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Abstract
Liquid crystals are of great interest in the field of nonlinear optics due to their efficient response to low intensity light fields. Here we present a new, to the best of our knowledge, mechanism of a nonlinear optical response which is observed for a dye-doped dual-frequency nematic liquid crystal. The local increase in temperature caused by the absorption of light beam in the liquid crystal medium leads to spatial variation and inversion of the sign of the dielectric anisotropy. When an alternating current electric field with a frequency close to the cross-over frequency is applied to the liquid crystal cell, the planar orientation sustains at the beam periphery, but elastic deformation occurs in the irradiation region. In the case of a dye dopant with negative absorption dichroism, a first-order orientational transition with large bistability region is obtained.
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1. Introduction
Liquid crystals represent unique optical materials with a large anisotropy of refractive indices, optical transparency, and high responsiveness to external stimuli [1]. A lot of electro-optical and display technologies are based on the reorientation of the liquid crystalline director, that is, the direction of the average orientation of the long molecular axes, under the action of an electric field. In particular, dual-frequency nematic liquid crystals (DFNLCs) [2] with positive dielectric anisotropy $\Delta \varepsilon$ below a certain cross-over frequency $f_c$ and negative above $f_c$ are widely considered for fast electro-optical modulators [3–6] and other photonic applications [7–10].
Nematic liquid crystals (NLCs) are responsive not only to constant (dc) or low-frequency electric field (ac), but also to light fields. Nonlinear optical properties of liquid crystals are intensively studied due to their evident applications in optics and photonics: adoptive phase elements, waveguides, image processing, beam steering, tunable metamaterials and plasmonic photonic structures, etc. [11–15].
Non-electronic optical nonlinearity is typically associated with a reorientation of the NLC director and a change in the nematic order parameter [11–13]. The nature of the orientational processes depends on the particular NLC composition and experimental conditions. In optically transparent liquid crystals, the director reorientation due to the dielectric anisotropy at light frequencies dominates [16]. This allows one to obtain the light-induced Fréedericksz transition, i.e. an orientational transition of the second order. When a NLC contains or consists of light absorbing molecules, other orientational effects are revealed. The most known is the Janossy effect, i.e. the enhancement of the optical response in dye-doped NLCs [17,18]. In this case, the director rotation is caused by a variation in intermolecular forces due to the excitation of dye molecules in the NLC bulk. In NLCs containing azobenzene high-molar-mass dopants, the first-order orientational transition is realized because of the relationship between the director reorientation and fractions of azobenzene isomers forming light-induced torque [19,20].
The combination of light and low-frequency electric fields leads to new orientational effects. Light irradiation can lead to the appearance of a dc electric field in the NLC bulk or affect its spatial distribution [21–24]. Similarly, light beam can modulate the properties of photoconductive substrates in a NLC cell ensuring the appearance or enhancement of a dc electric field capable of reorienting the director [25–29]. The mutual action of ac electric and light fields in certain geometries leads to a modification of the nonlinear optical response, which results in the first-order orientational transition [30–36].
In each case, the light polarization is an important parameter of the light-induced orientational response of NLCs. When a light beam directly affects NLC molecules, only the light beam component with extraordinary polarization is responsible for the director reorientation. For an "indirect" response, the situation is more complex and depends on its particular orientational mechanisms [11,12]. For example, the ratio of extraordinary and ordinary light wave intensities causes the absorption variation of an NLC film doped with a dichroic dye and, consequently, affects the linear and nonlinear optical responses.
Here, we describe a new mechanism for the DFNLC nonlinear optical response. The main idea is based on a strong variation of the dielectric anisotropy with temperature. If an electric field with a frequency below $f_c$ is applied to the DFNLC cell where $\Delta \varepsilon$ is negative, it stabilizes the planar alignment (Fig. 1). The light beam focused on the sample leads to a temperature increase up to a value at which $\Delta \varepsilon$ becomes positive, providing reorientation of the director in an irradiated area. The implementation of a dye dopant with negative dichroism into the bulk of DFNLC allows one to achieve a positive feedback of the light-induced reorientation, which leads to an abrupt first-order orientational transition with pronounced bistability.
