1. Introduction

The magneto-optical Kerr effect (MOKE) has proven to be an instrumental tool for probing the static and dynamic magnetization of ferromagnetic structures. In recent years, MOKE has been used to study magnetic microstructures not only for potential applications in data storage, quantum information processing, sensing, and logic [1–5], but also to probe fundamental aspects of magnetism and spintronics [6–12].

As compared to other techniques such as magnetic force microscopy, scanning transmission X-ray microscopy, Lorentz transmission electron microscopy, scanning Hall probe microscopy, and spin polarized scanning tunneling microscopy, MOKE microscopy has a unique combination of advantages: a simple, low-cost setup that can operate in a wide range of environments, and can provide independent information about all three magnetization components on timescales from statics to ultra-fast dynamics. In cases where the spatial resolution of optical microscopy is acceptable, MOKE provides a powerful technique for mapping magnetization.

To probe magnetic configurations with spatial variation on the microscale, it is important to have the capability to image multiple components of the magnetization vector. MOKE microscopy provides an avenue to do this, by controlling the angle of incidence of a probe laser. However, independent extraction of all three components is hampered by the need to perform several sequential measurements in different geometries, and then post-process the results to separate the components. One approach to overcome this problem has been to construct a custom photodetector using quadrant photodiodes allowing for measurements in multiple effective geometries at once [13–15]. Here, instead, we present a microscopy approach based on standard optical components in which mechanical modulation of the beam profile simultaneously encodes the three magnetization components separately in frequency and phase space. We demonstrate the technique by measuring magnetization dynamics in a prototypical magnetic microstructure: a vortex state in a thin, micron-scale disk. Specifically, we map out the equilibrium change in magnetization as the vortex is shifted by a quasi-static magnetic field, and the non-equilibrium magnetization dynamics following a fast magnetic field step. By comparing to the standard, sequential measurement approach, we find that the method used here provides improved separation of the magnetization components with similar signal-to-noise ratio, in just a single scan.

2. Methods and theory

2.1 Optomechanical multiplexing

In this section we will explain how our method encodes the three magnetization components in the probe laser beam.

To first order, MOKE is linearly proportional to the magnetization $\mathbf {M}$ of the medium and, upon reflection, gives rise to a rotation of linearly polarized light through an angle

Measurements of $\theta _{K}$ with different orientations of $\mathbf {k}$ yield information about the three components of $\mathbf {M}= M_x \hat {x} + M_y\hat {y}+M_z \hat {z}$. For example, a measurement with $\mathbf {k}=k_{z}\hat {z}$ yields $\theta _{K} \propto M_{z}$, and a measurement with $\mathbf {k}=k_{x}\hat {x}+k_{z}\hat {z}$ yields $\theta _{K}$ that is a mixture of $M_{x}$ and $M_{z}$. In principle, one should be able to extract $M_{x}$ and $M_{z}$ separately from these two measurements. This is not practical, however, due to the need for sequential measurements with different geometries, and the dependence of $C$ on $\mathbf {k}$.

Components of $\mathbf {M}(\mathbf {x})$ can be probed locally by measuring $\theta _{K}$ with a laser focused onto the sample at position $\mathbf {x}$ through a microscope objective. By altering the profile of the beam relative to the entrance aperture of the objective, the average incident $\langle \mathbf {k} \rangle$ can be varied to probe different components of $\mathbf {M}$. For instance, Fig. 1(a) shows a focused cone of polarized light with $\langle \mathbf {k} \rangle = \langle k_{z} \rangle \hat {z}$ normal to the surface. In this case, the reflected $\theta _{K} \propto M_z$. On the other hand, by blocking one half of the incoming beam profile, the resulting focus cone (Fig. 1(b)) has $\langle \mathbf {k} \rangle$ (green arrow) with components in both $\hat {x}$ and $\hat {z}$ directions, for a $\theta _{K}$ that depends on $M_{x}$ and $M_{z}$. Similarly, one can block the beam in the orthogonal direction to obtain information about $M_y$ and $M_{z}$. This method provides a straightforward way to image the components of $\mathbf {M}$, and has been used to measure hysteresis loops [18–21], and track the dynamics of a micron size ferromagnetic disk [22,23]. This method, however, suffers from the problem that it is not practical to separate $M_{x}$ and $M_{y}$ from $M_{z}$, and multiple sequential measurements are required to probe the different components. Previous work has aimed to avoid these problems by constructing a detector based on quadrant photodiodes which can simultaneously record different sections of the beam profile, and hence perform simultaneous measurements with different $\langle \mathbf {k} \rangle$ [13,14]. Here, we present a method that relies on standard photodetectors, using a mechanical modulation scheme to encode the three magnetization components separately into a single beam.

