Magneto-optic vector magnetometry for sensor applications
Introduction
Magneto-optic (MO) vector magnetometry is a powerful measurement technique based on a polarization change of light reflecting or transmitting by structure containing magnetic layers. MO effects have been recently extensively applied to study thin-film magnetic phenomena as magnetic anisotropy, magnetization reversal, oscillatory interlayer coupling, and magnetization dynamics. In addition, the magneto-optic effects find applications as sensors of magnetic field and electric current [1], [2], [3], [4], [5], [6], [7], [8], [9], [10].
The MO response has been found to be significantly affected by the quadratic or second-order magneto-optic effects [11], [12], [13]. Moreover, the quadratic MO effects are very sensitive to crystal symmetry and its anisotropy, i.e., oscillations of MO response as a function of the crystal orientation, has been modeled and experimentally observed [14], [15]. Similarly, Gridnev et al. [16] and Petukhov et al. [17] observed anisotropy of third-order MO effects.
The aim of this article is to describe how the magneto-optic response depends on components of the magnetization vector. We show that design of the precise magneto-optical sensors requires considering of vectorial MO response and higher-order MO effects.
In Section 2, the permittivity tensor of a magnetically ordered medium is defined up to third-order MO effects including symmetry consideration. Section 3 deals with the MO response in terms of the Jones reflection and transmission matrices. Influence of the magnetization components on the magneto-optic response and MO sensor applications is discussed in Section 4. Several advantages of magneto-optic sensor applications are presented.
Section snippets
Magneto-optic tensors
Magneto-optic properties of the magnetized crystal are completely described using the permittivity tensor , which can be expanded into a series: where is the permittivity tensor without magnetic ordering (the magnetization ). Kijk, Gijkl, and Hijklm are the components of the linear, quadratic, and cubic magneto-optic tensors, respectively [18]. The Einstein summation convention over coordinates x, y, z is
Magneto-optic response
In terms of the tensor components defined by Eq. (1) the MO response, i.e., the Jones reflection and transmission matrices, are calculated using Yeh’s matrix algebra [22], [23] in the form: rij and tij (i,j=s,p) denote the amplitude reflection and transmission coefficients, respectively. For example the rps coefficient is defined as the ratio of the x component (s-polarization) of the reflected light and the p component of the incident one. The coefficients are
MO vector magnetometry
In this section, we summarize the linear, quadratic (second-order), and cubic (third-order) MO effects for an interface and a single MO layer. Observable quantity dependence on the magnetization components MP, ML and MT is of considerable practical interest for the magneto-optic vector magnetometry [13], [24], [25]. The influence of the magnetization components on the amplitude reflection and transmission coefficients, rij, tij,i,j=s,p, and the MO rotation θ, and ellipticity ϵ is presented
Conclusion
Dependence of the magneto-optic response on components of the magnetization vector has been discussed including higher-order MO terms. The second-order (quadratic) and third-order (cubic) magneto-optic effects have to be considered for MO sensors and their adjustment and calibration.
Acknowledgements
Partial support from the Grant Agency of Czech Republic #202/01/0077, #202/03/0776 and from the projects KONTAKT ME 507, ME 508 (Ministry of Education, Youth and Sports, Czech Republic) is acknowledged.
K. Postava received his MSc degree in optics and optoelectronics at Palacky University in 1993 and PhD degree in optics and optoelectronics from INSA Toulouse (France) and Palacky University (Czech Republic) in 1997. He is currently as a lecturer at Technical University in Ostrava. His research area cowers optics and magneto-optics of thin films, spectroscopic ellipsometry, and vectorial magnetometry. He is a member of SPIE and IUPAP.
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K. Postava received his MSc degree in optics and optoelectronics at Palacky University in 1993 and PhD degree in optics and optoelectronics from INSA Toulouse (France) and Palacky University (Czech Republic) in 1997. He is currently as a lecturer at Technical University in Ostrava. His research area cowers optics and magneto-optics of thin films, spectroscopic ellipsometry, and vectorial magnetometry. He is a member of SPIE and IUPAP.
T. Yamaguchi graduated from Science University of Tokyo in 1965 and received his Doctor degree of engineering from The University of Tokyo in 1973. Currently, he is a professor of Research Institute of Electronics, Shizuoka University. His research interests include optical evaluation of surfaces and thin films based on the spectroscopic ellipsometry. He is a member of The Japan Society of Applied Physics, The Surface Science Society of Japan and The Optical Society of Japan.
J. Pištora received his MSc degree in theory of electromagnetic field from Czech Technical University in 1977 and PhD degree in experimental physics from Charles University in 1984. Currently, he is Professor of applied physics at Technical University in Ostrava. His research interests include magneto-optics, periodical structures and sensors. He is a member of SPIE, OSA, IEEE, ICO, EOS.