1. Introduction
Along with well-known display applications, liquid crystals (LCs) are of great interest for various photonic devices. These are electrically controlled beam-steering devices and switched gratings [
1,
2], lenses and arrays of high-aperture lenses [
3,
4,
5,
6,
7], liquid crystal metasurfaces [
8,
9,
10], as well as liquid crystal light amplifiers and microlasers [
11,
12,
13,
14,
15,
16]. In many cases, the operation of these devices presupposes an effective waveguide mode of light propagation. The waveguide regime in the LC layers has long been investigated in connection with liquid crystal optical switches and nematicons [
16,
17,
18,
19]. Recently, the waveguide mode was also proposed for the display applications [
20]. However, in recent decades, this regime has also become attractive from the point of view of laser generation in liquid crystal layers [
14,
15,
16]. For example, [
16] reports a significant decrease in the generation threshold in the LC lasers when using the in-plane geometry utilizing the waveguide mode. In [
14], a relatively low-threshold laser effect in the waveguide mode was obtained using a liquid crystal structure with a deformed lying helix. However, a deformed lying helix structure occurs only in the presence of an electric field, which in turn inevitably implies the existence of electrodes that bound the LC layer and, accordingly, waveguide energy losses in the electrodes. We believe that study of the losses that can appear in LCs for light propagating in the waveguide mode in order to decrease them is of crucial significance for in-plane low-threshold LC lasing.
As a rule, thin transparent indium tin oxide (ITO) coatings are used as electrodes. However, despite numerous studies of liquid crystals in waveguide mode, there is practically no information in the literature on the effect of ITO electrode parameters on light energy losses. This partly can be explained by the complexity of experimenting with the LC layers in the waveguide mode. It is rather difficult and expensive to vary, for example, the geometric parameters of ITO coatings in liquid crystal experiments. In addition, for the alignment of the LCs, additional thin layers are needed that can “shield” the LC layer from ITO electrodes, which makes it difficult to study the effect of ITO layers.
In this work, we try to reveal in detail the mechanisms of light loss associated with the presence of ITO coatings. The effect of ITO electrodes on the propagation of light through the LC layer in the waveguide mode is investigated numerically. The numerical approach makes it easy to both exclude and add thin layers that align LCs, thus allowing the impact of both ITO coatings and alignment layers to be assessed. As a numerical method, we use the finite difference time-domain (FDTD) method to solve Maxwell’s equations in the time domain. At the same time, we build the ITO dielectric model on the basis of experimental optical transmittance spectra obtained in a wide spectral range for industrial ITO-coated glasses used in LC display technologies. The use of the FDTD method, taking into account the spectral dispersion of ITO, makes it possible, in our opinion, to obtain the most realistic picture of the light propagation in the waveguide mode and the associated losses.
2. Specific Features of the Used FDTD Software and Lorentz Model
The FDTD method is currently one of the most popular for numerical modeling of complex optical systems. The method is based on the numerical solution of Maxwell’s equations in the time domain. The latter suggests that the relationship between electric and magnetic displacement and the corresponding field strengths also has a representation in the time domain. In the case of materials with a spectral dispersion of the dielectric permittivity, this relationship is not simple, and an appropriate model with time-domain equations is required. One of the most popular and versatile models that allows simulation of both dielectrics and metals taking into account the spectral dispersion is the Lorentz model [
21,
22]. This model, extended to describe optically anisotropic materials, is used in our FDTD software created by one of the authors (SPP) of this work. Below we provide information that makes it easier to understand the approach we use and the results of the simulation.
In the frequency domain, the components of the polarization vector
P of the material are expressed as a superposition of contributions from individual oscillators with an oscillator force
fm,j:
where Γ
m is the damping factor characterizing the width of the
m-th oscillator absorption band,
ω0,m is the resonant frequency of the
m-th oscillator,
ωp is the plasmonic frequency,
is the oscillator index corresponding to one of the principal axes of an anisotropic material,
Ej is the j-component of the electric field vector,
ε0 is the free space dielectric permittivity.
Note that, since in our case the LC materials are optically anisotropic and can be characterized by an inhomogeneous director distribution, the oscillators parameters are set independently for the three principal directions corresponding to the main axes of the local dielectric tensor. Since the main axes generally do not coincide with the xyz axes of the laboratory coordinate system, a coordinate transformation is required to find the corresponding components of the polarization vector and the components of the permittivity tensor in the laboratory coordinate system. Below, to simplify the presentation, the equations are written only for one of the spatial components of the polarization vector and the electric field under the assumption of the coincidence of the main axes and axes of the laboratory system, omitting the index j.
