On the fractal characteristics of a stabilised Newton method

In this report, we present a complete theory for the fractal that is obtained when applying Newton's Method to find the roots of a complex cubic. We show that a modified Newton's Method improves convergence and does not yield a fractal, but basins of attraction with smooth borders. Extensions to higher-order polynomials and the numerical relevance of this fractal analysis are discussed

Fractal Characteristics of Newton's Method on Polynomials

In this report, we present a simple geometric generation principle for the fractal that is obtained when applying Newton's method to find the roots of a general complex polynomial with real coefficients. For the case of symmetric polynomials $z^{\nu}-1$ , the generation mechanism is derived from first principles. We discuss the case of a general cubic and are able to give a description of the arising fractal structure depending on the coefficients of the cubic. Special cases are analysed and their characteristics, including scale factors and an approximate fractal dimension, are derived. The theoretical results are confirmed via computational experiments. An application of the theory in turbulence modelling is presented

Electron propagation in crossed magnetic and electric fields

Laser-atom interaction can be an efficient mechanism for the production of coherent electrons. We analyze the dynamics of monoenergetic electrons in the presence of uniform, perpendicular magnetic and electric fields. The Green function technique is used to derive analytic results for the field--induced quantum mechanical drift motion of i) single electrons and ii) a dilute Fermi gas of electrons. The method yields the drift current and, at the same time it allows us to quantitatively establish the broadening of the (magnetic) Landau levels due to the electric field: Level number k is split into k+1 sublevels that render the $k$th oscillator eigenstate in energy space. Adjacent Landau levels will overlap if the electric field exceeds a critical strength. Our observations are relevant for quantum Hall configurations whenever electric field effects should be taken into account.Comment: 11 pages, 2 figures, submitte

Four-path interference and uncertainty principle in photodetachment microscopy

We study the quantal motion of electrons emitted by a pointlike monochromatic isotropic source into parallel uniform electric and magnetic fields. The two-path interference pattern in the emerging electron wave due to the electric force is modified by the magnetic lens effect which periodically focuses the beam into narrow filaments along the symmetry axis. There, four classical paths interfere. With increasing electron energy, the current distribution changes from a quantum regime governed by the uncertainty principle, to an intricate spatial pattern that yields to a semiclassical analysis.Comment: submitted to Europhysics Letter

Resonant and non-resonant Tunneling through a double barrier

An explicit expression is obtained for the phase-time corresponding to tunneling of a (non-relativistic) particle through two rectangular barriers, both in the case of resonant and in the case of non-resonant tunneling. It is shown that the behavior of the transmission coefficient and of the tunneling phase-time near a resonance is given by expressions with "Breit-Wigner type" denominators. By contrast, it is shown that, when the tunneling probability is low (but not negligible), the non-resonant tunneling time depends on the barrier width and on the distance between the barriers only in a very weak (exponentially decreasing) way: This can imply in various cases, as well-known, the highly Superluminal tunneling associated with the so-called "generalized Hartman Effect"; but we are now able to improve and modify the mathematical description of such an effect, and to compare more in detail our results with the experimental data for non-resonant tunneling of photons. Finally, as a second example, the tunneling phase-time is calculated, and compared with the available experimental results, in the case of the quantum-mechanical tunneling of neutrons through two barrier-filters at the resonance energy of the set-up.Comment: replaced with some improvements in the text and in the references: pdf (11 pages) produced from a source-file in Word; including one Figur

Direct measurement of plasmon propagation lengths on lithographically defined metallic waveguides on GaAs

We present optical investigations of rectangular surface plasmon polariton waveguides lithographically defined on GaAs substrates. The plasmon propagation length is directly determined using a confocal microscope, with independent polarization control in both excitation and detection channels. Surface plasmon polaritons are launched along the waveguide using a lithographically defined defect at one end. At the remote end of the waveguide they scatter into the far-field, where they are imaged using a CCD camera. By monitoring the length dependence of the intensity of scattered light from the waveguide end, we directly extract the propagation length, obtaining values ranging from LSPP = 10-40 {\mu}m depending on the waveguide width (w=2-5 {\mu}m) and excitation wavelength (760-920 nm). Results are in good accord with theoretical expectations demonstrating the high quality of the lithographically defined structures. The results obtained are of strong relevance for the development of future semiconductor based integrated plasmonic technologies

Solar occultation with SCIAMACHY: algorithm description and first validation

International audienceThis presentation concentrates on solar occultation measurements with the spaceborne spectrometer SCIAMACHY in the UV-Vis wavelength range. Solar occultation measurements provide unique information about the vertical distribution of atmospheric constituents. For retrieval of vertical trace gas concentration profiles, an algorithm has been developed based on the optimal estimation method. The forward model is capable to simulate the extinction signals of different species as they occur in atmospheric transmission spectra obtained from occultation measurements. Furthermore, correction algorithms have been implemented to address shortcomings of the tangent height pre-processing and inhomogeneities of measured solar spectra. First results of O3 and NO2 vertical profile retrievals have been validated with data from ozone sondes and satellite based occultation instruments. The validation shows very promising results for SCIAMACHY O3 and NO2 values between 15 to 35 km with errors in the order of 10% and 15%, respectively