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Some Features of Magneto-optics of Cholesteric Liquid
Crystals
Journal:
Journal of Modern Optics
Manuscript ID
Draft
Manuscript Type:
Regular Paper
Date Submitted by the
Author:
n/a
Complete List of Authors:
Gevorgyan, A; Far Eastern Federal University
Golik, S. S.; Far Eastern Fed Univ
Vanyushkin , Nikolay ; Far Eastern Federal University
Borovsky, Anton; Far Eastern Federal University
Gharagulyan, Hermine; Institute of Chemical Physics NAS RA
Sarukhanyan, Tatevik; Yerevan State University
Hautyunyan, Meruzhan; Yerevan State University
Matinyan, Gvidon; Armenian National Agrarian University
General JMO Keywords:
Optical and photonic materials (inc. metamaterials)
Keywords:
cholesteric liquid crystal, Weil semimetals, magneto-optical activity,
eigen polarization, ellipticity, rotation
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Journal of Modern Optics
For Peer Review Only
Some Features of Magneto-optics of Cholesteric Liquid Crystals
A.H. Gevorgyan1*, S.S. Golik1,2, N.A. Vanyushkin1, A.V. Borovsky1, H. Gharagulyan3, T.M.
Sarukhanyan4, M.Z. Harutyunyan4, G.K. Matinyan5
1 School of Natural Sciences, Far Eastern Federal University, 10 Ajax Bay, Russky Island,
690922 Vladivostok, Russia
2 Institute of Automation and Control Processes, Far East Branch, Russian Academy of
Sciences, 690041 Vladivostok, Russia
3 Dept of Physics, Yerevan State University, 1 Alex Manoogian Str., 0025 Yerevan, Armenia
4 Institute of Chemical Physics NAS RA, 5/2, P. Sevak Str., 0014 Yerevan, Armenia
5 Dept of Agrarian Engineering, Armenian National Agrarian University, 74, Teryan Str., 009
Yerevan, Armenia
* Correspondence: agevorgyan@ysu.am
Abstract: We investigated the features of the optical properties of magnetoactive cholesteric
liquid crystals (CLCs) in an external static magnetic field. The boundary-value problem of light
transmission through a finite layer of a magnetoactive CLC at normal incidence of light and
reflection of light from a half-space is considered. We also investigated the evolution of the
reflection with a change in the azimuth and ellipticity of the incident light. The features of the
reflection spectra, optical rotation, and polarization ellipticity, as well as the ellipticity spectrum
of the eigen polarizations, are investigated, too. The peculiarities of the evolution of the ellipticity
of the eigen polarization with a change in the magneto-optical activity parameter are considered.
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Key words: cholesteric liquid crystal, Weil semimetals, magneto-optical activity, eigen
polarization, ellipticity, rotation
1. Introduction
In recent years, there has been renewed interest in magneto-optics of cholesteric liquid crystals
(CLCs), which is due to, in particular, the following circumstances. First, due to the considerable
interest in photonic crystals (PCs) with tunable parameters [1,2]. Second, CLCs in an external
magnetic field combine the natural optical activity due to the structure of the medium and the
magneto-optical activity, the joint presence of which leads to interesting features, in particular,
to the nonreciprocity, including the nonreciprocal transmission (reflection and absorption), and
to Faraday chiral anisotropy [3-5]. Third, at large values of the magneto-optical activity
parameter, a complex band structure appears in CLCs even at normal light incidence,
magnetically induced transmission is observed and the total field excited in the medium has an
elliptical polarization rather than a linear one, as in the absence of external magnetic fields [4-9]
(see analogical investigations for 1D magnetic PCs [10-13]).
CLCs are self-organized one-dimensional PCs formed by rod-shaped molecules that are attached
to chiral molecules in such a way that they form a helical structure [14]. The periodic structure
of a CLC leads to the appearance of a polarization-sensitive photonic band gap (PBG). CLC
layers have a variety of practical applications such as temperature and pressure sensors, gas
sensors, super-twisted nematic liquid crystal displays, tunable band-pass filters, and rewritable
color recorders; thermal printing for electronic paper; reflective displays without polarizer; and
prototypes of reflective smart windows, optical diodes, there are also applications in lasers, etc.
