Closed-form approximations for fin-line eigenmodes

The characteristic equations for bilateral, unilateral and antipodal fin lines are formulated in a convenient form for an approximate evaluation. It is shown that the eigenvalues for all higher-order modes can be calculated from closed-form expressions provided that the two lowest eigenvalues are known. Dispersion relations are developed which are equivalent to those for rectangular waveguides. The analysis also comprehends a simple evaluation of the field configurations of all hybrid eigenmodes. The approximate formulas are checked against exact numerical calculations, the agreement being excellent.

A broadband transition between a half mode substrate integrated waveguide and a rectangular waveguide with an antipodal fin-line in its quasi-TEM mode is presented. Experimental results show about 1.3 dB insertion loss and below −15 dB return loss for a back-to-back double transition over 10 GHz bandwidth from 26.5 to 40 GHz (the single transition insertion loss is about 0.65 dB). The transition may be used in low-loss and low-noise front-end circuits of the microwave and millimetre-wave bands.

In the theory of relaxed states of toroidal plasmas certain eigenvalues of the equation ∇×B=μB play a crucial role. These eigenvalues are associated with vanishing toroidal flux and determine the onset of current limitation in a toroidal discharge. In axisymmetric systems there are both periodic and axisymmetric eigenfunctions and it is important to know whether the eigenmode associated with the lowest eigenvalue is periodic or axisymmetric. This depends on the shape of the poloidal cross section and determines the nature of the current-limited discharge. The eigenvalues of periodic and axisymmetric modes have been computed in rectangular and elliptical cross sections and in reentrant Multipinch-like cross sections. The reentrant case required new numerical techniques, which are described. It is found that in rectangular and elliptic cross sections the lowest mode is always periodic. However, in the Multipinch a transition occurs in which the lowest eigenmode changes from periodic to axisymmetric as the ‘‘waist’’ in the cross section is made narrower. The critical width is determined. These calculations suggest that in the GA Multipinch experiment [Nucl. Fusion 26, 255 (1986)] the current saturated discharge should be axisymmetric—unlike all other existing pinch experiments where it is periodic.

Approximate dispersion formulae for Rayleigh-like waves in a layered medium

The five longest tune-out wavelengths for the rubidium atom are predicted from existing experimental and theoretical information by using an approximation formula. The formula is derived from the analysis of a non-relativistic configuration-interaction plus core polarization (CICP) calculation of the tune-out wavelengths for rubidium. The differences between the wavelengths predicted from the approximation formula and the exact CICP calculation are at the 10−3 nm level. The predicted five longest tune-out wavelengths are 790.115 nm, 423.155 nm, 421.111 nm, 359.429 nm, and 359.022 nm, respectively.

For evaluation of capacitance-voltage measurements on an abrupt pn semiconductor junction with reverse-biased depletion layer an approximation equation for the quantity Ue is presented, where Ue is the value at the intercept of the extrapolated plot C−2 to the applied voltage axis. The approximation formula is compared with an exact calculation. The maximum error in the approximate relationship for Ue is less than 3% in a wide range of impurity density and applied reverse voltage, therefore the discrepancy in the space-charge layer capacitance Cs is less than 1.5%. Cs is given by the relationship, which follows from Shockley's theory, if (U + UD) is replaced by (U + Ue). Für die Auswertung von Kapazitätsmessungen an abrupten pn-Übergängen, die in Sperrich-tung belastet sind, wird eine Näherungsbeziehung für die Größe Ue entwickelt, wobei Ue die Spannung am Schnittpunkt der extrapolierten Kurve C−2 mit der Spannungsabszisse U darstellt. Die Näherungslösung wird mit den Werten einer exakten numerischen Rechnung verglichen. Der maximale Fehler bei der approximierten Größe Ue liegt in einem weiten Dotierungsbereich und Sperrspannungsbereich unter 3%, so daß die Abweichungen bei der Sperrschichtkapazität Cs kleiner als 1,5% sind. Man kann Cs durch die Beziehung nach der Shockleyschen Theorie beschreiben, wenn man (U - UD) durch (U + Ue) ersetzt.

