# Terahertz Phase Retrieval from Multi-plane Intensity Measurements

Vivek Kumar¹,², Pitambar Mukherjee², Frédéric Fauquet², Sylvain Gigan¹, Patrick Mounaix²

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.

**Abstract**— Phase imaging in the terahertz (THz) regime poses both fundamental and practical challenges, necessitating innovative computational imaging strategies. In such direction, interferometric reconstruction and digital holography have been widely implemented; however, these techniques are inherently limited by their reliance on phase-stable reference beams, making them susceptible to noise and system instabilities. In this work, we address such limitations and present theoretical and experimental demonstrations of a reference-free phase retrieval approach based solely on direct intensity measurements. We investigate two experimental configurations for phase retrieval. The first employs a lensless imaging approach based on recording axial diffraction patterns at multiple planes. The second configuration captures defocused intensity patterns around the focal plane of a 4f imaging system. In both cases, we implement a nonlinear convex optimization framework to recover the complex phase from the stack of intensity measurements. Our proposed approach offers a robust and computationally efficient solution to the phase retrieval problem, effectively overcoming the limitations of slow convergence and stagnation commonly associated with traditional multi-plane methods. Such results pave the way for the broader adoption of computational THz imaging algorithms in diverse applications, including material characterization, biomedical diagnostics, and security screening.

## I. INTRODUCTION

THz spectral range within the electromagnetic spectrum possesses non-destructive and non-ionizing properties, demonstrating high advantages for applications in biomedicine, security, and quality control [1]. Such characteristics mentioned have fueled significant research initiatives, resulting in the evolution of THz imaging techniques that utilize state-of-the-art time-domain spectroscopy (TDS), including near-field [2], synthetic aperture [3] and ghost imaging through complex media [4], [5], providing capabilities that are mostly unavailable in other spectral range. However, these techniques predominantly rely on THz radiation generated by femtosecond laser pulses, which introduces inherent complexity and limitations to achieving bright THz sources. Additionally, the absence of matrix detectors requires raster scanning of samples, significantly extending data acquisition and processing times. Furthermore, the source stability of such imaging techniques remains challenging, often requiring controlled laboratory environments for optimal performance.

Recent advancements in THz instrumentation have led to the development of commercial THz equipments with enhanced performance, improved reliability, and greater ease of operation. In this context, continuous-wave (CW) THz radiation can now be efficiently generated and detected using modern electronic devices that are compact, cost-effective, and straightforward to implement. However, most CW imaging techniques are limited to capturing only the intensity

distribution of an imaging sample. Typically, phase information improves the imaging contrast for transparent samples and provides critical insights, such as the relative optical depth profile and internal density variations, which are otherwise inaccessible. To fully exploit the potential of THz imaging, phase retrieval techniques are essential, as they encode topographic details of the sample [6], [7]. In such a direction, CW THz phase imaging has been extensively investigated, leading to the development of digital holography [8] and ptychographic imaging [9] modalities. However, these approaches are often hindered by reference arm phase instability, introducing unwanted phase noise and degrading imaging quality.

Here, we present a THz phase retrieval approach that can be directly implemented without requiring significant modifications to existing imaging hardware. We demonstrate two imaging configurations in which multiple transmitted intensity patterns are recorded through the sample. A nonlinear convex optimization framework is then employed to accurately reconstruct the phase with high fidelity, enabling enhanced imaging capabilities without complex imaging system alterations.

## II. METHODOLOGY

Figure 1 shows a schematic that illustrates the recording and reconstruction stages of THz phase retrieval from the multiple-plane intensity measurements. We propose two imaging configurations for capturing multiple intensity patterns. The first configuration employs a lensless detection scheme (Figure 1a), where the initial measurement plane is at a certain distance from the imaging sample. In this setup, the diffracted field is recorded sequentially by capturing images at various axial positions, with a uniform interval of $\Delta z$ between successive planes. The second configuration (Figure 1b) involves collecting multiple intensity patterns around the focal plane of a 4f-imaging system while maintaining the interval $\Delta z$ between the recorded planes.

In both configurations, wave propagation is modelled using the angular spectrum method to solve the Rayleigh-

<figure><img src="image_1.png" /><figcaption>**Fig. 1:** THz phase imaging scheme using multi-plane intensity measurements- conceptual overview of the experimentally implemented methodology. (a) Lens-less imaging configuration. (b) 4f imaging configuration.</figcaption></figure>
Sommerfeld equation [3], expressed as:

$$E(x, y, z_{\pm}) = F^{-1} \left\{ \mathcal{F}\{E(x, y, z_0)\} e^{ikz_{\pm} \sqrt{1-(\lambda f_x)^2 - (\lambda f_y)^2}} \right\} \quad (1)$$

where $\lambda$ represents the incident THz wavelength, $k = 2\pi/\lambda$ is the wavenumber, and $z_{\pm}$ denotes the free-space propagation distance. The terms $E(x, y, z_0)$ and $E(x, y, z_{\pm})$ correspond to the complex fields at the input and output planes, respectively, while $(x, y)$ and $(f_x, f_y)$ denote the spatial and frequency coordinates. The operators $\mathcal{F}$ and $\mathcal{F}^{-1}$ represent the forward and inverse Fourier transforms, respectively. For data acquisition, we record a set of intensity patterns, $I_n = |E_n(x, y, z_{\pm})|^2$, where $n = 1, 2, ...$ $N$ represents the number of intensity patterns, by sequentially translating the camera along the optical axis. Furthermore, we formulate the phase retrieval problem as a nonlinear convex optimization task, where the objective is to minimize the cost function, $F(E(x, y))$, between the theoretically computed and experimentally measured complex wavefield for all the planes.

$$F(E(x, y)) = \sum_{N} \sum_{x} \sum_{y} \left\| I_n - \left| \xi_n(E(x, y)) \right|^2 \right\|_2^2 \quad (2)$$

Here, $\xi_n$ represents the forward model operator, as defined in equation (1), which governs the propagation of the complex wavefield in free space.

We solve this nonlinear optimization problem using a gradient descent approach incorporating both the Hessian and gradient to determine the optimal search direction. Additionally, the learning step size is dynamically adjusted through a backtracking line search algorithm, ensuring efficient convergence and improved stability in the phase retrieval process.

### III. RESULT

In our experimental demonstration, we use a polypropylene plastic imaging object shaped like an “Arrow,” illuminated by a collimated beam at 3.5 THz, generated using a quantum cascade laser source, as shown in Figure 1. The resulting scattered field is captured across multiple planes using a microbolometer camera (INO’s MICROXCAM-384i) with a pixel pitch of 35 µm. In the lens-less configuration, the object-to-camera distance is initially set to 6.5 mm. The camera, mounted on a motorized translation stage, is incrementally translated along the optical axis in 0.2 mm steps, capturing the diffracted intensity patterns at each plane. In the second configuration, a stack of defocused intensity patterns is recorded around the focal plane of a 4f system, again with a step size of 0.2 mm between successive planes. A relevant case example of the retrieved spatial phase distribution is shown in Figure 2. The spatial phase profile obtained from the lens-less configuration is shown in Figure 2a, while the result from the defocused intensity stack is shown in Figure 2b. For comparison, we also include the phase reconstruction using the standard Gerchberg–Saxton algorithm in Figure 2c. Our full theoretical and experimental investigation will be reported in detail.

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<figure><img src="image_2.png" /><figcaption>Fig. 2: THz spatial phase retrieval in (a) Lens-less imaging configuration. (b) 4f imaging configuration. (c) using Gerchberg-Saxton algorithm.</figcaption></figure>