Tailoring the magnetic field induced by the first higher order mode of an optical fiber

Xiaoqiang Zhang; Xiaoqiang Zhang; Guanghao Rui; Yong Xu; Yong Xu; Fan Zhang; Fan Zhang; Yinchang Du; Yinchang Du; Mingtao Lian; Mingtao Lian; Anting Wang; Hai Ming; Weishneg Zhao; Weishneg Zhao

doi:10.1364/OE.382293

1. Introduction

Over the past few years, driven by the fast development of optical magnetic recording [1,2], confocal and magnetic resonance microscopy [3] and spin dynamics [4,5], tailoring the focal field in magnetic materials is of great importance [6–12]. To customize the light-induced magnetic field in the focal region, pupil field distribution needs to be modulated carefully. Recently, longitudinal magnetizations [6,7] and transverse magnetizations have been generated by tailoring the incident light [8–10], and magnetizations with arbitrary three-dimensional orientations are realized as well [11–14]. However, in these studies, the incident pupil field with complex amplitude, phase and polarization variations should be considered accurately, resulting in complicated and the components employed for the system are costly configurations [6,8,12,15 16,]. For example, two counter-propagating beams in a 4π configuration are tightly focused [8–13], or a vectorial optical field generator (VOF-Gen) consisting of two spatial light modulators is used to tailor the pupil field [15,16]. Besides, the most promising applications of tailoring focal field are optical magnetic recording [1,5,17], where laser pulses are used to trigger the magnetization switching. As a result, a 4π configuration has no superiority, for it is almost impossible to control the two counter-propagating pulses in phase at all times. For example, as for a commercial femtosecond laser with a 100 MHz repetition rate, a wavelength of 1.55 μm, and pulse duration of 100 fs, the distance of two adjacent pulses is about 3 m, and the travel distance of one pulse in 100 femtoseconds is only about 3×10⁻⁵ m. So, it is very difficult to control two pulses in phase. In 2013, according to the study by Banzer et al, a purely transverse angular momentum, which is named as a photonic wheel, is generated in the focal plane by tailoring the polarization of an odd Hermite-Gaussian mode [18]. To extend the flexibility of the photonic wheel and generate a magnetic field with arbitrary orientation, in this paper, the amplitude, phase, polarization and field distribution of the first higher order mode of an optical fiber are tailored, and according to the inverse Faraday effect (IFE), a magnetic field with arbitrary orientation is generated in the focal region. In addition, the configuration is simple, and easy to realize.

In a traditional step-index fiber, it is well known that a slightest perturbation to the cylindrical symmetry of the fiber will induce the so-called linearly polarized (LP) beams [19,20]. Thus, in a few-mode optical fiber laser, through turning the polarization controller, the first higher-order LP₁₁ mode, where two lobes with inhomogeneous phase distribution exist, can be gotten, and the electric fields of these two lobes have an opposite instantaneous direction caused by the inhomogeneous phase distribution [21,22]. At present, many methods have been reported to generated a single LP₁₁ mode, e.g. a few-mode fiber Bragg grating [23], a mode selective coupler [24], and a long-period fiber grating [25]. Compared with the 4π configuration to generate a magnetic field with arbitrary orientation, which requires that the instantaneous electric field vector of the two counterpropagating beams have opposite directions, the instantaneous electric field vector of the two lobes of LP₁₁ mode has opposite directions natively [26,27], and it is more suitable for the generation of a magnetic field with arbitrary orientation. To achieve this, in this paper, before focusing the two lobes of a LP₁₁ mode by a high numerical aperture (NA) objective lens, the amplitude, phase, polarization and field distribution of the two lobes are modulated carefully. We believe that such an arbitrary orientation state of magnetic field can be applied in the field of optical magnetic recording, confocal and magnetic resonance microscopy and spin dynamics [28].

