Accessing electromagnetic properties of matters with cylindrical vector beams

Cylindrical vector beam (CVB) is a structured lightwave characterized by its topologically nontrivial nature of polarization. The unique electromagnetic field configuration of CVBs has been exploited to optical tweezers, laser accelerations, and so on. However, use of CVBs outside optics in, for example, condensed matter physics and chemistry has not progressed. In this paper, we propose potential applications of CVBs to those fields based on a general argument on their absorption by matters. We show that pulse azimuthal CVBs around terahertz (THz) or far-infrared frequencies can be a unique and powerful mean for time-resolved spectroscopy of magnetic properties of matters and claim that an azimuthal electric field of a pulse CVB would be a novel way of studying and controlling edge currents in topological materials. We also demonstrate how powerful CVBs will be as a tool for Floquet engineering of nonequilibrium states of matters.


I. INTRODUCTION
Developments of optics witnessed in the past decades enabled the realization of a high-intensity, ultra-short pulse laser 1,2 . The high temporal resolution due to the short pulse width allows us to directly access to nonequilibrium phenomena like chemical reactions and photo-induced phenomena in matters. The high intensity opened new fields of photo-induced physics 3 , ultrafast magnetism 4 , and femtochemistry 5 .
Moreover, the intense ultra-short light becomes vital for generating intense terahertz (THz) pulses 6 . Since THz is the typical energy scale of various collective phenomena 7-10 in solids like superconductivity, phonons, and magnons and is also crucial for understanding biomolecules [11][12][13][14] , the developing THz optics shares a central position in optical physics.
Existing studies of optical phenomena in matters have been mainly done with simple Gaussian beams, and other kinds of lasers developed in the past decades have not been paid much attention in condensed matter physics and chemistry. Among those modern variants of lasers, here we focus on so-called structured lights. In particular, we discuss applications of cylindrical vector beams (CVBs) [15][16][17] . Because of their unique focusing property, CVBs potentially have substantial impacts on studies of optical phenomena in matters. Nevertheless, unlike other structured lights like optical vortices 18,19 , applications of CVBs outside optics are almost unexplored.
We show that using CVBs instead of conventional Gaussian beams can be a quite powerful way of studying magnetic properties of matters in, especially, THz or far-infrared frequency ranges. By preparing samples sufficiently smaller than the wavelength, which can be easily achieved nowadays, we can control the applying electric and magnetic fields almost independently. In particular, we can avoid strong electric dipole absorption of matters which often overwhelms magnetic absorption we are interested 20 .
Using THz CVBs would be particularly effective for time-resolved measurements. The relatively short timescale of THz CVBs offers a high temporal resolution, and the characteristic spatial profile of focused CVBs allows us to probe electric and magnetic properties of matters selectively. We will give several applications of CVBs utilizing these properties; time-resolved electron spin resonance (ESR) of conducting materials, study of multiferroic materials in an unconventional way, electron paramagnetic resonance (EPR) study of the dynamics of biomolecules in their living environment. In addition to them, we consider how to exploit the characteristic azimuthal electric field configuration of CVBs for probing and controlling edge currents in topological materials 21,22 .
We also discuss applications of focused CVBs for Floquet engineering [23][24][25] . We demonstrate that the highly controllable nature of the electromagnetic fields of CVBs offers a building block for realizing desired nonequilibrium physics in periodically driven matters. We take a simple model of quantum magnets and compare the Floquet engineering with CVBs and that with circularly polarized lasers 26,27 .
The rest of this paper is organized as follows. In the second section, we review CVBs. In the third section, we study how small the electric field absorption can be by replacing Gaussian beams with CVBs. We give potential applications of CVBs for condensed matter physics and chemistry/biology. In the forth section, we deal with the electric field component of CVBs and show that the electric field absorption of a CVB can be a way of probing and controlling edge states of topological matter. The fifth section is about Floquet engineering. We show that CVBs have clear advantages over conventional lasers for designing nonequilibrium states of matters. The final section is devoted to the conclusion.

