Geometric Degrees Of Freedom Research Articles

Inverse design of coherent supercontinuum generation using free-form nanophotonic waveguides

Many key functionalities of optical frequency combs, such as self-referencing and broad spectral access, rely on coherent supercontinuum generation (SCG). While nanophotonic waveguides have emerged as a compact and power-efficient platform for SCG, their geometric degrees of freedom have not been fully utilized due to the underlying complex nonlinear and noise-dependent stochastic physics. Here, we introduce inverse design to unlock free-form waveguides for coherent SCG. The efficacy of our design is numerically and experimentally demonstrated on Si3N4 waveguides, producing flat and coherent spectra from visible to mid-infrared wavelengths. Our work has direct applications in developing chip-based broadband light sources for spectroscopy, metrology, and sensing across multiple spectral regimes.

Towards active learning: A stopping criterion for the sequential sampling of grain boundary degrees of freedom

Many materials processes and properties depend on the anisotropy of the energy of grain boundaries, i.e. on the fact that this energy is a function of the five geometric degrees of freedom (DOF) of the interface. To access this parameter space in an efficient way and to discover energy cusps in unexplored regions, a method was recently established, which combines atomistic simulations with statistical methods (Kroll et al., 2022). This sequential sampling technique is now extended in the spirit of an active learning algorithm by adding a criterion to decide when the sampling has advanced enough to stop. In this instance, two parameters to analyse the sampling results on the fly are introduced: the number of cusps, which correspond to the most interesting and important regions of the energy landscape, and the maximum change of energy between two sequential iterations. Monitoring these two quantities provides valuable insight into how the subspaces are energetically structured. The combination of both parameters provides the necessary information to evaluate the sampling of the 2D subspaces of grain boundary plane inclinations of even non-periodic, low angle grain boundaries. With a reasonable number of data points in the initial design, only a few appropriately chosen sequential iterations already improve the accuracy of the sampling substantially and unknown cusps can be found within a few additional sequential steps.

Fourier synthesis dispersion engineering of photonic crystal microrings for broadband frequency combs

Dispersion engineering of microring resonators is crucial for optical frequency comb applications, to achieve targeted bandwidths and powers of individual comb teeth. However, conventional microrings only present two geometric degrees of freedom – width and thickness – which limits the degree to which dispersion can be controlled. We present a technique where we tune individual resonance frequencies for arbitrary dispersion tailoring. Using a photonic crystal microring resonator that induces coupling to both directions of propagation within the ring, we investigate an intuitive design based on Fourier synthesis. Here, the desired photonic crystal spatial profile is obtained through a Fourier relationship with the targeted modal frequency shifts, where each modal shift is determined based on the corresponding effective index modulation of the ring. Experimentally, we demonstrate several distinct dispersion profiles over dozens of modes in transverse magnetic polarization. In contrast, we find that the transverse electric polarization requires a more advanced model that accounts for the discontinuity of the field at the modulated interface. Finally, we present simulations showing arbitrary frequency comb spectral envelope tailoring using our Fourier synthesis approach.

Connection dynamics of reduced five-dimensional Kaluza-Klein theory and its deparametrization

The connection dynamics of the 5-dimensional Kaluza-Klein theory reduced on 4-dimensional spacetime is obtained by performing the Hamiltonian analysis and canonical transformations. Deparametrization is achieved in the spherically symmetric model of the theory without introducing additional matter fields beyond the 5-dimensional gravity. Thus the physical time evolution and the physical spatial coordinate can be provided by the geometrical degrees of freedom in the higher dimensional spacetime.

Pure shape dynamics: general framework

We put forward a general framework for describing relational physical theories, which we call pure shape dynamics (PSD). Elaborating on the original insights brought about by the shape dynamics programme, PSD’s novel take on relationalism is its insistence on describing any dynamical system by means of the intrinsic geometry of its associated curve in the suitable relational configuration space of the theory, namely shape space, whereby the corresponding equation of state of the curve expresses the ratio of change of one of its geometric degrees of freedom with respect to another one. The mathematical structure underlying the equation of state is a local section over a natural generalisation of the unit tangent bundle, which we call shape phase space.

