THz Fourier Ptychographic Imaging

• THz Fourier Ptychographic Imaging is a high-resolution method that synthesizes a large numerical aperture via controlled multi-angle illuminations and computational Fourier stitching.
• It employs iterative phase retrieval algorithms and precise calibration to reconstruct quantitative amplitude and phase images with sub-diffraction resolution.
• Experimental results demonstrate synthetic NA improvements from 0.2 to 0.7, achieving practical resolutions of 60–100 micrometers for advanced materials characterization.

Terahertz (THz) Fourier Ptychographic Imaging is a high-resolution imaging methodology that overcomes the spatial-frequency and resolution limitations inherent in conventional THz imaging architectures by synthesizing a large numerical aperture (NA) within the THz regime. This technique utilizes controlled multi-angle plane-wave illuminations combined with computational Fourier domain stitching and phase retrieval to reconstruct quantitative amplitude and phase images, achieving sub-diffraction resolution without extensive hardware modifications. By integrating iterative phase-retrieval algorithms with robust illumination calibration and optionally exploiting the spectral domain, contemporary THz Fourier ptychographic frameworks enable advanced imaging performance suitable for materials characterization, non-destructive evaluation, and hyperspectral analysis in complex media (Mukherjee et al., 4 Dec 2025, Kumar et al., 4 Oct 2024, Guerboukha et al., 2017).

1. Principle and Theoretical Foundation

Conventional THz imaging systems are fundamentally limited by the low numerical aperture (NA) of THz optics, resulting in poor diffraction-limited resolution, typically $\sim\lambda/(2\,\mathrm{NA})$ (e.g., $\approx180~\mu$m at $1$ THz for $\mathrm{NA} \sim 0.2$). Fourier ptychographic imaging in the THz spectral domain circumvents this limitation by sequentially illuminating the sample with plane waves at a discrete set of incident angles $(\theta_x, \theta_y)$ (Mukherjee et al., 4 Dec 2025). Each oblique illumination imparts a linear phase ramp to the object field $O(x,y)$, rigidly shifting its spatial-frequency (Fourier) spectrum by $$\Delta k=(k_{xn},k_{yn})=(2\pi/\lambda)[\sin\theta_x,\,\sin\theta_y]$$. A low-NA collection lens acquires only a localized patch of the shifted spectrum, but by stitching complementary patches from a series of $N$ illuminations in the Fourier domain, a much wider synthetic aperture is constructed:

$$\mathrm{NA}_{\mathrm{synth}} = \mathrm{NA}_{\mathrm{obj}} + \sin\theta_{\max}$$

and

$$k_{\max,\mathrm{synth}} = \frac{2\pi}{\lambda}[\mathrm{NA}_{\mathrm{obj}}+\sin\theta_{\max}].$$

This process effectively expands the spatial-frequency support and enables resolution surpassing hardware diffraction limits. In time-domain modalities, the k-space/frequency duality can be further exploited—broadband spectral sweeps probe the radial spatial-frequency axis, reducing mechanical scan requirements (Guerboukha et al., 2017). In the synthetic aperture formalism, angled THz pulses shift the pupil function in k-space, and the aggregate union across multiple angles forms a composite passband supporting higher spatial frequencies (Kumar et al., 4 Oct 2024).

2. Optical Systems and Data Acquisition

THz Fourier ptychographic platforms are configured to support precision angle-resolved illuminations and robust Fourier domain coverage. Mukherjee et al. employed a continuous-wave quantum-cascade laser (CW-QCL) at $3.5$ THz ($\lambda\approx85.7~\mu$m, $\sim13$ mW output) shaped by parabolic mirrors and spatial filtering (Mukherjee et al., 4 Dec 2025). Controlled angular scanning was realized via motorized kinematic mirrors (Thorlabs Z912) orchestrated along a spiral trajectory in k-space, covering up to $\sin\theta_{\max} \approx 0.5$ with $N=30$ discrete plane-wave illuminations. Collection was performed by a 4f lens system with NA $\approx 0.2$ and detected by a microbolometric THz camera (INO MICROXCAM-384i; $384\times288$ pixels, $35~\mu$m pitch).

To calibrate mechanical and thermal uncertainties in the angle control, Fourier transforms of raw intensity images were employed to detect autocorrelation arcs in k-space, yielding sub-pixel accurate estimates of the true illumination k-vectors via circular-edge detection routines. Acquisition protocols involved single-second exposures per frame, yielding total scan times of $\sim30$ seconds per dataset. Field-of-view was experiment-dependent: e.g., $17.5\times17.5$ mm$^2$ (simulation, $500\times500$ grid), $4.1\times3.2$ mm$^2$ (coffee-stirrer, $117\times91$ grid), and $10.1\times10.1$ mm$^2$ (banknote, $288\times288$ grid) (Mukherjee et al., 4 Dec 2025).

