Magneto-Optical Kerr Effect (MOKE)

• MOKE is defined as the change in polarization and intensity of reflected light due to magnetic order, enabling contactless detection of various magnetic states.
• Different MOKE geometries (polar, longitudinal, transverse, quadratic) allow precise identification of magnetic anisotropies and nontrivial spin textures in materials.
• Advanced experimental techniques use high-sensitivity spectroscopy and imaging to quantitatively extract magneto-optical constants and study complex magnetic phenomena.

The magneto-optical Kerr effect (MOKE) describes the change in polarization state and intensity of reflected electromagnetic radiation—most often visible or infrared light—upon interaction with a magnetically ordered material. MOKE arises from off-diagonal terms in the dielectric tensor induced by magnetic order and is a widely utilized, contactless probe of diverse ferro-, ferri-, and antiferromagnetic states, spin accumulations, and topologically nontrivial magnetic textures across metals, insulators, and semiconductors. Originating as a spectral fingerprint of time-reversal symmetry breaking, MOKE encompasses multiple geometries (polar, longitudinal, transverse, and orthogonal), exhibits linear and nonlinear (quadratic) response to magnetization, and manifests through multiple physical mechanisms—including noncollinear spin structures, A-type ordering, and Berry curvature multipoles—which extend far beyond conventional ferromagnetic spin–orbit-driven scenarios.

1. Fundamental Theory and Mechanisms

The MOKE is governed by the optical dielectric tensor $\epsilon_{ij}(\omega)$, which develops magneto-optical off-diagonal components in the presence of a magnetic order parameter. The linear expansions for small magnetization $M$ take the form

$$\epsilon_{ij} = \epsilon_{ij}^{(0)} + \sum_k K_{ijk} M_k + \sum_{kl} G_{ijkl} M_k M_l + \ldots$$

where $K_{ijk}$ yields the linear (first-order) MOKE and $G_{ijkl}$ the quadratic MOKE (QMOKE) (Silber et al., 2019).

The key observable—the complex Kerr angle—$$\Phi_K(\omega) = \theta_K(\omega) + i\varepsilon_K(\omega)$$—relates to the Fresnel reflection matrix $r_{ij}$ of the sample, commonly through

$$\Phi_K(\omega) \approx \frac{r_{ps}(\omega)}{r_{pp}(\omega)}$$

for the p-polarized configuration (where $r_{ps}$ gives the cross-polarization and $r_{pp}$ the co-polarized amplitude) (Jan et al., 2017, Silber et al., 2019). At normal incidence (polar geometry), and for weak magneto-optic coupling (dipolar), the off-diagonal dielectric tensor elements produce a Kerr rotation

$$\theta_K(\omega) \approx \mathrm{Re}\left[ \frac{\epsilon_{xy}(\omega)}{\epsilon_{xx}(\omega) - 1} \right],$$

and Kerr ellipticity

$$\varepsilon_K(\omega) \approx \mathrm{Im}\left[ \frac{\epsilon_{xy}(\omega)}{\epsilon_{xx}(\omega) - 1} \right].$$

In the limits of small off-diagonal conductivity, the more general relation is

$$\theta_K(\omega) + i\varepsilon_K(\omega) = -\frac{\sigma_{xy}(\omega)} {\sigma_{xx}(\omega) \sqrt{1 + i\frac{4\pi}{\omega}\sigma_{xx}(\omega)}}$$

where $\sigma_{ij}(\omega)$ are the components of the optical conductivity tensor (Pan et al., 8 Dec 2025, Ortiz et al., 2021, Farhang et al., 13 Jul 2025, Sunko et al., 22 Apr 2025).

Magneto-optical response depends critically on the symmetry of both the crystal structure and magnetic ordering. In conventional (dipolar) MOKE, the off-diagonal term is proportional to the component of the magnetization projected along the relevant light-matter interaction geometry (e.g., in polar MOKE, $M_z$). In systems with lower symmetry or higher-order effects, quadratic terms and Berry curvature multipoles induce additional Kerr signals, including those orthogonal to $M$ (Pan et al., 13 Dec 2024).

Nontrivial mechanisms for MOKE include scalar spin chirality in noncoplanar spin textures, which can drive a finite $\sigma_{xy}$ and Kerr effect even in the absence of net magnetization ($M$) and spin-orbit coupling, via emergent real-space Berry phases (Farhang et al., 13 Jul 2025).

