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  • Open Access

Acoustic, electromagnetic, and elastic space-time media that compute their own inverse

Dirk-Jan van Manen*, Johannes Aichele, and Jonas Müller

Marc Serra-Garcia

Kees Wapenaar

  • Department of Earth and Planetary Sciences, ETH Zurich, Sonneggstrasse 5, 8092 Zurich, Switzerland

  • AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands

  • Department of Geoscience and Engineering, TU Delft, Stevinweg 1, 2628 CN Delft, The Netherlands

  • *Contact author: vdirk@ethz.ch
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Phys. Rev. Research 8, 033358 – Published 23 September, 2026

DOI: https://doi.org/10.1103/hghj-fgsw

Abstract

Time materials are materials with acoustic, electromagnetic, or elastic properties that change while waves are propagating through them. Waves can scatter from such temporal inhomogeneities, but due to causality, they cannot reflect back in time. As a result, time-scattered wave fields do not resemble space-scattered wave fields—which are spatially mirrored—but instead look like copies of the original signal traveling back in time. Consequently, they can form an ingredient for physical solvers of inverse problems that rely on time-reversal transformations. Here, we investigate the use of time reflections for calculating so-called focusing wave fields. We establish a one-to-one mapping between focusing wave fields and the transmitted wave fields produced by time scattering. This mapping is then used to construct a space-time medium where the time scattering anticipates the space scattering and “computes” the exact inverse transmission response for the space scattering. We demonstrate the construction with a simple numerical example of a single pulse encountering a series of time boundaries before reaching a spatial inhomogeneity. The time boundaries scatter the single pulse into a focusing wave field that subsequently focuses through the spatial inhomogeneity. With certain mild assumptions, the transmitted wave has both the same wave shape and amplitude as the original pulse, yielding a transmission coefficient of unity.

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Phys. Rev. Research 8, 033358– Published 23 September, 2026
  • Received 12 December 2023
  • Accepted 25 August 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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