reported, as
in early review papers by Cox in 1979 [2] and by Scheffer in 1983 [3], and
more recently by Sims in 2016 [4].

Modeling the dielectric function of dyes in LCs at optical frequencies can be
a powerful tool for the interpretation of measured transmittance spectra and
for predicting the transmittance and reflectance as function of wavelength.
Characterization of the dielectric function of dyes in solutions is a
challenging task in general, as dyes molecules often show the tendency to
associate causing the transmittance to depend nonlinearly on concentration and
absorption peaks to shift when mixing two dyes [5]. However, at low
concentrations, one can assume the linear response.

An overview of most popular models for dielectric functions and their
applicabilities can be found in [6]. Among those models, the classical Lorentz
model assumes that the motion of the electrons, bound to atomic nuclei, can be
modeled as a simple damped harmonic oscillator, subject to the external
electromagnetic fields. The dielectric function $\varepsilon$ can then be
expressed as (e.g. [7])

$\varepsilon^{LO}(\omega)=\varepsilon_{\infty}+\sum_{k=1}^{N}\frac{A_{k}\omega_{p}^{2}}{\omega_{0k}^{2}-\omega^{2}-i\gamma_{k}\omega},$
(1.1)

where $\varepsilon_{\infty}$ is the high frequency dielectric constant,
$\omega_{p}$ is the plasma frequency, $A_{k}$ is the strength of the $k$th
oscillator with resonant frequency $\omega_{0k}$ and the damping coefficient
$\gamma_{k}$. The damping coefficient $\gamma_{k}$ is proportional to the
width of the absorption peak. Taking the limit of of $\gamma_{k}\rightarrow 0$
in Eq. (1.1) gives the Sellmeier equation,

$\varepsilon^{S}(\omega)=\varepsilon_{\infty}+\sum_{k=1}^{N}\frac{A_{k}\omega_{p}^{2}}{\omega_{0k}^{2}-\omega^{2}},$
(1.2)

which is often used to model the dispersion relations for transparent
materials away from the absorption peaks. The Lorentz oscillator model
satisfies the Kramers-Kronig (KK) causality relations. Some authors [8, 9]
have used Lorentz oscillator models for dispersion relations for azo-dyes. The
Lorentz oscillator model can produce realistic absorption frequencies and
amplitudes. However, as the Lorentz oscillator model tends to predict slower
decays, the fitting to the absorption data is poor away from the absorbance
peaks [9, 10].

In 1992, Brendel and Bormann (BB) proposed a dielectric function model for
amorphous solids in the infrared (IR) range [11]. It is written as a
convolution of a Gaussian function with the Lorentz oscillator

$\varepsilon^{BB}(\omega)=\varepsilon_{\infty}+\sum_{k=1}^{N}\chi_{k}^{BB}(\omega),$
(1.3)

where

$\chi_{k}^{BB}(\omega)=\frac{1}{\sqrt{2\pi}\sigma_{k}}\int_{-\infty}^{\infty}\exp\left(-\frac{(x-\omega_{0k})^{2}}{2\sigma_{k}^{2}}\right)\frac{A_{k}\omega_{p}^{2}}{(x^{2}-\omega^{2})-i\omega\gamma_{k}}dx.$
(1.4)

It was interpreted as the superposition of damped harmonic oscillators whose
resonance frequencies follow a Gaussian distribution centered at $\omega_{0k}$
with standard deviation $\sigma_{k}$. Some authors [10, 12, 13] used the
Brendel and Bormann model, sometimes referred to as Voigt model, for modeling
the dielectric functions of dyes and showed improved accuracy over the Lorentz
oscillators. Subsequently it was shown that the BB model is not causal as it
fails to satisfy $\chi_{k}^{BB}(\omega)\neq\overline{\chi_{k}^{BB}(-\omega)}$
in addition to a divergent behavior near $\omega=0$ [14].

Meneses et. al. [15] in 2006 proposed the causal Gaussian oscillator model, to
model the dielectric function of binary lead silicate glasses,

$\varepsilon^{CG}(\omega)=\varepsilon_{\infty}+\sum_{k=1}^{N}\chi_{k}^{CG}(\omega),$
(1.5)

where the imaginary part of $\chi_{k}^{CG}$ is given by

$\chi_{k}^{CG\prime\prime}(\omega)=A_{k}\left[\exp\left(-\frac{(\omega-\omega_{0k})^{2}}{2\sigma_{k}^{2}}\right)-\exp\left(-\frac{(\omega+\omega_{0k})^{2}}{2\sigma_{k}^{2}}\right)\right],$
(1.6)

and the real part, which guarantees satisfying the KK relations, is given in
terms of the Dawson function $D(x)$ as

$\chi_{k}^{CG\prime}(\omega)=\frac{2A_{k}}{\sqrt{\pi}}\left[D\left(\frac{\omega+\omega_{0k}}{\sqrt{2}\sigma_{k}}\right)-D\left(\frac{\omega-\omega_{0k}}{\sqrt{2}\sigma_{k}}\right)\right],$
(1.7)

where $D(x)=e^{-x^{2}}\int_{0}^{x}e^{t^{2}}dt$. The causal Gaussian oscillator
model can be obtained by removing the divergent factor from the BB oscillator
model and setting $\gamma_{k}=0$. It has been used to characterize silica
glass in the far IR region [16], and Titanium dioxide from near-UV to near-IR
[17].

We point out that in Ref. [18], the authors used the renormalized ellipsometry
to determine the device parameters and tilt angles of a guest-host LC system.
In particular, they have used a combined Cauchy equation and Gaussian model
for the complex refractive indices of a guest-host LC system. Their assumed
dielectric function does not obey the KK relations. Nonetheless, they showed
that the extinction coefficient of the guest-host LC system can be
characterized by a Gaussian function.

