Theoretical guideline for generation of an ultralong magnetization needle and a super-long conveyed spherical magnetization chain

Weichao Yan; Zhongquan Nie; Xueru Zhang; Yuxiao Wang; Yinglin Song

doi:10.1364/OE.25.022268

1. Introduction

Over the last two decades, cylindrical vector beams are of increasing interest for optical trapping [1–3], laser machining [4], optical data storage [5], and optical imaging [6]. Specially, tightly focusing properties of cylindrical vector beams have received intensive researches [7–11]. To achieve a super-resolution spot, a lot of methods are used, such as amplitude modulation [12,13], phase modulation [14] and absorbance modulation [15]. Besides, there are also many ways to produce a super-long longitudinally polarized optical needle with lateral sub-wavelength scale and an extra-long longitudinally polarized optical chain with three dimentional (3D) super resolution in a tight focusing system [16–18].

When a magneto-optic (MO) film is inserted in the focal plane of a tight focusing system, light induced magnetization field is generated through the inverse Faraday effect (IFE) [19]. A pure longitudinally polarized magnetization spot is produced by tightly focusing an azimuthally polarized vortex beam instead of a circularly polarized one [20, 21]. The lateral size and the longitudinal length are 0.53λ and 1.29λ, respectively [21]. Further, a longitudinally polarized spherical magnetization spot with 3D super resolution (0.43λ) is generated by tight focusing of radially polarized vortex beams in a 4π tight focusing system [22]. On the other hand, a long (12λ) longitudinally polarized magnetization needle with lateral super resolution (0.43λ) is generated by tightly focusing an azimuthally polarized beam modulated by an annular vortex binary filter [23]. Moreover, through a specially designed filter with both azimuthally and radially modulated annular phase, an ultra-long longitudinal magnetization needle (28λ) with narrower lateral size (0.27λ) can be obtained [24]. Meanwhile, there are a lot of methods to produce a long longitudinally polarized magnetization chain with 3D super resolution. A sub-wavelength (0.416λ) and extra-long (12λ) longitudinally polarized magnetization chain is achieved by tight focusing of an azimuthally polarized Bessel-Gaussian beam modulated by an optimized vortex binary filter [25]. With the purpose of achieving a long spherical magnetization chain, two counter-propagating azimuthally polarized beams through narrow amplitude modulation are focused into the MO film in 4π high NA lenses [26]. To realize both a magnetization needle and a magnetization chain simultaneously in a tight focusing system, an azimuthally polarized vortex multi-Gaussian beam with a controlable main ring is utilized [27]. Except for the magnetization spot, the magnetization needle and the magnetization chain, spherical magnetization arrays with 3D super resolution are also obtained [28]. These spherical 3D super-resolution (λ³/22) spot arrays of pure longitudinally polarized magnetization are not only realized, the location, the number and the distance of the adjacent magnetization spots can be also adjusted freely. So many rich magnetization patterns with super resolution have potential applications in all-optical magnetic recording (AOMR) [29], atomic trapping [30], image encryption [31,32], confocal and magnetic resonance microscopy [33].

Through optimizing radii of two annuli, the long longitudinally polarized magnetization needle with lateral super resolution and the extra-long longitudinally polarized magnetization chain imposed with sub-wavelength scale are achieved [23, 25]. By making use of the specially designed azimuthally and radially modulated annular phase, the ultra-long longitudinally polarized magnetization needle with narrower lateral size is garnered [24]. By optimizing the central radius of the narrow annulus, the long spherical magnetization chain is obtained [26]. The above methods to produce the magnetization needle and the spherical magnetization chain are based on parameter optimization, and the optimization process is complicated and time-consuming. Naturally, a question arises: is it possible to produce them by a simple analytical method? Our answer is yes. Compared with the numerical parameter optimization method, it is very direct to find the properties of the magnetization needle and the magnetization chain. In present paper, we adopt the Gaussian annulus model described in [34], but with the azimuthally polarized vortex beam instead of the radially polarized one. Through a simple derivation, we present analytical formulas of the magnetization needle and the conveyed magnetization chain as the angular width is far less than the central angle of the incident beam. Meanwhile, the validity of these analytical formulas is proved, compared to numerical solutions. Therefore, these theoretical results provide a beneficial guideline to achieve a super-long magnetization needle with sub-wavelength scale and an ultralong spherical magnetization chain with 3D super resolution. The paper is organized as follows. In section 2, schematic diagram for a tight focusing system is described. The analytical formula of the magnetization needle for a single high NA lens and the analytical expression of the conveyed magnetization chain for 4π high NA lenses are deduced. Besides, the validity of these analytical formulas is discussed. In section 3, we conclude our work.

