# Terahertz Synthetic Aperture-based Spatiotemporal Imaging

Vivek Kumar¹, Pitambar Mukherjee², Lorenzo Valzania¹, Amaury Badon³, Patrick Mounaix², Sylvain Gigan¹

¹Laboratoire Kastler Brossel, ENS-Université PSL, CNRS, Sorbonne Université, Collège de France, 24 rue Lhomond, 75005 Paris, France

²IMS Laboratory, University of Bordeaux, UMR CNRS 5218, 351 Cours de la Libération Bâtiment A31, 33405 Talence, France

³Laboratoire Photonique Numérique et Nanosciences (LP2N), UMR 5298, University of Bordeaux, F-33400 Talence, France

**Abstract**— Terahertz (THz) microscopy has garnered significant attention due to the unique properties of THz radiation, making it appealing for chemical sensing, biomedical diagnostics, and imaging of complex media applications. However, conventional THz imaging systems encounter fundamental resolution limitations due to the Rayleigh diffraction limit. To overcome this challenge, we propose an innovative theoretical and numerical framework by adopting Fourier synthetic aperture principles in the THz spectral range. Our approach synthesizes high spatial frequency components by illuminating the sample with broadband THz pulses from multiple angles and capturing the time-resolved scattered fields through state-of-the-art time-domain spectroscopy (TDS). Our method offers a field-sensitive and experimentally viable solution for surpassing the diffraction limit in THz imaging. It paves the way for high-resolution, label-free material characterization and noninvasive imaging across various domains, including materials science, biomedical imaging, and security screening.

## I. INTRODUCTION

THz imaging has become a powerful tool for sensing the chemical and material composition of samples, establishing THz science as a rapidly evolving discipline. However, current THz imaging systems remain constrained by key limitations, most notably, the trade-off between field-of-view and spatial resolution [1]. It is primarily due to the Rayleigh diffraction limit, which constrains spatial resolution in the far field to ~180 µm at 1 THz (λ ≈ 0.3 mm), thereby limiting the space-bandwidth product and hampering applications that require fine details in structurally complex media analysis. To overcome such limitations, approaches like digital holography [2], ptychography [3], near-field imaging [4] and wavefront shaping [5-7] have been explored. While these techniques have shown resolution gains, they also face practical barriers: holography often relies on monochromatic sources and lacks broadband imaging capability; wavefront shaping requires complex, bulky setups due to limited THz modulation technologies; and near-field THz imaging, though subwavelength in resolution, suffers from experimental complexity and slow image acquisition.

Synthetic aperture imaging offers a compelling alternative by coherently integrating multiple diffraction-limited measurements, each capturing a subset of spatial frequency components, to synthetically extend the imaging system's spatial bandwidth. Building on this, we present a spatiotemporal coherent synthetic aperture framework in the THz domain that surpasses conventional resolution limits. By illuminating the sample with broadband THz pulses from

varying angles, we effectively scan an extended spatial frequency bandpass filter across Fourier space [1]. The temporally resolved signals captured via TDS are then propagated through our computational pipeline to reconstruct the full-field complex response -both amplitude and phase.

## II. METHOD

Figure 1 outlines our methodology, where broadband THz pulses illuminate the target object at multiple incident angles, and time-resolved scattered fields are captured via raster scanning using a single-pixel TDS detector at the image plane. In this way, a thin semi-transparent object of the coherent

<figure><img src="image_1.png" /><figcaption>Fig. 1: Fourier Synthetic Aperture-based THz Imaging scheme—conceptual overview of the numerically simulated methodology.</figcaption></figure>

transfer function, $\tilde{T}(x, y; \omega)$ is illuminated by a plane wave: $e^{i(k_{xa}x+k_{ya}y)}f(\omega)$, with the incident wave vectors $\vec{k}_a \equiv (k_{xa}, k_{ya})$ and $f(\omega)$ the THz broadband pulse spectrum. The transmitted field in its spatial frequency spectrum at a particular angular frequency $\omega$ can be expressed as,

$$ \tilde{E}^{+}(k_x - k_{xa}, k_y - k_{ya}; \omega) = \mathcal{F}_{x \to k}^{2D}\{\tilde{T}(x, y; \omega) \cdot e^{i(k_{xa}x + k_{ya}y)}f(\omega)\} \quad (1) $$

where $\mathcal{F}_{x \to k}^{2D}\{-\}$ represents the 2D spatial Fourier transform, and ‘.’ denotes point-wise multiplication. The final spatiotemporal field at the imaging plane after propagating through the pupil aperture can be written as,

$$ E_{out}(x, y; t) = \mathcal{F}_{\omega \to t}^{-1D} \left[ \mathcal{F}_{k \to x}^{-2D} \left\{ \tilde{E}^{+}(k_x - k_{xa}, k_y - k_{ya}; \omega) \cdot O_{p}^{(\omega)}(k_x, k_y) \right\} \right] \quad (2) $$

where $O_p^{(\omega)}$ denotes the pupil aperture due to pinhole, $\mathcal{F}_{k \to x}^{-2D}\{-\}$ represents 2D spatial inverse Fourier transform and $\mathcal{F}_{\omega \to t}^{-1D}[-]$ is inverse time-Fourier transform.

