All-optical demonstration of a scalable super-resolved magnetic vortex core

Xiaofei Liu; Weichao Yan; Zhongquan Nie; Zhongquan Nie; Yue Liang; Ensi Cao; Yuxiao Wang; Zehui Jiang; Yinglin Song; Xueru Zhang; Xueru Zhang

doi:10.1364/OE.454079

1. Introduction

Magnetic vortex core (MVC), as a kind of flux-closed magnetic polarization topology, is characterized by a curling in-plane (azimuthal) magnetization and an out-of-plane (longitudinal) core [1–3]. It has been considered as a promising candidate for an all-in-one information transfer and storage cell in future non-volatile magnetic memory, where both the doughnut curly vortex and solid perpendicular core work as an information carrier and a recording unit, respectively [4]. Therefore, a full understanding of formation process and manipulation mechanism of the well-defined MVC becomes a major requirement, which have moved into the focus of interest as well. To this end, according to the micromagnetic theory or by resorting to the external magnetic/electric field/spin current, numerous attempts have been devoted to shedding light on the enriched static and dynamic behaviors of MVC in the extensive magnetic order systems [4,5–11]. In spite of these remarkable progresses, it is still highly desirable to seek for powerful and alternative approaches for efficiently generating and controlling the exotic nucleation of vortices (namely, MVC).

In the wake of the recent dramatic developments in the control of light fields [12–15], the ability to trigger multifunctional magnetization patterns has been demonstrated in an all-optical manner, making the high-efficiency optomagnetic recording and storage possible [16–21]. In this respect, based on the inverse Faraday effect (IFE) [22–24], it is capable of yielding the versatile light-induced magnetization fields with stretchable spatial resolutions, tunable configuration paradigms, or steerable polarization orientations by tightly focusing the amplitude/phase/polarization-encoded incident beams [25–44]. Yet, these works were essentially interested in the all-optical realization of spatially super-resolved multi-structured magnetization fields with regular 3D/transverse/longitudinal polarization states and overlooked those with 3D curling polarization orientations. To alleviate such an issue partly, Helseth put forward the use of either a radially polarized beam or a two-beam configuration upon focusing to imprint magnetic vortices (MVs) in an isotropic magneto-optical medium (MOM) [45]. In principle, the MV can only be exploited to convey the designated information while fails to favor the data storage. It is thus of urgent necessity to formulate an efficacious strategy for producing super-resolved MVC with flexibly tunable capabilities, which remains to be unexplored until now.

In this connection, we here revisit a feasible all-optical pathway to activate the scalable MVC for the first time. This is achieved by tightly focusing the on-demand tailored radially polarized doughnut Gaussian beams (RP-DGBs) onto the specified MOM in the single/4π focusing lenses systems. In doing so, we succeed in creating single MV, single MVC, and extensive tunable MVC needle/chain within the ultra-small domains throughout the entire focal plane. The underlying reason to cause these magnetic polarization topologies may lie in the polarization filter effect of optimally designed multi-ring filter (MRF) in combination with the coherent interferences of three orthogonal focal magnetization components. The paper is arranged as follows: in section 2, the focused electric fields and induced magnetizations of the RP-DGBs with cascaded composite phase/amplitude modulations are deduced according to the vectorial diffraction theory and the IFE. We present in section 3 that the super-resolved MV, the 3D super-resolved MVC, and the super-resolved needle-like/chain-like MVC textures can be formed by choosing both the different focusing systems and wavefront encodings. Moreover, the associated physical mechanisms are also revealed. Finally, we give the conclusions in section 4.

2. Theoretical analysis of light-induced magnetic vortex core

The schematic configuration of a typical 4π focusing setup is shown in Fig. 1. Two incident RP-DGBs firstly transmit through the first-order SPPs and the MRFs in turn, and are subsequently focused by two high numerical aperture (NA) objective lenses (OLs) facing each other onto the nonabsorbing isotropic MOM. The SPP makes the wave-front phase along the azimuthal direction encircling the optical axis vary by 2π, whereas the MRF composed of multiple belts in the radial direction has either alternating π phase difference between adjacent rings or null transmittance. Mathematically, their synthetic transmittance function can be represented as t(θ, φ) = p(θ)e^iφ. After the RP-DGBs with the assigned wavefront encodings are strongly focused by the OLs, the counter-propagating electric fields in the vicinity of focus can be calculated by leveraging the vectoial diffraction theory [46–49],

(1)$${\displaystyle{{\mathbf E}_l}\left( {x,\;y,\;z} \right) = \left[ \begin{array}{l} {E_{lx}}\\ {E_{ly}}\\ {E_{lz}} \end{array} \right]\textrm{ = }\frac{{ - iA}}{\pi }\int_0^\alpha {\int_0^{2\pi } {\sin \theta \sqrt {\cos \theta } t(\theta ,\;\varphi )} } {l_0}\left( \theta \right){e^{i{k_0}\left( {r\sin \theta \cos \left( {\varphi - \phi } \right) + z\cos \theta } \right)}}\left[ \begin{array}{l} \cos \theta \cos \varphi \\ \cos \theta \sin \varphi \\ \;\;\;\;\sin \theta \end{array} \right]d\varphi d\theta ,$}$$

