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

This paper introduces a physics-informed, Fourier-space reconstruction method using an auto-differentiable micromagnetic framework to accurately recover complex magnetization textures and simultaneously determine unknown sensor-sample distances from nitrogen-vacancy magnetometry data.

The Gist

The Big Picture: Solving a Magnetic Puzzle

Imagine you are trying to figure out what a hidden object looks like, but you can only see the shadow it casts on the wall. In physics, this is exactly what scientists face when they try to understand the magnetic "texture" inside a material.

They use a super-sensitive tool called a Nitrogen-Vacancy (NV) magnetometer. Think of this tool as a tiny, atomic-scale camera that takes a picture of the magnetic "shadow" (the stray field) hovering above a material.

The Problem:
The shadow is ambiguous. Just like a shadow of a ball and a shadow of a cube could look identical from a specific angle, many different magnetic shapes inside the material could create the exact same shadow on the surface. Furthermore, the camera isn't always at a fixed distance; it might be hovering slightly higher or lower than we think, which distorts the shadow even more.

The Solution:
The authors of this paper created a new "smart solver." Instead of just guessing the shape based on the shadow, they built a system that asks: "If I assume this shape is real, does it obey the laws of physics?"

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The Analogy: The "Physics-First" Detective

To understand their method, let's imagine a detective trying to reconstruct a crime scene from a blurry photo.

1. The Old Way (Data-Only)

A traditional detective looks at the blurry photo and tries to draw anything that fits the pixels.

• Result: They might draw a monster, a tree, or a car. It fits the pixels, but it looks weird and impossible in real life.
• In Physics: This leads to "unphysical" magnetic maps that look like static noise or fragmented messes. They fit the data but break the laws of nature.

2. The New Way (Physics-Informed)

The authors' detective is a Physics-First Detective. Before drawing a single line, they have a strict rulebook: "The suspect must be made of solid matter, not floating ghosts, and must follow the laws of gravity."

• The Rulebook: In this paper, the "rulebook" is Micromagnetic Energy. This is a mathematical formula that describes how magnetic atoms naturally want to behave (they want to be close to their neighbors, they want to point in specific directions, etc.).
• The Process: The detective tries to draw a shape.
  • If the shape requires the atoms to do something impossible (like pointing in two directions at once), the "Physics Rulebook" screams, "No! That costs too much energy!"
  • The detective adjusts the drawing until it fits the blurry photo AND obeys the rulebook.

The "Smart Camera" Trick

There is a second major problem: We don't know exactly how far the camera is from the object.

In the real world, the NV sensor is a tiny diamond chip. The magnetic material is a flake. There might be a layer of rust (oxidation) or glue between them. We don't know the exact distance. If we guess the distance wrong, the reconstructed image will be blurry or distorted.

The Paper's Innovation:
The authors didn't just guess the distance; they made the distance a variable in their math.

• Imagine the detective is also trying to figure out how far away the camera was.
• They run a simulation where they can slide the camera up and down.
• They ask: "At what distance does the shadow look most like the photo, while still obeying the physics rules?"
• Result: The system automatically calculates the perfect distance (about 80 nanometers in their experiment) without anyone needing to measure it physically.

How It Works (The "Magic" Behind the Scenes)

The paper uses a clever mathematical shortcut called Fourier Space.

• The Analogy: Imagine you are trying to send a message across a noisy room. Instead of shouting the whole sentence word-by-word (which is slow and gets distorted), you translate the sentence into a specific musical chord. You send the chord, and the receiver translates it back.
• In the Paper: The magnetic field is complex. Calculating it point-by-point is slow. By translating the problem into "Fourier Space" (like the musical chord), the computer can solve the math incredibly fast using a technique called FFT (Fast Fourier Transform).
• The "Auto-Differentiable" Part: The entire system is built on PyTorch (a popular AI framework). This means the computer can automatically calculate how to tweak the magnetic shape and the sensor distance to get a better result, step-by-step, just like a video game character learning to jump higher by trying, failing, and adjusting.

The Results: What Did They Find?

