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The light-induced magnetization distributions for a high numerical aperture focusing configuration with an azimuthally polarized Bessel-Gaussian beam modulated by optimized vortex binary filters are investigated based on the inverse Faraday effect. It is found that, by adjusting the radii of different rings of the single/ cascaded vortex binary filters, super-long (12λ) and sub-wavelength (0.416λ) longitudinal magnetization chain with single/dual channels can be achieved in the focal region. Such well-behaved magnetization trait is attributed to the mutual effect between the optical polarization singularities of the azimuthally polarized beam and single/cascaded spiral optical elements. In addition, we find that the displacement distance of the longitudinal magnetization chain is proportional to the phase difference between the inner circle and outer ring of the vortex binary filters, thus giving rise to the steerable magnetization chain. It is expected that the research outcomes can be applied in multiple atoms trapping and transport, multilayer magneto-optical data storage, fabrication of magnetic lattices for spin wave operation and development of ultra-compact optomagnetic devices.
© 2015 Optical Society of America
1. Introduction
The increasing interest in developing ultra-compact optomagnetic devices has recently invoked intensive research attention on light-induced schemes, which are capable of steering longitudinal magnetization at the sub-wavelength scale in the magnetic materials. It has been demonstrated that, in particular, the circularly polarized beams in the case of a high numerical aperture (NA) focusing configuration allow for a sub-wavelength magnetic confinement via the inverse Faraday effect (IFE) [1–4 ], facilitating longitudinal magnetization recording in all-optical magnetic recording (AOMR) [5–7 ]. Despite these intriguing and outstanding characteristics, however, an appreciable portion of doughnut-shaped transverse magnetization is accompanied that is unfavorable to high density AOMR [8, 9 ].
Because of excellent polarization symmetries and vector characteristics, the tightly focusing cylindrical vector beams has displaced the circularly polarized ones as a promising alternative avenue to achieve pure longitudinal magnetization with sub-wavelength resolution. On the one hand, we lately demonstrated the possibility, unlike the magnetic vortices induced by strongly focusing a radially polarized beam [10], of creating spherical and sub-wavelength longitudinal magnetization (0.43λ) by 4π tightly focusing radially polarized hollow Gaussian vortex beams [11]. On the other hand, Jiang et al garnered sub-diffraction-limited pure longitudinal magnetization (0.508λ) throughout the entire focal plane of a high NA objective under vortex phase-modulated azimuthally polarized beams illumination [12]. Afterwards, further melioration was reported to prolong the longitudinal magnetization extension (7.48λ) along the optical axis and simultaneously sharpen the transverse dimension (0.38λ) by means of an annular vortex binary filter, which corresponds to an aspect ratio of 20 [13]. Recently, by applying both azimuthally and radially modulated annular phase filters, Ma et al exhibited that an ultra-long longitudinal magnetization needle (28λ) with a narrower lateral size (0.27λ) can be reached, and the relevant aspect ratio reaches up to 103 [14]. The well-defined pure longitudinal magnetization needle with high aspect ratio in principle provides the ability to trap atoms [15, 16 ] and implement high density AOMR, but they might be less useful for a desired number of atoms trapping and transport, as well as multilayer magnetic-optical recoding and storage.
In addition to the fundamental exploration of atomic trapping by means of light-induced longitudinal magnetization needle with the sub-wavelength scale, magnetic traps for cold atoms were also developed applying the fictitious magnetic field (magnetization) induced by the circularly polarized light field in conjunction with the external magnetic bias field [17–19 ]. However, a majority of magnetic potential wells so far lack the tunability of atomic trapping location once being generated. For this purpose, a kind of robust magnetic chain-shaped arrays, which is termed as magnetic conveyer belt, has been proposed in an external magnetic bias field to transport trapped atoms to very precisely controlled positions near a surface [20, 21 ]. Despite all this, there are several possible limitations for such a scheme. First, the patterns of the magnetic conveyer belt totally depend on the external field, rather than vector optical field, leading to less flexibility. Second, the polarization of the magnetic chain is not pure longitudinal and the transverse size of the relevant magnetic spots is not confined to the sub-wavelength dimension, which is adverse to high density magneto-optical storage except for multiple atomic trapping. In these regards, it is highly desirable to achieve and steer sub-wavelength pure longitudinal magnetic array (here call it as magnetization chain), which is in analogous to the versatile optical chain [22–24 ], induced by vector optical fields instead of external magnetic bias fields.
