Bernhard J. Hoenders

Photonic bandgap optimization in inverted fcc photonic crystals

We present results of photonic band-structure calculations for inverted photonic crystal structures. We consider a structure of air spheres in a dielectric background, arranged in an fcc lattice, with a cylindrical tunnel connecting each pair of neighboring spheres. We derive (semi)analytical expressions for the Fourier coefficients of the dielectric susceptibility, which are used as input in a standard plane-wave expansion method. We optimize the width of the photonic bandgap by applying a gradient search method and varying two geometrical parameters in the system: the ratios R/a and Rc/R, where a is the lattice constant, R is the sphere radius, and Rc is the cylinder radius. It follows from our calculations that the maximal gap width in this type of photonic-crystal structure with air spheres and cylinders in silicon is Δω/ω0=9.59%.

On the Calculation of the Exact Number of Zeroes of a Set of Equations

The number of simple zeroes common to a set of nonlinear equations is calculated exactly and analytically in terms of an integral taken over the boundary of the domain of interest. The integrand consists only of simple algebraic quantities containing the functions involved as well as their derivatives up to second order. The numerical feasibility is shown by some computed examples.ZusammenfassungDie genaue Anzahl der einfachen Nullstellen eines Systems nichtlinearer Gleichungen wird analytisch durch ein Integral über die Berandung des interessierenden Bereichs dargestellt. Der Integrand besteht aus einfachen algebraischen Größen, die die Funktionen samt ihren Ableitungen bis zur zweiten Ordnung enthalten. Die numerische Anwendbarkeit wird mit durchgerechneten Beispielen belegt.

SCATTERING BY A RANDOM SURFACE OF RECTANGULAR GROOVES

The results of a preliminary theoretical investigation of the statistical properties of electromagnetic radiation scattered by a randomly grooved surface are presented. The contrast of the intensity pattern is calculated for both near-and far-field geometries, and it is shown that the full probability density of intensity fluctuations can be obtained exactly in the far-field specular direction.

The determination of the location of the global maximum of a function in the presence of several local extrema

The global maximum of a function can be determined by using information about the number of stationary points in the domain of interest. This information is obtained by evaluating an integral that equals the exact number of stationary points of the function. The integral is based on work by Kronecker and Picard at the end of the nineteenth century. The numerical feasibility of the method is shown by two computed examples, i.e., estimation problems from statistics and optical communication theory. In these examples the global maximum of the likelihood function is obtained by using the total number of stationary points as revealed by the computed integral.

Fully relativistic treatment of electron-optical image formation based on the Dirac equation

The theory of image formation for an electron microscope is based on the non-relativistic Schrodinger equation, whereas present-day electron microscopes operate with acceleration voltages of the order of one hundred to several hundreds of kilovolts, in which case relativistic effects become important. We present a fully relativistic theory of image formation, based on the appropriate Dirac equation. It is shown that, within certain approximations, always valid for todays electron microscopes, a very simple expression for the current density can still be derived in terms of wave functions that are solutions to the relativistically covariant Klein-Gordon equation. The following paper presents the analysis of the often stated possibility to obtain the relativistically correct current density from the non-relativistic current density just by replacing the values of the non-relativistic momentum by the correct relativistic expression.

Analytic calculation of the radiance in an anisotropically scattering turbid medium close to a source

We present a method to calculate the radiance due to an isotropic point source in an infinite, homogeneous, anisotropically scattering medium. The method is an extension of a well known method for the case of isotropic scattering. Its basic mathematical ingredient is the Fourier transform. Its great advantage is that it also works very close to the source and not just far away from it, as is the case with most other methods. In principle, the method works for any phase function that can be expanded in a finite number of Legendre polynomials. Here, the simple example of linear anisotropic scattering is worked out numerically and the result is compared with Monte Carlo simulation. Good agreement is found between the two.

On the determination of the number and multiplicity of zeros of a function

It is shown that certain simple integrals determine the number of zeros with a certain multiplicity of a function of one variable in an arbitrary interval. Several typical numerical examples are given.ZusammenfassungWir zeigen, daß man mit Hilfe gewisser Integrale die Anzahl der Nullstellen einer gegebenen Vielfachheit für eine Funktion einer Veränderlichen in einem beliebigen Intervall bestimmen kann. Das Vorgeben wird durch typische Beispiele erläutert.

THE NON-RADIATING COMPONENT OF THE FIELD GENERATED BY A FINITE MONOCHROMATIC SCALAR SOURCE DISTRIBUTION

We separate the field generated by a spherically symmetric bounded scalar monochromatic source into a radiative and non-radiative part. The non-radiative part is obtained by projecting the total field on the space spanned by the non-radiating inhomogeneous modes, i.e. the modes which satisfy the inhomogeneous wave equation. Using residue techniques, introduced by Cauchy, we obtain an explicit analytical expression for the non-radiating component. We also identify the part of the source distribution which corresponds to this non-radiating part. The analysis is based on the scalar wave equation.

The Fully Relativistic Foundation of Linear Transfer Theory in Electron Optics Based on the Dirac Equation

The fully relativistic quantum mechanical treatment of paraxial electron-optical image formation initiated in the previous paper (this issue) is worked out and leads to a rigorous foundation of the linear transfer theory. Moreover, the status of the relativistic scaling laws for mass and wavelength, as advocated in the literature, is elucidated.

Spectral line shapes in linear absorption and two-dimensional spectroscopy with skewed frequency distributions

The effect of Gaussian dynamics on the line shapes in linear absorption and two-dimensional correlation spectroscopy is well understood as the second-order cumulant expansion provides exact spectra. Gaussian solvent dynamics can be well analyzed using slope line analysis of two-dimensional correlation spectra as a function of the waiting time between pump and probe fields. Non-Gaussian effects are not as well understood, even though these effects are common in nature. The interpretation of the spectra, thus far, relies on complex case to case analysis. We investigate spectra resulting from two physical mechanisms for non-Gaussian dynamics, one relying on the anharmonicity of the bath and the other on non-linear couplings between bath coordinates. These results are compared with outcomes from a simpler log-normal dynamics model. We find that the skewed spectral line shapes in all cases can be analyzed in terms of the log-normal model, with a minimal number of free parameters. The effect of log-normal dynamics on the spectral line shapes is analyzed in terms of frequency correlation functions, maxline slope analysis, and anti-diagonal linewidths. A triangular line shape is a telltale signature of the skewness induced by log-normal dynamics. We find that maxline slope analysis, as for Gaussian dynamics, is a good measure of the solvent dynamics for log-normal dynamics.