Longitudinal magnetization superoscillation enabled by high-order azimuthally polarized Laguerre-Gaussian vortex modes

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

doi:10.1364/OE.434190

1. Introduction

The ever-increasing demands of integrated miniaturized opto-magnetic devices require light-induced magnetization to be concentrated into the smallest possible domain [1]. However, the diffractive nature of light generally leads to the lateral spatial resolution of derived magnetization subjected to the inherent limit, which is on the order of half a wavelength (∼λ/2). In this connection, the inverse Faraday effect (IFE) [2,3] conceived in 1960s has recently resurged as a promising alternative for all-optically controlling multifunctional magnetizations beyond the diffraction limit in the far field [4,5]. It is widely recognized that the tightly focused circularly polarized beam acting upon the isotropic magneto-optical materials could serve as an effective stimulus to trigger a sub-wavelength longitudinal magnetization [6–10]. And yet a considerable proportion of doughnut-shaped transverse magnetization is accompanied in the meantime that will degrade the spatial resolution of the resultant magnetization. To mitigate this issue, numerous strategies, including subtle vectorial polarization engineering and optimized phase/amplitude encoding, have been attempted to not only achieve the versatile magnetization with controllable polarization orientations and spatial textures, but also compress it into a super-resolution domain [11–25]. Despite these prominent features, the spatial dimension of the light-induced magnetization by resorting to the ingenious fashions aforementioned is all inferior to the ultimate limit of 0.36λ/NA (NA is the numerical aperture of the objective lens), which is in particular not conducive to ultrahigh-density all-optical magnetic recording/storage [26–29].

Optical superoscillation is an exotic phenomenon that represents the local oscillation frequency of bandlimited function being higher than its fastest Fourier component (smaller than 0.36λ/NA), thus offering a feasible enabling technology for super-resolution focusing and imaging without evanescent waves [30–33]. Until now, various rich optical super-oscillation behaviors have been demonstrated by means of various well-designed approaches, such as diffraction grating [34], optical eigenmodes [35,36], super-oscillatory lens [37–39], and multi-ring phase/amplitude elements [40–42]. Actually, in addition to the optical superoscillation, such a new conceptual paradigm has been generalized to the electronic [43], acoustic [44], and even magnetic realms [45]. As such, it is able to acquire magnetization superoscillation driven by external fields for extensive magnetism-related applications. In this regard, Qiu et al. first came up with a powerful pathway to create a novel three-dimensional (3D) supercritical light-induced magnetic holography based on the super-oscillation design in combination with multibeam interference in the 4π system [46]. An extremely sharp magnetization hotspot with lateral and axial sizes in turn reached to 0.357λ and 0.255λ can be attainable. Under such a circumstance, they succeed in realizing a 3D deep super-resolved (λ³/59) purely longitudinal magnetization. The availability of this scheme, however, suffers from twofold possible problems. For one thing, in spite of the moderate side lobe (SL< 30%), the energy conversion efficiency (ECE) of principle super-oscillation magnetization spot is fairly low (< 1%), which will greatly weaken both the signal to noise ratio and image contrast of magnetization resonance imaging [47]. For another, the engineering and alignment of multibeam superposition in the 4π setup might be a challenged issue, thereby failing to implement the mass-production and high-tolerance magnetization recording/storage [26,28].

Motivated by these problems, we here propose a simple and efficient methodology to accomplish a longitudinal magnetization superoscillation (< 0.36λ/NA) in the focal region through tightly focusing the azimuthally polarized Laguerre–Gaussian (AP-LG) modes superimposed with the first-order vortex phase. It is further found that the transverse full width at half maximum (FWHM), ECE, and SL strength of the super-oscillation longitudinal magnetization can be mediated by picking both the radial mode index and truncation parameter, In principle, the underlying mechanism behind these fantastic magnetization phenomena resides in the partial construction and destruction interferences caused by the π phase shift between the different rings of the high-order incident beams. This paper is organized as follows: we derive in section 2 both the transversally polarized optical field and the relevant longitudinal magnetization according to the vectorial diffraction formula and the IEF. In section 3, we study in detail the focused magnetization behaviors of 1-order, 3-order and 5-order AP-LG vortex beams for the case of different truncation parameters. Additionally, the prime conditions to realize high-quality longitudinal magnetization superoscillation are discussed and the corresponding physical mechanism is elucidated. Finally, the conclusions are given in section 4.

