|  Dynamic Kerr and Pockels Electro-Optics of Liquid Crystals in Nanopores for
Active Photonic Metamaterials†
---|---
|  Andriy V. Kityk,∗a Marcjan Nowak,a Manuela Reben,b Piotr Pawlik,c Monika
Lelonek,d Anatoliy Andrushchak,e Yaroslav Shchur,f Nazariy Andrushchak,g and
Patrick Huber∗hij
|  Photonic metamaterials with properties unattainable in base materials are
already beginning to revolutionize optical component design. However, their
exceptional characteristics are often static, as artificially engineered into
the material during the fabrication process. This limits their application for
in-operando adjustable optical devices and active optics in general. Here, for
a hybrid material consisting of a liquid crystal-infused nanoporous solid, we
demonstrate active and dynamic control of its meta-optics by applying
alternating electric fields parallel to the long axes of its cylindrical
pores. First-harmonic Pockels and second-harmonic Kerr birefringence
responses, strongly depending on the excitation frequency- and temperature,
are observed in a frequency range from 50 Hz to 50 kHz. This peculiar behavior
is quantitatively traced by a Landau-De Gennes free energy analysis to an
order-disorder orientational transition of the rod-like mesogens and
intimately related changes in the molecular mobilities and polar anchoring at
the solid walls on the single-pore, meta-atomic scale. Thus, our study
evidences that liquid crystal-infused nanopores exhibit integrated multi-
physical couplings and reversible phase changes that make them particularly
promising for the design of photonic metamaterials with thermo-electrically
tunable birefringence in the emerging field of spacetime metamaterials aiming
at a full spatio-temporal control of light.

##

††footnotetext: a Faculty of Electrical Engineering, Czestochowa University of
Technology, Al. Armii Krajowej 17, 42-200 Czestochowa, Poland, E-mail:
<EMAIL_ADDRESS>b Faculty of Materials Science and
Ceramics, AGH-University of Science and Technology, al. Mickiewicza 30, 30-059
Cracow, Poland. ††footnotetext: c Faculty of Production Engineering and
Materials Science, Czestochowa University of Technology, Al. Armii Krajowej
19, 42-200 Czestochowa, Poland. ††footnotetext: d SmartMembranes GmbH,
Heinrich-Damerow-Str. 4, 06120 Halle(Saale), Germany. ††footnotetext: e
Institute of Telecommunications, Radioelectronics and Electronic Engineering,
Lviv Polytechnic National University, 12 Bandery St., Lviv 79046, Ukraine.
††footnotetext: f Institute for Condensed Matter Physics, 1 Svientsitskii
str., 79011 Lviv, Ukraine. ††footnotetext: g SPC SoftPartners, 97 Konovalca
str., 79057 Lviv, Ukraine. ††footnotetext: h Hamburg University of Technology,
Institute for Materials and X-Ray Physics, 21073 Hamburg, Germany,
<EMAIL_ADDRESS>††footnotetext: i Deutsches Elektronen-
Synchrotron DESY, Centre for X-Ray and Nano Science CXNS, 22607 Hamburg,
Germany. ††footnotetext: j Hamburg University, Centre for Hybrid
Nanostructures CHyN, 22607 Hamburg, Germany.

##  1 Introduction

2-D and 3-D metamaterials offer exciting opportunities for the manipulation of
light propagation and light-matter interactions 1, 2, 3. Although
metamaterials have proven to be a versatile platform for passive optics,
active controlling of their optical properties is still a challenging task.

Here phase-change materials, in particular employing inorganic, hard matter
with an amorphous to crystalline transition open up many opportunities for
implementing externally tunable metamaterials as the reversible transition is
accompanied by a considerable change of the refractive index 4, 5, 6. Also
embedding liquid crystals (LCs) in nanoporous solids allows one to combine
self-assembly and phase transition behavior of organic, soft matter with a
scaffold structure that provides mechanical robustness, and thus to fabricate
soft-hard hybrids with tunable multi-physical couplings in terms of self-
organization, mechanical, electrical and optical properties 7, 8, 9, 10, 11.

If the pore size is below the visible light wavelength the single pores act as
meta-atoms. The effective optics is then determined by the pore shape, size
and orientation. Moreover, the pore filling allows one to adjust the light-
matter interaction on a meta-atomic scale 12, 9, 13, 11. Effective optical
properties not achievable in base materials are possible 14, 15, 16, 12, 17,
18, 2, 19, 20, 11, specifically adjustable birefringence and thus
polarization-dependent light propagation speeds 9. However, the resulting
optics is also here often static as dictated by the fabrication process and
can not easily be dynamically adapted, as it is demanded for the emerging
field of active and spacetime photonic metamaterials aimed at a full spatio-
temporal control of light 21, 3, 22, 6. Here, we follow the strategy to
overcome this static behavior by applying external electrical fields to liquid
crystals confined in anodic aluminum oxide nanopores.

