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Abstract
By rotating the four-section π-shifted phase plate in the transverse plane relatively to the axes of the elliptical beam of 800-nm, 1.1-mJ, 35-fs pulse propagating in air, we switch between the regime of four parallel plasma channels and the regime of spatial symmetry breakup followed by on-axis plasma channel formation identified on the burnt paper images of the beam. Relaxation of the π-phase shift for 45° phase plate rotation is demonstrated explicitly in 3D+time carrier wave resolved numerical simulations yielding the initial step-like phase distribution degradation along the plasma region. This degradation becomes negligible as the angle between the ellipse major axis and the π-phase break line decreases to 15°.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Filamentation of femtosecond laser pulses in air [1] is governed by Kerr self-focusing and plasma defocusing [2] and allows numerous applications like remote sensing by supercontinuum, broadband terahertz emission, triggering electric discharge [3–7]. The modulational instability in cubic media [8] as well as plasma effect [9–11] form the stochastic pattern of multiple filamentation if the pulse peak power exceeds the critical power for self-focusing $P_{\textrm {cr}}$ by an order of magnitude or more (in air $P_{\textrm {cr}}=10$ GW [12]). The clamped intensity [13] is a robust quantity of a single filament, which increases due to filament fusion by not more than a factor of 1.88 even in a 4-J 60-fs pulse [7]. This slight growth of the clamped intensity ensures an order of magnitude increase in the plasma density due to strongly nonlinear dependence of the ionization rate on the laser intensity [14]. Fusion of multiple (up to 100) stochastic filaments in subpetawatt beam leads to an order of magnitude enhancement of the plasma fluorescence signal in both the experiment and carrier-resolved simulations [7,15–18].
Separation rather than fusion of filaments is required if one needs 3D waveguide writing with controlled quality [19] or if a bunch of plasma strings is going to be used as a THz antenna array [20,21]. In both cases plasma channels of femtosecond filaments should be equally spaced to impose corresponding structural modifications in the medium [22,23] or to efficiently direct the energy towards a desired angular sector [21,24].
To separate the filaments and to ensure robust non-interacting extended plasma channels, the initial laser beam should be regularized [25–27] by division into the sub-beamlets with the power of ${\gtrsim }P_{\textrm {cr}}$ [28,29]. This method was successfully implemented by inserting absorbing mesh into 10-cm-wide beam, suppression of stochastic filamentation and formation of non-interrupted plasma filaments on 15-m path in the corridor air [30].
However, the placement of such absorbing element at the laser system output leads to significant losses in the pulse energy, up to $\sim$70% for the amplitude mask used in [31]. Besides, if the geometrical focusing is applied to the regularized beam, the filaments, separated by the amplitude mask, will fuse [11,25,32] in the vicinity of the geometrical focus producing a superfilament [7,15,16,33].
Phase plate inserted into the initial beam breaks the reservoir into $\pi$-shifted beamlets and initiates non-interacting filaments from the spatial intensity distribution close to $\textrm {TEM}_{11}$ mode with the phase of neighboring lobes shifted by $\pi$. The ultimate advantages of the phase plate usage are semi-parallel non-interacting multiple filaments in the vicinity of the geometrical focus and minimized energy loss after transmission through transparent dielectric [34–36]. These regularized filaments were applied for waveguiding of radio-frequency [37,38] and optical [34,35] radiation. It was found that if a phase plate is inserted into the actual subterawatt loosely focused beam the transverse fluence distribution lacks the rigid boundary between the lobes, i. e. there is the radiation flux through a $\pi$-phase break [39].
In this paper we rotate the four-section $\pi$-shifted phase plate in the transverse plane relatively to the axes of the elliptical 800 nm beam (Fig. 1). The burnt paper images of the beam in the focal plane show the spatial symmetry breakup depending on the phase plate rotation angle. Through the experiment and carrier wave resolved numerical simulations of $(x,y,z,t)$ light field transformation in air we identify the several-millimeter-long plasma zone with off-axis channels originating from phase plate sections as well as 1-mm-long on-axis plasma channel formed due to the phase distribution degradation into the smooth one with much less transverse gradient as the propagation distance increases along the filament zone.
Fig. 1. A scheme of the relative position of the four-section $\pi$-phase plate and the elliptical beam used in our experiment: parallel configuration with zero angle (a) and inclined configuration with $45^\circ$ angle (b). Green solid line with an arrow shows the diagonal coordinate $\xi$ along which we trace the $\pi$-phase shift degradation.