2. Theoretical analysis of the longitudinal magnetization superoscillation

Figure 1(a) conceptually shows the schematic configuration to yield a light-induced super-oscillation longitudinal magnetization (see the top left inset) in the focal regime. An incident linearly polarized femtosecond laser (FSL) first transmits through a beam expander (L₁ and L₂) and is then modulated by a spatial light modulator (SLM), where the desired amplitude and phase patterns can be encoded together to create the LG vortex modes with different radial mode indexes [48]. The linearly polarized LG beams can be converted into the AP ones with the aid of the vortex waveplate (VW), which are further focused by a high NA objective lens (OL) onto an isotropic magneto-optical material (MOM). It should be noted that the additional vortex phase could produce the constructive interference between the transverse optical field components, thus giving rise to the transversely polarized super-oscillation optical field and associated longitudinal magnetization superoscillation. Essentially, the key to inducing magnetization superoscillation rests in the realization of associated optical pattern limited by its fastest Fourier component (∼0.36λ/NA). The magnetization superosillation can be described as the magnetization induced by the optical field whose main lobe region oscillates faster than the fastest Fourier component of focused light field. Since the fastest Fourier component is related intimately to a focused annular shaped beam stemming from the outer edge of the OL, it seems to be of great possibility to be able to overwhelm such a limitation (< 0.36λ/NA, the criterion of magnetization superoscillation) by strongly focusing the high-order LG vortex modes, which is caused by the controllable coherent interferences between their different rings [36]. In addition, it should be noted that the higher the beam order is, the better the focusing performance is [49]. In view of these facts, we here adopt the high-order AP-LG vortex modes as an ideal exciter to induce the super-oscillation longitudinal magnetization. Mathematically, the AP-LG modes with different radial mode indexes in the pupil plane are given by [50],

(1)$${{\textbf E}_0}({\theta ,\;\varphi } )= \left[ \begin{array}{l} {E_x}\\ {E_y} \end{array} \right]\textrm{ = }\frac{{\sin \theta {\beta ^2}}}{{{{\sin }^2}\alpha }}\textrm{exp} \left[ { - {{\left( {\beta \frac{{\sin \theta }}{{\sin \alpha }}} \right)}^2}} \right]L_p^1\left[ {2{{\left( {\beta \frac{{\sin \theta }}{{\sin \alpha }}} \right)}^2}} \right]\left[ \begin{array}{l} - \sin \varphi \\ \;\;\cos \varphi \end{array} \right], $$

where α is the maximum value of the convergent angle θ, and φ = arctan (y/x) is the polar angle in the initial plane. β is the truncation parameter defining as the ratio of the pupil radius to the incident beam waist in front of the focusing objective lens. For a first-order AP-LG beam incidence, β should exceed 1 because the outer ring of the first-order AP-LG beam will be completely blocked by the optical pupil if β ≤ 1. However, the intensity distribution of the higher-order AP-LG beams is partially truncated by the optical pupil according to β. $L_p^1$ is the generalized Laguerre polynomial, which can be written as,

(2)$$L_p^1(t )= \sum\limits_{k = 0}^p {\frac{{({p + 1} )!{{({ - t} )}^k}}}{{({k + 1} )!k!({p - k} )!}}} {\kern 1pt} {\kern 1pt} {\kern 1pt} . $$

Fig. 1. (a) The schematic to yield the light-induced super-oscillation longitudinal magnetization in the focal plane; the illustration at the top left represents a sketch map on super-oscillation magnetization. The dimension in the illustration is 2λ×2λ. (b)-(d) the intensity patterns (color bar) and polarization states (red and blue arrows) of the AP-LG₁₁, AP-LG₃₁ and AP-LG₅₁ beams in the pupil plane.

