We propose a feasible strategy for firstly constructing diffraction-limited light-induced magnetization spot arrays capable of dynamically controlling transverse polarization orientation of each spot. To achieve this goal, we subtly design a tailored incident light comprised of two sorts of beams and sufficiently demonstrate tit’s production through phase modulation of a radially polarized beam. Via tightly focusing counter-propagating composite illuminating beams in a 4π optical microscopic configuration, two orthogonally polarized focal fields with π/2 phase difference between them are formed, inducing a three-dimensional (3D) super-resolved transverse magnetization spot in the magnetic-optical (MO) film. Exploiting the ideal of the multi-zone plate (MZP) filter, we further achieve versatile magnetization spot arrays with controllable in-plane polarization direction in each spot. Such well-defined magnetization behavior is attributed to not merely the coherent interference of vectorial optical waves, but also non-overlapping superposition of localized focal fields. Our achievable outcomes pave the way for practical applications in spintronics and multi-value MO parallelized storage.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Since the effect of the giant magnetoresistance  was discovered, the perpendicular magnetic recording  has roused intensive research interests due to the practical application in high density information storage [3, 4]. Although the reading sensitivity is well distinct, the recording speed is restricted by electric signal frequency. To dramatically enhance this ability, the ultrafast response of the light-induced magnetization field [5, 6] in the MO film was investigated experimentally, facilitating the development of all-optical magnetic recording (AOMR) [7, 31] considered as a key component for the next generation storage technology. But, it suffers from magnetic bit with several micrometers. Further improving the storage density, the interaction between the light field and the magnetic material under tight focusing condition quickly attracts broad attention [8–11]. In this respect, numerous works about diverse light-induced magnetization structures are reported. By optimizing amplitude or phase modulation, a subwavelength long magnetization needle below the diffraction limit (λ/2) can be generated in a high numerical aperture (NA) lens [12–14]. Despite the lateral super-resolved feature is attained, the longitudinal size is excessively elongated. To handle this shortage, Z. Nie et al. garner a spherical 3D super-resolution magnetization spot (<λ3/8) by tightly focusing two counter-propagating radially polarized vortex hollow Gaussian beams in the 4π configuration . Yet, this strategy is not highly efficient because of only one magnetic bit in each recording process. In response to this predicament, a 3D super-resolved magnetization chain is produced by tightly focusing two modulated vectorial beams in the high NA lenses [16–18]. Though plenty of magnetization spots is garnered, the magnetization morphology is distributed along the z axis, which is not flexible and convenient for data storage. To overcome this imperfection, magnetization spot arrays with 3D super-resolution in each spot was recently pioneered by Z. Nie et al. based on the tight focusing and coherent interference of two reconfigured vectorial beams [19, 20].
There is no doubt that fruitful intriguing traits are realized, however, these efforts are devoted to the longitudinal magnetization carrying only two states of up and down, which may constrain the degree of freedom for data storage. Until recently, S. Wang et al. successfully achieve a magnetization oriented in-plane polarization with near-unity purity by focusing two counter-propagating reconfigured vector beams . The polarization direction of the produced magnetization spot can be continuously changed in the transverse plane, which is controlled by the polarization orientation of linearly polarized light. Nevertheless, this method is limited to control the polarization orientation of one magnetization spot instead of magnetization spot arrays, which is low efficiency for the era of big data. In order to boost this capacity, there is a stringent demand to develop multi-value MO parallelized storage, analogous to the combination of multi-dimensional optical storage  and multifocal optical storage . Toward this aim, there are threefold obstacles that have hitherto remained challenging in optomagnetization realm. Firstly, the magnetization spot should be confined in an ultra-small volume below the diffraction limit (λ3/8), which is essential to increase the storage density. Secondly, the magnetization spot should be extended to magnetization spot arrays without affecting the characteristic of the super-resolution, enhancing the processing speed as well as power efficiency of the opto-magnetic recording and reading. Last but not the least, the polarization orientation for each magnetization spot should be adjusted independently, enabling to provide multiple storage values instead of only two ones. It is noted that a great many diverse states undoubtedly take one step toward the potential application in multi-value MO parallelized storage.
In our work, to the best of our knowledge, we firstly report a facile methodology for 3D super-resolved magnetization spot arrays with steerable polarization orientation in each spot by employing reconfigurable incident beams in the 4π apparatus. In essence, they are achieved by coherent interference of the light fields at the exit pupil plane and no overlapping between localized focal fields located at different spatial positions. We hope that, these elegant results will open prosperous perspectives for multi-value MO parallelized storage. The paper is organized as follows. In section 2, raytracing models for tightly focusing two kinds of beams and schematic diagram for light-induced magnetization in the typical 4π setup are described. In section 3, the focal field for each part of the incident light is calculated to confirm the validity of raytracing models. By optimizing the corresponding the parameter of the phase modulation, the 3D super-resolved transverse magnetization spot with strongest intensity is obtained. When the concept of MZP filter is adopted [24,25], the magnetization spot arrays in possession of adjustable transverse polarization direction in each spot are achieved. In section 4, we conclude our work.