Fig. 1. Schematic representation of the DFNLC director orientation upon light beam irradiation. The blue dashes show the orientation of the local director $\textbf {n}$ in the middle plane of the NLC cell (director reorientation angle is $\psi _m$). $I$, $\Delta T$, and $\Delta \varepsilon$ curves illustrate the spatial distributions of the incident light beam intensity, the local temperature increase, and the dielectric anisotropy variation. $\textbf {G}$ is the applied electric field at frequency $f$.
2. Experimental
To study the nonlinear optical response of DFNLC, the following mixtures were prepared. As a dual-frequency nematic matrix, DP002-113 (HCCH, China) was used. It has a relatively high clearing temperature $T_{NI} =$ 97.7 $^{\circ }$C and refractive indices $n_e$ = 1.603 and $n_o$ = 1.486 for extraordinary and ordinary waves (at $\lambda$ = 546 nm and temperature $T$ = 25 $^{\circ }$C). The cross-over frequency is $f_c =$ 15.8 kHz at $T$ = 25 $^{\circ }$C. The DFNLC was doped with 0.1 wt% of KD-312 anthraquinone dye (NIOPIK, Russia) (Fig. 2(a)) or 0.03 wt % of Methyl Red (MR) azobenzene dye (Aldrich, USA) (Fig. 2(b)).
Fig. 2. Dye additives (a),(c) KD-312 and (b),(d) Methyl Red. (a),(b) Chemical structure. (c),(d) Polarized absorption spectra.
Absorption coefficients for extraordinary and ordinary waves measured at $\lambda =$ 532 nm with a spectrophotometer (Shimadzu UV-2600, Japan) were $\alpha _{par} =$ 8 cm$^{-1}$ and $\alpha _{perp} =$ 32 cm$^{-1}$ for DFNLC+KD-312 (Fig. 2(c)), and $\alpha _{par} =$ 80 cm$^{-1}$ and $\alpha _{perp} =$17 cm$^{-1}$ for DFNLC+MR (Fig. 2(d)), respectively. The pretilt angles estimated by the "crystal rotation" method [37] were 6$^\circ$ for each sample.
The investigation of the DFNLC response to an electric field was carried out in a commercial planar cell with thickness of $L$ = 5.0 $\mu$m using a Berek compensator. Optical textures in crossed polarizers were observed using a polarized optical microscope (Nikon LV100N Pol, Japan) equipped with a heating stage (TMS-93 stage temperature controller and THMS 600 microscope stage, UK).
The nonlinear optical response of DFNLC was studied using the setup shown in Fig. 3. For this, planar cells were assembled from two ITO-coated glass plates separated by 105-$\mu$m-thick Teflon stripes. The inner surfaces of the substrates were pretreated with polyimide (P84, HP Polymer, USA) followed by rubbing in opposite directions.
Fig. 3. Experimental setup. $\lambda$/2 – half-wave plate, LED – red light emitting diode, P and A – polarizer and analyzer. $\textbf {E}$ and $\textbf {n}_0$ are the directions of light polarization and undeformed director.
A focused light beam (with linear polarization $\textbf {E}$) of a continuous-wave (CW) laser with a wavelength of $\lambda =$ 532 nm was focused onto the DFNLC cell at normal incidence. The plane of light beam polarization was rotated by the $\lambda$/2 phase plate. The beam waist measured by the "knife-edge" method [38] was $w =$35 $\mu$m. The incident light beam power $P$ was measured with a Hioki 3664 power meter (Japan). A sinusoidal electric signal with voltage amplitude $U_a$ and frequency $f$ was applied to the LC cell from a waveform generator (Agilent 33220A, USA). Sample temperature $T_0$ was controlled with a home-made heating stage with an accuracy of $\pm$0.2 $^\circ$C and kept slightly above or equal to room temperature ($T_r$ = 26.5 $^\circ$C). The optical textures of the irradiated areas were observed using an imaging circuit consisting of a light source (a light emitting diode with a luminescence peak at $\lambda _{led} =$ 620 nm), polarizer P and analyzer A, a red filter absorbing laser illumination, and a USB camera equipped with a 16x long-working-distance objective. The light beam profile was observed and recorded with a USB-camera on a semi-transparent screen installed at a distance of 56 cm from the sample.
Nonlinear phase shift $|\Delta \Phi |$, i.e. the phase difference on beam axis and its periphery, was estimated from the number of concentric rings in the far-field aberrational pattern as $\left | {\Delta \Phi } \right | = 2\pi N = 2\pi \left | {{n_{nl}}} \right |L/\lambda$, where $n_{nl}$ is the averaged over the cell thickness nonlinear refractive index caused by the director reorientation on the beam axis [39].