2. Experimental setup
In the experiment [Fig. 2(a)] laser pulses at the central wavelength of 800 nm with the energy up to 1.1 mJ and the duration of 35 fs were generated by 1-kHz Ti:Sapphire laser system Legend Elite-Duo, Coherent, Inc. The pulse peak power was about 3–4$P_{\textrm {cr}}$, where $P_{\textrm {cr}}\approx 10$ GW is the critical power for self-focusing in air for a femtosecond pulse with the duration 35 fs [12]. The initial beam fluence distribution was elliptical with the ratio between the lengths of the semi-major and the semi-minor axes equal to 1.5 as well as about 1 cm beam diameter measured along the semi-major axis [Fig. 2(c)]. This elliptical beam was sent to the four-section phase plate with approximately 9 mm in diameter and 1.6 mm in thickness, so that the ellipse center coincides with the phase plate center. Each section of the phase plate contains about one critical power for self-focusing and can produce a single filament. Since not only the peak power, but also the transverse size is almost the same for all four sections into which the beam is subdivided, the four filaments appear at the same propagation distance [40] and create regularized plasma channels.
Fig. 2. (a) Experimental setup. (b) The profile of the $\pi$-phase break, recorded by the profilometer. (c) Initial elliptical beam distribution measured by CCD with $e^{-2}$ fluence level marked by white contour line. (d) The difference between the phase $\varphi$ initiated by the phase break (b) and $\pi$ in dependence on the wavelength (blue curve). Red curve represents the spectrum of the femtosecond pulse at the laser system output. Vertical red dashed lines indicates wavelengths at which the spectral intensity is 0.05 of the maximum. Horizontal blue dashed lines indicates the difference $\varphi -\pi$ at these wavelengths.
Our 1-kHz laser system provides high reproducibility of the spatial beam profile with shot to shot energy fluctuation less than 1.5%. Good beam pointing stability ensures stable shot-to-shot position of the beam relatively to the phase plate. In the experiment the phase plate is rotated around the beam propagation axis. This axis contains both the beam center and the phase plate center. The change of the transverse fluence distribution with the phase plate rotation originates from the asymmetry of the beam itself but not from the laser system shot-to-shot instability.
Neighboring sections of a phase plate have different thickness. The light transmitted through these sections is supposed to acquire either $\pi$ or 0 phase growth. We verify the particular difference (which is supposed to be $\pi$) by measuring the exact profile of the phase plate in the vicinity of the boundary between the sections by the profilometer Dektak 8 Stylus Profiler (Veeco Inc.), see Fig. 2(b). The difference in sections’ thickness is $(806\pm 3)$ nm and the width of the boundary, where this difference is accumulated, is 3 $\mu$m. Using the refractive index data for the material of our phase plate (O’Hara FTL10 glass), we calculate that exact $\pi$ phase shift appears for 744 nm wavelength. Taking into account the spectral width of our input pulse, the deviation of any spectral component from $\pi$ is not more than 0.012 rad [Fig. 2(d)].
After the $\pi$-phase plate the laser radiation is focused by the lens with the focal length $f=15$ cm to form several filaments in the vicinity of the focus. The plasma region was monitored by the side imaging of its incoherent optical emission. Both plasma channels fluorescence and scattered white light from the filaments (the supercontinuum spectrum measured after the end of filamentation is spanned to $\sim$400 nm) increments to the incoherent emission. Images of the regularized filaments were captured by a 4$\times$ microscopy objective and a CCD camera. A short wavelength pass filter (700FL, Andover Inc) with a transmission less than 1% from 750 nm to 875 nm was positioned in front of the objective to eliminate the scattered fundamental wavelength. The beam distribution inside the high-intensity filamentation zone was recorded by burning the photopaper, which was secured on the holder able to move with a micron precision along the focal region. A single shot laser burn spot was obtained during the experiment by quickly sweeping the photopaper across the propagation direction.
Our photopaper was selected to ensure the direct record of the beam distribution inside the high-intensity filament. Other methods to record beam distribution, e.g. a CCD-camera, require additional attenuation of the beam intensity by the wedge and neutral (or bandpass) filters [41]. In our experiment the geometrical focusing is comparatively tight, the intensity is high, and the wedge would be damaged by the first several laser shots. We note that the photopaper provides only rough information about the beam distribution as compared with a CCD-camera and distinguishes the high-intensity part of the beam only. One may compare beam distributions after the $\pi$-phase mask recorded by a photopaper (see Section 4 or Fig. 4 in [36]) with the ones taken by a CCD-camera (Fig. 2 of [39], where the wide background of the radiation is clearly seen).
