Magneto-optic Kerr effect CCD imaging with polarization modulation technique

Because of their unique magnetic domain structure and their dynamical behavior,^1–3 magnetic microstructures of ferromagnetic materials are of great interest,^4–6 especially for application to emergent data-storage devices. In order to understand the static and dynamical behavior of magnetic domain structures, it is important to study their spatial distribution and spatio-temporal dynamics. Thus, establishing a technique for visualizing the spatio-resolved magnetization in magnetic microstructures is necessary.

It is well known that magneto-optic Kerr effect (MOKE) microscopy^7,8 is a powerful tool to study magnetic domains and their dynamics. The polarization change due to the MOKE is a consequence of surface reflection from ferromagnetic materials with complex refractive index N and complex magneto-optic constant Q due to spin-orbit interaction in these materials.^9–12 Numerous studies using MOKE microscopy have been carried out to characterize the magnetic properties of ferromagnetic thin films.^13–16 Recently, MOKE microscopes with array detectors, such as a charge-coupled-device (CCD), which are useful to obtain the spatial distribution of magnetization, have been developed and utilized.^17–23 However, there are no reports about the quantitative estimation of Q from MOKE images. Fast characterization of the complex magneto-optic constant Q using the MOKE imaging system with CCD cameras will open up new research possibilities for characterizing magnetic microstructures.

In this paper, we report the spatio-resolved quantitative evaluation of Q from longitudinal MOKE (L-MOKE) measurement with a CCD camera. We utilized the rotating compensator polarimetry technique, a conventional technique in ellipsometry, for quantitative determination of the polarization state of light in terms of the orientation and ellipticity angles. From the dependence of the orientation and ellipticity angles on the angle of incidence in the L-MOKE configuration, the value of Q of the magnetic materials can be derived quantitatively. We have successfully obtained the spatial image of Q using our technique.

Figure 1 shows the schematic setup of the L-MOKE measurement using the rotating compensator technique. The polarization state of reflected light from the surface of the sample changes with respect to the polarization state of incident light due to the L-MOKE. The s- and p-polarized electric-field amplitudes reflected from the surface (⁠$Ers$⁠, $Erp$⁠) are described in terms of the incident light (⁠$Eis$⁠, $Eip$⁠) and the complex Fresnel reflection coefficients, r_jk(j,k = s,p), as follows:

and

In this work, we consider the incident light to be completely linearly s-polarized; therefore, $Eip=0$⁠. The Fresnel coefficients r_ss and r_ps with N and Q are given by¹⁰ 

and

where $ϕi$ and $ϕt$ are the angle of incidence and the complex angle of refraction, respectively. The term $cosϕt$ is approximated to unity (⁠$cosϕt≅1$⁠) in our experimental condition. In the case when the incident light is linearly s-polarized, the initial orientation angle θ is 90° and the initial ellipticity angle ε is zero before reflection from the surface of the magnetic materials. Both θ and ε represent the polarization state of the light. After reflection, both θ and ε change, and are expressed in terms of N and Q as

In this work, we determine the parameter Q for a given N from Ref. 10 using Eq. (5).

Determination of θ and ε in our experimental system is carried out as follows. As shown in Fig. 1, the reflected light from the surface of the sample passes through the rotating compensator and the linear polarizer before detection by the CCD camera. In this system, the detected signal intensity I, is described as a function of the angle of the compensator, $θm$⁠, as,^24,25

where S₁, S₂, and S₃ are the three Stokes parameters which represent the polarization state of the light. One can determine the parameters through Fourier analysis of Eq. (6). The Stokes parameters are related to θ and ε, and are given by,

and

The parameter Q is obtained by substituting Eqs. (7) and (8) in Eq. (5).

In this work, we used two forms of the Ni₈₁ Fe₁₉ sample with a thickness of 30 nm deposited on a silicon substrate by electron beam evaporation. One is a Ni₈₁ Fe₁₉ thin film sample, and the other is a rectangular Ni₈₁ Fe₁₉ microstructure sample with a dimension of 15 μm × 15 μm (see Fig. 2(a)). We fabricated the rectangular Ni₈₁ Fe₁₉ microstructure sample using a standard lift-off process.