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In Eq. (2), p is the radial mode index of the AP-LG beams, which can be characterized as p+1 concentric ring on the initial cross section. Obviously, the incoming light field reduces to a single-ring AP-LG mode when p = 0. Otherwise, if we take p ≥ 1 into account, multi-ring AP-LG modes appear. As an illustrated example, we here propose three high-order AP-LG beams with p = 1, 3, and 5 (which are referred as AP-LG₁₁, AP-LG₃₁ and AP-LG₅₁ for convenience) to fulfill the longitudinal magnetization superoscillation. The intensity and polarization patterns of AP-LG₁₁, AP-LG₃₁ and AP-LG₅₁ beams in the pupil plane are demonstrated in Figs. 1(b)-(d). As expected, the AP-LG₁₁, AP-LG₃₁ and AP-LG₅₁ beams feature 2, 4 and 6 bright ring-shaped intensity distributions with π phase shift (red and blue arrows indicate the reverse polarization states) between the adjacent rings. It is also interesting to note that a complete dark ring is sandwiched between the adjacent bright rings. Accordingly, the high-order AP-LG beams are able to be equivalent to a LG₀₁ beam transmitting through the self-contained multi-ring amplitude/phase hybrid filters [51]. In addition, the positions of both the bright and dark rings can be flexibly adjusted by changing the β. Therefore, it is expected to give rise to some peculiar magnetization behaviors (i. e. magnetization superoscillation) thanks to the constructive and destructive interferences between different rings when high-order AP-LG vortex beams are tightly focused.

Prior to deriving the light-induced magnetization, it is crucial to calculate the corresponding electric field components. According to the vectorial diffraction theory [52–54], the focal field of the high-order AP-LG vortex beams can be deduced as,

(3)$$\begin{aligned} {\textbf E} &= \left[ \begin{array}{l} {E_x}\\ {E_y}\\ {E_z} \end{array} \right] ={-} iA\int_0^\alpha {\int_0^{2\pi } {\sqrt {\cos \theta } \frac{{{{\sin }^2}\theta {\beta ^2}}}{{{{\sin }^2}\alpha }}\textrm{exp} \left[ { - {{\left( {\beta \frac{{\sin \theta }}{{\sin \alpha }}} \right)}^2}} \right]} } \\ &\quad\times L_p^1\left[ {2{{\left( {\beta \frac{{\sin \theta }}{{\sin \alpha }}} \right)}^2}} \right]\left[ \begin{array}{l} - \sin \theta \\ \;\;\cos \theta \\ \;\;\;\;0 \end{array} \right]{e^{i[{\varphi + k({r\sin \theta \cos ({\varphi - \phi } )+ z\cos \theta } )} ]}}d\varphi d\theta \end{aligned}, $$

where A is the amplitude constant, k = 2π/λ is the wavenumber in free space, λ is the wavelength of the incident beam, and (r, ϕ, z) are the cylindrical coordinates in the image space. After simple coordinate transformation and a series of calculations, the focal electric field components in Eq. (3) can be rewritten as,

