Metallische Nanoantennen: Frequenzverdopplung und photochemische Reaktionen auf kleinen Skalen

Diese Arbeit beinhaltet experimentelle und theoretische Untersuchungen der optischen Frequenzverdopplung (second-harmonic generation, kurz SHG) an metallischen Nanopartikeln. Frequenzverdopplung bedeutet, daß ein bei der Frequenz omega angeregtes Nanopartikel Strahlung der Frequenz 2*omega emittiert. Dieser Effekt tritt nicht nur in Materialien mit nichtzentrosymmetrischer Kristallstruktur, sondern auch an der Oberfläche von Metallen auf. Deshalb läßt er sich gut mit plasmonischen Feldüberhöhungen an metallischen Nanoantennen verbinden. Die Frequenzverdopplung wird an verschiedenen Nanostrukturen wie dreieckförmigen, stäbchenförmigen und vor allem kegelförmigen Nanopartikeln experimentell untersucht, welche aufgrund ihrer scharfen Spitzen starke SHG-Signale emittieren. Besonders die Kegel sind interessant: Bei Anregung mit einem fokussierten, radial polarisierten Strahl dominiert je nach Kegelgröße und Umgebungsmedium ein SHG-Signal entweder von der Spitze oder von der Bodenkante des Kegels. Diese an den Kegeln gemessenen Resultate werden durch theoretische Untersuchungen untermauert. In diesen Rechnungen werden die plasmonischen Feldüberhöhungen und die sich daraus ergebende Frequenzverdopplung für einen Kegel mit verschiedenen Parametern modelliert. An einem einzelnen Kegel gewonnene Resultate werden auch mit den Fällen eines kugelförmigen und eines stäbchenförmigen Partikels verglichen. Ein weiterer Gegenstand der theoretischen Untersuchungen ist die Superposition der zweiten Harmonischen von mehreren emittierenden Nanopartikeln zu einem Feldmaximum. Dabei wird eine kreisförmige Anordnung von 8 Nanostäbchen bzw. Nanokegeln von einer radial polarisierten Mode angeregt. Die Superposition der emittierten zweiten Harmonischen ergibt ein Feldmaximum innerhalb der Anordnung der Emitter. Durch eine Verkippung des anregenden Strahls kann dieser Fokus im Raum bewegt werden. Letztere Untersuchung ist insbesondere interessant im Hinblick auf lokalisierte photochemische Reaktionen, die durch das frequenzverdoppelte Licht von Nanopartikeln ausgelöst werden sollen. Mit chemischen Substanzen, die bei omega transparent, bei 2*omega aber photoreaktiv sind, wäre im Nahfeld dieser Nanoantennen eine starke Lokalisierung der Reaktion auf Bereiche kleiner als 100~nm möglich. Anhand von Photolacken und Polymermatrizen mit diesen Eigenschaften wird experimentell untersucht, ob frequenzverdoppeltes Licht überhaupt solche Reaktionen auslösen kann oder ob die photochemische Reaktionen überwiegend durch direkte Zwei-Photonen-Absorption des anregenden Lichts ausgelöst werden. Die Ergebnisse zeigen allerdings, daß die Zwei-Photonen-Absorption dominant ist. Durch die Zwei-Photonen-Absorption im Nahfeld von Partikeln ist aber dennoch eine vergleichbare Lokalisierung der Reaktion möglich.:1. Einführung 1.1 Frequenzverdopplung an Nanopartikeln 1.2 Photochemisches Schreiben auf kleinen Längenskalen 2. Theoretische Grundlagen 2.1 Nichtlineare optische Effffekte zweiter Ordnung 2.2 Frequenzverdopplung in Metallen 2.3 Frequenzverdopplung bei metallischen Nanopartikeln 2.4 Überlagerung der Strahlung mehrerer frequenzverdoppelter Dipole 2.5 Core-Shell-Nanopartikel mit nichtzentrosymmetrischem Kern 3. Experimenteller Aufbau 3.1 Beleuchtung der Proben und Detektionspfad 3.2 Objektiv und Probenhalter 3.3 Realisierung der radial polarisierten Mode 4. Messungen der zweiten Harmonischen an Nanostrukturen 4.1 Einzelne kugel- und stäbchenförmige Goldnanopartikel 4.2 Nanodreiecke (Fischer-Pattern) 4.3 Nanokegel 4.4 Nanostäbchen-Teppiche 4.5 Zusammenfassung 5. Nichtlinear-optisches photochemisches Schreiben auf kleinen Längenskalen 5.1 Photochemische Reaktionen auf der Sub-100nm-Skala 5.2 Erste Versuche an Photolacken 5.3 Photochemisches Schreiben auf azobenzolhaltigen PMMA-Copolymerschichten 5.4 Photochemisches Schreiben auf azosulfonathaltigen PMMA-Copolymerschichten 5.5 Ausblick 6. Zusammenfassung und Ausblick Anhang A. Darstellung der radialen Mode