The choice of KD-312 and MR dopants is explained by their good solubility in DFNLC and distinct dichroism of the opposite sign. KD-312 does not induce the Janossy effect in the DFNLC. In the DFNLC+MR sample, the orientational self-defocusing is observed in the milliwatt power range only at oblique incidence of the light beam. It is also necessary to note that, in our case, the MR dye does not induce the orientational nonlinear response shown in Refs. [24,40] as it requires specific orienting materials to be used. Thus, we expect light-induced heating to dominate at normal incidence of the laser beam for both samples.
3. Results
Let us start with the DFNLC electro-optical response. A planar cell with a thickness of $L =$ 5 $\mu$m was examined by polarized microscope equipped with a heating stage. Optical retardance at $\lambda =$ 550 nm was measured with a Berek compensator depending on various frequencies $f$ of the applied electric field and sample temperatures $T$ (Fig. 4). The applied voltage $U_a =$ 4 V was chosen to be below the threshold of electrohydrodynamic instability [41,42].
Fig. 4. Optical retardance $R$ as a function of frequency $f$ and temperature $T$ in a 5-$\mu$m-thick DFNLC cell measured with a Berek compensator. Applied voltage is $U_a$ = 4 V. (a) Frequency $f$ at different sample temperatures $T$, (1), 31$^\circ$C; (2), 34$^\circ$C; (3), 36$^\circ$C; (4), 38$^\circ$C. (b) Temperature T at different frequency $f$, (1), 18 kHz; (2), 20 kHz; (3), 24 kHz.
When the dielectric anisotropy $\Delta \varepsilon$ is negative, i.e. at $f{ {\ >\ }}{f_c}$, the electric field stabilizes the planar alignment of the NLC cell (Fig. 4(a)). As $f$ decreases, the dielectric anisotropy $\Delta \varepsilon$ becomes positive and the NLC director begins to reorient, decreasing $R$, at
where ${K_{11}}$ is the splay elastic constant. A small increase in temperature of a few degrees Celsius significantly shifts the crossover frequency $f_c$ (Fig. 4(a), curves 2-4). For example, at $f =$ 24 kHz, the DFNLC film is homogeneous at $T =$ 34 $^\circ$C, but it is significantly reoriented at $T =$ 36 $^\circ$C. Note that the maximum value of retardance $R$ barely decreases with temperature, indicating that the nematic order parameter changes little. The effect of temperature-induced reorientation is shown more clearly in Fig. 4(b). The dielectric anisotropy increases with temperature [2] and the director reorients, reducing the optical retardance of the sample when it exceeds $\Delta {\varepsilon _{th}}(f,T)$. Such behavior can be considered as a temperature-induced Fréedericksz transition.This effect was realized by local laser heating in thick DFNLC films. The combined action of ac electric ($U_a$ = 4 V, $f$ = 18 kHz) and light ($P$ = 6 mW) fields leads to director deformation, which is visualized as an interference ring pattern in crossed polarizers (Figs. 5(a)–5(c)). A light beam with extraordinary polarization strongly diverges and an aberrational pattern is formed in the far field (Fig. 5(d)). The steady-state pattern develops within a few minutes depending on the light power and relaxes within one minute. A slight displacement of the sample with respect to the light beam darkens one part of the aberrational pattern in the displacement direction, which indicates a decrease on the refractive index of the extraordinary wave at the beam axis [39]. A light beam of the ordinary polarization passes through the sample without extra broadening (Fig. 5(e)). This indicates that the director is reoriented in the $xz$-plane, which causes the variation of the extraordinary refractive index $n_{par}$, while $n_{perp}$ remains the same. This also justifies that the effect of "thermal" nonlinearity [43] caused by the reduction of the nematic order parameter is negligibly low. The texture of the deformation region is larger and contains more interference rings in the case of ordinary polarization due to the larger value of the absorption coefficient (Fig. 5(b)). An increase in the frequency $f$ leads to a shrinkage of the deformation area due to a stronger stabilization of the director at the beam periphery (Fig. 5(c)), while aberrational rings become more pronounced (Fig. 5(f)).
