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Abstract
We propose and demonstrate a method of generating two fundamental terahertz cylindrical vector beams (THz-CVBs), namely the azimuthally- and radially-polarized THz pulses, from a spintronic THz emitter. We begin by presenting that the spintronic emitter generates the HE21 mode, a quadrupole like polarization distribution, when placed between two magnets with opposing polarity. By providing an appropriate mode conversion using a triangular Si prism, we show both from experiment and numerical calculation that we obtain azimuthal and radial THz vector beams. The proposed method facilitates the access of CVBs and paves the way toward sophisticated polarization control in the THz regime.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In the last half-century, we have seen the growth and development of laser techniques that have enabled sophisticated manipulation of optical pulses, including the generation of vector beams with spatially-dependent polarization vectors [1–3]. The cylindrical vector beam (CVB) [4,5], which has an axially-symmetric polarization pattern, is one of the fascinating examples owing to its unique focal properties. When tightly focused, the two fundamental CVB modes, the azimuthal polarization ($\textrm {TE}_{\textrm {01}}$ mode) and the radial polarization ($\textrm {TM}_{\textrm {01}}$ mode), give rise to longitudinal magnetic and electric field components respectively, with the spot size of the longitudinal component being smaller than the diffraction limit for typical scalar beams with linear or elliptical polarizations. In the optical region, these extraordinary characteristics at the focal point have led to many important applications, including spectroscopy, nanoscale imaging, optical trapping, and laser processing.
Expanding the access of CVB and focal longitudinal fields down to the terahertz (THz) region has also become of great interest as the THz technologies have progressed over the years. The focal longitudinal THz electric field has seen application to particle accelerations [6], and the enhancement of coupling to cylindrical metallic waveguides [7,8]. It has also been expected that the tighter focusing of THz-CVBs can be exploited to improve the spatial resolution of THz imaging. Not to mention, various elementary excitations in solids lie in the THz range [9], and thus their coupling to THz-CVBs is indeed intriguing. In fact, it has been recently discussed that the longitudinal magnetic component of focal THz-CVBs has potential applications toward more efficient spectroscopy of magnetic structures of matter [10,11]. And to this date, various active and passive methods of generating THz-CVBs have been proposed. The list of techniques includes circular and customized photoconductive antennas [7,12–14], mode conversion using segmented THz waveplates [15], velocity-mismatched optical rectification [16], segmented nonlinear crystals [8,17], air-plasma filaments [18], and laser-thin foil interaction [19].
In this paper, we propose a new approach toward accessing THz-CVBs of radial and azimuthal polarizations and concomitant focal longitudinal fields using a spintronic THz emitter [20–23], which is a thin-film stack of ferromagnetic and nonmagnetic metals. The spintronic emitter has been one of the most extensively studied THz sources in the last few years, thanks to phonon absorption-free emission and high efficiency compared to conventional nonlinear crystals. The key characteristic of the spintronic emitter is that it exploits the spin degrees of freedom via the inverse spin Hall effect. As a consequence, the polarization is determined by the direction of the magnetic moment in the emitter, and it has been recently discussed that such property can be exploited to control, in principle, the polarization beam pattern by tailoring the external magnetic field, as exemplified by a quadrupole-like polarization pattern of THz electric field obtained by placing the spintronic emitter between a pair of magnets with opposing polarity [24]. Here, we demonstrate the generation of radial and azimuthal THz polarizations from a spintronic emitter. By placing a spintronic emitter between two magnets with opposing polarity, we first generate a $\textrm {HE}_{\textrm {21}}$ polarization mode. Succeedingly, we demonstrate that the generated $\textrm {HE}_{\textrm {21}}$ mode can be easily converted to radial and azimuthal polarizations using a triangular Si prism which effectively works as an achromatic half-wave plate. Combined with numerical simulations, our experimental results confirm the generation of the THz-CVBs.