Fig. 1: Dynamic electro-optics of a liquid crystal-infused nanoporous photonic
metamaterial. (a) A linearly polarized laser beam is split up into two waves,
i.e., the extraordinary (red) and ordinary wave (blue), when it traverses a
material consisting of liquid crystal-infused cylindrical nanopores. The
optical anisotropy, as illustrated by the ellipsoid of refractive indices
(indicatrix), results in different propagation speeds and thus in a phase
difference (retardation) $R$ between the two transmitted waves after passing
the birefringent medium. Here the extraordinary beam is slower than the
ordinary beam resulting in a positive phase retardation $R$ of the
electromagnetic waves after passing the pore array. This corresponds to a
prolate indicatrix and a ”positive" optical birefringence, i.e., $\Delta
n=n_{\rm e}-n_{\rm o}>0$. If the liquid-crystalline filling is exposed to an
external electrical field $E(\Omega)$ alternating with a circular frequency
$\Omega$, it responses with molecular reorientations and the phase shift of
the transmitted wave components consists of two frequencies, a first-harmonic,
$R(1\Omega)$ and a second-harmonic response, $R(2\Omega)$ corresponding to a
linear, Pockels and a quadratic, Kerr electro-optic effect, respectively. Note
that for clarity only the electrical fields of the electromagnetic waves are
shown.

Hereby, we have to consider that in many previous experimental and theoretical
studies 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37 it has been
shown that the self-assembly behavior and molecular mobility can be
substantially modified in spatially confined geometries. More importantly, the
impacts of these confinement effects on the electro-optics have been barely
explored so far. Therefore, part of our study will also explore the phase
behavior of the confined liquid crystal and its impact on the thermo-optical
and electro-optical couplings in nanoconfinement.

In the following we provide a brief introduction into electro-optic effects
and present then our experimental study aimed at integrating these
functionalities in a nanoporous composite. The electro-optic effect was first
reported by John Kerr in 1875 38. While examining optically isotropic media he
noticed electric field-induced optical birefringence (anisotropy), the
magnitude of which increased quadratically with the electric field strength.
This behavior is known nowadays as Kerr’s law. Eight years later, Röntgen and
Kundt independently reported electro-optic effects in quartz and tourmaline,
i.e., in anisotropic piezoelectric crystalline materials 39. In contrast to
the Kerr effect, the electrically induced birefringence changed linearly with
the applied electric field. In a number of later experiments it was
demonstrated that the linear electro-optic effect is observable only in non-
centrosymmetric crystals and it was properly described by Pockels in 1893
within a phenomenological theory based on crystal symmetry and tensor analysis
39.

Thus, in crystalline materials both linear or quadratic electro-optic effects
are essentially related to anisotropic effective optical properties. The
corresponding tensor analysis shows that inversion symmetry implies that a
linear response is forbidden. For this reason the Pockels effect is in
particular observable in piezoelectric crystals with broken inversion
symmetry. Moreover, it may be found in poled glasses or ceramics 40 and
likewise in other noncrystalline polar materials as well as chiral LC
materials, for example in tilted chiral smectics (ferroelectric LCs) 41, 42,
43, 36.

The Kerr effect, in contrast, is observable also in centrosymmetric media,
including isotropic liquids. It is particularly large in liquids with polar
molecules, such as e.g. nitrobenzene, being widely applied in electro-optical
light shutters and modulators 44.

Both linear Pockels and quadratic Kerr effects are caused by the rotational
diffusion of the molecules, the dipoles of which are able to follow to the
applied electric field. Liquids consisting of small polar molecules are
characterized by rather fast electro-optic response times, usually in the
range of nanoseconds or even below 45, 46, whereas the dynamics slows down for
large molecules 47, 48.

From the application point of view both electro-optic effects provide means
for a refractive index modulation driven by an applied electric field. This
offers an elegant possibility to integrate the modulation of phase retardation
or light intensity into a material. They are thus the base for a large variety
of applications in science and technology aimed at the spatio-temporal control
of light 39.

Here we report a dynamic electro-optical study (electro-optical time response)
on the rod-like nematic LC 7CB embedded into cylindrical nanopores of anodic
alumina oxide (AAO) membranes. Linear and quadratic electro-optical effects
are explored via first and second harmonic response, respectively, measured by
a lock-in technique in a frequency window from 50 Hz to 50 kHz and a broad
temperature range encompassing paranematic and confined nematic states. These
electro-optic measurements are complemented by simultaneous polarimetric
measurements of the optical retardation (birefringence) exploring the
temperature evolution of the static orientational order in the region of the
paranematic-to-nematic transition. The obtained results are analyzed within a
phenomenological Landau-De Gennes model accounting for the interaction of the
nematic order parameter and the applied electric field.

##  2 Experimental

###  2.1 Materials and sample preparation

The nematogen liquid crystal 7CB (4-heptyl-4-cyano biphenyl, see Fig. 2) was
purchased from Merck AG. A characteristic feature of 7CB is a simple phase
diagram in the bulk state. It exhibits only a single first order phase
transition during cooling from the isotropic (I) to the nematic (N) phase at
$T_{\rm IN}$ = 315.5 K 49 before solidification. The anodic aluminum oxide
(membrane thickness $h$ =130 $\mu$m) was synthesized by means of a double-
stage electrochemical etching process of pure aluminum in an oxalic acid (30%)
electrolyte. The resulting mesoporous AAO matrices consist of a hexagonally
arranged, parallel array of slightly conical channels (mean pore diameter 65
nm), see Fig. 2. The conicity of the pores results from the unidirectional
etching process 50.

The nematic 7CB melt was embedded into the cylindrical nanopores by
capillarity-driven spontaneous imbibition 51. To build an electro-optical cell
for the dynamic electro-optic and the polarimetric measurements the AAO:7CB
composite membrane was clamped between two transparent ITO glass electrodes,
see Fig. 2. The electro-optic cell was placed into a temperature cell operated
by a temperature controller (Lakeshore-340) with a control accuracy of 0.01 K.

