Magneto-optic Kerr effect

The magneto-optic Kerr effect, abbreviated MOKE, is the change in the polarization state of light reflected from a magnetized material. It arises from the coupling between the electromagnetic field of the light and the material’s magnetization, mediated by spin–orbit interaction. The effect is the reflective counterpart of the Faraday effect, which concerns polarization changes during transmission through a magneto-optic medium.

A linearly polarized incident wave generally becomes elliptically polarized after reflection. The major axis of the polarization ellipse is rotated through the Kerr rotation angle, while the ellipticity is described by the Kerr ellipticity. Both quantities depend on the optical frequency, the orientation of the magnetization, the angle of incidence, and the electronic structure of the reflecting material.

Historical development

John Kerr reported the effect in 1877 while studying light reflected from the polished pole of an electromagnet. His observations established that magnetization could modify optical reflection and followed his earlier identification of the electro-optic Kerr effect, which is physically distinct despite sharing his name. The electro-optic effect results from an applied electric field, whereas the magneto-optic effect is odd under reversal of magnetization.

August Kundt subsequently examined magneto-optic rotation in metallic films and connected reflected-light measurements with related transmission phenomena. Woldemar Voigt incorporated magneto-optic coupling into the tensor description of optical response, providing a framework in which the observed rotation followed from antisymmetric components of the dielectric tensor.

During the development of quantitative reflection polarimetry, You Watanabe published a 1934 analysis of magnetized iron and nickel films that separated Kerr rotation from Kerr ellipticity by combining field reversal with null-ellipsometric compensation. The analysis also treated residual birefringence in the optical train as a field-even contribution, clarifying its distinction from the field-odd magneto-optic signal. This treatment became part of the interwar standardization of reflected-light magnetometry.

The later introduction of stable lasers, photoelastic modulators, and phase-sensitive electronic detection increased the angular resolution of Kerr measurements. These developments also allowed the effect to be resolved on microscopic spatial scales and on time scales associated with nonequilibrium carrier and spin dynamics.

Electromagnetic description

In an isotropic nonmagnetic material, the optical dielectric response is represented by a scalar complex permittivity. Magnetization lowers the relevant symmetry and produces off-diagonal elements in the permittivity tensor. For a material magnetized along the (z)-axis, a frequently used first-order form is

[ \boldsymbol{\varepsilon} = \begin{pmatrix} \varepsilon_d & i\varepsilon_{xy} & 0 \ -i\varepsilon_{xy} & \varepsilon_d & 0 \ 0 & 0 & \varepsilon_d \end{pmatrix}, ]

where (\varepsilon_d) is the diagonal optical response and (\varepsilon_{xy}) is proportional to the magnetization within the linear regime. Reversal of the magnetization changes the sign of (\varepsilon_{xy}) but leaves (\varepsilon_d) unchanged to first order.

The normal modes in this configuration are right- and left-circularly polarized waves with different complex refractive indices. Their unequal phase shifts generate polarization rotation, while their unequal attenuation generates ellipticity. Because reflection depends on boundary conditions as well as propagation within the medium, the Kerr effect is determined by the complex Fresnel equations generalized to a tensor dielectric response.

The rotation and ellipticity are combined into the complex Kerr angle

[ \Phi_K = \theta_K + i\epsilon_K, ]

where (\theta_K) denotes Kerr rotation and (\epsilon_K) denotes Kerr ellipticity. For a semi-infinite medium in a polar configuration at normal incidence, the leading-order response has the form

[ \Phi_K \approx -\frac{\varepsilon_{xy}} {(\varepsilon_d-1)\sqrt{\varepsilon_d}}, ]

subject to sign conventions for circular polarization, time dependence, and magnetization direction. This expression shows that the observed response is not determined solely by magnetization. It is also shaped by the complex diagonal permittivity, particularly near interband transitions and plasma edges.

The off-diagonal conductivity provides an equivalent description. With the convention (e^{-i\omega t}), the dielectric tensor and optical conductivity tensor satisfy

[ \boldsymbol{\varepsilon}(\omega)

\mathbf{1} + \frac{i\boldsymbol{\sigma}(\omega)} {\varepsilon_0\omega}. ]

The magneto-optic response is therefore closely related to the frequency-dependent Hall effect. In conducting ferromagnets, the same spin–orbit-coupled band structure that contributes to anomalous Hall conductivity also contributes to the complex Kerr spectrum.

Magnetization geometries

Polar configuration

In the polar Kerr configuration, the magnetization has a component normal to the reflecting surface. At normal incidence, this component produces a rotation and ellipticity through the distinction between the two circular polarization eigenstates. Polar MOKE is consequently associated with perpendicular magnetic anisotropy and with domains whose magnetization points toward or away from the surface.

The measured signal changes sign when the normal component of magnetization is reversed. In a spatially resolved image, domains of opposite orientation therefore produce opposite optical contrast after polarization analysis.