2. Theoretical analysis

2.1. Schematic diagram for a tight focusing system

As illustrated in Fig. 1, two counter-propagating azimuthally polarized beams modulated by two identical spiral phase plates are focused into a MO film in a 4π tight focusing system. Through the IFE induced by the focused electric field, the magnetization field is produced in the MO film. In this system, two spiral phase plates are situated at the back apertures of two high NA objective lenses. The spiral phase plate has a function of delaying phase along the azimuthal direction. In mathematics, the modulation function of the spiral phase plate can be expressed by T (ϕ) = exp(jmϕ), where m is a topological charge. Besides, the MO film is located at the confocal plane of the two objective lenses in which the MO film is perpendicular to the optical axis. It is noted that a single-lens high NA focusing system is naturally formed by removing the right arm.

Fig. 1 Schematic illustration of a 4π tight focusing system. A MO film lies in the confocal plane of the configuration, which is illuminated by two counter-propagating azimuthally polarized vortex beams. AP1 and AP2 denote two counter-propagating azimuthally polarized beams, SPP1 and SPP2 signify two identical spiral phase plates, L1 and L2 respresent two high NA objective lenses.

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2.2. Generation of the magnetization needle for a single-lens high NA focusing system

Based on the vector diffraction theory [35], the focused electric field for a single high NA lens in the cylindrical coordinate system can be written [23]

E = [\begin{matrix} E_{r} \\ E_{φ} \\ E_{z} \end{matrix}] = A j^{m} \int_{0}^{θ_{\max}} T (φ) [\begin{matrix} V_{1} \\ j V_{2} \\ 0 \end{matrix}] \exp (j k z \cos θ) E_{0} (θ) \sqrt{\cos θ} \sin θ d θ,

where

V_{1} = J_{m - 1} (k r \sin θ) + J_{m + 1} (k r \sin θ),

V_{2} = J_{m - 1} (k r \sin θ) - J_{m + 1} (k r \sin θ),

and

T (φ) = \exp (j m φ) .

Here, A is a constant. r, φ, z are the cylindrical coordinates in the focal space. J_m−1 and J_m+1 are m − 1 order and m + 1 order Bessel functions of the first kind, respectively. θ_max is the maximum converging semi-angle, which is referred to the high NA objective lens by θ_max = arcsin(NA). It is highlighted that, we regard the incident light as a narrow annulus beam and adopt a Gaussian annulus model in [34]. E₀(θ) denotes the distribution of the Gaussian annulus

{(\sqrt{π} Δ θ)}^{- 1} \exp [- {(θ - θ_{0})}^{2} / Δ θ^{2}]

, where θ₀ is the central angle and Δθ is the angular width. These integrals may be evaluated numerically. In fact, Eq. (1) can simplify further in a first approximation and each of integrals can be taken by a first term, as given by [34]

E_{r} = A j^{m} T (φ) V_{1} \exp (- z^{2} / z_{0}^{2} + j k z \cos θ_{0}) \sqrt{\cos θ_{0}} \sin θ_{0},

E_{φ} = A j^{m + 1} T (φ) V_{2} \exp (- z^{2} / z_{0}^{2} + j k z \cos θ_{0}) \sqrt{\cos θ_{0}} \sin θ_{0},

where z₀ = 2/(kΔθ sin θ₀). It is specially noted that Eqs. (4a) and (4b) are acquired when Δθ ≪ θ₀. The total intensity of the transverse electric field is obtained by

I_{t} = {| E_{r} |}^{2} + {| E_{φ} |}^{2} = 2 {| A |}^{2} \cos θ_{0} \sin^{2} θ_{0} \exp (- 2 z^{2} / z_{0}^{2}) [J_{m - 1}^{2} (k r \sin θ_{0}) + J_{m + 1}^{2} (k r \sin θ_{0})] .