In another vein, calculations based on density functional theory or molecular
dynamics were carried out to predict the polarizabilities of dyes [10] and the
alignment of dyes in LCs [19, 20]. These calculations require knowledge of the
specific molecular structures to find the minimum potentials and their
frequency dependence.

In this work, the characterization of the dielectric functions of our samples
is solely based on the transmittance spectra, without the knowledge of their
molecular structures. Specifically, we model the dielectric function of each
layer in our multilayer LC samples in the near-UV/vis range by either the
Sellmeier equation, if they appear transparent or by the causal Gaussian
oscillator model, if they have absorption. We show that the dielectric
function of the dye and liquid crystal mixture can be accurately described by
the causal Gaussian oscillator model.

The rest of the paper is organized as follows. In section 2, we present the
mathematical models and methods which lead to the determination of the
parameters in the dielectric function models. Section 3 describes our
experimental setup for obtaining the transmittance spectra of our samples. In
section 4, we show the characterization results of dielectric functions of all
components in our guest-host LC sample. We give our conclusions in section 5.

##  2 Mathematical Models and Methods

In this section, we first briefly discuss the eigenvalue problem associated
with light propagation in a lossy uniaxial material, explain the setup of our
experiments and discuss the validity of applying the averaged transmittance
model of a multilayer structure to our problem. Then we present the least
squares approach used to determine the parameters in the dielectric function
models where we minimize the sum of the squared differences between the
experimental measurements and the results of theoretical models.

A uniform uniaxial material is birefringent, and the polarization of plane
polarized light will, in general, change as it propagates through the material
unless the polarization is along one of two eigen directions associated with
the wave propagation direction. The eigenvalue problem in term of the
displacement field $\mathbf{D}$, derived from Maxwell equation, is given by
(cf. [7])

$(k^{2}\mathbf{I}-\mathbf{kk})\varepsilon_{r}^{-1}\mathbf{D}=k_{0}^{2}\mathbf{D},$
(2.1)

where $\mathbf{I}$ is the identity tensor,
$\mathbf{k}=k^{\prime}\mathbf{\hat{k}}^{\prime}+ik^{\prime\prime}\mathbf{\hat{k}}^{\prime\prime}$
is the complex wave vector, $k^{2}=\mathbf{k}\cdot\mathbf{k}$ $=k^{\prime
2}-k^{\prime\prime
2}+2ik^{\prime}k^{\prime\prime}(\mathbf{\hat{k}}^{\prime}\cdot\mathbf{\hat{k}}^{\prime\prime})$
is the dot product, $\mathbf{kk}$ is the tensor product, and
$k_{0}^{2}=\omega^{2}\mu_{0}\varepsilon_{0}$, where $\mu_{0}$ is the
permeability, $\varepsilon_{0}$ is permittivity of free space, and $\omega$ is
the angular frequency. For uniaxial absorbing materials, the complex relative
dielectric tensor can be written as

$\varepsilon_{r}=\varepsilon_{\perp}\mathbf{I}+(\varepsilon_{\parallel}-\varepsilon_{\perp})\mathbf{\hat{n}\hat{n},}$
(2.2)

where $\varepsilon_{\parallel}$ is the dielectric constant for fields parallel
to the symmetry axis $\mathbf{\hat{n}}$, and $\varepsilon_{\perp}$ is the
dielectric constant for fields in the plane perpendicular to
$\mathbf{\hat{n}}$. In our LC samples, the symmetry axis $\mathbf{\hat{n}}$
coincides with the director of the nematic liquid crystal.

Given the directions of $\mathbf{\hat{k}}^{\prime}$ and
$\mathbf{\hat{k}^{\prime\prime}}$ and $\varepsilon_{r}$, one can solve for the
eigenvalues $k^{\prime}$, $k^{\prime\prime}$, and for the corresponding
eigenvector $\mathbf{D}$. One eigen-pair is given by

$k_{1}^{2}=k_{0}^{2}\varepsilon_{\perp},\mathbf{D}_{1}\parallel\mathbf{\hat{n}}\times\mathbf{k}_{1},$
(2.3)

while the other is given, implicitly, by

$k_{2}^{2}\varepsilon_{\perp}+\Delta\varepsilon(\mathbf{\hat{n}\cdot
k}_{2})^{2}=k_{0}^{2}\varepsilon_{\parallel}\varepsilon_{\perp},$ (2.4)

and

$\mathbf{D}_{2}\parallel\mathbf{k}_{2}\times(\mathbf{\hat{n}}\times\mathbf{k}_{2}).$
(2.5)

Similar results can be found in [21].

Suppose light propagating in an isotropic lossless medium is normally incident
on a nematic LC cell where the director $\mathbf{\hat{n}}$ is along the
$\mathbf{\hat{x}}$ direction. Boundary conditions require that, in the cell,
$\mathbf{\hat{k}}=\mathbf{\hat{k}}^{\prime}=\mathbf{\hat{k}}^{\prime\prime}=\mathbf{\hat{z}}$.
If the polarization of plane polarized incident light, and hence the direction
$\mathbf{\hat{E}}$ of the electric field of the incident light is
perpendicular to $\mathbf{\hat{n}}$, we have, by Eq. (2.3),

$k_{1}^{2}=k_{0}^{2}\varepsilon_{\perp},\mathbf{D}_{1}\parallel\mathbf{E}_{1}\parallel\mathbf{\
\hat{y}.}$ (2.6)

That is, the polarization of the light will remain in the
$\mathbf{\hat{y}}$-direction, and the wave number in the sample is only
dependent on $\varepsilon_{\perp}$. If the linear polarization is such that
$\mathbf{\hat{E}}$ is parallel to $\mathbf{\hat{n}}$, we obtain from Eqs.
(2.4)-(2.5),

$k_{2}^{2}=k_{0}^{2}\varepsilon_{\parallel},\mathbf{D}_{2}\parallel\mathbf{E}_{2}\parallel\mathbf{\hat{x}.}$
(2.7)

That is, the polarization of the light will remain in the
$\mathbf{\hat{x}}$-direction, and the wave number in the sample is only
dependent on $\varepsilon_{\parallel}$. Those correspond to the two cases
depicted in Fig. 1, and indicate the geometries considered in this work.