Fig. 2: Electro-optic experiment and detection scheme. (a, b) Scanning
electron micrographs of the AAO membrane ($d$=65 nm) used as host matrix. (c)
The electro-optic sample cell consists of an AAO membrane, filled with (d) the
calamatic LC 7CB, stacked between two ITO-glass electrodes. (f) The setup to
measure simultaneously the optical retardation and the dynamic electro-optic
response combines a photo-elastic modulation polarimeter and a lock-in
amplifier for the detection of the first (1$\Omega$) and second (2$\Omega$)
harmonics corresponding to the linear (Pockels) and quadratic (Kerr) electro-
optic effects, respectively. It consists of a He-Ne - helium-neon laser (633
nm), PEM - photo-elastic modulator, P - polarizer, A - analyzer, BC - Babinet
compensator, TC - controlled temperature cell, S - sample (AAO:7CB membrane
stacked between ITO-glass electrodes), PD - broad band diode photodetector.
(e) Dependence of the light intensity vs. the phase retardation: the modulated
retardation converted to the modulated light intensity. An electro-optic sine-
wave-retardation, $\Delta_{i}(E)\sin(\Omega_{i}t+\varphi_{i})$
($\Omega_{i}=\Omega$, 2$\Omega_{2}$) is converted in an undistorted way into a
modulated intensity, provided that the static retardation $\Delta_{0}=(2n\pm
1/2)\pi$ $(n=0,1,2,..)$ and $\Delta_{i}(E)\ll\pi$.

###  2.2 Electro-optic setup

In Fig. 2 the laboratory setup for simultaneous optical retardation and
electro-optic measurements is shown. It combines a photo-elastic modulation
polarimeter and lock-in amplifier technique for a reference registration of
the first (1$\Omega$) and second (2$\Omega$) harmonics of the electro-optical
response. Since the AAO membrane consists of an array of parallel tubular
nanopores both the empty AAO matrices as well as the LC nanocomposites exhibit
effectively an uniaxial optical anisotropy with one optical axis parallel to
the long pore axis 25, 27, 26. This optical uniaxiality remains unchanged both
as a function of temperature or under an applied electric field. Thus the
observation of changes in the optical indicatrix is only possible in a tilted
membrane geometry, see Fig. 1. Accordingly, the sample was tilted with respect
to the normal incidence of the laser beam by $\sim$ 40 degree about the
incident laser beam direction, see Fig. 1 and 2.

The measuring process switched between two detection modes, the polarimetry
and the electro-optic response mode, respectively. The polarimetry mode is
active when the photo-elastic modulator (PEM, $f=$ 50 kHz, PEM-90 Hind
Instruments) was turned on. The optical retardation, $\Delta$, is defined as
$\Delta n\,L$, where $\Delta n$ is the effective birefringence in the
direction of light propagation and $L$ is the length of the pathway of light
traveling through the anisotropic medium. In this mode a simultaneous
reference detection of the first ($r_{1f}$) and second ($r_{2f}$) harmonics of
modulated light (hereafter termed PEM harmonics) by means of the lock-in
amplifier (Stanford Research, SR-830) yielded the value of the optical
retardation $\Delta=\arctan(kr_{1f}/r_{2f})$. Here the factor $k$, which is
defined by a ratio of Bessel functions $J_{2}(A_{0})/J_{1}(A_{0})$ at the
retardation PEM amplitude ($A_{0}=$ 0.383 $\lambda$) and the photodetector
(PD) frequency response characteristic, was determined within a calibration
procedure. Similar polarimetry setups have been described in earlier studies,
for additional details see Refs. 42, 25, 52, 53, 9, 54.

###  2.3 Dynamic electro-optic analysis

In the electro-optic response mode the PEM was turned off. An alternating
electric field $E_{0}\sin{(\Omega t)}$ ($E_{0}\sim$3000 V/cm) was applied
parallel to the nanopore axis by applying electric potentials between the
transparent ITO electrodes. This induces an electro-optic retardation. In the
low frequency limit ($\Omega\tau\ll 1$) the linear electro-optic effect
implies that the electro-induced retardation $\Delta_{e}\propto
E_{0}\sin(\Omega t)$, i.e., its phase modulation is of the same frequency as
the exciting electrical field, $\Omega$. For the quadratic electro-optic
effect, in contrast, $\Delta_{e}\propto E^{2}_{0}\sin^{2}(\Omega
t)=E^{2}_{0}(1-\cos(2\Omega t))/2=E^{2}_{0}/2+(E^{2}_{0}/2)\sin(2\Omega
t+\pi/2)$ the electric field-induced retardation is characterized by the
static component ($\propto E^{2}_{0}$) and a dynamic component of doubled
frequency, $2\Omega$ that is phase shifted by $\pi/2$ and $\propto E^{2}_{0}$.
At higher frequencies relaxation processes emerge ($\Omega\tau\sim 1$) leading
to a phase shift $\varphi$ in the electro-optic response. In the polarizer-
analyzer system the electric field-induced dynamic retardation is converted
into a modulated light intensity, which is detected by the photodetector PD
and subsequently analyzed by the lock-in amplifier SR-830 with respect to the
applied reference voltage extracting thus the amplitude
($\rho_{{}_{\Omega_{i}}}$) and phase ($\varphi_{i}$) of its first ($1\Omega$)
and second ($2\Omega$) harmonic components, of the complex electro-optic
responses, I.e. $R^{*}_{1\Omega}=\rho_{{}_{1\Omega}}\exp(i\varphi_{1})$ and
$R^{*}_{2\Omega}=\rho_{{}_{2\Omega}}\exp(i(\varphi_{2}+\pi/2))=i\rho_{{}_{2\Omega}}\exp(i\varphi_{2})$,
respectively.

