A scalable hp-adaptive finite element software with applications in fiber optics

In this dissertation, we present a scalable parallel version of hp3D—a finite element (FE) software for analysis and discretization of complex three-dimensional multiphysics applications. The developed software supports hybrid MPI/OpenMP parallelism for large-scale computation on modern manycore architectures. The focus of the effort lies on the development and optimization of the parallel software infrastructure underlying all distributed computation. We discuss the challenges of designing efficient data structures for isotropic and anisotropic hp-adaptive meshes with tetrahedral, hexahedral, prismatic, and pyramidal elements supporting discretization of the exact sequence energy spaces. While the code supports standard Galerkin methods, special emphasis is given to systems arising from discretization with the discontinuous Petrov–Galerkin (DPG) method. The method guarantees discrete stability by employing locally optimal test functions, and it has a built-in error indicator which we exploit to guide mesh adaptivity. In addition to interfacing with third-party packages for various tasks, we have developed our own tools including a parallel nested dissection solver suitable for scalable FE computation of waveguide geometries. We present weak-scaling results with up to 24576 CPU cores and numerical simulations with more than one billion degrees of freedom. The new software capabilities enable solution of challenging wave propagation problems with important applications in acoustics, elastodynamics, and electromagnetics. These applications are difficult to solve in the high-frequency regime because the FE discretization suffers from significant numerical pollution errors that increase with the wavenumber. It is critical to control these errors to obtain a stable and accurate method. We study the pollution effect for waveguide problems with more than 8000 wavelengths in the context of robust DPG FE discretizations for the time-harmonic Maxwell equations. We also discuss adaptive refinement strategies for multi-mode fiber waveguides where the propagating transverse modes must be resolved sufficiently. Our study shows the applicability of the DPG error indicator to this class of problems. Finally, we present both modeling and computational advancements to a unique three-dimensional DPG FE model for the simulation of laser amplification in a fiber amplifier. Fiber laser amplifiers are of interest in communication technology, medical applications, military defense capabilities, and various other fields. Silica fiber amplifiers can achieve high-power operation with great efficiency. At high optical intensities, multi-mode amplifiers suffer from undesired thermal coupling effects which pose a major obstacle in power-scaling of such devices. Our nonlinear 3D vectorial model is based on the time-harmonic Maxwell equations, and it incorporates both amplification via an active dopant and thermal effects via coupling with the heat equation. The model supports co-, counter-, and bi-directional pumping configurations, as well as inhomogeneous and anisotropic material properties. The high-fidelity simulation comes at the cost of a high-order FE discretization with many degrees of freedom per wavelength. To make the computation more feasible, we have developed a novel longitudinal model rescaling, using artificial material parameters with the goal of preserving certain quantities of interest. Numerical tests demonstrate the applicability and utility of this scaled model in the simulation of an ytterbium-doped, step-index fiber amplifier that experiences laser amplification and heating.Computational Science, Engineering, and Mathematic

Simulation of Optical Fiber Amplifier Gain Using Equivalent Short Fibers

Electromagnetic wave propagation in optical fiber amplifiers obeys Maxwell equations. Using coupled mode theory, the full Maxwell system within an optical fiber amplifier is reduced to a simpler model. The simpler model is made more efficient through a new scale model, referred to as an equivalent short fiber, which captures some of the essential characteristics of a longer fiber. The equivalent short fiber can be viewed as a fiber made using artificial (unphysical) material properties that in some sense compensates for its reduced length. The computations can be accelerated by a factor approximately equal to the ratio of the original length to the reduced length of the equivalent fiber. Computations using models of two commercially available fibers -- one doped with ytterbium, and the other with thulium -- show the practical utility of the concept. Extensive numerical studies are conducted to assess when the equivalent short fiber model is useful and when it is not

Adaptive multilevel solvers for the discontinuous Petrov–Galerkin method with an emphasis on high-frequency wave propagation problems