2. Raytracing models for tight focusing and schematic diagram for light-induced magnetization
Figure 1(a) describes raytracing models for the tight focusing of radially polarized beams and radially polarized beams imposed with π-phase-step filters  along the y axis in the 4π configuration. The blue and red arrows signify the polarization and propagation directions, respectively. Figure 1(a1) depicts that the incident beams are radially polarized. There is a π phase difference between the left and right parts. A longitudinally polarized component Ez is formed due to the constructive interference of the longitudinally polarized components and the destructive interference of the transversely polarized components. Figure 1(a2) illustrates that the incident beams are radially polarized beams modulated by π-phase-step filters along the y axis. A polarized component along the x axis Ex is emerged owing to the constructive interference of the polarized components along the x axis and the destructive interference of the other two orthogonally polarized components. Compared with Ez in Fig. 1(a1), Ex has an additional π/2 phase delay introduced by the incident beams. When these two orthogonally polarized focal components are combined, a focal spot with spin orientation along the y axis is generated in the focus. If an isotropy MO film is placed in the focal plane, a magnetization spot with the polarization orientation along the y axis My can be produced in principle, according to the inverse Faraday effect (IFE) .
For this purpose, Fig. 1(b) exhibits that two counter-propagating incident beams are focused to the MO film, which situates at the confocal plane of high NA objective lenses. The illuminating waves are phase-modulated radially polarized beams, representing the linear superposition of the two sorts of beams depicted in Fig. 1(a). In mathematics, the transmittance functions for the left and right phase filters can be expressed asEq. (1), the positive and negative formulas stand for the complex amplitudes for the left and right filters, respectively. The mark “sgn” is a typical function of taking the symbol. γ and δ indicate the amplitude factors of two kinds of beams shown in Fig. 1(a) and are in the range of [−1, 1], which meet γ2 + δ2 = 1. In addition, j represents the aforementioned π/2 phase delay, φ is the azimuthal angle with respect to the x axis and sgn(cos φ) denotes the π-phase-step filter along the y axis. Specially, −γ chosen for the right filter is because of the inverse instantaneous polarization with respect to the left parts for radially polarized beams in the 4π system .Eq. (2)EL(R) implies the left/right focal field (positive for the left lens and negative for the right lens), where A0 is the amplitude constant and l(θ) indicates the amplitude distribution, which can be assumed as Bessel-Gaussian beam l(θ) = exp[−(β0 sin θ/sin α)2] J1(2β0 sin θ/sin α). Here, β0 denotes the ratio between the pupil radius and the beam waist, and α = arcsin(NA) signifies the maximal convergence angle determined by the NA. Throughout this work, we choose β0 = 1 and α = π/2 (NA = 1). As a result, the total focal field can be given by Fig. 1(b), the thickness ranges from several nanometers to sub-micrometers [31, 32]. When a femtosecond laser pulse imprints the MO film, the response time is within picosecond order of magnitude . The induced static magnetization in the MO material is the vector product of the electric field, which is obtained by  Eq. (3) into Eq. (4), the distribution of light-induced magnetization field can be calculated. Based on these mathematical formulas, all the simulation results can be obtained by complex integral operations or simple mathematic computation using powerful Matlab software in the following sections.
3. Numerical simulation
3.1. Focal field for each part of the incident beam in the 4π system
As mentioned in the last section, the first and second parts hidden in Eq. (1) represent two kinds of beams shown in Fig. 1(a). Based on Eqs. (1)–(3), the focal field for the first part of the incident beam is given in Fig. 2. |Ex|2 is not depicted because of the very low scale (less than 1%). |Ey|2 is outside the black circle, whereas |Ez|2 mainly concentrates inside the black circle, see Figs. 2(a) and (b). The maximum value of the transverse component is 11% and that of the longitudinal component reaches up to 100%. Figure 2(c) shows that the phase angle of Ex inside the black circle alternates in [0°, −180°] along the azimuthal direction. There is a phase singularity in the focus, which yields the vanishing intensity of Fig. 2(a). Figure 2(d) shows that the phase angle of Ez is 0° and very homogeneous inside the black circle. Consequently, for tightly focusing the first part of the incident beam, the longitudinal component determines the intensity distribution in the focal region, which is in accord with the raytracing model of Fig. 1(a1). Similarly, we also calculate the focal field for the second part of the incident light. |Ey|2 and |Ez|2 are not illustrated in Fig. 3 owing to their very low proportions (less than 1%). Figure 3(a) shows that |Ex|2 mainly focuses inside the black circle and the maximal value nearly comes up to 100%. Figure 3(b) exhibits that the phase angle is −90° and well uniform in the focal region. As a result, for tightly focusing the second part of the incident beam, |Ex|2 dominates the intensity distribution in the focal region, which is consistent with the raytracing model of Fig. 1(a2). Comparing the phase angle of Ez in Fig. 2 with that of Ex in Fig. 3, there is a π/2 phase difference between them in the focal region. According to the IFE in the MO film, a magnetization spot with polarization direction along the y axis can be produced through focus.