2. Methods
2.1 Generation of $\textrm {HE}_{\textrm {21}}$ mode and its mode conversion
We first begin our discussion by illustrating the basic principle of spintronic THz emitter and the generation of the radial and azimuthal THz-CVBs. As we show in Fig. 1(a), a typical spintronic emitter comprises two thin films of ferromagnetic and nonmagnetic metal layers. When we excite the emitter under external magnetic field using a near-infrared femtosecond pulse entering along the $z$-axis, the electrons in the ferromagnetic layer are excited to states above the Fermi energy with spin polarization. Due to the different transport properties of ferromagnetic and nonmagnetic layers, a net spin current in $z$-direction is induced inside the heterostructure [25]. The spin current $\mathbf{j}_{\mathrm {s}}$ is then converted to a transient charge current $\mathbf{j}_{\mathrm {c}}$ by the inverse spin Hall effect,
where $\gamma$ is the spin Hall angle, and $\hat {\mathbf{m}}$ denotes the direction of magnetization that can be controlled by the orientation of external magnetic field.
Fig. 1. (a) THz generation from a spintronic THz emitter. Transient spin current $\mathbf{j}_{\mathrm {s}}$ generated upon photoexcitation converts into charge current $\mathbf{j}_{\mathrm {c}}$ via the inverse spin Hall effect, which radiates THz electric field with the polarization perpendicular to the magnetization $\hat {\mathbf{m}}$. (b) Schematic of the experimental setup. (c) and (d) Schematic presentation of generating azimuthal and radial polarization by converting $\textrm {HE}_{\textrm {21}}$ mode with different orientations, respectively.
Equation (1) tells that the polarization of emitted THz pulse is determined by the direction of magnetization in the magnetic layer. It has been confirmed experimentally that when we place the spintronic emitter between two magnets with aligned polarity, i.e., poles of the two magnets pointed toward the same direction, the polarization of emitted THz is perpendicular to the magnetic field regardless of the pump polarization. Further, flipping one of the magnets to have opposing polarity results in the emergence of a quadrupole-like vector beam pattern as demonstrated in Ref. [24]. This distinctive spatial distribution of polarization can be regarded as the $\textrm {HE}_{\textrm {21}}$ mode.
The mode conversion from the $\textrm {HE}_{\textrm {21}}$ mode to radial and azimuthal modes can be achieved by imposing a $\pi$-phase shift of the $y$-polarized component. To this end, we exploit in this study the Fresnel’s phase shift property at the total reflection. When an incident light propagating inside an optically-denser medium is reflected at the total reflection condition, the reflected light acquires a phase shift between the $s$- and $p$-polarized components. The relative phase shift $\delta (=\delta _s - \delta _p)$, where $\delta _s (\delta _p)$ is the amount of phase shift for $s$- ($p$-) polarization, is given from the Fresnel’s formulas of reflection and refraction [26],
where $\theta _i$ is the incident angle. Conventionally, the phase shift at the total-reflection interface described in Eq. (2) has been utilized in the polarization conversion from linear to elliptical polarization in e.g. the Fresnel rhomb. Here, we prepared a triangular prism made of high-grade Si with a mirror-polished surface [27]. Since the refractive index of Si is flat at $n = 3.4$ in the current frequency range [28,29], the total reflection at the incident angle $\theta _i = 45^{\circ }$ gives approximately $95^{\circ }$ of phase shift between $x$- ($p$-) and $y$- ($s$-) polarization. Therefore, the Si triangular prism has been employed in previous experiments as an alternative to a quarter-wave plate to generate circularly-polarized THz pulses [27]. In this work, we take advantage of this phase-shifting property as an achromatic half-wave prism (HWP) with two total reflections in the prism to generate the desired $\pi$-phase shift. This enables us to achieve achromatic mode conversion for a wide bandwidth, which is crucial in our case of generating single-cycle THz-CVBs.Fig. 2. Numerical simulation results of spatial polarization distributions. The arrows indicate the polarization direction of electric-field unit vector at each position. (a) $\textrm {HE}_{\textrm {21}}$-type polarization distribution inside the spintronic THz emitter when placed between opposing magnets parallel to $x$-direction (corresponding to Fig. 1(c)) and (b) polarization profile after imposing a $\pi$-phase shift to Fig. 2(a). (c) $\textrm {HE}_{\textrm {21}}$-type polarization distribution inside the spintronic THz emitter when placed between opposing magnets rotated $45^{\circ }$ from Fig. 2(a) (corresponding to Fig. 1(d)) and (d) polarization profile after imposing a $\pi$-phase shift to Fig. 2(c).