In these two scenarios, we can then use the following recursive relations to
calculate the transmission and reflection coefficients for light propagation
through an isotropic multilayer structure at normal incidence, with light
incident from medium 1 and exiting to medium $m$ (see, e.g. [22]),

$\displaystyle t_{1:m}$
$\displaystyle=\frac{t_{12}t_{2:m}e^{ik_{0}n_{2}d_{2}}}{1+r_{12}r_{2:m}e^{2ik_{0}n_{2}d_{2}}},$
(2.8) $\displaystyle r_{1:m}$
$\displaystyle=\frac{r_{12}+r_{2:m}e^{2ik_{0}n_{2}d_{2}}}{1+r_{12}r_{2:m}e^{2ik_{0}n_{2}d_{2}}},$
(2.9)

where

$\displaystyle r_{ij}$
$\displaystyle=\frac{n_{i}-n_{j}}{n_{i}+n_{j}},t_{ij}=\frac{2n_{i}}{n_{i}+n_{j}},$
(2.10) $\displaystyle n_{i}$
$\displaystyle=\sqrt{\varepsilon_{i}},{i=1,\cdots,m}.$ (2.11)

Here $n_{i}$ is the complex refractive index whose real and imaginary parts
are referred to as the index of refraction and the extinction coefficient,
respectively. The transmittance and reflectance of the multilayer structure,
with $n_{1}=n_{m}$, can be readily obtained from

$T_{1:m}=|t_{1:m}|^{2},R_{1:m}=|r_{1:m}|^{2}.$ (2.12)

A general result for transmittance and reflectance of anisotropic absorbing
multilayer structure can be found in [23]. We note that a decrease in the
transmittance of a multilayer structure can be due to absorption in the layers
and/or to reflections at the interfaces. In the case of absorption, the
refractive index has a nonzero imaginary part, originating in the imaginary
part of $\varepsilon$, and in the case of reflections, there is a mismatch of
the refractive indices of adjacent layers.

The transmittance and reflectance from Eq. (2.12) of a thick ($d\gg\lambda)$
single- or multilayer structure can show large and rapid oscillations as
function wavelength due to interference and multiple reflections. These become
increasingly difficult to observe experimentally as the sample thickness
increases, due to the finite resolution of spectrometers and sample
inhomogeneities. In this limit, ignoring multireflection interference, or,
equivalently, by assuming random phase, the average transmittance
$\mathcal{T}_{1:m}$ and reflectance $\mathcal{R}_{1:m}$ for a multilayer
structure at normal incidence are given by

$\displaystyle\mathcal{T}_{1:m}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{T}_{12}\mathcal{T}_{2:m}e^{-2k_{0}\text{Im}(n_{2})d_{2}}}{1-\mathcal{R}_{21}\mathcal{R}_{2:m}e^{-4k_{0}\text{Im}(n_{2})d_{2}}},$
(2.13) $\displaystyle\mathcal{R}_{1:m}$ $\displaystyle=$
$\displaystyle\mathcal{R}_{12}+\frac{\mathcal{T}_{12}\mathcal{R}_{2:m}\mathcal{T}_{21}e^{-4k_{0}\text{Im}(n_{2})d_{2}}}{1-\mathcal{R}_{21}\mathcal{R}_{2:m}e^{-4k_{0}\text{Im}(n_{2})d_{2}}},$
(2.14)

where

$\mathcal{R}_{ij}=\left|\frac{n_{i}-n_{j}}{n_{i}+n_{j}}\right|^{2},\mathcal{T}_{ij}=\left|\frac{2n_{i}}{n_{i}+n_{j}}\right|^{2}.$
(2.15)

A similar expression for a single etalon with $m=3$ can be found in [24]. In
our study, we have set $n_{1}=n_{m}=n_{air}=1$.

By assuming a specific form for the dielectric function of each layer, the
parameters in the dielectric functions can be found by minimizing the sum of
the squares of the errors between the theoretical transmittance of a
multilayer structure and the experimental data. Specifically, the minimization
problem can be written as

$\min\sum_{i=1}^{M}[\left(y_{\parallel}(\omega_{i})-\mathcal{T}_{1:m}(\omega_{i};n_{1\parallel},\cdots,n_{m\parallel})\right)^{2}+\left(y_{\perp}(\omega_{i})-\mathcal{T}_{1:m}(\omega_{i};n_{1\perp},\cdots,n_{m\perp})\right)^{2}],$
(2.16)

where $M$ is the total number of data points, $y_{\parallel}(\omega_{i})$ and
$y_{\perp}(\omega_{i})$ are the locally averaged transmittance data at
$\omega_{i}$ when the polarization of the incident light is parallel and
perpendicular to the LC director, respectively. The two principal refractive
indices of each layer are given by

$n_{j\parallel}=\sqrt{\varepsilon_{j\parallel}(\varepsilon_{\infty\parallel}^{j},{\bar{A}_{\parallel}^{j},\bar{\omega}_{0}^{j},\bar{\sigma}}^{j})},n_{j\perp}=\sqrt{\varepsilon_{j\perp}({\varepsilon_{\infty\perp}^{j},\bar{A}_{\perp}^{j},\bar{\omega}_{0}^{j},\bar{\sigma}}^{j})},j=1,\cdots,m,$
(2.17)

and the dielectric functions $\varepsilon_{j\parallel}$ and
$\varepsilon_{j\perp}$ are modeled either as a Sellmeier equation in Eq.
(1.2), or as a causal Gaussian oscillator model in Eqs. (1.5)-(1.7), where
$\varepsilon_{\infty\parallel}^{j},\varepsilon_{\infty\perp}^{j},{\bar{A}_{\parallel}^{j},\bar{A}_{\perp}^{j},\bar{\omega}_{0}^{j},\bar{\sigma}}^{j}$
are the fitting parameters. The lengths of the vector parameters
${\bar{A}_{\parallel}^{j},\bar{A}_{\perp}^{j},\bar{\omega}_{0}^{j},\bar{\sigma}}^{j}$are
equal to the number of oscillators. We have assumed that the peak locations
$\bar{\omega}_{0}^{j}$ and their widths $\bar{\sigma}^{j}$ along the two
principal directions are the same, but the amplitudes
$\bar{A}_{\parallel}^{j},\bar{A}_{\perp}^{j}$ are different, as suggested by
the experimental data.

