Abstract
In this chapter the magneto-optical effects that are relevant for magnetic microscopy on layered structures are reviewed. This includes the conventional effects which occur at visible light frequencies as well as the X-ray-based effects. Before the phenomenology and physical origin of the different effects is presented, we firstly collect some optical and electromagnetic basics that are relevant for an understanding of the magneto-optical effects.
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Notes
- 1.
In the classical oscillator model, this deviation can be readily understood: due to the anisotropy of the binding forces, the charge will be displaced in the direction of the weakest restraint for a given force component and not in the direction of \(\varvec{E}\) which drives the charge. The field, induced in the material, will thus be oriented differently from \(\varvec{E}\) [166].
- 2.
The reason is that the Larmor-precession (\(<\)100 GHz) of the atomic moments cannot follow the high frequencies of the magnetic light vector, i.e. at optical frequencies the spins are too ‘slow‘ to follow the alternating magnetic field of the electromagnetic wave.
- 3.
For linearly- (or plane-) polarized light the orientation of the electric field is constant, while its magnitude and sign vary in time.
- 4.
It may appear somewhat counterintuitive that a phase lead is described by a negative phase angle. This is owed to the fact that we use the notation (\(\varvec{k}\cdot \varvec{z} - \omega t\)) for the phase [see (2.7)]. Had we used (\( \omega t - \varvec{k}\cdot \varvec{z}\)), the signs for the phase angles would have been interchanged.
- 5.
The absolute index of refraction \(n = c_\mathrm {0}/v\) is defined by the ratio of the speed \(c_\mathrm {0}\) of an electromagnetic wave in vacuum to the speed \(v\) in matter, see (2.10).
- 6.
The two attenuation coefficients may be frequency-dependent and vary in different ways with the frequency. If white light enters the crystal, the crystal will in general appear colored with the color depending on the vibrational direction of the incident light. For uniaxial crystals there are two characteristic colors, leading to the term ‘dichroism’.
- 7.
Today, retarders are also made of polyvinyl alcohol sheets that have been stretched so as to align their long-chain organic molecules. A permanently birefringent substance is created in this way even though the material is not crystalline. The foil is cemented between two glass plates by a filler with suitable refraction index.
- 8.
In magneto-optical microscopy the light beams behind the objective lens emerge in parallel bundles, so that the polarization vectors of each beam can be assumed to be co-planar to the polarization elements on the observation side (compensator and analyser), as well as to the image plane. All polarization aspects can therefore be treated two-dimensionally in a plane perpendicular to the optical axis of the microscope.
- 9.
The term “cubic crystal” is used in the sense that the crystal symmetry is cubic neglecting magnetization.
- 10.
Another widely used convention [177] postulates a zero trace for the second matrix. This amounts in a ferromagnet (because of \(\left| \varvec{m}\right| = 1\)) to adding an isotropic term, which is shifted into the first matrix in (2.58).
- 11.
As mentioned in Sect. 2.2.2, the Faraday rotation is non-reciprocal: if the light passes the material again in reversed direction, the rotation does not cancel but is rather doubled. This is different to the circular birefringence of optically active media. The reason is that the Faraday rotation is tied to the direction of the magnetization. In Sect. 2.4.2 we will see that the magnitude of rotation is proportional to the projection of \(\varvec{M}\) on the direction of propagation.
- 12.
This is the situation illustrated in Fig. 2.5d. The rotation angle \(\theta \) is defined by \(\tan \theta = \frac{E_{\mathrm {y}}}{E_{\mathrm {x}}} = \frac{\sin (-\rho (z)/2)}{\cos (\rho (z)/2)} = -\tan \frac{\rho (z)}{2}\), so that \(\theta = -\frac{\rho (z)}{2}\).
- 13.
The analysis of the Kerr effect at oblique light incidence, which is generally required to obtain a Kerr signal on in-plane domains (see Sect. 2.4.2), is more complicated [170]: Here the wave in the medium is a mixture of linearly and circularly polarized eigenmodes. For normal incidence, however, and with the magnetization either parallel or perpendicular to the surface, the symmetry is high enough so that these polarizations do give pure eigenmodes in the medium.
- 14.
- 15.
In case of the Voigt effect, orthogonally magnetized domains lead to the strongest contrast as shown in Sect. 2.5.
- 16.
The phenomenology is thus opposite to that of the magnetic circular birefringence and dichroism: there birefringence causes a rotation, whereas the dichroism leads to ellipticity (compare Fig. 2.11).
- 17.
In materials with a strong Kerr ellipticity (like Permalloy) it is, consequently, the other way round: here the Kerr effect is mainly adjusted by the compensator, whereas the Voigt effect requires an opening of the analyser.
- 18.
Note that in thin films the domains walls are of the Néel type with an in-plane rotation of magnetization. The same applies to the Néel cap of vortex walls in low-anisotropy bulk material [111]. In case of high-anisotropy materials, perpendicular components at domain walls might be present. But here the wall width is far below resolution anyway, so that the polar Kerr effect does not play a role.
- 19.
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Kuch, W., Schäfer, R., Fischer, P., Hillebrecht, F.U. (2015). Magneto-Optical Effects. In: Magnetic Microscopy of Layered Structures. Springer Series in Surface Sciences, vol 57. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44532-7_2
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DOI: https://doi.org/10.1007/978-3-662-44532-7_2
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