2. Methods

Figure 1(a) sketches the geometry to generate an arbitrarily oriented magnetic field, where an incident LP₁₁ beam from a few-mode fiber laser travels through two identical half wave plates, two identical quarter wave plates, and two optical attenuators. As shown in Fig. 1(a), these six optical elements are merged together, and then the laser is focused on a magnetic-optic film (MOF) by a high NA objective lens. The inset in Fig. 1(a) shows the incident LP₁₁ odd mode with y-polarization, where the two lobes of LP₁₁ mode have a π phase difference, the two identical quarter-wave plates, where they are merged with their fast-axes being oriented under 45° and −45° with respect to the bonding axis, and the generated circularly polarized LP₁₁ mode. In the inset, η is defined as the angle between the fast axis orientations of the QWP and HWP. According to the Richards-Wolf diffraction theory, the tightly focused optical fields in the focal region can be calculated as follows [29]:

(1)$${\textbf E}({r,\phi ,z} )={-} \frac{{ikf}}{{2\pi }}\int_0^{{\theta _{\max }}} {\int_0^{2\pi } {{{\textbf E}_\varOmega }({\theta ,\varphi } )} } {e^{ikr\sin \theta \cos ({\varphi - \phi } )+ ikz\cos \theta }}\sqrt {\cos \theta } \sin \theta d\theta d\varphi ,$$

where E_Ω(θ,φ) is the electric field on the spherical surface Ω, and it can be written as [30,31]

(2)$${{\textbf E}_\varOmega }({\theta ,\varphi } )= A({\theta ,\varphi } )\left( {\begin{array}{cc} {{{\cos }^2}\varphi \cos \theta + {{\sin }^2}\varphi }&{\cos \varphi \sin \varphi ({\cos \theta - 1} )}\\ {\cos \varphi \sin \varphi ({\cos \theta - 1} )}&{{\sin^2}\varphi \cos \theta + {{\cos }^2}\varphi }\\ {\sin \theta \cos \varphi }&{\sin \theta \sin \varphi } \end{array}} \right)\left( {\begin{array}{c} {{p_x}}\\ {{p_y}} \end{array}} \right).$$

Fig. 1. (a) Schematic illustration of the setup to generate an arbitrarily oriented magnetic field. HWP: half wave plate; QWP: quarter wave plate; A: attenuators; MOF: magnetic-optic film. η is the angle between the fast axis orientations of the QWP and HWP. (b) The orientation distribution of the induced magnetic field in Q point.

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In Eq. 2, A(θ,φ) is the pupil apodization function of the incident beam, p_x and p_y determine the states of the focused beam, and they can be expressed as

(3)$$\left( {\begin{array}{c} {{p_x}}\\ {{p_y}} \end{array}} \right) = \frac{t}{{\sqrt {1 + {\chi ^2}} }}\left( {\begin{array}{c} 1\\ {\chi {e^{i\delta }}} \end{array}} \right).$$

Here t is the transmittance, and it is determined by the attenuators, χ is the ellipticity of the polarization ellipse, and χ = tan(2η) [33]. δ is phase retardation between these two orthometric components, which is determined by the phase plates, and as for a QWP, δ = ±π/2. We can select different wave plates to alter the phase retardation [32]. It should be noted that the two attenuators in Fig. 1(a) also serve to phase compensation effect, resulting in the two lobes of LP₁₁ mode with the same phase.

In a step-index fiber, LP₁₁ mode has four different patterns, and one of the modes is an odd mode with y polarized which can be given as [24,33]

(4)$${{\textbf E}_{11}}({r,\varphi } )= {E_0}{J_1}({{U_{11}}{r \mathord{\left/ {\vphantom {r {{R_0}}}} \right.} {{R_0}}}} )\sin \varphi {\overrightarrow e _y},$$

where E₀ is const amplitude, J₁ is the first order Bessel function, U₁₁ is the normalized transvers phase parameters, and R₀ is the core radius of the fiber. Since most objective lenses obey the sine condition [30], we can get r = f·sinθ. Therefore, the pupil apodization function can be achieved via

(5)$$A({\theta ,\varphi } )= {E_0}{J_1}\left( {\frac{{{U_{11}} \cdot \sin \theta }}{{\sin {\theta_{\max }}}}} \right)\sin \varphi .$$

According to the foregoing analysis, the tightly focused optical fields in the focal region can be eventually obtained by

