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Topological Classification of Non-Normalizable Vector Fields
Authors:
Philipp Gessler,
Alessandro Pignedoli,
Alexander Neuhaus,
Frank-J. Meyer zu Heringdorf,
Maria Azhar,
Karin Everschor-Sitte
Abstract:
Topological classification of physical vector fields conventionally relies on field normalization and homotopy-based invariants. However, when field amplitudes vanish, normalization becomes ill-defined, preventing a direct topological characterization. Here, we introduce a general framework for the topological classification of non-normalizable $n$-dimensional vector fields with compactifiable bas…
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Topological classification of physical vector fields conventionally relies on field normalization and homotopy-based invariants. However, when field amplitudes vanish, normalization becomes ill-defined, preventing a direct topological characterization. Here, we introduce a general framework for the topological classification of non-normalizable $n$-dimensional vector fields with compactifiable base spaces by transforming them into $(n+1)$-dimensional normalized vector fields. This construction extends homotopy-based classification to fields containing amplitude zeros. We explicitly demonstrate the approach for one-, two-, and three-dimensional non-normalized vector fields and derive the corresponding topological invariants. The resulting topological charges are robust under continuous deformations and can change only when the embedding structure becomes singular. Our framework provides a unified route to the topological characterization of non-normalizable fields and opens the door to the study of topological phenomena in a broad range of systems, including magnetic textures, ferroelectrics, electromagnetic fields, and wave systems.
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Submitted 28 July, 2026;
originally announced July 2026.
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Analysis of the Topology of a Plasmonic Target-Skyrmion Texture
Authors:
Alexander Neuhaus,
Pascal Dreher,
Philipp Gessler,
Bettina Frank,
Timothy J. Davis,
Harald Giessen,
Karin Everschor-Sitte,
Frank-J. Meyer zu Heringdorf
Abstract:
Topological concepts are frequently used to describe structured optical fields, including plasmonic near fields. Topological descriptions in terms of skyrmion numbers implicitly assume the compactness of the underlying manifold. Even when skyrmion-like textures appear locally, the compactness is usually not fulfilled in extended optical fields. Here, we use photoemission electron microscopy to inv…
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Topological concepts are frequently used to describe structured optical fields, including plasmonic near fields. Topological descriptions in terms of skyrmion numbers implicitly assume the compactness of the underlying manifold. Even when skyrmion-like textures appear locally, the compactness is usually not fulfilled in extended optical fields. Here, we use photoemission electron microscopy to investigate a plasmonic nano-focus that exhibits a sequence of radially extending alternating skyrmion and antiskyrmion textures. The full spatio-temporal reconstruction of the electric field vectors and their topology is accessible by vector polarimetry. The experiments confirm the expected oscillatory behavior of the skyrmion number and demonstrate that a global skyrmion number cannot be assigned in such non-compact fields.
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Submitted 16 July, 2026;
originally announced July 2026.
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Linking extended vector wave fields with momentum space topology
Authors:
A. Neuhaus,
P. Gessler,
P. Dreher,
D. Janoschka,
A. Rödl,
M. Manten,
Th. Bauer,
M. Azhar,
B. Frank,
T. J. Davis,
H. Giessen,
K. Everschor-Sitte,
F. Meyer zu Heringdorf
Abstract:
Topology describes properties of physical systems that remain constant under continuous deformations. For infinite vector waves, global topological invariants in position space are typically associated with periodic patterns. We demonstrate that even for aperiodic Helmholtz-decomposable wave fields, possessing only the wave's intrinsic periodicity, a topological invariant can be found in momentum…
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Topology describes properties of physical systems that remain constant under continuous deformations. For infinite vector waves, global topological invariants in position space are typically associated with periodic patterns. We demonstrate that even for aperiodic Helmholtz-decomposable wave fields, possessing only the wave's intrinsic periodicity, a topological invariant can be found in momentum space. This invariant, the linking number, represents a Berry phase. By utilizing electromagnetic and hydrodynamic surface waves, we confirm the robustness of the linking number against deformations, and experimentally observe discrete transitions between distinct topological sectors. The linking number captures the topology of vector wave fields across both continuous and discrete momentum spaces. Our work introduces a unified topological framework for vector wave fields, enabling their classification via a global invariant.
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Submitted 29 April, 2026;
originally announced April 2026.