3. Numerical model
For numerical simulations we adopt the approach described in [31,42]. Particularly, Eq. (2) in [31] represents Forward Maxwell equation [43,44] in three-dimensional Fourier domain. Forward Maxwell equation is a paraxial approximation of Unidirectional pulse propagation equation [44] with the non-paraxial terms omitted both in linear and nonlinear parts. Equation (3) in [31] describes nonlinear medium response which consists of the third-order Kerr polarization, effective absorption current due to medium ionization (which is calculated according to the tunnel model), and transient photocurrent. According to the initial conditions of the experiment, the light field distribution on the lens is Gaussian in temporal domain $T(\tau )=e^{-\tau ^2/2\tau _0^2}$ with the duration at $e^{-1}$ intensity level $2\tau _0=42$ fs. The time $\tau$ is given in the frame moving with the pulse group velocity. Geometrical focusing distance of the lens is $f=15$ cm. We consider that the lens is positioned at $z=-f$. This position is the starting point of the simulations, where the transverse spatial distribution of the beam in $(x,y)$ space is elliptical with semi-axis ratio $\varepsilon =1.5$. The ellipse is rotated by an angle $\alpha$ relatively to the horizontal $x$-axis. The center of the ellipse remains at the same position $(x=0,y=0)$. By ellipse rotation we simulate the rotation of the phase plate relatively to the laser beam in the experiment. An ideal $\pi$-phase break is located along the $x$ and $y$ axes, and the phase distribution $\varphi (x,y)$ is shown in Fig. 1. In overall, the initial field in time-domain Fourier representation can be written as
The self-focusing effect does not crucially affect propagation in our case of relatively tight focusing, and the collapse arrest is determined by the balance between external focusing and defocusing induced by the free electron photocurrent. The first propagation step $\Delta z = 14.5\textrm { cm}$ from focusing lens at $z=-15\textrm { cm}$ to $z=z_0=-0.5$ cm is performed in the linear regime. For this initial step the contribution of the nonlinearity to the light field transformation is negligibly small. Quantitatively this negligible effect of the nonlinearity is confirmed by $B$-integral estimation for the Gaussian beam. Indeed, $B$-integral from $z=-15\textrm { cm}$ to $z=-0.5\textrm { cm}$ is less than 0.1. Within the several millimeter long plasma zone the propagation step size varies between 0.2 and 18.7 $\mu$m inversely proportional to the peak intensity.
4. Results and discussion
Side images of the filament plasma zone registered at zero angle of the phase plate rotation show two parallel strings corresponding to the orthogonal projection of the rectangular parallelepiped, the ribs of which are the four plasma channels produced by the four-section phase plate [Fig. 3(c)]. At zero degree the parallelepiped of filaments is turned to the CCD with the facet [Fig. 3(a)], and only two filaments are seen [red plasma channels in Fig. 3(b)], and the other two filaments [grey plasma channels in Fig. 3(b)] are hidden by them. After $45^\circ$ rotation of the phase plate the CCD should observe [see Fig. 3(f)] from a sense of geometry one closer rib [red plasma channel in Fig. 3(e)] and the two remote ribs [green plasma channels in Fig. 3(e)] with only one hidden rib corresponding to the $45^\circ$ rotated parallelepiped [Fig. 3(e)]. Three filaments can be clearly distinguished for $100\textrm { }\mu \textrm {m}\le z\le 300\textrm { }\mu \textrm {m}$ [Fig. 3(f)]. In the experiment such view results in the appearance of the third intense string and several weaker plasma strips due to natural asymmetry of the elliptical beam [Fig. 2(c)].
Fig. 3. (a,d) The orientation of CCD camera, beam, and phase mask axes for side imaging of plasma fluorescence and scattered white light. (b,e) Sketch of the array of filaments. (c,f) Side-view of plasma channels fluorescence and scattered white light captured by CCD camera in false color scale. Transverse coordinate $X$ is parallel to the minor axis of the beam ellipse. The upper row corresponds to $\alpha = 0^\circ$ orientation of the phase mask, the lower row—to $\alpha = 45^\circ$.