Spectral-broadband implementations rely on THz time-domain spectroscopy (THz-TDS) and single-pixel photoconductive antenna (PCA) detectors traversing angular positions on a Fourier-plane circle, with frequency sweeps replacing radial spatial-frequency scanning (Guerboukha et al., 2017).

3. Computational Reconstruction Algorithms

The reconstruction of high-resolution amplitude/phase images in Fourier ptychographic THz systems relies on inverse algorithms that integrate the multi-angle, multi-spectral dataset. For each illumination index $n$, the forward model for the collected intensity data is

$I_n(x,y) = |\psi_n(x,y) * p(x,y)|^2,$

where the exit field is $\psi_n(x,y)=O(x,y)\exp[i(k_{xn}x+k_{yn}y)]$. In Fourier space,

$$\mathcal{F}\{I_n\}(k) = [Õ(k-\Delta k_n)P(k)] \ast [Õ^*(k-\Delta k_n)P^*(k)],$$

where $Õ=\mathcal{F}\{O\}$ and $P=\mathcal{F}\{p\}$.

A joint-object/pupil iterative phase-retrieval protocol such as ePIE is iteratively applied. At each iteration, shifted spectrum patches are extracted, imposed with the measured amplitude, and updated according to:

$$\Delta Õ(k-\Delta k_n) = \alpha\,P^*(k)[E_{\mathrm{upd}}(k) - E_{\mathrm{proj}}(k)]\,/\, (|P(k)|^2+\epsilon_1)$$

$$Õ^{(t)}(k-\Delta k_n) = Õ^{(t-1)}(k-\Delta k_n) + \Delta Õ(k-\Delta k_n)$$

$$\Delta P(k)= \beta\,Õ^*(k-\Delta k_n)[E_{\mathrm{upd}}(k) - E_{\mathrm{proj}}(k)]\,/\, (|Õ(k-\Delta k_n)|^2+\epsilon_2)$$

with regularization and support enforcement outside the known NA (Mukherjee et al., 4 Dec 2025). Empirical parameter settings $\alpha,\,\beta \approx 1$, $\epsilon_1,\,\epsilon_2\approx10^{-3}$ yield robust convergence, achieving SSIM $\approx 0.8$ in $\sim 60$ iterations (simulation) or $0.85$ (experiment), with computation times $\sim 90$ s on consumer hardware.

For synthetic aperture THz imaging in the pulsed domain, the object $T(x,y;\omega)$ is optimally recovered by minimizing:

$$\hat{T} = \arg\min_{T}\sum_{a=1}^A\sum_{x,y}\left\|E_{\mathrm{out}}^{(a)}(x,y;\omega) - \Psi_a\{T\}(x,y;\omega)\right\|^2+\mathcal{R}[T]$$

where $\Psi_a\{\cdot\}$ encodes the full forward simulation for each angle, and $\mathcal{R}$ is a regularizer. Accelerated field-sensitive gradient descent (e.g., Nesterov’s algorithm) is employed for convergence (Kumar et al., 4 Oct 2024).

Hybrid approaches with single-pixel detectors perform “hybrid inverse transforms” wherein spectral and angular sweeps replace full raster scanning. The image is recovered via:

$$\widetilde{S}(x,y) = \int_0^{2\pi} d\theta \int_{\nu_{\min}}^{\nu_{\max}} d\nu \; \frac{p_0}{cF} \frac{U(p_0,\theta;\nu)}{U_{\mathrm{ref}}(\nu)} \exp \left[j\frac{2 \pi \nu}{cF} p_0(x\cos\theta + y\sin\theta)\right]$$

For amplitude imaging, $U_{\mathrm{ref}}(\nu)$ is set to $\frac{j\nu}{cF} U(0,0;\nu)$; for phase imaging, $\nu^1(j\nu/cF)U(0,0;\nu)$ (Guerboukha et al., 2017).

4. Experimental Performance and Resolution Metrics

THz Fourier ptychographic imaging has demonstrated significant resolution enhancement and quantitative imaging capabilities. In Mukherjee et al., the baseline optical Rayleigh limit of $260~\mu$m (NA $\approx 0.2$, $\lambda\approx85~\mu$m) is surpassed by constructing a synthetic NA $\approx 0.7$, leading to $\sim 60~\mu$m theoretical resolution. Empirical results support practical resolution of $100~\mu$m slits spaced by $140~\mu$m (simulation) and $230~\mu$m prongs separated by $150~\mu$m (experiment). Spatial-frequency bandwidth is expanded by $6.5\times$; synthetic NA increases $2.5\times$. SSIM reaches $0.8$–$0.85$ under both simulated and experimental conditions. SNR robustness is reported down to $15$ dB (Mukherjee et al., 4 Dec 2025).