II. CYLINDRICAL VECTOR BEAM
Exploring applications of topological lightwaves like optical vortices 18,19 and CVBs 15-17 is one of the cen-arXiv:1811.10617v1 [physics.optics] 26 Nov 2018 tral issues of modern optics. Both beams have vanishing intensity along the propagation axis with different physical origins. In the case of optical vortices, their scotoma comes from the spiral-shaped phase structure. The azimuthally inhomogeneous wavefront carries a nonvanishing orbital angular momentum (not a spin angular momentum). On the other hand, CVBs are azimuthally homogeneous. The topological nature of CVBs originates from the spatial profile of the polarization vector. That is, the scotoma is nothing but the vortex core of the inplane components of the polarization vector of CVBs. Both beams are experimentally realized in the broad range of frequencies by using holograms or structured filters 17,19 .
There are a number of applications of optical vortices in optics, condensed matter physics, biology, and so on [28][29][30][31][32][33][34][35] including the stimulated emission depletion microscopy 28 which is awarded the Novel prize in chemistry in 2014. Compared to optical vortices, however, the use of CVBs outside optics is, not well explored 36 .
Let us give a mathematical description of CVBs following Ref. 15 . A CVB is obtained by solving the Maxwell's equation in a vacuum in the cylindrical coordinate (ρ, φ, s). Here, ρ is the radial coordinate, φ is the azimuthal angle around the cylindrical axis, and s is the coordinate along the cylindrical axis. On the focal plane, an azimuthal CVB (in the following, we mostly omit "azimuthal") with the wave number k and the frequency ω = ck is given by: where c is the speed of light in a vacuum, J n (x) is the Bessel function, and l 0 (θ) is the pupil apodization function determined by the optical system 15 . Other components of electromagnetic fields are zero. The size of the pupil α is given by α = sin −1 (NA/n) where n and NA are the refractive index and the numerical aperture of the lens respectively. The constant A determines the field amplitude. In Fig. 1, we show the schematics of a tightly focused CVB and its field configuration near the focus. Notable features of focused CVBs are (i) vanishing amplitude of electric fields at the center (ρ = 0) and (ii) non-vanishing "longitudinal" magnetic field there. As a result, near the focus, the longitudinal magnetic field becomes the dominant component of the CVB. Therefore, if we shine a CVB on a tiny sample placed at the focus of the beam [ Fig. 1(b)], we can approximately regard the system to be under an oscillating longitudinal field with optical frequency with electric fields kept small 37 . We note that as the focusing goes tighter, the longitudinal field becomes more prominent.
From the argument above, to make use of the longitudinal field of a focused CVB, we have to prepare a sample much smaller than the wavelength of the incident beam. For visible lights, this requirement is hardly satisfied, but for (sub-) THz or far-infrared lights, it is not so hard. In these days, by using focused ion beam equipment, it is possible to prepare a sample with its size 100 nm to µm. Therefore, if we use the (sub-) THz CVB (wavelength of the order of 100 µm to mm) we could measure the magnetic field absorption while suppressing the electric field one strongly. In the next section, we quantitatively discuss this point.

III. LONGITUDINAL MAGNETIC FIELD: MAGNETIC FIELD SPECTROSCOPY
In this section, we discuss how useful the longitudinal magnetic field of focused CVBs for spectroscopic studies of magnetic properties of matters. We will see that CVBs in the THz region is particularly useful as their relatively long wavelength enables us to strongly suppress the electric field absorption while keeping the high temporal resolution of the measurements.