To confirm our working principle presented here, we performed numerical calculations of the resulting polarization distribution at the spintronic emitter and its focal fields. First, we examined the electric-field polarization inside the spintronic THz emitter and after the mode conversion by calculating the distribution of the external magnetic field. Details of the magnetic-field calculation are presented in Appendix A. The result of numerically-calculated polarization distributions is shown in Fig. 2. By placing the magnet pair with opposing polarity, we can visibly identify that the $\textrm {HE}_{\textrm {21}}$-type polarization distribution is created in the spintronic THz emitter, where the principal axis of the $\textrm {HE}_{\textrm {21}}$ mode is determined by the angle of the magnet pair. Furthermore, application of $\pi$-phase shift converts the two $\textrm {HE}_{\textrm {21}}$-type distribution to azimuthal and radial polarizations respectively as shown in Figs. 2(c) and 2(d).
Next, we calculated the focal electric field based on the Richards–Wolf diffractorial method [30,31]. A brief description on the calculation is given in Appendix B. In Fig. 3, we present spatial patterns of focal electric fields (which correspond to Figs. 1(c) and 1(d)) obtained by converting the numerically-calculated output from the spintronic source illustrated in Fig. 2. In this calculation, the base frequency is fixed at $\omega /2\pi = 1\,\textrm{THz}$ and other parameters are set to match the experimental conditions presented in the subsequent section. This simulation confirms that the generated polarization pattern can be classified as azimuthal and radial CVBs, with characteristic focal features of donut-shaped transverse electric field distributions and finite longitudinal electric field component for radial polarization. We also notice that the transverse component presents an asymmetry that is possibly due to the placement of a pair of magnets. It is worth noting that the focal field without the mode conversion by Si HWP, i.e., the tight focusing of $\textrm {HE}_{\textrm {21}}$ mode does not lead to finite longitudinal electric field component, which contradicts the observation in Ref. [24].
Fig. 3. Numerical simulation results of focal electric field from the spintronic THz emitter when the magnet pair has opposing polarity. The frequency used in the calculation is set at $\omega /2\pi = 1\,\textrm{THz}$. (a)-(c) Spatial patterns of (a) transverse and (b) longitudinal components for the azimuthal polarization mode (corresponding to Fig. 1(c)), and (c) the line profile of longitudinal component at the white dashed line in (b). (d)-(f) Spatial patterns of (d) transverse and (e) longitudinal components for the radial polarization mode (corresponding to Fig. 1(d)), and (f) the line profile of longitudinal component at the white dashed line in (e).
2.2 Experimental setup
Fig. 4. Schematic of the experimental setup. The generated THz pulse propagates through the triangular Si prism for the mode conversion, and gets focused by a $90^{\circ }$ off-axis parabolic mirror (OAP) into the electro-optic crystal (EOX). THz time-domain spectroscopy is performed by scanning the delay stage inserted in the path of the pump pulse. The probe pulse enters EOX in the back-reflection geometry, making it possible to scan the spatial profile of the focal THz electric field with translation stage without changing the time delay.
The experimental setup in our current study is schematically shown in Fig. 4. As a fundamental light source, we used a Ti:sapphire-based regenerative amplifier system with a pulse energy of 2 mJ, central wavelength of 800 nm, pulse width of 45 fs, and repetition rate of 1 kHz. The laser output was divided into two pulses: a pump pulse for the THz emission, and a probe pulse for the electro-optic sampling of the THz time-domain waveform, respectively. The pump pulse was collimated into the $1/e^{2}$ diameter of 12 mm and the pump energy of 1200 µJ at maximum and irradiated onto the spintronic emitter placed on a metal sample holder with the radiance aperture of 22 mm. The emitted THz pulse was guided to the triangular Si HWP and acquired a $\pi$-phase shift after two internal reflections. The transmitted THz pulse was then focused onto an electro-optic crystal for the electro-optic sampling [32,33] by a $90^{\circ }$ off-axis parabolic mirror with a diameter and a parental focal length of 25 mm. The electro-optic sampling of the transverse electric-field components was performed using (110)-faced gallium phosphide (GaP) with a thickness of 380 µm. For measurements of the transverse electric field, we inserted a pair of wire grid polarizers to perform polarization-resolved measurement by separating into horizontal ($x$) and vertical ($y$) components. For the detection of the longitudinal THz electric field or $z$ component, we used instead (100)-faced GaP with a thickness of 500 µm.