The transverse beam distribution recorded on the photopaper, which was translated along the plasma zone, is shown in Fig. 4, the upper row corresponds to zero angle $\alpha =0$ between the ellipse semi-major axis and the phase break, the lower row corresponds to $\alpha =45^\circ$ (see Fig. 1). The relative ellipse and phase plate positions corresponding to the parallel and inclined configurations are shown in Figs. 1(a) and 1(b), respectively. For the parallel configuration of the ellipse axis and the line of phase breakup ($\alpha =0$), the filaments are clearly separated both before and after the geometrical focus. This corresponds to the beam background breakup and regularization of multiple filamentation.
Fig. 4. Transverse beam distribution recorded by photopaper at different propagation distances along the plasma zone. Top row corresponds to the parallel configuration with zero angle $\alpha = 0$ between the ellipse orientation axis and the $\pi$-phase break, and bottom row corresponds to the inclined configuration with $\alpha =45^\circ.$
In contrast, for $\alpha =45^\circ$, the filaments stay well separated only before the geometrical focus. The on-axis filament appears in close vicinity of the geometrical focus and survives during further propagation. So, at the propagation length $z=1.5$ mm one can see the string-like structure in the beam center (Fig. 4(c)–4(e), lower row). The formation of the filament on the beam axis is similar to the on-axis emission experimentally observed in the case of superfilamentation [15] and fusion of multiple filaments regularized by the absorption mesh with four holes [16]. Thus, in the conditions of our experiment with the high-quality beam and phase mask the $\pi$-phase shift can be broken, which is best pronounced for the angle of $\alpha =45^\circ$ between the ellipse axis and the phase break line.
To exclude the effect of non-perfect phase mask on the $\pi$-phase break degradation, we perform full 3D+time carrier wave resolved simulations of multiple filamentation in the conditions of our experiment. We assumed that the mask has the ideal $\pi$-phase break in the wideband wavelength interval, see Eq. (1). The simulations reproduce the observed $\pi$-phase break degradation effects in reasonable qualitative agreement (Fig. 5).
Fig. 5. Transverse distributions of the beam fluence obtained in 3D+time numerical simulations at different propagation distances $z$. Top row corresponds to the zero angle $\alpha$ between the ellipse orientation axis and the $\pi$-phase break, and bottom row corresponds to $\alpha =45^\circ.$
The perfect regularization of filamentation with the reservoir breakup along the zero intensity lines, which are defined by the equations $x+0y=0$ and $0x+y=0$, occurs for $\alpha =0$ (Fig. 5, upper row). The zero intensity lines [white cross centered at $(x=0,y=0)$] replicates the $\pi$-phase break of the mask [Fig. 1(a)]. In contrast, the cross-like structure is strongly perturbed in the case of $\alpha =45^\circ$ (Fig. 5, lower row). The relatively bright string-like structure forms in the beam center and survives at least to the propagation distance $z=1.5$ mm. Besides, the energy flow takes place between the beamlets at the beam periphery; this energy flow is clearly seen at $z=0.5$ mm where the perturbed cross-like structure disappears (Figs. 5(c)–5(e), lower row). Thus, the observed relaxation of the $\pi$-phase break is of general origin and is not connected with specific features of our phase mask or transverse laser pulse structure. The perfect phase mask and the absence of noise make the central hot spot pronounced weaker in the simulations than in the experiment. Its intensity and size are much less than the ones of the main filaments’. In the experiment, all the five beamlets leaving burns on the photopaper seem comparable. We associate this with the limited sensitivity while recording the beam profile with photopaper.
The $\pi$-phase break in the initial conditions [see Eq. (1)] is introduced for the neighboring lobes in the pairs of quadrants $\{x,y\}$ and $\{-x,y\}$ or $\{x,y\}$ and $\{x,-y\}$, while between the lobes in the quadrants $\{x,y\}$ and $\{-x,-y\}$ the phase difference in the point of their intersection $\{x=0,y=0\}$ is zero. So, we study the evolution with the propagation distance $z$ of the phase $\varphi$ along the transverse coordinate $\xi$, which is represented by a diagonal $y=-x$ for $x \le 0$ and by a diagonal $y=x$ for $x\;>\;0$ [see Fig. 1(a)]. The initial $\sim$40-nm-wide spectrum of the pulse [Fig. 2(d)] is additionally broadened due to the nonlinear interaction with the medium. Therefore, the transverse phase of the pulse is not well defined, and we calculated the dependence $\varphi (\xi )$ for the filtered fundamental (800 nm) harmonic denoted as $E_{\textrm {800 nm}}$.
