Generation of sub-wavelength longitudinal magnetization needle and multiple longitudinal spots using circularly polarized beam through an annular walsh function filter

Abstract

This numerical work demonstrates the possibility of generating sub wavelength scale magnetization spot of long focal depth and axially separated multiple magnetization segments using annular Walsh function filter. Numerical result calculated on the basis of vector diffraction theory (VDT) and Inverse Faraday Effect (IFE)  shows upon suitably choosing the parameters of Walsh function filter such as order and annular obstruction, one can generate and tune the longitudinal magnetization needle and multiple magnetization spots in the focal plane for the input circularly polarized Bessel Gaussian beam. The generated sub wavelength magnetization structures are highly applicable in fields such as all-optical magnetic recording, multiple trapping and transportation of magnetic particles, magnetic resonance microscopy and multilayer ultra-high-density magnetic storage.

Introduction

Generation of sub-wavelength longitudinal magnetic probe as well as multiple magnetic spots in magneto-optical material (OM) films are at great interest due to its novel applications in all-optical magnetic recording (AOMR), high density magneto optical data storage, magnetic resonance nanoscopy [1, 2–3]. In recent years, magnetization dynamics as well as switching in the magnetic material by ultra-fast laser pulse has been demonstrated by many researchers [4, 5, 6, 7–8]. In 2007, stanciu et al., first experimentally demonstrate AOMR without any applied magnetic field [1]. In 2008, L.E. Helseth numerically investigate the impact of strongly focused electromagnetic wave in E x E* media and derived vectorial magnetization distribution induced by the inverse Faraday effect (IFE)[9]. In 2008,Y.Zhang et al., numerically demonstrated the formation of sub wavelength scale magnetization domain in an OM film owing to the IFE upon tight focusing of circularly polarized beam [10].Followed by this several articles demonstrating the formation of pure longitudinal magnetization spots with ultra long focal depth using both azimuthally and circularly polarized beams are reported recently [1, 10, 11, 12, 13, 14, 15, 16, 17, 18–19]. Apart from this several recent articles more number of articles are reported to generation of the sub-wavelength longitudinal magnetization chain of spots by using different types of pupil engineering [20, 21, 22, 23, 24, 25, 26, 27–28]. For example, 4π configuration is used to generates sub-wavelength longitudinal magnetization needle/chain of spots by using circularly as well as azimuthally polarized beam as an input beam [20, 26, 27–28]. However, 4π configuration are highly affected by aberrations and coherence mismatching due to misalignment of two objective lenses [29–30]. On the other hand, M. Udhayakumar et al., proposed a simple method to demonstrate the longitudinal magnetization needle/chain of spots by using complex phase filter in the input pupil with single objective lens [3, 4–5, 23–24]. In recent years, Walsh function filters are used in the input pupil to manipulate the point spread function(PSF).In 1923, J.L. Walsh defined as the Walsh function is a closed set of normal orthogonal functions over the interval (0, 1) except at a finite number of zero crossings within the specified domain[31]. In 1976, L.N. Hazra et al., numerically analyzed the application of Walsh function in generation of optimum apodizers [32]. In 1977, M. De et al., rectify the problems of optical imagery through Walsh filter [33–34]. In 2007, L.N. Hazra et al., used Walsh filter as a pupil filter to tailoring the resolution of the microscopic imaging [35]. In 2013, P. Mukherjee et al., briefly analyzed the focusing properties of annular Walsh filter [36]. The same group also analyzed the self-similar properties of Walsh function filters [37, 38–39]. In 2018, F. Machado et al., demonstrated the Multiple-plane imaging by using Walsh zone plates [40]. Recently, we theoretically generates ultra-long multiple optical focal hole structures using the annular Walsh function filter as an pupil plane filter [41]. Based on IFE and Richard and Wolf’s VDT, here we numerically analyzed the effect of different parameters of Walsh function filters, pupil to beam ratio of input circularly polarized beam and the NA of the objective for the for the input circularly polarized beam tightly focused with high NA objective. On the basis of numerical results we optimized the focusing (pupil to beam ratio, NA of lens, annular obstruction of Walsh function filtes) to obtain usable magnetization structures. Here, we properly modulate the parameters of annular Walsh filter such as order and annular obstruction, to generate different size of longitudinal optical needle and multiple longitudinal magnetization spots in the focal plane.