To obtain the spatial profile of the focal THz electric field, we employed the back-reflection geometry for the electro-optic sampling adapted from Ref. [24], and scanned around the beam axis with the step width of 0.04 mm along the direction parallel to the $x$-direction. This configuration allowed us to scan the sampling position of the probe pulse using an electronically-controlled translation stage without changing the total optical path length. Since the measured sampling pulse includes the contribution of surface reflection which does not contribute to the electro-optic sampling, we performed a correction of the sampling pulse intensity when estimating the THz electric field strength. To obtain the THz electric field strength in free space, we also took into consideration the reflection of incident THz pulse at the surface of the electro-optic crystal. It is noteworthy that the Fresnel’s reflection coefficients for the transverse and longitudinal electric field are different by a factor of GaP’s refractive index [12,24,34].
As for the spintronic THz source, we fabricated a Co$_{20}$Fe$_{60}$B$_{20}$(2 nm)/Pt (2 nm) heterostructure using radio frequency (RF) magnetron sputtering on a large-area quartz plate (27×32 mm2) with a thickness of 500 µm. The composition and thickness of the metallic bilayer were set based on the optimal condition for the THz pulse generation in accordance with previous studies [20]. We applied an external magnetic field onto the emitter by placing a pair of disk neodymium magnets (N40) with a diameter of 25 mm and a thickness of 5 mm. Each magnet was placed next to the metal sample holder with a separation of 2 mm between the sample holder and the face of the magnet. The magnets were placed in the opposing configuration, and the field strength measured with a Hall probe was 200 mT at the surface of the magnets and completely diminished at the center of the sample holder.
3. Results and discussions
We begin by examining the magnetic field configuration which generates the azimuthally-polarized THz vector beam. In Fig. 5, we summarize the spatial profiles of electric field and their Fourier-transformed power spectra decomposed into transverse ($x$ and $y$) and longitudinal ($z$) polarizations.
We first observe that for transverse components, the $y$-component (Figs. 5(b) and 1(e)) dominates the $x$-component (Figs. 5(a) and 1(d)) of THz electric field. The transverse $y$-polarized THz component presents two lobes with flipped polarity of THz time-domain waveform as we scan across the focal point and vanishes at the beam axis, which shows a fair correspondence to the donut-shaped structure presented in Fig. 3. These features, combined with negligibly small longitudinal components (Figs. 5(c) and 5(f)), are consistent with the behavior of the azimuthal polarization, where in this case a finite longitudinal magnetic field is expected to be present. The behavior of small $z$-component that behaves similar to the transverse $y$-component is expected to be caused by the projection of transverse electric field which may arise from small tilting of the detection crystal. This effect will be treated in detail in the later discussion. The strengths of peak $y$-component electric field reached up to 5.4 kV/cm and 5.8 kV/cm at the negative and positive scanning positions, respectively. From these results, we can also estimate the longitudinal magnetic field strength using the ratio between the longitudinal and transverse components that can be determined analytically for ideal CVBs. In the present setup, the calculated ratio becomes $0.35$, and since the average of the transverse electric field 5.6 kV/cm corresponds to 1.87 mT, the estimated longitudinal magnetic field reaches $0.35 \times 1.87\,\textrm{mT} = {0.65}\,\textrm{mT}$.
Fig. 5. Scanned spatial profile of time- and frequency-domain THz amplitudes for azimuthal polarization ($\textrm {TE}_{\textrm {01}}$ mode). (a)-(c) Transverse ($x$, $y$), and longitudinal ($z$) components of time-domain THz electric field and (d)-(f) the corresponding frequency-domain intensity obtained by Fourier transform. Color maps for the longitudinal component are scaled for visibility (Scale factors are denoted inside the parentheses.). For the frequency-domain intensity plots, the line profile at 1 THz (corresponding to the white-dashed line) is presented on the right side of each graph.
Next, we change the magnetic field orientation by $45^{\circ }$ to generate the radially-polarized THz beam. In Fig. 6, we present the experimental result of polarization-resolved measurement of focal THz electric field similarly to Fig. 5. The change in direction of magnetic polarization results in the swapping of transverse polarization component, with the $x$-component (Fig. 6(a) and 6(d)) presenting peak values of 5.8 kV/cm and 6.1 kV/cm at the negative and positive scan regions, respectively. Moreover, as we expect for the radial polarization, the longitudinal electric-field component (Figs. 6(c) and 6(f)) is clearly detected at the focal point. The time-domain longitudinal electric field and its Fourier-transformed intensity spectrum at the focal point are presented in Figs. 7(a) and 7(b). The longitudinal electric field strength of electro-optic sampling using the (100)-faced GaP crystal measures 2.4 kV/cm at the focus. The ratio of longitudinal electric field strength to the (maximum) transverse component is $2.4/6.1 \simeq 0.39$, presenting a good agreement with an ideal radial polarization where the ratio becomes $0.35$ as mentioned previously.