[3-5,15-30]. The CLC parameters can be easily controlled by external electric or magnetic fields,
temperature gradient, infrared radiation, etc. Appropriate manipulation of the parameters of the
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CLC layer makes it possible to fine-tune its PBG and other characteristics. The polarization-
sensitive PBG lies between the wavelengths and , where p is the pitch of the
helical structure, and are the ordinary and extraordinary refractive indices of
a locally uniaxial structure. Note that CLC suspensions of various micro- or nanoparticles or dye
molecules have recently become the subject of renewed interest because they combine the
fluidity and anisotropy of liquid crystals with the special properties of micro- or nanoparticles or
dye molecules. The presence of nanoparticles (either ferroelectric, or ferromagnetic, or magneto-
optical media) or dye molecules in the structure of a CLC leads to a significant change in its local
parameters (both dielectric and magnetic, as well as magnetoactive), such as: a change in the
phase transition temperature of the isotropic phase − liquid crystal phase; a significant change in
the width of the PBG and frequency localization; a change in the number of PBGs; a change in
the coefficients of the elasticity of CLC; a significant increase in the tunability of the CLC, etc.
(see [31,32] and some references therein).
However, there are a number of important topics that have not been covered yet. That is, the
problems of eigen polarizations (EPs, about eigen polarizations see below), the features of
reflection of light with EPs, the rotation of the plane of polarization and the ellipticity of
polarization, and some others remain open. This work will be devoted to these issues, namely, to
the study of the features of the light reflection spectra at different values of the magneto-optical
activity parameter and different polarizations of the incident light; to the evolution of the EP
ellipticity spectra of the CLC layer, again with a change in the magneto-optical activity
parameter; and also, to the study of the features of the optical rotation and polarization ellipticity
of the light transmitted through the CLC layer.
2. Models and methodology
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Let us consider the reflection and transmission of light through a planar CLC layer, that has
magneto-optical activity in an external magnetic field directed along the axis of the CLC helix
(Figure 1). We will assume that the tensors of dielectric permittivity and magnetic permeability
have the form:
, , (1)
where
are the principal values of the local dielectric
permittivity tensor in the presence of an external magnetic field, g is the parameter of magneto-
optical activity, a = 2π / p, p is the pitch of the helix in the presence of an external magnetic field.
We consider the case of normal incidence that is when the light propagates along the z-axis of
the helix.
Figure 1. The geometry of the problem. The large ellipsoids represent the anisotropic molecules,
which are rotating continuously forming a helicoidal structure along the z-axis.
The exact analytical solution of Maxwell's equations for CLCs in an external magnetic field when
light propagates along its axis is known [4-6]. According to [4-6], the dispersion equation has the
form:
, (2)
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where
,
is the angular frequency and c is the speed of light in a vacuum.
Using the eigenvalues of the wave numbers found from (2), we solve the problem of reflection
and transmission in the case of a planar CLC layer of finite thickness. We assume that the optical
axis of the CLC, which coincides with the z-axis, is perpendicular to the boundaries of the layer.
The CLC layer is bordered on both sides by isotropic half-spaces with the same refractive indices
equal to ns. The boundary conditions, consisting in the continuity of the tangential components
of the electric and magnetic fields, represent a system of eight complex linear equations with
eight complex unknowns. By solving this boundary value problem, we can determine the values
for the components of the reflected
and transmitted
fields, and, therefore, calculate
the energy reflection coefficients
, transmission
,, absorption ,
rotation of the plane of polarization
ellipticity of polarization
etc. Here
, and are the x and y components of the electric
field of the transmitted wave.