For a conventional THz imaging system operating at $\omega$ and with a finite numerical aperture (NA), the pupil aperture is represented as a circular area formed by $k_x^2 + k_y^2 < (\frac{NA\omega}{2\pi c})^2$ in
the spatial-frequency space. Consequently, TDS images at the image plane are inherently limited in spatial-frequency content by this aperture constraint. Synthetic aperture imaging leverages the fact that the wavevector offset ($-k_{xa}$, $-k_{ya}$) in the transmitted field is directly associated with the illumination angle. Varying this angle effectively shifts the pupil function in Fourier space, allowing access to higher spatial frequency components [8].

Further, to recover the object's complex transfer function, we formulate a least-squares minimization problem that reduces the discrepancy between the measured field, $\tilde{E}_{out}^a(x, y; \omega)$, and the modelled field, $\xi^a$, derived from the forward model in Eqs. (1-2). The fitness function at frequency $\omega$ is defined as follows:

$$F(\tilde{T}_r) = \arg\min_{\tilde{T}_r} \sum_{a}^{\mathcal{A}} \sum_{x} \sum_{y} |\tilde{E}_{out}^a(x, y; \omega) - \xi^a\{\tilde{T}_r(x, y; \omega)\}|^2 \quad (2)$$

where 'a' corresponds to the input beam having spatial wave vector $\vec{k}_a \equiv (k_{xa}, k_{ya})$, $\mathcal{A}$ is the total number of illumination angles and $\xi^a\{-\}$ is the forward model operator that predicts the spatial field distribution on the image plane after illuminating the sample with the THz pulse of input wavevector $\vec{k}_a$.

### III. RESULTS

Figure 2 shows the efficiency of our full-field approach in retrieving high-resolution spatiotemporal images, emphasizing phase-sensitive reconstruction of objects exhibiting complex phase structures. The test object (Fig. 2a) introduces modulation to the incident THz field incident on various angles. The resulting low-resolution, diffraction-limited spatiotemporal images, shown in Fig. 2b, are recorded via TDS at the image plane. Our reconstruction algorithm yields high-fidelity phase images, as demonstrated in Fig. 2c, showing the precision of our time-resolved imaging framework. Furthermore, Fig. 2d presents temporal waveforms corresponding to two spatial locations (indicated by red and

<figure><img src="image_2.png" /><figcaption>Fig. 2: Time-resolved retrieval of the phase image object. (a) Phase object under investigation. (b) Measured diffraction-limited, low-resolution field images at the peak of THz pulse obtained using a conventional imaging system. (c) Reconstructed phase images at the peak of the THz pulse. (d) Temporal evolution of reconstructed THz pulses for two different pixels (red and blue stars in panel (a)). (e) Line-cut plot for phase imaging object (red dashed line) and reconstructed image (cyan solid line in panel (a)).</figcaption></figure>

blue stars), revealing distinct 0 and $\pi$ phase shifts in the reconstructed pulses, highlighting our method's capability to resolve temporal dynamics at the pixel level. The resolution enhancement achieved by our method is quantitatively illustrated in Fig. 2e through a comparison of line profiles from the conventional phase image and the reconstructed output.

As a further demonstration, we extend our methodology to characterizing structurally complex and heterogeneous samples. To this end, we simulate a semi-transparent target composed of three materials- Teflon, Topas, and HDPE, with a

<figure><img src="image_3.png" /><figcaption>Fig. 3. Hyperspectral imaging for material characterization.</figcaption></figure>

(a) Schematic of the 100 µm thick imaging object composed of Teflon, Topas, and HDPE, showing the spatial distribution of the refractive index at 1 THz. (b) Distorted spatial variation in the refractive index profile at 1 THz recorded using a diffraction-limited imaging system.

(c) Reconstructed spatial refractive index profiles at 1 THz.

thickness of $\Delta = 100$ µm, as shown in Fig. 3a. Under conventional diffraction-limited imaging, the retrieved refractive index distribution at 1 THz (Fig. 3b) appears distorted, with poor material discrimination. In contrast, our reconstruction (Fig. 3c) yields a hyperspectral image with enhanced spatial resolution and clear delineation of the constituent materials, showcasing the potential of our approach for high-precision, label-free material characterization in the THz domain.

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