(6)$$\begin{aligned}{\textbf E}({r,\phi ,z} ) & ={-} \frac{{ikf}}{{2\pi }}\int_0^{{\theta _{\max }}} \left[ \int_0^\pi {\textbf E}_\varOmega^u({\theta ,\varphi } ){e^{ikr\sin \theta \cos ({\varphi - \phi } )+ ikz\cos \theta }}d\varphi\right.\\ &\quad \left.+ \int_\pi^{2\pi } {{\textbf E}_\varOmega^l({\theta ,\varphi } ){e^{ikr\sin \theta \cos ({\varphi - \phi } )+ ikz\cos \theta }}d\varphi } \right] \sqrt {\cos \theta } \sin \theta d\theta\end{aligned}$$

where the E_Ω(θ,φ) with the superscript symbols u and l represents the upper and lower lobes of the LP₁₁ mode, where they have opposite polarization. According to the IFE, the magnetization field induced in the focal region can be expressed as [6]

(7)$${\textbf F} = i\gamma [{{\textbf E} \times {{\textbf E}^\ast }} ],$$

where γ and E* are the magneto-optical susceptibility and the conjugate of the electric field E, respectively.

3. Results and discussions

As for a traditional few-mode fiber with core radius R₀= 4.5 μm, the refractive indices of the core and cladding are 1.4558 and 1.4490, respectively. According to the dispersion equation, it can be calculated that the normalized transverse phase parameters U₁₁ = 3.1658 [34]. Based on the above theory, the polarization and amplitude of the incident LP₁₁ mode are tailored carefully, and manipulating and shaping the orientations of the magnetization accurately in the focal region are realized. Throughout this manuscript, the other parameters we select during the simulation are listed as follows: NA = 0.92, λ = 1064 nm, f = 1.7 mm. First, as shown in Fig. 2, we generate a transverse magnetization within the focal region. During the simulation, the angle between the fast axis orientations of the QWP and HWP is η^u= η^l= 22.5°, the two attenuators have the same value t^u = t^l = 1, and the phase difference of the upper and lower lobes is δ^u=π/2 and δ^l= -π/2, respectively. As a result, the electric field of the upper and lower lobs has the same magnitude with left circular polarization and right circular polarization, respectively. In Fig. 2, the upper row shows the three components of electric energy in the focal plane, and E_t is the total electric field, where |E_t|² = |E_x|²+|E_y|²+|E_z|². These values are normalized to the maximum of |E_t|². The lower row shows the magnetization distributions of the three independent components, and the total magnetization F_t, where ${F_t} = \sqrt {F_x^2 + F_y^2 + F_z^2}$. The magnetization is also normalized to the maximum of F_t. From Fig. 2(b3), we can find no longitudinal magnetization exists, and a pure transverse magnetization is realized. The inset in Fig. 2(a4) shows the polarization of the upper and lower lobes of LP₁₁ mode, and the green and blue circles indicate the left-hand state and the right-hand state, respectively. We can find the spot size of the focus is about 0.5λ. In Fig. 2(b4), the black arrows show the direction of generated transverse magnetization, and cutting through F_t along the y axis, we can find the direction of the transverse magnetization points to -y direction. This phenomenon can be understood like this when tightly focusing the two circular polarized lobes of LP₁₁ mode which have the same intensity distributions, the two longitudinal components of the induced magnetic fields by these two lobes will have same intensities, while their directions are inversed. Consequently, their transverse components add up to a purely transverse in the focal plane [18]. If the two circular polarized lobes of LP₁₁ mode have different amplitudes, the two longitudinal components of the induced magnetic fields by these two lobes will have different intensities and directions. As a result, the direction of the induced magnetic field can be tailored. In Fig. 2(b4), we can see that the spot size of the induced magnetic field is similar to the focus, and it is about 0.5λ as well. In fact, the spot size of the focus is relative to the NA of the objective lens. So, if a lens with a larger NA is selected, the spot size of the induced magnetic field can be smaller.