A. Suppressed electric field absorption for CVB
Electromagnetic absorption of matters is characterized by electrical conductivity and magnetic susceptibility. In this subsection, to demonstrate that using CVBs instead of Gaussian beams results in the strong suppression of the electric field absorption, we examine a high-frequency ESR measurement of electric conductors as a prototypical example of a system with substantial electric field absorption.
In electron spin resonance (ESR) 38,39 , the imaginary part of the magnetic susceptibility, which is relevant for the magnetic field absorption, has a sharp peak at the frequency of magnetic dipole transitions. In the simplest case where the spin-splitting is caused by the Zeeman effect of the external static magnetic field H 0 , the resonance frequency ω 0 is, using the gyromagnetic ratio γ, given by γH 0 . From the peak positions and line widths of ESR absorption spectra, we obtain the resonance frequency, spin relaxation time, magnetic anisotropy, and so on 38,39 .
In electric conductors, in addition to the magnetic field absorption, we always have strong electric field absorption determined by the real part of the electric conductivity. In many cases, the electric field absorption is order of magnitude stronger than the magnetic field one. Hence, detecting magnetic contribution to the absorption spectrum is, in conducting systems, challenging. A possible prescription is a subtraction of the electric field contribution by using, for example, the phenomenological Drudelaw 40 to find out the magnetic contribution. However, the absorption spectra of real materials are far more complicated, and the actual absorption has deviations from the Drude-law (we call those deviations microstructures in the following), making it hard or impossible to separate the two contributions.
Typical frequency of the oscillating field in ESR measurements is gigahertz (GHz). Recently, however, highfrequency ESR using (sub-) Tera Hz (THz) fields is becoming important [41][42][43] . High-frequency ESR has several advantages. It extends the applicability of ESR to matters a sizable zero-field spin splitting 44,45 in, for example, antiferromagnets and spin liquids 46 . Moreover, the use of high-frequency (for example THz) fields improves the temporal resolution of the time-dependent ESR for dynamical properties of free radicals in the target.
At high frequency, however, the absorption by conduction electrons mentioned above becomes more serious. The dominant electric field absorption in combined with microstructures in the electric conductivity is an obstacle for measuring ESR signals. For example, when there is a strong spin-orbit-coupling (SOC), so-called the electron dipole spin resonance (EDSR) takes place 47,48 . As a result, the high and broad peaks originating from the EDSR mechanism 49 form microstructures and hide ESR peaks. Even without SOC, microstructures reflecting materials details could be problematic. Indeed, there has been no report of THz ESR for conducting materials so far.
Combining the electric and magnetic field absorption (coming from, for example, ESR), as a whole, we can write the electromagnetic absorption of a matter as 50 Here σ is the real part of the electrical conductivity and χ is the imaginary part of the magnetic susceptibility. The subscripts i, j are for the spatial coordinates. Let us assume that our sample has the radius of R λ (λ is the wavelength of the beam) and is placed at the focus of the CVB. We expand E φ (ρ, φ) in Eq. (1) in terms of ρk satisfying ρk < R/λ 1 and obtain We see that the leading order term is O(kρ), and that of the absorption (2) is O(k 2 ρ 2 ). The overall absorption is obtained by integrating Eq. (2) over the sample area. Then, the leading order becomes of the order of O(R 2 /λ 2 ). On the contrary, if we use conventional Gaussian beams, since they are approximately plane waves near the focus, the leading order is independent of the ratio R/λ. Hence, compared to Gaussian beams, the electric absorption of a focused CVB would be suppressed by the factor of O(R 2 /λ 2 ).
To make the contrast with Gaussian beams better, we define an "effective conductivity" σ eff (ω) = R 2 λ 2 σ (ω). Then, using a focused CVB instead of a Gaussian beam is effectively to replace σ by σ eff . If the sample size is 1 µm, and the wavelength is 300 µm (1THz), the effective conductivity σ eff becomes 10 −5 smaller than the original electric conductivity 51 .
Below we use the ratio of the two contributions in Eq. (2) as a measure of the relative strength of the magnetic field absorption. Here B CVB and E CVB are field amplitudes of the CVB. We are ignoring the ρ component of the magnetic field and its absorption since it is geometrically suppressed just as E φ . By using the effective conductivity, we can rewrite Eq. (4) as where E G and B G are field amplitudes of the corresponding Gaussian beams, and we used the relation B G = E G /c to reach to the final expression. We also replace σ eff (ω) by its DC value σ eff 0 since the electric conductivity in the THz region is almost the same as the DC value in most cases 52 .
As an example, let us consider a two-dimensional (2D) electric conductor under a static in-plane magnetic field H 0 which induces a net magnetization of conductionelection spins. We are going to apply the azimuthal CVB from the out-of-plane direction and measure the ESR signals coming from the conduction electrons. To estimate the order of the magnetic field absorption by conduction electrons, we use a phenomenological equation of motion of magnetization dynamics, Bloch equation 53 .
By solving the Bloch equation (see Appendix A), the imaginary part of the magnetic susceptibility is derived to be where T 2 is the transverse relaxation time of the magnetization, and χ 0 is the static susceptibility (real part of the susceptibility at ω = 0). The resonance frequency and thus the energy of the Zeeman coupling with the external field are assumed to be around THz. This means that the external Zeeman field is quite strong, larger than ten Tesla. At the resonance ω = γH 0 ≡ ω 0 , we have χ = χ 0 (ω 0 T 2 )/2. If we take the transverse relaxation time to be about a nanosecond (typical timescale of the magnetization dynamics), the peak height will be χ 10 3 χ 0 /2. Putting this expression in Eq. (5), in the SI unit, we obtain P ∼ 0.02 χ0 The dimensionless factor P gives a criterion of whether the magnetic field absorption peaks are detectable or not.
In order the position and linewidth of magnetic resonance peaks to be obtained, their height should be larger than the noises or those from the microstructures in the electric field absorption. In the later section, we will give a quantitative argument for the microstructure from EDSR absorption but here we give a rough estimate of the criterion, just assuming that the microstructure has the amplitude 1% of σ 0 with no regard to its origin. Under this assumption, the system must satisfy P > 0.01, Remember that if we use the Gaussian beam in the first place, the same parameters lead to the criterion σ 0 ≤ 0.3(R 2 /λ 2 ) ∼ 3×10 −6 [S/m]. In the latter case, almost only insulators can satisfy that. On the other hand, the former can be satisfied by various semiconductors (see Fig. 2). By using CVBs, therefore, we could observe THz time-resolved ESR of conduction electrons in semiconductors.
As is evident from the argument above, the criterion for the electric conductivity depends on the magnetic susceptibility and the magnitude of the conductivity microstructures. Since magnetic materials can have much larger magnetic susceptibility than that of a vacuum χ = 4π × 10 −7 [H/m], there would exist many magnetic semiconductors where CVBs become a powerful probe of their magnetic properties.