 

Fig. 1. Two different geometries of incident light focused through a microscope objective. (a) a Gaussian beam centered on the objective aperture produces a full cone of light with $\langle \textbf {k} \rangle =\langle k_z \rangle \hat {z}$ normal to the surface (green arrow). (b) Blocking half of the incoming beam produces a half-cone of incident light, with $\langle \textbf {k} \rangle$ at an angle $\alpha$ to the surface normal.

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In this technique, we use optomechanical choppers to modulate the laser beam profile, hence modulating $\langle \mathbf {k} \rangle$ in time. Figure 2(a) shows the calculated intensity profile of a Gaussian beam with standard deviation $\sigma$ expanded to fill an aperture of a standard optomechanical chopper wheel (Thorlabs MC1F10), as the wheel rotates. Focusing this beam onto the sample using a microscope objective results in the time-dependent $\langle \mathbf {k} \rangle$ shown in Fig. 2(b). The $\langle k_{x} \rangle$ and $\langle k_{z} \rangle$ components oscillate nearly sinusoidally, and out of phase, as the chopper wheel cuts through the expanded Gaussian beam with frequency $f_{x}$. Optimal results are obtained when the oscillations are maximally sinusoidal, which occurs when the beam diameter $2\sigma$ is very close to the aperture width. The $\langle k_{y} \rangle$ component is nearly zero, by symmetry. A second chopper, arranged as shown in Fig. 2(c) to cut through the beam in the orthogonal direction at frequency $f_{y}$ similarly provides oscillating components of $\langle k_{y} \rangle$ and $\langle k_{z} \rangle$. With two optical choppers in place, the reflected light from the microscope objective yields $\theta _{K}(t) = C \mathbf {M}\cdot \langle \mathbf {k}(t) \rangle$. For simplicity, we will neglect the $\mathbf {k}$-dependence of $C$, which would alter the measured amplitude of the different magnetization components but would not mix them. Taking the Fourier transform of $\theta _{K}(t)$, we obtain $\tilde {\theta }_{K}(f) = C\left [M_{x} \langle \tilde {k}_{x}(f) \rangle + M_{y} \langle \tilde {k}_{y}(f) \rangle +M_{z} \langle \tilde {k}_{z}(f) \rangle \right ]$, in terms of the Fourier transforms of $k_{x}(t)$, $k_{y}(t)$ and $k_{z}(t)$, which are shown in Fig. 2(d). At $f_{x}$, the real part of $\langle \tilde {k}_{x} \rangle$ is a maximum, while the real parts of $\langle \tilde {k}_{y} \rangle$ and $\langle \tilde {k}_{z} \rangle$ are near zero. On the other hand, the imaginary part of $\langle \tilde {k}_{z} \rangle$ is peaked at $f_{x}$, with the imaginary part of the other components near zero. Similarly, at $f_{y}$, peaks are seen in the real part of $\langle \tilde {k}_{y} \rangle$ and the imaginary part of $\langle \tilde {k}_{z} \rangle$.