Please note that a retardation-to-intensity conversion function in the system
with crossed polarizer P and analyzer A is defined as $\propto
I_{0}\sin^{2}(\Delta/2)=I_{0}(1-\cos\Delta)/2$, see sketch in Fig. 2, where
$I_{0}$ is the incident light intensity of the coherent He-Ne laser light
($\lambda=$633 nm). The total retardation $\Delta$, in general, represents a
superposition of the static retardation, $\Delta_{0}$, induced by the sample
(S) and the Babinet compensator (BC), as well as the dynamic electro-optic
component, $\Delta_{i}(E_{0})\sin(\Omega_{i}t+\varphi_{i})$. Accordingly, the
dynamic retardation is converted to the modulated light intensity in an
undistorted way (linear conversion), if the static retardation
$\Delta_{0}=(2n\pm 1/2)\pi$ $(n=0,1,2,..)$ whereas $\Delta_{i}(E_{0})\ll\pi$.
Taking into account that the sample birefringence changes with temperature the
BC serves here to adjust precisely the static phase shift to the condition
mentioned above. The corresponding manual adjustment, which corresponds to the
condition that the first PEM harmonic $r_{1f}$ reaches a maximum (i.e.
$r_{2f}=0$), was made precisely in the polarimetry mode at each temperature
set point prior to the electro-optic response measurements.

The magnitude of the first PEM harmonic, $r_{1f}(A_{0}=0.383\lambda)$, which
is proportional to the incident light intensity $I_{0}$, has also been used as
reference value for the normalization of the electro-optic response amplitudes
$\rho_{{}_{1\Omega}}$ and $\rho_{{}_{2\Omega}}$. The electro-optic retardation
amplitude $\Delta_{i}(E_{0})$, on the other hand, did not exceed in our
experiments one angular minute. Thus both requirements for linear retardation-
to-intensity conversion indeed have been perfectly fulfilled. Note that the
amplitudes $\rho_{{}_{1\Omega}}$ and $\rho_{{}_{2\Omega}}$ are normalized by a
value of $10^{-6}r_{1\omega}(0.383\lambda)$ as reference magnitude, that is
extracted in polarimetry mode for a PEM retardation amplitude set to
$0.383\lambda$.

##  3 Results and discussion

###  3.1 Temperature-dependent static collective orientational order and
dynamic electro-optics

Fig. 3: Temperature-dependent static nematic order in comparison with dynamic
electro-optics and the corresponding free-energy modeling in an AAO:7CB
photonic metamaterial. (a) Temperature dependence of the measured optical
retardation and its theoretical description by a free energy model (KKLZ-
model). (b) Temperature dependence of the measured quadratic electro-optic
response (Kerr effect) and its best fit based with the KKLZ model. Inset in
(b) voltage dependence of the second-harmonic electro-optic response at a
temperature above and below the paranametic-to-nematic phase transition,
$T_{\rm PN}$, respectively. The yellow and green backgrounds indicate the
nematic and paranematic temperature ranges, respectively.

The optical retardation, $R(T)$ characterizes the evolution of the static
orientational ordering inside the nanochannels ($R\propto S$, where $S$ is the
nematic order parameter) as a function of temperature, see Fig. 3. In the
paranematic phase ($T>T_{PN}$) $R(T)$ is small indicating an almost isotropic
confined LC. However, the $R(T)$-dependence exhibits a characteristic
pretransitional tail typical of nematic precursor ordering in the interfacial
region of tubular nanochannels 25. A nearly step-like increase at $T_{PN}$
indicates a discontinuous evolution of the paranematic-to-nematic ordering
caused by an abrupt appearance of nematic molecular arrangement in the core
region of the pore filling, see Fig. 3 25.

Fig. 4: Temperature dependence of the static nematic order in comparison to
the resulting dynamic electro-optics of the AAO:7CB photonic metamaterial.
Temperature dependences of the amplitudes of (a,b) the first- and (c,d)
second-harmonic electro-optic responses at a driving frequency of 420 Hz and a
voltage amplitude of 40 V. (b,d) Electro-optic response in the vicinity of the
paranematic-to-nematic transition. The temperature dependence of the optical
retardation, $R$ is presented for comparison. Insets in (b) and (d), voltage
dependences of the amplitude of the first and second harmonic electro-optic
response, respectively, measured at fixed temperatures in the paranematic and
nematic states. The yellow and green backgrounds indicate the nematic and
paranematic temperature ranges, respectively.