(2)$${\displaystyle{{\mathbf E}_l}\left( {x,\;y,\;-z} \right) = \left[ \begin{array}{l} {E_{rx}}\\ {E_{ry}}\\ {E_{rz}} \end{array} \right]\textrm{ = }\frac{{ - iA}}{\pi }\int_0^\alpha {\int_0^{2\pi } {\sin \theta \sqrt {\cos \theta } t(\theta ,\;\varphi )} } {l_0}\left( \theta \right){e^{i{k_0}\left( {r\sin \theta \cos \left( {\varphi - \phi } \right) - z\cos \theta } \right)}}\left[ \begin{array}{l} \cos \theta \cos \varphi \\ \cos \theta \sin \varphi \\ \;\;\;\;\sin \theta \end{array} \right]d\varphi d\theta ,$}$$

(3)$${{\mathbf E}_{tot}} = {{\mathbf E}_l}({x,\;y,\;z} )+ {{\mathbf E}_r}({x,\;y,\; - z} ).$$

Fig. 1. The schematic diagram to generate scalable super-resolved MVC patterns. RPDGB: radially polarized doughnut Gaussian beam; SPP: spiral phase plate; MRF: multi-ring filter; OL: objective lens; MOM: magneto-optical medium; MVC: magnetic vortex core. Insert: the uniform ultra-long MVC needle/chain (red dotted box) and the 3D polarization of single MVC at z = 0 (red solid box).

Download Full Size | PDF

In Eqs. (1)– (3), E_l, E_r and E_tot are the left-side, right-side and total focused electric fields, respectively. α is the maximum convergence angle determined by the high NA OLs, A is the amplitude constant, k₀ is the wave vector in free space, $r = \sqrt {{x^2} + {y^2}}$ and $\phi \textrm{ = actan(}{y / x}\textrm{)}$ are the radial and azimuthal coordinates in the focal regime, and l₀(θ) depicts the apodization function of the RP-DGB, which is given by [44,50],

(4)$${l_0}(\theta ) = \frac{{\beta \sin \theta }}{{\sin \alpha }}\exp \left( { - \frac{{{\beta^2}{{\sin }^2}\theta }}{{{{\sin }^2}\alpha }}} \right),$$

where β represents the truncation parameter, which is defined as the ratio of the pupil radius to the incident beam waist. When the total focused field irradiates onto the MOM, the light-induced magnetization related to the IFE can be written as [22–24],

(5)$${\mathbf M}({r,\varphi ,z} )= j\gamma {{\mathbf E}_{tot}} \times {\mathbf E}_{tot}^ \ast ,$$

in which γ is the magneto-optical constant and ${\mathbf E}_{tot}^ \ast $ is the complex conjugate of ${{\mathbf E}_{tot}}$. To see the MVC more clearly, we will further derive the magnetization quantitatively following subsequent two aspects.

First, given that both the SPP and MRF are absent, the whole focusing behavior reduces to the situation that the tight focusing of radially polarized beams. In this case, it is well known that the electric fields in the focal plane consist of related radial and longitudinal components. Unambiguously, the resultant magnetization is purely azimuthal according to Eq. (5), thus indicating the formation of MV [45]. Such peculiar magnetization fields in the 4π focusing geometry read as,

(6)$${{\mathbf M}_\phi }({r,\;\phi ,\;z} )= 16\gamma A_0^2{R_0}({r,\;\phi ,\;z} ){R_1}({r,\;\phi ,\;z} ){{\mathbf e}_\phi },$$

where

(7)$${R_0}({r,\;\phi ,\;z} )\textrm{ = }\int_0^\alpha {{{\sin }^2}} \theta \sqrt {\cos \theta } {l_0}(\theta ){J_0}({{k_0}r\sin \theta } )\cos ({{k_0}z\cos \theta } )d\theta ,$$

(8)$${R_1}({r,\;\phi ,\;z} )\textrm{ = }\int_0^\alpha {\sin ({2\theta } )} \sqrt {\cos \theta } {l_0}(\theta ){J_1}({{k_0}r\sin \theta } )\cos ({{k_0}z\cos \theta } )d\theta ,$$

where J_n(x) signifies the nth-order Bessel function of the first kind.

To further trigger the MVC, it is the key to giving birth to a solid longitudinal magnetization together with a dark azimuthal one. Toward this end, we make use of the SPP to tailor the radially polarized beam and can garner the radially, azimuthally and longitudinally polarized light fields all at once in the focal plane [51]. It is thus able to produce associated radial, azimuthal, and longitudinal magnetizations based on the IFE in the single lens focusing geometry [31]. When two counter-propagating radially polarized beams superimpose coherently, it is possible to eliminate the radial magnetization, leaving behind the other two components. Under such a circumstance, the induced magnetization of RP-DGBs in the confocal plane with the 4π focusing can be expressed as,

(9)$${\mathbf M}({r,\phi ,z} )= \left[ \begin{array}{l} {M_r}\\ {M_\phi }\\ {M_z} \end{array} \right] = 8\gamma A_0^2\left[ \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0\\ 2{\textrm{Re}} ({{I_1}({r,\;\phi ,\;z} )\ast I_3^ \ast ({r,\;\phi ,\;z} )} )\\ - {\textrm{Re}} ({{I_1}({r,\;\phi ,\;z} )\ast I_2^ \ast ({r,\;\phi ,\;z} )} )\end{array} \right],$$

in which

(10)$${I_1}({r,\phi ,z} )= \frac{1}{2}\int_0^\alpha {\sin 2\theta \sqrt {\cos \theta } } p(\theta ){l_0}(\theta )[{{J_2}({{k_0}r\sin \theta } )- {J_0}({{k_0}r\sin \theta } )} ]\cos ({{k_0}z\cos \theta } )d\theta ,$$