3.2. Light-induced magnetization in the 4π setup
In light of Eqs. (1)–(4), in addition to the numerical aperture and amplitude distribution, it is turned out that either γ or δ influences the production intensity of magnetization field. So for this reason, the capacity of the light-induced magnetization field in the focus is investigated. Figure 4 displays that the magnetization component My increases firstly in a slow speed and then decreases fast, as the parameter γ increases. When γ varies around 0.7 close to , the magnetization intensity attains the extreme value. In such case, the intensities of the two sorts of beams are equal. This outcome agrees with the result of raytracing models in Fig. 1(a). According to IFE in the MO film, when the amplitude factors of two kinds of beams are identical, the intensities of the two orthogonally polarized focal components are equal, giving rise to the strongest magnetization intensity. On account of the optimizing outcome, we choose in the next simulation.
Figure 5 shows that the distributions of each magnetization field component in three orthogonal planes are depicted. It is easily found from Figs. 5(a1), (a2) and (a3) that Mx is outside the focus and the maximum value is less than 15%. Mx is only in the x – y plane and the distributions in the other two orthogonal planes are very low (less than 1%). On the contrary, My mainly concentrates on the focus and the maximal value reaches up to 100%, seen from Figs. 5(b1), (b2) and (b3). Figure. 5(b1) shows that the maximal sidelobe value is about 10% in the x – y plane, which is so low that it can be neglected. Although the maximal sidelobe values in the z – x plane and the z – y plane reach up to 45% and 30%, respectively. Such large sidelobe effects are not preferable to some applications such as high density AOMR as well as the confocal and magnetic resonance microscopy [35, 36]. It is well to be reminded that the large sidelobe value can be reduced by choosing high-order beams properly . Similar to the result of Mx, Figs. 5(c1), (c2) and (c3) show that Mz focuses outside the focus and the maximum value is less than 15%. The majority of Mx is in the z – y plane and the maximum values in the other two planes are less than 1%. As a result, the light-induced magnetization component My determines intensity distribution in the focal region.
To further investigate the characteristic of super-resolution for the magnetization, the distributions of predominant magnetization component My along each axis are shown in Fig. 6. The calculated full widths at half maximum along the x axis (Wx), the y axis (Wy) and the z axis (Wz) are 0.46λ, 0.53λ and 0.37λ, respectively. The volume of the magnetization spot can be calculated as , where the magnetization spot is approximated as an ellipsoidand. A super-resolved voxel of about λ3/21 is obtained, far smaller than the diffraction limit of λ3/8. The calculated value is close to the result in  and smaller than the outcome of . Although our result is bigger than the supercritical resolved outcome in , the controlling of two beams are more feasible and simple than that of six beams in experiment. In addition, the lateral sidelobe value is less than 10% and so low that it can be neglected in practical applications. The maximum value of the longitudinal sidelobe value reaches up to 31%, but the sidelobe is away from the focal plane and can be reduced by amplitude modulation .
In order to describe the manipulation of the magnetization spot visibly, we map the phase distributions for two phase filters based on Eq. (1) and the magnetization spot with transverse polarization orientation based on Eq. (4). Figure. 7(a) shows that the phase distribution of the left filter is divided into two parts along the horizontal direction. The phase of the left part is very uniform and keeps at 1.75π. For the right part, the phase angle is also homogenous and stays at 0.25π. There is a constant phase difference of 1.5π between these two parts. It is the phase difference that determines the generation of the pure My near the focus. The above two values are particular cases and can be taken other values with maintaining phase difference of 1.5π between them. Similarly, the phase angles of the left and right parts for the right filter are also uniform, as shown in Fig. 7(b). The phase angles for these two parts are 1.25π and 0.75π, respectively. The phase difference is 0.5π. The phase distributions of these two phase filters resemble π-phase-step filters, and can be easily encoded by pure phase spatial light modulator . Figure. 7(c) shows that the bright magnetization spot is formed in the focal plane and the polarization direction is mainly along the y axis. In the mainlobe region, the maximum of My reaches up to 100%. Meanwhile, Mx appears in the sidelobe area and the calculated maximal value is 15%. For practical magnetization storage, we only focus on the mainlobe region. Therefore, we can neglect the effect introduced by this undesired component My. When these two phase filters revolve, the magnetization spot can rotate correspondingly (See Visualization 1). The rotation of the phase filters can be easily realized by phase encoding of pure phase spatial light modulator. In other words, the magnetic spot with in-plane magnetization can be governed by the two pure phase filters in front of lenses. Compared with the polarization modulation in , our method is more simple and accessible in experiment.