In this article, we theoretically propose a novel method to yield sub-wavelength pure longitudinal magnetization chain in the focal region with single/double channels through tightly focusing an azimuthally polarized Bessel Gaussian (BG) beam that is modulated by specially designed vortex binary filters. In addition, the desired magnetic chain-shaped patterns can be flexibly controlled in a premeditated manner by tuning the phase difference (Δφ) between the inner circle and outer ring of the optimized vortex binary filters. The mechanisms for the well-defined magnetization distributions in principle are attributed to the interaction between the optical polarization singularities of the azimuthally polarized beam and single/cascaded spiral binary elements. This paper is organized as follows: we describe in section 2 the vector diffraction formulas and IFE under the vortex phase-modulated azimuthally polarized beam illumination. We give in section 3 the numerical studies of the magnetization distributions in the focal region for the azimuthally polarized BG vortex beam in a high NA objective focusing configuration, leading to sub-wavelength pure longitudinal magnetization chain in the focal region with single/double channels. Moreover, the dependence of displacement distance of the longitudinal magnetization chain on the values of Δφ is dissected. Finally, we conclude our work in section 4.
2. Theoretical analysis
The schematic diagram is shown in Fig. 1(a) . An incoming azimuthally polarized BG beam travels through a vortex binary filter (which is called as single vortex binary filter for comparison) and then is focused on an isotropic and non-magnetically ordered magnetic-optic (MO) film by a high NA lens. The proposed single vortex binary filter consists of three rings with disparate transmittances in the radial orientation, as well as the gradually varied phase of each ring from 0 to 2π in the azimuthal direction, as depicted in Fig. 1(b). It should be stressed that phase difference between the inner circle and the outer ring can be flexibly regulated depending on rich magnetization profiles. Additionally, R 1 and R 2, respectively, corresponding to the radii of the inner circle and the middle ring, are alterable parameters to dominate the well-behaved magnetization distribution. However, R 0 is a fixed constant that represents the radius of the outermost ring and that of the NA lens. In this study, the single vortex binary filter is a pivotal element to induce the desired longitudinal magnetization chain. Mathematically, the transmittance function of light beam transmitting through this special mask can be given by
where Δφ denotes the phase difference between inner circle and exterior ring. On the basis of the vector diffraction theory [25, 26 ], the electric field of the azimuthally polarized beam in the focal region can be expressed as [27–29 ]withwhere θ max = arcsin(NA/n 0) is the maximum focal angle determined by the NA and the refractive index n 0 in image space; k is the wave number in free space; Jn is the n order Bessel function of the first kind; A 0 is a constant coefficient; l 0(θ) denotes the pupil function obeying the BG distribution [27, 30 ]. represents phase function in the focal region, which is written as
Fig. 1 (a) Schematic setup to generate the sub-wavelength longitudinal magnetization chain. (b) Sketch diagram of a vortex binary filter, which is composed of a binary filter and a spiral phase plate.
It is apparent from Eqs. (2)-(4) that the focal field with pure transverse polarization can be formed as a result of null longitudinal component. In principle, a transversally polarized field is responsible for yielding pure longitudinal magnetization.