 

Fig. 2. Choppers’ role in determining the $\langle \textbf {k} \rangle$ of the beam profiles. (a) Geometries of a Gaussian beam spot cut by a chopper wheel aperture in different rotation angles $\phi$ of the wheel with $N$ apertures. (b) Calculated $\langle \mathbf {k} \rangle$ components of the Gaussian beam geometries in (a). (c) Two chopper wheels are in position to cut the beam spot (shown in red) in two orthogonal directions; chopper with frequency $f_{y}$ ($f_{x}$) cuts through the beam in y-direction (x-direction). (d) Calculated real and imaginary parts of the Fourier transforms of wavevector $\tilde {\textbf {k}}$ for the frequency range of choppers.

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The distinct peaks in Fig. 2(d) allow for the separate extraction of $M_{x}$, $M_{y}$, and $M_{z}$ by demodulating $\theta _{K}$. Specifically, $M_{x} \propto \Re {[\tilde {\theta }_{K}(f_{x})]}$, $M_{y} \propto \Re {[\tilde {\theta }_{K}(f_{y})]}$, and $M_{z} \propto \Im {[\tilde {\theta }_{K}(f_{x})+\tilde {\theta }_{K}(f_{y})]}$. In practice, this is achieved by detecting $\theta _{K}$ with a pair of dual-phase lock-in amplifiers, at frequencies $f_{x}$ and $f_{y}$. The two phase components from the lock-in at $f_{x}$ yield a signal proportional to $M_{x}$ and $M_{z}$, while the two phase components from the lock-in at $f_{y}$ yield signal proportional to $M_{y}$ and $M_{z}$. The two independent measurements of $M_{z}$ may then be added together. The constants of proportionality for the $x$ and $y$ measurements should be nearly identical, but will differ from that for the $z$ measurement depending on the $\mathbf {k}$-dependence of $C$ and numerical aperture of the objective.

2.2 Samples and setup

To demonstrate the 3D MOKE microscopy technique described above, measurements are performed on a $2.2~\mu$m diameter, $40$ nm thick permalloy disk. In a soft ferromagnetic thin film patterned into a disk shape of micrometer size or below, the balance between magnetostatic and exchange energy results in a vortex ground state [24–26]. The vortex magnetization state is characterized by in-plane magnetization curling around a central core, where the magnetization is oriented out-of-plane over a 10-nm region (see Fig. 3(a)). An applied, in-plane magnetic field shifts the equilibrium position of the vortex core, resulting in non-trivial dynamics of this 3D, nanoscale magnetization configuration.

 

Fig. 3. Schematic of a vortex state and the sample (not to scale). (a) Magnetic vortex state in a 2 $\mu$m-diameter permalloy disk. Arrows indicate the direction of magnetization on the disk. (b) Illustration of the sample. The gray disk represents a 2.2-$\mu$m-diameter, 40-nm-thick permalloy disk deposited on the bottom of a glass cover slip. The yellow strip underneath of glass is the 130-nm-thick gold CPW which covers the permalloy disk. The red arrow, $i_{CPW}$ indicates the direction of the current through CPW, and the blue arrow on top of the gold, $\textbf {B}_{CPW}$ represents the direction of magnetic field produced by the current.

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Permalloy disks are fabricated on a glass cover slip via electron beam lithography, electron beam evaporation, and liftoff. A gold coplanar waveguide (CPW) is then fabricated on top of the disks and glass, using photolithography, with the geometry shown in Fig. 3(b).