2. MOKE Geometries and Spectroscopic Signatures

MOKE is classified by incident beam geometry and the alignment of the magnetization vector:

• Polar MOKE probes out-of-plane magnetization at (or near) normal incidence, sensitive primarily to $M_z$, and is associated with rotation of the polarization axis (Jan et al., 2017, Pan et al., 8 Dec 2025).
• Longitudinal MOKE captures in-plane ($M_x$ or $M_y$) components with oblique incidence, yielding changes primarily via the $M$ component in the plane of incidence (Jan et al., 2017, Greening et al., 1 Apr 2025).
• Transverse MOKE (TMOKE) detects reflectivity or intensity changes proportional to $M$ components orthogonal to both the light's plane of incidence and the surface normal, and is described by the normalized reflectivity difference: $\delta = 2 \frac{R(B) - R(-B)}{R(B) + R(-B)}$ with explicit analytical forms for $\delta(\omega)$ in the vicinity of narrow Zeeman-split optical resonances (Borovkova et al., 2018).
• Quadratic MOKE (QMOKE) arises from second-order terms in $M$, producing effects proportional to $M^2$ or products such as $M_x M_y$. QMOKE can dominate the Kerr response in certain phases (e.g., austenite of Ni$_2$MnGa, cubic Fe) and is parameterized by tensors $G_s$, $2G_{44}$, and their anisotropy $\Delta G$ (Silber et al., 2019, Jan et al., 2017).
• Orthogonal MOKE geometry leverages Berry curvature multipoles to enable detection of magnetization components perpendicular to the Poynting vector, producing rotation signals (with $C_3$-symmetry angular dependence) even for in-plane $M$ under normal incidence (Pan et al., 13 Dec 2024).
• Spin accumulation MOKE (SA-MOKE) enables detection of buried ferromagnetic domains by probing the Kerr signal generated by pump-pulse-induced spin accumulation diffusing through thick capping layers (Rodriguez et al., 17 Nov 2025).

Geometry	Sensitivity	Probe direction	Kerr signal depends on
Polar	$M_z$	Normal incidence	$\epsilon_{xy}$
Longitudinal	$M_x$ or $M_y$	Oblique incidence	$\epsilon_{xz}$ or $\epsilon_{yz}$
Transverse	$M$ transverse	Oblique incidence	Reflectivity change
Quadratic	$M_iM_j$	Any	$\epsilon_{ij}^{(2)}$
Orthogonal	$M$ in-plane	Normal incidence	Berry multipole/cubic symmetry

3. Advanced Theoretical Mechanisms and Material Classes

The conventional association of MOKE with ferromagnetic order and spin–orbit coupling is now superseded by several diverse mechanisms:

• Noncollinear Antiferromagnets and Scalar Spin Chirality: Systems with real-space scalar spin chirality $$X_{ijk} = \vec{S}_i \cdot (\vec{S}_j \times \vec{S}_k)$$ (e.g., Co$_{1/3}$TaS$_2$) display large, SOC-independent polar Kerr effects, with conductivity $\sigma_{xy}(\omega)$ derived from the Berry phase acquired by conduction electrons traversing noncoplanar spin textures (Farhang et al., 13 Jul 2025).
• Altermagnetic Insulators: In hematite (α-Fe$_2$O$_3$), MOKE is driven dominantly by the Néel vector $N$, appearing at linear order in SOC, while contributions from net magnetization and magnetic field are two orders of magnitude smaller (Pan et al., 8 Dec 2025).
• A-type Antiferromagnets: In layered AFMs (e.g., MnBi$_2$Te$_4$), despite bulk $\mathcal{T}S_{1/2}$ symmetry forbidding magneto-optic gyrotropy in transmission, surface sensitivity of MOKE allows finite Kerr signals without net magnetization. The effect is described by an antisymmetric surface dielectric tensor and robust against thickness and field variations (Sunko et al., 22 Apr 2025).
• 2D Heterostructures: Ferroelectric–antiferromagnetic van der Waals heterostructures (CrI$_3$/In$_2$Se$_3$/CrI$_3$) activate MOKE via simultaneous breaking of inversion ($\mathcal{I}$, by polarization) and time-reversal ($\mathcal{T}$, by AFM), enabling controllable, switchable Kerr signals with potential for ultra-compact memory applications (Ding et al., 2023). Similar symmetry-derived MOKE occurs in noncollinear 2D AFMs that break $\mathcal{TI}$ (e.g., HfFeCl$_6$) (Zhou et al., 21 Mar 2025).
• Fully Compensated Ferrimagnets: Even at net zero magnetization, a strong MOKE can arise from highly spin-polarized conduction bands (e.g., Mn$_2$Ru$_x$Ga), reflecting helicity-dependent Drude response tied to a dominant sublattice (Fleischer et al., 2018).

