A Fourier-Space Approach to Physics-Informed Magnetization Reconstruction from Nitrogen-Vacancy Measurements

Abstract

Reconstructing complex magnetization textures from nitrogen-vacancy (NV) magnetometry stray-field measurements presents a challenging inverse problem. In this work, we introduce a physics-informed method that addresses this by incorporating the full micromagnetic energy directly into the variational formulation. Built on a PyTorch backend, our forward model integrates an auto-differentiable finite-differences micromagnetic framework with FFT-based stray-field calculations and Fourier-space upward continuation. This enables efficient gradient-based optimization via the adjoint method and allows the sensor-sample distance to be treated as an optimization parameter. By doing so, we eliminate the experimental uncertainty arising from unknown NV implantation depths and surface oxidation layers. Validation on synthetic data demonstrates high-fidelity reconstruction of spin textures and precise sensor height estimation. Furthermore, when applied to NV measurements of the van der Waals ferromagnet $Fe_{3-x}GaTe_2$, the method reconstructs the previously unknown NV-sample distance and physically plausible magnetization textures, which accurately reproduce the experimental observations.

Figures (8)

• Figure 1: Physics-informed reconstruction of magnetization textures from NV magnetometry. The schematic illustrates the workflow for overcoming the ill-posed nature of stray-field imaging. (a) NV magnetometry measures the stray field projection $H^\mathrm{meas}$ at a unkown distance $d_\mathrm{NV}$ from the sample. (b) Multiple distinct magnetization configurations can produce a nearly identical stray field, making direct inversion unreliable. (c) Our physics-informed framework solves this by coupling a differentiable forward model with a joint objective function. This objective balances data fidelity with micromagnetic energy. Through iterative gradient-based optimization, the framework simultaneously reconstructs the magnetization $\mathbf{m}$ and refines the sensor distance $d_\mathrm{NV}$. This yields the final parameters $\mathbf{m}^\ast$ and $d_\mathrm{NV}^\ast$ as an energetically favorable configuration that remains consistent with the experimental observations.
• Figure 2: Synthetic data generation. Individual components ($m_x, m_y, m_z$) of reference magnetization $\mathbf{m}_\mathrm{ref}$ and the corresponding stray field map $H^\mathrm{meas}$ that an NV sensor would sense at a sensor distance of $d_\mathrm{NV,ref} = 80nm$, including additive Gaussian noise scaled to 3% of the RMS signal amplitude.
• Figure 3: L-curve analysis for synthetic measurement data. (a) The characteristic L-Curve illustrates the trade-off between field mismatch and total energy, identifying the optimal regularization parameter $\lambda_{\mathrm{opt}} = 2.2 \times 10^{17}$. (b) Converged sensor height $d_{\mathrm{NV}}^{\ast}$ as a function of the regularization parameter $\lambda$.
• Figure 4: Impact of regularization strength on magnetization reconstruction accuracy. In the unconstrained case, the reconstruction is highly diffuse and exhibits a large deviation from the ground truth. The optimal parameter, $\lambda_{\mathrm{opt}} = 2.2 \times 10^{17}$, providing the best balance between data fidelity and physical consistency also corresponds to the reconstruction with the minimum L2 error norm, $\|\mathbf{m}^\ast - \mathbf{m}_{\mathrm{ref}}\|_2$.
• Figure 5: Experimental NV magnetometry measurements of a 100 nm thick $\mathrm{Fe}_\mathrm{3-x}\mathrm{GaTe}_\mathrm{2}$ thin film. The measurements are displayed from left to right with scan areas of: (1) $800 \times 800$ nm$^2$ ($100 \times 100$ pixels); (2) $1000 \times 1000$ nm$^2$ ($150 \times 150$ pixels); and (3) $5000 \times 5000$ nm$^2$ ($200 \times 200$ pixels). The exact NV sensor-sample distances are unknown. The first field map $H^\mathrm{meas}$ (left) was used for the magnetization reconstruction in this work.