Example reconstructions include phase and amplitude recovery in slits and prongs, topographical mapping (phase-to-height), and detection of hidden watermarks in currency (subsurface imaging). Numerical demonstrations in synthetic-aperture pulsed systems recover phase objects with correlation lengths down to $150~\mu$m, multislice scatterers, and hyperspectral material maps matching known dispersions (Kumar et al., 4 Oct 2024).

Measurement scaling in hybrid inverse transform modalities reduces acquisition complexity from $O(N^2)$ (raster scan) or $O(N^2)$ (optical Fourier ptychography) to $O(N)\times L$ (linear scan times frequency bins), supporting near-real-time operation in well-designed systems (Guerboukha et al., 2017).

5. Applications Across Domains

THz Fourier ptychographic imaging excels in scenarios requiring sub-wavelength resolution, phase sensitivity, and non-invasive contrast. Key application domains include:

• Materials characterization: Quantitative mapping of polymer thickness, phase shifts in plastics and coatings, and refractive-index retrieval via hyperspectral imaging (Mukherjee et al., 4 Dec 2025, Kumar et al., 4 Oct 2024).
• Spectroscopy: Multi-frequency or spectrally-resolved ptychography enables advanced chemical sensing within the THz regime.
• Nondestructive evaluation: Detection of buried or subsurface defects in composites, ceramics, and multi-layer structures (Mukherjee et al., 4 Dec 2025).
• Security and forensics: Imaging of currency watermarks (such as subsurface "Europa" patterns), inspection of printed documents for tampering or forgery (Mukherjee et al., 4 Dec 2025).
• Complex media imaging: Time-resolved synthetic-aperture methods enable reconstruction of spatial/temporal features in highly inhomogeneous samples (Kumar et al., 4 Oct 2024).

6. Limitations, Trade-Offs, and Future Directions

Current limitations center on mechanical scan rates (motorized mirror frame-rate), pixel pitch constraints (detector $35~\mu$m vs. $\lambda=85~\mu$m), algorithmic computation time, and robustness to aberrations. Emerging THz spatial light modulators (SLM) or phased arrays may considerably reduce acquisition times by replacing mechanical tilt stages. Detector arrays with $<35~\mu$m pitch will improve sampling of high spatial frequencies. GPU acceleration and deep-learning priors are plausible strategies to accelerate computation and enhance robustness, and multiplexed coded-aperture illumination could further improve k-space coverage with fewer measurements. Aberration modeling via enhanced pupil-function priors or iterative schemes is critical where system imperfections are non-negligible (Mukherjee et al., 4 Dec 2025, Guerboukha et al., 2017).

Hybrid spectral/angular protocols offer compressed acquisition, but the conflation of hyperspectral detail in the final images may limit spectrally resolved analyses unless the transform is adapted (Guerboukha et al., 2017). Acquisition speed can be further improved by fast delay lines or arrays of photoconductive detectors (Guerboukha et al., 2017).

7. Related Methodologies and Comparative Perspectives

THz Fourier ptychography unifies and extends several prior concepts in optical and THz imaging:

• Visible-light Fourier ptychography: Originates with angle-resolved illumination in the visible regime, now extended to THz frequencies with unique spectral and time-domain opportunities (Mukherjee et al., 4 Dec 2025).
• Synthetic aperture and THz-TDS: Synthetic aperture methods, as implemented in Fourier-based THz-TDS (Kumar et al., 4 Oct 2024), leverage multi-frequency and multi-angle measurements for broader k-space support, echoing ptychographic approaches.
• Compression-less hybrid imaging: Guerboukha et al. introduced a hybrid image reconstruction framework leveraging k-space/frequency duality in broadband Fourier optics, enabling linear measurement scaling ($O(N)$) for both amplitude and phase-contrast imaging (Guerboukha et al., 2017).
• Multislice beam propagation: Advanced algorithms recover multiple inhomogeneous slices in the sample, extending phase retrieval to complex layered objects in THz microscopy (Kumar et al., 4 Oct 2024).

A plausible implication is that future THz imaging systems will increasingly rely on computational synthetic aperture expansion, integrating single-pixel and array detectors, fast modulation hardware, and deep algorithmic stacks to reach real-time, high-fidelity, quantitative phase imaging across a wide range of applications.