Designing Geometric Degrees of Freedom in ReO3‐Type Coordination Polymers

AbstractEngineering the interplay of structural degrees of freedom that couple to external stimuli such as temperature and pressure is a powerful approach for material design. New structural degrees of freedom expand the potential of the concept, and coordination polymers as a chemically versatile material platform offer fascinating possibilities to address this challenge. Here, we report a new class of perovskite‐like AB2X6 coordination polymers based on a [BX3] − ReO3‐type host network ([Mn(C2N3)3] −), in which the spatial orientation of divalent A2+ cations ([R3N(CH2)nNR3]2+) with separated charge centers that bridge adjacent ReO3‐cavities is introduced as a new geometric degree of freedom. Herringbone and head‐to‐tail order pattern of [R3N(CH2)nNR3]2+ cations are obtained by varying the separator length n and, together with distortions of the pseudocubic [BX3] − network, they determine the materials’ stimuli‐responsive behavior such as counterintuitive large negative compressibility and uniaxial negative thermal expansion. This new family of coordination polymers highlights the chemists’ capabilities of designing matter on a molecular level to address macroscopic material functionality and underpins the opportunities of the design of structural degrees of freedom as a conceptual framework for rational material synthesis in the future.

Efficient Prediction of Grain Boundary Energies from Atomistic Simulations via Sequential Design

AbstractWith the goal of improving data based materials design, it is shown that by a sequential design of experiment scheme the process of generating and learning from the data can be combined to discover the relevant sections of the parameter space. The application is the energy of grain boundaries as a function of their geometric degrees of freedom, calculated from a simple model, or via atomistic simulations. The challenge is to predict the deep cusps of the energy, which are located at irregular intervals of the geometric parameters. Existing sampling approaches either use large sets of datapoints or a priori knowledge of the cusps' positions. By contrast, the authors' technique can find unknown cusps automatically with a minimal amount of datapoints. Key point is a Kriging interpolator with Matérn kernel to estimate the energy function. Using the jackknife variance, the next point in the sequential design is a compromise between sampling the region of largest fluctuations and avoiding a clustering of datapoints. In this way, the cusps of the energy can be found within only a few iterations, and refined as desired. This approach will be advantageous for any application with strong, localized fluctuations in the values of the unknown function.

Junction conditions for generalized hybrid metric-Palatini gravity with applications

The generalized hybrid metric-Palatini gravity is a theory of gravitation that has an action composed of a Lagrangian given by $f(R,\mathcal{R})$, where $f$ is a function of the metric Ricci scalar $R$ and a new Ricci scalar $\mathcal{R}$ formed from a Palatini connection, plus a matter Lagrangian. This theory can be rewritten by trading the new geometric degrees of freedom that appear in $f(R,\mathcal{R})$ into two scalar fields, $\ensuremath{\varphi}$ and $\ensuremath{\psi}$, yielding a dynamically equivalent scalar-tensor theory. Given a spacetime theory, the next important step is to find solutions within it. To construct appropriate solutions it is often necessary to know the junction conditions between two spacetime regions at a separation hypersurface $\mathrm{\ensuremath{\Sigma}}$, with each spacetime region being an independent solution of the theory. The junction conditions for the generalized hybrid metric-Palatini gravity are found here, both in the geometric representation and in the scalar-tensor representation, and in addition, for each representation, the junction conditions for a matching with a thin-shell of matter and for a smooth matching at the separation hypersurface are worked out. These junction conditions are then applied to three configurations, namely, a star, a quasistar with a black hole, and a wormhole. The star is made of a Minkowski interior, a thin shell at the interface with all the matter energy conditions being satisfied, and a Schwarzschild exterior with mass $M$, and unlike general relativity where the matching can be performed at any radius ${r}_{\mathrm{\ensuremath{\Sigma}}}$, for this theory the matching can only be performed at a specific value of the shell radius, namely ${r}_{\mathrm{\ensuremath{\Sigma}}}=\frac{9M}{4}$, that corresponds to the general relativistic Buchdahl radius. The quasistar with a black hole is made of an interior Schwarzschild black hole surrounded by a thick shell that matches smoothly to a mass $M$ Schwarzschild exterior at the light ring radius ${r}_{\mathrm{\ensuremath{\Sigma}}e}=3M$, and with the matter energy conditions being satisfied for the whole spacetime. The wormhole is made of some interior with matter that contains the throat, a thin shell at the interface, and a Schwarzschild-AdS exterior with mass $M$ and negative cosmological constant $\mathrm{\ensuremath{\Lambda}}$, with the matter null energy condition being obeyed everywhere within the wormhole.

