Revisiting Gouy phase

The aim of this paper is to highlight the added value of the generalized Gouy phase shift introduced by Siegman. Although suited for optical systems study, including those more complex than free space, we note that it did not meet the use that it deserves so far. The analysis of the whole of the ideas and analytical approaches associated to the important concept of the Gouy phase proves its effectiveness.

Usually, the resonance condition is systematically built on the basis of the equivalent empty cavity. Unfortunately, this approach does not cover some of the useful parameters of the real resonator. By means of the generalized Gouy phase and the self-consistent complex parameter q, we derive here a new approach for the calculation of the resonance condition for the real cavity. Moreover, the use of the generalized Gouy phase clearly simplifies the study of resonators, while making it possible to avoid the use of the Huygens’ Fresnel integral.     

Growth and study of single crystals of (Na,K)NbO<Subscript>3</Subscript> solid solutions

(Na, K)NbO₃ crystals with a perovskite structure and a KNbO₃ content up to 40 mol % were grown from flux with the use of the solvent NaBO₂. The dielectric measurements of the crystals grown revealed phase transitions that had never been observed before in ceramic samples.     

Multistep numerical methods for functional-differential-algebraic equations

This paper deals with constructing multistep numerical methods for differential delay equations under additional algebraic constraints. Theorems on the orders of convergence of these methods are proved both for functional-differential equations with algebraic constraints and for singular functional-differential equations.     

A local search approach to a circle cutting problem arising in the motor cycle industry

This paper is concerned with the development of a customized circle packing algorithm for a manufacturer of sprockets for the motor cycle industry. Practical constraints mean that the problem differs somewhat from those tackled elsewhere in the literature. In particular, the layouts need to conform to a given structure. This is achieved by using a local search algorithm with an appropriate starting solution and a series of neighbourhoods designed to preserve the layout structure. Empirical evidence based on real data shows that the quality of the resulting solutions closely matches that of cutting patterns currently produced by human experts. Computation times average around 20–30?s per order as compared to several hours for an equivalent manual solution.     

On quadratic differentials on multiply connected domains that are perfect squares

The paper considers the problem on extremal decomposition of a multiply connected domain G ⊂ ℂ in the case where the associated quadratic differential is a perfect square. It is shown that in the case considered, the value of the functional for this extremal decomposition is the least one in a certain class of decompositions. Bibliography: 10 titles. Dedicated to the 100th anniversary of G. M. Goluzin’s birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 113–133.     

Homogenization of the periodic magnetic Schrödinger operator with degenerate lower edge of the spectrum

Homogenization of the periodic magnetic Schrödinger operator with degenerate lower edge of the spectrum is considered. The standard problem of homogenization leads to a degenerate limiting operator. To obtain more precise information, one needs a certain modification of the problem considered. Bibliography: 10 titles.     

Investigation of the structural elements of a fullerene within the static-fluctuation approximation of the Hubbard model

The anticommutator Green’s functions, correlation functions, and ground-state energies for closed atomic structures that consist of five or six atoms and are structural elements of a fullerene are calculated within the static-fluctuation approximation of the Hubbard model. The results of the calculations are analyzed and compared with the data available in the literature.     

Thermographic inspection of TBC coated gas turbine blades: Discrimination between coating over-thicknesses and adhesion defects

When performing in-field tests on thermal barrier coatings (TBCs) deposited on gas turbine blades, the main problem encountered is the difficulty to correctly interpret the experimental data. The coating thickness changes, which are not regarded as anomalies, are one of the most common sources of false alarms. They are very challenging to recognize because, unlike stains, eroded areas and optical property variations, they are not detectable in the visible band. Furthermore, their thermal signal is similar to the one produced by an adhesion defect. In this paper, a procedure to reliably discriminate thickness changes and real defects is proposed. The approach is based on the analysis of the apparent effusivity profile. Preliminary results are presented and discussed.