Theory:

Based on the VDT, the electric field for right circularly polarized beam in cylindrical coordinate system is expressed as [10, 42–43]

1

$$E_{\mathit{circ}}^R(r,\varphi ,z)=\left[\begin{array}{c}-(I_0+I_2{e}^{i2\varphi })\\ -i(I_0-I_2{e}^{i2\varphi })\\ 2i{I}_1{e}^{i\varphi }\\ \end{array}\right]$$ where$$\begin{array}{c}{I}_0=P(\theta )(1+cos\theta )\sqrt{cos\theta }sin\theta J_0(krsin\theta )exp(ikzcos\theta )d\theta \\ I_1=P(\theta )\sqrt{cos\theta }{sin}^2\theta J_1(krsin\theta )exp(ikzcos\theta )d\theta \\ I_2=P(\theta )(1-cos\theta )\sqrt{cos\theta }sin\theta J_2(krsin\theta )exp(ikzcos\theta )d\theta \\ \end{array}$$

Here, k is wave number given by 2π/λ, α = arcsinNA is the maximum convergence semi angle of the objective lens, J_n—nth Bessel function of the first kind, ϕ and θ – denotes the azimuthal angle tangential angle, P(θ)- pupil apodization function of the incident beam.

Figure 1 shows the schematic representation of the proposed model. Here the OM film in the focal plane are perpendicular to the z-axis. For simplicity, we treated as the free electrons in the OM film as collisionless plasma and beam’s fluctuating magnetic field is neglected. Therefore, the induced magnetization field M by IFE inside the OM is proportional to iE × E.^*, which can be given by[10]$$M(r,\varphi ,z)=i\gamma E\times E^{\ast }$$

2

$$M_n^R(r,\varphi ,0)\equiv \frac{M^R(r,\varphi ,0)}{\gamma }=\left[\begin{array}{c}-4I_1(I_0+I_2)sin\varphi \\ 4I_1(I_0+I_2)cos\varphi \\ 2(I_0^2-I_2^2\\ \end{array}\right]$$where, γ is an optic-magneto constant. “*” –denotes the complex conjugate of the electric field.

[See PDF for image]

Fig. 1

The schematic diagrams for generation of magnetization multiple spots inside the OM film. a-d represents the circularly polarized beam, annular Walsh function filter, high NA lens, magneto optical film, respectively

Here P(θ) is the pupil apodization function and for the incident Bessel Gaussian beam it is expressed as [43]

3

$$P(\theta )=exp\left[-{\beta }^2{\left(\frac{sin(\theta )}{sin\alpha }\right)}^2\right]J_1\left(2\beta \frac{sin(\theta )}{sin\alpha }\right)\times T(\theta )$$β is the pupil to beam ratio and J₁(x) is the Bessel function of first order. T(θ)- denotes the phase transmission function of an annular Walsh filter which is given by

4

$$T(\theta )=\left\{\begin{array}{c}\begin{array}{cc}0,& 0\le \theta <\epsilon \end{array}\\ \begin{array}{cc}{\psi }_k^{\epsilon }(\theta ),& \epsilon \le \theta <\alpha \end{array}\\ \end{array}\right)$$

Here, ε-is the value of the central annular obstruction and k-is the order of the annular Walsh function filter.

The annular Walsh function can given as [36]

5

$${\psi }_k^{\epsilon }(\theta )=\prod _{s=0}^{v-1}\text{sgn}\left\{cos\left[K_s{2}^s\pi \frac{\left({\theta }^2-{\epsilon }^2\right)}{\left(1-{\epsilon }^2\right)}\right]\right\}$$

K_s are the bits,0 or 1 of the binary numerical for k, and (2^v)is the power of 2 that just exceeds k, for all θ in (ε,1).Where

6

$$\text{sgn}(y)=\left\{\begin{array}{c}\begin{array}{cc}+1,& y>0\end{array}\\ \begin{array}{cc}0,& y=0\end{array}\\ \begin{array}{cc}-1,& y<0\end{array}\\ \end{array}\right)$$

The Walsh order k as

7

$$k=\sum _{s=0}^{\upsilon -1}{K}_s{2}^s$$

The zero crossings locations for ${\mathrm{\Psi }}_k^{\epsilon }(\theta )$, n = 0,1,…,(M-1) are given by[34]

8

$${\theta }_i=\sqrt{\frac{[(M-i)]{\epsilon }^2+i}{M}}\times \alpha$$θ₀ = ε and θ_M = 1 are the inner and outer radii of the pre mentioned filter. The set of (M-1) zero crossing locations, θ_i,i = 1,2,…(M-1) consist of all phase transiting locations. ε-is the value of the central annular obstruction.α-is the maximum convergence semi angle of the objective lens.

Figure 2 represent the annular Walsh filter for different annular obstruction with order k = 3. From Fig. 2, black color represents the central annular obstruction, pink and blue colors represents the phase transition value 0 and + π, respectively.