This dissertation focuses on the development of fast and efficient solution schemes for the simulation of challenging problems in wave propagation phenomena. In particular, emphasis is given on high frequency acoustic and electromagnetic problems which are characterized by localized solutions. This kind of simulations are essential in various applications, such as ultrasonic testing, laser scanning and modeling of optical laser amplifiers. In wave simulations, the computational cost of any numerical method, is directly related to the frequency. In the high-frequency regime very fine meshes have to be used in order to satisfy the Nyquist criterion and overcome the pollution effect. This often leads to prohibitively expensive problems. Numerical methods based on standard Galerkin discretizations lack pre-asymptotic discrete stability and therefore adaptive mesh refinement strategies are usually inefficient. Additionally, the indefinite nature of the wave operator makes state of the art preconditioning techniques, such as multigrid, unreliable. In this work, a promising alternative approach is followed within the framework of the discontinuous Petrov–Galerkin (DPG) method. The DPG method offers numerous advantages for our problems of interest. First and foremost, it offers mesh and frequency independent discrete stability even in the pre-asymptotic region. This is made possible by computing, on the fly, an optimal test space as a function of the trial space. Secondly, it provides a built-in local error indicator that can be used to drive adaptive refinements. Combining these two properties together, reliable adaptive refinement strategies are possible which can be initiated from very coarse meshes. Lastly, the DPG method can be viewed as a minimum residual method, and therefore it always delivers symmetric (Hermitian) positive definite stiffness matrix. This is a desirable advantage when it comes to the design of iterative solution algorithms. Conjugate Gradient based solvers can be employed which can be accelerated by domain decomposition (one- or multi- level) preconditioners for symmetric positive definite systems. Driven by the aforementioned properties of the DPG method, an adaptive multigrid preconditioning technology is developed that is applicable for a wide range of boundary value problems. Unlike standard multigrid techniques, our preconditioner involves trace spaces defined on the mesh skeleton, and it is suitable for adaptive hp-meshes. Integration of the iterative solver within the DPG adaptive procedure turns out to be crucial in the simulation of high frequency wave problems. A collection of numerical experiments for the solution of linear acoustics and Maxwell equations demonstrate the efficiency of this technology, where under certain circumstances uniform convergence with respect to the mesh size, the polynomial order and the frequency can be achieved. The construction is complemented with theoretical estimates for the condition number in the one-level setting.Computational Science, Engineering, and Mathematic

Analyse des Solitonengehaltes von optischen Impulsen in Glasfasern

Der Solitonengehalt von Lichtimpulsen in einer Glasfaser wird untersucht. Dabei wird ein neu entwickeltes Verfahren angewendet, welches auf der spektralen Analyse der Schwebunsstrukturen beruht. Dieses Verfahren ist in der Lage, den allemeinen Solitonengehalt zu bestimmen, sogar für nichtintegrable Systeme. Dies war bisher nur Näherungsweise möglich. Aus der vorgestellten Analyse wird ein Messprinzip abgeleitet, mit dem sich der Solitonengehalt bestimmen lässt. Dies wird anhand einer Beispielmessung demonstriert

A 920 km optical fiber link for frequency metrology at the 19th decimal place

With residual uncertainties at the 10^-18 level, modern atomic frequency standards constitute extremely precise measurement devices. Besides frequency and time metrology, they provide valuable tools to investigate the validity of Einstein's theory of general relativity, to test a possible time variation of the fundamental constants, and to verify predictions of quantum electrodynamics. Furthermore, applications as diverse as geodesy, satellite navigation, and very long base-line interferometry may benefit from steadily improving precision of both microwave and optical atomic clocks. Clocks ticking at optical frequencies slice time into much finer intervals than microwave clocks and thus provide increased stability. It is expected that this will result in a redefinition of the second in the International System of Units (SI). However, any frequency measurement is based on a comparison to a second, ideally more precise frequency. A single clock, as highly developed as it may be, is useless if it is not accessible for applications. Unfortunately, the most precise optical clocks or frequency standards can not be readily transported. Hence, in order to link the increasing number of world-wide precision laboratories engaged in state-of-the-art optical frequency standards, a suitable infrastructure is of crucial importance. Today, the stabilities of current satellite based dissemination techniques using global satellite navigation systems (such as GPS, GLONASS) or two way satellite time and frequency transfer reach an uncertainty level of 10^-15 after one day of comparison . While this is sufficient for the comparison of most microwave clock systems, the exploitation of the full potential of optical clocks requires more advanced techniques. This work demonstrates that the transmission of an optical carrier phase via telecommunication fiber links can provide a highly accurate means for clock comparisons reaching continental scales: Two 920 km long fibers are used to connect MPQ (Max-Planck- Institut für Quantenoptik, Garching, Germany) and PTB (Physikalisch-Technische Bundesanstalt, Braunschweig, Germany) separated by a geographical distance of 600 km. The fibers run in a cable duct next to a gas pipeline and are actively compensated for fluctuations of their optical path length that lead to frequency offsets via the Doppler effect. Together with specially designed and remotely controllable in-line amplication this enables the transfer of an ultra-stable optical signal across a large part of Germany with a stability of 5 x 10^-15 after one second, reaching 10^-18 after less than 1000 seconds of integration time. Any frequency deviation induced by the transmission can be constrained to be smaller than 4 x 10^-19. As a first application, the fiber link was used to measure the 1S-2S two photon transition frequency in atomic hydrogen at MPQ referenced to PTB's primary Cs-fountain clock (CSF1). Hydrogen allows for precise theoretical analysis and the named transition possesses a narrow natural line width of 1.3 Hz. Hence, this experiment constitutes a very accurate test bed for quantum electrodynamics and has been performed at MPQ with ever increasing accuracy. The latest measurement has reached a level of precision at which satellite-based referencing to a remote primary clock is limiting the experiment. Using the fiber link, a frequency measurement can be carried out directly since the transmission via the optical carrier phase provides orders of magnitude better stability than state-of-the-art microwave clocks. The achieved results demonstrate that high-precision optical frequency dissemination via optical fibers can be employed in real world applications. Embedded in an existing telecommunication network and passing several urban agglomerations the fiber link now permanently connects MPQ and PTB and is operated routinely. It represents far more than a proof-of-principle experiment conducted under optimized laboratory conditions. Rather it constitutes a solution for the topical issue of remote optical clock comparison. This opens a variety of applications in fundamental physics such as tests of general and special relativity as well as quantum electrodynamics. Beyond that, such a link will enable clock-based, relativistic geodesy at the sub-decimeter level. Further applications in navigation, geology, dynamic ocean topography and seismology are currently being discussed. In the future, this link will serve as a backbone of a Europe-wide optical frequency dissemination network