We note that one can’t model all the layers collectively as an effective
single layer, which disregards the arrangement of the layers, since, in
general, the transmittance of the multilayer structure may sensitively depends
on the order of the layers. It is also impossible to determine the refractive
indices of all layers simultaneously from a single transmittance measurement
of all layers, as the solutions won’t be unique. Therefore, ideally, we should
determine the dielectric response of each layer separately, but this is often
not practically viable. However, we can measure, for example, the
transmittance of a single glass layer first, and then that of a single glass
layer coated with indium tin oxide (ITO) layer, and next that of a polyimide
(PI) alignment layer on top of the glass and ITO layers. These three
measurements enable us to determine the dielectric properties of glass, ITO,
and PI iteratively, one at a time. To determine the dielectric properties of
dye-LC mixtures, we assume the effect of a dye guest on the dielectric
response of the mixture is an additive contribution to that of the pure LC
host. With this in mind, we first use the transmittance spectrum of pure LC in
the cell to determine the dielectric function of LC host, and then use the
transmittance spectrum of dye-LC mixture in the cell to determine the
contribution of dye to the dielectric function of dye-LC mixture.

We have implemented the Levenberg$-$Marquardt algorithm to solve the above
minimization problems. It is important to note that the function to be
minimized, Eq. (2.16), has multiple local minima. To obtain the most faithful
fitting results, we have carefully selected the initial parameters which give
a crude agreement of the experimental and model transmittances. Specifically,
the absorption peak positions $\omega_{0k}$ were chosen to be in proximity of
the dips of the transmittance spectra. The number of oscillators used in
modeling each layer is determined by whether or not the addition of one more
oscillator improves the fitting significantly. The goodness of fit is assessed
from maximum pointwise errors, rather than an averaged error.

##  3 Experiment

Our liquid crystal cells typically use two glass substrates with each
substrate coated with a conductive ITO layer enabling the application of an
electric field across the cell. The PI alignment layer, next to the ITO layer,
is buffed antiparallel on the two substrates to impose the homogeneous
boundary conditions on the nematic directors. The thicknesses of the glass,
ITO and PI layers are approximately $1$mm, $100$ nm and $100$ nm respectively.
The thickness of the cell gap is approximately $20\mu m$. The empty cells are
filled with pure LCs or with dye-LC mixtures, with $1\%$ or $0.5\%$ dye
concentrations.

As discussed in Section 2, we separately measured the transmittance spectra of
the following samples: a single glass, glass with ITO layer, glass with ITO
and PI layers, empty cell, pure LC in the cell, and dye-LC mixtures in the
cell. To measure the transmittance spectra of each sample, we use a
spectrometer with a polarizer. The sample is aligned so that light is normal
incident on the cell. The polarizer is oriented so that light is polarized
parallel, as shown in Fig. 1(a), or perpendicular, in Fig. 1(b), to the
nematic director.

The transmission spectra are measured using an Ocean Optics HR400CG-UV-NIR
High-Resolution Spectrometer with resolution $\sim$ $0.2-0.3$ nm with an Ocean
Insight DH-2000-BAL light source covering the range of $300$ nm to $900$ nm.

Since the resolution of the spectrometer is high and sample is thin, the
transmittance shows some oscillations due to multiple reflections and
interference. As it is preferable to work with a smooth spectra for fitting
purposes, local averaging of the transmittance was carried out before the
fitting procedure. Measurement errors were estimated by comparing repeated
spectral measurements on single cells. Variations due to differences in cell
properties were estimated by comparing the spectra of six empty cells. The
combined variations gave an estimated uncertainty of approximately $5\%$ in
the wavelength range $300$ to $900$ nm.

(a) (b)

Figure 1: Schematic of the experimental setup. Light propagating in the
$\mathbf{\hat{z}}-$ direction is normally incident on the cell. The LC
director $\mathbf{\hat{n}}$ is in the $\mathbf{\hat{x}}-$ direction. The
transmittance is measured when the polarization of the light is (a) parallel
to the $\hat{\mathbf{n}}$ , and (b) perpendicular to $\hat{\mathbf{n}}$. The
sample between the polarizer and the photodetector represents the LC cell
where a dye-LC mixture is contained in two glass(2,8) substrates coated with
ITO (3,7) and PI (4,6) layers. Layers labeled 1 and 9 are air. The sample may
also be a single glass, glass coated with ITO, glass coated with ITO and PI
for separate measurements.

##  4 Results

In this section, we first characterize the dielectric function of each
substrate layer using the causal Gaussian oscillator model for glass and the
Sellmeier equation for ITO and PI in section 4.1. Knowing the refractive
indices of substrate layers, we then proceed to characterize the dielectric
function of the pure LC and two dye-doped liquid crystal mixtures in section
4.2.

###  4.1 Dielectric functions of the substrate layers

The transmittance spectra of the substrate layers: glass only, glass with ITO,
and glass with ITO and PI, are shown in Fig. 2(a). The transmittance spectra
of the three layers do not show significant polarization dependence, thus we
regard all substrate layers as optically isotropic.

