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Abstract
We report a new paradigm for achieving magnetization spot arrays with controllable three-dimensional (3D) orientations. Toward this aim, we subtly design a tailored incident beam containing three parts and further demonstrate that the designed incident beam is phase-modulated radial polarization. Based on the raytracing model under tight focusing condition and the inverse Faraday effect on the magneto-optic (MO) film, the magnetization field components along the y-axis and z-axis directions are generated through the focus. In particular, we are able to garner orientation-tunable 3D magnetization under different numerical apertures of the focusing objectives by adjusting the ratios between the three parts of incident beam. Apart from a single magnetization spot, magnetization spot arrays capable of dynamically controlling 3D orientation in each spot can also be achieved by multi-zone plate (MZP) phase filter. Such a robust magnetization pattern is attributed to not only the constructive interferences of three orthogonal focal field components, but also the position translation of each magnetization spot resulting from shifting phase of the MZP phase filter. It is expected that the research outcomes can be beneficial to spintronics, magnetic encryption and multi-value MO parallelized storage.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The discovery of the giant magnetoresistance [1] in magnetic materials has tremendously facilitated the manipulation capability of magnetization driven by tempting prospective applications in all-optical magnetic recording [2–4], confocal and magnetic resonance microscopy [5,6], atom trapping [7,8], magnetic encryption [9–11] and multi-value high density magneto-optical storage [12–14]. In this respect, all-optical magnetization control through the inverse Faraday effect (IFE) [15] on the magneto-optic (MO) film conceived in 1960s is widely considered as a key component for the next generation magnetic storage technology. Compared with the traditional magnetic storage [16,17] and electron beam lithography [18], this technology has unbeatable advantages: ultrafast speed, low power consumption, high density, high reliability, erasability, large capacity and low cost [19,20].
Tightly focusing a circularly polarized beam in high numerical aperture (NA) lens can produce a longitudinal magnetization spot with half-wavelength size in the focal plane [21]. However, apart from the longitudinal component of the magnetization field, a certain amount of transverse magnetization component inevitably appear in a high NA objective lens focusing geometry, which largely degrades the purity of the longitudinal magnetization in the perpendicular magnetic storage [22,23]. As a versatile endeavor, polarization and phase singularities of the incident beam [24,25] are introduced to tackle this obstacle. Via the polarization and phase modulations, researchers have further demonstrated multifunctional longitudinal magnetization structures such as magnetization vortices [26], axial extended magnetization spot [22], magnetization needle [27,28], spherical magnetization spot [29], magnetization chain [30,31], magnetization spot arrays [9,32,33] and non-diffracting magnetization [34].
As a counterpart of longitudinal magnetization field, the purely transverse magnetization is highly desired for horizontal magnetic storage. For this purpose, a magnetization spot oriented in-plane with near-unity purity was produced by tightly focusing linearly and radially polarized beams in a $4\pi$ microscopic configuration [35]. In order to promote multi-value MO parallelized storage, we proposed a viable route to give rise to transverse magnetization spot arrays with controllable in-plane orientation in each spot [36]. Whereas the above reported control over magnetization orientation is restricted to either the longitudinal direction or in the transverse plane, and the magnetization orientation should be adequately exploited for further enlarging the MO storage capacity. Therefore, the achievement of magnetization field along arbitrary orientation is urgently needed. As yet, several subtle approaches such as the raytracing model (RM) method [37], symmetry-breaking method [38], the reversing electric dipole method [39] and anisotropy-mediated spin-orbit coupling method [40], have been attempted to induce a magnetization spot with 3D orientation. Apart from a single magnetization spot, magnetization needles with controllable 3D orientations were produced by reversing electric dipole arrays radiation [41]. However, the required incident light was extremely complicated beam involving the phase, amplitude and polarization modulations, and has to be produced by a bulky optical system [42].
In this work, we report a facile methodology for magnetization spot arrays with steerable 3D orientation in each spot by employing a reconfigurable incident beam in a single objective lens focusing geometry. Unlike the previous reports involving complicated modulations, the magnetization orientation along any arbitrary direction is solely controlled by the pure phase distribution. Our solution opens up a novel all-optical route to efficiently steer the 3D orientation of the magnetization field and promotes the development of multistate magnetization storage.