According to the energy consideration, the magnetization field induced by the IFE in the isotropic magnetically ordered material can be expressed as [21]

M = j γ E \times E^{*},

where E is the focused electric field, E^* stands for its conjugate and γ is the coupling efficiency proportional to the magneto-optical susceptibility χ of the material. Substituting Eqs. (4a) and (4b) into Eq. (6), the light induced magnetization field can be written as

\begin{array}{l} M = j γ (E_{r} E_{φ}^{*} - E_{φ} E_{r}^{*}) e_{z} \\ = 2 γ {| A |}^{2} \cos θ_{0} \sin^{2} θ_{0} \exp (- 2 z^{2} / z_{0}^{2}) [J_{m - 1}^{2} (k r \sin θ_{0}) - J_{m + 1}^{2} (k r \sin θ_{0})] e_{z} . \end{array}

It is easily found from Eqs. (5) and (7) that a longitudinally polarized magnetization needle is induced by a transversely polarized optical needle. Both the transversely polarized optical needle and the longitudinally polarized magnetization needle exhibit an oscillatory behavior with sidelobes along the transverse direction, while they follow a Gaussian distribution in the longitudinal direction.

The lateral sizes of the transverse optical needle and the magnetization needle for m = 1 can be also calculated by Eqs. (5) and (7), respectively. It is no surprise that their full widths at half maximum (FWHMs) are independent of the angular width (Δθ), as reported in [34] for the radially polarized beam. They are only related to the values of θ₀, as shown in Fig. 2. For comparison, the size of the longitudinal optical needle for the radially polarized beam [36, 37], which is calculated by FWHM = 0.36λ/sin θ₀, is also plotted in Fig. 2. As seen from Fig. 2, all FWHMs decrease when θ₀ increases. In other words, the lateral sizes are mainly due to the high-frequency component of the incident light [38]. Specially, FWHM of the magnetization needle is smallest, that of the transverse optical needle is largest and that of the longitudinal optical needle is mediate. As θ₀ approaches 90 degrees, the theoretically limited FWHMs for the transverse optical needle, the longitudinal optical needle and the magnetization needle are about 0.37λ, 0.36λ and 0.35λ, respectively.

Fig. 2 FWHMs vs various central angles for the transverse optical needle, the longitudinal optical needle and the magnetization needle.

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Their longitudinal length can be calculated by the term $\exp (- 2 z^{2} / z_{0}^{2}) = 1 / 2$ in Eqs. (5) and (7), and the depth of focus (DOF) can be given by DOF = z₀ (2ln2)^1/2 = λ(2ln2)^1/2/(πΔθ sin θ₀), as demonstrated in [34]. Therefore, the DOF is inversely proportional to the multiplication of sin θ₀ and Δθ. In terms of physics, the magnetization needle is longer, when the radius of the annulus is smaller and the width of the annulus is narrower, as illustrated in Fig. 3. Moreover, it is Δθ rather than θ₀ that determines the DOF. In the limit of an infinitesimally thin annulus of the incident beam, the longitudinal length becomes infinite.

Fig. 3 DOFs of the magnetization needles vs Δθ for θ₀ = 45 degrees, 60 degrees, 75 degrees.

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The lateral FWHMs of the optical needle and the magnetization needle for |m| ≠ 1 are a little complicated and can not be expressed by a simple equation. For visualization, the optical needle with dual channels and the corresponding light induced magnetization needle are shown in Fig. 4. When m = 2, a pure longitudinal magnetization needle is induced via a pure transverse optical needle with dual channels, as seen from Figs. 4(a) and (b). The magnetization needle with dual channels is the projection of a hollow magnetization cylinder onto the r-z plane. In view of the properties of the term $J_{m - 1}^{2} (k r \sin θ_{0}) - J_{m + 1}^{2} (k r \sin θ_{0})$ in Eq. (7), the magnetization needle with main dual channels are formed, as depicted in Fig. 4(b). Noted that the distance between two channels shows a proportional relation with m, as demonstrated in Fig. 5. Here the “distance” refers to the space between maximum intensity locations of the dual channels. Through this figure, it is easily found that the distance decreases as the value of θ₀ increases and the decreasing speed of the distance increases for a larger m. Besides, the average increasing distance is about 1λ, when m adds by 2. Specially, the direction of the magnetization field is reversed when m is negative.

