Simultaneous Determination of Refractive Index and Thickness of Submicron Optical Polymer Films from Transmission Spectra

High-transparency polymers, called optical polymers (OPs), are used in many thin-film devices, for which the knowledge of film thickness (h) and refractive index (n) is generally required. Spectrophotometry is a cost-effective, simple and fast non-destructive method often used to determine these parameters simultaneously, but its application is limited to films where h > 500 nm. Here, a simple spectrophotometric method is reported to obtain simultaneously the n and h of a sub-micron OP film (down to values of a few tenths of a nm) from its transmission spectrum. The method is valid for any OP where the n dispersion curve follows a two-coefficient Cauchy function and complies with a certain equation involving n at two different wavelengths. Remarkably, such an equation is determined through the analysis of n data for a wide set of commercial OPs, and its general validity is demonstrated. Films of various OPs (pristine or doped with fluorescent compounds), typically used in applications such as thin-film organic lasers, are prepared, and n and h are simultaneously determined with the proposed procedure. The success of the method is confirmed with variable-angle spectroscopic ellipsometry.

A Bimodal Burst Energy Distribution of a Repeating Fast Radio Burst Source

Fast radio bursts (FRBs) are cosmic sources that emit millisecond-duration radio pulses with a wide range of luminosities and yet unknown origin(s) (Petroff et al. 2019; cordes et al. 2019). A subset of FRBs were found to repeat, the prototype of which is the first precisely-located FRB 121102 (Spitler et al. 2016), residing in a dwarf galaxy at redshift z=0.193 (Chatterjee 2017; Tendulkar et al. 2017). The source has been observed by most major telescopes and shows non-Poisson clustering of bursts over time, the hitherto highest burst rate, and a burst isotropic equivalent energy largely consistent with a power-law (Law et al. 2017; zhang et al. 2018; Gourdji et al. 2019), all of which are crucial characteristics to be compared to non-repeating sources. However, due to sensitivity limits, no true energy distribution of any FRB is known. Here we report the detection of 1652 independent bursts, more than quadruple the total of all previously published ones combined, in a total of 59.5 observing hours spanning 47 days using the Five-hundred-meter Aperture Spherical radio Telescope (FAST). The peak burst rate of 122 hr-1is by far the highest ever observed of any FRB. A characteristic peak in the isotropic equivalent energy distribution is found to be ~4.8×1037 erg at 1.25 GHz, suggesting a possible threshold for producing abundant coherent radio bursts from FRBs. The burst energy distribution is optimally described by a bimodal distribution consisting of a log-normal function plus a Cauchy function. While no periodicity was found between 1 ms and 1000 s, and the majority of the burst arrival times are consistent with being random, there exists a visible peak in the waiting time distribution at about 3.4 ms, corresponding to significant clustering. Our results start to reveal the stochastic nature of abundant weaker bursts, which could be present in other FRB sources, apparently repeating or not. FRB generation mechanisms must be efficient and economical. Expensive triggers and/or contrived conditions for burst production seem unlikely.

ON ASYMPTOTIC PROPERTIES OF THE CAUCHY FUNCTION FOR AUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATION OF NEUTRAL TYPE

We investigate stability of a linear autonomous functional differential equation of neutral type. The basis of the study is the well-known explicit solution representation formula including an integral operator, the kernel of which is called the Cauchy function of the equation under study. It is shown that the definitions of Lyapunov, asymptotic and exponential stabilities can be formulated without loss of generality in terms of the corresponding properties of the Cauchy function. The conclusion is drawn that stability with respect to initial data depends on the functional space which the initial function belongs to, and, as a consequence, that there is the need to indicate this space in the definition of stability. It is shown that, along with the concept of asymptotic stability, a certain stronger property should be introduced, which we call strong asymptotic stability. The main study is devoted to stability with respect to initial function from spaces of integrable functions. Special attention is paid to the study of asymptotic and exponential stability. We use the following known properties of the Cauchy function of an equation of neutral type: this function is piecewise continuous, and its jumps are determined by a Cauchy problem for a linear difference equation. We obtain that the strong asymptotic stability of the equation under consideration for initial data from the space L1 is equivalent to an exponential estimate of the Cauchy function and; moreover, we show that these properties are equivalent to the exponential stability with respect to initial data from the spaces Lp for all p from 1 to infinity inclusive. However, we show that strong asymptotic stability with respect to the initial data from the space Lp for p greater than one may not coincide with exponential stability.