(11)$${I_2}({r,\phi ,z} )= \frac{1}{2}\int_0^\alpha {\sin 2\theta \sqrt {\cos \theta } } p(\theta ){l_0}(\theta )[{{J_2}({{k_0}r\sin \theta } )\textrm{ + }{J_0}({{k_0}r\sin \theta } )} ]\cos ({{k_0}z\cos \theta } )d\theta ,$$

(12)$${I_3}({r,\phi ,z} )= \int_0^\alpha {{{\sin }^2}\theta \sqrt {\cos \theta } } p(\theta ){l_0}(\theta ){J_1}({{k_0}r\sin \theta } )\cos ({{k_0}z\cos \theta } )d\theta .$$

It is seen from Eqs. (9)–(12) that, as expected, the radial magnetization disappears completely, whereas the longitudinal and azimuthal counterparts reside in the in-focus and off-focus regions, respectively. This renders the MVC possible. More interestingly, the specific MRF can be engineered skillfully to modulate the incoming RP-DGBs, which is beneficial to further prolong the desired MVC homogenously along the optical axis (i.e. the needle-like and chain-like MVC patterns), as demonstrated in the inset of Fig. 1.

3. Results and discussions

3.1 Light-induced magnetic vortices

To start with, we examine the magnetization distributions mediated by the RP-DGB without any modulation in both the single and 4π OLs focusing systems, respectively, as showcased in Fig. 2. It is quite clear that the induced magnetization is completely made up of azimuthal component regardless of the focusing configuration, which is akin to that reported in [45]. For the one-side focusing, the total magnetization (M_t) forms a hollow ellipsoid throughout the entire focal plane (Fig. 2(a)), while the azimuthal one (M_ϕ) exhibits a pair of cigar-shaped patterns with opposite polarity (Fig. 2(b)). On the other hand, when two RP-DGBs are collected by double face-to-face high NA OLs, both the M_t and M_ϕ are subjected to considerable squeezing along the optical axis accompanied with strong side lobes. This leads to the appearance of a sub-wavelength doughnut texture and a couple of conversely polar compressed magnetizations (Figs. 2(c) and 2(d)), which is ascribed to the constructive interference of two incident vecorial beams coming from two propagation directions [52].

Fig. 2. Light-induced magnetic vortices. Total(M_t, (a) and (c)) and azimuthal (M_ϕ, (b) and (d)) 3D iso-magnetic surfaces located at 0.5 (red) and–0.5 (green) of the maximum magnetization in the single ((a) and (b)) and 4π ((c) and (d)) objective lens focusing systems. Note that the insets in (a)-(d) are their own magnetization contour profile in the r-z plane (4λ×4λ). The line scans of the total and azimuthal magnetizations along the r and z axes in the single (e) and 4π (f) objective lens focusing systems. M_tr (red solid curves) and M_tz (blue solid curves): the M_t along the r and z axes, respectively; M_ϕr (black dash curves) and M_ϕz (magenta dash curves): the M_ϕ along the r and z axes, respectively; (g) The 3D polarization distribution of magnetic vortices at z = 0 and (h) the related 2D projection ((2λ×2λ)); the colors and lengths of arrows represent the magnitude of magnetization (i. e, long and yellow arrow is the strongest magnetization). The simulation parameters are set as NA = 0.95, β = 1.5, and λ = 532nm, which remain unchanged in the following other unless otherwise specified.

Download Full Size | PDF

Furthermore, we plot the line scans of the total and azimuthal magnetizations along the r and z axes in Figs. 2(e) and 2(f). The transverse full width at half maximum (FWHM) values of their hollow area and each bright lobe are calculated to be in turn 0.486λ and 0.654λ for two proposed focusing systems (red solid and black dash curves in Figs. 2(e) and 2(f)). However, the axial dimension (0.368λ) within the magnetization lobes under 4π focusing (blue solid and magenta dash curves in Fig. 2(e)) is 4 times narrower than that obtained with the single lens focusing (blue solid and magenta dash curves in Fig. 2(f)) [53].To get more insights into the derived magnetization, it is of vital importance to delineate its polarization pattern, as revealed in Fig. 2(g). We observe that the 3D magnetization direction is azimuthal anticlockwise all the time, which is also echoed by the closed flux domain structure in the associated 2D projection (Fig. 2(h)), thereby indicating the creation of light-induced super-resolved doughnut MVs. In practice, such an ultra-small singular curling magnetization polarization could be considered as information carries in high-density optomagnetic recording and storage, as each vortex can take along two magnetic bits [4,7].

3.2 Light-induced single super-resolved magnetic vortex core

In addition to the MV, it is highly desired to all-optically yield the MVC. Namely, the induced magnetization constitutes both the bright longitudinal core and flux-closure dark azimuthal components. For this purpose, the radially polarized beams superposed with a first-order optical vortex can be adopted as the driving sources. For comparison, we depict the induced magnetization with the single lens focusing in Fig. 3. It is observed that three orthogonal magnetization components (M_r, M_ϕ, and M_z in Figs. 3(b)–3(d)) coexist and thus the total magnetization (M_t in Fig. 3(a), 3(e) and 3(f)) takes on a tiny focal shift in the lateral direction. This depends primarily upon the interplay between the optical vortex and the vectorial polarization [29]. Although the magnetization at the focal regime (z = 0) is totally dominated by both the longitudinal and azimuthal components (magenta and blue curves in Fig. 3(f)), the radial component occupies appreciable portions at the out-of-focus region (e.g. z = 0.5λ), even up to maximum value of 31% (red curve in Fig. 3(f)). As a result, we fail to implement the MVC in the whole focal area under such a condition. This fact can also be disclosed by the 3D magnetization polarization (Fig. 3(g)) and its 2D projection (insert). It is apparent that the polarization orients to the z axis at r = z = 0, whereas it lies in the synthetic direction of azimuthal and radial axes rather than the purely azimuthal orientation around the central focus. That is to say, such an intricate 3D polarization doesn’t belong to the well-known MVC because of the non-negligible M_r. Therefore, the key to producing the MVC is to reduce even eliminate the redundant M_r.

