🔬 optics

The 2D approximation quickly breaks down in reflection ptychography

This paper establishes that the standard two-dimensional thin-sample model is often invalid for reflection ptychography due to significantly stricter thickness constraints compared to transmission geometries, but demonstrates that incorporating a three-dimensional weak-scattering description into the forward model resolves resulting artifacts and enables quantitative depth-sensitive reconstructions.

Original authors: Sander Senhorst, Stefan Witte, Wim Coene

Published 2026-04-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sander Senhorst, Stefan Witte, Wim Coene

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to take a perfect photograph of a complex, multi-layered cake using a special camera that only sees light patterns, not the cake itself. This is essentially what ptychography does: it shines a laser-like beam on a sample, scans it around, and uses the resulting diffraction patterns to mathematically reconstruct a high-resolution image of the object.

For years, scientists have been using a "shortcut" to do this. They assume the object is so thin that it's basically a flat, 2D sheet (like a piece of paper). This works great for transmission (shooting light through the object). But when scientists tried to use this same shortcut for reflection (shooting light off a surface, like looking in a mirror), things started to go wrong.

This paper is the "warning label" and the "fix-it guide" for that shortcut. Here is the breakdown in simple terms:

1. The Problem: The "Flat Earth" Assumption

In the old way of thinking, scientists treated the sample like a flat piece of paper. They assumed that when the light hits it, bounces off, and comes back, the light didn't care how "tall" the object was.

The Analogy: Imagine you are throwing a ball at a wall. If the wall is a flat sheet of paper, the ball bounces back predictably. But if the wall is actually a thick, multi-layered sandwich, the ball hits the top layer, then the middle, then the bottom, and bounces back at slightly different times and angles.

The paper argues that in reflection ptychography (especially with extreme ultraviolet light used in chip manufacturing), the sample is rarely just a flat sheet. It has depth. When you ignore that depth, your mathematical model gets confused. It's like trying to navigate a 3D city using a 2D map; you might get lost, or end up in a building that doesn't exist.

2. The Discovery: Reflection is Picky

The authors did the math to figure out exactly how thick a sample can be before this "flat sheet" shortcut breaks.

The Analogy: Think of the light waves as ripples in a pond.

Transmission (Looking through): The ripples move straight through. The "rules" for how thick the water can be before the ripples get messy are fairly lenient.
Reflection (Looking at a mirror): The ripples hit the surface and bounce back. Because of the angle of the bounce, the "rules" become incredibly strict.

The paper found that for reflection, the sample needs to be 10 to 100 times thinner than what is allowed in transmission before the image starts to distort. If the sample is too thick, the math predicts the image will look weird, especially at certain "magic angles" where the light waves cancel each other out (called Bragg minima).

3. The Consequence: Ghosts and Glitches

When you use the old, flat model on a thick, reflective sample, the computer tries to force the data into a 2D shape. Since the data actually comes from a 3D object, the computer invents "ghosts" or distortions to make the math work.

The Analogy: Imagine trying to describe a 3D sculpture using only a 2D shadow. If the sculpture is complex, the shadow might look like a monster with extra limbs that aren't really there. In the paper's simulations, when the sample thickness hit a specific "destructive" point, the reconstructed image of the object would literally vanish or turn into a weird derivative (like a blurry outline) because the math hit a "zero" point.

4. The Solution: The "3D Glasses"

The authors didn't just point out the problem; they fixed it. They developed a new mathematical model that acknowledges the sample has depth.

The Analogy: Instead of using a 2D map, they gave the computer 3D glasses.

They updated the "forward model" (the part of the math that predicts how light should behave) to include the time it takes light to travel through the thickness of the sample.
The Result: When they used this new model, the "ghosts" disappeared. The images became sharp and accurate, even for thicker samples.
Bonus: Because the model now understands depth, it can actually measure the thickness of the sample just by looking at the reflection data. It's like looking at a reflection and being able to tell exactly how thick the glass is without touching it.

Why Does This Matter?

This is huge for the world of lithography (making computer chips).

Modern chips are built in tall, stacked layers.
Scientists need to inspect these stacks using reflection ptychography.
If they keep using the old "flat" math, they might think a chip layer is defective when it's actually fine, or vice versa.
This paper tells them: "Stop using the flat math. Use the 3D math, or your measurements will be wrong."

Summary

The Old Way: Treat reflective samples as flat sheets. (Works for thin things, fails for thick things).
The Problem: Reflection is much more sensitive to thickness than transmission. The "flat" math breaks down quickly, creating fake artifacts.
The Fix: Use a new 3D math model that accounts for depth.
The Payoff: You get accurate images of thick, reflective samples and can even measure their thickness automatically.

In short, the paper says: "Stop pretending reflective samples are flat pancakes. They are lasagnas, and if you want a good picture, you need to model the layers."

1. Problem Statement

Ptychography is a powerful coherent diffractive imaging (CDI) technique used to reconstruct sample structures with diffraction-limited resolution. While the standard mathematical framework for ptychography assumes a two-dimensional (2D) thin-sample model (where the sample is treated as a phase/amplitude mask with no depth dependence), this approximation has been widely applied to reflection geometries without rigorous validation.

In reflection ptychography (particularly at Extreme Ultraviolet, EUV, wavelengths around 13.5 nm), samples are often thick stacks of materials. The authors identify a critical gap:

The 2D model assumes the exit wave is a simple product of the probe and a 2D object function.
In reality, reflection involves complex 3D wave propagation, interference between layers, and depth-dependent phase shifts.
Previous experimental demonstrations have not established the validity limits of the 2D approximation. Consequently, reconstructions may suffer from unexplained artifacts, particularly near Bragg minima, leading to incorrect interpretations of surface topography or material properties.