Further, all calculations were performed for a magnetically active CLC layer with the following
parameters: ε1 = 0.8, ε2 = 0.35, the helix of the CLC layer is right-handed, its pitch is p = 400 nm,
and the CLC layer thickness is d = 5p. The choice of these parameters is dictated by the fact that
the effects revealed below are more pronounced for the given values of the ambient parameters.
Naturally, the question arises, how realistic are CLCs with these parameters? Such CLC
structures can be created artificially. As noted above, CLC suspensions with various micro- and
nanoparticles have recently become the subject of renewed interest. By dissolving nanoparticles
with the appropriate parameters of the dielectric constant in the CLC, it is possible to obtain a
structure of the CLC type with the above parameters for the dielectric constant. CLC-like
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structures can also be created artificially using an appropriate anisotropic material as a source of
physical vapor deposition [33]. Further, in our calculations, the parameter g changes in the range
(0 ÷ 0.7). According to [34], such an interval of variation of the parameter g is achieved when
the external magnetic field Bext changes in the interval (1 ÷ 5) T, e.g., in CeF3, which is quite
achievable under modern conditions. However, according to [35], such values for g in Weyl
semimetals are also obtained for much smaller values of Bext. At first, we consider the case
, that is the CLC layer is in a vacuum. Then, to minimize the influence of dielectric boundaries,
we also consider the case , i.e., the CLC layer is sandwiched between two semi-infinite
isotropic spaces with refractive indices equal to the average refractive index of the CLC layer.
Here, first, we note that the value = 0.76 does not practically correspond to the real
situation, and second, we have considered some spectra for EPs, that depend on the wavelength
of the incident light and can vary at large intervals. EPs are two polarizations of the incident light,
which do not change when passing through the system under study [36,37]. Our calculations
performed for the "practical case scenario", that is, in particular, for the case , when the
CLC is in a vacuum, and the incident light has, for example, right-hand circular polarization,
show that the main effects considered in the work naturally depend on the external environment,
but are not critical. On the other hand, such a formulation of the problem, namely, the
consideration of CLCs with values for ε1 and ε2 less than unity, with large values for the magneto-
optical activity parameter g, and consideration of the case of the minimal influence of dielectric
boundaries allows one to reveal interesting new manifestations in the magneto-optics of CLCs,
and also gives more a complete understanding of some of the features of CLC magneto-optics.
When considering the reflection of light from a half-space, we have one boundary and, therefore,
only four equations of continuity for the tangential components of the electric and magnetic
fields. Consequently, not all eigenmodes will be excited (equation (2) is of the fourth order, and
therefore has four solutions). Of these four modes, we choose two, whose imaginary parts of the
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wavenumbers are positive (with the introduction of weak absorption). Indeed, in this case, the
amplitudes of these modes decrease exponentially as they propagate deep into the medium. In
the opposite case, these amplitudes would increase indefinitely, which is incompatible with the
finite amplitude of the incident wave.
3. Results and discussion
In this section we present the simulation results: by solving the dispersion equation (2), we obtain
the dependences of the wave vectors of the medium on the wavelength, then, by solving the
boundary value problem, we proceed to the representation of the spectra of reflection,
transmission, rotation of the plane of polarization and ellipticity of polarization, azimuth and
ellipticity of the EP, as well as the evolution of the spectra of ellipticity of the EP with a change
in the magneto-optical activity parameter g.