(4)$${\textbf E} = \left[ \begin{array}{l} {E_r}\\ {E_\phi }\\ {E_z} \end{array} \right] = \left[ \begin{array}{l} - A{e^{i\phi }}({{I_2} + {I_0}} )\\ A{e^{i\phi }}({{I_2} - {I_0}} )\\ \;\;\;\;\;\;\;0 \end{array} \right]$$

with

(5)$${I_m} = \int_0^\alpha {\sqrt {\cos \theta } \frac{{{{\sin }^2}\theta {\beta ^2}}}{{{{\sin }^2}\alpha }}\textrm{exp} \left[ { - {{\left( {\beta \frac{{\sin \theta }}{{\sin \alpha }}} \right)}^2}} \right]} L_p^1\left[ {2{{\left( {\beta \frac{{\sin \theta }}{{\sin \alpha }}} \right)}^2}} \right]{J_m}({kr\sin \theta } ){e^{ikz\cos \theta }}d\theta, $$

where J_m (m = 0 and 2) is the mth-order Bessel function of the first kind. Furthermore, when the focused optical field in Eq. (4) impinges on the given MOM, the light-induced magnetization via the IFE reads [2,3],

(6)$${\bf M}({r,\varphi ,z} )= i\gamma {\bf E} \times {{\bf E}^ \ast }, $$

in which γ denotes a magneto-optical constant, E is the electric field and E*is its complex conjugate. Substituting Eq. (4) into Eq. (6), the resultant magnetization field is reduced to,

(7)$${\bf M} = 2{A^2}\gamma {\textrm{Re}} ({I_0^2 - I_2^2} ){{\bf e}_{\bf z}}, $$

where e_z is a unit vector along the z axis. Clearly, the longitudinal magnetization in Eq. (7) is fully triggered by the transversely polarized optical fields in Eq. (4). What’s more, it is capable of creating the all-optical magnetization superoscillation in the focal region by cautiously adjusting both the radial mode index p and truncated parameter β of AP-LG_p₁ vortex beams under tight focusing.

3. Results and discussions

3.1 Dependence of light-induced magnetization in the focus on both the p and β

To reveal the essential conditions for realizing the light-induced longitudinal magnetization superoscillation, we depict the dependence of focal magnetization (M(0, 0)) on both the p and β of three high-order AP-LG beams, as plotted in Fig. 2. The calculation parameters are chosen as NA = 0.95, A = 1, p = 1, 3, 5, and 1 ≤ β ≤ 4.5. It is observed from Fig. 2 that a strongest peak magnetization (> 0.36λ/NA) appears at β = 2.732 (see downward black arrow) for the double-ring AP-LG vortex beam (p = 1, black curve). This means that the constructive interference takes place between the adjacent rings because of the outer ring moving to the optical pupil with respect to the larger β [55]. On the contrary, it gives rise to a null magnetization when the β drops to 1.424 (see upward black arrow), which is ascribed to the occurrence of destructive interference between the adjacent rings coming from the π phase difference [53,56]. Besides, it is paramount to pay attention to the intermediate scenario of 1.424 < β < 2.732, where the incomplete construction and destruction interferences between the adjacent rings may occur synergistically [57]. That is, it is quite likely to cause the fastest Fourier magnetization component (∼0.36λ/NA), and even the magnetization superoscillation (< 0.36λ/NA) by carefully adjusting the β in this range. It should point out, however, that such shrinkage of the magnetization domain are at the expense of ECE and SL. When the AP-LG₃₁ (AP-LG₅₁) vortex modes are adopted, it is apparent that the truncation parameter to produce both the peak and null magnetizations shifts to larger values of β = 3.545 (4.252) and β = 2.308 (2.980) compared with the AP-LG₁₁ vortex mode. This indicates that more light beams begin to transfer toward each ring near the edge of the pupil plane, thus leading to sharper focal magnetization spots with admissible ECE and SL [36]. We here define the ECE as the ratio of energy of light-induced magnetization field in the focal plane to the total energy of the incident beam, as well as the SL can be represented as the nearest positive ring to the central magnetization spot, In fact, more complicated coherent interferences coming from more beam belts will also contribute to the magnetization superoscillation in the focal regime by selecting proper β values (e. g. 2.308 < β < 3.545 (2.980 < β < 4.252) for the AP-LG₃₁ (AP-LG₅₁) vortex beam. Incidentally, there is more than a pair of β values that meet the demands to achieve both the peak and null magnetizations for the AP-LG₃₁ and AP-LG₅₁ vortex beams. However, their associated peak values don’t reach to the maxima, which may bring about poorer energy conversion in magnetism-based applications. We therefore put away these situations in the following studies.