2. Theoretical analysis
As shown in Fig. 1, RMs for focusing three kinds of well-designed radially polarized beams are depicted. In Fig. 1(a), a longitudinally polarized component ($E_z$) can be produced by tightly focusing radially polarized beam ($\mathbf {e}_{\rho }$), which is due to the constructive interference of the longitudinal components and the destructive interference of the radial components [43]. The radially polarized beam is modulated by $\pi$-phase-step filter [44] along the $x$-axis, as illustrated in Fig. 1(b). As we know, "sgn" is a symbolic function. Here, sgn($x$) represents $\pi$-phase-step filter along the $x$-axis. A polarized component along the $x$-axis ($E_x$) can be generated in the focus owing to the constructive interference of $E_x$ and the destructive interference of the other two polarized components. Specially, an additional $\pi /2$ phase delay is added by the imaginary number $j$. Similarly, Fig. 1(c) describes that a polarized component along the $y$-axis ($E_y$) can be obtained in the focus, when the radially polarized beam is modulated by $\pi$-phase-step filter along the $y$-axis (sgn($y$)). According to the general definition of light-induced magnetization field in isotropic magnetically ordered MO film, the magnetization field is proportional to the vector product between two complex conjugate electric fields [15]. Therefore, magnetization fields along the $y$-axis ($M_y$) and $z$-axis ($M_z$) can be produced in Fig. 1(d)-(e), respectively. Combining $M_y$ with $M_z$, magnetization field along 3D space is achieved in Fig. 1(f). By altering amplitude factors of the three well-engineered radially polarized beams, the ratios between three mutually orthogonal focal field components can be adjusted, thereby allowing for dynamic control over the 3D orientation of the magnetization spot.
Fig. 1. Schematic diagram of producing magnetization distribution with 3D orientation. RMs for tightly focusing radially polarized beam (a), radially polarized beam imposed with $\pi$-phase-step filter along the $y$-axis and $\pi /2$ phase delay (b), and radially polarized beam imposed with $\pi$-phase-step filter along the $y$-axis (c). Generation of magnetization orientation along the $y$-axis (d), the $z$-axis (e) and 3D direction (f), respectively.
Though the qualitative magnetization field can be analyzed by the RM, its accurate distribution should be calculated by the Richards and Wolf diffraction theory [45,46]. The focal field for radially polarized beam can be written in Cartesian base vectors:
3. Single magnetization spot with 3D orientation
As discussed in Fig. 1, if the incident beam comprises of three kinds of well-defined radially polarized beams, magnetization field along 3D orientation can be produced. For this purpose, a special MZP phase is designed to achieve this goal. Figure 2(a) shows that the MZP phase is equally divided into $N$ fan-shaped areas with an angular width $\Delta \varphi =2\pi /N$, where $N$ is a positive integer number. Each fan-shaped area is then further divided into three smaller fan-shaped subareas, as shown in Fig. 2(b). The weights of the angular widths are expressed by $\gamma :\delta :\beta$, which differ from those of the previous report [48]. The phase encodings filled in these three parts are $1$, $j\textrm {sgn}(x)$ and $\textrm {sgn}(y)$, which correspond three kinds of beams in Fig. 1(a)-(c). In principle, the orientation of the magnetization spot can be dynamically controlled in the 3D space via adjusting the weights of three parts in the incident beam.
Fig. 2. Configuration of the designed MZP phase to generate a single 3D oriented magnetization spot. (a) The phase filter is divided to $N$ fan-shaped areas. (b) A single fan-shaped area consists of three subareas.