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The Heisenberg-RIXS instrument at the European XFEL
Authors:
Justine Schlappa,
Giacomo Ghiringhelli,
Benjamin E. Van Kuiken,
Martin Teichmann,
Piter S. Miedema,
Jan Torben Delitz,
Natalia Gerasimova,
Serguei Molodtsov,
Luigi Adriano,
Bernard Baranasic,
Carsten Broers,
Robert Carley,
Patrick Gessler,
Nahid Ghodrati,
David Hickin,
Le Phuong Hoang,
Manuel Izquierdo,
Laurent Mercadier,
Giuseppe Mercurio,
Sergii Parchenko,
Marijan Stupar,
Zhong Yin,
Leonardo Martinelli,
Giacomo Merzoni,
Ying Ying Peng
, et al. (22 additional authors not shown)
Abstract:
Resonant Inelastic X-ray Scattering (RIXS) is an ideal X-ray spectroscopy method to push the combination of energy and time resolutions to the Fourier transform ultimate limit, because it is unaffected by the core-hole lifetime energy broadening. And in pump-probe experiments the interaction time is made very short by the same core-hole lifetime. RIXS is very photon hungry so it takes great advant…
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Resonant Inelastic X-ray Scattering (RIXS) is an ideal X-ray spectroscopy method to push the combination of energy and time resolutions to the Fourier transform ultimate limit, because it is unaffected by the core-hole lifetime energy broadening. And in pump-probe experiments the interaction time is made very short by the same core-hole lifetime. RIXS is very photon hungry so it takes great advantage from high repetition rate pulsed X-ray sources like the European XFEL. The hRIXS instrument is designed for RIXS experiments in the soft X-ray range with energy resolution approaching the Fourier and the Heisenberg limits. It is based on a spherical grating with variable line spacing (VLS) and a position-sensitive 2D detector. Initially, two gratings are installed to adequately cover the whole photon energy range. With optimized spot size on the sample and small pixel detector the energy resolution can be better than 40 meV at any photon energy below 1000 eV. At the SCS instrument of the European XFEL the spectrometer can be easily positioned thanks to air-pads on a high-quality floor, allowing the scattering angle to be continuously adjusted over the 65-145 deg range. It can be coupled to two different sample interaction chamber, one for liquid jets and one for solids, each equipped at the state-of-the-art and compatible for optical laser pumping in collinear geometry. The measured performances, in terms of energy resolution and count rate on the detector, closely match design expectations. hRIXS is open to public users since the summer of 2022.
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Submitted 13 March, 2024;
originally announced March 2024.
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The 1-Megapixel pnCCD Detector for the Small Quantum Systems Instrument at the European XFEL: System and Operation Aspects
Authors:
Markus Kuster,
Karim Ahmed,
Kai-Erik Ballak,
Cyril Danilevski,
Marko Ekmedžić,
Bruno Fernandes,
Patrick Gessler,
Robert Hartmann,
Steffen Hauf,
Peter Holl,
Michael Meyer,
Jacobo Montaño,
Astrid Münnich,
Yevheniy Ovcharenko,
Nils Rennhack,
Tonn Rüter,
Daniela Rupp,
Dieter Schlosser,
Kiana Setoodehnia,
Rüdiger Schmitt,
Lothar Strüder,
Rico Mayro P. Tanyag,
Anatoli Ulmer,
Hazem Yousef
Abstract:
The X-ray free-electron lasers that became available during the last decade, like the European XFEL (EuXFEL), place high demands on their instrumentation. Especially at low photon energies below $1\,\text{keV}$, detectors with high sensitivity, and consequently low noise and high quantum efficiency, are required to enable facility users to fully exploit the scientific potential of the photon sourc…
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The X-ray free-electron lasers that became available during the last decade, like the European XFEL (EuXFEL), place high demands on their instrumentation. Especially at low photon energies below $1\,\text{keV}$, detectors with high sensitivity, and consequently low noise and high quantum efficiency, are required to enable facility users to fully exploit the scientific potential of the photon source. A 1-Megapixel pnCCD detector with a $1024\times 1024$ pixel format has been installed and commissioned for imaging applications at the Nano-Sized Quantum System (NQS) station of the Small Quantum System (SQS) instrument at EuXFEL. The instrument is currently operating in the energy range between $0.5$ and $3\,\text{keV}$ and the NQS station is designed for investigations of the interaction of intense FEL pulses with clusters, nano-particles and small bio-molecules, by combining photo-ion and photo-electron spectroscopy with coherent diffraction imaging techniques. The core of the imaging detector is a pn-type charge coupled device (pnCCD) with a pixel pitch of $75\,μ\text{m}\times 75\,μ\text{m}$. Depending on the experimental scenario, the pnCCD enables imaging of single photons thanks to its very low electronic noise of $3$e$^-$ and high quantum efficiency. Here we present an overview on the EuXFEL pnCCD detector and the results from the commissioning and first user operation at the SQS experiment in June 2019. The detailed descriptions of the detector design and capabilities, its implementation at EuXFEL both mechanically and from the controls side as well as important data correction steps aim to provide useful background for users planning and analyzing experiments at EuXFEL and may serve as a benchmark for comparing and planning future endstations at other FELs.
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Submitted 23 November, 2020;
originally announced November 2020.