Fig. 2. (a) Focal field distributions through tightly focusing the polarization and amplitude tailored LP₁₁ mode under t^u = t^l = 1, η^u= η^l= 22.5°, δ^u=π/2 and δ^l= -π/2, and the electric energies are normalized to the maximum value of |E_t|². (b) Calculated magnetization components, which are normalized to the maximum value of F_t, in the focal plane. The inset in a4 shows the polarization of the upper and lower lobes of LP₁₁ mode, respectively.

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To clarify the detailed distribution of the light-induced magnetization in the focal plane, the 3D magnetization strength at the focal plane is plotted as shown in Fig. 3. In Fig. 3, the arrows show the orientation of the induced magnetization. We can find the direction of F_t is nearly parallel to the horizontal plane, and the direction in the center stays in the -y direction clearly.

Fig. 3. The 3D magnetization strength at the focal plane for the case of t^u = t^l = 1, η^u= η^l= 22.5°, δ^u=π/2 and δ^l= -π/2, and the arrows indicate the orientation of the magnetization.

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By turning the transmittance of these two attenuators, the amplitudes of the two circular polarized lobes of LP₁₁ mode can be tailored. Figure 4 shows the field distributions and induced magnetization in the focal plane with t^u = 1, t^l = 1/3, η^u= η^l= 22.5°, δ^u=π/2 and δ^l= -π/2. As shown in Fig. 4(a1)-(a3), the transverse components of electric energy are weakened in the focal plane, while the longitudinal component is enhanced. As a result, a longitudinal magnetization appears as shown in Fig. 4(b3), and the intensity of the two-transverse magnetizations F_x and F_y is weakened as shown in Fig. 4(b1)-(b2). It is easy to conclude that the magnetization will have an angle to the x-y plane, and in the following, we will show that the magnetization will have arbitrary orientations to the x-y plane. The inset in Fig. 4(a4) shows the polarization of the upper and lower lobes of LP₁₁ mode, respectively, and the lower lobe has a smaller amplitude. Figure 5 shows the 3D magnetization strength at the focal plane, and the arrows show the orientation of the induced magnetization. Compared with Fig. 3, we can find the orientation of F_t is rotated toward the vertical direction, and this is due to the appearance of F_z, which is relative to the difference of the amplitude of the two lobes.

Fig. 4. (a) Focal field distributions through tightly focusing the polarization and amplitude tailored LP₁₁ mode under t^u = 1, t^l = 1/3, η^u= η^l= 22.5°, δ^u=π/2 and δ^l= -π/2, and the electric energies are normalized to the maximum value of |E_t|². (b) Calculated magnetization components, which are normalized to the maximum value of F_t, in the focal plane. The inset in a4 shows the polarization of the upper and lower lobes of LP₁₁ mode, respectively.

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Fig. 5. The 3D magnetization strength at the focal plane for the case of t^u = 1, t^l = 1/3, η^u= η^l= 22.5°, δ^u=π/2 and δ^l= -π/2, and the arrows indicate the orientation of the magnetization.

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To quantitatively analyze the orientation of the induced magnetization, the azimuthal angle of the magnetization in the focal region is surveyed. Figure 6 shows the azimuthal angle of induced magnetization, which cuts through the center of F_t along the y axis. Here, two directional angles α₀ and β₀ are defined as shown in Fig. 1(b), and $\sin {\alpha _0} = {{{F_y}} \mathord{\left/ {\vphantom {{{F_y}} {\sqrt {F_x^2 + F_y^2} }}} \right.} {\sqrt {F_x^2 + F_y^2} }}$, $\sin {\beta _0} = {{{F_z}} \mathord{\left/ {\vphantom {{{F_z}} {{F_t}}}} \right.} {{F_t}}}$. Figure 6(a) shows the azimuthal angle of the induced magnetization under t^u = t^l = 1, η^u= η^l= 22.5°, δ^u=π/2 and δ^l= -π/2, and we can find a pure transverse magnetization along -y direction is realized, when the value of y ranges from about −0.5λ to 1.5λ. Figure 6(b) shows the azimuthal angle of induced magnetization F_t along the y axis when t^u = 1, t^l = 1/3, η^u= η^l= −22.5°, δ^u=π/2 and δ^l= -π/2. Remarkably, we can find α₀= -π/2 all the time, as can be seen from the two solid lines, and β₀ is decreasing from π/2 to -π/4, when the value of y ranges from about −0.3λ to 0.8λ. We also survey the condition when t^u = 1/3, t^l = 1, η^u= η^l= 22.5°, δ^u=π/2 and δ^l= -π/2, as shown in Fig. 6(c). It can be found that the angle α₀= -π/2 as well while the angle β₀ is increasing from -π/2 to π/4, when the value of y ranges from about −0.3λ to 0.8λ. The insets in Fig. 6 show the azimuthal angle of the induced magnetization. From the above analysis, we can find by tuning the transmittance of these two attenuators, the intensities of these three components of induced magnetization are tuned, and by changing the MOF’s position, β₀ of the entire azimuthal angle can be gotten.