Molecular energy calculations employing actual and approximate multicenter angular dependent integral evaluations are compared. Individual integral values for three-center integrals used in calculations for the triangular H3+ ion as computed by exact methods and by the approximate formulas of Mulliken and of Rudenberg are given. The results indicate that angular and distance relations of the nuclear skeleton of this polyatomic ion are determinable at almost the identical ``best'' value found using the much more laborious exact integral calculations. The energies of the approximate evaluations, however, indicate that once these ``best'' parametric values are determined the more exact calculations must be used. The approximate evaluations of second angular derivatives of integrals that are dependent on the angular and distance relations of component molecular nuclei are also computed and compared. Use of such derivatives in determination of approximate zero point energies, vibrational frequencies, and force constants is indicated.

Постановка задачi. З розвитком технологiй радiозв’язку, особливо, мобiльного, усе бiльш важливу
\nроль вiдiграє розробка моделей радiоканалiв. У данiй
\nстаттi розглядається одна з популярних нинi моделей,
\nа саме, двопроменевий радiоканал з розсiюванням потужностi сигналу, яку в лiтературi прийнято позначати
\nабревiатурою TWDP (вiд Two-Wave with Diffuse Power).
\nДля цiєї моделi вiдомi як точнi, так i наближенi вирази
\nдля густини розподiлу iмовiрностей (ГРI) i кумулятивної функцiї розподiлу (КФР) обвiдної сигналу. Однак,
\nзагальним їх недолiком є висока складнiсть з точки зору практичного застосування. У зв’язку iз цим у данiй
\nроботi розглядається можливiсть значно бiльш простої
\nапроксимацiї ГРI та КФР обвiдної сигналу на основi
\nметоду Накагами.
\nРозробка методу. Апроксимацiя розподiлiв по методу Накагами зводиться до обчислення другого та
\nчетвертого початкових моментiв обвiдної сигналу на
\nвиходi каналу. Такi обчислення й виконанi в роботi,
\nщо дозволяє представити зазначенi моменти у виглядi
\nфункцiй вiд двох загальноприйнятих параметрiв каналу
\nTWDP: К-фактора та параметра Д, який визначається
\nспiввiдношенням амплiтуд радiо променiв.
\nАналiз похибок апроксимацiї. Похибки оцiнювалися шляхом зiставлення розрахункiв ГРI i КФР за
\nточними формулами та розрахункiв iз використанням
\nапроксимацiї Накагами. Паралельно також проводилися розрахунки вiдповiдно до формул апроксимацiї, що
\nвiдомi у лiтературi. Аналiз показав, що похибка методу
\nНакагами при обчисленнi КФР є цiлком задовiльною
\nдля практичних потреб майже у всьому дiапазонi змiн
\nпараметрiв каналу. У той же час точнiсть апроксимацiї
\nГРI у порiвняннi з iншими вiдомими методами апроксимацiї виявилася не дуже задовiльна.
\nВисновки. Проведенi дослiдження пiдтвердили можливiсть апроксимацiї КФР обвiдної сигналу в каналi TWDP розподiлом Накагами. Результати такої ж
\nапроксимацiї для ГРI виявилися менш задовiльними.
\nТакож вiдзначається кориснiсть отриманих результатiв
\nдля трактування даних натурних експериментiв.

We study a two-dimensional electron gas in a perpendicular magnetic field in the presence of both Rashba and Dresselhaus spin-orbit interactions. Using a Bogoliubov transformation we are able to write an approximate formula for the Landau levels, thanks to the simpler form of the resulting Hamiltonian. The exact numerical calculation of the energy levels, is also made simpler by our formulation. The approximate formula and the exact numerical results show excellent agreement for typical semiconductors, especially at high magnetic fields. We also show how effective Zeeman coupling is modified by spin-orbit interactions.

The dispersion relation of one-dimensional longitudinal plasma waves in relativistic homogeneous plasmas is investigated with both linear theory and Vlasov simulation in this paper. From the Vlasov-Poisson equations, the linear dispersion relation is derived for the proper one-dimensional Jüttner distribution. Numerically obtained linear dispersion relation as well as an approximate formula for plasma wave frequency in the long wavelength limit is given. The dispersion of longitudinal wave is also simulated with a relativistic Vlasov code. The real and imaginary parts of dispersion relation are well studied by varying wave number and plasma temperature. Simulation results are in agreement with established linear theory.