Fig. 5. Cross-polarized optical images of the irradiated area and the corresponding far-field patterns formed upon light beam irradiation of DFNLC+0.1% KD-312. (a)-(c) Cross-polarized optical images of the irradiated area. (d)-(f) Corresponding far-field patterns formed upon light beam irradiation. (a), (d), (c), (f) Extraordinary polarization. (b), (e) Ordinary polarization. Power $P$ is (a), (b), (d), (e), 6 mW; (c), (f), 9 mW. An ac applied voltage $U_a =$ 4 V and frequency $f$ is, (a), (b), (d), (e), 18 kHz; (c), (f), 24 kHz. Sample temperature is $T_0 =$ 26.5$^\circ$C. $\textbf {E}_{inc}$ and $\textbf {n}_0$ show the polarization direction of the incident light beam and the undeformed NLC director; P and A are the polarizer and analyzer axes.
The dependencies of the nonlinear phase shift $|\Delta \Phi |$ on the light beam power $P$ or on the on-axis intensity $I=2P/\pi {w^2}$ show a threshold behavior with abrupt transitions (Fig. 6(a)). At $U_a$ = 4 V and $f$ = 18 kHz, the electric field stabilizes the planar orientation at $T$ = 26.5 $^\circ$C (see line (1) in Fig. 4(b)), but with a light-induced increase in temperature to $T \sim$ 30 $^\circ$C at $P = P_{th1}$, the dielectric anisotropy becomes sufficient for elastic deformation under the action of an applied electric field. The reverse transition occurs at the value $P_{th2}$, which is lower than the threshold $P_{th1}$ of the forward transition. This is due to the higher light absorption in the reoriented NLC state, which provides a higher temperature rise at lower light powers. Thus, two stable director configurations are possible in the range of ${P_{th2}} < P < {P_{th1}}$. This behavior is considered as the first-order orientational transition [32].
Fig. 6. Steady-state nonlinear phase shift $|\Delta \Phi |/2\pi$ as a function of the light beam power $P$ (intensity $I$) and applied voltage $U_a$, for DFNLC+0.1% KD-312. (a) Light beam power $P$, at an applied voltage $U_a =$ 4 V. (b) Applied voltage $U_a$, at the constant light beam power $P =$ 7 mW. The ac field frequency $f$ is (1), 18 kHz; (2), 20 kHz; (3), 24 kHz. The polarization of incident light beam is along $x$-axis. Circles indicate the light power increase, while crosses stand for the decrease. Sample temperature is $T_0 =$ 26.5$^\circ$C.
Note that the time of approaching the steady-state director deformation from a homogeneous orientation significantly increases up to ten minutes in the vicinity of thresholds, but the relaxation time remains almost unchanged. The relaxation constant $t_0 \approx 20$ s was estimated from the exponential approximation of the nonlinear phase shift after an abrupt decrease in the light power by $\sim$ 50 times using the formula $|\Delta \Phi (t)|=|\Delta \Phi _0|exp(-t/t_0)$, where $|\Delta \Phi _0|$ is the steady-state value.
Threshold powers increase with frequency, and the relative bistability region also enlarges: $R_b =$ $\left ( {{P_{th1}} - {P_{th2}}} \right )/{P_{th1}} =$ 0.13 for $f$ = 18 kHz and $R_b =$ 0.20 for $f$ = 24 kHz. Similar dependencies of the nonlinear phase shift were obtained by varying the voltage at the constant light beam power $P =$ 7mW (Fig. 6(b)). In this case, the regions of relative bistability are negligibly small for $f$ = 18 kHz and 20 kHz, while a large $R_b =$ 0.11 for $f =$ 24 kHz. Note that with a further increase in the frequency $f$, the light-induced orientational response is accompanied by a "thermal" nonlinearity [43], when the aberrational pattern is partially formed due to the order parameter decrease.
An increase in sample temperature leads to suppression of the bistability region and a decrease in the threshold powers (Fig. 7). Here the first-order transition transforms into the second-order one. Note that the variation of environmental temperature by $\sim$ 3 $^\circ$C dramatically changes the orientational behavior under the action of light beam.
Fig. 7. Steady-state nonlinear phase shift $|\Delta \Phi |/2\pi$ as a function of the light beam power $P$ (intensity $I$), for DFNLC+0.1%KD-312. Applied voltage $U_a =$ 4 V, $f$ = 24 kHz. The polarization of incident light beam is along $x$-axis. Circles indicate the light power increase, crosses stand for the decrease. Sample temperature $T_0$ is (1), 26.5$^\circ$C; (2), 27.6$^\circ$C; (3), 28.6$^\circ$C; (4), 29.6$^\circ$C.