For convenience, we consider conducting electrons within magnetic-optic film as a collisionless plasma in which the electrons can migrate freely [31]. Under such a circumstance, the magnetization field of the film induced by IFE can be expressed as [1–4, 7–13 ]
with γ is a magneto-optical constant, E is the electric field and E* denotes its complex conjugate. By substituting Eqs. (2)-(4) into Eq. (5), one getswhere e z represents the unit vector along the optical axis. As expected, the resultant magnetization is totally longitudinal. This well-defined behavior is distinguished from that under the circularly polarized beam illumination due to the depolarization effect by a high NA objective.3. Magnetization distributions of the azimuthally polarized BG beam
3.1 Sub-diffraction-limited longitudinal magnetization chain with single channel
To create the light-induced longitudinal magnetization chain with single channel, the proposed single vortex binary filter is optimized with a global search optimization algorithm to make the destructive interference occurs at the dark region while the constructive interference takes place at the high intensity light wall in the form of a periodic array. The optimal parameters characterized by the disparate radii of the vortex mask are R 1 = 0.225R 0 and R 2 = 0.943R 0, respectively. Moreover, we consider other parameters as NA = 0.95, A 0 = 1 and n 0 = 1. Equations (2)-(6) are adopted to illustrate flexible functionality of the prescribed single vortex binary element for redistributing the focal profiles. Figures 2(a) and 2(b1)-2(b4) give, respectively, the total electric field intensity distribution for Δφ = 0 and the light-induced magnetization field at disparate Δφ values. As demonstrated in Fig. 2(a), an optical chain-shaped pattern is produced along the axial orientation and increasingly weakened from the geometric focus. Apparently, it can be inferred from Eq. (2) that the optical chain is thoroughly transverse, which is alien to the counterpart generated by tightly focusing radially polarized beams [22, 24 ]. Since this pure transversally polarized optical chain can be treated as an array of either dark focusing spots or bright spots in a recurrent fashion, two types of particles with disparate refractive index (lower or higher than the ambient) can be trapped simultaneously at the various nodal regions [32–34 ].
Fig. 2 (a) Transversally polarized optical chain with single channel for Δφ = 0, and (b1)-(b4) longitudinally polarized magnetic chain with single channel at various Δφ values in the r-z plane
As such, a light-induced magnetization chain with single channel is also realized and stably shifted along the axial direction depending on the values of Δφ, as shown in Fig. 2(b1)-2(b4). For sure, this intriguing magnetization chain is pure longitudinal throughout the entire focal plane. As a result, we can draw a conclusion that a pure longitudinal magnetization chain is induced by the pure transversally polarized optical chain. In principle, this outstanding magnetic pattern is ascribed to the mutual effect between the polarization singularity of the azimuthally polarized beam and the single vortex binary element. Moreover, it is observed from Fig. 2(b) that the extension of this chain distribution reaches up to around 12λ in the axial direction. More importantly, by tuning the values of Δφ, the light-induced magnetization array is gradually transported towards the negative axial direction, as shown in Fig. 2(b1)-2(b4). Namely, the bright magnetization spot can be converted into dark magnetic trap alternatively with the increments of the Δφ, which is propitious to manipulate and transport multiple magnetic particles (e. g. atoms) at several dark focusing region in sequence [20, 21 ]. In addition to this remarkable trait, the pure longitudinal magnetization chain with controllable and flexible displacement is also of potential significance for multilayer magneto-optical storage and the fabrication of an integrated and ultra-compact magnetic device. It should be stressed that, as a demonstration of Δφ = 0 (see Fig. (b1)), the peak of the bright magnetization spot away from the geometric focus accounts for around 55% of maximum of the principle lobe. In spite of moderate strength, however, these magnetization patterns still work regarding some applications mentioned above.