B. Discussion
The magnetic absorption does not necessarily come from conduction-electron spins. The magnetic degrees of freedom can be magnetic impurities, localized moments, and so on. Moreover, the origin of the magnetic excitation is not limited to the Zeeman splitting. In particular, it can be zero-field splitting due to magnetic anisotropy. As we discuss later, in antiferromagnetic systems, the splitting from magnetic anisotropy is essential. If the ratio of the wavelength and the system size satisfies (R/λ) 2 = 10 −5 , various semiconductors including silicon, gallium arcenide, and cadmium sulfide become "insulating" in terms of the effective conductivity. We show the typical ranges of electric conductivity of several semiconductors taken from a literature 54 .
Although we have focused on electric absorption by conduction electrons in conductors, the same argument applies to dielectric absorption by electric dipole moments in insulating systems like multiferroics or polar liquids. By using CVBs, we can suppress dielectric contribution to the electromagnetic field absorption. In the next subsection, we consider applications for both conducting and dielectric systems.
We again note that we are assuming pulse CVBs, having time-resolved measurements primarily in mind. If a powerful THz light source generating continuous waves will be developed, we can use that to perform frequencydomain measurements. However, in this case CVBs may not be the best option. If continuous waves are available, we may use resonators to form standing waves and exploit their spatial property to suppress the electronic absorption. In the lower frequency region, this approach has been established 55,56 , and ESR measurements of heavy fermion metals 57,58 are achieved at frequencies up to 360 GHz 56 .

C. Applications
As long as the sample size is small, the suppression of the electric field absorption for CVBs is true for any systems. Below we provide several examples where focused pulse CVBs at (sub-) THz frequency would be useful.

Magnetic resonance in conducting systems
ESR of conduction electrons studied in Sec. III A is a prototypical example where electric absorption is harmful. As we mentioned above, experimentally the magnetic absorption can be from either conduction electrons or localized magnetic moments. In any case, the success of the measurement relies on the relative magnitude of the magnetic absorption and the electric one corresponding to the microstructures in electric conductivity. In this part, we give more detailed arguments on applications for magnetic resonance measurements. We firstly discuss the effect of SOC in more details and then consider resonances in systems with large zero-field splittings.
EDSR is a primary source of the conductivity microstructures in electric conductors with SOC. Let us take a 2D electron system with a Rashba-type SOC 59 to make the situation more clear. The Hamiltonian of the SOC is given by where σ are Pauli matrices with matrix indices α and β representing electron spins and d p is the SOC vector. The operator c p,α (c † p,α ) is the annihilation (creation) operator of electrons with a momentum p in the second quantization form 40 . The SOC vector d p depends on the momentum of the electron and is taken to be, for example, d p ∝n × p wheren is the unit vector in the out-of-plane direction. As a result, the SOC term can be regarded as a Zeeman coupling with the momentum-dependent field and thus leads to the different spin-splitting for each (crystal) momentum. The momentum-induced splitting makes the system active to electric fields, resulting in the strong and broad EDSR absorption which smears out the ESR contributions.
The actual magnitude of the EDSR absorption strongly depends on the electron density of the system. To make the argument more quantitative, we consider the following one-dimensional tight-binding model where the second term is the Zeeman coupling with the external field, and the third one is the Rashba SOC. Here t 0 is the electron hopping, g is the g-factor, µ B is the Bohr magneton, and λ R is the SOC constant. The operators c † i,α and c i,α create and annihilate electrons with spin α at the lattice site i. The Hamiltonian (8) is a special case of the model studied in Ref. 49 . Although this is a one-dimensional system, we can consider a collection of such conducting chains and calculate the standard electric conductivity. We assume λ R /t 0 = gµ B B/t 0 = 10 −3 and take the intra-and inter-chain lattice constants to be a = a ⊥ = 0.5 nm. As the electron hopping t 0 is typically order of electron volt, the magnitudes of λ R and gµ B B are of THz. According to the calculation in Ref. 49 , in this case, the EDSR contribution to electric conductivity for ω ∼ gµ B B takes the range of 10 −8 to 10 −1 [S/m] depending on the electron density. Comparing this with Fig. 2, we notice that the EDSR contribution is in the "insulating" region of σ eff relevant to CVBs. Therefore, by using CVBs, at least for a simple tight-binding model, absorption through the EDSR mechanism is suppressed enough, making it possible for us to observe ESR signals even in the presence of EDSR. We point out that the assumption on the magnitude of the microstructures used in the previous section is consistent with the calculation above.
So far we have assumed simple magnetic resonance of paramagnetic or ferromagnetic systems whose resonance frequency is determined by the external static magnetic field. Hence, to lift the resonance frequency to be in the THz region, we have to apply a very large magnetic field over ten Tesla. On the other hand, in the case of conductors with localized moments, the resonance frequency can be very high even without an external field. For example, in antiferromagnets, due to the mechanism so-called the exchange enhancement, the frequency of the antiferromagnetic resonance (AFMR) coming from a magnetic anisotropy is often lifted to the (sub-) THz region 44,45 . Specifically speaking, when the magnetic moments are coupled with each other through the antiferromagnetic exchange coupling J ex and under the influence of the uniaxial anisotropy A, according to the Bloch equation for antiferromagnets, the resonance frequency is given by ω res = √ J ex A. We note that in the case of paramagnetic or ferromagnetic conductors, the advantage of measurements with CVBs compared to the low-frequency ones with, for example, resonators is apparent only in their temporal resolution. On the contrary, in the case of antiferromagnets, due to the large zero-field splitting, using high-frequency lights is, in the first place, necessary to measure their magnetic resonance. Therefore, CVBs will have broad applications for AFMRs in the presence of electric absorption through, for instance, EDSR.