[See PDF for image]

Fig. 2

shows schematic represent of annular Walsh function filter for ε = 0.2,0.3,0.4,0.5, 0.6,0.7, 0.8,0.9 with Walsh order k = 3

Results and discussion

Based on Eq. (2), the magnetization of focal segment is calculated using the parameters λ = 1,n = 1,k = 2π/λ. For all over simulations, the length unit is λ and intensity distribution is normalized to unity. Figure 3ac depicts magnetization distribution in r-z plane with NA = 0.95,k = 2,β = 2 for three different annular obstruction values such as ε = 0.3,0.6.0.9 respectively. Their corresponding axial (in z-axis) and radial (r-axis) distributions are depicted in Fig. 3d-I, respectively. Setting annular obstruction ε as 0.3,is found to generate long depth longitudinal magnetization probe with confined size as demonstrated in Fig. 3a, d, g. From Fig. 3d and g the generated magnetization probe is found to have focal depth around ~ 4.0λ as well as full width half maximum(FWHM) of spot size as ~ 0.39λ. It is observed further increasing the annular obstruction as 0.6,creates magnetization probe with much improved depth and confined size as mentioned Fig. 3b, e, h. When compare Fig. 3e with Fig. 3d, the depth of focus is slightly increased as ~ 0.8λ as shown in Fig. 3e. At the same time, FWHM of spot size decreased as ~ 0.37λ as shown in Fig. 3h. Further increasing the annular obstruction value as high as 0.9,a super long magnetization probe is generated at the focal plane as shown in Fig. 3c-i.The generated magnetization probe reached the DOF as high as ~ 5.8λ with corresponding FWHM still as small as ~ 0.39λ and are shown in Fig. 3f, i respectively. From Fig. 3g-i, the generated total magnetization needle (black color) is observed to be only dominated by M_z component (green color) and not depends on the M_r and M_ϕ components. These type of pure longitudinal magnetization needles are highly appreciable in high density all-optical magnetic recording and high density magnetic storage [1, 3, 44, 45, 46–47].

[See PDF for image]

Fig. 3

a-c shows the 2D of magnetization distribution in r-z plane with fixed parameters NA = 0.95, k = 2, β = 2 for ε = 0.3, 0.6, 0.9, respectively. d-i denotes their corresponding 1D distributions through z-axis (r = 0), r-axis (z = z_max), respectively

In the following case we investigate the order of Walsh function filter with parameters fixed as NA = 0.95, k = 10, β = 1.2 and ε = 0.3, 0.6, 0.9, respectively. Fig. 4a–c shows the focal segments contains three axial magnetization spots in the scale range of − 1.2λ to 1.2λ.We also noted that from Fig. 4d and f), each spot in the focal segment obtains the DOF as 0.6λ and FWHM as 0.57λ. Further increasing ε from 0.3 to 0.6, the magnetization spots in the segment axially elongated and it's DOF improved as 0.8λ where as it's FWHM confined to 0.42λ. Finally increasing the annular obstruction value as high as 0.9, the three focal spots are totally splitting and extends axially up to 8λ range. From Fig. 4f and i, the generated each magnetization needle is found to have DOF as 1.46λ and FWHM as 0.37λ. Here here we conclude that for higher Walsh order (ε > 2), axially splitted magnetization structures are generated at the axial plane.

[See PDF for image]

Fig. 4

a-c shows the 2D magnetization distribution in r-z plane with fixed parameters NA = 0.95, k = 10, β = 1.2 for ε = 0.3, 0.6, 0.9, respectively. d-i denotes their corresponding 1D distributions through z-axis(r = 0), r-axis (z = z_max), respectively

In the following case we studied the magnetization patterns for Walsh order further increased as varying from k = 25 with annular obstruction ε = 0.3, 0.6, 0.9 respectively.Fig. 5a–g shows the focal segments containing eight magnetization spots in the scale range 0f −  3λ to 3λ.The DOF of each spot in the focal segment is noted as 0.65λ from Fig. 5d with it's corresponding FWHM as 0.59λ (Fig. 5g). Further increasing ε as 0.6,is found to generate magnetization segment containing ten axially splitted magnetization spots in the range of − 4.5λ to 4.5λ. We also noted from Fig. 5e and h, the generated magnetization segment bears DOF as 9λ and FWHM as 0.51λ.Tuning the annular obstruction from 0.6 to 0.9, created fully splitted multiple longitudinal magnetization spots as shown in Fig. 5c.The scale range also increasing up to − 10λ to 10λ. From Fig. 5f, the generated focal segment is observed to have DOF as 20λ and the corresponding FWHM is found to reduced as 0.37λ Fig. 5i. These type of sub wavelength scale multiple magnetization spots are useful in multilayer magneto-optical data storage, multiple particle trapping and transportation, magnetic resonance microscopy etc., [2, 45–46].