Fig. 6. Scanned spatial profile of time- and frequency-domain THz amplitudes for radial polarization ($\textrm {TM}_{\textrm {01}}$ mode). (a-c) Transverse ($x$, $y$), and longitudinal ($z$) components of time-domain THz electric field and (d-f) the corresponding frequency-domain intensity obtained by Fourier transform. Color maps for the longitudinal component are scaled for visibility (Scale factors are denoted inside the parentheses.). For the frequency-domain intensity plots, the line profile at 1 THz (corresponding to the white-dashed line) is presented on the right side of each graph.
Fig. 7. Longitudinal electric field measured at the focal point using (100)-faced GaP crystal. (a) Time-domain waveform of the longitudinal THz electric field and (b) corresponding frequency-domain intensity spectrum. (c) The peak electric field amplitude measured with a wire-grid polarizer normalized by the peak amplitude of the radially-polarized THz beam. The lines between the points are guide to the eye.
Concerning the measurement of the longitudinal electric field, it is important to rule out the possibility of an artifact by the transverse component, since a small tilt of the detection crystal can result in observing the projection of transverse electric field as a signal of the longitudinal component. To this end, we also experimented with the same setup except for the pair of magnets which were placed in the magnetically-aligned configuration. In this case, we obtain a typical linear polarization with an essentially negligible longitudinal component, and therefore we can acquire information of how much the transverse electric field gets projected as the longitudinal component. The transverse electric field for linear polarization measured with the (110)-faced GaP crystal reached 17.7 kV/cm at the same photoexcitation intensity. When we utilized the (100)-faced crystal instead for detection, we measured the projected transverse component corresponding to 1.3 kV/cm. Seeing that the transverse component of the radial polarization at the focus drops to $\leq {2}\,\textrm{kV/cm}$, the contribution of the transverse component onto the (100)-faced GaP caused by the misalignment is at least an order of magnitude smaller than the longitudinal THz electric field detected above. Therefore, we can safely exclude the possibility of an artifact from the perspective of field strength comparison.
The finite longitudinal electric field and the isotropic nature of the radially-polarized THz beam were further confirmed by observing the longitudinal electric field component with the insertion of a wire-grid polarizer on the path of the THz field. In this setup, we expect the wire-grid to function as a mode converter from radial polarization to $\textrm {HG}_{\textrm {10}}$ mode with its principal axis determined by the angle of metal wire grids. And since $\textrm {HG}_{\textrm {10}}$ presents a finite longitudinal electric field component at the focal point, we can predict the strength of the longitudinal electric field to stay constant at any wire-grid angle. In Fig. 7(c), we present the polarizer-angle dependence of peak longitudinal electric field from $\textrm {HG}_{\textrm {10}}$ mode normalized by the peak electric field of the radial polarization. The ratio of the two longitudinal-field amplitude follows the $1/2$ line for all angles. This correspondence, in fact, matches the calculation result based on the Richards–Wolf theory of vectorial diffraction [30,31] (See Appendix B for a brief discussion.). In addition, the result in Fig. 7(c) presents a slight deviation and finite ellipticity in the electric field, which can be attributed to the asymmetry of the magnetic field setup. Such behavior has also been evidenced in the numerical simulation as shown in Fig. 3. Overall, this observation corroborates our finding of longitudinal THz electric field with our scheme of generating radially-polarized vector beam pattern.