1. Synthetic Test: They tested it on a computer-generated "fake" magnetic world where they knew the answer. The method perfectly reconstructed the magnetic shape and the sensor distance.
2. Real World Test: They applied it to a real material called Fe3-xGaTe2 (a type of van der Waals magnet).
   • They didn't know the sensor distance.
   • They didn't know the exact magnetic texture.
   • The Outcome: The method successfully reconstructed a realistic magnetic texture (showing how the magnetic atoms are arranged) and told them the sensor was hovering about 80 nanometers away. This matched the physical reality of the experiment.

Why Does This Matter?

This is a huge step forward because:

• No More Guessing: Scientists no longer need to guess the distance between their sensor and the sample. The math figures it out.
• Real Physics: It stops scientists from seeing "ghosts" in the data. The results are guaranteed to be physically possible.
• New Material Discovery: By seeing the magnetic texture so clearly, scientists can better understand how new materials work, which is crucial for building faster, smaller, and more efficient computers and sensors in the future.

In a nutshell: The authors built a "smart mirror" that doesn't just reflect the magnetic shadow; it uses the laws of physics to fill in the missing details and even figures out how far away the mirror is, all automatically.

Technical Summary

1. Problem Statement

Reconstructing complex 3D magnetization textures from Nitrogen-Vacancy (NV) magnetometry stray-field measurements is a classic ill-posed inverse problem.

• Non-Uniqueness: Infinitely many distinct magnetization configurations can produce identical stray field maps at a sensor distance.
• Experimental Uncertainty: A critical parameter, the distance between the NV sensor and the sample surface ($d_{NV}$), is often unknown due to factors like NV implantation depth, surface oxidation layers, and physical gaps.
• Limitations of Existing Methods:
  • Deconvolution methods (e.g., qMFM) rely on instrument calibration and often only reconstruct surface charge equivalents.
  • Variational methods (e.g., Tikhonov regularization) use mathematical smoothness constraints that act as qualitative proxies for physics but lack quantitative rigor.
  • Deep Learning approaches often require training data or initial guesses and struggle with fully 3D textures without separating data-driven estimation from physics-based refinement.

2. Methodology

The authors propose a self-consistent, physics-informed reconstruction framework that integrates micromagnetic physics directly into the optimization loop.

A. Forward Model (Differentiable Simulation)

The core of the method is a fully differentiable forward model built on PyTorch and the magnum.np library:

• Micromagnetic Solver: Uses a finite-difference approach to simulate the magnetization ($m$) and calculate the demagnetizing field ($H_{dem}$).
• FFT-Based Stray Field: Calculates stray fields using Fast Fourier Transforms (FFT) with a demagnetization tensor kernel, achieving $O(N \log N)$ complexity.
• Fourier-Space Upward Continuation: A key innovation is the derivation of an exact transfer function in 2D Fourier space. This corrects for the vertical averaging inherent in finite-difference simulations (where the field is averaged over a cell height $\Delta z$) and extrapolates the field to an arbitrary sensor height $d_{NV}$.
  • The transfer function is: $\tilde{H}_{dem}(d_{NV}) = \langle\tilde{H}_{dem}\rangle \cdot \left(\frac{k\Delta z}{1 - e^{-k\Delta z}}\right) \cdot e^{-k(d_{NV}-z_0)}$.
• Projection: The vector field is projected onto the NV center's quantization axis ($n_{NV} \approx 55^\circ$) to generate a scalar field map comparable to experimental data.

B. Physics-Informed Loss Functional

The reconstruction is formulated as a variational optimization problem minimizing the loss function $J(m, d_{NV})$:
$$J(m, d_{NV}) = \underbrace{|H_{dem}(m, d_{NV}) - H_{meas}|}_{\text{Data Fidelity } (L_{data})} + \lambda \underbrace{E_{total}(m)}_{\text{Physical Regularization}}$$

• Data Fidelity: Minimizes the mismatch between the simulated and measured stray fields.
• Physical Regularization: Instead of heuristic smoothness terms, the method penalizes the total micromagnetic energy ($E_{total}$), which includes:
  • Exchange energy ($E_{ex}$)
  • Demagnetization energy ($E_{dem}$)
  • Uniaxial anisotropy ($E_{aniu}$)
  • Interfacial Dzyaloshinskii–Moriya interaction ($E_{dmii}$)
• Joint Optimization: Both the magnetization field $m$ and the sensor distance $d_{NV}$ are optimized simultaneously using gradient-based methods (Riemannian Adam optimizer to enforce $|m|=1$).