Geometrical Degrees of Freedom for Cellular Structures Generation: A New Classification Paradigm

Cellular structures (CSs) have been used extensively in recent years, as they offer a unique range of design freedoms. They can be deployed to create parts that can be lightweight by introducing controlled porous features, while still retaining or improving their mechanical, thermal, or even vibrational properties. Recent advancements in additive manufacturing (AM) technologies have helped to increase the feasibility and adoption of cellular structures. The layer-by-layer manufacturing approach offered by AM is ideal for fabricating CSs, with the cost of such parts being largely independent of complexity. There is a growing body of literature concerning CSs made via AM; this presents an opportunity to review the state-of-the-art in this domain and to showcase opportunities in design and manufacturing. This review will propose a novel way of classifying cellular structures by isolating their Geometrical Degrees of Freedom (GDoFs) and will explore the recent innovations in additively manufactured CSs. Based on the present work, the design inputs that are common in CSs generation will be highlighted. Furthermore, the work explores examples of how design inputs have been used to drive the design domain through various case studies. Finally, the review will highlight the manufacturability limitations of CSs in AM.

Design and manufacturing of a cylinder head by laser powder bed fusion

Nowadays the reduction of fuel consumption is the most important challenge for engine development. Laser Powder Bed Fusion (LPBF) as an additive manufacturing process with its implied design freedom makes it possible to reduce combustion emissions, increase engine efficiency and respond specifically to customer requirements. In this work, the outstanding degree of freedom of LPBF is exploited for a new design concept for a cylinder head of an engine in automotive industry in order to address further reduction of fuel consumption through improved cooling performance. Within the given installation space and under consideration of the existing mounting and connection positions, the design concept was developed, subsequently manufactured and reworked. In order to develop a design that meets the LPBF requirements, both the potential of the LPBF using the geometric degrees of freedom as effectively as possible and the limitations are considered. The opposing design drivers between exploitation of potentials and limitations in LPBF are achieved by an iterative design process supported by an own developed design tool. With the new design concept, a reduction of the metal temperature in the cylinder head of approx. 25 K is achieved, verified by simulation.

Dual-Polarized All-Metallic Metagratings For Perfect Anomalous Reflection

We theoretically formulate and experimentally demonstrate the design of metagratings (MGs) composed of periodic rectangular grooves in a metallic medium, intended for perfect anomalous reflection. Using mode matching, a semianalytical scheme for analysis and synthesis of such MGs, containing multiple, arbitrarily arranged, grooves per period, is derived. Following the typical MG design approach, we use this formalism to identify the relevant Floquet-Bloch (FB) modes and conveniently formulate constraints for suppression of spurious scattering, directly tying the structure's geometrical degrees of freedom (DOFs) to the desired functionality. Solving this set of constraints, in turn, yields a detailed fabrication-ready MG design, without any full-wave optimization. Besides providing means to realize highly-efficient beam deflection with all-metallic formations, we show that the rectangular (two-dimensional) groove configuration enables \emph{simultaneous} manipulation of both transverse electric (TE) and transverse magnetic (TM) polarized fields, unavailable to date with common, printed-circuit-board-based, microwave MGs. In addition, we highlight a physical limitation on the TE-polarization performance, preventing the ability to achieve perfect anomalous reflection in any desired angle. These capabilities are verified using three MG prototypes, produced with standard computer numerical control (CNC) machines, demonstrating both single- and dual-polarized control of multiple diffraction modes. These results enable the use of MGs for a broader range of applications, where dual-polarized control is required, or all-metallic devices are preferable (e.g., spaceborne systems or at high operating frequencies).

Application of Constrained Optimization Techniques in Optimal Shape Design of a Freezer to Dosing Line Splitter for Ice Cream Production