To further offer more insight into the eminent feature of the demonstrated pure longitudinal magnetization chain, let us highlight the magnetization resolution in the lateral direction and the moving behavior of the array in the axial direction. Figure 3(a) depicts the magnetization distribution in the focal region for the case of Δφ = 0. It is found that the FWHM value of the bright magnetic spot in the transverse direction is determined to be 0.518λ, which is slightly smaller than the intrinsic diffraction limit of this objective λ/2NA = 0.526λ. Moreover, the peak of the lateral side lobe accounts for merely 14.3% of the maximum of the main lobe under such a circumstance. What we should pay more attention to is that although the magnetization strength gradually diminishes from the geometric focus, the FWHM values of bright magnetic spot at different locations are nearly identical. On the other hand, we plot the magnetization along the axial direction when the values of Δφ vary from 0 to 1.5π, as revealed in Fig. 3(b). As analyzed above, the magnetization chain shifts periodically and the displacement separation is dependent of the value of Δφ, owing to intricate interference between different parts of the incoming BG beam along the axial direction. To be specific, the displacement distance of the sub-diffraction-limited longitudinal magnetization chain with single channel as a whole is approximately z0 = 0.345λ, when Δφ is ranging from 0 to 0.5π. Ideally, the phase difference Δφ is able to change continually by use of phase-only spatial light modulator [35], allowing for transport of a desired number of atoms in a controllable and prescribed fashion. Considering another aspect of application, such a light-induced longitudinal magnetization chain with sub-diffraction-limited scale is also beneficial to multilayer high density magneto-optical recording and storage. Essentially, different from the studies in [20, 21 ], the desirable magnetization chain associated with the vector light field not only possesses pure longitudinal polarization, but also offers sub-diffraction-limited transverse confinement
Fig. 3 Magnetization distribution (a) in the focal region at Δφ = 0 and (b) along the optical axis at various Δφ values
3.2 Sub-wavelength longitudinal magnetization chain with dual channels
Now that we have illustrated its capability to trap and transport multiple atoms for the sub-diffraction-limited longitudinal magnetization chain with single channel, we demonstrate next how a novel sub-wavelength longitudinal magnetization chain with dual channels behaves. For this purpose, we focus our discussion on the results from tightly focusing of the azimuthally polarized BG beam modulated by a so-called cascaded vortex binary filter, which is composed of a filter with the obstructed middle ring and cascaded spiral phase plates. Different from the transmission function of the single vortex binary filter, Eq. (1) and Eq. (4) in the case of the cascaded vortex binary filter can be, in turn, rewritten as
Analogously, optimized parameters from numerical simulation is found to be R 1 = 0.267R 0 and R 2 = 0.933R 0, respectively. Combining Eq. (2), Eq. (6) with Eq. (7), the beam-shaping nature of this cascaded element is studied comprehensively. In Fig. 4(a) , we plot the intensity distribution on the y-z cross-section for the case of Δφ = 0. It is observed that a dark hole with a long depth of focus (12λ) is achieved near the focus, resulting from the mutual effect between polarization singularities of the azimuthally polarized beam and cascaded optical vortices. From closer inspection of this well-defined dark channel, we find that each unit cell is regarded as two symmetrical hollow regions on the optical axis sandwiching a bottle-shaped profile, which is in analogous to a bottle-hollow beam [36]. It is important to note that there are two fundamental discrepancies comparing our results with that shown in [36]. On the one hand, the bottle-hollow beams, reported here, are pure transverse (radial + azimuthal) while the counterparts reported by Ye et al are either radial or azimuthal. On the other hand, the generated bottle-hollow beams possess a periodic chain-shaped pattern. Besides, one can find from Fig. 4(a) that two periodic optical arrays with transverse polarization locate on two opposite sides of optical axis ( ± 0.95λ). In contrast to the field pattern in Fig. 2(a), however, both of the demonstrated optical chains look seemingly unapparent as a result of awful cleanness of the dark spots [37].
Fig. 4 (a) Transversally polarized optical chain with dual channels for Δφ = 0, and (b1)-(b4) longitudinally polarized magnetic chain with dual channels at various Δφ values in the r-z plane
Figures 4(b1)-4(b4) depict, respectively, the magnetization distributions for the case of disparate Δφ values based on the IFE. As such, a periodic bottle-hollow magnetization channel is presented in the vicinity of the focus. This pure longitudinal bottle-hollow magnetization channel, in fact, not only allows for atoms confinement in adjustable containers but also delivers a larger number of magnetic particles to predefined location with excellent diffraction confinement. More interestingly, we garner two identical longitudinal magnetization chains situated the locations of 0.95λ from the optical axis. Apparently, the lengths of both magnetization arrays are also extended to 12λ along the optical axis. Notably, in this light-induced magnetization array with double channels, each bright magnetization spot is akin to a crescent distribution while each magnetic trap behaves as a triangular profile. Also, by tuning phase differences between the inner circle and outer ring Δφ, both the bottle-hollow magnetization channel and the longitudinal magnetization chain with double channels migrate synchronously along the optical axis in a designated way. Thus well-formed magnetization pattern is propitious to steer and transport a great number of magnetic particles situated diverse lateral locations, as well as to store multiserial and multilayer magneto-optical data with 1.9λ apart from each other. From an experimental perspective, the possibility of achieving such magnetic chain-shaped structures using the cylindrical vector beams is particularly fascinating, largely due to the versatile patterning capability of the state-of-the-art spatial light modulator. In principle it is possible to further achieve pure longitudinal magnetization chain with more channels by applying more spiral phase plates, for instance, the triple vortex binary filter instead of the cascaded one. Indeed, this light–induced pure longitudinal magnetization chain with more channels is phenomenologically analogous to a periodic lattice of magnetic traps fabricated by patterning a multilayered magnetic film [38, 39 ], which provides a potential guidance to fabricate magnetic lattices for spin wave operation in addition to multiple atoms steering and delivery along with multilayer magneto-optical data recording and storage.
