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Orthogonality, Lommel integrals and cross product zeros of linear combinations of Bessel functions
(2015)
The cylindrical Bessel differential equation and the spherical Bessel differential equation in the interval R\(\leq\)r\(\leq\)\(\gamma\)R with Neumann boundary conditions are considered. The eigenfunctions are linear combinations of the Bessel function \(\Phi\)\(_{n,ν}\)(r) = Y'\(_{ν}\) (\(\lambda\)\(_{n,ν}\))J\(_{ν}\)(\(\lambda\)\(_{n,ν}\) r/R) - J'\(_{ν}\)(\(\lambda\)\(_{n,ν}\))Y\(_{ν}\)(\(\lambda\)\(_{n,ν}\)r/R) or linear combinations of the spherical Bessel functions \(\psi\)\(_{m,ν}\)(r) = y'\(_{ν}\)(\(\lambda\)\(_{m,ν}\))j\(_{ν}\)(\(\lambda\)\(_{m,ν}\)r/R) - j'\(_{ν}\)(\(\lambda\)\(_{m,ν}\))y\(_{ν}\)(\(\lambda\)\(_{m,ν}\)r/R). The orthogonality relations with analytical expressions for the normalization constant are given. Explicit expressions for the Lommel integrals in terms of Lommel functions are derived. The cross product zeros Y'\(_{ν}\)\(\lambda\)\(_{n,ν}\))J'\(_{ν}\)(\(\gamma\)\(\lambda\)\(_{n,ν}\))- J'\(_{ν}\)(\(\lambda\)\(_{n,ν}\))Y'\(_{ν}\)(\(\gamma\)\(\lambda\)\(_{n,ν}\)) = 0 and y'\(_{ν}\)(\(\lambda\)\(_{m,ν}\))j'\(_{ν}\)(\(\gamma\)\(\lambda\)\(_{m,ν}\)) - j'\(_{ν}\)(\(\lambda\)\(_{m,ν}\))y'\(_{ν}\)(\(\gamma\)\(\lambda\)\(_{m,ν}\)) = 0 are considered in the complex plane for real as well as complex values of the index ν and approximations for the exceptional zero \(\lambda\)\(_{1,ν}\) are obtained. A numerical scheme based on the discretization of the twodimensional and three-dimensional Laplace operator with Neumann boundary conditions is presented. Explicit representations of the radial part of the Laplace operator in form of a tridiagonal matrix allow the simple computation of the cross product zeros.
Purpose
Non-invasive end-points for interventional trials and tailored treatment regimes in chronic obstructive pulmonary disease (COPD) for monitoring regionally different manifestations of lung disease instead of global assessment of lung function with spirometry would be valuable. Proton nuclear magnetic resonance imaging (1H-MRI) allows for a radiation-free assessment of regional structure and function. The aim of this study was to evaluate the short-term reproducibility of a comprehensive morpho-functional lungMRI protocol in COPD.
Materials and Methods
20 prospectively enrolled COPD patients (GOLD I-IV) underwent 1H-MRI of the lung at 1.5T on two consecutive days, including sequences for morphology, 4D contrast-enhanced perfusion, and respiratory mechanics. Image quality and COPD-related morphological and functional changes were evaluated in consensus by three chest radiologists using a dedicated MRI-based visual scoring system. Test-retest reliability was calculated per each individual lung lobe for the extent of large airway (bronchiectasis, wall thickening, mucus plugging) and small airway abnormalities (tree in bud, peripheral bronchiectasis, mucus plugging), consolidations, nodules, parenchymal defects and perfusion defects. The presence of tracheal narrowing, dystelectasis, pleural effusion, pulmonary trunk ectasia, right ventricular enlargement and, finally, motion patterns of diaphragma and chest wall were addressed.
Results
Median global scores [10(Q1:8.00; Q3:16.00) vs. 11(Q1:6.00; Q3:15.00)] as well as category subscores were similar between both timepoints, and kappa statistics indicated "almost perfect" global agreement (\(\kappa\)= 0.86, 95%CI = 0.81-0.91). Most subscores showed at least "substantial" agreement of MRI1 and MRI2 (\(\kappa\)= 0.64-1.00), whereas the agreement for the diagnosis of dystelectasis/effusion (\(\kappa\)= 0.42, 95%CI = 0.00-0.93) was "moderate" and of tracheal abnormalities (\(\kappa\)= 0.21, 95%CI = 0.00-0.75) "fair". Most MRI acquisitions showed at least diagnostic quality at MRI1 (276 of 278) and MRI2 (259 of 264).
Conclusion
Morpho-functional 1H-MRI can be obtained with reproducible image quality and high short-term test-retest reliability for COPD-related morphological and functional changes of the lung. This underlines its potential value for the monitoring of regional lung characteristics in COPD trials.
Efficient and fast on-demand single photon sources have been sought after as critical components of quantum information science. We report an efficient and tunable single photon source based on an InAs quantum dot (QD) embedded in a photonic crystal cavity coupled with a highly curved \(\mu\)-fibre. Exploiting evanescent coupling between the \(\mu\)-fibre and the cavity, a high collection efficiency of 23% and Purcell-enhanced spontaneous emissions are observed. In our scheme, the spectral position of a resonance can be tuned by as much as 1.5 nm by adjusting the contact position of the \(\mu\)-fibre, which increases the spectral coupling probability between the QD and the cavity mode. Taking advantage of the high photon count rate and the tunability, the collection efficiencies and the decay rates are systematically investigated as a function of the QD-cavity detuning.