The setup we used to measure dynamics of the vortex state in both regimes is shown in Fig. 4. A flip mirror is used to switch between pulsed and continuous-wave (CW) lasers, for the two different measurements. An optical expander (not shown) is used to increase the beam diameter. The laser then passes through the two choppers as described above. After the beam passes through a linear polarizer and a beam splitter, a $100$x oil-immersion objective (numerical aperture $1.25$) focuses the beam onto the sample, which is mounted on a three-axis piezo nanopositioning stage. The reflected light is collected from the beamsplitter after recollimation by the microscope objective. The polarization rotation is measured by passing the beam through a half-wave plate and Wollaston prism to separate orthogonal polarization components, which are then detected by a balanced photodiode bridge. To increase the sensitivity of the measurements and eliminate edge artifacts, a modulation of the magnetic field is introduced at $f_{m}=15$ kHz. (The specific modulation in the two cases are outlined by blue dashed and green connections, and will be described in more detail below.) The signal from the photodetector is then amplified and filtered using a preamplifier, and the component of the signal at $f_{m}$ is extracted by a lock-in amplifier (lock-in I). The time constant on lock-in I is sufficiently fast ($640~\mu$s) that the output is still modulated at $f_{x}$ and $f_{y}$. The output of lock-in I is then sent to both lock-in II and lock-in III for the demodulation that separates the $M_{x}$, $M_{y}$, and $M_{z}$ signals as described above.

 

Fig. 4. Schematic of the setup. The connections shown in blue dashed (green) are used for measuring the equilibrium (non-equilibrium) response.

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Lock-ins II and III output two orthogonal phases of the signal at their reference frequency; in order to separate $M_{x,\;y}$ from $M_{z}$, we must identify the correct phase offset to obtain the desired orthogonal phases. Because both phases generally have non-zero signal, the usual method of adjusting the phase to maximize the signal does not yield the correct phase offset. Instead, we block one of the photodiodes in the detector, and add an additional modulation to the CW laser at $f_{m} = 15$ kHz. With the rest of the setup the same, both lock-in II and III output a signal that is merely proportional to the laser power at the detector. This signal has the same phase offset as the $M_{z}$ component, with no signal in the orthogonal phase. Thus this phase offset can be found by maximizing the signal.

To further optimize the phase offsets, we can make use of the fact that our measurement produces two independent measurements of $M_{z}$, $M_{z,1}$ and $M_{z,2}$. If the phase offset in either lock-in is incorrect, then the $M_{z}$ component will be mixed with the $M_{x}$ component on lock-in II and the $M_{y}$ component on lock-in III, and hence the measured $M_{z}$ signals will not be the same. We can fine-tune the phase offsets by performing a measurement (either a spatial map of the equilibrium magnetization response, or a time delay scan of the non-equilibrium dynamics), and then search the 2-D space of phase of offsets in the neighborhood of the previous optimum to minimize the quantity $\langle M_{z,1}-M_{z,2} \rangle$, where the brackets represent the root-mean-square.

2.3 Vortex magnetostatics and dynamics

To demonstrate the technique for 3D magnetization mapping, we implement measurement of vortex state dynamics in two regimes: equilibrium and non-equilibrium. The energy of the vortex state in an in-plane magnetic field $\mathbf {B}$ can be approximated as

2.4 Method I: imaging of equilibrium magnetization response

By modulating the applied magnetic field on timescales slow compared to the vortex energy relaxation time, we measure the change in the equilibrium configuration of $\mathbf {M}$ in response to a change in magnetic field. For this measurement, a square-wave current at $f_{m}=15$ kHz is driven through the CPW by a function generator, as shown by the blue dashed connection in Fig. 4 (the green connections are not used here). Near the surface of the CPW, the current produces a magnetic field $B_{\pm }= \pm 1.9$ mT in the $\hat {x}$ direction, as shown in Fig. 3. The equilibrium position of the vortex core then switches between $\mathbf {x}_{\pm } = c \chi _{0} B_{\pm } \hat {y}$. With the reference from the function generator at $f_{m}$, lock-in I outputs signal proportional to $\Delta \mathbf {M}(\mathbf {x}) = \mathbf {M}(\mathbf {x}; B_{+})-\mathbf {M}(\mathbf {x}; B_{-})$, with the different components of $\Delta \mathbf {M}$ encoded in frequency and phase as described above.

