Dual-Wavelength FPT Topography

• Dual-wavelength FPT is an advanced quantitative phase-imaging technique that combines complex field reconstructions from two illumination wavelengths to extend the unambiguous height range.
• It employs rigorous joint phase retrieval and robust unwrapping strategies to achieve sub-micrometer lateral resolution and nanometer height precision in semiconductor and industrial metrology.
• Practical guidelines, such as proper wavelength selection, numerical aperture matching, and noise regularization, are critical for accurate topographic reconstruction of quasi-2D surfaces.

Dual-Wavelength Fourier Ptychographic Topography (FPT) is an advanced quantitative phase-imaging technique that extends the unambiguous height range of reflection-mode Fourier ptychographic microscopy by synthesizing height information from complex field reconstructions at two distinct illumination wavelengths. This approach enables height mapping far beyond the intrinsic $\lambda/2$ phase-wrapping limit of single-wavelength FPT, providing sub-micrometer lateral resolution and nanometer-scale height precision on samples of interest in semiconductor and industrial metrology. The methodology incorporates a rigorous model of multi-wavelength illumination, joint phase retrieval, and robust unwrapping strategies to achieve accurate, high-fidelity quantitative topography on quasi-2D surfaces, particularly where phase wrapping and aspect ratio effects challenge conventional single-wavelength methods (Shen et al., 9 Dec 2025, Gao et al., 15 Dec 2024).

1. Theoretical Foundations and Forward Model

Dual-wavelength FPT inherits its core physical principles from reflective Fourier ptychographic microscopy (FPM), which acquires a series of intensity images under varied angles of coherent plane-wave illumination at a fixed wavelength. For a sample with height profile $h(\mathbf{r})$ and albedo $a(\mathbf{r})$, the reflection-mode FPT forward model at wavelength $\lambda$ is:

$$O_\lambda(\mathbf{r}) = a(\mathbf{r})\,\exp\!\big[i\,\varphi(\mathbf{r})\bigr], \qquad \varphi(\mathbf{r}) = \frac{4\pi}{\lambda}\,h(\mathbf{r})$$

The $4\pi$ term arises from the round-trip optical path at $45^\circ$ incidence. Each illumination angle imparts a spatial carrier $$S_n(\mathbf{r}) = \exp[i\,\mathbf{k}_n \cdot \mathbf{r}]$$. Following pupil filtering and detector imaging, the forward-modeled intensity is:

$$I^s_{n,\lambda}(\mathbf{r}) = \left| \mathscr{F}^{-1} \left\{\mathscr{F}\bigl[ O_\lambda S_n \bigr] P \right\} \right|^2$$

where $P(\mathbf{k})$ denotes the pupil function, which also captures defocus and aberrations. The model is extended for dual-wavelength acquisition by sequentially or simultaneously illuminating with two wavelengths, reconstructing independent complex fields, and leveraging their phase relationship for extended-range topography (Shen et al., 9 Dec 2025, Gao et al., 15 Dec 2024).

In the presence of multi-layer or buried interfaces, as in semiconductor manufacturing, the forward model is further generalized to account for multiple coherently superposed wavefronts corresponding to each interface, with the measured intensity formulated as an incoherent sum over modes after filtering cross-interference terms. This enables robust interface-specific reconstruction and accurate topographic mapping when multiple reflection paths contribute to the detected signal (Gao et al., 15 Dec 2024).

2. Synthetic Wavelength and Height Range Extension

A single-wavelength FPT system can only recover phase unambiguously over a $[\pm \lambda/4]$ optical path difference due to $2\pi$ phase wrapping, leading to an unambiguous height range of $\lambda/2$. Dual-wavelength FPT extends this by introducing a synthetic wavelength $\lambda_s$ defined as:

$$\lambda_s = \frac{\lambda_1 \lambda_2}{|\lambda_1 - \lambda_2|}$$

Given wrapped phase maps $\hat{\varphi}_1 = \mathcal{W}(4\pi h/\lambda_1)$, $\hat{\varphi}_2 = \mathcal{W}(4\pi h/\lambda_2)$ for wavelengths $\lambda_1, \lambda_2$, their difference encodes the phase at $\lambda_s$:

$$\varphi_s = \varphi_1 - \varphi_2 = \frac{4\pi h}{\lambda_s} \pmod{2\pi}$$

This increases the unambiguous range to $\lambda_s/2 \gg \lambda_{1,2}/2$, contingent on $\lambda_1$ and $\lambda_2$ spacing. However, direct phase differencing amplifies noise proportionally to $\lambda_s/|\lambda_1-\lambda_2|$, necessitating robust unwrapping mechanisms (Shen et al., 9 Dec 2025).

