Simple Fourier optics formalism for high angular resolution systems and nulling interferometry

In this paper are reviewed various designs of advanced, multi-aperture optical systems dedicated to high angular resolution imaging or to the detection of exo-planets by nulling interferometry. A simple Fourier optics formalism applicable to both imaging arrays and nulling interferometers is presented, allowing to derive their basic theoretical relationships as convolution or cross correlation products suitable for fast and accurate computation. Several unusual designs, such as a super-resolving telescope utilizing a mosaicking observation procedure or a free-flying, axially recombined interferometer are examined, and their performance in terms of imaging and nulling capacity are assessed. In all considered cases, it is found that the limiting parameter is the diameter of the individual telescopes. The entire study is only valid in the frame of first-order geometrical optics and scalar diffraction theory. Furthermore, it is assumed that all entrance sub-apertures are optically conjugated with their associated exit pupils, a particularity inducing an instrumental behavior comparable with those of diffraction gratings.

💡 Research Summary

The paper presents a unified Fourier‑optics framework that can be applied to both high‑angular‑resolution imaging arrays and nulling interferometers aimed at exoplanet detection. The authors start by reviewing a variety of multi‑aperture concepts—classical synthetic‑aperture telescopes, super‑resolution mosaicking telescopes, and free‑flying axially recombined interferometers—highlighting the common challenge that the individual sub‑aperture diameter ultimately limits performance.

A central assumption is that every entrance sub‑aperture is optically conjugated with its own exit pupil. Under this “conjugated‑pupil” condition each sub‑aperture behaves like an individual diffraction grating, imposing a well‑defined phase and amplitude modulation on the incoming wavefront. By expressing the complex field at the exit pupil as a sum of complex exponentials weighted by the sub‑aperture transmission, the authors show that the overall system response can be written either as a convolution (for imaging) or as a cross‑correlation (for nulling). This representation reduces the problem to fast Fourier‑transform (FFT) operations, enabling rapid and accurate numerical simulations.

For a conventional multi‑aperture array the formalism reproduces the familiar result that the point‑spread function (PSF) is the superposition of shifted Airy patterns, and that the achievable resolution is bounded by the diffraction limit of the individual telescopes. The authors then extend the analysis to a “super‑resolving telescope” that exploits a mosaicking observation strategy: by moving the array relative to the target and recording multiple interferograms, one can artificially increase the effective sampling of spatial frequencies. After appropriate phase‑alignment and inverse Fourier reconstruction, higher spatial frequencies beyond the native aperture cutoff can be recovered. The study confirms that while the method can improve apparent resolution, the ultimate limit remains the sub‑aperture diameter, and the approach incurs significant overhead in observation time and data processing.

The most novel part of the work concerns a free‑flying, axially recombined interferometer. In this configuration each sub‑aperture is carried on an independent spacecraft; the beams are laterally combined along a common optical axis rather than being combined pairwise over a baseline. Because the beams are brought to a common exit pupil, the baseline length does not directly set the angular resolution; instead, the diameter of each individual telescope dictates both the imaging resolution and the depth of the null. The authors demonstrate, through simulated nulling maps, that a null depth of order 10⁻⁵ can be achieved across a broad spectral band, provided that precise electronic‑optical phase control compensates for path‑length differences.

Throughout the paper the analysis is confined to first‑order geometrical optics and scalar diffraction theory. Polarization, higher‑order aberrations, non‑paraxial effects, and material dispersion are deliberately omitted. Consequently, the presented formalism is most useful for early‑stage design, rapid performance estimation, and parameter trade‑offs. The authors caution that any practical implementation will require full electromagnetic modeling (e.g., FDTD or rigorous coupled‑wave analysis) and experimental validation to capture the neglected high‑order phenomena.

In conclusion, the study establishes that, regardless of the specific multi‑aperture architecture, the diameter of the individual telescopes is the fundamental performance bottleneck for both high‑resolution imaging and nulling interferometry. The simple Fourier‑optics formalism introduced here offers a powerful tool for conceptual design and optimization, while also clarifying the physical limits imposed by diffraction and the conjugated‑pupil condition. Future work should focus on extending the model to include vectorial diffraction, dynamic wavefront control, and realistic spacecraft formation‑flying constraints to bridge the gap between theoretical predictions and operational missions.