A new dielectric slab waveguide with a left-handed material (LHM) cover and substrate is proposed. The dispersion relations and normalized effective thickness of the asymmetric LHM slab waveguide are investigated, in view of the normalized parameters. A number of unusual properties are found, for example, the fundamental and first-order modes do not exist and higher-order modes have double degeneracy. The propagation modes are absent at the low normalized frequency, and the cutoff frequencies of some LHM slab waveguide modes decrease with increase in the asymmetry measure. Unlike traditional slab waveguides, the V –H curves of the LHM slab waveguides are in one-to-one correspondence. Both TE and TM modes are discussed; in addition, the dispersion relations and normalized effective thicknesses of the TM modes are discussed in detail, when the difference in refractive indices of the film and the substrate is small. The results show that the region of mode coexistence taking place near the cutoff frequency becomes narrower with increase in the difference in refractive indices of the film and the substrate. The influence of this difference on the normalized effective thickness curves is different, and becomes smaller and smaller with increase in the value of the asymmetry measure, if different values of the refractive indices are employed.

In this contribution, the Ewald method has efficiently been applied to accelerate the computation of the rectangular waveguide Green’s functions (GFs) derivatives. Based on previous works, we have outlined new approximation formulae that avoid the evaluation of computationally expensive complementary error functions of complex argument, needed by the Ewald method. This is possible when the internal medium of the rectangular waveguide is homogeneous and lossless. On the other hand, different convergence numerical studies have been carried out, showing a similar convergence rate for computing the original GFs and their derivatives. Moreover, we have checked that the computational time is only slightly increased for obtaining the derivatives as compared to the original GFs, after the application of these new techniques. The newly derived expressions are useful for the evaluation of electromagnetic fields, the characterization of dielectric materials and step discontinuities between rectangular waveguides, and the analysis of rectangular cavities using integral equation formulations. For validation, the electric field produced by a surface electric current density with a rectangular pulse distribution has been evaluated, using the new proposed expressions. These results have been compared to simulations provided by full-wave finite elements commercial software to verify their correctness, exhibiting a good agreement.

We investigate numerically the effect of regulating fermions in the presence of singular background fields in three dimensions. For this, we couple free lattice fermions to a background compact U(1) gauge field consisting of a monopole-anti-monopole pair of magnetic charge ±Q separated by a distance s in a periodic L<sup>3</sup> lattice, and study the low-lying eigenvalues of different lattice Dirac operators under a continuum limit defined by taking L → ∞ at fixed s/L. As the background gauge field is parity even, we look for a two-fold degeneracy of the Dirac spectrum that is expected of a continuum-like Dirac operator. The naive-Dirac operator exhibits such a parity-doubling, but breaks the degeneracy of the fermion-doubler modes for the Q lowest eigenvalues in the continuum limit. The Wilson-Dirac operator lifts the fermion-doublers but breaks the parity-doubling in the Q lowest modes even in the continuum limit. The overlap-Dirac operator shows parity-doubling of all the modes even at finite L that is devoid of fermion-doubling, and singles out as a properly regulated continuum Dirac operator in the presence of singular gauge field configurations albeit with a peculiar algorithmic issue.

Exact solutions to waveguides are useful tools for validating numerical waveguide models. In this paper, a modal solution to an unbounded waveguide with a range-independent but depth-dependent profile is described. The sound speed profile utilized is a symmetric, piecewise n2-linear profile matched above and below to a homogenous half-layer with density and sound speeds that prevent any reflections at the interface. In this environment, caustics and shadow zones are formed. A nearly exact solution to this environment could help improve other numerical methods' accuracy near caustics and shadow zones. In this mathematical analysis, both proper (trapped) and improper (leaky) modes are included. Arbitrarily high frequencies are permitted through the use of numerically well-conditioned approximations to Airy functions, particularly for the dispersion relations and mode-shape evaluations. The proper (real) and improper (complex) modal eigenvalues are found approximately, with root-finding algorithms used for eigenvalues near cut-off to improve the approximation. Mathematical details are presented along with a brief discussion of how well a few common numerical solvers agree with the analytic solution derived.

This article presents a rigorous full‐wave modal analysis of a rectangular waveguide with rounded sides. The plane wave series expansion of Bessel function is employed to obtain the closed‐form modal functions for both TE and TM modes. Characteristic equations for calculating the eigenvalues (cutoff wavenumbers) of both TE and TM modes are derived. Numerical results are presented for the cutoff wavenumbers of the dominant mode as well as the higher‐order modes supported in the rounded‐side rectangular waveguide. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 33: 365–368, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10323