Figure 2 shows the reflection spectra at different values of g. The light incident on the layer has
right- (curve 1) and left- (curve 2) circular polarizations, as well as linear polarizations along the
x-axis (curve 3) and y-axis (curve 4) at . The left column (Figure 2 a, c, e, g, i, k, m)
represents the spectra of light reflection from the CLC half-space, and the right column (Figure
2 b, d, f, h, j, l, n) are the spectra of light reflection from a CLC layer of finite thickness. As it is
known [3,4], an external magnetic field leads to a shift of the PBG, and both at g> 0 and at g <0
this shift is of the same order and is directed towards short wavelengths; therefore, this shift is
quadratic in g and with an increase in the parameter g the PBG undergoes a blue shift. For small
values of the parameter g, this shift is insignificant and can be neglected. However, for large
values of parameter g, this shift can no longer be neglected. With an increase in the parameter g,
the frequency width of the PBG also changes (it expands). With a further increase in g, the further
shifts of PBG towards short waves take place (Figure 2 g, h, i, j). Then, with a further increase
in the parameter g, starting from its certain value, a new PBG appears in the long-wavelength
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part of the spectrum (Figure 2 g, h, i, j). The long-wavelength boundary of this new PBG (the
second PBG) approaches infinity λ → ∞. This new PBG is sensitive to the polarization of the
incident light and depends on the direction of the external magnetic field. Thus, if g> 0, then light
with left-handed circular polarization undergoes diffraction reflection, and in the case, g <0 light
with right-hand circular polarization undergoes diffraction reflection. With a further increase in
the parameter g, the short-wavelength boundary of the new PBG undergoes a blue shift and,
starting from a certain value of the parameter g, this new (second) PBG occupies the entire
spectrum (Figure 2 k, l, m, n).
0
0.2
0.4
0.6
0.8
1
100 200 300 400 500
g=0
3
(a)
1
2
4
R
l(nm)
0
0.2
0.4
0.6
0.8
1
100 200 300 400 500
g=0
3
(b)
1
2
4
0
0.2
0.4
0.6
0.8
1
100 200 300 400 500
g=0.25
3
(c)
1
2
4
0
0.2
0.4
0.6
0.8
1
100 200 300 400 500
g=0.25
3
(d)
1
2
4
0
0.2
0.4
0.6
0.8
1
100 200 300 400 500
g=−0.25
3
(e)
1
2
4
0
0.2
0.4
0.6
0.8
1
100 200 300 400 500
g=−0.25
3
(f)
1
2
4
0
0.2
0.4
0.6
0.8
1
100 200 300 400 500 600 700 800
g=0.58
3
(g)
1
2
4
0
0.2
0.4
0.6
0.8
1
100 200 300 400 500 600 700 800
g=0.58
3
(h)
1
2
4
0
0.2
0.4
0.6
0.8
1
100 200 300 400 500 600 700 800
g=−0.58
3
(i)
1
2
4
0
0.2
0.4
0.6
0.8
1
100 200 300 400 500 600 700 800
g=−0.58
3
(j)
1
2
4
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Figure 2. Reflection spectra for different values of the parameter g.
Now we investigate the influence of azimuth and ellipticity of the incident light on the reflection.
By representing the incident light field components in the form and
, where is the incident light ellipticity and is its azimuth, we investigate the
influence of incident light azimuth and ellipticity on the reflection at different values of the
parameter of magnetooptical activity g. Figure 3 shows the evolution of the reflection with a
change in the parameters and at different values of g and for wavelengths in the center of
PBG: (a) g=0, l=300 nm; (b) g=0.58, l=200 nm; and (c) g=−0.58, l=200 nm. As can be seen
from these evolutions, the change in the azimuth and the ellipticity of the incident light at
wavelengths inside the PBG is significant, it changes in the interval (0 ÷ 1). At wavelengths
outside the PBG, this interval is relatively small and approaches zero far from the PBG. Further,
as it is seen in Figure 1a and 1c, for each value of azimuth ψ, a change in ellipticity ϵ from a
value of −1 to +1 leads to a change in the reflection from 0 to 1, i.e., at a fixed azimuth, a change
in ellipticity can modulate the reflection with a depth modulation equal to unity. When the
ellipticity is fixed, the azimuth changes from a value of −900 to a value of +900, the reflection
changes in a relatively small interval. In this case, the modulation depth is of the order of (0.2–