Our strategy is as follows: we start with the characterization of the
dielectric function of glass with the transmittance spectrum of a single slab
of glass. Next, we obtain the dielectric function of ITO by fitting the
transmittance spectrum of glass with ITO, given we have already known the
refractive index of glass. Lastly, the dielectric function of the PI is found
by fitting the transmittance spectrum of glass with ITO and PI layers, given
the dielectric functions of glass and ITO. The results are shown in detail
below.

####  4.1.1 glass

Glass is generally taken to be transparent in the visible range. The
dispersion relation was generally modeled with Sellmeier equation in [25, 26].

In this work, the dielectric function of glass is characterized solely based
on the transmittance spectrum of a single glass slab in the range of $300-900$
nm. In Eqs. (2.13)-(2.15), we set $m=3,n_{1}=n_{3}=1$, and solve the
minimization problem Eq. (2.16) to find $n_{2}=n_{glass}$. We first attempted
to use the Sellmeier equation for $\varepsilon_{glass}$. Here, the one-term
Sellmeier equation was fitted with parameters
$\varepsilon_{\infty}=1.5302,A=0.6001,\omega_{0}=2\pi c/309.19nm$, and the
predicted transmittance of the 3-layer structure is shown as the green dot-
dashed curve in Fig. 2(b). We observed that the Sellmeier equation cannot
accurately capture the sharp corner in the transmittance spectrum near $300$
nm due to the nature of polynomial decay away from from the absorption peak;
it also fails to model the observed slow decrease of transmittance between
about $600$ and $900$ nm. In addition, since the peak of the oscillator is
located at $309$ nm, the dielectric constant diverges near the peak, which is
nonphysical.

We therefore used the causal Gaussian oscillator model for
$\varepsilon_{glass}$. We have used two such oscillators, with one oscillator
peak at $195$ nm for the strong absorption in the near-UV and one at $987.31$
nm for the slow decrease in the transmittance over visible to near-IR. The
causal Gaussian oscillator model produces excellent agreement between the
theoretical transmittance of the 3-layer structure and the experimental data
as shown in Fig. 2(b). The average relative error in the range of $400-900$ nm
is $0.8\%.$ The real and imaginary parts of the refractive indices of glass,
calculated via the causal Gaussian oscillator model with parameters given in
Table 1, are plotted in Fig. 3. The real part of the refractive index of the
glass is about $1.53-1.54$. A more sophisticated model combining Sellmeier and
causal Gaussian can be used [27] to produce larger variation in the refractive
indices with respect to wavelength. However additional information would be
needed to decide which model is preferable, hence we do not pursue this topic
in this work. The refractive index of the glass given by the causal Gaussian
oscillator model is used for the subsequent characterizations of dielectric
functions of the remaining layers of the LC cell.

####  4.1.2 ITO layer

For modeling of the dielectric function of the ITO, in the literature, either
a combination of a single Drude oscillator for free carriers and double
Lorentz oscillators for strong absorption in the near-UV [28], or a combined
Drude, causal Gaussian oscillators and a Cauchy term [29] were used.

Here, the dielectric function of ITO is characterized based on the
transmittance spectrum of glass coated with an ITO layer in the range of
$300-900$ nm. In Eqs. (2.13)$-$(2.15), we set $m=4,n_{1}=n_{4}=1$, and
$n_{2}=n_{glass}^{cG}$, which is obtained from the causal Gaussian oscillator
model in Sec. 4.1.1. We then solve the minimization problem Eq. (2.16) to find
$n_{3}=n_{ITO}$. By comparing the transmittance of glass with ITO with that of
glass with ITO and PI in Fig. 2(a), we observe that the transmittance of the
5- layer structure in the near-UV is higher than that of the 4-layer
structure. Absorption in the near-UV from the ITO or from the PI won’t be able
to produce such a spectrum. This increase in the transmittance with more
layers can only be explained by index matching. Therefore, we model the
dielectric functions of both the ITO layer and PI layer with the Sellmeier
equation. For the ITO, the Sellmeier equation with two peaks, one at $275.01$
nm and the other at $1000.01$ nm, produces excellent agreement between the
theoretical prediction of transmittance of the 4-layer structure and the data,
as shown in Fig. 2(c), with the average relative error $1\%$ in the range of
$400-900$nm. Fig. 3(b) shows the refractive index of ITO calculated from the
Sellmeier equation with parameters given in Table 1. In the range of $350-900$
nm, the refractive index for ITO decreases from $2.2$ to $1.6$, which is
consistent with that in the literature [30, 31].

####  4.1.3 PI layer

Buffed polyimide polymer layers can be optically anisotropic, with an average
refractive index in the visible of about $1.5-1.8$ depending on chemical
structure and morphology [32, 33]. Our cells do not show appreciable
birefringence, so we model the PI layer with a single principal dielectric
function.

The dielectric function of PI is determined using the transmittance spectrum
of glass with ITO and PI layers from $300-900$ nm. In Eqs. (2.13)$-$(2.15), we
set $m=5,n_{1}=n_{5}=1$, $n_{2}=n_{glass}^{cG}$ obtained from Sec. 4.1.1 and
$n_{3}=n_{ITO}^{S}$ obtained from Sec. 4.1.2. We then solve the minimization
problem Eq. (2.16) to find $n_{4}=n_{PI}$. As discussed in 4.1.3, we model the
dielectric function of PI with the Sellmeier equation. A two-term Sellmeier
equation is fitted with one peak located at $199.32$ nm, and the other in the
IR at $1135.22$ nm. The corresponding theoretical transmittance of the 5-layer
structure shows reasonable overall agreement with experimental data, with
slightly larger discrepancy in the $300-500$ nm region, as shown in Fig. 2(d).
The average relative error in the range of $400-900$nm is $1\%$. Surprisingly,
the contribution of the PI layer to the transmittance varies dramatically from
one piece of substrate to another. Therefore, we only aim to get a reasonable
fitting here.  Fig. 3(b) shows the refractive index of PI calculated from the
Sellmeier equation with parameters given in Table 1. The refractive index of
PI decreases from $1.95$ to $1.54$ in the $350$ to $900$ nm range, in
agreement with reported values in [34, 35, 36].