Fig. 3. The magnetization distributions induced by radially polarized vortex beam with the single lens focusing. (a)-(d) The total magnetization (M_t), radial (M_r), azimuthal (M_ϕ) and longitudinal (M_z) components in the r-z plane (4λ×4λ). (e)The 3D iso-magnetic surfaces of M_t located at 0.5 of the maximum magnetization. (f) The line scans of M_t, M_r, M_ϕ and M_z along the r axis at z = 0.5λ. (g)The 3D magnetization polarization at z = 0.5λ and the related 2D projection ((2λ×2λ)).

Download Full Size | PDF

To cope with this issue, the best option to date consists in focusing the radially polarized vortex beams in a coherent manner through two objective lenses facing each other [31,52]. By doing so, it can be seen that the radial magnetization (M_r) does indeed vanish completely and thus the total magnetization (M_t) consists of solid longitudinal (M_z) and dark azimuthal (M_ϕ) components, as depicted in Figs. 4(a)-(d). Moreover, closer inspection finds that the magnetization is totally dictated by the M_z in the out-of-plane direction, while it hinges on the cooperative contribution of both the M_ϕ and M_z along the in-plane direction, which indicates the possibility of inducing the expected MCV. In this regard, the corresponding 3D iso-magnetic surfaces are plotted in Figs. 4(e)-(g), from which we observe that both the M_t and M_z manifest themselves as two symmetrical side lobes sandwiching a main lobe in the focal area. On the whole, however, the M_t enlarges horizontally to some extent relative to the M_z. This is because the M_ϕ contributes to a fair portion away from the focal region. The mutual fusion of the M_ϕ and M_z also makes the MVC phenomenon unclear.

Fig. 4. The magnetic vortex core induced by radially polarized vortex beams with the 4π OLs focusing. (a)-(d) The total magnetization (M_t), radial (M_r), azimuthal (M_ϕ) and longitudinal (M_z) components in the r-z plane (4λ×4λ). (e)-(g) The 3D iso-magnetic surfaces of M_t located at 0.5 (red) and–0.5 (green) of the maximum magnetization.

Download Full Size | PDF

In order to further uncover the light-trigger MVC visually, we study the 3D polarization distribution and related 2D projection of M_t in the focal area in Figs. 5(a) and (b). It is evident that, in the vicinity of the focus, the total magnetization directions alter gradually clockwise, and the angle between adjacent magnetizations becomes increasingly larger when they are confined in-plane. More intriguingly, the magnetization within a small domain eventually turns out-of-plane and parallel to the propagation direction. Unarguably, such a continuous azimuthal curling magnetization together with a central perpendicular spot underlies the so-called MVC [1–4]. It is worth noting that this kind of exotic 3D magnetic polarization differs from recently reported magnetic twisting in essence. The former refers to magnetization polarization (or spin) curling, whereas the latter highlights the magnetization texture curling [44]. In real applications, the MVC could empower the information transfer and storage all in one, which is in turn mastered by the azimuthal curling and longitudinal core. It is thus of equal importance to confirm the 3D extension of turned-up magnetization core (M_z). As depicted in Fig. 5 (c), we can calculate the FWHM values of longitudinal magnetization spot in the transverse and axial directions to be 0.495λ and 0.349λ, respectively. This corresponds to a voxel size of λ³/22 (smaller than the diffraction limit of λ³/8), indicating the formation of 3D super-resolved MVC. Essentially, such an ultra-small curling magnetization configuration is attributed to not only the perfect destruction interference of the radial component formed by two counter-propagating RP-DGBs throughout the entire focal volume, but also the construction interferences of both azimuthal and longitudinal components along the optical axis. It should be noted that this situation distinguishes from that reported in [31], in which the azimuthal magnetization rather than the radial one is completely counteracted, and the local longitudinal component forms in the focal region.

Fig. 5. The magnetic vortex core induced by radially polarized vortex beams with the 4π OLs focusing. (a) and (b) The 3D magnetization polarization at z = 0 and the related 2D projection ((2λ×2λ)). (c) The line scans of core (M_z) along the r (black curve) and z (red curve) axes

Download Full Size | PDF

3.3 Light-induced scalable super-resolved magnetic vortex core

For the multiple-bit magnetic information transfer and storage, the use of single MVC frequently brings about poor processing efficiency and high error rate [17,54]. Accordingly, the unitary MVC requires to be multiplexed into extended and scalable counterparts. The current routine approach for this target is to iteratively engineer amplitude/phase elements at the back aperture of the objective lens. In this context, we here take advantage of a self-designed MRF to tailor the radially polarized vortex beams for generating needle-like and chain-like MVCs at one time. The prescribed MRF consists of six amplitude/phase modulated belts embracing four π-phase-shift alternant rings surrounded by two dark ones, whose transmission function p(θ) is given by,