To understand the expected $\Delta \mathbf {M}$ from a magnetic vortex, we perform micromagnetic simulations using the object-oriented micromagnetic framework (OOMMF) [29], as in [19]. Figure 5 shows the simulated components of $\mathbf {M}$ and $\Delta \mathbf {M}$ near the center of a vortex state in a $2.2~\mu$m diameter, 40 nm thick permalloy disk with a magnetocrystalline anisotropy $K_u=$ 8$\times 10^3$ J/m$^3$ along $\hat {y}$, and applied magnetic fields $B_{+}$ and $B_{-}$. Although we expect to have zero magnetocrystalline anisotropy, in reality a non-zero uniaxial anisotropy may be induced during the film deposition or by strain [30,31].

 

Fig. 5. Simulated maps of the magnetization components of a vortex state in external magnetic field. The simulation is for a 2.2 $\mu$m-diameter, 40-nm thick permalloy disk with a uniaxial anisotropy $K_u=$ 8$\times 10^3$ J/m$^3$ along $\hat {y}$. Colorbars are normalized to saturation magnetization $M_s$, and the colorbars in the bottom row display a reduced range to make smaller changes in magnetization visible, apart from the motion of the core itself. Here $\Delta \textbf {M} (\textbf {x})=\mathbf {M}(\mathbf {x}; B_{+})-\mathbf {M}(\mathbf {x}; B_{-})$. The white arrows indicate the vortex core position. An external magnetic field $B_+$ ($B_-$) in $\hat {x}$ direction, moves the vortex core to the negative (positive) in $\hat {y}$ direction.

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The measured components of $\Delta \mathbf {M}(\mathbf {x})$ will be the difference of the magnetization components (the $\Delta M_{x}$, $\Delta M_{y}$, and $\Delta M_{z}$ in Fig. 5) convolved with the focused laser spot profile, as shown in Fig. 6(a), 6(b), and 6(c). (The focused laser spot profile is calculated from the spatial Fourier transform of the beam profile at the objective entrance aperture.) $\Delta M_{x}$ is characterized by a single spot at the center of the disk. As the vortex core translates right and left of the center, $M_{x}$ at the center of the disk changes sign, yielding a large $\Delta M_{x}$. $\Delta M_{y}$ shows a four-lobed pattern as $M_{y}$ tilts in the different quadrants of the disk when the vortex moves. Finally, $\Delta M_{z}$ shows two lobes of opposite sign above and below the center of the disk with four additional fainter lobes adjacent to the two central lobes. Although the vortex core translates left and right, the resulting change in magnetization is not visible due to the small size of the core. Instead, the simulated $\Delta M_{z}$ is dominated by the smaller amplitude, yet larger length scale deformation of $M_{z}$ due to the uniaxial anisotropy. There is an additional contribution to $\Delta M_{z}$ near the disk edges as the applied field causes the magnetization to begin to rotate out of plane at the disk edge.

 

Fig. 6. Micromagnetic simulation results of a 2.2 $\mu$m-diameter, 40-nm thick permalloy disk with uniaxial anisotropy $K_u=$ 8$\times$10$^3$ J/m$^3$ along the y-direction. The dotted circles in panels (a)–(c) indicate the outline of the permalloy disk. In simulations the differential magnetization components ($\Delta \textbf {M}$) of the disk are convolved with the focused beam profile. (a) $\Delta M_x$, (b) $\Delta M_y$, and (c) $\Delta M_z$ for a vortex state in equilibrium at $B = \pm 1.9$ mT. Scale bar: 400 nm. (d) Simulated $\Delta M_{y}$ vs. $\Delta M_{x}$ over time with the probe focused 200 nm from the center of the disk, showing gyrotropic precession following a step in magnetic field from $B=0$ mT to $B=+1.9$ mT. (e) $\Delta M_{z}$ vs. time for the same simulation as in (d).

