Based on the vector diffraction theory and the inverse Faraday effect, we numerically study the light-induced magnetization near the focus of a 4π high numerical aperture focusing configuration under the illumination of two counter- propagating radially polarized hollow Gaussian vortex beams. The simulated results demonstrate that, by selecting higher-order vortex beam modes (e.g. with – the beam order) and proper truncation parameter (e.g. with – the ratio of the pupil radius to the incident beam waist), spherical and sub-wavelength longitudinal magnetization can be generated in the vicinity of focus. Such special magnetization feature is attributed to not only the interaction between optical vortices and the radially polarized beams, but also the completely destructive interference of azimuthal components and the constructive interference of the longitudinal component of the two counter-propagating radially polarized vortex beams. This spherical and sub-wavelength longitudinal magnetization distribution may be of interest for applications in all-optical magnetic recording and confocal and magnetic resonance microscopy.
© 2015 Optical Society of America
In the past decade, the interaction between optical field and magnetic-optical materials has received considerable attention due to its potential applications in all-optical magnetic recording (AOMR) [1–3], atomic trapping , confocal and magnetic resonance microscopy [5, 6], etc. To further extend those fascinating applications, it is highly desirable to obtain light-induced magnetization with sub-wavelength resolution (magnetic superresolution), which is in analogous to the optical superresolution. Previous reports have shown that the strongly focused circular polarized beam under proper amplitude and/or phase modulation [7, 8] is a versatile approach to achieve magnetic superresolution in the focal plane via inverse Faraday effect (IFE) [2, 3, 9–12]. Two-zone amplitude-only and phase-only filters were designed to improve the lateral magnetization resolution at the expense of accompanying undesirable transverse magnetization . Also, sub-wavelength longitudinally magnetic probe can be generated by the use of high numerical aperture (NA) lens axicon and a pure-phase filter . Furthermore, Helseth demonstrated that arbitrary small magnetization distributions near the focus can be achieved by focusing evanescent vector waves . Despite extensive effects, however, most of the previous reports were mainly concentrated on sub-wavelength magnetization in the focal plane, where the pure longitudinal sub-wavelength magnetization remains largely unexplored. Moreover, to the best of our knowledge, the super-resolution magnetization along the optical axis has not been reported.
Recently, Jiang et al demonstrated the possibility of generating sub-wavelength pure longitudinal magnetization throughout the entire focal plane using a high NA objective lens with an azimuthally polarized vortex beam . Later, the same group showed that an ultra-long pure longitudinal magnetization needle can be generated by tightly focusing an azimuthally polarized beam through an annular vortex binary filter . Under these circumstances, the mutual effect between polarization singularity (azimuthal polarization) and optical vortices can give rise to pure longitudinal magnetization through the IFE. Moreover, special attention in  should be noted that a flat-top magnetization with a large portion of the transverse magnetic field is generated under the illumination of a radially polarized vortex beam. Also, it is demonstrated that in single-lens high NA focusing geometries a radially polarized beam is confined to induce pure azimuthal magnetization . It would be highly interesting to develop an alternative way to generate the pure longitudinal super-resolution magnetization. It is known that, in a 4π high NA focusing system, the radially polarized beams under proper choice of the incoming field or special amplitude modulation can yield a spherical optical focal spot at sub-wavelength scale [17–21]. Thus, a question arises: is it possible to remove the unfavorable azimuthal magnetization, as well as achieve spherical and sub-wavelength longitudinal magnetization near the focus of a 4π high NA focusing configuration when a radially polarized beam is superposed with an optical vortex? Our answer is “yes.”
In this paper, we propose a novel method to generate a spherical and sub-wavelength magnetic spot (0.43λ) near the focus with pure longitudinal polarization using a 4π high NA focusing system and two counter-propagating radially polarized hollow Gaussian (HG) vortex beams. Here, the HG beams, which have been extensively researched from both theoretical [22–24] and experimental [25–27] perspectives, with radial polarization are selected to interact with the optical vortex, which is propitious to control the aspect ratio of the longitudinal magnetization. Moreover, the creation of the desired magnetic pattern is generated by a combination of the perfectly destructive interference of azimuthal components and the constructive interference of the longitudinal component created by the two counter-propagating radially polarized vortex beams. This paper is organized as follows: we describe in section 2 the vector diffraction formulas and IEF in a 4π high NA focusing configuration. We present in section 3 the numerical studies of the magnetization distributions near the focal point for the radially polarized HG vortex beams in a single-objective and a 4π high NA focusing systems, respectively. This leads to spherical and super-resolution magnetization near the focus with pure longitudinal polarization. Also, the mechanisms for the well-defined magnetization distribution are elucidated in several aspects. Finally, we conclude our work in section 4.