In order to further demonstrate the feasibility of the phase design, the focusing electric fields are illustrated in Fig. 3. As expected, it is found from Fig. 3(a) that the designed MZP phase consists of many equal fan-shaped areas each of which has three unequal smaller fan-shaped subareas, which is consistent with the illustration of Fig. 2. Under the modulation of the designed phase filter, three orthogonal polarized focusing fields are generated in Fig. 3(b)-(d). For easy analysis, we only map electric fields distributed in the $y$-$z$ plane. The phases in the focus for $E_x$, $E_y$ and $E_z$ are $0.51\pi$, $1.99\pi$ and $0$, respectively. It is easily found that the phase difference between $E_x$ and $E_y$ is $1.54\pi$, which is close to $1.5\pi$. In this sense, a circularly polarized field is produced by combining $E_x$ with $E_y$, which is consistent with the result in Fig. 1(e). According to the IFE, a longitudinal magnetization field $M_z$ can be induced in the focal region. The phase difference in the focus between $E_x$ and $E_z$ is $0.51\pi$, which is in close proximity to $0.5\pi$. Similarly, a magnetization field along the $y$-axis $M_y$ is generated. The phase difference in the focus between $E_y$ and $E_z$ is $1.99\pi$, which is nearly close to $2\pi$. Almost no magnetization field is induced by these two orthogonal polarized fields. To summary up, the total magnetization orientation is the vector superposition of $M_z$ and $M_y$. In Fig. 4, the corresponding induced magnetization distributions in the $y-z$ plane and along the axial directions are depicted. For simplicity, we only map the orientation distributed in the y-z plane, which is a representative manifestation of 3D orientation because of rotated symmetry along the optical axis. In Fig. 4(a), we can see that the magnetization direction angle with respect to the $z$-axis is about -45 degrees. Moreover, magnetization intensity distributions along the $z$-axis and $y$ axis are plotted in Fig. 4(b). The values of full width at half maximum (FWHM) along these two axes are $1.810\lambda$ and $0.735\lambda$, respectively.
Fig. 3. Distributions of focusing electric fields when the parameters are chosen as $N=50$, $\delta =\beta =1$ and $\gamma =2$. (a) The required phase distribution of the MZP filter, (b)-(d) the amplitude and phase (the inset) distributions of $E_x$, $E_y$ and $E_z$ in the $y$-$z$ plane.
Fig. 4. (a) The distribution of the induced magnetization field in the $y$-$z$ plane, (c) scans of magnetization intensity along the cross-sections ($y$ and $z$ axes). The chosen parameters are the same as Fig.3.
We note that when the weights of three fan-shaped subareas vary, the orientation of the magnetization spot in the $y$-$z$ plane can alter correspondingly. Hence, we further reveal the relation between the orientation direction with respect to the $z$-axis $\theta _M=\arctan (M_y/M_z)$ and the corresponding weights. For simplicity, the parameters $\delta$ and $\beta$ are set as 1, and only the parameter $\gamma$ can change. Here, the settings of $\delta =\beta =1$ and $\gamma$ in the range of [0, 4] are just for convenient analysis and customary consideration. These three values are not fixed and can be arbitrary values. We only focus on the relative values between them and the absolute values are not significant. The magnetization orientation is determined by their relative values. In Fig. 5, the variation of the orientation angle with the parameter $\gamma$ for different NAs is plotted. For comparison, the result based on the theoretical approximation $\theta '_M=\arctan (\eta \gamma )$ arising from the ratio between the transverse and longitudinal magnetization orientation components is also plotted in Fig. 5. Here, we assume that the amplitude of the longitudinal magnetization component is equal to 1 and $\gamma$ denotes that of the transverse component. Besides, $\eta$ is a fitting parameter for different NAs. It is found from Fig. 5(a)-(d) that the orientation angle increases with the increase of $\gamma$. Meanwhile, the orientation angle increases with the higher $NA$ value when we keep $\gamma$ as the same value. Specially, the maximum of the magnetization orientation is less than 80 degrees and can not reach to 90 degrees. Therefore, this result seems that the arbitrary orientation can not be achieved. In fact, this outcome comes from the setting of $\delta =\beta =1$. if $\beta =0$, a focal polarized component along the $y$-axis vanishes in the focus and only the other two orthogonal components exist in the focus. Therefore, a magnetization field along the $y$-axis is induced by these orthogonal components and the maximal orientation angle (90 degrees) is achieved. Moreover, the fitting parameter $\eta$ increases with the increase of the $NA$. In fact, this result can be explained in a quite easy way. When the $NA$ of the tight focusing lens increases, the weight of the longitudinal component for radially polarized beam correspondingly increases. According to the IFE on the MO film, the transverse component of the magnetization spot increases in the corresponding way. In order to clearly demonstrate this physical relations, the variation of the fitting parameter $\eta$ with the $NA$ value is plotted in Fig. 6. It is easily found that the fitting parameter $\eta$ increases with the ascending $NA$ value.