Fig. 4 (a) and (b), the light needle with dual channels and the corresponding magnetization needle for m = 2. The rest parameters are chosen as θ₀ = 75 degrees and Δθ = 0.01.

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Fig. 5 The distance between dual channels of magnetization needle vs θ₀ for different m.

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In fact, the long magnetization needle in [23] can be owing to only one narrow annulus because of the partial destructive interference between two annuli. According to the above theoretical analysis, the DOF and the FWHM can be estimated as 7.54λ and 0.38λ, respectively. These values perfectly fit the results obtained from [23]. Furthermore, the estimated values of the DOF and the FWHM are also in accord with the results in [27]. The radio of the side lobe for the magnetization needle can be also determined by the term $J_{0}^{2} (k r \sin θ_{0}) - J_{2}^{2} (k r \sin θ_{0})$ in Eq. (7). The calculated ratio is about 20%, which is nearly identical to the value of [23, 27]. However, the longitudinal homogeneity is not well due to the Gaussian envelope of the magnetization field, which is not desired in many applications. This problem can be solved by inserting a designed filter with misplaced helical phases, as demonstrated in [24, 39]. Well longitudinal homogeneity along with a longer DOF are achieved without affecting the lateral super resolution.

The results above are based on the approximate analytical solutions Eqs. (4a) and (4b). In order to validate these analytical solutions, it is essential to compare with the results from the numerical integral solutions Eq. (1). To deduce the analytical formulas, Δθ ≪ θ₀ is necessary. Therefore, the narrow angular width is chosen in the range of [0.01, 0.1]. Meanwhile, the central angle ranges from 45 degrees to 75 degrees due to the tight focusing condition.

In Fig. 6, differences between properties of the magnetization needles with the single channel based on these two groups of solutions are described. The differences between lateral FWHMs are depicted in Fig. 6(a). It is obvious that the value of Δθ has minor effect on the difference between the lateral FWHMs. On the other hand, the difference with an oscillating property becomes larger, when the value of θ₀ increases. In all, the difference of the lateral FWHM is less than 3%, herein it can be neglected. The differences of longitudinal DOFs are illustrated in Fig. 6(b). It is shown that the difference is less than 1% when the value of θ₀ is in the range of [45, 60] degrees or Δθ ranges from 0.01 to 0.04. In other areas, the difference from two kinds of solutions increases with the increasings of θ₀ and Δθ. Neverthless, the maximum value of difference is less than 5%. As a result, the difference is less than 5% for the case of magnetization needle. Therefore, the validity of the analytical solutions is proved by the comparison with our caculations.

Fig. 6 The differences based on the numerical and analytical results. In (a), the differences between the lateral FWHMs of magnetization needles with a single channel. In (b), the differences between the longitudinal DOFs of magnetization needles with a single channel.

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2.3. Generation of the magnetization chain for a 4π high NA focusing system

For a 4π tight focusing system, the electric field in the focal region can be expressed as

E (r, φ, z) = E_{1} (r, φ, z) + \exp (j ϕ_{0}) E_{2} (- r, φ, - z),

where E₁ and E₂ denote the electric fields focused by the left objective lens and the right objective lens, respectively. The negative signs in r and z represent the opposite directions of instantaneous polarization and propagation with respect to E₁. The term exp(jϕ₀) is introduced as an additional phase difference between two arms in the 4π high NA focusing system. Substituting the approximate analytical solutions Eqs. (4a) and (4b) into Eq. (8), the focused electric field for an odd topological charge (m = 2n + 1) can be given by

E_{r} = 2 A j^{m} T (φ) \exp (j ϕ_{0} / 2) \sqrt{\cos θ_{0}} \sin θ_{0} V_{1} \exp (- z^{2} / z_{0}^{2}) \cos (k z \cos θ_{0} + ϕ_{0} / 2),

E_{φ} = 2 A j^{m + 1} T (φ) \exp (j ϕ_{0} / 2) \sqrt{\cos θ_{0}} \sin θ_{0} V_{2} \exp (- z^{2} / z_{0}^{2}) \cos (k z \cos θ_{0} + ϕ_{0} / 2) .