2. Theoretical analysis
Figure 1 sketches the geometry of a typical 4π focusing configuration consisting of two high NA objectives illuminated by two counter-propagating radially polarized beams. Two spiral phase plates (SPPs) are then located in front of the high NA objectives. The SPP is a kind of phase encoding element that delays the phase of the incident beam from 0 to 2π with the ray azimuthal position at the cross section of the beam. Mathematically, the phase transmittance of the SPP with topological charge 1 is described as . The instantaneous electric field vectors are characterized by the green arrows. It should be noted that a magneto-optic film with IFE effect is placed at the confocal plane of the two objectives where the magneto-optic (MO) film is perpendicular to the optical axis.
To gain the light-induced magnetization of the tightly focused radially polarized vortex beams under the circumstance of the 4π focusing system, we need to calculate the corresponding electric field distributions of the single objective. Based on vector diffraction theory [28–33], the electric field vector of the radially polarized vortex beam near focus can be expressed as22–27]Eq. (3) reduces to the equation for a fundamental Gaussian beam. From Eq. (3), it is obvious that both β and n play a key role in determining the field distributions of the HG beams.
Therefore, the electric fields near the focus of the 4π focusing system can be written asFig. 1), and the negative sign of z in E2 indicates their counter-propagation. We consider that the magnetic-optic film in the vicinity of the focus is an isotropic and non-magnetically ordered material. Magnetization of the film induced by the IFE can be given byEqs. (2)-(4) into Eq. (5), the induced magnetization field by the tightly focused radially polarized HG vortex beams is obtained, which is, normalized to :Eq. (6), one can readily see that the light-induced magnetization is a complex three-dimensional distribution. This situation stems from the appearance of the radial component in Eq. (1) that is generated in the presence of SPP [34, 35]. Here, the magnitude of total magnetization is calculated as . According to Eqs. (1)-(6), the induced magnetization of the radially polarized HG vortex beams with various orders can be explored theoretically in both the cases of single-lens and the 4π high NA focusing system, respectively.
3. Magnetization field distribution of the radially polarized HG vortex beams
3.1 Light-induced magnetization for a single-lens high NA focusing system
For the purpose of comparison, the magnetization distributions in the single-lens high NA focusing system are calculated. In the calculation, we set , , and . Based on Eqs. (1)-(6), the normalized magnetization distributions of the radially polarized HG vortex beams with the orders of are demonstrated in Fig. 2. As shown in Fig. 2, one can readily see that the total magnetization near the focus is mainly dominated by longitudinal and azimuthal components. It should be emphasized that here the radial magnetization distribution is not presented since the nonzero value only appears in the out-of-focus region [14, 36]. The azimuthal component has always hollow shape with null magnetization on the optical axis because of in Eqs. (1) and (6). However, the longitudinal component has a magnetization maximum on the optical axis because the in Eqs. (1) and (6) shows its maximum at . Apparently, the peak amplitude of total magnetization on the focal plane is only determined by the longitudinal component. On the contrary, the off-focus magnetization mainly depends on the azimuthal and radial components. As a result, a perfect magnetic spot could not be generated provided that the total magnetization is not mainly contributed by the longitudinal component, as demonstrated in Figs. 2(b1) and 2(c1). Otherwise, one can obtain bright magnetization at focal plane (see Fig. 2(a1)).
Figures 3(a)-3(c) reveal the magnetization distributions on the focal plane for respectively, from which it is found that perfect magnetization is only achieved in Fig. 3(a). Under the circumstances of and , objective with fails to tightly focus the radially polarized HG vortex beams in the focal plane. It is observed that the interaction between beam vortices and the radially polarized beams with different orders can effectively vary the resultant magnetization field distributions. As shown in Figs. 3(a)-3(c), the larger the order is, the larger the azimuthal component is, as well as the worse the magnetization is. More specifically, in the case of the Gaussian beam (), a flat-topped magnetic focal spot with a prevailing longitudinal component in the focal plane is generated in Figs. 3(a) and Fig. 2(a), as well as accompanied by a high ratio of doughnut-shaped azimuthal magnetization (60%). Meanwhile, we can attain the lateral and axial FWHM of the magnetic focal spot is approximately 0.7λ and 1.46λ, respectively. On the other hand, when higher beam orders are presented ( and ), the total magnetic focal spot will split at the focal plane, which is shown as black curves in Figs. 3(b) and 3(c) or contour plots Figs. 2(b1) and 2(c1). Namely, the magnetization of the focal point is no longer the peak magnetization along the r axis. This behavior depends largely on the larger azimuthal magnetization component (80%). Such a tremendous azimuthal magnetization will be bad for high density AOMR. Therefore, it is helpful for practical applications provided that the azimuthal magnetization component can be dramatically suppressed or eliminated. In the following, an alternative avenue, using a 4π high NA focusing system, is utilized to deal with it.