The orientational nonlinearity has also been found for DFNLC doped with other types of dyes. In particular, we have observed the orientational optical response of the DFNLC+MR sample. The aberrational patterns and optical textures of the irradiated areas in crossed polarizers are similar to DFNLC+KD-312. Orientational transitions were found to be of the second order (Fig. 8). The increase in frequency shifts the threshold power in the same way as for the DFNLC+KD-312 cell. Note that the dependencies are relatively smooth: rotating the director leads to a decrease in light absorption, which limits the increase of $T$ and, consequently, $\Delta \varepsilon$, with the light power. The threshold powers are comparable to those of DFNLC+KD-312, whereas the absorption coefficient $\alpha _{par}$ is an order of magnitude higher. This peculiarity can be explained by a decrease in absorption during the trans-cis photoisomerization of the dopant molecules.
Fig. 8. Steady-state nonlinear phase shift $|\Delta \Phi |/2\pi$ as a function of the light beam power $P$ (intensity $I$) and applied voltage $U_a$, for DFNLC+0.03% MR. (a) Light beam power $P$, at an applied voltage $U_a =$ 4 V. (b) Applied voltage $U_a$, at the constant light beam power $P$ = 7 mW. The polarization of incident light beam is along $x$-axis. The ac field frequency $f$ is (1), 18 kHz; (2), 24 kHz. Circles indicate the light power increase, while crosses stand for the decrease. Sample temperature is $T_0 =$ 26.5$^\circ$C.
To finalize, we have observed the orientational optical response caused by a mapping of the dielectric properties of the DFNLC film. Namely, light heating enhances low-frequency dielectric anisotropy causing director reorientation and the decrease in refractive index of extraordinary light wave. The presence of the dye dopant with negative dichroism leads to a positive feedback between the director deformation and light absorption, which allows one to obtain the first-order transitions with relatively large bistability regions. The nonlinearity coefficient of this Kerr-like nonlinearity is ${n_2} = {n_{nl}}/I = \pi N\lambda {w^2}/2PL \approx$ 10$^{-3}$ cm$^2$/W.
4. Discussion
Our experimental results directly show that the combination of light and electric field is required for the observed orientational response of dye-doped DFNLCs. For theoretical consideration, we do not take into account the direct light-induced reorientation because of low light beam intensities in our experiment. We cannot fully exclude the contribution of the Janossy effect to the described orientational transitions for the DFNLC+MR sample, but we neglect it for the sake of simplicity.
Thus, we assume that the elastic deformation is caused only by an applied electric field, while the laser beam with a plane wavefront does not cause an orientational torque, but affects the temperature due to the light absorption. Director orientation is affected by elastic torques and the torque produced by the low-frequency electric field ${\bf {G}} = {{\bf {e}}_z}{U_a}\sin (2\pi ft)/L$ (${\bf {e}}_z$ is a unit vector along $z$-axis). In this case, the torque balance equation [11] is
The dielectric anisotropy $\Delta \varepsilon$ at frequency $f$ changes its sign at a certain sample temperature $T_c$. And vice versa, there is the crossover frequency $f_c$ for a particular $T$. Thus, $\Delta \varepsilon$ can be approximated as
Then we take into account the temperature increase caused by the light absorption. Assuming that the temperature increase $\Delta T = T - {T_0}$ is proportional to the intensity $I = 2P/\pi {w^2}$ on the light beam axis and the absorption coefficient, $\alpha (\psi ) = {\alpha _{par}}{\cos ^2}\psi + {\alpha _{perp}}{\sin ^2}\psi$ [44], we rewrite the Eq. (3) as
Substituting Eq. (4) into Eq. (2) and averaging over time (under the assumption that the characteristic director reorientation time is much greater than $1/f$), we obtain
To solve Eq. (5), we use sinusoidal approximation of the director reorientation angle $\psi \left ( {\eta,t} \right ) = {\psi _m}\sin \eta \left ( t \right )$ along the $z$-axis, where $\psi _m$ is the maximum director tilt angle achieved at $z = L/2$. Multiplying by sin$\eta$ and integrating over $0<\eta <\pi$, we get
Relationship between the director tilt angle and the nonlinear phase shift can be expressed as [19]