The charge carrier lifetime is an important parameter in solar cells as it defines, together with the mobility, the diffusion length of the charge carriers, thus directly determining the optimal active layer thickness of a device. Herein, we report on charge carrier lifetime values in bromine doped planar methylammonium lead iodide (MAPbI\(_3\)) solar cells determined by transient photovoltage. The corresponding charge carrier density has been derived from charge carrier extraction. We found increased lifetime values in solar cells incorporating bromine compared to pure MAPbI\(_3\) by a factor of ~2.75 at an illumination intensity corresponding to 1 sun. In the bromine containing solar cells we additionally observe an anomalously high value of extracted charge, which we deduce to originate from mobile ions.
We report a giant thermal shift of 2.1 MHz/K related to the excited-state zero-field splitting in the silicon vacancy centers in 4H silicon carbide. It is obtained from the indirect observation of the optically detected magnetic resonance in the excited state using the ground state as an ancilla. Alternatively, relative variations of the zero-field splitting for small temperature differences can be detected without application of radiofrequency fields, by simply monitoring the photoluminescence intensity in the vicinity of the level anticrossing. This effect results in an all-optical thermometry technique with temperature sensitivity of 100 mK/Hz\(^{1/2}\) for a detection volume of approximately 10\(^{−6}\) mm\(^3\). In contrast, the zero-field splitting in the ground state does not reveal detectable temperature shift. Using these properties, an integrated magnetic field and temperature sensor can be implemented on the same center.
We uncover the fine structure of a silicon vacancy in isotopically purified silicon carbide (4H-\(^{28}\)SiC) and reveal not yet considered terms in the spin Hamiltonian, originated from the trigonal pyramidal symmetry of this spin-3/2 color center. These terms give rise to additional spin transitions, which would be otherwise forbidden, and lead to a level anticrossing in an external magnetic field. We observe a sharp variation of the photoluminescence intensity in the vicinity of this level anticrossing, which can be used for a purely all-optical sensing of the magnetic field. We achieve dc magnetic field sensitivity better than 100 nT/√Hz within a volume of 3×10\(^{−7}\)mm\(^3\) at room temperature and demonstrate that this contactless method is robust at high temperatures up to at least 500 K. As our approach does not require application of radio-frequency fields, it is scalable to much larger volumes. For an optimized light-trapping waveguide of 3 mm\(^3\), the projection noise limit is below 100 fT/√Hz.
Analysis of Triplet Exciton Loss Pathways in PTB7:PC\(_{71}\)BM Bulk Heterojunction Solar Cells
(2016)
A strategy for increasing the conversion efficiency of organic photovoltaics has been to increase the VOC by tuning the energy levels of donor and acceptor components. However, this opens up a new loss pathway from an interfacial charge transfer state to a triplet exciton (TE) state called electron back transfer (EBT), which is detrimental to device performance. To test this hypothesis, we study triplet formation in the high performing PTB7:PC\(_{71}\)BM blend system and determine the impact of the morphology-optimizing additive 1,8-diiodoctane (DIO). Using photoluminescence and spin-sensitive optically detected magnetic resonance (ODMR) measurements at low temperature, we find that TEs form on PC\(_{71}\)BM via intersystem crossing from singlet excitons and on PTB7 via EBT mechanism. For DIO blends with smaller fullerene domains, an increased density of PTB7 TEs is observed. The EBT process is found to be significant only at very low temperature. At 300 K, no triplets are detected via ODMR, and electrically detected magnetic resonance on optimized solar cells indicates that TEs are only present on the fullerenes. We conclude that in PTB7:PC\(_{71}\)BM devices, TE formation via EBT is impacted by fullerene domain size at low temperature, but at room temperature, EBT does not represent a dominant loss pathway.
The second-order correlation function g\(^2\)(\(\tau\) = 0), input-output curves and pulse duration of the emission from a microcavity exciton-polariton system subsequent to picosecond-pulsed excitation are measured for different temperatures. At low temperatures a two-threshold behaviour emerges, which has been attributed to the onset of polariton lasing and conventional lasing at the first and the second threshold, respectively. We observe that polariton lasing is stable up to temperatures comparable with the exciton binding energy. At higher temperatures a single threshold displays the direct transition from thermal emission to photon lasing.
The photoluminescence of annealed Cd\(_x\)Zn\(_{1-x}\)Se quantum dots (QDs) under the influence of an external magnetic field has been studied in this paper. Post-growth annealing was performed for different annealing times. Above a critical annealing time, the QD luminescence shows a pronounced red-shift of the Zeeman split magnetic subcomponents. This observation is in contrast to the blue-shift caused by the diamagnetic behavior that is usually observed in non-magnetic QDs. We attribute our finding to the paramagnetism caused by the mixing of heavy and light hole states. Hence, post-growth thermal annealing treatment might be employed to render undoped epitaxial QDs intrinsically magnetic in a controlled manner. Two theoretical models were developed: a few-particle model to account for excitonic complex effects and a multiband calculation that describes the valence band hybridization. Contrasting the two models allowed us to unambiguously elucidate the nature of such an effect.
Graphene's peculiar electronic band structure makes it of interest for new electronic and spintronic approaches. However, potential applications suffer from quantization effects when the spatial extension reaches the nanoscale. We show by photoelectron spectroscopy on nanoscaled model systems (disc-shaped, planar polyacenes) that the two-dimensional band structure is transformed into discrete states which follow the momentum dependence of the graphene Bloch states. Based on a simple model of quantum wells, we show how the band structure of graphene emerges from localized states, and we compare this result with ab initio calculations which describe the orbital structure.