2. Methodology

The authors developed a rigorous theoretical framework and verified it through numerical simulations.

A. Theoretical Development

3D Weak-Scattering Model: They extended the ptychographic theory to a fully three-dimensional description using the first Born approximation.
Reciprocal Space Analysis: Instead of real-space convolution, they analyzed the problem in reciprocal space (Fourier domain) using the Ewald sphere concept.
- They defined the scattered field as a 3D convolution of the probe and object functions, constrained to the Ewald sphere (representing elastic scattering).
- They introduced a "sifting" delta function ( $\delta_{ES}$ ) to enforce that only wavevectors satisfying the dispersion relation ( $|\mathbf{k}| = k$ ) contribute to the far-field.
Derivation of Approximation Criteria:
- They compared the exact 3D model against the standard 2D model to determine when a 2D approximation ( $O'$ and $P'$ ) could mimic the 3D reality.
- They analyzed the variation of the axial Fourier component ( $\xi_z$ ) of the object function.
- They derived explicit thickness criteria ( $t_{max}$ ) based on the range of sampled axial spatial frequencies ( $\Delta \xi_z$ ).
- They distinguished between "Strict" approximations (requiring constant phase shift over the entire aperture) and "Relaxed" approximations (where the phase shift is constant only over the detector or illumination range).

B. Simulation Verification

Ground Truth Model: They simulated a "2+1D" reflection scenario: a sample with lateral patterning (Siemens star) and a specific axial thickness ( $t$ ), modeled as two scattering layers separated by distance $t$ .
Reconstruction: They reconstructed these simulated datasets using a standard 2D ptychographic solver (ignoring depth) and compared the results to the ground truth.
Optimization: They also tested a 2+1D forward model that explicitly includes depth-dependent propagation to see if it could recover the true thickness and correct distortions.

3. Key Contributions

A. Theoretical Limits on Sample Thickness

The paper derives that reflection geometries impose far stricter thin-sample conditions than transmission geometries.

Transmission: The sampled axial frequency range is dominated by the curvature of the Ewald sphere. The thickness limit scales with $1/NA^2$ .
Reflection: The sampled axial frequency range is dominated by the rotation of the Ewald sphere (due to the grazing incidence angle). The thickness limit scales with $1/\sin(\theta_i)$ .
Quantitative Impact: For a representative EUV geometry ( $\theta_i = 60^\circ$ $θ_{i} = 6 0^{\circ}$ , $NA_i=0.05$ $N A_{i} = 0.05$ , $NA_d=0.3$ $N A_{d} = 0.3$ ), the allowable thickness for a valid 2D reconstruction is reduced by one to two orders of magnitude compared to transmission.
- Example: While transmission might tolerate thicknesses up to $\sim 800\lambda$ , reflection may require thicknesses $< 12\lambda$ (relaxed) or even $< 1.7\lambda$ (strict) to avoid artifacts.

B. Identification of Artifacts

The theory predicts that 2D reconstructions will fail when the sample thickness causes the axial Fourier component to vary significantly across the aperture.

Bragg Minima: The most severe distortions occur near destructive Bragg conditions (where the specular reflection is minimized). Here, the zeroth-order term vanishes, and the reconstruction becomes dominated by higher-order variations, leading to severe artifacts (e.g., the reconstructed object appearing as a derivative of the true object).
Periodic Errors: The reconstruction error oscillates periodically with thickness, corresponding to the Bragg interference period ( $\lambda/2\cos\theta$ ).

C. Depth-Sensitive Reconstruction

The authors demonstrate that incorporating the correct depth-dependent propagation into the forward model (the 2+1D approach) resolves these distortions.

This approach allows for the recovery of the sample thickness directly from the diffraction data without needing to vary the incidence angle or use spectral diversity.
Simulations showed that the thickness could be optimized to an accuracy of $\pm 2.7$ m $\lambda$ (approx. 50 pm for 18.5 nm wavelength).

4. Results

Validation of Theory: Simulations confirmed that 2D reconstructions exhibit thickness-dependent artifacts exactly as predicted by the derived theory.
Artifact Characteristics:
- At low thicknesses, 2D reconstructions are accurate.
- As thickness increases, residual loss in optimization increases periodically.
- At destructive Bragg conditions ( $t \approx 0.5\lambda, 1.5\lambda, \dots$ ), the reconstructed object function loses contrast and exhibits derivative-like features.
Correction Success: When using the 2+1D model, the distortions were eliminated, and the true lateral structure and thickness were recovered, even when the initial thickness guess was incorrect (provided the guess was within a convergence basin of $\sim 0.2\lambda$ ).

5. Significance

Practical Validity Limits: The paper establishes clear, quantitative boundaries for when 2D reflection ptychography is valid. It warns that for many modern lithography stacks (which are often thicker than a few wavelengths), standard 2D models may produce non-physical artifacts that could be misinterpreted as surface features.
Metrology for EUV Lithography: Given the push for smaller features and taller stacks in EUV lithography, this work provides a critical tool for metrology. It suggests that ignoring depth effects in reflection ptychography could lead to significant measurement errors.
Path to 3D Imaging: The work identifies a pathway toward quantitative, depth-sensitive 3D reconstructions using reflection ptychography alone. By modeling the depth, researchers can not only correct artifacts but also measure sample thickness with nanometer-scale precision, opening new possibilities for inspecting complex multilayer structures without destructive cross-sectioning.

In summary, the paper fundamentally challenges the routine application of 2D models to reflection ptychography, proving that the approximation breaks down much faster in reflection than in transmission, and offers a robust 3D modeling solution to overcome these limitations.