Figures 6 and 7 show the dependencies $|E_{800 nm}|^2 (\xi)$ (solid red curve) and $\varphi (\xi )=\arg [E_{\textrm {800 nm}}(\xi )]$ (dotted black curve) at different propagation distances $z$ for the angle between the phase break on the plates and the major ellipse axis $\alpha =0^\circ$ and $45^\circ$ respectively. Each black circle in the light field transverse phase distribution shown in the right-hand side column of Figs. 6 and 7 corresponds to a node of our numerical grid. In the symmetrical case of $\alpha =0^\circ$ the phase break at $\xi =0$ remains equal to $\pi$ with propagation (Fig. 6). As the phase plate is rotated by $\alpha =45^\circ$, the symmetry is broken. The phase $\pi$-step break with the size of $\pi$ survives till 5 mm before the geometrical focus (Fig. 7, $z=-5$ mm, $\alpha =45^\circ$). Further in the propagation the $\pi$-phase break deteriorates and almost vanishes behind the geometrical focus ($z\ge 2$ mm).
Fig. 6. The evolution of the field (the left-hand side column, red solid curve) and the phase $\varphi (\xi )=\arg [E_{\textrm {800 nm}}(\xi )]$ (the right-hand side column, black curve marked by circles) in the transverse beam section across the diagonal coordinate $\xi$. In the right column, the axes are magnified for a better view, and red curves are the same as in the left column. Each row corresponds to the propagation distance $z$ evolving forwardly in the focal zone. The angle between the ellipse major axis and the phase plate axis is $\alpha =0^\circ .$
Fig. 7. Same as in Fig. 6 for the angle $\alpha =45^\circ$ between the ellipse major axis and the phase plate axis.
Our experiment and simulations show, that for $\alpha =45^\circ$ case with the maximum asymmetry the beamlets are not clearly separated and can exchange the energy. The distinguished hot spot forms in the beam center $\{x=0,y=0\}$, see the lower rows of Figs. 4 and 5. The intensity in the beam center can overcome the nonlinear ionization threshold of about $\sim$50 TW/cm$^2$ and thus form the plasma channel with the center in $\{x=0,y=0\}$. Since this on-axis plasma channel might play a parasitic role for THz antennas and plasma waveguides formation, we have studied this phenomenon in detail.
So, we simulated filamentation for different initial angles $\alpha$ between the ellipse major axis and the phase break line to study the physical origin of the $\pi$-shifted phase break degradation. Figure 8(a) shows the peak plasma density $N_{\textrm {peak}}$, which is reached in the off-axis plasma channels divided by the phase mask, and the plasma density on the beam axis $N_{\textrm {axis}}$ for different angles $\alpha$ from $0$ to $45^\circ$. The maximum plasma density $N_{\textrm {peak}}(z)$ depends on the angle $\alpha$ weakly. This corresponds to the localization of the major portion of the pulse energy in the beamlets initiated by the mask. In contrast, the dependence $N_{\textrm {axis}}(z)$ demonstrates two-order of magnitude increase with the growth of the angle $\alpha$ from $15^\circ$ to $45^\circ$ [in the case of $\alpha =0$ the function $N_{\textrm {axis}}(z)$ equals zero and is not shown in Fig. 8(a)]. For the angles $\alpha\;>\;45^\circ$ the dependencies $N_{\textrm {axis}}(z)$ reproduce the ones for the case $90^\circ - \alpha$ completely due to the symmetry of the system.
Fig. 8. (a) The variation of the peak plasma density $N_{\textrm {peak}}$ (left axis) and on-axis plasma density $N_{\textrm {axis}}$ (right axis) with propagation distance $z$ for different angles $\alpha$ between the ellipse orientation and $\pi$-phase break line. (b) The maximum on-axis plasma density $N_{\textrm {axis}}^{\textrm {max}}=\max _z[N_{\textrm {axis}}(z)]$ plotted as the function of the ellipse orientation angle $\alpha$ (filled red circles and dashed curve given by Eq. (3); left axis). Asymmetry parameter $\gamma$ characterizing the difference between the energies contained in neighboring $\pi$-shifted regions within the beam (solid line, right axis).