We note that the fitting parameters of the Sellmeier equations for the ITO and
PI layers are independent of the thicknesses of the ITO and PI layers, since
we have considered the averaged transmittance which depends on the values of
the (real) refractive indices but not on the cell thickness.

The goodness of fit for the dielectric functions of the three substrate layers
was further tested by comparing the experimental measurements of empty cells
with the theoretical prediction of the transmittance of the 9-layer structure
with
$m=9,n_{1}=n_{5}=n_{9}=1,n_{2}=n_{8}=n_{glass}^{cG},n_{3}=n_{7}=n_{ITO}^{S},n_{4}=n_{6}=n_{PI}^{S}$,
using fitted model parameters in Table 1. The experimental data is plotted as
the average over the measurements of six different empty cells, with upper and
lower bounds from the six measurements at each wavelength. Our theoretical
prediction of the empty cell shows reasonable agreement with these
measurements, with a larger discrepancy near $300-500$ nm as shown in Fig. 4.
This is mainly due to the imperfection of the PI fitting at near-UV. We note
that since the fitting for the PI didn’t show excellent agreement between
theory and measurement in the near-UV, the error in the PI fitting will carry
over to the fitting of the LC and subsequent dye-LC mixtures. Even though the
relative error in the near-UV made in the previous steps carried forward, it
is not significant enough to invalidate the subsequent steps. If a highly
accurate characterization in the near-UV is needed, more accurate measurements
will also be required.

(a) (b)

(c) (d)

Figure 2:  Transmittance of substrate layers. (a) shows the averaged experimental transmittance spectra of glass only, glass with ITO, and glass with ITO and PI. (b, c, d) show the averaged experimental data and theoretical prediction of multilayer structures using dielectric functions of substrate layers with fitting parameters in Table 1, for glass only, glass coated with ITO, and glass coated with ITO and PI, respectively.  | $\varepsilon_{\infty}$ |  | $A_{k}$ | $\omega_{0k}$ |  $\sigma_{k}$ (THz)
---|---|---|---|---|---
Glass(Gaussian) | $2.3104$ | $k=1$ | $0.2107$ | $2\pi c/195.02nm$ | $34851$
|  | $k=2$ | $0.00001$ | $2\pi c/987.31nm$ | $3127$
ITO(Sellmeier) | $1.1221$ | $k=1$ | $1.4299$ | $2\pi c/275.01nm$ | $-$
|  | $k=2$ | $0.0641$ | $2\pi c/1000.01nm$ | $-$
PI (Sellmeier) | $2.4779$ | $k=1$ | $0.7831$ | $2\pi c/199.32nm$ | $-$
|  | $k=2$ | $0.5697$ | $2\pi c/1315.22nm$ | $-$
Table 1: Fitted parameters in the dielectric functions of glass, ITO and PI.
Here $c$ is the speed of light in air.

(a) (b)

Figure 3: (a) Refractive index of the glass substrate. (b) Refractive indices
of the ITO and PI layers.  Figure 4: Average of measured transmittance spectra
of six empty cells together calculated values using dielectric functions of
substrate layers with fitted parameters in Table 1. The solid red curve is the
measured transmittance averaged over six cells, bounded by the upper and lower
extreme values (dotted curves) among the six measurements at each wavelength.

###  4.2 Dielectric functions of dye-doped liquid crystal mixtures

Liquid crystals are typically birefringent, as are oriented dichroic dyes.
When the transmittance of a mixture is measured with the polarization of
incident light either parallel or perpendicular to the LC director, only one
principal refractive index contributes to the transmittance. The alignment of
LCs and dye mixtures was confirmed by applying a voltage across the cell.
Since the LC has a negative dielectric anisotropy at low frequency, no change
in orientation is expected, and none was observed.

We assume that dielectric functions of the LC and of the dye can be treated as
additive. This is valid when the mixture is homogeneous and the concentration
of dye is low. The experimental data suggest that the locations of the
absorption peaks and their broadenings are nearly the same for both
polarizations and only the magnitudes differ. The two principal dielectric
functions of dye-doped LC mixtures are then given as

$\displaystyle\varepsilon_{\parallel}^{LC+dye}$ $\displaystyle=$
$\displaystyle\varepsilon_{\infty\parallel}^{LC}+\sum_{k=1}^{2}\chi_{k}^{S}(\omega;A_{k\parallel}^{LC},\omega_{0k}^{LC})+c_{dye}\sum_{k=1}^{N_{dye}}\chi_{k}^{cG}(\omega;A_{k\parallel}^{dye},\omega_{0k}^{dye},\sigma_{k}^{dye}),$
(4.1) $\displaystyle\varepsilon_{\perp}^{LC+dye}$ $\displaystyle=$
$\displaystyle\varepsilon_{\infty\perp}^{LC}+\sum_{k=1}^{2}\chi_{k}^{S}(\omega;A_{k\perp}^{LC},\omega_{0k}^{LC})+c_{dye}\sum_{k=1}^{N_{dye}}\chi_{k}^{cG}(\omega;A_{k\perp}^{dye},\omega_{0k}^{dye},\sigma_{k}^{dye}),$
(4.2)

where $c_{dye}$ denotes the concentration of the dye, and $N_{dye}$ the number
of oscillators associated with the dye. The first sum accounts for the
contribution from the LC host and the second from the dye guest. In Eqs.
(2.14)-(2.13), we set
$m=9,n_{1}=n_{9}=1,n_{2}=n_{8}=n_{glass}^{cG},n_{3}=n_{7}=n_{ITO}^{S},n_{4}=n_{6}=n_{PI}^{S}$,
and try to find $n_{5}$ which minimizes the sum of squared errors of the model
and data. In what follows, we first obtain the pure LC contribution of the
first two terms before the dye is introduced.