For each contribution in the frequency-domain expression (1) there is the following equation in the time domain:
In our software, Equation (2) is solved in the representation of displacement currents
Jm(
t)
= dPm(
t)
/dt as:
where the electric field
E(
t) is related to the electric displacement
D(
t) as:
where
P(
t) is time dependent polarization defined by contributions from all the oscillators,
ε∞ is the dielectric permittivity at a conditionally infinite frequency.
The power of the Lorentz model is that, for any material with a spectral dispersion defined in the frequency domain, we have ready-to-use equations in the time domain for updating the field vectors D and E in accordance with the FDTD algorithm.
The relative dielectric permittivity in the frequency-domain representation is
and for the real and imaginary parts of the permittivity one can obtain:
where
Am ≡
fmωp2.
The features of our software interface are such that, instead of optical frequency values, damping factor, and
Am, we use free space wavelengths (
λ), real part (
n) of the complex refractive index, and absorption coefficients (
k) connected to the imaginary part (
κ) of the complex refractive index as
k = 4
πκ/λ, which is more familiar to those who are engaged in optical spectroscopy in the visible spectral range. In this case, the relationship of these quantities with the complex dielectric constant is as follows:
and the parameters of the Lorentz bands in (6) and (7) are given by the values
km,
λ0,m, Δ
λm,
n∞, based on the following definitions:
where
c is the free space light speed,
km is the value of the absorption coefficient for the
m-th oscillator at a resonant wavelength
λ0,m = 2
πc/ω0,m, and
n∞ is real part of the refractive index at a conditionally infinite frequency, since the contribution to the real part of the dielectric constant from the oscillator at the resonant frequency according to (6) is zero.
Since the Γ
m characterizes the width of the absorption band, in our software it is accordingly defined in terms of an effective width Δ
λ absorption band as follows:
Thus, the individual oscillators in the software interface are defined by three parameters: (i) resonant absorption coefficient km, (ii) resonant wavelength λ0,m, (iii) effective absorption bandwidth Δλm.
To account for optical anisotropy, the three independent quantities km and n∞ are defined for the three main axes of the dielectric tensor. The main axes in the general case may not coincide with the axes of the laboratory coordinate system, and their orientation may vary in space. Therefore, to find the corresponding components of polarization, currents, electric field, and electric displacement in the laboratory system, an appropriate coordinate transformation is used. This allows us to simulate the LC systems with an inhomogeneous director distribution, although in this paper we restrict ourselves to considering homogeneously aligned LC layers in which the principal axes coincide with the axes of the laboratory xyz coordinate system, and ITO is considered as an optically isotropic material.
3. ITO Experimental Transmittance Spectra and Lorentz Model
Figure 1 (curve 1) shows the experimental spectrum of the ITO film on the glass substrate. Industrial ITO-coated soda-lime glass for display technology is used as the sample. In this case, the ITO film has a fairly low electrical resistance and, accordingly, a relatively high thickness of 170 ± 15 nm, measured by us both using an electron microscope and Linnik interferometer.
The transmittance spectrum of the ITO-coated glass substrate is measured using a Fourier-transform spectrometer of local design in a range of 450–2000 nm, as well as using an fiber optic spectrometer (Avaspec ULS2048 by Avantes, Netherlands) in a range of 360–1000 nm. The obtained data (
Figure 1) are in good agreement with literature [
23]. The spectra show pronounced Fabry–Perot oscillations with positions of spectral maxima depending on the ITO refractive index and thickness. In our case, these positions are well correlated with the data in [
23].
To determine the spectral dependence of the complex permittivity, we built the Lorentz model, which also is used in the FDTD simulations. The creation of the model is based on calculations of the transmittance spectrum of the ITO film on the glass substrate, taking into account the spectral dispersion of the glass [
24] when varying the set and parameters of the Lorentz oscillators until the difference between the calculated and experimental spectra comes within the measurement error of the latter,
Figure 1. The parameters of our model according to expressions (6)–(10) are presented in
Table 1.
The above set of oscillators does not necessarily correctly describe the absorption outside the spectral interval of our interest (360–2000 nm). This set is also not the only possible one for the correct description of the measured spectra of the transmittance. This, however, is not a problem affecting the reliability of numerical calculations, because the transmittance and reflectance are uniquely defined by the integral spectral dependence of the complex permittivity in the desired spectral range. It is natural that, in a restricted spectral range, the complex permittivity can be approximated within a given error by a superposition of form (5) with different sets of oscillators.