How is this thermotropic orientational order connected with the electro-optic
response to an alternating external electrical field $\vec{E}$, see Fig. 1. In
Fig. 4 we show the amplitudes of the first [$\rho_{{}_{1\Omega}}$] and second
[$\rho_{{}_{2\Omega}}$] harmonics of the electro-optic response vs.
temperature $T$ at a fixed driving frequency of 420 Hz and a voltage amplitude
of 40 V. Whereas the observation of a finite quadratic Kerr amplitude is
expected as a response of the confined LC, we observe also a linear Pockels
effect, at first glance symmetry forbidden, since a non-centrosymmetric
structure of the nanocomposite is not apparent. The existence of both distinct
electro-optic effects is unambiguously corroborated by the voltage
dependencies of the amplitudes of the first and second harmonic amplitude,
$\rho_{{}_{1\Omega}}(U)$ and $\rho_{{}_{2\Omega}}(U^{2})$. As expected they
scale linearly and quadratically on the amplitude of the applied voltage and
thus to the applied external electrical field, respectively.

We trace the linear electro-optic response of the confined mesogens to a
preferential unipolar interfacial ordering that breaks the macroscopic
centrosymmetry of the bulk isotropic or nematic ordering. However, given the
cylindrical pore wall geometry this would still result in centrosymmetric
structures. Thus we suggest that the conicity of the channels, as dictated by
the fabrication process 50, results in a macroscopic non-centrosymmetric
polarizability along the channel axis and thus causes the Pockels effect. It
is interesting to mention that a similar impact of conicity on second harmonic
generation has recently been reported for chromophores embedded in conical
silica nanochannels 55.

In Fig. 5 we show the frequency dispersion of the real and imaginary parts of
the first ($R^{*}_{1\Omega}=\rho_{{}_{1\Omega}}\exp(i\varphi_{1})$, left) and
second ($iR^{*}_{2\Omega}=-\rho_{{}_{2\Omega}}\exp(i\varphi_{2})$, right)
harmonics of the electro-optic response measured at different temperatures.
The dynamic electro-optic measurements are limited in our experiments to a
frequency window of 0.05-50 kHz. Whereas its upper frequency limit is given by
the instrumental constrains of the lock-in technique, the lower one originates
from an ionic conductivity rising strongly at lower frequencies. It results in
additional phase shifts and losses in the low frequency electro-optic
responses below $\sim$200 Hz, especially at higher temperatures, see mark in
Fig. 5. The ionic conductivity influences both the linear and quadratic
electro-optic responses and thus distorts the low-frequency dipolar dynamics.
At driving frequencies above 400 Hz, however, the ionic conductivity is
considerably suppressed. Therefore, we usually omit the frequency region below
400 Hz in our discussion.

In the paranematic phase $T>T_{PN}$ linear and quadratic electro-optic
responses indicate evidently quite different frequency dispersions, compare
the panels a) and i) in Fig. 5. The linear response, symmetry forbidden in the
bulk, exhibits a slow relaxation of Debye-like type observed in the frequency
dispersions of its real ($\textnormal{Re}R^{*}_{1\Omega}$) and imaginary
($\textnormal{Im}R^{*}_{1\Omega}$) parts, see panel (a) and (b) in Fig. 5. A
characteristic relaxation maximum in the
$\textnormal{Im}R^{*}_{1\Omega}(\omega)$-curve appears at a frequency
$f_{r}=2\pi/\omega_{r}$ of about 0.5 kHz.

The quadratic electro-optic response, by contrast, remains practically
constant in the paranematic phase in the frequency range up to 50 kHz
indicating thus a much faster relaxation with characteristic frequencies
beyond the _f_ -window of the electro-optic measurements.

Fig. 5: Frequency-dependent electro-optics of an AAO:7CB photonic metamaterial
above and below the paranematic-to-nematic phase transition. Frequency
dependence of the real and imaginary parts of (left) the first
($R^{*}_{1\Omega}=\rho_{{}_{1\Omega}}\exp(i\varphi_{1})$, and (right) the
second ($iR^{*}_{2\Omega}=-\rho_{{}_{2\Omega}}\exp(i\varphi_{2})$, right)
harmonic electro-optic response measured at different temperatures in the
confined paranematic (green background) and nematic states (yellow
background).

These distinct electro-optical dynamics originates in the heterogeneous
molecular dynamics of molecular matter confined in nanopores. It has been
demonstrated in a series of experimental studies that molecules show a slow
molecular dynamics, in particular relaxation behavior near the LC-pore wall
interface and a fast relaxation in the core region of the pore filling 56, 57,
58, 59, 60.

Thus the fast quadratic Kerr electro-optic response is dominated by the
molecular dynamics originating from the isotropic (centrosymmetric) core
region, whereas the linear Pockels effect ows his slow dynamics to the slow
interfacial dipolar relaxations, in agreement with our conclusions above
regarding its origin.

Cooperative molecular ordering in LCs, which accompanies phase
transformations, may lead to an additional slowing down of the relaxation
dynamics or even its anomalous behavior in the region of phase transformations
43, 52. Therefore, a peculiar behavior can be found for bulk liquid-
crystalline materials. In the isotropic phase far above the clearing point
nematic LCs (with dipolar mesogens) behave like dipolar liquids with a
quadratic electro-optic response that diverges upon approaching the transition
temperature to the nematic phase. According to the Landau-de Gennes free
energy model for LC phase transitions 61, 62 the magnitude of the Kerr effect,
as quantified by the Kerr constant, growths as $(T-T^{*})^{-\gamma}$ where
$T^{*}$ denotes the second-order pretransitional temperature (supercooling
limit of the isotropic phase) and $\gamma$ is a critical exponent equal to 1
in mean-field approximation. This anomalous behavior was first observed by
Tsvetkov and Ryumtsev 63 and demonstrated in subsequent studies, see e.g. the
Refs. 64, 65, 66.