Multiferroics
As we noted, electric absorption can also come from electric dipoles. In this and the next parts, we consider insulating systems with dielectric loss.
In a class of matters, there appear multiple long-range orders at the same time, and it is called multiferroics. In particular, the coexistence of ferromagnetic and ferroelectric orders is intensively studied. The magnetization and electric polarization in multiferroic materials mutually affect 60-67 with each other, providing a way of controlling the ferroelectricity with a magnetic field and the magnetization with an electric field. As electric polarization in multiferroic materials depends on the configuration of localized magnetic moments, the elementary magnetic excitation of multiferroic materials, called the electromagnon, is active to both electric and magnetic components of the incident beams.
If we use Gaussian beams, the excitation is caused by both electric and magnetic fields. If we know the magneto-electric coupling of the target beforehand, we could avoid the complication by combining several measurements. For example, experimentally it is sometimes possible to identify electric and magnetic contributions by using beams with several different polarizations 65 . However, this is not always the case, and, in particular, is not applicable to newly synthesized materials.
As a proof of concept, let us take a simple toy to make things concrete. We consider a pair of classical spins S 1 and S 2 under a linearly polarized beam B(t) = Bŷ cos(ωt), E(t) = Eẑ sin(ωt). We assume that the electric field couples with spins through the spincurrent mechanism 61 ; p = λ c e 1,2 × (S 1 × S 2 ) where e 1,2 =x is the unit vector in the direction connecting these spins.
We are going to study the laser-driven dynamics of the spin S 1 (t). If there exists a strong static magnetic field in the x direction, both spins stay almost fully-polarized in that direction. As long as we are interested in the small-amplitude dynamics of S 1 , therefore, we can approximately ignore that of S 2 and take S 2 =x. In this case, the Hamiltonian for the spin S 1 can be written as Here, the static magnetic field H 0 incorporates the interaction between S 1 and S 2 .
As shown in the appendix C, the time evolution of S y 1 (t) is given as where γ = gµ B / is the gyromagnetic ratio. We see that the spin dynamics comes from both Zeeman and magneto-electric couplings with the beam. In the present case, as we know the functional form of the dynamics Eq. (10), we can read out magnetic and electric contributions by measuring the time-evolution while changing parameters like ω and H 0 .
However, in general we do not even know the actual mechanism of the magneto-electric coupling so that examining parameter dependence is no help for understanding the field-induced spin/polarization dynamics. With focused CVBs, we can independently apply the electric and magnetic field to the target. We can selectively activate the magnetization/polarization dynamics to study the effect of electric and magnetic fields without any prior knowledge of the spin-polarization coupling of the target. The use of CVBs for studying multiferroics will, therefore, largely streamline the characterization of such materials.