Figure 2(b) shows the experimental setup of our L-MOKE imaging system. We used a picosecond laser diode (wavelength: 406 nm) as a light source (LS). The light beam from LS is focused on a rotating glass diffuser D with a rotating frequency of 41 Hz by using lens L1, to reduce the effect of speckle on the L-MOKE image.²⁰ After collimating the light beam by lens L2, the light beam passes through polarizer P1 and a half-wave plate (HWP) so that the light is s-polarized. After reflection by a beam splitter BS, the light beam irradiates the sample via lens L3 and an objective lens (OL) with a numerical aperture of 0.8. The light beam was focused on the back focal plane of OL by L3 for Köhler illumination. The incident angle $ϕi$ is controlled ranging from -45° to +45° by moving L3 mounted on a translation stage. The magnetic field is applied to the sample using an electromagnet (EM), which is controlled by an external voltage source. The reflected light beam from the surface of the sample was collected by OL and passed through BS. The reflected light beam passed through a quarter-wave plate (QWP) mounted on a hollow-shaft motor with a rotating frequency (f_m) of 3 Hz and polarizer P2. The transmission axis of P2 was set to the direction of p-polarization. The reflected light was imaged in a CCD camera using lens L4, where its intensity was measured. In addition, we controlled the exposure timing of the CCD camera by an external trigger input with a repetition frequency of f_Shutter = (29/10) f_m = 8.7 Hz. The exposure time of the CCD camera is set to be about 45 ms.

The method of data accumulation and analysis in our L-MOKE measurement system is shown in Fig. 3(a). In our system, we acquire the image by the CCD camera with a time interval of (10/(29f_m)). The angle of the QWP at the 30th image is identical with that at the 1st image. In other words, the 29 intensity data sets with different $θm$ are repeatedly obtained with a time interval of 10/f_m. Thus, we can accumulate and average the intensity signal by repeating the experiments N_repeat times to increase the signal-to-noise ratio (SNR). Note that f_Shutter/f_m is not an integer, which means that $θm$ (⁠$0≤θm<360°$⁠) is not proportional to the frame number N_frame. Therefore, we reorder the sequential data of the intensity images recorded by the CCD camera to those corresponding to the sequential evenly-spaced angles of the QWP as shown in the bottom panel of Fig. 3(a) in order to carry out the Fourier analysis.

In Fig. 3(b), we show the intensity of the 29 images, where we integrate the signal intensity obtained at all the CCD pixels as a function of the elapsed time. The integrated intensity after reordering in accordance with the angle of the QWP at each measuring condition is shown in Fig. 3(c). We carry out the Fourier transform of the data in Fig. 3(c), and derive the amplitude of the $nθm$ frequency components, which is plotted as a function of n in Fig. 3(d). The relatively large amplitudes are observed at n = 2 and 4. Indeed, the intensity data in Fig. 3(c) can be well characterized from Eq. (6) by using the $2θm$ and $4θm$ frequency components as shown in the solid curve from which we can obtain the three Stokes parameters. Finally, since we obtain θ and ε from the amplitudes and phases of these frequency components as described in the previous section (Eqs. (7) and (8)), the value of Q is evaluated from θ and ε by Eq. (5).

In the first experiment, we carried out the L-MOKE measurements of the Ni₈₁ Fe₁₉ thin film sample at different angles of incidence $ϕi$ to verify the precision of our measurement system. In this measurement, because we used a thin film sample without any spatial patterns, all the light imaged onto the CCD camera was reflected from Ni₈₁ Fe₁₉ under the same magnetization direction. We integrated the signal intensity at all the CCD pixels to quantitatively estimate θ and ε of the Ni₈₁ Fe₁₉ thin film. We obtained two values of θ and ε by applying the external magnetic fields of ± 8.6 mT, where the sample is uniformly magnetized along the direction of the external magnetic field. We subtracted one value from the other one, and then evaluated the Kerr rotation angles, $θK$ and $εK$ as follows; $θK=(θ+−θ−)/2$ and $εK=(ε+−ε−)/2$⁠, where $θ+(−)$ and $ε+(−)$ are the orientation and ellipticity angles obtained at +(-) 8.6 mT, respectively.