3.2 Longitudinal magnetization superoscillation induced by high-order AP-LG modes

To confirm the super-oscillation ability of magnetization spots created by the structured vortex modes, the versatile longitudinal magnetization patterns of AP-LG₁₁, AP-LG₃₁ and AP-LG₅₁ vortex beams with different preset β values together with their profiles along transverse axis are demonstrated in Figs. 3 and 4, respectively. For comparison, we first take into account the double-ring AP-LG vortex beams (p = 1) acting upon the isotropic MOMs. It is found that, as the β decreases from 2.732 to 1.576, both the central solid spot and related strength of the normalized magnetizations diminish sequentially (from 100% to 12.6%) in the focal region, whereas the SL rise sharply (from 22.3% to 246.8%) relative to the main lobes (Figs. 3(a1)-(a4)). Otherwise, we can observe from Figs. 3(b1)-(b4) that the focal magnetization features a needlelike texture in the r-z plane for the β = 2.732 [12,58,59]. However, the light-induced magnetization needles are subjected to violation and shorten at the same time as the diminution of the β. In particular, the longitudinal magnetization forms two symmetric bright domains on the optical axis sandwiching a very dim region when the β = 1.576. It should be noticed that we here only focus on the magnetization behaviors in the focal plane (z = 0) instead of the r-z plane, the latter is therefore not considered hereafter.

Fig. 2. The magnetization of high-order AP-LG vortex modes with various radial mode indexes at the focus as a function of the truncation parameter β

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Fig. 3. Multifunctional longitudinal magnetization patterns induced by the high-order AP-LG vortex modes with the radial modes index p and the truncation parameter β. (a1)-(a4) and (b1)-(b4) are β = 2.732, 2.051, and 1.753, and 1.576 when p = 1in the x-y plane and r-z plane, respectively; (c1)-(c4) are β = 3.545, 2.854, 2.506, and 2.411 when p = 3 in the x-y plane; (c1)-(c4) are β = 4.252, 3.489, 3.234, and 3.078 when p = 5 in the x-y plane.

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Fig. 4. The normalized longitudinal magnetization profiles along the r axis induced by the high-order AP-LG vortex modes with the radial mode index p and the truncation parameter β. (a) p = 1 and β = 2.732, 2.051, 1.753, and 1.576; (b) p = 3 and β = 3.545, 2.854, 2.506, and 2.411; (c) p = 5 and β = 4.252, 3.489, 3.234, and 3.078.

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Furthermore, we plot the focal magnetization profiles of the AP-LG₁₁ vortex beam along the r axis in Fig. 4(a), from which the lateral FWHM values of the central spot are calculated to be 0.478λ, 0380λ, 0.335λ and 0.271λ for the β = 2.732, 2.051, 1.753, and1.576 respectively. We divide the whole magnetization variations into three stages for clarity. First, the peak sub-wavelength magnetization (>0.36λ/NA) can be achieved for the β = 2.732, which is then translated to the magnetization produced by the fastest Fourier light field component (∼0.36λ/NA) when the β drops to 2.051. The FWHM values of the magnetization fields proceed to reduce below 0.36λ/NA with the further decrease of the β, thus resulting in the formation of the longitudinal magnetization superoscillation. Apart from the super-oscillation magnetization hotspot, another two significant problems, commonly accompanied in the super-oscillation focusing, are noticeable SL near the central focus and poor ECE of the principal lobe. To be specific, the corresponding SL and ECE values are in turn 22.3%, 36.8%, 73.6% and 246.8% as well as 22.9%, 10.4%, 3.8% and 0.9% for the β = 2.732, 2.051, 1.753, and 1.576, respectively. Overall, for the AP-LG₁₁ vortex beam illumination, the smaller the β is, the higher the magnetization resolution is, whereas the larger the SL is and the lower the ECE is. Although the sharper FWHM can be produced in some cases, the ultrahigh SL and bad ECE will cause huge crosstalk [28,47]. It is thus of vital importance to compromise these three ingredients to meet the practical magnetism-based applications.