In interface-multiplexed platforms, post-reconstruction phase maps $\varphi^{(m)}(\mathbf{r})$ for each wavelength are combined via the same synthetic-wavelength relation to retrieve relative height maps, calibrated as needed by comparison with external references such as atomic force microscopy (AFM) (Gao et al., 15 Dec 2024).

3. Reconstruction Algorithms and Unwrapping Strategies

The phase retrieval problem in dual-wavelength FPT entails solving for the complex object field $O_\lambda(\mathbf{r})$ and pupil $P(\mathbf{k})$ that best explain the measured intensity images for each illumination condition and wavelength. The optimization minimizes:

$$\min_{O, P} \sum_{n=1}^N \left\| \sqrt{I^s_{n,\lambda}} - \sqrt{I^m_{n,\lambda}} \right\|_2^2 + R(O) + R(P)$$

where $R$ incorporates $\ell_2$, isotropic TV, and circular TV regularization terms. Circular TV penalizes angular discontinuities on $S^1$ and avoids introducing spurious edges at phase boundaries (Shen et al., 9 Dec 2025).

For dual-wavelength synthesis, a two-stage unwrapping is employed:

1. Wrapped-number search: For each pixel, integer pairs $(k_1, k_2) \in \mathbb{Z}^2$ are enumerated to find the pair $(K_1, K_2)$ minimizing $|h_1(\mathbf{r}; k_1) - h_2(\mathbf{r}; k_2)|$, where $h_i$ are candidate heights from each wavelength. This enforces per-pixel cross-wavelength height consistency without noise-amplifying phase subtraction.
2. Global Convex Refinement: A convex optimization with isotropic TV and soft-constraint penalties further regularizes the initial height map $h^*(\mathbf{r})$, encouraging spatial smoothness while preserving true topographic steps and ensuring estimates remain within feasible bounds given by $h_{\min}, h_{\max}$. Solvers like primal–dual splitting or ADMM efficiently yield the final map (Shen et al., 9 Dec 2025).

For interface-multiplexed reflection FPT, iterative algorithms such as Douglas–Rachford and ePIE variants are utilized, leveraging autocorrelation filtering to separate interface contributions and employing multi-slice angular-spectrum propagation to reconstruct both probe and object simultaneously across wavelengths (Gao et al., 15 Dec 2024).

4. Experimental Validation and Performance Analysis

Simulation and experimental validation confirm the height-range extension and quantitative accuracy of dual-wavelength FPT. Representative experimental setups involve RGB LED arrays or spatially overlapped lasers (e.g., 633 nm and 532 nm), scanned either rasterwise or with overlapping grid patterns. Imaging is performed with high-resolution, large-FOV detectors (e.g., 2048×2048 CCD, 0.08 m working distance, or 4k CMOS at 2.74 µm pitch) and objectives with numerical apertures up to 0.28 (Shen et al., 9 Dec 2025, Gao et al., 15 Dec 2024).

Key performance metrics include:

Metric	Value/Range (Experimental)	Context
Lateral resolution	~4 nm (10–90% edge width)	Limited by synthetic NA and wavelength (Gao et al., 15 Dec 2024)
Axial (height) precision	≤10 nm (vs. AFM reference)	Noise-limited, depends on phase stability (Gao et al., 15 Dec 2024)
Unambiguous height range	Up to $\lambda_s/2$ (micron-scale)	Extended versus $\lambda/2$ limit (Shen et al., 9 Dec 2025)
Field of view	~10 μm × 10 μm	Determined by scan range and optics (Gao et al., 15 Dec 2024)

Simulations using full-wave Modified Born Series solvers on structured silicon (heights up to ±0.6 μm, widths 1–2 μm, AR up to 0.6) show RMS height errors <50 nm when AR < 0.5. In experimental FPT setups, dual-wavelength surface profiles track Zygo-interferometer reference measurements across ±1 μm and preserve sub-micron lateral features. Direct comparison against AFM or interferometry provides independent calibration and validates accuracy across a broad range (Shen et al., 9 Dec 2025, Gao et al., 15 Dec 2024).