Figures 4(a) and (b) show the dependence of $θK$ and $εK$⁠, on the angle of incidence, respectively. We repeatedly carried out the same measurement to estimate the standard errors, which are shown as error bars in Figs. 4(a) and (b). Both of them show clear dependence on the angle of incidence. The solid curves in Figs. 4(a) and (b) are curves fitted to the experimental data based on Eq. (5) as a function of $ϕi$⁠. In the fitting, we used a literature value of N = 1.45 - 2.03i at 406 nm from Ref. 10. From the fitting, the real and imaginary parts of Q are quantitatively estimated to be 0.009±0.001 and 0.003±0.001, respectively. The good agreement between experimental data and the fitting curves means that our measurement system is useful for measuring the change in the L-MOKE signal.

Next, we obtained the spatial image of |Q| in the rectangular Ni₈₁ Fe₁₉ microstructure sample as shown in Fig. 5(a). This image is derived from similar data analysis corresponding to Fig. 4, while in Fig. 5, we took the image only at the angle of incidence of $ϕi$=45° and used Eq. (5) to obtain the spatial variations of |Q|. To increase the SNR, we averaged the data with 71 repeated experiments and partially integrated the signals at spatial grid of 10 pixels × 10 pixels. The |Q| values show high contrast between the sample and the substrate to the right of the sample, whereas, the contrast is low between the sample and the substrate to the left of the sample. This is because that the intensity of incident light beam on the left area is weaker compared to that on the right area due to a slight misalignment of the setup, resulting in decrease of the SNR. Figure 5(b) shows the cross-section profiles of Fig. 5(a) along with the solid line in Fig. 5(a). The dotted vertical lines in Fig. 5(b) correspond to the interfaces between the Ni₈₁ Fe₁₉ thin film and the Si substrate. We can clearly observe the difference in |Q| between the Ni₈₁ Fe₁₉ and Si, which is estimated to be 0.008. The value of |Q| for the rectangular Ni₈₁ Fe₁₉ structure is consistent with that obtained from the Ni₈₁ Fe₁₉ thin film. This result proves that the measurement of quantitative |Q| image is achieved by using our measurement system.

Finally, we briefly mention the advantages of our developed L-MOKE imaging system. First, our table-top system has the potential to be easily extended to experiments such as pump and probe imaging spectroscopy. In addition, our system would allow us fast visualization of the magnetic domain structures including the complex dielectric tensor in the magnetic microstructures. Thus, we believe that our developed L-MOKE imaging system is a powerful tool for characterizing the material properties of magnetic microstructures, and can be utilized to reveal the dynamic behavior of the magnetic microstructure for device applications.

In conclusion, we have reported in this paper the L-MOKE imaging system with a CCD camera using the rotating compensator polarimetry technique. Precise management of the CCD exposure timing which is linked to the angle of the compensator enables us to obtain the L-MOKE images with high SNR. We succeeded in quantitatively estimating the complex magneto-optic constant Q of the Ni₈₁ Fe₁₉ sample by fitting the dependence of the L-MOKE signal on the angle of incidence, and achieved the imaging of |Q| for the rectangular Ni₈₁ Fe₁₉ microstructure sample. The quantitative assessment of Q, which is specific to the magnetic materials, will be useful to determine spatial magnetic components. In particular, our table-top system provides a novel platform for characterizing the material properties of various magnetic materials.

We thank Prof. T. Tachizaki and M. Sozawa for experimental help. This work was partly supported by JSPS KAKENHI Grant Number JP26249052, and a Grant-in-Aid from the Ministry of Education, Culture, Sports, Science, and Technology, Japan for the Photon Frontier Network Program.