When it comes to the higher-order vortex modes, such as the AP-LG₃₁ and AP-LG₅₁ vortex beams, a couple of fascinating phenomena of the light-induced magnetizations near the focus are manifested in Fig. 3(c) and (d). Firstly, it is obvious that multi-ring magnetization patterns encircling central hotspots reserve all the time for the given β (2.308 < β < 3.545 (2.980 < β < 4.252) for the AP-LG₃₁ (AP-LG₅₁) vortex beam). Moreover, similar to the AP-LG₁₁ vortex mode, both the strength and FWHM of the central hotspots decrease as the decline of the β. In contrast, the relevant SL could moderate to some extent as a result of the brighter magnetization loops staying away from the focus. Figures 4 (b) and (c) delineate the magnetization profiles in the focal plane for the AP-LG₃₁ and AP-LG₅₁ vortex beams, respectively. Their transverse FWHM values of central spots are determined to be 0.451λ, 0380λ, 0.328λ, 0.287λ for the β = 3.545, 2.854, 2.506, and 2.411 as well as 0.442λ, 0380λ, 0.351λ, and 0.307λ for the β = 4.252, 3.489, 3.234, and 3.078, respectively. Likewise, the focal textures transform from the peak magnetization, via the magnetization produced by the fastest Fourier light field component, toward the super-oscillation magnetization for the adequate β regardless of the beam order. It needs to mention that the higher-order AP-LG vortex modes can create a smaller peak magnetization spot than can the double-ring AP-LG vortex one. Besides, we calculate both the SL and ECE are, respectively, 10.8%, 13.9%, 31.8%, and 77.6% (8.6%, 10.2%, 15.1%, and 36.5%) as well as 20.2%, 10.3%, 2.5%, and 0.7% (19.5%, 10.5%, 4.3%, and 0.7%) for the AP-LG₃₁(AP-LG₅₁) vortex mode. Compared to the AP-LG₁₁ mode, the magnetization fields induced by tightly focused AP-LG₃₁ and AP-LG₅₁ vortex beams present analogous super-oscillation behaviors and ECE but much weaker SL. In this respect, the higher-order AP-LG vortex modes seem to be preferable to the wide-ranging realistic applications. For clarity, the dependence of FWHM, ECE, and SL on the truncation parameter β for AP-LG₁₁, AP-LG₃₁ and AP-LG₅₁ vortex modes is shown in Table 1.

Table 1. The dependence of FWHM, ECE, and SL on the truncation parameter β for AP-LG₁₁, AP-LG₃₁ and AP-LG₅₁ vortex modes

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It is nontrivial to know whether one can garner the robust longitudinal magnetization superoscillation (e. g. FWHM < 0.36λ/NA, ECE > 5%, and SL < 30%) in ultra-high density MO recording/storage applications [26–29]. To this end, the effects of both the p (1, 3, 5) and β (< 4.5) on the lateral FWHM (black circle curves), ECE (red triangle curves) and SL (blue square curves) of tightly focused AP-LG vortex beams are given in Figs. 5(a)-(c). Note that here we choose the initial β to be slightly larger than that of the null magnetization (e.g. β > 1.424 (2.308, 2.980) for the AP-LG₁₁ (AP-LG₃₁, AP-LG₅₁) vortex mode), as the three indicators aforementioned make no sense at all at such specific locations. At first sight, each indicator appears to take on similar profiles irrelevant to both the p and β (Figs. 5(a)-(c)). From closer inspection of these profiles, we find that, for the AP-LG₁₁ vortex mode (Fig. 5(a)), the FWHM can reach to the super-oscillation criteria (horizontal violet dashed line) only if the β is no more than a critical value β_s = 2.051(vertical violet dashed line). However, it is nearly impossible to achieve ECE > 5% and SL < 30% simultaneously in these cases. Specifically, although the FWHM and ECE conform to the requirements of robust magnetization superoscillation in the range of 1.8< β < 2.051, the high SL of 74% emerges. When the β is further reduced to 1.5, we are able to vastly squeeze the FWHM to 0.209λ, but along with an extra-high SL (894%) and extremely low ECE (0.1%). Therefore, the AP-LG₁₁ vortex mode is an unsuitable candidate to realize high-quality super-oscillation magnetization.