####  4.2.1 liquid crystal host

The ordinary refractive index $n_{o}$ of the LC typically ranges from
$1.50-1.57$ while extraordinary refractive index $n_{e}$ can range from
$1.5-1.9$ depending on molecular structure, temperature and wavelength [37].
Many models, Cauchy or Sellmeier, have been proposed to model the wavelength
and temperature dependence of the refractive indices of LCs [38, 39, 40].

In this work, the characterization of the dielectric function of the LC host
is based on the transmittance spectra of the cell containing pure LC host from
$300-900$ nm. The LC appears to be transparent at visible wavelengths, but
with appreciable reduction of the transmittance in the near-UV. In order to
understand the origin of this reduction, we carried out a further set of
experiments, where three pure LC cells with the same glass substrates but with
different thicknesses, $5\mu m,20\mu m,47\mu m$, were prepared and
transmittance spectra were measured. The results show that the transmittances
of the LC cells are apparently independent of gap thickness, which implies
that the LC host is nonabsorptive. This prompts us to use the Sellmeier
equation for the dielectric function of LC host. A two-term Sellmeier equation
with one peak located at $125.01$ nm and the other at $289.62$nm was obtained
from the fit. Fig. 5(a) shows reasonable agreement between the theoretical
predictions and experimental measurements for both polarizations; it is
expanded in the visible range in Fig. 5(b) to allow better visual assessment.
The average relative error is about $1\%$ in the range of $400-900$ nm. The
larger discrepancy of fitting and data occurs at short wavelengths in the
near-UV, where larger experimental measurement errors likely occur. Figure 6
shows the two principal refractive indices of the host LC calculated with
Sellmeier with parameters given in Table 2. Our LC host is a multicomponent
mixture, with manufacturer provided refractive indices $n_{e}=1.55$ and
$n_{0}=1.47$ without any wavelength dependence information. Our results show
reasonable agreement with the manufacture’s data.

####  4.2.2 dichroic dye and liquid crystal mixtures

Lastly, we report the characterization of the dielectric functions of two dye-
LC guest-host mixtures. The concentration of dye in the dye-LC mixture is
approximately $1\%$. The dielectric functions of the dye-LC mixtures are
modeled as Eqs. (4.1) -(4.2). The first two terms in both equations have
already been acquired in Sect. 4.2.1; here we only need to obtain the last
term which only contains the dye contribution.

The first dye, referred to as dye 1032, is orange in color. We have used four
causal Gaussian oscillators for the contribution of this dye. One is peaked at
$304.07$ nm for the slight absorption near $300$ nm, and two are at $455.08$
and $495.89$nm for the strong absorption near $465$nm, and one at $683.51$ nm
for slight absorption at longer wavelength. We note that a single oscillator
near the $465$nm is not sufficient to accurately fit both the breadth and
shape of the transmittance spectrum. The second dye, referred to as dye 4102,
is blue in color. The transmittance spectrum clearly shows two peaks between
$600-750$ nm, as shown in Fig. 7(b). We have tried to use three causal
Gaussian oscillators with one in the near-UV and two in the visible, but these
can’t reproduce the shape of transmittance spectrum near $600$ nm. We have
then used four causal Gaussian oscillators, one is peaked at $291.23$ nm for
the absorption in the near-UV, and three in the visible at $594.28$, $645.31$
and $688.63$ nm, respectively. The fit results for LC mixture with dyes 1032
and 4102 are shown in Fig. 7(a) and (b), respectively, showing excellent
agreement of the experimental measurements and theoretical predictions. The
average relative errors are $2\%$ in the range of 400-900 nm. Figure 8 shows
the corresponding refractive indices of dye-LC mixtures calculated from the
combined Sellmeier and causal Gaussian oscillator models with parameters given
in Table 2. Both dyes show only weak absorption for perpendicular polarization
in the visible; the values of $A_{\perp k}$ are much smaller than those of
$A_{\parallel k}$. This is an indication of a good alignment of dye molecules
with the nematic director.

(a) (b)

Figure 5: (a) Transmittance spectra of an LC sample without dye in a cell. (b)
Expanded region of (a) in the visible region. Solid red curves show
experimental results, and dashed blue curves indicate predictions from model
with two Sellmeier oscillators for ${\bm{\varepsilon}}_{LC}^{S}$ with fitted
parameters in Table 2.

(a)

Figure 6: Refractive indices of nematic LC for polarization (a) parallel and perpendicular to the LC director. | $\varepsilon_{\infty\parallel}=1.1649$ | $\varepsilon_{\infty\perp}=1.0149$ |  |
---|---|---|---|---
liquid crystal | $A_{\parallel k}$ | $A_{\perp k}$ | $\omega_{0k}$ |  $\sigma_{k}$ (THz)
$k=1$ | 1.0671 | 1.0341 | $2\pi c/125.01nm$ | $-$
$k=2$ | 0.0932 | 0.0482 | $2\pi c/289.62nm$ | $-$
dye 1032 |  |  |  |
$k=1$ | 0.5211 | 0.2701 | $2\pi c/304.07nm$ | $12896$
$k=2$ | 0.5098 | 0.0001 | $2\pi c/455.08nm$ | $16593$
$k=3$ | 0.3207 | 0.0210 | $2\pi c/495.89nm$ | $25654$
$k=4$ | 0.0191 | 0.0298 | $2\pi c/683.51nm$ | $7236$
dye 4102 |  |  |  |
$k=1$ | 0.4001 | 0.3799 | $2\pi c/291.23nm$ | $15375$
$k=2$ | 0.2234 | 0.0288 | $2\pi c/594.28nm$ | $12708$
$k=3$ | 0.5089 | 0.0794 | $2\pi c/645.31nm$ | $18050$
$k=4$ | 0.4119 | 0.0686 | $2\pi c/688.63nm$ | $40319$
Table 2: Fitted parameters in the combined Sellmeier and causal Gaussian
oscillator model with contributions from liquid crystal, dye 1032 and dye
4102. Here $c$ is the speed of light in air.