(13)$$p(\theta )\textrm{ = }\left\{ \begin{array}{l} 1,\;\;\;\;\;\;\;\;\;\;\;\;\;\textrm{for}\;\;{\theta_1} \le \theta < {\theta_2},\;\;{\theta_3} \le \theta < {\theta_4},\\ - 1,\;\;\;\;\;\;\;\;\;\;\;\textrm{for}\;\;{\theta_2} \le \theta < {\theta_3},\;\;{\theta_4} \le \theta < {\theta_5},\\ 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\textrm{for}\;\;\;0 \le \theta < {\theta_1},\;\;\;{\theta_5} \le \theta < \alpha .\;\; \end{array} \right.$$

Based on the analysis above, we aim to optimize the five angles θ_i (i = 1,…,5) corresponding to five radial positions to enhance the longitudinal and azimuthal magnetizations and remove the radial component within an ultra-long axial coverage. The next key question is how to ascertain the best angles by a powerful optimization method. As a feasible choice, the particle swarm optimization is adopted to devise the MRF for achieving uniform super-resolved MVC needle and chain [55]. Of course, the angle optimization process is not unique. As a reasonable example, we suggest set of angles found from all the optimization parameters: θ₁= 50.41^°, θ₂= 50.65^°, θ₃= 50.90^°, θ₄= 51.13^°, and θ₅= 52.49^°.

As a contrast, we present the focal magnetization of RP-DGBs modulated by the SPP and MRF together with the single lens focusing in Fig. 6. One can see that the M_r disappears almost (Fig. 6(a)), which is clearly distinguishable from that in Fig. 3. This is thanks to the distinct polarization filter function of MRF [48]. More impressively, the M_ϕ, M_z and M_t (Fig. 6(b)–6(d)) feature an ultra-long depth of focus (DOF: 40λ) with very uniform extension (> 92%). To further mirror this uniformity, the 3D polarization distributions at z = 0, ±20λ are displayed in Fig. 6(e), from which we observe that they share the nearly same magnetic vortex core at the predefined three sites. Besides, it is noticeable that the non-trivial MVs with reverse curling direction appear at their peripheries. Such a non-diffracting character can also be unfolded by the magnetic isosurface in Fig. 6(f), where the internal extended MVC is enfolded by the relevant MVs. In addition, if the MRF is replaced by a narrow annular belt (NAB) with the angle ranging from θ=50.41^° to θ=52.49^° (the transmittance is one), the DOF of MVC is shortened vastly (Fig. 6(g)). Although the transverse FWHM value (0.447λ, black and red curves) of the core (M_z) is immune to this replacement, the longitudinal DOF value suffers from a severe degradation (Fig. 6(h)). To be exact, the DOF value decreases from 40λ (blue curve) to 8λ (magenta curve) given that the uniformity exceeds 92%. As a matter of fact, the DOF can be adjusted flexibly by changing the ring number and radius of the MRFs. Therefore, we are able to acquire scalable super-resolved MVC needle with high uniformity by engineering specific MRFs in the single lens focusing setup.

Fig. 6. The needle-like magnetic vortex core induced by modulated radially polarized vortex beam with the single lens focusing. (a)-(d) The radial (M_r), azimuthal (M_ϕ), longitudinal (M_z) and total magnetizations (M_t) in the r-z plane (4λ×40λ). (d) The 3D magnetization polarization distribution at z =0, ±20λ. (e) The 3D iso-magnetic surfaces of M_t located at 0.5 of the maximum magnetization. (g) The total magnetization (M_t) modulated by the NAB in the r-z plane (4λ×40λ). (h) The line scans of core (M_z) modulated by the MRF (black and blue curves) and NAB (red and magenta curves) along the r and z axes.

Download Full Size | PDF

The whole scenario changes drastically when two RP-DGBs modulated by the same SPPs and MRFs propagate face to face in the 4π focusing system, as exhibited in Fig. 7. The M_r cancels each other from two side (Fig. 7(a)), whereas the M_ϕ, M_z and M_t (Fig. 7(b)–7(d)) turn into numerous uniform bright-dark alternating periodic (0.8λ period) textures with ultra-long axial extension (40λ), which is caused by the destruction interferences of the M_r along with the construction interferences of the other two components apart from the extraordinary polarization filter role of the MRF. Likewise, each isolated magnetic configuration at z = 0, ±20λ is comprised of central MVC and peripheral reverse MV (Fig. 7(e)). Such a chain-like MVC can also be visualized in its magnetic isosurface (Fig. 7(f)). However, the number of MVC having super-high uniformity more than 92% reduces from 49 to 11 if the NAB (like Fig. 6(g)) is employed instead of the MRF (Fig. 7(g)). Now we are in position to investigate the 3D domain of the core (M_z) importing either the MRF or the NAB, as seen in Fig. (h). Their transverse FWHM values are collectively determined to be 0.440λ (black and red curves in the upper row). Although both the axial scalable lengths of highly uniform chain-like MVC has huge difference (blue and magenta curves in the lower row), the axial FWHM value of the central single core is as the same as 0.411λ. Furthermore, it needs to be mentioned that such 3D super-resolved (λ³/24) MVC chain can also be shifted controllably through exerting an additional phase difference between the left and right sides incident ports [32,56], thus enabling the conveying, exchange and storage of multiple-bit magnetic information in a high-efficiency fashion.