Design of multiple branches splitting of equal mass flow rate in complex rheological flows like ice cream near melting point temperature can be a challenging task. Pulsations in flow rate due to air pumping process and small fluctuations in temperature affecting flow rheology can determine a consistent difference in internal pipe velocity distribution, resulting in a significant difference in the distribution of ice cream dosage. Computational sciences and engineering techniques have allowed a major change in the way products and equipment can be engineered, as a computational model simulating physical processes can be more easily obtained, rather than making prototypes and performing multiple experiments. Among such techniques, optimal shape design (OSD) represents an interesting approach. In OSD, the essential element with respect to classical numerical simulations in fixed geometrical configurations relays on the introduction a certain amount of geometrical degrees of freedom as a part of the unknowns. This implies that the geometry is not completely defined, but part of it is allowed to move dynamically in order to minimize or maximize an objective function. From a mathematical point of view, OSD is a branch of differentiable optimization and more precisely the application of optimal control for distributed systems. OSD is still today numerically difficult to implement, because it relies on a computer intensive activity and moreover because the concept of “optimal” is a compromise between shapes that are good with respect to several criteria. In this work, the applications of a multivariate constrained optimization algorithm is proposed in the case of a mechanical ice cream 1 to 5 splitting system, required to distribute in an evenly way from one freezer into five dosing valves. Results allowed to design a retro-fitting system on an existing production plant reducing the dosing error down to 3% on the average.

Bi-functional meta-device with full energy utilization in co- and cross-polarization fields

In high contrast to traditional geometric meta-surfaces, which can only operate on the cross-polarized field, we propose a bi-functional meta-device with full energy utilization in both co- and cross-polarized fields. In this paper, co-polarized refraction and cross-polarized focusing are integrated in one single meta-surface by taking advantage of compound propagation and geometric phases. A multi-layer element structure with various geometric degrees of freedom is designed to realize simultaneous and independent phase modulation of both co- and cross-polarized transmitted waves. The derived criterion is verified in the microwave region, and the proof-of-concept experiments indicate that the proposed meta-surface can have distinct functionalities with orthogonal polarizations in the transmitted field with a specific circularly polarized illumination. This scheme for bi-functional meta-surface construction provides an effective method to manipulate both co- and cross-polarized output fields simultaneously, which can extremely enhance the modulation efficiency of output energy. Different output wavefronts can be conveniently switched by filtering only the polarization state of the output end, simplifying the operational complexity of the bi-functional meta-system. This criterion paves the way for flexible manipulation of electromagnetic waves with a higher degree of freedom and facilitates a larger diversity of applications across the whole frequency spectrum.

Holographic relationships in Lovelock type brane gravity

We show that the Lovelock type brane gravity is naturally holographic by providing a correspondence between bulk and surface terms that appear in the Lovelock-type brane gravity action functional. We prove the existence of relationships between the and allowing to be determined completely by . In the same spirit, we provide relationships among the various conserved tensors that this theory possesses. We further comment briefly on the correspondence between geometric degrees of freedom in both bulk and surface space.

Matter Growth in Imperfect Fluid Cosmology

Extensions of Einstein’s General Relativity (GR) can formally be given a GR structure in which additional geometric degrees of freedom are mapped on an effective energy-momentum tensor. The corresponding effective cosmic medium can then be modeled as an imperfect fluid within GR. The imperfect fluid structure allows us to include, on a phenomenological basis, anisotropic stresses and energy fluxes which are considered as potential signatures for deviations from the cosmological standard Λ -cold-dark-matter ( Λ CDM) model. As an example, we consider the dynamics of a scalar-tensor extension of the standard model, the e Φ Λ CDM model. We constrain the magnitudes of anisotropic pressure and energy flux with the help of redshift-space distortion (RSD) data for the matter growth function f σ 8 .

Schwarzschild-de Sitter black hole in canonical quantization

We solve Wheeler-De Witt (WDW) metric probability wave equation on the apparent horizon hypersurface of the Schwarzschild de Sitter (SdS) black hole. To do so we choose radial dependent mass function M(r) for its internal regions in the presence of a dynamical massless quantum matter scalar field ψ(r) and calculate canonical supper hamiltonian constraint on t-constant hypersurface near the horizon r = M(r). In this case M(r) become geometrical degrees of freedom while ψ(r) is matter degrees of freedom of the apparent horizon. However our solution is obtained versus the quantum harmonic oscillator which defined against the well known hermit polynomials. In the latter case we obtain quantized mass of the SdS quantum black hole as in which Λ is the cosmological constant and n = 0,1, 2, ⋯ are quantum numbers of the hermit polynomials. This shows that a quantized SdS in its ground state has a nonzero value for the mass . Thus one can infer that the latter result satisfies Penrose hypotheses for cosmic censorship where a causal singularity may be covered by a horizon surface.

Connecting Composition-Driven Faceting with Facet-Driven Composition Modulation in GaAs-AlGaAs Core-Shell Nanowires.

