Dual-wavelength Fourier Ptychographic Topography

TL;DR

This work extends Fourier Ptychographic Topography (FPT) from a λ/2 unambiguous height limit by introducing dual-wavelength FPT, which uses a synthetic wavelength λ_s = λ_1 λ_2 / |λ_1−λ_2| to achieve an unambiguous range of λ_s/2 while preserving lateral resolution. A noise-robust per-pixel wrapped-number search paired with digital refocusing and a global circular-TV–regularized refinement enables accurate height reconstructions across extended topographies, validated by both simulations (Modified Born Series) and experiments on structured silicon. The study also introduces practical distance- and frequency-domain predictors—structure AR and ph-MTF—that quantify reconstruction fidelity and determine practical limits, showing robust performance up to AR ≈ 0.75 before wave-optical 3D effects undermine the surface-based forward model. Overall, dual-wavelength FPT delivers wide-field, high-resolution topography with extended height range, offering a scalable metrology tool for semiconductor and industrial inspection, and points toward integrating multi-wavelength strategies with 3D diffraction tomography for higher-AR structures.

Abstract

We introduce a dual-wavelength Fourier ptychographic topography (FPT) method that extends the lambda/2 height-range limit of single-wavelength FPT. By reconstructing complex fields at two illumination wavelengths and exploiting their phase difference, the method achieves an effective synthetic wavelength lambda_s and an unambiguous range of lambda_s/2 without reducing lateral resolution. A noise-robust wrapped-number search is used to select per-pixel integer pairs (k1, k2), and a global refinement with circular TV regularization and soft bounds improves stability and preserves height discontinuities. The approach is validated through rigorous scattering-model-based simulations and experiments on structured silicon samples, demonstrating accurate height recovery in regimes where single-wavelength FPT exhibits phase wrapping. We analyze the limits of the FPT forward model and identify aspect ratio (AR) and phase modulation transfer function (ph-MTF) as key predictors of reconstruction fidelity. Simulations and experiments show that increasing AR beyond a practical threshold causes loss of high-frequency phase transfer and destabilizes dual-wavelength unwrapping. Within this AR range, dual-wavelength FPT provides robust, high-resolution topography suitable for semiconductor and industrial metrology.

Figures (12)

• Figure 1: Dual-wavelength FPT pipeline. (a) The imaging system captures intensity measurements $I_{m,n}$ under programmable angular illumination using red and green LEDs. The object $O(x,y)$ and pupil function $P(u,v)$ are jointly reconstructed by minimizing the mismatch between the measured images $I_{m,n}$ and the simulated images $I_{s,n}$. Gradients $\partial L/\partial O$ and $\partial L/\partial P$ are used for alternating updates. Each wavelength $i$ produces a wrapped phase map $\hat{\varphi}_{i}$. (b) Dual-wavelength phase unwrapping algorithm. Sequential red and green illuminations generate wrapped phase maps $\hat{\varphi}_{r}$ and $\hat{\varphi}_{g}$ with Algorithm \ref{['alg:FPT']}, each corresponding to multiple candidate height values $h_{k_r}$ and $h_{k_g}$ due to the $2\pi$ ambiguity. By evaluating all candidate index pairs $(k_r, k_g)$ at each pixel, a consistent pair $(k_r', k_g')$ is selected to enable robust unwrapping and height recovery across wavelengths, as shown in Algorithm \ref{['alg:dwFPT']}.
• Figure 2: Comparison of conventional TV and cTV on wrapped phase data. (a) True phase $\varphi$ (top left) and wrapped phase $\hat{\varphi}$ (bottom left), together with their gradient magnitudes computed by conventional TV (top right) and cTV (bottom right). While conventional TV produces strong artificial edges at the $\pm\pi$ wrapping boundaries, cTV correctly interprets them as continuous on $S^1$ and suppresses such spurious gradients. (b) 1D cross sections: the wrapped phase (red) exhibits jumps at $\pm\pi$ (top), which lead to sharp peaks in the conventional TV profile (bottom, red). In contrast, cTV (blue) only responds to the true object boundary, yielding physically consistent edge peaks under the $2\pi$ periodicity.
• Figure 3: Dual-wavelength FPT reconstruction simulation results. (a) The ground truth height map of the test sample is composed of structured silicon features with varying topography. (b) Simulated intensity images generated using the MBS model under red and green illumination. (c) Wrapped phase reconstructions from single-wavelength FPT under red and green LEDs show phase aliased due to changes in height. (d) Phase cross-section from single-wavelength FPT under red and green LEDs. (e) Reconstructed height map using the dual-wavelength phase unwrapping algorithm, which accurately recovers continuous surface profiles. (f) Height cross-section comparison between the ground truth and the reconstruction demonstrating high quantitative agreement.
• Figure 4: Dual-wavelength phase-unwrapping mechanism and performance comparison. (a) Schematic of the wrapped-number search. Each wrapped phase $\varphi_1$ and $\varphi_2$ maps to multiple candidate heights differing by integer multiples of $2\pi$. Cross-wavelength consistency is used to identify the correct physical height (orange line). (b) Error matrix over integer-pair combinations $(k_1,k_2)$. The optimal pair (green box) is the global minimum of the consistency metric, yielding the most reliable height estimate. (c) Quantitative comparison of three unwrapping strategies: the baseline "Basic" phase difference method, PUMA, and the proposed approach. The baseline method amplifies noise and deviates significantly from the ground truth. PUMA improves accuracy but is computationally expensive and retains larger residual errors. The proposed method closely matches the ground truth across all regions with substantially lower noise sensitivity.
• Figure 5: Experimental validation of single-wavelength FPT. (a) Bright-field reflectance image of a USAF resolution target used to assess amplitude resolution. (b) Ground-truth height profile of a calibrated 100 nm step structure measured by a Zygo interferometer. (c) Reconstructed amplitude images under red and green LED illumination, demonstrating recovery of high-frequency features. (d) Reconstructed phase maps for the red and green wavelengths, with corresponding line profiles showing agreement with the Zygo reference. Red dotted lines: expected height of the structures.