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2.5 Method II: measuring time-resolved magnetization dynamics

In order to measure the non-equilibrium dynamics of the vortex core, we probe the magnetization using a pulsed laser synchronized to a series of magnetic field pulses, as shown by the green connections in Fig. 4. A pulsed laser with a repetition rate of 5 MHz is introduced via the flip mirror. A digital delay generator (DDG) that is triggered by the laser pulses drives current pulses through the CPW with a duration of 35 ns and rise/fall time of several nanoseconds. This produces magnetic field pulses with amplitude 2 mT along $\hat {x}$. By adjusting the delay of the pulses from the DDG, we sweep the time $t$ between the beginning of the field pulses and the arrival of the laser pulses. To introduce modulation at $f_{m}$, the magnetic field pulses are inhibited using the square-wave output of the function generator. In this case then, the output of lock-in I is proportional to $\Delta \mathbf {M} = \mathbf {M}(t)-\mathbf {M}_{0}$, where $\mathbf {M}(t)$ is the magnetization at time $t$ after the magnetic field pulse, and $\mathbf {M}_{0}$ is the equilibrium magnetization with no field pulse applied. Here, we focus the laser close to the center of the disk. At this position, a small displacement of the vortex core $\mathbf {x}=(x,\;y)$ yields $\Delta M_{x} \propto y$ and $\Delta M_{y} \propto x$, providing a direct measurement of the vortex core trajectory. The $\Delta M_{z}$ component also oscillates as the vortex core precesses, with details depending on the precise position of the laser spot. Figures 6(d) and 6(e) show the simulated $\Delta \mathbf {M}(t)$ for the disk with uniaxial anisotropy, following a step to nonzero magnetic field in the x-direction, where the laser spot is positioned 200 nm from the disk center. As expected from Thiele’s equation of motion, the trajectory of the vortex core revealed by $\Delta M_{x}$ and $\Delta M_{y}$ spirals around the new equilibrium position away from zero. The elliptical nature of the spirals is caused by the uniaxial anisotropy along $\hat {y}$. At this focus position, the $\Delta M_{z}$ component also shows oscillations at the gyrotropic frequency. (The fast oscillations at the beginning are caused by confined spin waves excited by the field step. We do not generate these excitations in the experiment due to the slower rise time of the magnetic field step.)

3. Results

3.1 Equilibrium response

In this section, we present the results of the equilibrium magnetization response, and compare to a more standard approach based on a static beam blocker to change the probe beam profile [19].

The images obtained using the multiplexed method described here (top row of Fig. 7) are in good agreement with the simulation results. These four images are all from a single scan, with the two independent measurements of $\Delta M_{z}$ shown separately. The images are normalized by the time-averaged power at the detector to account for variations in sample reflectivity. The $\Delta M_{x}$ component (Fig. 7(a)) shows an elongated spot centered on the disk (the outline of the disk is shown by the dotted circle), and the $\Delta M_{y}$ component (Fig. 7(b)) shows the expected four-lobed structure. In the $\Delta M_{z}$ component (Figs. 7(c) and 7(d)), the two central lobes are clearly visible, with four fainter adjacent lobes as seen in the simulations including uniaxial anisotropy. One unexplained discrepancy is the lack of the crescent-shaped response at the top and bottom edges of the disk. The two measurements of $\Delta M_{z}$ are in good agreement with each other: Fig. 7(e) shows the difference of the two images.

 

Fig. 7. Experimental results for the vortex equilibrium response imaging. The dotted circles indicate the outline of the permalloy disk. The top row images ((a)–(d)) $\Delta M_x$, $\Delta M_y$, $\Delta M_{z_1}$, and $\Delta M_{z_2}$ are obtained from a single scan in multiplexed method. (e) is the difference of the two $\Delta M_{z}$ measured separately in multiplexed method. Bottom row images ((f)–(h)) are $\Delta M_x$, $\Delta M_y$, and $\Delta M_z$ measured from three sequential scans in the beam-blocker method. Scale bar: 500 nm.