To illustrate the focusing behavior more clearly, the magnetization profile in the focal plane for the case of Δφ = π is depicted in Fig. 5(a) , from which we can find that two symmetry bright magnetization spots situate on two opposite sides of optical axis. Under such a circumstance, the FWHM values of both the bright magnetic spots in the lateral direction are calculated to be 0.416λ, and the largest side lobe with a peak value around 22% of the principal lobe peak value accompanies. Actually, such a sub-wavelength longitudinal magnetization spot is in favor of high density AOMR. It should be pointed out that when we choose Δφ = 0, four magnetization spots with comparatively dullish strength appear at the focal region, as shown in Fig. 4(b1). Also, Fig. 5(b) gives the magnetization distribution along r = 0.95λ axis at various Δφ values. Similar to the performances in Fig. 2(b), the displacement separation of both the sub-wavelength longitudinal magnetization chain with dual channels and the bottle-hollow magnetization channel are around z0 = 0.345λ, when Δφ is ranging from 0 to 0.5π. This indicates that the movement of light-induced magnetization array is independent of both spiral phase plate and the radii of each ring, resulting from the phase difference between inner circle and exterior ring. For the purpose of quantification, we further give the dependence of the displacement distance (z 0) of the longitudinal magnetization chain with single/double channels on Δφ in the range of from 0 to 2π, as revealed in Fig. 6 . Obviously, the z 0 increases linearly with the increase of Δφ. Namely, the values of z 0 can be adjusted continuously as required. In particular, the displacement distance z 0 reaches up to nearly a wavelength scale. Therefore, the use of binary optics is of significant versatility and flexibility to steer the light-induced sub-wavelength longitudinal magnetization chain in contrast to the magnetic conveyer belt depending on the applied magnetic field [20, 21 ].
Fig. 5 Magnetization distribution (a) in the focal region at Δφ = π and (b) along r = 0.95λ axis at various Δφ values
4. Conclusion
In summary, the achievement of light-induced longitudinal magnetization chains via IFE by tightly focusing an azimuthally polarized BG beam with middle ring blocked vortex binary filters is exhibited. By tuning the radii of each ring of the single vortex binary filter, a pure transversally polarized optical chain and a sub-diffraction-limited (0.518λ) pure longitudinal magnetization chain (12λ) with single channel can be generated simultaneously in the focal region. In order to extend the flexible functionalities of the light-induced magnetization profiles, a novel cascaded vortex binary filter is proposed to realize a periodic bottle-hollow magnetization channel and a sub-wavelength (0.416λ) longitudinal magnetization chain (12λ) with double channels. As such, we conceive the possibility of creating pure longitudinal magnetization chain with more channels with more spiral phase plates. Essentially, such intriguing magnetization patterns are attributed to the mutual effect between the polarization singularities of the azimuthally polarized beam and single/cascaded spiral optical elements Moreover, it is demonstrated that, by varying the phase difference between inner circle and exterior ring of the single/double vortex binary filters, the sub-wavelength longitudinal magnetization chain either single channel or double channels can be harnessed in a prescribed fashion. The present findings in this paper offer a promising guidance to steer and transport multiple magnetic particles, store multilayer magneto-optical data, develop ultra-compact optomagnetic devices, as well as fabricate magnetic lattices for spin wave operation.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (11474077, 91227113, 11404283 and 11374079), National Natural Science Foundation of Guangdong, China (2014A030307028), and the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (2013LYM_0053).
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