3.2 Light-induced magnetization for a 4π high NA focusing system
In view of the discussion above, our main objective in this subsection is to remove the azimuthal magnetization of the radially polarized HG vortex beams by the use of a 4π high NA focusing system. Also, the reduction in magnetic spot size in the axial orientation can be achieved in this fashion. Therefore, the possibility of creating three-dimensional and super-resolution longitudinal magnetization in vicinity of the focus may be realized.
Toward this goal, we demonstrate the magnetization distributions in the 4π high NA focusing configuration with an increasing order of the radially polarized HG vortex beams. Figures 4(a1)-4(c1), 4(a2)-4(c2), and 4(a3)-4(c3) show the contour profiles of the total magnetization, the longitudinal component, and the azimuthal component of the radially polarized HG vortex beams with beam orders of , respectively. It is observed from Fig. 4(a1)-4(c1) that excellent magnetization can be attained regardless of any beam orders. Particularly, compared with the magnetization patterns of the single-lens focusing system shown in Fig. 2(a3)-2(c3), the azimuthal magnetization components disappear completely in the focal plane as exhibited in Figs. 4(a3)-4(c3), which is beneficial to reduce the magnetization spot size. Therefore, the induced total magnetization near focus is primarily determined by the corresponding longitudinal component as demonstrated in Figs. 4(a2)-4(c2). More intuitive evidences are illustrated in Figs. 5(a)-5(c), which depict the magnetization profiles in the focal plane and on the optical axis for the zero-order and higher-order radially polarized HG vortex beams. It can be seen that the total magnetization (black curve) is generally in good agreement with the longitudinal component (blue curve). This fascinating feature, the disappearance of the azimuthal component, in the 4π focusing configuration stems from the opposite-phase coherent superposition of two counter-propagating radially polarized vortex beams as shown in Fig. 1.
From Figs. 5(a)-5(c) we can calculate the FWHM values of the magnetization spot in the transverse direction are 0.565λ, 0.468λ, and 0.41λ for vortex HG beams with radial polarization and respectively. Analogously, the FWHM values of the magnetization focal spot in the axial direction are calculated to be 0.324λ, 0.378λ, and 0.488λ for respectively. It is obvious that, for zero-order beam mode, the FWHM along optical axis is much smaller than the diffraction limit of the objective , whereas the FWHM along transverse direction is larger than the corresponding the diffraction limit. Therefore, the three-dimensional super-resolution in this case can’t be achieved. However this goal can be reached for the higher-order beam modes. In contrast to the results in the single-lens focusing system, not only the zero-order beam but also the higher-order radially polarized vortex beams realize better magnetization performance. As an illustration, the magnetization spot size in the transverse orientation for zero-order beam is 19.3% smaller than that obtained in single-lens focusing system, as well as in the axial orientation provides a rapidly reduction by a factor 4. Physically, the reduced FWHM values of the magnetization corresponding to the generated outcomes in the single objective-lens focusing system result from the following two aspects. On the one hand, the azimuthal component of the magnetization originating from arbitrary directions will bring about perfectly destructive interference caused by π phase difference between the two counter-propagating incident radially polarized HG vortex beams. Obviously, null azimuthal component is propitious to contract the magnetization spot size. On the other hand, the longitudinal component of the magnetization will cause completely constructive interference in the vicinity of the confocal point along the optical axis, which is responsible for limiting the elongation of the magnetization spot along the optical axis. Moreover, it should be noted that as the beam orders increase, the FWHM values of the magnetization focal spot along the transverse axis decrease while the corresponding FWHM values in the optical axis increase. The appearance of this variation is due to the mutual effect between beam vortices and the radially polarized HG fields with different beam orders. This is of great significance to achieve spherical and super-resolution longitudinal magnetization near focus.