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To evaluate the benefits of our method, we will compare to a set of three measurements using a fixed beam blocker (BB) to set the probe laser profile, and otherwise similar experimental parameters. In this case, we remove the two choppers from the setup, and instead introduce a sinusoidal modulation of the CW laser current at $f=230$ Hz, referenced to lock-in II. (lock-in III is not used.) The modulation is set to maintain the same average power emitted by the laser (that is, the power oscillates between zero and double the original CW power). By centering the full Gaussian beam profile on the microscope objective, $\langle \mathbf {k} \rangle = \langle k_{z} \rangle \hat {z}$, as in Fig. 1(a) and the resulting $\theta _{k}\propto \Delta M_{z}$. By blocking half of the beam profile vertically or horizontally, $\theta _{K}$ is sensitive to $M_{x}$ or $M_{y}$, though with a portion of the $M_{z}$ signal mixed in, and we refer to these measurements as $\Delta M_{x}$ and $\Delta M_{y}$ respectively. The bottom row in Fig. 7 shows the three consecutive scans of $\Delta M_x$, $\Delta M_y$, and $\Delta M_z$ using the beam-blocker method. Again, the images are normalized by the time-averaged, reflected power at the detector. The same lock-in time constants were used for both sets of images.

Comparing the images obtained with the beam blocker method to those obtained with the multiplexed method, we see that the beam blocker method produces images that are distorted, and with lower signal amplitude. In the $\Delta M_{x}$ image obtained with the BB method (Fig. 7(f)), instead of an elongated spot centered on the disk, we observe a less elongated spot off-center. This occurs because of the mixing of the $\Delta M_{x}$ with $\Delta M_{z}$ components. One of the two strong lobes in the $\Delta M_{z}$ image adds to the $\Delta M_{x}$ signal, and the other lobe tends to cancel it out. Likewise, in the $\Delta M_{y}$ image (Fig. 7(g)), some of the lobes appear stronger and others weaker due to mixing with the $\Delta M_{z}$ image. Without already knowing what to expect in these images, it would be difficult to use the measured $\Delta M_{z}$ image to correct this mixing. The degree of mixing depends on the particular spatial profile of the probe beam, and the $\mathbf {k}$-dependence of the parameter $C$. The multiplexed method does not suffer from this problem, as the components are already well-separated. Further, despite the additional complexity of the multiplexed method, the images have higher signal amplitude for a given average power level at the detector than with the BB method, with comparable signal-to-noise ratio SNR $=50$. This occurs because of the differential nature of the multiplexed measurement.

These images provide a method for measuring the displacement susceptibility $\chi _{0}$ of the vortex. By applying an additional static magnetic field in the $x$ or $y$ direction, the features in the images are displaced. Most clearly, the spot in the $\Delta M_{x}$ image translates along with the vortex core. By measuring this shift as a function of in-plane static field, we obtain $\chi _{0} = 70$ nm/mT for displacement in the x-direction and $\chi _{0} = 80$ nm/mT for displacement in the y-direction. The anisotropy in $\chi _{0}$ is likely due to the uniaxial anisotropy discussed above.

3.2 Time-resolved dynamics

In this section, we demonstrate the multiplexed method as applied to measurement of the vortex trajectory in response to a fast magnetic field pulse. As described above, a train of current pulses is driven through the CPW to generate a time-dependent magnetic field that drives the vortex out of equilibrium. The pulsed magnetic field, shown in Fig. 8(a), begins at zero, then at time $t=0$ steps to $B=2$ mT with a rise time of $\sim 1$ ns. The field then remains at that value until $t=35$ ns when it returns to zero. The field remains at zero for 165 ns (the remainder of the $200$ ns repetition period). When the magnetic field steps, the equilibrium position of the vortex core changes by about 200 nm in a time much less than the vortex relaxation time, leading to non-equilibrium dynamics.

 

Fig. 8. Experimental results for non-equilibrium dynamics of the vortex core. (a) $\Delta M_x$ (blue) and $\Delta M_y$ (red) vs. time, in response to the magnetic field profile (yellow). (b) $\Delta M_{z_1}$ from lock-in II (blue), and $\Delta M_{z_2}$ from lock-in III (red) vs. time. (c) Vortex core trajectory from the data in (a).