[See PDF for image]

Fig. 5

a–c shows the 2D magnetization distribution in r-z plane with fixed parameters NA = 0.95, k = 25, β = 1.2 for ε = 0.3, 0.6, 0.9, respectively. d–i denotes their corresponding 1D distributions through z-axis (r = 0), r-axis (z = z_max), respectively

In the next case we numerically analyzed the effect of order of the Walsh filter with k = 5, 15, 25 for ε = 0.9. From Fig. 6a-g, we observed by keeping the Walsh order as 5 generated two axially extended magnetization needle in the range of − 3λ to 3λ. We numerically calculating the DOF of each spot as 2λ with FWHM as 0.40λ as shown in Fig. 6d and g. Increasing the Walsh order from 5 to 15, generated six equal needle of magnetization spots in the range of − 7λ to 7λ as shown in Fig. 6b. Each and every magnetization spots in the focus is found to have DOF as 0.85λ and FWHM as 0.40λ as shown in Fig. 6e and h). Further the Walsh order as high as 25 is observed to generate eight equal sized magnetization spots in the longitudinal direction as shown in Fig. 6c. The axial scale range is also found to increase upto − 10λ to 10λ with DOF measured as 20λ as shown in Fig. 6f. At the same time FWHM of the magnetization spot is found to reduced as 0.37λ. This type of tunable magnetization spots are highly useful in multiple magnetic particle manipulation.

[See PDF for image]

Fig. 6

a-c shows the 2D magnetization distribution in r-z plane with fixed parameters NA = 0.95, ε = 0.9, β = 1.2 for k = 5, 15, 25, respectively. d–i denotes their corresponding 1D distributions through z-axis (r = 0), r-axis (z = z_max), respectively

In the following studing, we investigated the effect of NA on the formation of multiple magnet -ization segment. Here NA of the lens are varied as NA = 0.75, 0.85, 0.95. From Fig. 7a–g, for NA as 0.75, one can generates 10 equal sized magnetization spots in the scale range of − 35λ to 35λ. From Fig. 7d and g, the axial extend of the focal segment is measured as 66λ with FWHM of 0.40λ. The each and every spots in the segment found to have DOF as 6.2λ. Changing the NA value from 0.75 to 0.85, generates 8 equi-sized multiple magnetization spots in the range of − 20λ to 20λ.Each and every spots having DOF as 7λ and FWHM as 0.36λ.To compare Fig. 7g with h, FWHM decrease upto 0.2λ.Increasing the NA as 0.95 can generates 8 equidistant equi-sized multiple magnetization spots in the range of − 10λ to 10λ.Each and every magnetization spots in the scale range having DOF as 2.5λ with FWHM of 0.37λ as shown in Fig. 7f and i. To compare Fig. 7g and i, FWHM of each and every spot reduce upto 0.3λ such a tunable multiple magnetization longitudinal spots are useful in magnetic trapping of different number of particles. It should also be noted that the majority of the proposed methods used either a Digital Micromirror Device (DMD) or a Spatial Light Modulator (SLM) or pure phase filters to accomplish the same thing. Despite the fact that DMD and SLM have the advantage of dynamically programming the incident field to generate and control multiple focal spots on the focal plane of the objective lens, their efficiency has been found to be poor. Using a dedicated Phase mask, on the other hand, is more efficient, but it necessitates high precision fabrication technology and their parameters are fixed [47, 48, 49–50] 0.4pi microscopy is commonly used to create such a chain of magnetization spots. However, it has been reported that aberration has a significant impact on the performance of 4Pi microscopy [20–21, 29–30]. However, the method proposed here is much simpler and easier to implement because the practical realization of an aperiodic annular Walsh function filter is simple with the help of high-efficiency spatial light modulators [51].

[See PDF for image]

Fig. 7

a-c shows the 2D magnetization distribution in r-z plane with fixed parameters k = 25, β = 1.2 for NA = 0.75, 0.85, 0.95, respectively. d-i denotes their corresponding 1D distributions through z-axis(r = 0), r-axis(z = z_max), respectively

Conclusion

On the basis of VDT and IFE, here we numerically analyzed the effect of different parameters of annular Walsh function filters such as Walsh order and annular obstruction. Apart from this the effect of pupil to beam ratio of input beam input circularly polarized Bessel Gaussian beam and NA of the objectives are also analyzed and optimized to achieve novel magnetization segments highly usable for applications such as all-optical magnetic recording, multiple trapping and transportation of magnetic particles, magnetic resonance microscopy and multilayer ultra-high-density magnetic storage.

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