Fig. 5. The effects of the β on the FWHM (black dotted line), ECE (red triangular line) and SL (blue square line) of the longitudinal magnetizations induced by the AP-LG vortex modes with different radial mode indexes p. (a) AP-LG₁₁, (b) AP-LG₃₁, (c) AP-LG₅₁. The horizontal and vertical violet dashed lines are the critical FWHM and β of magnetization superoscillation, respectively.

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The situation is totally different when leveraging the higher-order (AP-LG₃₁ and AP-LG₅₁) vortex modes, as demonstrated in Figs. 5(b)-(c). Clearly, it allows us to access to the magnetization superoscillation (horizontal violet dashed line) if the β falls beneath the threshold (β_s = 2.854 and 3.489, vertical violet dashed lines). More importantly, the self-defined robust magnetization superoscillation is available given that the β resides in the opportune scopes (2.612 < β < 2.854 and 3.251 < β < 3.489). For example, the FWHM, ECE and SL of the magnetization hotspot are, respectively, determined to be 0.356λ, 6%, and 19% in the case of β = 2.65 by tightly focusing the AP-LG₃₁ vortex beams. As such, the AP-LG₅₁ vortex counterpart performs the robust super-oscillation behavior as well. These facts indicate that the higher-order AP-LG vortex modes may serve as superior solutions to induce the desired longitudinal magnetization superoscillation. It should be stressed that the zero-order (AP-LG₀₁) vortex mode fails to deliver the super-oscillation magnetization because of the elimination of coherent interference contributed by the single inner ring (not show here). In fact, it is believed that our proposed all-optical avenue is also compatible with the 4π microscopic setup [9,13,15,17–19,46] and multifunctional metalens engineering [37–39,60–62], thereby making the 3D super-oscillation magnetization hotspot or needle possible.

3.3 Physical mechanism of the triggered longitudinal magnetization superoscillation

Having considered the control of the longitudinal magnetization superoscillation in high-order AP-LG vortex modes with the peculiar β, we proceed with the demonstration of the physical mechanism involved. It is well known that the longitudinal magnetization superoscillation is in essence dictated by the transversely polarized super-oscillation optical field according to the IFE. As an illustration, we resort to the tightly focused optical fields of the AP-LG₃₁ vortex beam relative to the β to unravel the fundamental principle on the magnetization superoscillation (Fig. 6). More insights into this underlying mechanism can be gained from the coming two aspects. One the one hand, as for the β close to that of the null magnetization (β = 2.308), the transverse components of the focal optical fields near the focus will bring about incomplete destruction interference in spite of the π phase difference between the adjacent rings. This is due to the fact that the strength of the transverse components created by the inner rings is unequal to that formed by the outer rings [57]. In this case, sharp central hotspots reaching optical superosillation accompanied by dime intensity arise in the focal plane (Figs. 6(c) and (d)). On the other hand, unlike the β adjacent to that of peak magnetization (β = 3.545), the transverse components of the focal optical fields are unable to give birth to perfect construction interference (Figs. 6(a) and (b)) in the case of β < β_s = 2.854. Such an event originates from the outer ring of the AP-LG₃₁ vortex mode completely shifting into, even partly spilling over, the edges of the objective lens as the β decreases in the range of 2.308 < β < 2.854, which promotes the total focal optical fields to surpass the highest Fourier component (∼0.36λ/NA) and hence constitutes the so-called optical superoscillation [36,63]. Meanwhile, the infirm construction interference renders the central hotspot fairly faint owing to the brightest ring far from the focus, as exhibited in Figs. 6(c) and (d). All in all, the partial destruction and construction interferences caused by different rings of the AP-LG vortex modes are responsible for the redistribution of transversally polarized focused optical fields, enabling the realization of longitudinal magnetization superoscillation.