To obtain the above analytical equations, the odevity of the bessel function is fully utilized. Then, the total intensity of the electric field can be written as

\begin{array}{l} I_{t} = 4 {| A |}^{2} \cos θ_{0} \sin^{2} θ_{0} \exp (- 2 z^{2} / z_{0}^{2}) \\ [J_{m - 1}^{2} (k r \sin θ_{0}) + J_{m + 1}^{2} (k r \sin θ_{0})] \cos^{2} (k z \cos θ_{0} + ϕ_{0} / 2) . \end{array}

And substituting Eqs. (9a) and (9b) into Eq. (6), the induced magnetization field for the 4π high NA lenses can be expressed as

\begin{array}{l} M = 8 γ {| A |}^{2} \cos θ_{0} \sin^{2} θ_{0} \exp (- 2 z^{2} / z_{0}^{2}) \\ [J_{m - 1}^{2} (k r \sin θ_{0}) - J_{m + 1}^{2} (k r \sin θ_{0})] \cos^{2} (k z \cos θ_{0} + ϕ_{0} / 2) e_{z} . \end{array}

Meanwhile, the focused electric field for an even topological charge (m = 2n) can be given by

E_{r} = 2 A j^{m + 1} T (φ) \sqrt{\cos θ_{0}} \sin θ_{0} V_{1} \exp (- z^{2} / z_{0}^{2}) \sin (k z \cos θ_{0} + ϕ_{0} / 2),

E_{φ} = 2 A j^{m + 2} T (φ) \sqrt{\cos θ_{0}} \sin θ_{0} V_{2} \exp (- z^{2} / z_{0}^{2}) \sin (k z \cos θ_{0} + ϕ_{0} / 2) .

Thus, the total intensity of the electric field can be written as

\begin{array}{l} I_{t} = 4 {| A |}^{2} \cos θ_{0} \sin^{2} θ_{0} \exp (- 2 z^{2} / z_{0}^{2}) \\ [J_{m - 1}^{2} (k r \sin θ_{0}) + J_{m + 1}^{2} (k r \sin θ_{0})] \sin^{2} (k z \cos θ_{0} + ϕ_{0} / 2) . \end{array}

Similarly, substituting Eqs. (12a) and (12b) into Eq. (6), the induced magnetization field can be obtained

\begin{array}{l} M = 8 γ {| A |}^{2} \cos θ_{0} \sin^{2} θ_{0} \exp (- 2 z^{2} / z_{0}^{2}) \\ [J_{m - 1}^{2} (k r \sin θ_{0}) - J_{m + 1}^{2} (k r \sin θ_{0})] \sin^{2} (k z \cos θ_{0} + ϕ_{0} / 2) e_{z} . \end{array}

As seen from Eqs. (10) and (13), the focused electric fields for the odd-even topological charge are almost identical, except for the sine-cosine function modulation. Considering for induced magnetization fields for the topological charge with an odd-even number, the difference is also the sine-cosine function modulation. Therefore, it is only required to discuss the results for the topological charge with the odd number. It is found from Eqs. (10) and (11) that, a long optical chain and a long corresponding magnetization chain with the Gaussian envelope distribution are produced in the focal region.

The properties in a static state can be investigated, provided that ϕ₀ = 0. When the topological charge is selected to 1, a long magnetization chain with a single channel is formed. The lateral FWHM of a central magnetization spot is the same as the magnetization needle. The axial size of the central magnetization spot can be calculated by the term cos²(kz cos θ₀) = 1/2 in Eq. (11) and the value is π/(2k cos θ₀). The axial FWHM tends to infinity, as θ₀ approaches 90 degrees. Besides, the distance between adjacent magnetization spots can be calculated by cos²(kz cos θ₀) = 0 and the value is π/(k cos θ₀). It is easily found that, when the distance is twice as much as the axial FWHM, the magnetization needle cannot be formed. According to Eq. (11), the direction of magnetization field is reversed as the topological charge is selected to −1. When |m| ≠ 1, the magnetization chain with dual channels situated in opposite locations are formed, just as the case of the magnetization needle with dual channels. When the value of |m| increases, the distance between two channels increases, alike the case of Fig. 5.