Also here for the confined L-C state both the Pockels and the Kerr effect
exhibit an anomalous behavior in the vicinity of the paranematic-to-nematic
phase transition, see Figs. 3 and 4.

In the confined nematic phase, immediately below the transition temperature
$T_{PN}$, the relaxation dynamics of the quadratic electro-optic response
experiences a strong slowing down apparently caused by nematic ordering of the
entire core region. The corresponding broad relaxation band in the dispersion
curve $\textnormal{Im}iR^{*}_{2\Omega}(\omega)$ occurs in the frequency region
of 10-20 kHz, see Fig. 5(k-m). Interestingly, the slow relaxation of the
linear electro-optic response, which is still dominated in close vicinity of
$T_{PN}$ disappears at lower temperatures, see Fig. 5 (c-e). It is replaced by
a considerably faster response dynamics with characteristic relaxation rates
in the frequency range of 30-40 kHz as evidenced in panel g) and h) in Fig. 5.

Fig. 6: Illustration of temperature-dependent nematic order in the vicinity of
the isotropic-to-nematic phase transition. (left) Orientational ordering of
the rod-like mesogens inside the cylindrical channel in the paranematic state
($T>T_{PN}$), in the vicinity of the phase transition ($T\approx T_{PN}$) and
in the nematic state ($T<T_{PN}$). (right) Sketch of the phase boundary
(domain wall) separating the symmetry broken (non-centrosymmetric) interface
region (yellow) and the centrosymmetric core region (cyan).

The origin of this peculiarity deserves a more detailed discussion. According
to Bengoechea and Aliev 56, 67 and in agreement with subsequent dielectric
studies by Calus _et al._ 58, 59 slow relaxations of confined liquid crystals
suggest that the mobility of molecules in the surface layers is strongly
affected by boundary conditions. Near-interfacial layers are characterized by
a considerably larger effective viscosity compared to the bulk one which
solely slows down the dipolar relaxation and can thus not explain an abrupt
change in the response dynamics. Therefore, we suggest that rather an
alternative mechanism based on interphase boundary dynamics in an alternating
electric field is responsible for the peculiar dynamics.

The left sections of Fig. 6 sketch the orientational ordering of the rod-like
LC molecules inside a cylindrical channel in the paranematic state
($T>T_{PN}$), in the close vicinity of the phase transition ($T\approx
T_{PN}$) and in the confined nematic state ($T<T_{PN}$), respectively. The
right section sketches the corresponding phase boundary ("domain wall")
separating the symmetry broken (non-centrosymmetric) interfacial polar regions
and the centrosymmetric core region. An alternating electric field, applied
along the pore axis, results in an oscillating radial shift of the phase
boundary causing thereby an alternating optical birefringence, i.e., the
dynamic electro-optic effect.

Note that similar domain-like or soliton approaches have been successful in
the explanation of anomalous behavior of dielectric and elastic properties in
a number of ferroelectric and ferroelastic materials, see e.g. the Refs.68,
69, 70. The corresponding domain wall dynamics is due to its collective
character slow compared to the dynamics of individual molecules. This feature
is evidently presented in the frequency dispersions of the imaginary part of
the linear electro-optic response, $\textnormal{Im}R^{*}_{1\Omega}(\omega)$,
recorded in the paranematic phase: A characteristic relaxation band appears in
the frequency range of several hundred Hertz, see Fig. 5 (a-d). The electro-
optic contribution due to the alternating phase boundary dynamics is expected
to be substantial whenever the birefringence of neighboring phase regions,
i.e., the interfacial birefringence, $\Delta n_{i}$, and the core
birefringence, $\Delta n_{c}$ substantially differ from each other, see Fig.
6.

Analytically, an effective linear electro-optic coefficient
$r\propto\Delta(\Delta n)/\Delta E\propto(\Delta n_{i}-\Delta
n_{c})\cdot(\Delta x/\Delta E)$ results, where $\Delta x$ is the radial shift
of the phase boundary (domain wall) induced by the applied electric field
$\Delta E$. In the paranematic phase, $\Delta n_{i}\gg\Delta n_{c}=0$ thus the
domain wall contribution with slow relaxation dynamics is evidently presented
here, see panel a) and b) in Fig. 5. Far above $T_{PN}$ the interfacial polar
regions form just rare local islands. Their number grows however as the
temperature decreases transforming them into closed surface layers upon
approaching the paranematic-to-nematic transition temperature $T_{PN}$. This
behavior is characterized by a pretransitional tail, i.e., a slowly rising
retardation, evidently seen in the temperature dependence of the optical
retardation above $T_{PN}$, see Fig. 4.

Note that in the close vicinity of $T_{PN}$ the core region is still isotropic
($\Delta n_{c}=0$) thus due to the large interphase area the linear electro-
optic response is anomalous, see Fig. 5 (c, d) and Fig. 4 (a, b). An abrupt
almost step-like increase in the optical retardation right below $T_{PN}$
indicates a complete nematic ordering in the core region.