Electron paramagnetic resonance in absorbing media
So far, we have been considering condensed-matter applications of CVBs. The potential use of CVBs is, however, not limited to solid-state physics. In this part, we give an example for chemistry and biology; a pulse electron paramagnetic resonance (EPR) of (bio) molecules in absorbing media. EPR 68-70 is a primary tool of chemistry and biology where the absorption spectrum works as a fingerprint of unpaired electrons in the target. Specifically speaking, in the spin label (or spin probe) method of EPR, we introduce a stable radical as a spin marker and measure its EPR absorption to study macromolecules like a protein.
In order to understand the dynamical properties of molecules, we have to resort to the time-dependent EPR using pulse waves. Since the temporal resolution of the measurement is determined by the pulse width, it is advantageous to use high-frequency fields for EPR.
The problem is that when target molecules are dissolved in absorbing media (e.g., liquid water), lightwave at (sub-) THz region is strongly absorbed by the environment, and thus the EPR of solute molecules becomes difficult 71 . In biological or medical context, it is essential to study the nature of biological molecules in the living environment. Hence, the (sub-) THz absorption by an absorbing medium, especially liquid water, is a problem.
By using a pulse CVB, we can avoid exciting the oscillation modes of solvent polar molecules and perform the time-dependent EPR of solute molecules. Since the energy of spin triplet excitations of liquid water is high (of the order of eV and higher) 72 , the dominant absorption of the (sub-) THz CVBs would be, if exists, by the solute molecules like spin markers of the spin label/probe methods.
As a marker, commonly a nitroxide radical and molecules with that are used. The highly anisotropic nature of the g-factor and the hyperfine coupling constant of nitrogen atom allow high precision measurements and make nitroxides be the standard. However, if we are to use nitroxides as markers of the high-frequency pulse EPR, we have to apply a strong static magnetic field to lift the Zeeman splitting energy. Another option for high-frequency pulse EPR is markers with large zero-field splitting like single-molecule magnets (SMM) [73][74][75] . For example, if there are at least two magnetic ions in an SMM interacting with each other, the energy splitting among different spin multiplets naturally resides in the THz region. Even if an SMM contains only one magnetic ion, recently it is possible to design a molecular structure having a large magnetic anisotropy and thus a large zerofield splitting 76,77 . The use of CVBs, therefore, allows us to use variety of SMMs as spin markers of high-frequency pulse EPR.

IV. AZIMUTHAL ELECTRIC FIELD: IMAGING AND CONTROLLING CIRCULATING CURRENTS
So far, we have examined the use of the magnetic field component of the azimuthal CVBs. We discussed how their geometrical feature can be utilized to suppress the electric field absorption, and how to utilize that for various applications of measuring magnetic properties of matters. In this section, we consider the use of the electric field component of azimuthal CVBs which has been a nuisance in the previous section. As discussed in Sec. II, the longitudinal magnetic field is prominent only if the focusing is tight so that for a weakly focused CVBs, electric absorption is dominant. In this section, we are only interested in the azimuthal nature of the electric field so that we assume a weakly focused CVB in the following argument and neglect the magnetic field component. The purely azimuthal electric field is written in the following form: where C is a constant determining the field amplitude andê φ is the unit vector in the azimuthal direction. This field indeed has the azimuthal polarization and is obtained by superimposing two optical vortices. The spatial profile is controlled by the beam width w, which should be larger than the wavelength because of the diffraction limit. We note that unlike the previous sections, the arguments in this section are independent of the frequency/wavelength of the CVBs. In particular, the sample size has not to be small compared to the beam width in the following. The azimuthal nature of the electric field of CVBs allows the field to couple with circulating currents efficiently. Here we consider its application to visualize and control the edge circulating current in topological insulators 21,22 . Topological insulators are characterized by the coexistence of the insulating bulk and metallic edges/surfaces. Notable examples of topological insulators in 2D are quantum Hall insulators and quantum spin Hall insulators. The former has metallic edges with chiral transport and realized in, for example, 2D electron systems under a strong out-of-plane magnetic field. The latter is schematically a combination of a pair of the former in a way restoring the time-reversal symmetry. The edge states in the quantum spin Hall insulators are thus helical consisting of two counter-propagating modes with the opposite spins. The metallic edges in those materials naturally host circulating electric current transport. Both the quantum Hall insulators and quantum spin Hall insulators are experimentally studied well. The existence of edge states has been verified by the transport measurements 78 but its direct visualization is also important [79][80][81][82][83] .
Here we discuss how to use a CVB to visualize the edge states in topological insulators.