Fig. 6. The intensity profiles in the focal plane for the AP-LG₃₁ vortex beam with different β values. (a) β = 3.545; (b) β = 2.854; (c) β = 2.506; (d) β = 2.411. I (black curve), I_r (red curve) and I_ϕ (blue curve) are the total optical field, radially and azimuthally polarized field components, respectively. The illustrations within in the magenta boxes in Figs. 6(a)-(c) are total (top), radial (left bottom) and azimuthal (right bottom) intensity distributions of the AP-LG₃₁ vortex beam in the x-y plane. The dimension in the illustrations is 2λ×2λ.

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4. Conclusions

In conclusion, we have theoretically studied the longitudinal magnetization superoscillation characteristics by tightly focusing high-order AP-LG vortex modes through the IFE. It is found that the precise control over the intrinsic truncation parameters of the AP-LG₁₁, AP-LG₃₁ and AP-LG₅₁ vortex modes enables the light-induced magnetization in the focal plane converting from a peak sub-wavelength pattern, via the fastest Fourier component, toward a super-oscillation hotspot. Furthermore, we are able to garner highly sharp magnetization spot size (< 0.36λ/NA) with acceptable energy conversion efficiency (> 5%) and moderate side lobe (< 30%) only if the AP-LG₃₁ or AP-LG₅₁ vortex mode is adopted, demonstrating that the higher-order vortex beams are preferable to accomplish the robust super-oscillation magnetization. Essentially, these exotic magnetization phenomena depends upon the redistribution of transversely polarized focused light fields, owing to the partial construction and destruction interferences caused by the π phase shift between the different rings of the incident structured vortex beams. Our proposed method is relatively simple and nearly optimization-free, which underscores the fundamental interest of the super-oscillation longitudinal magnetization theory. More importantly, the obtainable results could be adapted to potential exciting applications in high-density all-optical magnetic recording/storage, super-resolution magnetic resonance imaging, atom trapping 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).

Disclosures

The authors declare no conflicts of interest.

Data availability

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

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Longitudinal magnetization superoscillation enabled by high-order azimuthally polarized Laguerre-Gaussian vortex modes

Abstract

1. Introduction

2. Theoretical analysis of the longitudinal magnetization superoscillation

3. Results and discussions

3.1 Dependence of light-induced magnetization in the focus on both the p and β

3.2 Longitudinal magnetization superoscillation induced by high-order AP-LG modes

3.3 Physical mechanism of the triggered longitudinal magnetization superoscillation

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Tables (1)

Equations (7)

Optics Express

β	AP-LG₁₁ mode			β	AP-LG₃₁ mode			β	AP-LG₅₁ mode
β	FWHM	ECE	SL	β	FWHM	ECE	SL	β	FWHM	ECE	SL
2.732	0.478λ	22.9%	22.3%	3.545	0.451λ	20.2%	10.8%	4.252	0.442λ	19.5%	8.6%
2.051	0380λ	10.4%	36.8%	2.854	0380λ	10.3%	13.9%	3.489	0380λ	10.5%	10.2%
1.753	0.335λ	3.8%	73.6%	2.506	0328λ	2.5%	31.8%	3.234	0.351λ	4.3%	15.1%
1.576	0.271λ	0.9%	246.8%	2.411	0.287λ	0.7%	77.6%	3.078	0.307λ	0.7%	36.5%