Fig. 5. Magnetization orientations of simulation result and theoretical fit as a function of the parameter $\gamma$ for $NA=0.50$ (a), $NA=0.60$ (b), $NA=0.80$ (c) and $NA=0.95$ (d). Here, $\eta$ is the fitting parameter for different NAs.
4. Magnetization spot arrays with 3D orientations
To multiplex a single magnetization with 3D orientation to prescribed spatial location, magnetization spot arrays with controllable 3D orientation can be further generated in principle. Toward this aim, a special MZP phase filter is designed, as shown in Fig. 7. First of all, the MZP phase filter in the pupil plane is divided into $N$ annuli each of which is subdivided into $M$ smaller annuli, where $N$ and $M$ are positive integers [49]. Then, each smaller annulus is divided into $L$ pieces along the azimuthal direction [48]. At last, each piece consists of three smaller pieces ($\gamma :\beta :\delta$). The rule of phase encoding is as follows. First, the shifting phase to determine the position of each magnetization spot is encoded in each smaller annulus. Each of $M$ smaller annuli has a different shifting phase distribution and $M$ denotes the number of magnetization spot arrays. Each of $N$ annuli has the same distribution of shifting phase and $N$ is used to control the uniformity of magnetization spot arrays. The shifting phase distribution can be expressed by
Fig. 7. Configuration of the designed MZP phase to generate magnetization spot arrays with 3D orientations. (a) The designed phase consists of $N$ annuli. (b) Each annulus includes $M$ smaller annuli. (c) Each smaller annulus is divided into $L$ pieces each of which is composed of three smaller pieces.
In the next, the intensities and magnetization orientations of magnetization spot arrays under the modulations of different MZP filters are illustrated in Fig. 8. It is seen from Fig. 8, four magnetization spot arrays are formed in the focal region. Comparing Fig. 8(a) and Fig. 8(b), we can conclude that the larger the values of $N$ and $L$, the more homogeneous the magnetization spot arrays [48]. In Fig. 8(b), four magnetization spots labeled by 1, 2, 3 and 4 in different positions are formed. The magnetization spot arrays are distributed in four corners of the square region ($6\lambda \times 6\lambda$). The distance between adjacent magnetization spots is $6\lambda$. The maximum intensities of these four magnetization spots are 0.95, 1.00, 0.94 and 0.94, respectively, and these values are nearly equal. The values of FWHM for these four magnetization arrays along the $y$-axis are $0.48\lambda$, $0.57\lambda$, $0.54\lambda$ and $0.54\lambda$, respectively. For the values of the FWHM along the $z$-axis, they are $1.53\lambda$, $1.54\lambda$, $1.8\lambda$ and $1.8\lambda$. Although the transverse and longitudinal sizes for these four magnetization spots are not equal, these values are close to the values of the single magnetization spot in section 3. Therefore, the uniformity in the intensities and spatial sizes of the magnetization spot arrays is well homogeneous. The homogeneity can be further improved by increasing the distance between the magnetization spot arrays. Moreover, the orientation of each magnetization spot is different and individually independent. In order to demonstrate the uniformity of the magnetization orientations, the purities of the magnetization spot arrays in Fig. 8(b) are given and depicted in Fig. 9. The calculation of the purity is based on the inner product between the magnetization orientation in the center of the spot and magnetization vector in the whole spot area, which is alike the definition in previous report [39]. It is seen from Fig. 9, the purities are not well uniform in these four spot arrays. And purities gradually decrease with the increase of the distance to the spot center. Fortunately, the smallest values of the purities for the whole spot arrays volumes are not less than 0.5. Especially in FWHM valume, the purities are not less than 0.75. Therefore, the purities of the magnetization orientations are enough homogeneous, which completely meet the requirement of magnetization storage. Depart from the homogeneity of the magnetization orientations, the dynamic control of the magnetization orientation in each spot is also depicted in Fig. 10.