A. Edge visualization
Let us take a disk-shaped 2D topological insulator (quantum spin Hall insulator) with helical edge modes. We consider of applying the azimuthal field Eq. (11) and measure the electric absorption. The edge current can be modeled as a spatially inhomogeneous electric conductivity: where R is the sample radius. The localization length ξ is determined by the bulk property and typically of the order of nm. The function Θ(x) is the Heaviside Theta function.
The field amplitude of Eq. (11) has its peak position determined by the beam width w. Then, we can expect that when the peak position matches with the position of the electric current, the electric absorption grows. That is, by measuring the electric absorption while changing the beam width, we can quantify where and how much the electric current is localized [see Fig. 3(a)]. In the following, we examine this expectation for the edge localized currents. As a reference, we also consider the case with homogeneous conductivity which corresponds to ordinary metals.
The electric field absorption is, as we have seen in Eq. (2), given by This integral can be analytically performed but the expression is lengthy and thus we omit that (see appendix B). In the lowest order of the localization length ξ, it simplifies to The analytical expression will be useful as a fitting function in the experiment. We show the w dependence of the absorption in Fig. 3(b). The spatial profile of the conductivity for the localization length ξ = 0.05R is shown as the inset. We see that there exists a rising edge at around w = 0.5R, and the absorption takes its maximum when w ∼ R as we expected. Contrary to that, as shown in Fig. 3(c), if the target is an ordinary metal with spatially homogeneous conductivity σ(ρ) ∝ Θ(R − ρ), there is no such rising edge and the w dependence is linear in the small w region. Also, the absorption peaks for the width w smaller than R. The linear dependence on w for w R is easily derived from Eq. (12) by taking ξ → ∞ which corresponds to σ(ρ) ∝ Θ(R − ρ). By reconstructing the spatial profile of the electric conductivity, or that of the edge modes from the lineshape of the w dependence, we can visualize the edge transport in topological insulators.

B. Orbital magnetization
The visualization of edge states with CVBs is interesting but is not conceptually new. In artificial systems like photonic or phononic crystals, the visualization of edge states is well-established 79,80 . Even in solids, there are several reports based on microscopes and SQUID techniques [81][82][83] . The advantage of using CVBs compared to those existing studies is that it also allows us to control the edge states, not just measuring that. In particular, by driving the circulating currents, we can control the orbital magnetization arising from them 84,85 : The total magnetization measured experimentally is a sum of the orbital and spin magnetizations 86 . Since we are considering a weakly focused CVBs, the longitudinal component of magnetic fields and the spin magnetization induced by that are negligible. Then, the orbital magnetization induced by the azimuthal electric field pulse is measurable just by looking at the out-of-plane component of the total magnetization with, for example, magneto-optical means. Hence, by using CVBs, we can control the magnetic property of the target material relying solely on the electric field.
We note that the laser-induced orbital magnetization does not require the system to be topological insulators. In metals, the induced magnetization will be much stronger than that in topological insulators due to the large bulk electric conductivity. In particular, in spinorbit-coupled systems, due to the coupling between the spin and orbital magnetization, it might be possible to control the spin magnetization through the laser-induced orbital magnetization. Further studies on their interplay is thus an interesting research direction.

C. Discussion
The candidate system for the experiment proposed above would be a thin film of 3D topological insulators such as Bi compounds and HgTe wells 21,22 . While the quantum Hall insulator is the simplest topological system, the chiral nature of the edge state would cause a complication. It would be indeed interesting to study the response of such chiral edge states in, for example, quantum Hall insulator or chiral topological superconductors to CVB pulses. The interplay of the chiral nature of edge mode and the azimuthal electric field pulse would result in nonreciprocal responses which may be useful as a fingerprint of chiral edge modes.
Although we did not discuss in detail, controlling circulating currents in metallic or superconducting rings would also be interesting. For example, by using the tightly focused CVB, we can insert magnetic fluxes in a metallic ring within the optical timescale and study its effect on electron transports such as Aharonov-Bohm effect. If we consider superconductors, with a CVB, we can control the electronic property of superconductors in the timescale much faster than the typical timescale of superconducting fluctuations. Such ultrafast dynamics of superconductors is a frontier of the field of superconductivity, and recently, interesting results like photoinduced superconductivity are reported 87,88 . The use of CVBs allows us to control the magnetic flux and circulating current in an ultrafast way, offering new methods of controlling the nature of superconductivity.