The generation of azimuthally- and radially-polarized THz beams has also been reported in segmented nonlinear crystal approach using (111)-faced GaP with customized orientations [17]. However, thanks to the much higher efficiency of THz generation from the spintronic THz emitter, we have achieved at least an order of magnitude improvement in the peak strength in terms of the longitudinal THz electric field generation, where the evaluated conversion efficiency from the pump pulse to the radially-polarized THz pulse becomes $5.5\times 10^{-7}$ . Using spintronic THz emitters is also advantageous since the THz bandwidth is not limited by phonon absorptions or phase matching, which are crucial factors for THz generation using nonlinear crystals. In addition, we believe that the polarization modification by controlling external magnetic field has better controllability of the polarization pattern because the resolution of polarization profile is in principle restricted by the magnetic domain size, which is typically in the scale of 10 nm [35], whereas in the segmented approach the physical size of diced crystals becomes the limit and requires more complex fine processing techniques. Moreover, further optimization of spintronic THz emitter with trilayer structure has demonstrated the capability of high field generation that reaches up to 300 kV/cm for linear polarization [22], and thus application of the proposed method may provide simple access to intense longitudinal electric and magnetic field while retaining wide bandwidth unlike in other methods such as longitudinal field generation using LiNbO$_3$ [34,36].
4. Conclusion
In summary, we have presented that single-cycle THz pulses with azimuthal and radial polarizations can be generated from the spintronic THz emitter. We revealed that the spintronic emitter placed between magnets with opposite polarities generate $\textrm {HE}_{\textrm {21}}$ mode, and demonstrated that the azimuthal and radial polarizations is obtained by providing a $\pi$-phase shift by achromatic Si prism. The focal THz electric field presented a donut-shaped transverse electric field with a finite longitudinal component emerging for the radial polarization, and further, the behavior of the THz vector beams was well-characterized by our numerical calculation of polarization distribution.
The unique characteristic of THz polarization tuning by magnetic field combined with the achromatic polarization conversion technique facilitates the access and control of CVBs in the THz regime. In addition, the generated focal longitudinal electric and magnetic field components offer numerous opportunities of application including THz spectroscopy and imaging, coupling to waveguides, and particle accelerations.
Appendix A: numerical simulation of the vector beam pattern generated from the spintronic THz emitter
We performed numerical calculations presented in Fig. 2 to confirm vector beam patterns obtained in our experimental setup. We assume here that the sample is placed in between two circular-disk magnets, and the sample magnetization follows the trace of magnetic flux applied externally. In the present case where the magnetostatics is concerned, the magnetic field distribution $\mathbf{H}(\mathbf{x})$ can be determined by a magnetic scalar potential $\phi _{\mathrm {m}}(\mathbf{x})$ which is the sum of contributions from the two magnets, via $\mathbf{H}(\mathbf{x}) = -\nabla {\phi _{\mathrm {m}}}(\mathbf{x})$. The scalar potential from a single disk magnet $\phi _0(\mathbf{x})$ can be described for a given magnetic dipole density $\tau$ as follows:
Here, the plus-minus sign corresponds to the north and south poles, respectively, and $\textrm{d} {\Omega }$ is the solid angle of the disk magnet observed from the given position $\mathbf{x}$. The solid angle of a circular disk was obtained from the analytical solution which has been derived in detail in Ref. [37]. After calculating $\mathbf{H}(\mathbf{x})$ and the magnetization $\hat {\mathbf{m}}(\mathbf{x}) = (\mathbf{H} / |\mathbf{H}|)$, we obtained the spatial profile of electric field polarization $\hat {\mathbf{e}}(\mathbf{x})$ shown in Figs. 2(a) and 2(c) based on Eq. (1). For the simulation presented in Figs. 2(b) and 2(d), a $\pi$-phase shift to the $y$-component was applied to account for the mode conversion in the Si triangular prism.Appendix B: calculation of the focal electric field
To scrutinize the principle of THz-CVB generation in detail, we performed a numerical calculation of the focal electric field presented in Fig. 3 based on the theory formulated by Richards and Wolf [30]. The analytical form of the focal electric field $\mathbf{E}(\rho , \varphi , z)$ written in the cylindrical coordinates $(\rho , \varphi , z)$ is given in basic textbooks (see e.g. Ref. [31])
Next, we briefly reflect on the comparison of longitudinal electric field strength between radial ($\textrm {TM}_{\textrm {01}}$) and $\textrm {HG}_{\textrm {10}}$ polarization modes [31]. An ideal radially-polarized electric field $\mathbf{E}_{\mathrm {radial}}$ can be written in terms of the linear combination of two $\textrm {HG}_{\textrm {10}}$ mode,
Funding
Core Research for Evolutional Science and Technology (JPMJCR19T3); Japan Society for the Promotion of Science (18H05324).
Acknowledgments
The authors acknowledge Yoshua Hirai and Morihiko Nishida for stimulating discussions.
Disclosures
The authors declare no conflicts of interest.
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