It should also be mentioned that, due to the Lorentzians peculiarity associated with the weak attenuation of the absorption on their “wings” (far from the maximum), in order to achieve the low absorption coefficients observed in the visible spectral range, we introduced a wide-band Lorentzian with a negative absorption coefficient (oscillator 3 in
Table 1). The spectral range of 360–2000 nm we are interested in is far from the maximum absorption of ITO in the ultraviolet (UV) region. Therefore, UV absorption is modeled by only a few narrow Lorentzians, providing the required normal spectral dispersion of the real part of the complex refractive index.
The model dependences of the spectra of the real and imaginary dielectric permittivity of ITO are shown in
Figure 2. The corresponding spectra of the real and imaginary parts of the complex refractive index are shown in
Figure 3. We will need these spectral dependences below when discussing the features associated with losses in the waveguide regime. It should be noted that, using these spectral dependences, the method of complex 4 × 4 matrices, known as the Berreman method [
25], can also be used to calculate the transmittance spectra of ITO films on glass substrates. These calculations were also performed, fully confirming the result in
Figure 1, obtained by the FDTD method.
In the spectral range of 360–1000 nm, the refractive index dependence
n(λ) in
Figure 3 is close to that for ITO in the same spectral range presented in [
26] (see data shown by square-shaped symbols). However, the values of the refractive index from [
26] are systematically higher compared to our data over the whole spectral range except for the short wavelengths range near 400 nm. The last can be associated not only with measurements errors (the most significant source of inaccuracy ~10% in our model is related to the measurements of the ITO-film thickness), but with the dependence of the ITO permittivity on the technological conditions of the ITO-film deposition.
Quite important is that, in the infrared (IR) region (at wavelengths greater than 1300 nm), the real part of the dielectric permittivity becomes negative (
Figure 2), and the refractive index
n (
Figure 3) is less than unit. This is due to the fact that in this spectral range, the contribution from long-wavelength oscillators 4–6 (
Table 1), due to anomalous dispersion, prevails over the contribution from the UV oscillators that provide normal spectral dispersion of the refractive index. Thus, in the IR region, the spectral behavior of ITO is similar to that of metals with the possibility of excitation of TM-polarized plasmonic states [
27].
4. FDTD Model of the Liquid Crystal Cell
Figure 4 shows the FDTD model of the optical system we studied. The model includes the LC layer placed between two ITO electrodes (ITO-1, ITO-2). The thickness of the LC layer ranged from 2 to 3 μm. The thickness of the ITO electrodes also varied from 0.1 to 0.2 μm. To study the influence of the alignment layers, additional layers are placed onto the ITO surface (see
Section 5.3), which exist in the experimental samples and serve to align the LC. To facilitate the interpretation of the data, we neglected the spectral dispersion of the principal values of the refractive indices of the LC, and we used the following values:
n|| = 1.7,
n⊥ = 1.52. These values are shown by straight horizontal lines in
Figure 3 for comparison with the corresponding values of the ITO refractive index (
n). We also neglected the absorption coefficient of the LC, which in the experimental samples is too small to make a meaningful contribution to the losses through the simulated length
L of the LC layer along the
x direction. The simulated waveguide length
L varied in the range from 5 to 20 μm and is limited to a maximal value of 20 μm, taking into account significant computing resources required for the simulations with a spatial resolution up to 10 nm.
Numerical studies are carried out for two fundamentally important LC alignments: (i) homeotropic (vertical), when the long axes of the LC molecules (director) are perpendicular to the ITO surface (z-axis); and (ii) planar, wherein the LC director is in the y-direction.
Due to the homogeneity of the optical system in the y-direction, the FDTD simulations are restricted by solving the two-dimensional problem.
A pulsed unidirectional light source is used,
Figure 4. The duration of the light pulse is 2 fs, which provides a sufficiently wide spectrum for virtual registration of the electromagnetic field in the spectral range from 360 to 2000 nm, corresponding to the spectral range of our ITO model. The light pulse is injected directly at the left boundary of the LC layer and, after passing the length
L along the
x direction, is registered by the virtual light sensor. The transmittance spectrum is calculated as the ratio of the average values of the Poynting vector on the sensor
z-line at the output of the layer to the corresponding values of the light source found as a result of the Fourier transform of the time dependences of the electromagnetic field. The time dependences of the electromagnetic field are recorded on a time interval that provides, according to the properties of the Fourier transform, a spectral resolution of about 2 nm in the visible range.
Thick quartz substrates, onto which the ITO electrodes are coated, are taken into account by a transparent medium with a refractive index
n0 of 1.46, which is the external medium with respect to the ITO-LC-ITO system (spectral dispersion is not taken into account to facilitate the interpretation of the data). To eliminate the reflections from the boundaries of the computational domain, the “uniaxial perfectly matched layers” (UPML) are used [
21,
22,
28].