Ternary III-V alloys of tunable bandgap are a foundation for engineering advanced optoelectronic devices based on quantum-confined structures including quantum wells, nanowires, and dots. In this context, core-shell nanowires provide useful geometric degrees of freedom in heterostructure design, but alloy segregation is frequently observed in epitaxial shells even in the absence of interface strain. High-resolution scanning transmission electron microscopy and laser-assisted atom probe tomography were used to investigate the driving forces of segregation in nonplanar GaAs-AlGaAs core-shell nanowires. Growth-temperature-dependent studies of Al-rich regions growing on radial {112} nanofacets suggest that facet-dependent bonding preferences drive the enrichment, rather than kinetically limited diffusion. Observations of the distinct interface faceting when pure AlAs is grown on GaAs confirm the preferential bonding of Al on {112} facets over {110} facets, explaining the decomposition behavior. Furthermore, three-dimensional composition profiles generated by atom probe tomography reveal the presence of Al-rich nanorings perpendicular to the growth direction; correlated electron microscopy shows that short zincblende insertions in a nanowire segment with predominantly wurtzite structure are enriched in Al, demonstrating that crystal phase engineering can be used to modulate composition. The findings suggest strategies to limit alloy decomposition and promote new geometries of quantum confined structures.

The role of the interface stiffness tensor on grain boundary dynamics

Grain boundary (GB) properties and associated anisotropies due to the boundary's geometric degrees of freedom (DOF) greatly influence many of the salient features of polycrystalline aggregates, and as a consequence the observable properties of the material. Through theoretical analysis and atomistic simulations, we show that when considering the GB plane normal DOF, the GB interface stiffness tensor plays a paramount role in a wide range of GB dynamical processes. As a demonstration, we examine the interface stiffness tensor of Σ3 GBs in nickel and show that the stiffness can be much larger in magnitude and more anisotropic than the GB energy itself. Moreover, it is found that a wide range of inclinations exhibit negative stiffness values, a signature of a structural instability. This is the first study to consider the complete spatial description of the GB stiffness tensor, where both angular variations describing the interface plane normal are explored. In broad terms, our results highlight the integral role that the stiffness tensor plays in GB interfacial phenomena, such as curvature-driven flow and faceting instabilities.

Information Graph Flow: A Geometric Approximation of Quantum and Statistical Systems

Given a quantum (or statistical) system with a very large number of degrees of freedom and a preferred tensor product factorization of the Hilbert space (or of a space of distributions) we describe how it can be approximated with a very low-dimensional field theory with geometric degrees of freedom. The geometric approximation procedure consists of three steps. The first step is to construct weighted graphs (we call information graphs) with vertices representing subsystems (e.g. qubits or random variables) and edges representing mutual information (or the flow of information) between subsystems. The second step is to deform the adjacency matrices of the information graphs to that of a (locally) low-dimensional lattice using the graph flow equations introduced in the paper. (Note that the graph flow produces very sparse adjacency matrices and thus might also be used, for example, in machine learning or network science where the task of graph sparsification is of a central importance.) The third step is to define an emergent metric and to derive an effective description of the metric and possibly other degrees of freedom. To illustrate the procedure we analyze (numerically and analytically) two information graph flows with geometric attractors (towards locally one- and two-dimensional lattices) and metric perturbations obeying a geometric flow equation. Our analysis also suggests a possible approach to (a non-perturbative) quantum gravity in which the geometry (a secondary object) emerges directly from a quantum state (a primary object) due to the flow of the information graphs.

Broadband ultra-flattened dispersion, ultra-low confinement loss and large effective mode area in an octagonal photonic quasi-crystal fiber.

In this work, an octagonal Penrose-type photonic quasi-crystal fiber (PQF) with dual-cladding is proposed. By optimizing three geometric degrees of freedom, the PQF exhibits ultra-flattened near-zero dispersion of 0.014±0.293 ps/nm/km, ultra-low order confinement loss of 10-4 dB/km, and large effective mode area of over 16.2 μm2 in a broadband of wavelength from 1.27 to 1.67μm, covering almost all optical communication bands. At the common communication wavelength 1.55μm, completely opposite trends of the dispersion and the confinement loss varying with the air-filling factor in the inner cladding are demonstrated. In addition, the robustness of optical properties including dispersion, confinement loss, and effective mode area in this PQF is discussed, assuming a deviation ±3% of all air holes.