SOLVING MINERALOGY PROBLEMS WITH THE HELP OF THE “ORIGIN" PACKAGE

Algorithms for solving typical mineralogical problems associated with quantitative x-ray spectral analysis and quantitative x-ray phase analysis using the program “Origin” are developed. The calculation of the areas and midpoint of spectral lines using the tabular processor of the program “Origin” is considered. Various approaches to determining the parameters of spectral lines using the least squares method using the standard functions of the program “Origin” were tested. The creation of a user function for approximation of diffraction maxima by the Cauchy function taking into account the doublet character of Ka series of x-rays is also considered. Various built-in algorithms for smoothing functions (based on averaging, polynomial approximation and Fourier analysis – synthesis) were tested to find weak diffraction maxima against strong noise; optimal schemes for the application of these algorithms were found. The considered algorithms can be applied in universities when processing the results of laboratory works on the topics "Analysis of spectra of emission of atoms", "Quantitative x-ray spectral analysis" and "Quantitative x-ray phase analysis".

Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem

In this study, the asymptotic behavior of the solutions to a boundary value problem for a third-order linear integro-differential equation with a small parameter at the two higher derivatives has been examined, under the condition that the roots of the additional characteristic equation are negative. Via the scheme of methods and algorithms pertaining to the qualitative study of singularly perturbed problems with initial jumps, a fundamental system of solutions, the Cauchy function, and the boundary functions of a homogeneous singularly perturbed differential equation are constructed. Analytical formulae for the solutions and asymptotic estimates of the singularly perturbed problem are obtained. Furthermore, a modified degenerate boundary value problem has been constructed, and it was stated that the solution of the original singularly perturbed boundary value problem tends to this modified problem’s solution.

Analysis of natural frequency of flexural vibrations of a single-span beam with the consideration of Timoshenko effect

This paper presents general solution of boundary value problem for constant cross-section Timoshenko beams with four typical boundary conditions. The authors have taken into consideration rotational inertia and shear strain by using the theory of influence by Cauchy function and characteristic series. The boundary value problem of transverse vibration has been formulated and solved. The characteristic equations considering the exact bending theory have been obtained for four cases: the clamped boundary conditions; a simply supported beam and clamped on the other side; a simply supported beam; a cantilever beam. The obtained estimators of fundamental natural frequency take into account mass and elastic characteristics of beams and Timoshenko effect. The results of calculations prove high convergence of the estimators to the exact values which were calculated by Timoshenko who used Bessel functions. Characteristic series having an alternating sign power series show good convergence. As it is shown in the paper, the error lower than 5% was obtained after taking into account only two first significant terms of the series. It was proved that neglecting the Timoshenko effect in case of short beams of rectangular section with the ratio of their length to their height equal 6 leads to the errors of calculated natural frequency: 5%÷12%.

Full width at half maximum of low-angle basal phyllosilicate X-ray diffraction reflections: fitted peaks vs. diffraction traces

ABSTRACTBernard Kübler measured illite ‘crystallinity’, the half-height width or full width at half maximum (FWHM) of the X-ray diffraction line of illite/mica at 10 Å, directly on the diffraction traces; this procedure has since been followed by the vast majority of workers. However, some workers have recently measured the FWHM of the fitted Pearson VII function rather than on the diffraction traces. The FWHM of this function for low-angle phyllosilicate diffraction peaks (FWHM*PVII) is almost consistently ‘broader’ than those measured directly on the diffraction trace profiles (FWHMtrace) by up to 0.08°Δ2θ for the broader peaks. The Pearson VII function shows gentle curvature (‘smoothing’) at its tops and fast fading of the tails relative to virtually all 10 Å diffraction traces. The broad FWHM*PVII results from the consequent lowering/’under-fitting’ of the peak tops and the upper tails and compensatory broadening/’over-fitting’ of the intermediate peak flanks. FWHM*PVII ‘contraction’ with respect to FWHMtrace and enhancement of the peak maximum is found on traces of muscovite strips. The fitting reliabilities of the Cauchy function are almost invariably better than those of the Pearson VII function. Their FWHM*Cauchy values are narrower for both the illite/mica 10 Å and the chlorite 7 Å reflections; although they still differ somewhat from the FWHMtrace, they are much closer, usually within 0.02°Δ2θ. This markedly lesser broadening of FWHM* of the Cauchy of the Pearson VII function is the result of its stronger top curvature and notably faster tail fading (less ‘smoothening’). For higher-angle mica peaks, the FWHM* values of the Pearson VII and Cauchy functions converge, usually differing only by 0.01–0.03°Δ2θ for the 5 Å peak, and even less for the 3.3 Å peak. It is therefore strongly recommended that FWHM values of the illite/mica 10 Å reflections be measured on the diffraction traces rather than on fitted functions. Where peak fitting is unavoidable (e.g. in order to separate the contributions of adjoining, partly resolved or unresolved reflections on broadened 10 Å reflections), Cauchy rather than Pearson VII functions should be used.