Fig. 6. The azimuthal angle of induced magnetization in the center of F_t. (a) t^u = t^l = 1, η^u= η^l= 22.5°, δ^u=π/2 and δ^l= -π/2. (b) t^u = 1, t^l = 1/3, η^u= η^l= 22.5°, δ^u=π/2 and δ^l= -π/2. (c) t^u = 1/3, t^l = 1, η^u= η^l= 22.5°, δ^u=π/2 and δ^l= -π/2. The insets show the direction angles of the induced magnetization.

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For further analysis, the phase retardation of the lower lobe of LP₁₁ mode under δ^l = -π/2, δ^l = -π/3, and δ^l = -π/6 is explored. As shown in Fig. 7, the phase retardation of the upper lobe of LP₁₁ mode has the same phase difference δ^u =π/2, the two attenuators have the same transmittance t^u = t^l = 1, and the ellipticity of these two lobes is χ^u= 1 and χ^l= 1/3, respectively. As we all know, when the two orthometric components of one beam have different amplitudes and phases, the beam will degenerate into an elliptically polarized beam. In Fig. 7, the first row shows the polarization of the upper and lower lobes, respectively. In Fig. 7, the polarization is indicated by the polarization ellipses, and the green ellipses indicate the left-hand states while the black ellipses indicate the right-hand states. The middle row in Fig. 7 shows the distributions of the magnetization, and we can find the evolution of the δ^l will cause a rotation of the polarization pattern, and the direction of transverse magnetization is altered as well. The last row in Fig. 7 shows the 3D magnetization strength at the focal plane, and it is clear to find the rotation of induced magnetization.

Fig. 7. The polarization states of the two lobes of incident LP₁₁ mode, the distributions of the magnetization, and the 3D magnetization strength at the focal plane when t^u = t^l = 1, χ^u= 1 and χ^l= 1/3, δ^u= π/2, (a) δ^l = -π/2, (b) δ^l = -π/3, (c) δ^l = -π/6.

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In addition, we also explore the induced magnetization when the polarizations of the two lobes are inversed. As shown in Fig. 8, t^u = t^l = 1, η^u= η^l= 22.5°, δ^u= -π/2, δ^l =π/2. We can find the focal field distribution is mirrored to the case t^u = t^l = 1, η^u= η^l= 22.5°, δ^u=π/2 and δ^l= -π/2, and the F_y is changed into a positive [(Fig. 8(b2)]. Cutting through F_t along the y axis, α₀=π/2 can be gotten invariably [Fig. 8(b4)]. So, if we want a magnetization with + y direction, we can change the direction of these two QWPs to turn the polarization of these two lobes of LP₁₁ mode. Figure 9 shows the 3D magnetization strength at the focal plane. Compared with the magnetic orientation in Fig. 3 and Fig. 5, the orientation of F_t is changed into + y direction in the center.

Fig. 8. Focal field distributions through tightly focusing the polarization and amplitude tailored LP₁₁ mode under t^u = t^l = 1, η^u= η^l= 22.5°, δ^u= -π/2 and δ^l =π/2, and the electric energies are normalized to the maximum value of |E_t|². (b) Calculated magnetization components, which are normalized to the maximum value of F_t, in the focal plane. The inset in (a4) shows the polarization of the upper and lower lobes of LP₁₁ mode, respectively.

