The scanning SQUID microscope (SSM) is a tool to visualize the magnetic field on a two-dimensional plane by scanning a superconducting pickup coil on the plane. The magnetic flux through the coil is detected by a SQUID magnetometer. Since SQUID is the most sensitive magnetometer available today, SSM applications in pure and applied sciences are promising. Recently, the development of SSM with a three-dimensional (3D) vector pickup coil [1] has opened new possibilities for efficient measurements of magnetic field and magnetization vectors in various situations, from nanoscale to macroscale. Especially applications to nanosciences, including electronics and spintronics, and macroscopic systems in biology and geology are of particular interest.
It should be noted that the SSM image is not an actual magnetic field image measured at the scanning plane but is a convolved one with the form factor of the pickup coil. Because the pickup coil itself is superconducting, the form factor becomes complex by the demagnetizing effect. In the case of 3D vector pickup coils, the form factor is more complicated than the conventional single-loop coil. On the other hand, the 3D vector pickup coil is an assembly of multiple coils, and the measured data includes more information than the conventional SSM. Therefore, by utilizing the increased information, more effective observations can be expected. Although the analysis of 3-D vector coils is a complex problem, we describe an attempt to solve these problems using modern numerical methods.
The problem with the numerical analysis of SSM images is that some of the magnetic field data is lost due to the convolution of the coil form factor. In our previous study [2], we proposed a method to suppress noise amplification caused by the lost data by adding a wave number-dependent correction factor for each Fourier component to smooth the image. In this study, we use the singular value decomposition of the inverse transformation matrix, a method frequently used in modern linear analysis. A remarkable feature of this method is that we can control the smoothness of the image by selecting singular values larger than a particular threshold value. This method has the great advantage of reducing the amount of numerical computations required for the actual inverse transformation. Furthermore, this method can handle multi-dimensional data in a unified manner because the inverse transformation matrix does not need to be square. We demonstrate how this method works for actual experimental data in addition to the model simulations. We also discuss the further possibility of applying it to vector magnetization measurement.
[1] T. D. Vu, T. H. Ho, S. Miyajima, M. Toji, Y. Ninomiya, H. Shishido, M. Maezawa, M. Hidaka, M. Hayashi, S. Kawamata and T. Ishida, Supercond. Sci. Technol. 32, 115006 (2019).
[2] M. Hayashi, H. Ebisawa, H. T. Huy, and T. Ishida, Appl. Phys. Lett. 100,182601 (2012).
This work is partially supported by Grants-in-Aid for Scientific Research (JP22K04246) from JSPS.