Catching hold of the pure phase modulation, we can employ the tactics of the two MZP filters introduced in [24, 25] to achieve magnetization spot arrays further. These two MZP filters are all divided into N fan-shaped areas along the azimuthal direction, and then each fan-shaped area is further divided into L smaller fan-shaped subareas. Each of these subareas is filled with phase information linked to the polarization direction and the position of each magnetization spot. In other words, the phase encoding of each subarea is the superposition of the rotated phase associated with the magnetization orientation and the phase shift responsible for the position of each magnetization spot. It is noted that L is the number of the magnetization spot arrays. Here, we give an example to elaborate our descriptions. In our simulation, we choose N = 50 and L = 4. Figure. 8 shows that four magnetization spots possessing different transverse orientation in each spot are formed in the focal plane. Resembling multifocal array with controllable polarization published in . the polarization direction for each magnetization spot is prescribed owing to the usage of MZP filters. Therefore, magnetization arrays are consistent with the purity of a single circular polarization-induced longitudinal magnetization field. The distance between adjacent magnetization spots is 4λ. The shapes of these magnetizaton spots are nearly the same. The maximum values for these four magnetization spots are 0.96, 1, 0.96 and 0.95, respectively and are almost equal. Therefore, these magnetization spot arrays in shape and intensity aspects are very homogeneous. More specially, when the number of fan-shaped area N is chosen to be a larger value or the adjacent spacing is bigger, the uniformity of the magnetization spot arrays can be better. By varying the phase encoding of each fan-shaped subarea, we can individually manipulate the polarization direction of each magnetization spot (see Visualization 2). In fact, the polarization direction of each spot varies in the range of [0°, 360°] with respect to the x axis. Compared to the longitudinal magnetization spot arrays with only two states of up and down [19, 20], a great variety of polarization directions tremendously boost storage capacity. Besides, it is pointed out that it is difficult to achieve this instructive magnetization pattern by utilizing the method of the polarization modulation in . The robust characteristics of the magnetization structure mainly arise from two critical factors. On the one hand, the formation of the transverse polarization direction is because of the interference effects of two counter-propagating incident beams in the focusing process of the 4π system. The focal electric fields for two kinds of beams correspond to two orthogonally polarized components. These two focal electric fields are out of π/2 phase, allowing to induce transverse magnetization according to the IFE in the MO film. On the other hand, dynamic control over the transverse polarization orientation for each magnetization spot is owing to nonoverlapping superposition of the separated focal fields confined in subwavelength scale. The MZP filters have the shiftable ability in the Fourier transformation, making the focal fields locate different positions. When the incident beams through the MZP filters, a 3D super-resolution focal spot is transformed to spot arrays with keeping the super-resolved feature and then induce magnetization spot arrays. The position and the polarization direction of each magnetization spot can be dynamically controlled by the diverse phase modulations of the MZP filters. Importantly, the roadmap that we proposed can give a guideline for magnetization manipulation and the excellent magnetization pattern that we achieved can be used to multi-value MO parallelized storage.
In summary, we have theoretically studied the light triggered magnetization by tightly focusing structured beams consisting of two kinds of beams in the 4π microscopic device. We find that both the amplitude factors of these two sorts of beams and the phase encodings of MZP filters play a pivotal role in determining the polarization orientation, resolution capability, intensity power of the focal magnetization field. As such, 3D super-resolved magnetization spot arrays together with adjustable in-plane polarization direction in each spot can be energetically generated. We further exemplify, as a demonstration, that the composite incident light is accessed by the phase modulated radially polarized beam. Additionally, the physical mechanisms for this well-behaved magnetization structure are elaborated in two aspects. We argue that the research results can open up broad applications in ultrahigh-density magneto-optic memory, multi-value MO parallelized data storage and multiple atoms trapping.
National Natural Science Foundation of China (NSFC) (11474078, 11604236, 61575139, 11374079); Youth Science Foundation of Heilongjiang University (NO. QL201601); Basic Scientific Research Operating Expense for Colleges and Universities of Heilongjiang Province & Special Fund for Heilongjiang University (NO. HDRCCX-201601).
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