4. Experimental Methodologies and Instrumentation

Sophisticated instrumentation has been developed for high-sensitivity, multidimensional, and ultrafast MOKE measurements:

• Fourier-space MOKE/Ellipsometry: By mapping the angular (Fourier) distribution of reflected intensity onto a camera and fitting to the linear-Kerr model, all components of the magnetization vector and optical constants can be extracted in a single measurement with high accuracy, including under single-shot acquisition (Sandoval et al., 2022).
• Reference-free sub-THz Kerr Spectroscopy: Use of a modified Martin–Puplett interferometer allows sub-milliradian concurrent measurement of Kerr rotation and ellipticity at 0.1–1 THz, facilitating high-field, cryogenic studies of quantum materials (Moshe et al., 27 Jul 2024).
• High-field, pulsed Measurements: Plug-and-play ferrule-based fiber fixtures, when combined with all-fiber Sagnac interferometers and phase-resolved lock-in amplification, enable MOKE measurements up to 43 T (beyond previous DC limits) with <0.1 mrad sensitivity and millisecond time resolution (Ikeda et al., 9 Sep 2025).
• Contrast Enhancement via Perfect Optical Absorption (POA): Engineered multilayer stacks achieve near-complete suppression of non-magnetic reflectance, amplifying the MOKE signal in ultrathin films (e.g., 20° Kerr rotation in 1 nm Co), supporting analyzer-free and real-time, sub-diffraction-limited imaging (Kim et al., 2019).
• Geometry-selective Suppression: Optical setups incorporating double-reflection and waveplate manipulation permit isolation/suppression of polar or longitudinal MOKE signals, even in birefringent thin-film heterostructures, delivering suppression factors >100 (Greening et al., 1 Apr 2025).
• Wide-field MOKE Microscopy: Köhler-illuminated, microscope-based systems enable direct imaging and quantification of spin–orbit torques and domain dynamics in microstructured and unpatterned films (Tsai et al., 2017, Chvykov et al., 2012).
• Spin Accumulation Deep Imaging: SA-MOKE uses optical pump–probe spin injection to image magnetic contrast through metallic overlayers up to 140 nm thick, exceeding the optical penetration depth by more than 10× (Rodriguez et al., 17 Nov 2025).

5. Materials, Spectroscopy, and Quantitative Relationships

MOKE spectroscopy enables extraction of magneto-optical constants, magnetic anisotropy, domain wall dynamics, and phase transitions across a broad material spectrum:

• Line Shapes and Resonance Regimes: In confined quantum wells or narrow optical resonances (e.g., (Cd,Mn)Te), TMOKE spectra transition from S-shaped linear lineshapes (for Zeeman splitting $\Delta \ll$ linewidth $\Gamma$) to saturated split peaks (for $\Delta \gg \Gamma$), with explicit analytical forms for $\delta(\omega)$ and peak amplitudes (Borovkova et al., 2018).
• Quadratic vs Linear Regimes: In bcc-Fe, QMOKE spectral parameters ($G_s$, $2G_{44}$, $\Delta G$) and their anisotropy are benchmarked against DFT-LDA calculation, capturing fine structure in Im $G_s$ and magnetic linear dichroism (Silber et al., 2019). In Ni$_2$MnGa, the austenite phase above 338 K is characterized by dominant quadratic Kerr signals (Jan et al., 2017).
• Magnitude and Sensitivity: Kerr rotation magnitudes vary from sub-milliradian (antiferromagnets, compensated ferrimagnets, topological insulators) to tens of degrees (POA-enhanced ultrathin ferromagnetic films) (Kim et al., 2019, Pan et al., 8 Dec 2025, De et al., 2020).
• Spectroscopic Selectivity: Material- and transition-specific resonance enhancement enables layer-selective or depth-resolved probing, e.g., utilizing wavelength-tuned exciton or interband transitions in semiconductors, 2D magnets, or noble metals with strong spin–orbit coupling (Borovkova et al., 2018, Ortiz et al., 2021, Ding et al., 2023).

