This Conceptual Design Report describes LUXE (Laser Und XFEL Experiment), an experimental campaign that aims to combine the high-quality and high-energy electron beam of the European XFEL with a powerful laser to explore the uncharted terrain of quantum electrodynamics characterised by both high energy and high intensity. We will reach this hitherto inaccessible regime of quantum physics by analysing high-energy electron-photon and photon-photon interactions in the extreme environment provided by an intense laser focus. The physics background and its relevance are presented in the science case which in turn leads to, and justifies, the ensuing plan for all aspects of the experiment: Our choice of experimental parameters allows (i) field strengths to be probed where the coupling to charges becomes non-perturbative and (ii) a precision to be achieved that permits a detailed comparison of the measured data with calculations. In addition, the high photon flux predicted will enable a sensitive search for new physics beyond the Standard Model. The initial phase of the experiment will employ an existing 40 TW laser, whereas the second phase will utilise an upgraded laser power of 350 TW. All expectations regarding the performance of the experimental set-up as well as the expected physics results are based on detailed numerical simulations throughout.

The dynamics of charged particles in electromagnetic fields is an essential component of understanding the most extreme environments in our Universe. In electromagnetic fields of sufficient magnitude, radiation emission dominates the particle motion and effects of nonlinear quantum electrodynamics (QED) are crucial, which triggers electron-positron pair cascades and counterintuitive particle-trapping phenomena. As a result of recent progress in laser technology, high-power lasers provide a platform to create and probe such fields in the laboratory. With new large-scale laser facilities on the horizon and the prospect of investigating these hitherto unexplored regimes, we review the basic physical processes of radiation reaction and QED in strong fields, how they are treated theoretically and in simulation, the new collective dynamics they unlock, recent experimental progress and plans, as well as possible applications for high-flux particle and radiation sources. CONTENTS42 VI. Applications 43 A. Radiation generation 43 1. Electron-beam driven radiation sources 43 2. Laser-driven radiation sources 44 B. Positron sources 46 C. Polarized particle beams 46 D. Ion acceleration 47 VII. Outlook 48 A. Open questions 48 1. Theoretical questions 48 2. Simulation developments 49 B. Experimental programs 50 VIII. Conclusions 50 Acknowledgments 51 List of commonly used symbols 51 References 52

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When a photon collides with a laser pulse, an electron-positron pair can be produced via the nonlinear Breit–Wheeler process. A simulation framework has been developed to calculate this process, which is based on a ponderomotive approach that includes strong-field quantum electrodynamical effects via the locally monochromatic approximation (LMA). Here we compare simulation predictions for a variety of observables, in different physical regimes, with numerical evaluation of exact analytical results from theory. For the case of a focussed laser background, we also compare simulation with a high-energy theory approximation. These comparisons are used to quantify the accuracy of the simulation approach in calculating harmonic structure, which appears in the lightfront momentum and angular spectra of outgoing particles, and the transition from multi-photon to all-order pair creation. Calculation of the total yield of pairs over a range of intensity parameters is also used to assess the accuracy of the locally constant field approximation (LCFA).

When a pulsed, few-cycle electromagnetic wave is focused by optics with f-number smaller than two, the frequency components it contains are focused to different regions of space, building up a complex electromagnetic field structure. Accurate numerical computation of this structure is essential for many applications such as the analysis, diagnostics, and control of high-intensity laser-matter interactions. However, straightforward use of finite-difference methods can impose unacceptably high demands on computational resources, owing to the necessity of resolving far-field and near-field zones at sufficiently high resolution to overcome numerical dispersion effects. Here, we present a procedure for fast computation of tight focusing by mapping a spherically curved far-field region to periodic space, where the field can be advanced by a dispersion-free spectral solver. In many cases of interest, the mapping reduces both run time and memory requirements by a factor of order 10, making it possible to carry out simulations on a desktop machine or a single node of a supercomputer. We provide an open-source C++ implementation with Python bindings and demonstrate its use for a desktop machine, where the routine provides the opportunity to use the resolution sufficient for handling the pulses with spectra spanning over several octaves. The described approach can facilitate the stability analysis of theoretical proposals, the studies based on statistical inferences, as well as the overall development and analysis of experiments with tightly-focused short laser pulses.