und des z-polarisierten Fokus B. Mehode der multiplen Multipole (MMP) C. Präparation der Proben Literaturverzeichnis Abbildungsverzeichnis Verzeichnis der Tabellen Verwendete Abkürzungen Liste der Veröffffentlichungen Danksagung ErklärungThis work includes experimental and theoretical investigations of second-harmonic generation (SHG) at metallic nanoparticles. SHG means that a nanoparticle that is excited at the frequency omega emits radiation at the frequency 2*omega. SHG does not only occur in materials with noncentrosymmetric structure, but also on metal surfaces. Hence, SHG can be combined well with plasmonic field enhancement at metallic nanoantennae. SHG is investigated experimentally at different nanostructures such as triangle-like, rod-like and especially cone-like nanoparticles. With their sharp tips these structures show a much stronger SHG signal than spherical nanoparticles. Especially the cones are interesting: Excited with a focused radially polarized beam, for different cone sizes and in different surrounding media either the signal from the tip or the signal from the bottom edge dominates. The measurement results from the cones are underpinned by theoretical investigations. In these calculations the plasmonic field enhancements and the resulting SHG are modeled for a cone with different parameters. The single-cone results are also compared with the cases of a spherical or rod-shaped particle. A further subject of the theoretical investigations is the superposition of the SHG radiation from a number of emitting nanoparticles to a field maximum. For that, a circular arrangement of 8 nanorods or nanocones is excited by a radially polarized beam. The superposition of the second-harmonic radiation fields yields a field maximum in the space between the emitters. A tilt of the exciting beam can move this focus in space. The latter item is of special interest concerning localised photochemical reactions induced by the second-harmonic light from nanoparticles. In the near field of these nanoantennae, a strong localisation of the reaction on regions smaller than 100 nm would be possible by using chemical substances being transparent at omega, but photoreactive at 2*omega. With photoresists and polymer matrices, experiments are carried out to investigate whether SHG light can trigger such reactions at all, or if these photochemical reactions are triggered predominantly by direct two-photon absorption of the exciting light. The results show that the two-photon absorption is the dominant process. Yet, through two-photon absorption in the near field of particles, the localisation of the reaction is still similar.:1. Einführung 1.1 Frequenzverdopplung an Nanopartikeln 1.2 Photochemisches Schreiben auf kleinen Längenskalen 2. Theoretische Grundlagen 2.1 Nichtlineare optische Effffekte zweiter Ordnung 2.2 Frequenzverdopplung in Metallen 2.3 Frequenzverdopplung bei metallischen Nanopartikeln 2.4 Überlagerung der Strahlung mehrerer frequenzverdoppelter Dipole 2.5 Core-Shell-Nanopartikel mit nichtzentrosymmetrischem Kern 3. Experimenteller Aufbau 3.1 Beleuchtung der Proben und Detektionspfad 3.2 Objektiv und Probenhalter 3.3 Realisierung der radial polarisierten Mode 4. Messungen der zweiten Harmonischen an Nanostrukturen 4.1 Einzelne kugel- und stäbchenförmige Goldnanopartikel 4.2 Nanodreiecke (Fischer-Pattern) 4.3 Nanokegel 4.4 Nanostäbchen-Teppiche 4.5 Zusammenfassung 5. Nichtlinear-optisches photochemisches Schreiben auf kleinen Längenskalen 5.1 Photochemische Reaktionen auf der Sub-100nm-Skala 5.2 Erste Versuche an Photolacken 5.3 Photochemisches Schreiben auf azobenzolhaltigen PMMA-Copolymerschichten 5.4 Photochemisches Schreiben auf azosulfonathaltigen PMMA-Copolymerschichten 5.5 Ausblick 6. Zusammenfassung und Ausblick Anhang A. Darstellung der radialen Mode und des z-polarisierten Fokus B. Mehode der multiplen Multipole (MMP) C. Präparation der Proben Literaturverzeichnis Abbildungsverzeichnis Verzeichnis der Tabellen Verwendete Abkürzungen Liste der Veröffffentlichungen Danksagung Erklärun