Figure 8(b) shows the dependence of the maximum on-axis plasma density $N_{\textrm {axis}}^{\textrm {max}}=\max _z[N_{\textrm {axis}}(z)]$ on the angle $\alpha$ in the range 0–90$^\circ$. One can see the increase in the asymmetry between the beamlets separated by the $\pi$-phase break with the growth of the angle $\alpha$ from $0$ to $45^\circ$. We estimated the value of this asymmetry $\gamma$ as the difference between the energies in neighboring beamlets:
The dependence of the on-axis plasma density $N_{\textrm {axis}}^{\textrm {max}}$ on the asymmetry parameter $\gamma$ can be fitted by the power law with the best fit for the power of 5:
see Fig. 8(b). According to Eq. (3) the $N_{\textrm {axis}}^{\textrm {max}}(\alpha )$ increases very slowly for the angles $\alpha \lesssim 15^\circ$. This provides the stability of the regularized filamentation against the small perturbation of the angle between the ellipse axis and the phase mask. The pragmatic advantage of this stability is in the low requirements for the adjustment of the phase mask relatively the beam: the deviation of the angle $\alpha$ from zero can be increased up to $\sim$15$^\circ$ without significant influence on the multiple filamentation pattern (i. e. the filament-like structure in the beam center will be very weak).5. Conclusions
We have shown experimentally and numerically that a high-quality four-section phase plate inserted into an elliptical beam of a high-peak-power 35-fs 1-kHz repetition rate laser pulse can produce either well regularized semi-parallel four plasma channels or provide the relaxation of the $\pi$-phase break between the neighboring allocated asymmetrical beamlets. Switching between the regime of four parallel plasma channels, corresponding to each phase plate section, and the regime of symmetry breakup followed by on-axis plasma channel formation, is introduced through rotation of the phase plate at $0^\circ$ or $45^\circ$ relatively to the elliptical beam major axis, respectively.
In 3D+time carrier wave resolved numerical simulations we rotated the phase plate relatively to the elliptical beam major axis with a step $5^\circ$ and show that the origin of the $\pi$-phase break degradation is the asymmetry between the neighboring $\pi$-phase shifted lobe areas. The plasma density as well as intensity in the on-axis beam channel increases with the phase plate rotation angle increase from $0^\circ$ to $45^\circ$ and decreases from $45^\circ$ to $90^\circ$. Such behavior of the light field with the rotation of the phase plate provides from the measurement point of view the periodical formation/annihilation of the filament in the beam center in addition to four filaments initiated by the phase plate subsections directly. The period of formation/annihilation is $90^\circ$. Four or five filaments can be recorded, for example, by the burns on the photopaper.
For the applications the important outcome is that the requirements on the adjustment of the phase plate relatively to the actual beam with inevitable ellipticity are quite reasonable: the phase break degradation is almost suppressed for the $15^\circ$ angle between the ellipse major axis and the $\pi$-phase break line. Therefore, the filament-assistant THz antenna or waveguide could be simply optimized by rotating the phase plate and its adjustment with relaxed accuracy.
Our work mainly focuses on the fundamental dynamics of the beam symmetry effect on the laser intensity distribution when a $\pi$-shift phase plate is used. During the practical application, long distance propagation might be more interested. Besides employment of high peak power laser system, expansion of input laser beam diameter will be required to extend our study to long distance. It is mainly because of the fact that the filament starting position is proportional to the diffraction length in principle. Another advantage of using large beam size is such that one could design sophisticated phase pattern to achieve more flexibility of the spatial distribution control of the filament array. Fortunately, very large size deformable mirror up to more than 400-mm diameter and sustainable to sub-PW laser pulse has already been available commercially [45], making direct beam phase modulation across large spatial dimension very promising.
Funding
National Key R&D Program of China (2018YFB0504400); Tianjin Research Program of Application Foundation and Advanced Technology of China (19JCYBJC16800); Fundamental Research Funds for the Central Universities; Russian Foundation for Basic Research (18-02-00954, 18-32-01000, 18-52-16020); Foundation for the Advancement of Theoretical Physics and Mathematics; Council on grants of the President of the Russian Federation (SP-2453.2018.2); Tianjin Special Program for Talent Development; Open Research Funds of the State Key Laboratory of High Field Laser Physics Shanghai Institute of Optics and Fine Mechanics (SIOM), 111 Project (B16027).
Acknowledgments
We thank Prof. See Leang Chin for fruitful discussions and advice.
Disclosures
The authors declare no conflicts of interest.
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