V. APPLICATION TO FLOQUET ENGINEERING
The highly controllable nature of CVBs offers an ideal playground for Floquet engineering 23-25 where we drive a system periodically with, for example, lasers to realize desired nonequilibrium states. In this section, we take a toy model of quantum magnets to demonstrate the advantage of CVBs for Floquet engineering over conventional lasers.
By superimposing radial and azimuthal CVBs, we can apply electric and magnetic field independently for systems smaller than the wavelength. We can freely change the relative angles, phases, and amplitudes of those fields. In the context of Floquet engineering, this property makes CVBs extremely powerful tools for designing nonequilibrium states of matters.
First, we briefly introduce the concept of Floquet engineering. In Floquet engineering, we are interested in the nonequilibrium states of matters under a periodic drive. The time-evolution in a time-periodic system is described by a Hamiltonian H(t) = mĤ m e imt where t is time andĤ m is the m-th Fourier component of H(t). For the time-evolution operator U (t 1 , t 2 ) = T t exp i t2 t1 H(t)dt with T t being the time-ordering operator, we introduce a "Floquet effective Hamiltonian" H F as where T is the period of the external drive. That is, at each stroboscopic time t = T, 2T, ... the system looks as if following the static Hamiltonian H F . When the driving frequency is sufficiently high, socalled the Floquet-Magnus expansion gives a simple formula of the Floquet effective Hamiltonian: This effective Hamiltonian enables us to predict the nonequilibrium dynamics under the drive in an intuitive way. For example, if we consider a Zeeman coupling between spins and a circularly polarized magnetic fields, Eq. (16) predicts an emergence of a large synthetic magnetic field perpendicular to the propagating axis of the beam 39,89,90 , and this is consistent with the known inverse-Faraday effect. Let us take a simple example, laser-induced multiferroicity 26 to show why Floquet engineering with CVBs is advantageous over with other means like circularly polarized beams. We consider a pair of quantum spins S 1 and S 2 placed along the x-axis and apply CVBs to them. The spins couple to magnetic fields through the Zeeman where we consider a focused azimuthal CVB applied along the z-axis; B(t) = B cos(ωt + δ)ẑ. We assume that spins couple to electric fields through the magneto-electric coupling of the spin-current mechanism H p = −g me p · E with p =x × (S 1 × S 2 ), and take the electric field to be in the following form E = E(cos θẑ + sin θŷ) cos(ωt). That is, we apply a focused radial CVB with frequency ω along the direction (0, sin θ, cos θ). If we take δ = π/2 and θ = π/2, it becomes equivalent to apply a linearly polarized laser propagating in the x direction.
The Floquet effective Hamiltonian is obtained from Eq. (16) as That is, the periodic drive by the focused CVBs results in the synthetic Dzyaloshinskii-Moriya (DM) type interaction 91,92 between the spins. In the appendix D, we show that this Floquet effective Hamiltonian indeed gives an approximate description of the spin dynamics. The leading order term of the Floquet Hamiltonian (17) contains only the synthetic DM interaction. This is in stark contrast to the previous work 26 dealing with the same model but using circularly polarized lasers applied along the z axis instead. There, in addition to the synthetic DM interaction, we inevitably have a synthetic Zeeman field which is undesirable for exploiting the synthetic DM interaction for designing nonequilibrium states.
We see that in Eq. (17), there are a number of controllable parameters for CVBs; relative phase δ, relative angle θ, field amplitudes E and B, frequency ω. Such high controllability is achieved by the unique geometrical feature of focused CVBs. Since the typical timescale of spin system is slower than THz, the high-frequency expansion is supposed to be a good approximation for the laser-driven dynamics. Therefore, using focused CVBs for Floquet engineering is a promising application.

VI. CONCLUSION
In this paper, we explored the use of the azimuthal cylindrical vector beam for studying and controlling electromagnetic properties of matters. Compared to the conventional Gaussian beams, the electric field contribution to optical absorption is strongly suppressed for small samples.
In the third section, we enumerated some applications; studies of magnetic resonances in conductors, simplified characterization of multiferroics, and high-frequency electron spin resonance of water solutions. Other examples may include the study of topological insulators, magnetic superconductors, and quantum dots/wells. Like the electron paramagnetic resonance in polar medium for biological purposes, the proposed spectroscopy using cylindrical vector beams has various applications even outside condensed matter physics. The high temporal resolution of THz time-domain spectroscopy with cylindrical vector beams would be a window to nonequilibrium magnetic properties of matters.
In the fourth section, we discussed the applications of the azimuthal electric field as well. It can couple to circulating currents and dynamically generate an orbital magnetization. We could use those properties to study circulating edge modes in (topological) materials. It is an open question whether the photo-induced orbital magnetization has any applications in the field of spintronics. Also, it would be interesting to use cylindrical vector beams for controlling other types of circulating currents like those in superconducting rings or mesoscopic systems.
In the fifth section, we discussed Floquet engineering. Taking a simple model of a driven spin, we show that using focused cylindrical vector beams has clear advantages over known schemes. The extremely high controllability of focused cylindrical vector beams would offer building blocks of artificially designed nonequilibrium states of matters.
The spatial properties of cylindrical vector beams have various applications. We can use them for suppressing electric field absorption, controlling circulating currents in matters, and designing electromagnetic fields. The requirement on the sample size can be easily satisfied for various condensed matter and chemistry uses. We expect that cylindrical vector beams would be powerful means for characterizing and controlling electromagnetic properties of matters at THz frequency.

VII. ACKNOWLEDGMENT
We thank Shunsuke Furuya for useful comments. H.