5. Limitations and Predictors of Performance

While dual-wavelength FPT substantially extends the topographic height range, several constraints emerge from the physics of ptychographic imaging and the nature of the forward model:

• Aspect Ratio (AR; $H/W$): As the depth-to-width ratio of surface features increases, geometric self-shadowing and wave-diffraction effects reduce optical visibility under high-angle illumination. Ray optics predict loss of visibility at $$\mathrm{AR}_{\rm geo} \approx 1/(2\,\mathrm{NA}_{\rm illum})$$, but full-wave simulations indicate phase transfer breaks down for $\mathrm{AR} \gtrsim 0.5-0.75$, causing phase unwrapping errors, loss of high-frequency detail, and gross topography errors.
• Phase Modulation Transfer Function (ph-MTF): The ph-MTF, $$\mathrm{phMTF}(f) = |\hat{\varphi}(f)| / |\varphi_{\rm true}(f)|$$, quantifies frequency-dependent phase transfer fidelity. For high aspect ratio or deep surface features, ph-MTF exhibits early roll-off, corresponding to the observed collapse in dual-wavelength unwrapping reliability.
• Noise Amplification: The synthetic wavelength $\lambda_s$ grows rapidly as $\lambda_1 \to \lambda_2$, but noise in the phase difference is amplified by $\lambda_s / |\lambda_1 - \lambda_2|$. This effect motivates the use of wrapped-number search and regularization rather than naive differencing.
• Surface Quasi-2D Assumption: The underlying forward model assumes a surface structure fully visible under all illumination angles. Volumetric or highly occluded samples, or those with AR outside the established range, are not reliably reconstructed by surface-based FPT (Shen et al., 9 Dec 2025).

A plausible implication is that for true 3D or highly nonplanar structures, alternative inversion strategies, e.g., tomographic or confocal approaches, may be preferable.

6. Practical Guidelines and Applications

Dual-wavelength FPT is particularly suited for quantitative topography of semiconductor wafers, patterned silicon, and other quasi-planar samples in manufacturing and industrial metrology, where feature height exceeds the single-wavelength range but volumetric thickness is limited. Practical system design involves:

• Wavelength Selection: Choose $\lambda_1, \lambda_2$ to yield a synthetic wavelength $\lambda_s$ that covers the expected dynamic range, typically with $\Delta\lambda/\lambda \sim 10\text{–}20\%$ to balance height range and noise.
• Numerical Aperture: Match the synthetic NA ($\mathrm{NA}_{\rm obj} + \mathrm{NA}_{\rm illum}$) to the lateral resolution target; $\mathrm{NA} \sim 0.5\text{–}0.6$ suffices for sub-micron features.
• Aspect Ratio Constraint: Restrict feature aspect ratios to $\mathrm{AR} \lesssim 0.5$ for robust, reliable unwrapping and high-fidelity height recovery. For AR approaching 0.75, unwrapping failures and loss of contrast become prevalent.
• Regularization Weights: Empirically, wrapped-phase TV weights $\beta_{\mathrm{arg}} \sim 10^{-2}\text{–} 10^{-1}$ and amplitude TV $\beta_{\mathrm{ampl}} \sim 10^{-3}\text{–}10^{-2}$, with post-processing TV and soft-bound parameters in the range of a few nm$^{-1}$ and $\rho \sim 10$, balance smoothing and edge preservation (Shen et al., 9 Dec 2025).
• Spectral Contrast: Element-specific reflection (e.g., different reflectivity of chromium features at 532 nm versus 633 nm) provides chemical specificity in amplitude channels, in addition to quantitative height mapping (Gao et al., 15 Dec 2024).

Applications are validated both on standard resolution targets and patterned silicon chips, yielding ≲10 nm axial precision and lateral resolution constrained by the objective and scan geometry, with performance verified against AFM and interferometric techniques (Shen et al., 9 Dec 2025, Gao et al., 15 Dec 2024).

7. Extensions, Future Directions, and Outlook

Advances in dual-wavelength FPT motivate several future directions:

• Multi-wavelength and Single-shot Implementation: Spectrum multiplexing with additional wavelength channels or tunable comb sources could further extend the attainable height range and permit element-specific topography in a single acquisition (Gao et al., 15 Dec 2024).
• Tabletop EUV Sources: Integration with extreme ultraviolet sources (∼13 nm) is anticipated to enable sub-10 nm spatial resolution, further extending the capabilities for nanoscale metrology in semiconductor manufacturing (Gao et al., 15 Dec 2024).
• Dynamic and Volumetric Imaging: Combining dual-wavelength or spectrum-multiplexed approaches with multi-angle acquisitions or real-time snapshot architectures may realize three-dimensional dynamic imaging, with possible integration into in-situ inspection workflows.
• Improvement of Noise Handling: Algorithmic developments in global refinement and robust phase unwrapping, particularly for handling higher aspect ratio features or samples with challenging reflectivity contrast, remain active areas of research.

The combination of robust unwrapping, synthetic wavelength engineering, and rigorous forward and inverse modeling positions dual-wavelength FPT as a key tool for quantitative surface metrology where high lateral resolution, nanometer-scale height sensitivity, and extended dynamic range are simultaneously required (Shen et al., 9 Dec 2025, Gao et al., 15 Dec 2024).