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Fig. 9. The 3D magnetization strength at the focal plane for the case of t^u = t^l = 1, η^u= η^l= 22.5°, δ^u= -π/2 and δ^l=π/2, and the arrows indicate the orientation of the magnetization.

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At last, we further study the magnetization when the field distribution of the LP₁₁ mode is rotated clockwise by an angle of τ = π/6 as shown in Fig. 10(a). As a comparison, the focal field distribution, the magnetization distribution, and the 3D magnetization strength at the focal plane are shown in Fig. 10(b)-(d). In this simulation, the parameters we selected are listed as follows: t^u = t^l = 1, η^u= η^l= 22.5°, δ^u=π/2 and δ^l= -π/2. We should stress that the direction of these half wave plates, quarter wave plates and attenuators in Fig. 1 is also rotated clockwise by an angle of τ = π/6. As expected, the focal field distribution and the induced magnetization distribution are rotated simultaneously. The arrows in Fig. 10(c) and (d) show the direction of induced magnetization, and we can find the direction of induced magnetization is rotated by 30 degrees. However, in Fig. 10(d), we can also find that the direction of induced magnetization in the focus is also parallel to the x-y plane. As we have mentioned above, by turning the polarization controller, the field distribution of the LP₁₁ mode can be controlled, and this method also provides a new way to tailor the induced magnetic field. At last but not least, we can summarize that not only the LP₁₁ odd mode with y-polarization, but also the other three LP₁₁ modes (LP₁₁ odd mode with x-polarization, LP₁₁ even mode with x-polarization, and LP₁₁ even mode with y-polarization,) can be applied to generate a magnetic field with arbitrary orientation. The only thing needs to do is rotating the direction of these half wave plates, quarter wave plates and attenuators in Fig. 1 as we have analyzed above. We would also like to underline that to generate a magnetic field with controllable orientation induced by the LP₁₁ mode, the LP₁₁ mode from an optical fiber export should be carefully calibrated. If the LP₁₁ mode could not be carefully calibrated, it will become difficult for passing through both identical upper and lower regimes divided by the transverse coordinate axis. To calibrate the output LP₁₁ mode, a spatial filter with two separate open apertures can be used [35].

Fig. 10. (a) Intensity distribution of the LP₁₁ mode with a rotation angle of τ =π/6; (b) the focal field distribution; (c) the induced magnetization distributions; (d) the 3D magnetization strength at the focal plane. The parameters in this simulation are listed as follows: t^u = t^l = 1, η^u= η^l= 22.5°, δ^u=π/2 and δ^l= -π/2.

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4. Conclusion

In conclusion, due to the advantage of the first higher-order LP₁₁ mode, which has two lobes with opposite instantaneous direction of electric fields all the time, we present a simple method to generate a magnetic field with arbitrary orientation. By tailoring the amplitude, phase, polarization and field distribution of a LP₁₁ mode carefully, the focal field distribution and the induced magnetization distribution by IFE are surveyed. The orientation of the induced magnetization is also analyzed quantitatively, and we find that in the focal region the azimuthal angle of the induced magnetization ranges from -π/2 to π/2, covering the entire azimuthal angle. In particular, further study shows that the direction of induced magnetization is regulated by rotating the field distribution of the LP₁₁ mode. Compared with the traditional setup to generate a magnetic field with arbitrary orientation, ours is efficient. We believe that such an arbitrary orientation state of magnetic field can be applied in the field of confocal and magnetic resonance microscopy and spin dynamics, especially for the use of optical magnetic recording.

Funding

China Postdoctoral Science Foundation (2019M650437); Beihang Hefei Innovation Research Institute Project (BHKX-19-01, BHKX-19-02).

Disclosures

The authors declare no conflicts of interest.

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Tailoring the magnetic field induced by the first higher order mode of an optical fiber

Abstract

1. Introduction

2. Methods

3. Results and discussions

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (10)

Equations (7)

Optics Express