Considerably below $T_{PN}$ $\Delta n_{i}\approx\Delta n_{c}$ thus the domain
like contribution, which is $\propto(\Delta n_{i}-\Delta n_{c})$, becomes
evidently suspended, see Fig. 5 (g, h). The remaining linear electro-optic
response is fast and originates from the orientational dynamics of individual
molecular dipoles in an alternating electric field. The surface induced
polarity in the interfacial molecular layer may be transferred via
intermolecular interactions towards the nematic core region of the pore
filling. Accordingly, a larger number of molecular dipoles may contribute to
the linear electro-optic response explaining its larger magnitude in the
confined nematic phase compared to the paranematic one.

###  3.2 Landau-De Gennes free energy analysis of confined liquid-crystalline
order and Kerr electro-optics

Whereas the description of the linear electro-optic response has qualitative
character, the Kerr effect in the AAO:7CB nanocomposite and the quadratic
electro-optic response may be described phenomenologically by employing the
Kutnjak-Kralj-Lahajnar-Zumer (KKLZ)-model 24, 71. The orientational order
inside the nanochannels results in an excess optical retardation $\Delta
R\propto S$ (see Fig. 3a, where $S=\frac{1}{2}\langle 3\cos^{2}\phi-1\rangle$
is the scalar order parameter, $\phi$ is the angle between the characteristic
axis of the molecules and a direction of preferred local molecular orientation
(the so-called director $\vec{n}$).

The KKLZ model extends phenomenological the Landau-de Gennes approach towards
spatially confined nematic phases by introducing a so-called nematic ordering
field $\sigma$ which couples bilinearly with the nematic order parameter 24,
71. For the untreated host AAO matrix the channel walls enforce planar
anchoring, whereas their elongated cylindrical geometry induces preferential
ordering of guest rod-like molecules parallel to the long channel axes. The
ordering field $\sigma$ is proportional to the anchoring strength and
inversely proportional to the pore diameter, i.e. it is defined by host-guest
interactions at the channel wall interface and its geometrical constraints.
The KKLZ free energy is expressed in terms of the dimensionless scaled order
parameter $q=S/S_{0}$, where $S_{0}=S(T_{IN})$ stands for the nematic order
parameter value at the isotropic-to-nematic transition temperature of the bulk
system, $T_{IN}$. The dimensionless free energy of the spatially confined
nematic LC is then represented as follows:

$f=f(q)+f(q,\sigma)+f(q,\kappa)+f(q,E),$ (1) $\displaystyle
f(q)=tq^{2}-2q^{3}+q^{4},$ (1a) $\displaystyle f(q,\sigma)=-q\sigma,$ (1b)
$\displaystyle f(q,\kappa)=\kappa q^{2},$ (1c) $\displaystyle
f(q,E)=-a_{1}qE^{2}-a_{2}q^{2}E^{2}.$ (1d)

where $t=(T-T^{*})/(T_{IN}-T^{*})$ is the dimensionless reduced temperature
and $T^{*}$ is the effective temperature. Here $f(q)$ represents the free
energy of the bulk system, $f(q,\sigma)$ describes the couplings between the
nematic order parameter $q$ and the geometric ordering field $\sigma$,
$f(q,\kappa)$ accounts for quenched disorder effects, $f(q,E)$ characterizes
the coupling between the order parameter $q$ and an external electric field
$E$ applied along the pore axis.

For symmetry reasons the electric field couples quadratically with the order
parameter, i.e. two terms $qE^{2}$ and $q^{2}E^{2}$ are presented in the
Landau-de Gennes free energy 72. The quenched disorder ($\kappa$-therm)
results in a downward shift of the effective transition temperature $t_{n}$,
whereas all other coupling terms rise their value as
$t_{n}=1+\sigma-\kappa+a_{1}E^{2}+a_{2}E^{2}$. Here we assume that the
electric coupling constants $a_{1}$ and $a_{2}$ are positive, i.e., an
application of the electric field lowers the free energy of the axial nematic
state making it energetically preferable. Moreover, the bilinear coupling term
$q\sigma$, likewise $qE^{2}$-term, induces a nematic ordering also above
$t_{n}$. The geometrical constraint results in a precursor behavior with
characteristic paranematic tails observed in the temperature dependence of the
nematic order parameter. In the vicinity of the paranematic-to-nematic
transition depending on a pore diameter the order parameter is characterized
by a discontinuous subcritical ($\sigma<1/2$) or a continuous overcritical
($\sigma>1/2$) evolution 24, 71, 25, 26, 27, 73.

Minimization of the free energy expansion (eq. 1) with respect to the order
parameter $q$ at $E=0$ gives its equilibrium value $q_{e}$. It scales linearly
with the measured excess retardation, $\Delta R$. A transfer from a reduced
dimensionless temperature scale to the normal allows one to calibrate the
model parameter $\Delta T^{*}=T_{IN}-T^{*}$ to 3.4 K, as it was derived in
previous polarimetric studies 27. A best fit in the region of the nematic-to-
isotropic (paranematic) phase transition yields $\sigma=$0.18 and
$\kappa=$0.15. Both values agree well with Ref.27 as extrapolations of
$\sigma$\- and $\kappa$-values obtained in a series of AAO:7CB nanocomposites
with host matrices of smaller pore diameters.