To gain insight on the magnetization field polarization, we plot in Fig. 6 the distributions of the average magnetization field for the single high NA lens focusing system and the 4π high NA focusing system, respectively. The transverse magnetization component is given by and the longitudinal magnetization component is determined by . The orientation of the polarization vector (blue arrows) is selected along the optical axis. As shown in Fig. 6(a), it can be seen that no matter near focus or in the periphery of the beam incoming radiation there is a prominent contribution of transversal magnetization components in the single-lens focusing system. Such three-dimensional polarization of the total magnetization field near focus can be understood as a considerable proportion for the azimuthal magnetization (see red curves in Figs. 3(a)-3(c)). As contrast, one can see from Fig. 6(b) that in the periphery of the beam incoming radiation transverse polarization is dominant, nevertheless in the both near-focus and near-axis regions, the magnetization filed has basically homogenous longitudinal polarization (see parallelogram region with red curve in Fig. 6(b)). Upon careful research we find this region of longitudinal polarization is practically coincided with that of the magnetization focal spot (see Fig. 4(c1)). As a result, in contrast to the usual single-lens focusing situation in which the azimuthal magnetization component accounts for an appreciable ratio, the objective with 4π focusing can generate sub-wavelength longitudinal magnetization. Such super-resolution net longitudinal magnetization in the vicinity of focus is a necessity for high capacity magnetic storage devices and all-optical magnetic recording. What’s more, it is noticeable that, compared with that in [14, 15], the pure longitudinal polarization only takes place near the focal plane and near the optical axis. In fact, it is enough completely since the magnetization in the periphery is not necessarily useful for high density AOMR.
As we all know, in confocal microscopy it is often desired to achieve high axial as well as transverse resolution [17, 18]. As such, for the confocal and magnetic resonance microscopy [5, 6] that scans the sub-wavelength magnetic cells in the three dimensions, the optimal goal is to obtain a magnetic focal spot with spherical symmetry, the smallest possible magnetic spot size, and minimal side-lobe levels. To this end, a spherical and sub-wavelength longitudinal magnetic focal spot can be formed by tuning the value of the truncation parameter β properly. In order to determine the β value, we plot the lateral/longitudinal FWHM as a function of the truncation parameter β with beam orders of 0, 1, and 4, as shown in Figs. 7(a) and 7(b). It is observed that the lateral FWHM value increases whereas the longitudinal FWHM value decreases as the β increases regardless of any beam order. Moreover, for an invariant value of the β, the smaller the beam order is, the smaller the FWHM along axial orientation is, whereas the larger the FWHM along transverse orientation is. More specifically, for the beam orders and , it is unambiguous from Figs. 7(a) and 7(b) that the FWHM along transverse orientation is always larger than that along axial orientation at any β value. Explicitly, spherical and sub-wavelength longitudinal magnetization distributions are unavailable when (the corresponding magnetization distributions of and are not shown here). Conversely, both FWHM values are equal near for the order. Therefore, under such a circumstance, a spherical and super-resolution longitudinal magnetic spot with FWHM size of around 0.43λ is achieved, as seen in Figs. 8(a) and (b). Such magnetization spot size reduction in the transversal as well as the axial direction is attributed to not only the mutual interaction between beam vortices and the radially polarized fields with various beam orders, but also the perfectly destructive interference of the azimuthal components and the constructive interference of the longitudinal component formed by the two counter-propagating incident beams. In addition to the central magnetic focal spot, side lobes appear as well. Revealed as black curve in Figs. 8(b), the peak of the transverse side lobe is as smaller as 19% of the maximum of the principal lobe. However, it is noteworthy that the side lobe of the magnetic spot along the z axis reaches up to 50%, which will degenerate the quality of nearly pure longitudinal magnetization field and lead to a great background noise . Such large side lobe is also not preferable to some applications such as high density AOMR and the confocal and magnetic resonance microscopy. Hence it is of great necessity to reduce the axial side lobe of the magnetization volume. Here, as described in , the amplitude of the side lobe can be further reduced if the NA of the objective is increased. Moreover, the inversion of the longitudinal magnetization can be easily realized by converting the helical direction of the optics vortices , which is outside the scope of the current study.
To summarize, based on the vector diffraction theory and the inverse Faraday effect, we have theoretically investigated the light-induced magnetization by focusing radially polarized HG vortex beams through a 4π high NA aperture focusing configuration. The focal magnetization distribution is calculated in detail for the radially polarized HG vortex beams with different orders. It is found that, compared with the circumstances of single-lens high NA focusing system, spherical and sub-wavelength (0.43λ) magnetization with longitudinal polarization near the focus can be generated by selecting appropriate parameters. Such excellent magnetization performance is attributed to not only the mutual effect between the radially polarized beams with various orders and optical vortices, but also the destructive interference of the azimuthal components as well as the constructive interference of the longitudinal component created by the two counter-propagating radially polarized cortex beams. Moreover, the longitudinal polarization characteristic of the calculated magnetization can be verified by calculating the averaged light-induced magnetization field. The results in this paper are of vital value in high density AOMR and the confocal and magnetic resonance microscopy.
This work was supported by the National Natural Science Foundation of China (11474077, 91227113 and 11374079), and the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (2013LYM_0053).
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