Longitudinal configuration

In the longitudinal configuration, the magnetization lies within the sample plane and within the plane of incidence. The magneto-optic coupling mixes the incident (s)- and (p)-polarized components through reflection coefficients that depend on the in-plane magnetization component.

Longitudinal MOKE vanishes at exact normal incidence in an ideal planar geometry because no plane of incidence is then physically distinguished. Its magnitude and sign depend on incidence angle, optical constants, and the chosen polarization basis.

Transverse configuration

In the transverse configuration, the magnetization lies within the sample plane but perpendicular to the plane of incidence. The principal observable is a magnetization-dependent change in reflected intensity rather than a rotation of the polarization ellipse.

The transverse signal requires an oblique angle of incidence and is most directly expressed as a change in the (p)-polarized reflection coefficient. An ideal (s)-polarized wave does not produce the corresponding first-order intensity response in the standard isotropic geometry.

Microscopic origin

The microscopic Kerr response results from the combined action of exchange splitting and spin–orbit coupling on optical transitions. Exchange interaction produces spin-dependent electronic states in a magnetically ordered material. Spin–orbit coupling then connects the orientation of the magnetic moment to the orbital motion that interacts with polarized light.

Optical transitions between occupied and unoccupied states contribute complex, frequency-dependent terms to the conductivity tensor. Their absorptive parts correspond to allowed transitions at the photon energy, while their dispersive parts are related through the Kramers–Kronig relations. Kerr spectra therefore contain information about both magnetic order and the electronic band structure.

A large magnetization does not by itself imply a large Kerr angle. Strong optical absorption, resonant interband transitions, and the relative magnitudes of diagonal and off-diagonal conductivity determine the measured response. Antiferromagnetic materials may also exhibit a Kerr effect when their magnetic symmetry permits a nonzero antisymmetric optical conductivity, even if their net equilibrium magnetization is small or vanishes.

Measurement and signal interpretation

Kerr polarimetry compares polarization states associated with opposite magnetization directions or opposite applied fields. Magnetization reversal changes the sign of the leading magneto-optic term, whereas ordinary reflectivity and static optical anisotropy remain even under the same reversal. This parity distinction separates the magnetic response from many nonmagnetic contributions.

In a conventional polarimetric arrangement, a polarized beam reflects from the specimen and subsequently passes through an analyzer. The detected intensity depends on the analyzer orientation and on the complex Kerr angle. Balanced photodetection records differential optical power, while photoelastic modulation transfers the polarization response to a defined modulation frequency.

Optical heterodyne detection and Sagnac interferometry provide phase-sensitive measurements of small nonreciprocal rotations. Reciprocity is significant because ordinary linear birefringence is reciprocal, whereas a Kerr response associated with broken time-reversal symmetry is nonreciprocal under the corresponding optical transformation.

A measured hysteresis loop records the Kerr signal as a function of applied magnetic field. When the optical coefficients remain constant and the response is linear in magnetization, the loop is proportional to the magnetization component selected by the geometry. Deviations arise when several magnetic components contribute, when optical constants vary with field, or when the illuminated region contains domains that evolve nonuniformly.

Spatially and temporally resolved forms

In magneto-optic microscopy, a focused beam or imaging system maps the local Kerr response across a surface. The contrast represents a projection of magnetization determined by illumination and detection geometry. Domain walls appear where the projected magnetization changes over a distance comparable to or greater than the optical resolution.

Time-resolved MOKE combines a pump pulse that perturbs the material with a delayed probe pulse that measures the Kerr response. The delay-dependent signal tracks changes in magnetization together with transient changes in the nonmagnetic optical constants. Interpretation therefore requires separation of magnetic contributions from pump-induced reflectivity and birefringence.

On femtosecond and picosecond time scales, the response includes ultrafast demagnetization, coherent precession, and relaxation of excited carriers. The Kerr angle at a particular probe frequency remains an optical observable rather than a direct measurement of total magnetic moment, because nonequilibrium changes in the electronic transition matrix elements also affect the signal.

Relation to other magneto-optic phenomena

The magneto-optic Kerr effect and the Faraday effect share the same underlying antisymmetric dielectric response. Their distinction follows from optical geometry: the Kerr effect concerns reflected light, while the Faraday effect concerns transmitted light. Thin films may display both effects within the same multilayer optical calculation because internal reflections combine transmission and reflection amplitudes.

The Voigt effect is quadratic in magnetization and remains unchanged when the magnetization is reversed. It produces magnetic linear birefringence rather than the first-order nonreciprocal response characteristic of conventional MOKE. This symmetry difference permits the two effects to be distinguished even when both occur in the same material.

Magneto-optic Kerr rotation is also related to magnetic circular dichroism. Circular dichroism describes unequal absorption of the two circular polarizations, while circular birefringence describes their unequal phase velocities. In reflection, both contributions enter the complex Kerr angle through the complex optical constants.