0.3) and less. However, an interesting situation occurs in the case when two PBGs arise. In this
case, at wavelengths in a short-wavelength PBG, a change in the azimuth ψ from a value of −900
0
0.2
0.4
0.6
0.8
1
100 200 300 400 500
g=0.7
3
(k)
1
2
4
0
0.2
0.4
0.6
0.8
1
100 200 300 400 500
g=0.7
3
(l)
1
2
4
0
0.2
0.4
0.6
0.8
1
100 200 300 400 500
g=−0.7
3
(m)
1
2
4
0
0.2
0.4
0.6
0.8
1
100 200 300 400 500
g=−0.7
3
(n)
1
2
4
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to a value of +900 leads to a change in the reflection in a significantly wide interval and the
modulation depth is of the order of (0.9 ÷ 1)․ In the absence of absorption, the changes in
reflection and transmission are equal in magnitude, and therefore the same can be said about
transmission. Thus, the CLC layer can be used as a practically ideal monochromator, by changing
the azimuth at fixed ellipticity or by changing the ellipticity at a fixed azimuth of the incident
light. And as it is well known, high-speed, power-efficient light modulation is in high demand
for a variety of photonic devices used as building blocks of displays and optical information
processors. These include tunable lenses, focusers, wave-front correctors, and correlators [38–
44]. Liquid crystal spatial light modulators are widely used as devices to modulate the amplitude,
phase, or polarization of light waves in space and time [45].
(a)
(b)
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(c)
Figure 3. The evolution of the reflection with a change in parameters and at . (a) g=0,
l=300 nm (in the center of the PBG); (b) g=0.58, l=200 nm (again in the center of the PBG);
and (c) g=−0.58, l=200 nm (again in the center of the PBG).
We now turn to EPs. EPs of CLCs layer at normal incidence of light are orthogonal, almost
circular polarizations (at small values of local birefringence). The picture changes significantly
at high values of local refraction and in the presence of magneto-optical activity.
Figure 4. The spectra of CLC layer EP ellipticity at .
Figure 4 shows the spectra of the EP ellipticity at various values of the parameter g but now at
. Here we have e2=−e1. As can be seen from Fig. 4, in the presence of an external
magnetic field, the ellipticity of the EP is characterized by strong dispersion, varying in
considerable intervals. It is easy to see that for g = 0.58 and for g = −0.58, the EPs differ from
0
0.2
0.4
0.6
0.8
1
100 300 500
l
(nm)
1
2
3
e1
1. g =0
2. g =0.58
3. g = −0.58
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each other and, moreover, if for g <0 the EP is quasi-circular polarization, then for g> 0 it differs
greatly from circular polarization.
Figure 5. Reflection spectra from the CLC layer at different values of the parameter g at
.
Figure 5 shows the reflection spectra from the CLC layer for the light reflection at g = 0.58 and
g = − 0.58. The light incident on the layer has right (curve 1) and left (curve 2) circular
polarizations, as well as polarizations that coincide with the first (curve 3) and second (curve 4)
EP. As can be seen from Figure 5b at g = −0.58, the reflection of the incident light for the first
and second EP practically coincides with the reflection of the incident light with right-hand and
left-hand circular polarizations (curves 1 and 3, 2 and 4, respectively, practically coincide).
Whereas, at g = 0.58 (Figure 5a), the reflection of incident light for the first and second EP differ
significantly from the reflection of incident light with right and left circular polarizations,
especially in the short-wavelength part of the spectrum (curve 3 differs significantly from curve
1, and curve 2 from curve 4, and this difference is much larger in the short-wavelength part of
the spectrum). This is due to the above-mentioned features of EPs.
For completeness, in Figure 6 we show the evolution of the ellipticity spectra of the EP with a
change in the parameter g.
0
0.2
0.4
0.6
0.8
1
100 300 500
l
(nm)
1
2
3
4
R
g=0.58
(a)
0
0.2
0.4
0.6
0.8
1
100 200 300 400 500 600
1
2
3
4
g= −0.58
(b)
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Figure 6. Evolution of the ellipticity spectra of the EP with a change in the parameter g at
.