Fig. 8. (a)-(b) Normalized intensities (color map) as well as 3D orientations (blue arrows) of magnetization spot arrays, when $M=4$, $N=20$ & $L=20$ as well as $M=4$, $N=50$ & $L=50$.
Different from the previous magnetization spot arrays with longitudinal or transverse orientations [22,35], the orientation here can vary in 3D space. Compared with polarization, amplitude and phase modulations to obtain 3D orientation [37–39], our pure phase encoding is relatively simple. Although the magnetization needles with controllable 3D orientations have been reported in Ref. [41], the 3D size is limited to the axial length of the long magnetization needle. It is pointed out that the 3D sizes of the magnetization spot arrays that we achieve are tremendously reduced, which is greatly beneficial for high-density magnetization storage. Though the 3D size of the magnetization spot in Ref. [42] is reduced, a bulky optical system is needed to produce such a complicated beam involving amplitute, phase and polarization modulations. In our work, only a pure phase filter (MZP) is used. The designed idea of the MZP phase filter that we used comes from the the corresponding works in Ref. [48,50], but the core idea to construct our magnetization outcomes is compeletely different. The achievement of 3D orientation is the result of three orthogonal polarized focal fields, which is coming from the tight focusing of three kinds of beams with different weights. In this sense, we only use the designed MZP phase filter to perform such a function. It is specially noted that a great variety of orientations tremendously boost storage capacity.
The well-defined magnetization behavior is contributed from two aspects. First of all, it is due to the superpositions of three kinds of well-designed radially polarized beams. The superpositions contribute to the control of 3D orientation by adjusting the amplitude weights of the three parts contained in the incident beam. Secondly, it is owning to the shifting phase. The designed MZP phase filter contains a number of shifting phases, which results in the magnetization spot arrays located at different positions by the Fourier transformation. Thus, when the incident beam is modulated by the MZP phase filter, magnetization spot arrays with 3D orientations are produced through the focus. The position and 3D orientation can be controlled by the diverse phase modulations of the MZP filter at will. Most importantly, the pathway of achieving the magnetization pattern can benefit other magnetization shapings. What is more, the research result of magnetization spot arrays with 3D orientations can broaden the magnetization manipulation and largely facilitate the development in multi-value MO parallelized data storage.
5. Conclusion
In summary, we have theoretically studied the light-induced magnetization field distributions by tightly focusing the configured incident beam in a single lens. According to the RMs under tight focusing condition and the IFE on the MO film, it is easily found that the intrinsic amplitude factors of the constituent beams perform a crucial role in determining the 3D orientation of the magnetization spot. Via adjusting the amplitude ratios between the three parts in the incident beam for different NAs, we can achieving the dynamic control over the 3D orientation of the magnetization spot. We also demonstrate that the structured incident light can be accessed by the radially polarized beam with phase modulation. Furthermore, magnetization spot arrays with dynamically controllable 3D orientation in each spot can be energetically generated by judiciously designing the MZP phase filter. Essentially, the physical mechanism of such an intriguing magnetization pattern is clearly elaborated from two aspects. The present findings can open up broad applications in ultrahigh-density MO memory, multi-value MO parallelized data storage and multiple atoms trapping.
Funding
National Natural Science Foundation of China (11604236, 11974258, 12004155, 61575139, 61865009, 61927813); Key Research and Development (R&D) Projects of Shanxi Province, China (201903D121127); Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2019L0151); Australia Research Council Industrial Transformation Training Centres scheme (IC180100005).
Disclosures
The authors declare no conflicts of interest.
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