C. Regularization Selection

The trade-off parameter $\lambda$ is determined using the L-Curve method, which plots data mismatch against total energy to find the "corner" representing the optimal balance between fitting the data and maintaining physical stability.

3. Key Contributions

1. End-to-End Physics Integration: Replaces heuristic regularizers with quantitative micromagnetic energy terms, creating a transparent optimization process that requires no training data.
2. Simultaneous Distance Estimation: The framework treats the sensor-sample distance ($d_{NV}$) as an optimization variable, eliminating the need for independent calibration and resolving experimental uncertainties regarding implantation depth and oxidation.
3. Fourier-Space Correction: Introduces an analytical transfer function to decouple the simulation's vertical discretization from the measurement geometry, allowing accurate modeling of thick samples with coarse grids while calculating fields at arbitrary heights.
4. Manifold Optimization: Enforces the physical constraint $|m|=1$ (unit magnetization) directly via Riemannian optimization on the unit sphere manifold, avoiding artificial penalty terms.

4. Results

Synthetic Validation

• The method successfully reconstructed complex spin textures (e.g., Néel walls) and the sensor height from synthetic data with additive Gaussian noise.
• Unconstrained reconstruction ($\lambda=0$) resulted in fragmented, unphysical states.
• Optimal reconstruction ($\lambda_{opt}$) achieved high fidelity, accurately recovering the ground-truth magnetization and converging the sensor height to the true value (80 nm).

Experimental Application ($Fe_{3-x}GaTe_2$)

• Applied to NV measurements of a van der Waals ferromagnet ($Fe_{3-x}GaTe_2$) at room temperature.
• Sensor Distance: The method converged to an optimal sensor distance of $d_{NV} \approx 77\text{--}80$ nm, a value stable across a range of regularization strengths. This effectively quantified the total separation (oxidation + gap + implantation depth).
• Magnetization Texture: Reconstructed physically plausible domain wall structures that accurately reproduced the experimental stray field observations.
• Regularization Sensitivity:
  • Low $\lambda$: Overfitting to noise, resulting in fine-scale, unphysical variations.
  • High $\lambda$: Over-smoothing, failing to capture specific experimental features; the optimizer artificially increased $d_{NV}$ to dampen the simulated field to match the data.
  • Optimal $\lambda$: Balanced energy minimization with data fidelity.

5. Significance and Limitations

Significance:

• Quantitative Imaging: Provides a robust tool for quantitative magnetic imaging without prior knowledge of sensor geometry.
• Material Characterization: The sensitivity of the solution to specific energy terms (e.g., DMI sign) suggests the method can be used to identify underlying physical mechanisms and material parameters.
• Efficiency: The use of FFT and GPU acceleration makes the approach computationally feasible for large-scale 3D problems.

Limitations:

• Model Dependency: The reconstruction quality depends heavily on the accuracy of the assumed physical model (material parameters, anisotropy axes). Mismatches can lead to artifacts (e.g., artificial fragmentation to compensate for unknown defects).
• Non-Uniqueness: If multiple physical models produce equally good fits to the data, the inverse problem remains non-unique, particularly when the sensor is far from the sample or spatial resolution is low.
• ODMR Artifacts: The method is sensitive to Optically Detected Magnetic Resonance (ODMR) sign ambiguity artifacts, which occur when the stray field opposes and exceeds the bias field, requiring careful data selection.

In conclusion, this work establishes a powerful, self-consistent framework for reconstructing nanoscale magnetic textures, bridging the gap between raw experimental data and quantitative micromagnetic understanding by treating the sensor distance as a solvable variable within a physics-constrained optimization.