Tensor Rank and Complexity

These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will be expressed in terms of tensor (border) rank, (border) symmetric rank and the asymptotic rank of certain tensors. We introduce the multilinear rank of a tensor as well, deal with the concept of tensor equivalence and study prehomogeneous vector spaces with the Castling transform. Moreover, we treat Apolarity Theory and use it to determine the symmetric rank (Waring rank) of some symmetric tensors.Comment: 44 page

Generation of synthetic EEG data for training algorithms supporting the diagnosis of major depressive disorder

IntroductionMajor depressive disorder (MDD) is the most common mental disorder worldwide, leading to impairment in quality and independence of life. Electroencephalography (EEG) biomarkers processed with machine learning (ML) algorithms have been explored for objective diagnoses with promising results. However, the generalizability of those models, a prerequisite for clinical application, is restricted by small datasets. One approach to train ML models with good generalizability is complementing the original with synthetic data produced by generative algorithms. Another advantage of synthetic data is the possibility of publishing the data for other researchers without risking patient data privacy. Synthetic EEG time-series have not yet been generated for two clinical populations like MDD patients and healthy controls.MethodsWe first reviewed 27 studies presenting EEG data augmentation with generative algorithms for classification tasks, like diagnosis, for the possibilities and shortcomings of recent methods. The subsequent empirical study generated EEG time-series based on two public datasets with 30/28 and 24/29 subjects (MDD/controls). To obtain baseline diagnostic accuracies, convolutional neural networks (CNN) were trained with time-series from each dataset. The data were synthesized with generative adversarial networks (GAN) consisting of CNNs. We evaluated the synthetic data qualitatively and quantitatively and finally used it for re-training the diagnostic model.ResultsThe reviewed studies improved their classification accuracies by between 1 and 40% with the synthetic data. Our own diagnostic accuracy improved up to 10% for one dataset but not significantly for the other. We found a rich repertoire of generative models in the reviewed literature, solving various technical issues. A major shortcoming in the field is the lack of meaningful evaluation metrics for synthetic data. The few studies analyzing the data in the frequency domain, including our own, show that only some features can be produced truthfully.DiscussionThe systematic review combined with our own investigation provides an overview of the available methods for generating EEG data for a classification task, their possibilities, and shortcomings. The approach is promising and the technical basis is set. For a broad application of these techniques in neuroscience research or clinical application, the methods need fine-tuning facilitated by domain expertise in (clinical) EEG research