where $J_0(x)$ is the Bessel function, $|\Delta {\Phi _0}| = \pi \delta nL/\lambda$, and $\delta n = 0.5(n_{par}^2 - n_{perp}^2){n_{par}}/n_{perp}^2$.Figure 9(a) shows typical dependencies of the nonlinear phase shift on the dimensionless light power at a constant applied voltage computed using Eqs. (6) and (7) with the parameter $m = 3$ corresponding to the absorption coefficients of DFNLC+KD312 sample. The trivial solution ${\psi _m} = 0$ is stable at $\tilde p \le {\tilde p_{th1}}$. In the region of ${\tilde p_{th2}} \le \tilde p \le {\tilde p_{th1}}$, two stable director configurations coexist. The elastic deformation ceases to be stable at $\tilde p \le {\tilde p_{th2}}$ (see line (1) in Fig. 9(a)). Increasing the dielectric anisotropy $\Delta \varepsilon {\rm {(f}}{\rm {,\ }}{T_0}{\rm {)}}$ decreases the bistability region as well as the transition thresholds. The relatively high positive $\Delta \varepsilon {\rm {(f}}{\rm {,\ }}{T_0}{\rm {)}}$ suppresses the first-order transitions. A similar trend also appears in the dependencies on the dimensionless voltage square at a constant light power (Fig. 9(b)). An increase in sample temperature as well as a decrease in frequency cause an enhancement of $\Delta \varepsilon {\rm {(f}}{\rm {,\ }}{T_0}{\rm {)}}$ (see Eq. (3)), thus, all dependencies are in good qualitative agreement with the experimental results.
Fig. 9. Calculated steady-state nonlinear phase shift $|\Delta \Phi |/2\pi$ as a function of the dimensionless light power $\tilde p$ and electric field square $\tilde u$. (a) Dimensionless light power $\tilde p$, at a constant electric field square $\tilde u = 1$ and $\Delta \varepsilon {\rm {(f}}{\rm {,\ }}{T_0}{\rm {)}}$ is (1), -0.5; (2), 0; (3), 0.5; (4), 0.8. (b) Electric field square $\tilde u$, at a constant parameter $\tilde p = 1$ and $\Delta \varepsilon {\rm {(f}}{\rm {,\ }}{T_0}{\rm {)}}$ is (1), -0.5; (2), 0; (3), 0.5; (4), 4.0. Solid lines are stable solutions, dashes lines are unstable solutions of Eq. (6).
Note that the first-order optical Fréedericksz transition in a homeotropic NLC doped by dye dopant with negative dichroism was first discussed by Ong and later experimentally observed in the collaboration with Simoni [46–48]. The advantage of the current approach is that the director reorientation occurs under the action of ac electric field, whereas the light beam locally modulates the dielectric properties of the material. This allows us to reach a relatively strong light-induced response with a low light beam power of only $\sim$10 mW regardless of the thickness of NLC film.
5. Conclusion
We have studied the effect of orientational nonlinearity in DFNLC films caused by the mapping of dielectric properties. The low-frequency dielectric anisotropy is modulated by local light heating. An increase in the dielectric anisotropy leads to a director reorientation under an applied electric field. The following change in the refractive index of the extraordinary light wave causes a strong light beam divergence with aberrational self-action. For DFNLC containing the dye dopant with negative dichroism, the positive feedback is realized, leading to the first-order orientational transitions accompanied by tunable bistability regime. Proposed phenomenological model describes well the observed effects.
Thus the presented method allows one to obtain the light-induced DFNLC film reorientation by a milliwatt power light beam. The optical response is also very sensitive to sample temperature, which allows one to use this method for creation of temperature-triggered light beam modulators.
This study also reveals a general approach to the fabrication of light-controllable NLC films, in which the dielectric properties are modulated by a light beam, while the structural transition is induced by an applied electric field. The dielectric properties can be mapped not only by local heating, but also by reversible chemical reactions such as photoisomerization. In the latter case, the orientational optical response can be isothermal.
Funding
Russian Science Foundation (22-23-00391).
Acknowledgments
A.V.E. and S.A.S. thank the Russian Science Foundation (Project No. 22-23-00391, https://rscf.ru/en/project/22-23-00391/) for the financial support. T.O. acknowledges the support from the Government of the Russian Federation through the ITMO Professorship and Fellowship program.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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