6. Applications and Future Directions

MOKE serves as a probe and control element for magnetization, spin current, and topological order in a vast range of advanced materials and devices:

• Vector Tomography and 3D Magnetometry: Simultaneous application of multiple MOKE geometries allows vector-resolved measurement of magnetization, critical for spintronics, quantum information, and magnetic sensing (Borovkova et al., 2018, Pan et al., 13 Dec 2024).
• Antiferromagnetic and Topological Devices: MOKE imaging of AFM domains in A-type layered systems, even at the zero net magnetization, provides a path for optically readable, nonvolatile AFM memory and logic (Sunko et al., 22 Apr 2025, Zhou et al., 21 Mar 2025, Pan et al., 8 Dec 2025).
• Spin Accumulation Sensing: Enhanced Kerr detection in gold and other spin–orbit materials yields sub-µrad sensitivity to spin density, benefiting ultrafast spintronic and opto-spintronics platforms (Ortiz et al., 2021, Rodriguez et al., 17 Nov 2025).
• Electrically Switchable MOKE: Heterostructures combining TIs and AFMs allow for large, electrically gated Kerr rotations controlled via Fermi level tuning, promising for on-chip isolators, modulators, and memory at THz frequencies (De et al., 2020).
• Optical Probing of Exotic Spin Textures: Topological MOKE in chiral and noncoplanar antiferromagnets permits stray-field-immune, ultrafast optical imaging and control of scalar spin chirality and related quantum phases (Farhang et al., 13 Jul 2025).
• High-frequency and Cryogenic MOKE: Extension to sub-THz and operation at sub-Kelvin and multi-tesla regimes expands the reach to superconductors, 2D materials, and quantum magnets (Moshe et al., 27 Jul 2024, Ikeda et al., 9 Sep 2025).
• Analyzer-free and Sub-diffraction Imaging: POA methodologies and advances in spin accumulation imaging facilitate domain structure detection beyond standard optical or electrical limits (Kim et al., 2019, Rodriguez et al., 17 Nov 2025).

7. Tabular Summary of Representative MOKE Regimes and Phenomena

Physical Regime	Key Mechanism/Parameter	MOKE Signature	Reference
Collinear ferromagnet	SOC-induced $\epsilon_{xy}$	Linear Kerr rotation proportional to $M$	(Jan et al., 2017)
Quadratic/cubic symmetry	$\bm{G}$ tensor, $M^2$	Anisotropic QMOKE, peak at $3d\rightarrow3d$ transitions	(Silber et al., 2019)
Noncoplanar AFM (scalar chirality)	Berry phase, $X_{ijk}$	Large Kerr effect without net $M$ and SOC	(Farhang et al., 13 Jul 2025)
Altermagnetic insulator	Néel-vector, linear-in-SOC term	Strong MOKE, $M$-independent, AFM domains optically resolved	(Pan et al., 8 Dec 2025)
A-type AFM (layered)	Surface-broken $\mathcal{T}S_{1/2}$	Surface-reflection Kerr even at $M=0$	(Sunko et al., 22 Apr 2025)
Topological insulator/AFM	Proximity-induced symmetry breaking	Cavity-resonant Kerr, electrically switchable ($>6°$)	(De et al., 2020)
2D noncollinear/chiral AFM	Broken $\mathcal{T}$, $\mathcal{TI}$	Distinct, switchable Kerr spectra by chirality state	(Zhou et al., 21 Mar 2025, Ding et al., 2023)
Perfect optical absorption	Multilayer interference, POA	$|\theta_K| \gg 1°$, analyzer-free contrast	(Kim et al., 2019)
Spin accumulation (deep MOKE)	Pump-induced transient, spin diffusion	Imaging through $>100$ nm metallic caps	(Rodriguez et al., 17 Nov 2025)
Fourier-space MOKE	Angular mapping, full vector fitting	Fast, vector-sensitive Kerr/ellipsometry extraction	(Sandoval et al., 2022)

MOKE stands as an indispensable, microscopically rich probe of time-reversal symmetry breaking and nontrivial magnetic structure, with growing reach into antiferromagnetic, compensated, topological, and spintronic material systems. Its geometry dependence, susceptibility to both linear and higher-order tensorial effects, and the depth of its spectroscopic and imaging capability continue to underpin progress across condensed matter physics and device engineering.