For purpose of achieving a spherical magnetization chain, the lateral FWHM and the axial FWHM of the central magnetization spot vs the central angle are investigated. The relationships between these FWHMs and the central angle are plotted in Fig. 7. The lateral FWHM decreases as the central angle increases, whereas the axial FWHM increases. When the angle equals 54.5 degrees, these two FWHMs are identical and the value is 0.43λ. Compared with the results obtained by global optimization [26], this kind of analytical method is very simple, quick and direct.

Fig. 7 Both the lateral FWHM and the axial FWHM of the central magnetization spot vs the central angle.

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If ϕ₀ ≠ 0, the magnetization chain can be conveyed in the −z direction. The displacement can be expressed by dz = −ϕ₀/(2k cos θ₀) coming from kz cos θ₀ + ϕ₀/2 = 0. The displacement vs ϕ₀ and θ₀ is shown in Fig. 8. It is found that, dz increases by increasings ϕ₀ and θ₀. Importantly, the displacement exhibits a linear relationship with ϕ₀.

Fig. 8 Displacement vs ϕ₀ and θ₀.

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Similar to the case of the magnetization needle, differences between properties of magnetization chains with the single channel based on those two groups of solutions are also studied. These results are shown in Fig. 9. The differences between the lateral FWHMs of the central magnetization spots are illustrated in Fig. 9(a). The tendency is similar to the case of the magnetization needle. The differences between the axial FWHMs of the central magnetization spots are displayed in Fig. 9(b). The difference is less than 1%, when the value of θ₀ is in the range of [45, 60] degrees or Δθ ranges from 0.01 to 0.06. In other areas, the difference increases as θ₀ and Δθ increase. The maximum value of difference is less than 3%. As a result, the difference is less than 3% for the case of the magnetization chain. Therefore, the validity of the analytical solutions is proved through the comparison.

Fig. 9 The differences based on the numerical solutions and analytical expressions. In (a), the differences between the lateral FWHMs of the central magnetization spots. In (b), the differences between the axial FWHMs of the central magnetization spots.

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3. Conclusion

In summary, we propose an azimuthally polarized vortex beam with a Gaussian annulus as an incident light. As the angular width is far less than the central angle of the incident beam, the analytical formulas of the magnetization needle and the conveyed magnetization chain are presented. By making use of these analytical formulas, we discuss the properties of the magnetization needle and the magnetization chain. Meanwhile, the difference between analytical formulas and numerical integral solutions is also investigated and the difference is limited to below 5%. Therefore, the validity of these analytical formulas is proved.

The analytical formulas that we deduce are beneficial to the design of the magnetization needle and the spherical magnetization chain. The super-long magnetization needle with sub-wavelength scale and the extra-long magnetization chain with 3D super-resolution can be used to all-optical magnetic recording (AOMR), atomic trapping, confocal and magnetic resonance microscopy.

Funding

National Natural Science Foundation of China (NSFC) (11374079,11474078,11604236).

Acknowledgments

Dr. Weichao Yan thanks Prof. Xueru Zhang, Prof. Yuxiao Wang and Dr. Zhongquan Nie for fruitful discussions. Dr. Weichao Yan thanks Dr. Xingzhi Wu and Dr. Zhongguo Li for language polishing in revised paper. Dr. Weichao Yan thanks Prof. Yinglin Song for sustained attention and vigorous support.

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Theoretical guideline for generation of an ultralong magnetization needle and a super-long conveyed spherical magnetization chain

Abstract

1. Introduction

2. Theoretical analysis

2.1. Schematic diagram for a tight focusing system

2.2. Generation of the magnetization needle for a single-lens high NA focusing system

2.3. Generation of the magnetization chain for a 4π high NA focusing system

3. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (9)

Equations (18)

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