(a) (b)

Figure 7: Transmittance spectra of mixture (a) $1\%$ dye 1032 in LC and (b)
$1\%$ dye 4102 in LC. Solid red curves are experimental data, and dashed blue
curves are calculated transmittance of structures using the combined Sellmeier
and causal Gaussian oscillator model for ${\bm{\varepsilon}}_{LC+dye}$ with
fitted parameters in Table 2.

(a) (b) (c) (d)

Figure 8: Top: Refractive indices of $1\%$ dye 1032 in LC. Bottom: Refractive
indices of $1\%$ dye 4102 in LC. Left: principle refractive indices for
polarization along the LC director; Right: principle refractive indices for
polarization perpendicular to LC director.

To test the usefulness of our model, we examine the goodness of the
theoretical prediction of $0.5\%$ dye concentration in LC. We set
$c_{dye}=0.5\%$ in Eqs. (4.1)$-$(4.2) and keep all the other parameters the
same to obtain the dielectric functions of $0.5\%$ dye-LC mixtures via Eqs.
(4.1)$-$(4.2). We then use Eq. (2.13) with $m=9$ to calculate the
transmittance spectra of the cell. The results are given in Fig. 9, showing
excellent agreement with experimental measurements $0.5\%$ dye in LC, with
average relative errors $3\%$ in the range of 400-900 nm. We are confident to
conclude that the dye and liquid crystal mixtures we have studied behave
linearly, which makes the modeling straightforward and we can effectively
predict the transmittance at any concentration of dye-LC mixture, in the low
concentration regime.

In Ref. [18], the assumed dielectric function has uncorrelated real and
imaginary parts. This allows the use of the Beer-Lambert law to obtain the
extinction coefficients in a simple way, disregarding their dependence on the
substrate layers. Our method, however, deals with the more general case when
the real and imaginary parts cannot be assumed to be independent and multiple
reflections are properly considered in all the layers.

(a) (b)

Figure 9: Transmittance spectra of $0.5\%$ dye (a) 1032 and (b) 4102 in LC.
Solid red curves are experimental data, and dashed blue curves are calculated
values using the combined Sellmeier and causal Gaussian oscillator model for
${\bm{\varepsilon}_{dye+LC}}$ with fitted parameters in Table 2 and scaled
amplitudes.

##  5 Conclusions

In this work, we have characterized the wavelength dependence of the
refractive indices of dichroic dye-liquid crystal guest-host systems from the
near-UV/vis ($300$ to $900$ nm) transmittance spectra. The dye-LC mixtures are
contained in planar cells, where the substrates are glass coated with ITO and
PI layers. The PI layer is buffed in the plane to impose in-plane orientation
of the LC director, which in turn aligns the dichroic dye in the same
direction. When linearly polarized light parallel or perpendicular to the LC
director is normally incident on the sample, only one principal refractive
index of the dye-LC system contributes to the transmittance of the sample.
These experimental arrangments allow characterization of the two principal
components of a uniaxial material independently.

We have used the Sellmeier equation for the dielectric functions of the
transparent layers away from their absorption peaks, and the causal Gaussian
oscillator model for layers showing appreciable absorption in the near-UV/vis.
Both models satisfy the Kramers-Kronig causality relations. The parameters in
the dielectric function of each layer were obtained by minimizing the sum of
squared errors between theoretical predictions and measurements of the
transmittance.

Experimental measurements on substrate layers show no polarization dependence,
hence all substrate layers are modeled as optically isotropic. The dielectric
function of glass is modeled by the causal Gaussian oscillators with one peak
in the near-UV for the strong absorption around $300$ nm and one in the near
IR for the slight decrease in the transmittance at long wavelengths. The
dielectric functions of the ITO and PI layers are modeled by Sellmeier model
with two oscillators: one near-UV and one near-IR.

To model the two principal dielectric functions of the dye-LC mixtures, we
have used the combined Sellmeier and causal Gaussian oscillator models. We
assume that each oscillator has the same resonance location and width, but
different amplitudes, in the two principal directions, as indicated by the
experimental data. The dielectric function of the liquid crystal host was
characterized first, before introduction of the dye. The dielectric function
of the pure LC host is modelled by the Sellmeier equations with two near-UV
oscillators. The dye contribution to the dielectric function of dye-LC
mixtures is modeled by the addition of causal Gaussian oscillators. For the
two dye-LC mixtures reported, we have used four oscillators for each dye. Each
dye contains one oscillator in the near-UV, and the others are placed at
visible wavelengths to account for the absorption.

The causal Gaussian oscillator model can accurately characterize the
dielectric function of dichroic dyes in liquid crystals in the near-UV/vis
range in the low concentration regime. We have also used the same approach to
model commercial dyes in isotropic solvents and obtained similar excellent
quantitative agreement between theory and experiment.

Knowing the dielectric functions of dichroic dyes in LCs allows further
analysis of different variation of the guest-host LC devices. Based on our
results we expect to be able to predict the transmittance and reflectance of
samples with inhomogeneous director fields, with different dye concentration
and arbitrary angles of incidence in similar multilayer structures. The
transmittance spectrum of the sample can be mapped onto the colors perceived
by the human eye, leading, ultimately to the optimal design of LC devices.

## Acknowledgments

The dye-mixtures are provided by AlphaMicron Inc. This work was supported by
Air Force contract FA8649-20-C-0011 as part of the STTR AF18B-T003
Electronically Dimmable Eye Protection Devices (EDEPD) program and by the
Office of Naval Research through the MURI on Photomechanical Material Systems
(ONR N00014-18-1-2624).

## Conflicts of interest

There are no conflicts of interest to declare.

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