The optical rotation of the plane of polarization occurs due to the difference in the phase
velocities of light with right and left circular polarizations. Far from the PBG, optical rotation in
the CLC occurs due to the chiral structure of the CLC. In the PBG, a wave with one circular
polarization experiences strong diffraction reflection, while a wave with another circular
polarization weakly interacts with the medium, as a result the optical rotation in the CLC in the
PBG turns out to be much larger than the intrinsic rotation caused by the optical activity of the
CLC [46]. In the presence of an external magnetic field, another mechanism of rotation of the
plane of polarization appears and it becomes important to study the features of the spectra of
optical rotation and ellipticity of polarization when linearly polarized light (polarized along the
x-axis) is incident on the CLC layer.
Figure 7 shows the spectra of ellipticity (curve 1) and optical rotation (curve 2) for linearly
polarized light incident on the CLC layer at various values of the parameter g. Comparison of the
curves in Figure 7 (a) at g = 0 with similar curves in [45] shows the coincidence of the main
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regularities (note that in our case ) and serves as an additional check of the
correctness of our approach. As can be seen from Figure 6, the external magnetic field has a
significant effect on optical rotation and polarization ellipticity. One should not expect a simple
summation of the intrinsic rotation of the CLC and the Faraday rotation, since the rotation in the
CLC is of a diffractive nature too, and besides this, the external magnetic field (at large values
of the magneto-optical parameter) itself, as was shown above, also significantly affects the band
structure of the CLC.
-1
-0.5
0
0.5
1
1.5
2
2.5
100 200 300 400 500
l
(nm)
e,
j
(rad)
g=0
1
2
(a)
-1
-0.5
0
0.5
1
1.5
100 200 300 400 500
l
(nm)
e,
j
(rad)
g=0.25
1
2
(b)
-1.5
0
1.5
3
4.5
6
100 200 300 400 500
l
(nm)
e,
j
(rad)
g= −0.25
1
2
(c)
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Figure 7. Spectra of ellipticity (curve 1) and optical rotation (curve 2) at different values of the
parameter g. d = 2p. .
4. Conclusion
The paper investigates the features of the optical properties of magnetoactive CLCs in an external
magnetic field. An external magnetic field leads to a displacement of the PBG, and this
displacement is quadratic in g. With an increase in g, the PBG shifts towards shorter wavelengths.
Then, with a further increase in the parameter g, a new PBG appears in the long-wavelength part
of the spectrum starting from a certain value of g. The long-wavelength boundary of this PBG
approaches infinity λ → ∞. This new PBG is sensitive to the polarization of the incident light
and depends on the direction of the external magnetic field. Thus, in this new PBG, for g> 0,
light with left-handed circular polarization undergoes diffraction reflection, and for g <0, light
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
100 200 300 400 500
l
(nm)
e,
j
(rad)
g=0.58
1
2
(d)
-1.5
-1
-0.5
0
0.5
100 200 300 400 500
l
(nm)
e,
j
(rad) g= −0.58
1
2
(e)
0
0.5
1
1.5
100 200 300 400 500
l
(nm)
e,
j
(rad) g=0.7
12
(f)
-1
-0.5
0
0.5
100 200 300 400 500
l
(nm)
e,
j
(rad) g= −0.7
1
2
(g)
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with right-hand circular polarization undergoes diffraction reflection. It is worth noting that a
complex band structure at normal incidence is also formed in CLCs based on metamaterials [47].
Acknowledgments
The authors (AHG, SSG, NAV, AVB) would like to thank the Ministry of Science and Higher
Education of the Russian Federation FZNS-2020-0003 No. 0657-2020-0003.
Disclosure statement
No potential conflict of interest was reported by the authors.
Funding
The authors (AHG, SSG, NAV, AVB) wish to express their gratitude to the Ministry of Science
and Higher Education of the Russian Federation FZNS-2020-0003 No. 0657-2020-0003.
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