Quantum tests of the universality of free fall

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Electromagnetic Manipulation of Individual Nano- and Microparticles

Gegenstand der vorliegenden Dissertation ist die Untersuchung von einzelnen nano- und mikrometergroßen Partikeln, zum Verständnis und zur Entwicklung von neuartigen nanooptischen Elementen, wie Lichtquellen und Sensoren, sowie Strukturen zum Aufsammeln und Leiten von Licht. Neben der Charakterisierung stehen dabei verschiedene Methoden zur elektromagnetischen Manipulation im Vordergrund, die auf eine Kontrolle der Position oder der Geometrie der Partikel ausgerichtet sind. Die gezielten Manipulationen werden verwendet, um vorausgewählte Partikel zu isolieren, modifizieren und transferieren. Dadurch können Partikel zu komplexeren photonischen Systemen kombiniert werden, welche die Funktionalität der einzelnen Bestandteile übertreffen. Der Hauptteil der Arbeit behandelt Experimente mit freischwebenden Partikeln in linearen Paul-Fallen. Durch die räumliche Isolation im elektrodynamischen Quadrupolfeld können Partikel mit reduzierter Wechselwirkung untersucht werden. Neben der spektroskopischen Charakterisierung von optisch aktiven Partikeln (farbstoffdotierte Polystyrol-Nanokügelchen, Cluster aus Nanodiamanten mit Stickstoff-Fehlstellen-Zentren, Cluster aus kolloidalen Quantenpunkten) sowie optischen Resonatoren (plasmonische Silber-Nanodrähte, sphärische Siliziumdioxid-Mikroresonatoren) werden neu entwickelte Methoden zur Manipulation vorgestellt, mit denen sich individuelle Partikel freischwebend kombinieren und elektromagnetisch koppeln sowie aus der Falle auf optischen Fasern zur weiteren Untersuchung bzw. zur Funktionalisierung photonischer Strukturen ablegen lassen. In einem weiteren Teil der Arbeit wird eine Methode zur Manipulation der Geometrie von plasmonischen Nanopartikeln vorgestellt. Dabei werden einzelne Goldkugeln auf einem Deckglas mit einem fokussierten Laserstrahl zum Schmelzen gebracht und verformt. Durch die kontrollierte und reversible Veränderung der Symmetrie lassen sich die lokalisierten Oberflächenplasmonen des Partikels gezielt beeinflußen.The topic of the present thesis is the investigation of single nano- and microsized particles for the understanding and design of novel nanooptical elements as light sources and sensors, as well as light collecting and guiding structures. In addition to particle characterization, the focus is on different methods for electromagnetic particle manipulation aimed at controlling the particle’s position or geometry. The specific manipulations are used for isolation, modification and transfer of preselected particles, enabling combination of particles into more complex photonic systems, which exceed the functionalities of the individual constituents. The main part of this work deals with experiments on levitated particles in linear Paul traps. Due to the spatial isolation in the electrodynamic quadrupole field, particles can be investigated with reduced environmental interaction. In addition to spectroscopic characterization of optically active particles (dye-doped polystyrene nanobeads, clusters of nanodiamonds with nitrogen vacancy defect centers, clusters of colloidal quantum dots) and particles with optical resonances (plasmonic silver nanowires, spherical silica microresonators) new manipulation methods are presented that enable assembly and electromagnetic coupling of individual, levitated particles as well as deposition of particles from the trap on optical fibers for further characterization or functionalization of photonic structures. In a further part of this work a method to manipulate the geometry of plasmonic nanoparticles is presented. Single gold nanospheres on a coverslip are melted and shaped with a focused laser beam. The localized surface plasmons can be influenced specifically by controlled and reversible changes of the particle symmetry