Invariant theory and scaling algorithms for maximum likelihood estimation

We uncover connections between maximum likelihood estimation in statistics and norm minimization over a group orbit in invariant theory. We focus on Gaussian transformation families, which include matrix normal models and Gaussian graphical models given by transitive directed acyclic graphs. We use stability under group actions to characterize boundedness of the likelihood, and existence and uniqueness of the maximum likelihood estimate. Our approach reveals promising consequences of the interplay between invariant theory and statistics. In particular, existing scaling algorithms from statistics can be used in invariant theory, and vice versa.Comment: 34 pages; minor changes in comparison to version 2. The discrete part on log-linear models from version 1 is contained in the companion paper arXiv:2012.0779

Polaron-Mediated Luminescence in Lithium Niobate and Lithium Tantalate and Its Domain Contrast

In this review article, we discuss photoluminescence phenomena mediated by polarons in lithium niobate (LNO). At first we present the fundamentals on polaron states in LNO and their energy levels, i.e., on free and bound electron polarons, on hole polarons as well as on bipolarons. We discuss the absorption measurements on reduced as well as on doped LNO that made the characterization of the formed polaron states possible by their absorption bands. Next, we proceed by reporting on the two polaron-mediated photoluminescence bands that have been observed in LNO: (1) A near-infrared luminescence band in the range of 1.5 eV shows a mono-exponential decay and a strong dependence on iron doping. This luminescence is emitted by bound polarons returning from an excited state to the ground state. (2) A luminescence band at visible wavelengths with a maximum at 2.6 eV shows a stretched-exponential decay and is strongly enhanced by optical damage resistant doping around the doping threshold. This luminescence stems from the recombination of free electron and hole polarons. The next major topic of this review are domain contrasts of the visible photoluminescence that have been observed after electrical poling of the substrate, as singly inverted domains show a slightly reduced and faster decaying luminescence. Subsequent annealing results in an exponential decrease of that domain contrast. We show that this contrast decay is strongly related to the mobility of lithium ions, thus confirming the role of polar defect complexes, including lithium vacancies, for these domain contrasts. Finally we discuss the extension of our investigations to lithium tantalate (LTO) samples. While the results on the domain contrast and its decay are similar to LNO, there are remarkable differences in their luminescence spectra

Adaptation Strategies for Personalized Gait Neuroprosthetics

Personalization of gait neuroprosthetics is paramount to ensure their efficacy for users, who experience severe limitations in mobility without an assistive device. Our goal is to develop assistive devices that collaborate with and are tailored to their users, while allowing them to use as much of their existing capabilities as possible. Currently, personalization of devices is challenging, and technological advances are required to achieve this goal. Therefore, this paper presents an overview of challenges and research directions regarding an interface with the peripheral nervous system, an interface with the central nervous system, and the requirements of interface computing architectures. The interface should be modular and adaptable, such that it can provide assistance where it is needed. Novel data processing technology should be developed to allow for real-time processing while accounting for signal variations in the human. Personalized biomechanical models and simulation techniques should be developed to predict assisted walking motions and interactions between the user and the device. Furthermore, the advantages of interfacing with both the brain and the spinal cord or the periphery should be further explored. Technological advances of interface computing architecture should focus on learning on the chip to achieve further personalization. Furthermore, energy consumption should be low to allow for longer use of the neuroprosthesis. In-memory processing combined with resistive random access memory is a promising technology for both. This paper discusses the aforementioned aspects to highlight new directions for future research in gait neuroprosthetics.Peer ReviewedPostprint (published version

Melting, bubble-like expansion and explosion of superheated plasmonic nanoparticles

We report on time-resolved coherent diffraction imaging of gas-phase silver nanoparticles, strongly heated via their plasmon resonance. The x-ray diffraction images reveal a broad range of phenomena for different excitation strengths, from simple melting over strong cavitation to explosive disintegration. Molecular dynamics simulations fully reproduce this behavior and show that the heating induces rather similar trajectories through the phase diagram in all cases, with the very different outcomes being due only to whether and where the stability limit of the metastable superheated liquid is crossed.Comment: 17 pages, 8 figures (including supplemental material