Fig. 7. The chain-like magnetic vortex core induced by modulated two radially polarized vortex beams in the 4π focusing system. (a)-(d) The radial (M_r), azimuthal (M_ϕ), longitudinal (M_z) and total magnetizations (M_t) in the r-z plane (4λ×40λ). (d) The 3D magnetization polarization distribution at z =0, ±20λ. (e) The 3D iso-magnetic surfaces of M_t located at 0.5 of the maximum magnetization. (g) The total magnetization (M_t) modulated by the NAB in the r-z plane (4λ×40λ). (h) The line scans of core (M_z) modulated by the MRF (black and blue curves) and NAB (red and magenta curves) along the r and z axes.

Download Full Size | PDF

4. Conclusions

We have theoretically demonstrated the generation of scalable super-resolved MVC by tightly focusing radially polarized beams with the synergistic modulation of SPP and MRF. It is found that both the additional encoded filters and adoptive focusing configurations play a pivotal role in identifying the spatial polarization orientation of focal magnetization. When the SPP is individually superimposed on the incident vectorial beams, we are able to yield single 3D super-resolved (λ³/22) MVC in the 4π focusing geometry, which is inaccessible with the single lens focusing, let alone the scenario (only MVs emerge) in the absence of SPP. Even more significantly, the light-induced single MVC could be further scalable with the cascaded use of SPP and MRF. On the one hand, we are capable to produce transverse super-resolved (0.447λ) and ultra-long (40λ) MVC needle with high uniformity (> 92%) in the single lens focusing system. On the other hand, the unitary MVC can be multiplexed into highly uniform 3D super-resolved (λ³/24) chain-like textures with the double face-to-face lenses focusing. Physically, the former is related to the fact that the MRF may serve as a continuous and extended polarization filter. The latter, however, hinges primarily upon the joint contribution of the special polarization filter functionality of MRF in combination with the coherent interferences of three orthogonal focal magnetization components. The findings presented here hold great value in the conveying, exchange and storage of multiple-bit magnetic information, the trapping and manipulation of multiple atoms, and spintronics.

Funding

National Natural Science Foundation of China (11604236, 11974258, 12004155, 61575139); Key Research and Development (R&D) Projects of Shanxi Province (201903D121127); Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2019L0151); the Natural Sciences Foundation in Shanxi Province (201901D111117); Science and Technology Foundation of Guizhou Province (ZK [2021]031).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono, “Magnetic Vortex Core Observation in Circular Dots of Permalloy,” Science 289(5481), 930–932 (2000). [CrossRef]

2. A. Wachowiak, J. Wiebe, M. Bode, O. Pietzsch, M. Morgenstern, and R. Wiesendanger, “Direct Observation of Internal Spin Structure of Magnetic Vortex Cores,” Science 298(5593), 577–580 (2002). [CrossRef]

3. J. Miltat and A. Thiaville, “Vortex Cores—Smaller Than Small,” Science 298(5593), 555 (2002). [CrossRef]

4. K. Yamada, S. Kasai, Y. Nakatani, K. Kobayashi, H. Kohno, A. Thiaville, and T. Ono, “Electrical switching of the vortex core in a magnetic disk,” Nat. Mater. 6(4), 270–273 (2007). [CrossRef]

5. R. Hertel and C. M. Schneider, “Exchange Explosions: Magnetization Dynamics during Vortex-Antivortex Annihilation,” Phys. Rev. Lett. 97(17), 177202 (2006). [CrossRef]

6. Y. Liu, S. Gliga, R. Hertel, and C. M. Schneider, “Current-induced magnetic vortex core switching in a Permalloy nanodisk,” Appl. Phys. Lett. 91(11), 112501 (2007). [CrossRef]

7. D. H. Kim, E. A. Rozhkova, I. V. Ulasov, S. D. Bader, T. Rajh, M. S. Lesniak, and V. Novosad, “Biofunctionalized magnetic-vortex microdiscs for targeted cancer-cell destruction,” Nat. Mater. 9(2), 165–171 (2010). [CrossRef]

8. M. Kammerer, M. Weigand, M. Curcic, M. Noske, M. Sproll, A. Vansteenkiste, B. V. Waeyenberge, H. Stolll, G. Woltersdorf, C. H. Back, and G. Schuetz, “Magnetic vortex core reversal by excitation of spin waves,” Nat. Commun. 2(1), 279 (2011). [CrossRef]

9. T. Tanigaki, Y. Takahashi, T. Shimakura, T. Akashi, R. Tsuneta, A. Sugawara, and D. Shindo, “Three-Dimensional Observation of Magnetic Vortex Cores in Stacked Ferromagnetic Discs,” Nano Lett. 15(2), 1309–1314 (2015). [CrossRef]

10. M. Möller, J. H. Gaida, S. Schäfer, and C. Ropers, “Few-nm tracking of current-driven magnetic vortex orbits using ultrafast Lorentz microscopy,” Commun. Phys. 3(1), 36 (2020). [CrossRef]

11. C. Holl, M. Knol, M. Pratzer, J. Chico, I. L. Fernandes, S. Lounis, and M. Morgenstern, “Probing the pinning strength of magnetic vortex cores with sub-nanometer resolution,” Nat. Commun. 11(1), 2833 (2020). [CrossRef]

12. R. H. Tong, Z. Dong, Y. H. Chen, F. Wang, Y. J. Cai, and T. Setälä, “Fast calculation of tightly focused random electromagnetic beams: controlling the focal field by spatial coherence,” Opt. Express 28(7), 9713–9727 (2020). [CrossRef]