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Figure 8 shows the measured vortex core dynamics, with the $\Delta M_{x}$ and $\Delta M_{y}$ components in Fig. 8(a) and the two $\Delta M_{z}$ measurements in Fig. 8(b). All four datasets are obtained simultaneously. The $\Delta M_{x}$ and $\Delta M_{y}$ components directly reveal the vortex core trajectory. At $t<0$, the vortex has completely relaxed to equilibrium in the approximately $150$ ns since the last pulse, and both components begin at zero. When the pulse arrives, both components begin oscillating about the new equilibrium at the gyrotropic frequency $f_{g}=125$ MHz, with similar amplitude and a phase difference $\approx \pi /2$. Because of the field step to $B_{x} = 2$ mT, we expect the new equilibrium to be at $y_{0} = \chi _{0} B_{x} = 160$ nm. Indeed we observe a significant offset in the oscillations in $\Delta M_{x}$, corresponding to an offset in the y-direction. When the pulse ends at $t=35$ ns, the equilibrium position shifts back to zero, and the vortex now undergoes oscillations about the original position. As expected, the $\Delta M_{z}$ signal also undergoes oscillations, with good agreement between the two measurements.

The trajectory of the vortex core can be visualized more clearly in Fig. 8(c) by plotting $\Delta M_{x}$ vs. $\Delta M_{y}$, with the axes given in terms of vortex core displacement, scaled by the known spacing between the two equilibrium points. At $t \,< \,0$, the core is stationary at $(0,0)$, as seen by the cluster of points at the origin. At $t=0$ ns, the vortex core begins oscillating about the new equilibrium at $(-50,160)$ nm, and spiraling in as energy is dissipated. Then at $t=35$ ns, the vortex switches to spiraling about the origin as it relaxes back to the original equilibrium. This measurement is performed with a time constant for lock-ins II and III set to $1$ s with 16 scans averaged together to reduce noise. With these parameters, the noise in the position measurement is $\approx 10$ nm, comparable to the size of the vortex core itself. The observed trajectory also shows the ellipticity expected for material with an induced uniaxial anisotropy, as in the simulation in Fig. 6(d). In the experiment, the gyrotropic precession is extended along a diagonal direction, likely indicating the anisotropy axis. We observe the same elongation regardless of the initial probe polarization direction, making it less likely that this could be caused by differing sensitivity of the $\Delta M_{x}$ and $\Delta M_{y}$ components.

4. Discussion

MOKE microscopy is a straightforward tool for mapping magnetization at a surface, yielding information about all three magnetization components. However, when the structure under study has a spatially complicated magnetization texture, it can be difficult to accurately separate out the different components to obtain the true three-dimensional magnetization. Here, we demonstrate a method to accomplish the separation of the three components with a minimal increase in experimental complexity, and no reduction in signal-to-noise ratio as compared to standard scanning MOKE microscopy. Many scanning, confocal MOKE experiments already use an optical chopper and lock-in amplifier to detect the small MOKE signals. Such a setup can be modified to multiplex two magnetization components simply by expanding the beam through the chopper. The third component can then be measured as well by the addition of one extra chopper.

Though we demonstrate the technique here by measuring magnetic vortex dynamics, the method is far more general. One could just as well measure static magnetization, removing the need for the additional modulation at $f_{m}$ and lock-in I. Or, by removing the polarization analyzer one may be able to extract the transverse MOKE signal from the frequency components of the reflected intensity. With the emergence of new magnetic materials and structures involving perpendicular magnetic anisotropy and the Dzyaloshinskii-Moriya interaction, there is a growing need to image magnetization in three dimensions [32–34]. The method we have demonstrated here provides a convenient approach for characterizing the complex magnetic textures that arise in these materials.