The solid line in Fig. 4a represents the best fit of experimental data points
obtained within the KKLZ model. The dotted line traces the thermo-optic
contribution (base line) obtained within the same fitting procedure. Overall,
the KKLZ model describes remarkably well the behavior of the optical
retardation in the paranematic state as well as in a broad temperature region
of paranematic-to-nematic phase transition. Deviations far below $T_{PN}$ can
be attributed to a saturation of the nematic ordering in the core region of
pore filling that are originates the limitation to fourth-order terms 26.

The amplitude of the quadratic electro-optic response, measured at an applied
field $E=U/h$ ($U$ is the applied voltage), scales linearly with the
difference $\Delta R(E)-\Delta R(0)$. $\Delta R(E)$ may be derived similarly
to $\Delta R(0)$ as described above, i.e., by the minimization of the free
energy expansion (eq. 1) with respect to the order parameter $q$ taking into
account the electrical coupling terms of the free energy defined by eq. 1c.
The quadratic electro-optic response derived in such a way,
$\rho_{{}_{2\Omega}}=k[\Delta R(E)-\Delta R(0)]\propto[q_{e}(E)-q_{e}(0)]$
($k$ is the scaling parameter) is shown in Fig. 4b. The measured quadratic
electro-optic response is presented for comparison. Here the coupling
constants $a_{1}$=1.0$\cdot$10-13 cm2/V2 and $a_{2}$=1.3$\cdot$10-12 cm2/V2
have been chosen in a way that provides the values of field induced optical
retardation close to those ones observed in the experiment.

The presented phenomenological model well reproduces basic features of the
anomalous electro-optic behavior, particularly the asymmetric peak of the
quadratic electro-optic response. A discrepancy between the theory and
experiment, evidently seen in the confined nematic state, may have several
reasons. Our phenomenological approach considers the static electro-optic
response ($\omega=$ 0), whereas the experiments explore the quasi-static
electro-optic response ($\omega/2\pi=\leavevmode\nobreak\ $420 Hz). The chosen
frequency can be considered as a compromise providing, on the one hand, a
reduced influence of the ionic conductivity, and on the other hand, still
insignificant dipolar relaxation losses. However, at least a weak influence of
the ionic conductivity on the measured electro-optic response, which is
considerably temperature-dependent, indeed is presented. Also spatial
inhomogeneities of the channel diameters (up to 10% radius variation) can
result in gradient field effects and likewise field depolarization at channel
interfaces.

##  4 Conclusion

We presented a study on the electro-optical dynamics of the rod-like liquid
crystal 7CB embedded into parallel arrays of cylindrical nanopores in anodic
aluminum oxide. Linear and quadratic electro-optical effects are explored via
first- and second-harmonic polarimetry responses that are measured by a lock-
in technique in a frequency window from 0.05 - 50 kHz and a broad temperature
range comprising a phase transition between a confined paranematic and nematic
state.

We find that a linear electro-optical response, symmetry forbidden in bulk
nematics, occurs in both states due to polar ordering at the nanopore walls.
The dynamics of the quadratic electro-optical response reveals anomalous
dynamical behavior at the paranematic-to-nematic phase transformation that can
be quantitatively described by a Landau-De Gennes free energy model. The
linear and quadratic electro-optical effects are characterized by
substantially different molecular relaxation rates caused apparently by a slow
and fast mesogen reorientation in the proximity of the pore wall and in the
pore center, respectively.

One can envision that the observed integrated electro-optics that sensitively
depends on multiphyiscal couplings is particularly interesting to induce
lateral phase shift gradients for electromagnetic waves in solid supports or
surfaces at will by external electrical field and temperature gradients. Such
active photonic metamaterials are demanded for the design of in-operando
adjustable optical components 74, 75, 76, 77, 78, 21.

Thus, similarly as it has been demonstrated for liquid-infused porous
structures with electrochemical actuation 79 and wetting changes 80 as well as
versatile multifunctional material behavior in general 81 our study indicates
that liquid crystal-infused nanoporous solids allow for a quite simple
fabrication of electro-active functional nanomaterials. The simple
functionalization approach by capillarity-driven spontaneous imbibition of the
liquid-crystalline melt in combination with tailorability of anodic aluminum
oxide 82, 83, 84, 36 and the availability of other self-organized, optically
transparent mesoporous media 85, 86, 60, 10 indicates also the versatile
character of the presented material fabrication process.

Finally, our study is a fine example, how the combination of soft and hard
matter opens up the possibility of using multi-scale self-assembly and phase
transitions to transport peculiar multi-physical couplings in geometrical
confinement from the nano- via the meso- to the macroscale in order to design
robust hybrid materials with integrated functionality, similar as found in
many biological composites 87, 88.

## Author Contributions

A.V.K., A.A., M.R., N.A., and P.H. conceived and designed the electro-optic
experiments. A.V.K, M.L., and P.P. fabricated the nanocomposites. A.V.K and
M.N. performed the optical experiments. A.V.K. and Y.S. analyzed the data.
A.V.K., Y.S., and P.H. interpreted the experiments. A.V.K. and P.H. wrote the
manuscript. All authors proofread and edited the manuscript.

## Acknowledgements

We received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skodowska-Curie grant agreement No.
778156. Support from resources for science in the years 2018-2022 granted for
the realization of the international co-financed project No. W13/H2020/2018
(Dec. MNiSW 3871/H2020/2018/2) is also acknowledged. We also profited from
support by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) within the collaborative reserach center SFB 986 ”Tailor-Made
Multi-Scale Materials Systems”, project number 192346071).

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