13. Q. Wang, C. H. Tu, Y. N. Li, and H. T. Wang, “Polarization singularities: Progress, fundamental physics, and prospects,” APL Photonics 6(4), 040901 (2021). [CrossRef]

14. Y. X. Zhang, X. F. Liu, H. Lin, D. Wang, E. S. Cao, S. D. Liu, Z. Q. Nie, and B. H. Jia, “Ultrafast multi-target control of tightly focused light fields,” Opto-Electron Adv0(0), 210026 (20XX). [CrossRef]

15. X. Liu, Y. Peng, S. J. Tu, J. Guan, C. F. Kuang, X. Liu, and X. Hao, “Generation of Arbitrary Longitudinal Polarization Vortices by Pupil Function Manipulation,” Adv. Photonics Res. 2(1), 2000087 (2021). [CrossRef]

16. C. Stanciu, F. Hansteen, A. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, “All-optical magnetic recording with circularly polarized light,” Phys. Rev. Lett. 99(4), 047601 (2007). [CrossRef]

17. M. Gu, X. Li, and Y. Cao, “Optical storage arrays: a perspective for future big data storage,” Light: Sci. Appl. 3(5), e177 (2014). [CrossRef]

18. A. Stupakiewicz, K. Szerenos, D. Afanasiev, A. Kirilyuk, and A. V. Kimel, “Ultrafast nonthermal photo-magnetic recording in a transparent medium,” Nature 542(7639), 71–74 (2017). [CrossRef]

19. A. Stupakiewicz, K. Szerenos, M. D. Davydova, K. A. Zvezdin, A. K. Zvezdin, A. Kirilyuk, and A. V. Kimel, “Selection rules for all-optical magnetic recording in iron garnet,” Nat. Commun. 10(1), 612 (2019). [CrossRef]

20. D. O. Ignatyeva, C. S. Davies, D. A. Sylgacheva, A. Tsukamoto, H. Yoshikawa, P. O. Kapralov, A. Kirilyuk, V. I. Belotelov, and A. V. Kimel, “Plasmonic layer-selective all-optical switching of magnetization with nanometer resolution,” Nat. Commun. 10(1), 4786 (2019). [CrossRef]

21. S. Wang, C. Wei, Y. Feng, H. Cao, W. Li, Y. Cao, B. Guan, A. Tsukamoto, A. Kirilyuk, A. V. Kimel, and X. Li, “Dual-shot dynamics and ultimate frequency of all-optical magnetic recording on GdFeCo,” Light: Sci. Appl. 10(1), 8 (2021). [CrossRef]

22. J. P. van der Ziel, P. S. Pershan, and L. D. Malmstrom, “Optically-induced magnetization resulting from the inverse faraday effect,” Phys. Rev. Lett. 15(5), 190–193 (1965). [CrossRef]

23. P. S. Pershan, J. P. Van der Ziel, and L. D. Malmstrom, “Theoretical discussion of the inverse Faraday effect, Raman scattering, and related phenomena,” Phys. Rev. 143(2), 574–583 (1966). [CrossRef]

24. L. E. Helseth, “Strongly focused electromagnetic waves in E × E* media,” Opt. Commun. 281(23), 5671–5673 (2008). [CrossRef]

25. S. Wang, X. Li, J. Zhou, and M. Gu, “All-optically configuring the inverse Faraday effect for nanoscale perpendicular magnetic recording,” Opt. Express 23(10), 13530–3536 (2015). [CrossRef]

26. Z. Nie, H. Lin, X. Liu, A. Zhai, Y. Tian, W. Wang, D. Li, W. Ding, X. Zhang, Y. Song, and B. Jia, “Three-dimensional super-resolution longitudinal magnetization spot arrays,” Light: Sci. Appl. 6(8), e17032 (2017). [CrossRef]

27. C. Hao, Z. Nie, H. Ye, H. Li, Y. Luo, R. Feng, X. Yu, F. Wen, Y. Zhang, C. Yu, J. Teng, B. Luk’yanchuk, and C. Qiu, “Three-dimensional supercritical resolved light-induced magnetic holography,” Sci. Adv. 3(10), e1701398 (2017). [CrossRef]

28. S. Lin, Z. Nie, W. Yan, Y. Liang, H. Lin, Q. Zhao, and B. Jia, “All-optical vectorial control of multistate magnetization through anisotropy-mediated spin-orbit coupling,” Nanophotonics 8(12), 2177–2188 (2019). [CrossRef]

29. Y. Jiang, X. Li, and M. Gu, “Generation of sub-diffraction-limited pure longitudinal magnetization by the inverse Faraday effect by tightly focusing an azimuthally polarized vortex beam,” Opt. Lett. 38(16), 2957 (2013). [CrossRef]

30. S. Wang, X. Li, J. Zhou, and M. Gu, “Ultralong pure longitudinal magnetization needle induced by annular vortex binary optics,” Opt. Lett. 39(17), 5022–5025 (2014). [CrossRef]

31. Z. Nie, W. Ding, D. Li, X. Zhang, Y. Wang, and Y. Song, “Spherical and sub-wavelength longitudinal magnetization generated by 4π tightly focusing radially polarized vortex beams,” Opt. Express 23(2), 690–701 (2015). [CrossRef]

32. Z. Nie, W. Ding, G. Shi, D. Li, X. Zhang, Y. Wang, and Y. Song, “Achievement and steering of light-induced sub-wavelength longitudinal magnetization chain,” Opt. Express 23(16), 21296–21305 (2015). [CrossRef]

33. L. Gong, L. Wang, Z. Zhu, X. Wang, H. Zhao, and B. Gu, “Generation and manipulation of super-resolution spherical magnetization chains,” Appl. Opt. 55(21), 5783–5789 (2016). [CrossRef]

34. W. Yan, Z. Nie, X. Zhang, Y. Wang, and Y. Song, “Theoretical guideline for generation of an ultralong magnetization needle and a super-long conveyed spherical magnetization chain,” Opt. Express 25(19), 22268–22279 (2017). [CrossRef]

35. S. Wang, Y. Cao, and X. Li, “Generation of uniformly oriented in-plane magnetization with near-unity purity in 4π microscopy,” Opt. Lett. 42(23), 5050–5053 (2017). [CrossRef]

36. W. Yan, Z. Nie, X. Liu, X. Zhang, Y. Wang, and Y. Song, “Creation of isotropic super-resolved magnetization with steerable orientation,” APL Photonics 3(11), 116101 (2018). [CrossRef]

37. W. Yan, Z. Nie, X. Liu, X. Zhang, Y. Wang, and Y. Song, “Arbitrarily spin-orientated and super-resolved focal spot,” Opt. Lett. 43(16), 3826–3829 (2018). [CrossRef]

38. S. Wang, J. Luo, Z. Zhu, Y. Cao, H. Wang, C. Xie, and X. Li, “All-optical generation of magnetization with arbitrary three-dimensional orientations,” Opt. Lett. 43(22), 5551–5554 (2018). [CrossRef]

39. W. Yan, Z. Nie, X. Liu, G. Lan, and Y. Song, “Dynamic control of transverse magnetization spot arrays,” Opt. Express 26(13), 16824–16835 (2018). [CrossRef]

40. M. Udhayakumar, K. Prabakaran, K. B. Rajesh, Z. Jaroszewicz, A. Belafhal, and D. Velauthapillai, “Generation of ultra-long pure magnetization needle and multiple spots by phase modulated doughnut Gaussian beam,” Opt. Laser Technol. 102, 40–46 (2018). [CrossRef]

41. X. Zhang, G. Rui, Y. Xu, F. Zhang, Y. Du, X. Lin, A. Wang, and W. Zhao, “Fully controllable three-dimensional light-induced longitudinal magnetization using a single objective lens,” Opt. Lett. 45(8), 2395–2398 (2020). [CrossRef]

42. W. Yan, S. Lin, H. Lin, Y. Shen, Z. Nie, B. Jia, and X. Deng, “Dynamic control of magnetization spot arrays with three-dimensional orientations,” Opt. Express 29(2), 961–973 (2021). [CrossRef]

43. X. Liu, W. Yan, Z. Nie, Y. Liang, Y. Wang, Z. Jiang, Y. Song, and X. Zhang, “Longitudinal magnetization superoscillation enabled by high-order azimuthally polarized Laguerre-Gaussian vortex modes,” Opt. Express 29(16), 26137–26149 (2021). [CrossRef]

44. X. Liu, W. Yan, Y. Liang, Z. Nie, Y. Wang, Z. Jiang, Y. Song, and X. Zhang, “Twisting Polarization-Tunable Subdiffraction-Limited Magnetization through Vectorial Beam Coupling,” Adv. Photonics Res. 3(1), 2100117 (2022). [CrossRef]

45. L. E. Helseth, “Light-induced magnetic vortices,” Opt. Lett. 36(6), 987–989 (2011). [CrossRef]

46. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358–379 (1959).

47. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef]

48. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008). [CrossRef]

49. Z. Nie, G. Shi, X. Zhang, Y. Wang, and Y. Song, “Generation of super-resolution longitudinally polarized beam with ultra-long depth of focus using radially polarized hollow Gaussian beam,” Opt. Commun. 331, 87–93 (2014). [CrossRef]

50. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009). [CrossRef]

51. H. Li, C. Ma, J. Wang, M. Tang, and X. Li, “Spin-orbit Hall effect in the tight focusing of a radially polarized vortex beam,” Opt. Express 29(24), 39419–39427 (2021). [CrossRef]

52. G. Chen, F. Song, and H. Wang, “Sharper focal spot generated by 4π focusing of higher-order Laguerre –Gaussian radially polarized beam,” Opt. Lett. 38(19), 3937–3940 (2013). [CrossRef]

53. N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett. 29(17), 1968–1970 (2004). [CrossRef]

54. L. D. Geng and Y. M. Jin, “Magnetic vortex racetrack memory,” J. Magn. Magn. Mater. 423, 84–89 (2017). [CrossRef]

55. C. Shi, Z. Nie, Y. Tian, C. Liu, Y. Zhao, and B. Jia, “Super-resolution longitudinally polarized light needle achieved by tightly focusing radially polarized beams,” Optoelectron. Lett. 14(1), 1–5 (2018). [CrossRef]

56. Z. Zhou, X. Liu, Z. Nie, S. Yang, Y. Gao, W. Wu, G. Lan, Z. Chai, and D. Kong, “Generation and manipulation of three-dimensional polarized optical chain and hollow dark channels,” Opt. Laser Technol. 144, 107408 (2021). [CrossRef]

All-optical demonstration of a scalable super-resolved magnetic vortex core

Abstract

1. Introduction

2. Theoretical analysis of light-induced magnetic vortex core

3. Results and discussions

3.1 Light-induced magnetic vortices

3.2 Light-induced single super-resolved magnetic vortex core

3.3 Light-induced scalable super-resolved magnetic vortex core

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (13)

Optics Express