Faddeeva Function Research Articles

Resonance Scattering Treatment with the Windowed Multipole Formalism

A new method for directly sampling the resonance upscattering effect is presented. Alternatives have relied on inefficient rejection sampling techniques or large tabular storage of relative velocities. None of these approaches, which require pointwise energy data, are particularly well suited to the windowed multipole cross-section representation. The new method, called multipole analytic resonance scattering, overcomes these limitations by inverse transform sampling from the target relative velocity distribution where the cross section is expressed in the multipole formalism. The closed-form relative speed distribution contains a novel special function we deem the incomplete Faddeeva function, and we present the first results on its efficient numerical evaluation.

A rational-expansion-based method to compute Gabor coefficients of 2D indicator functions supported on polygonal domain

We propose a method to compute Gabor coefficients of a two-dimensional (2D) indicator function supported on a polygonal domain by means of rational expansion of the Faddeeva function and by solving second-order linear difference equations. This method has the following three attractive features: (1) the problem of computing Gabor coefficients is formulated as the calculation of a sequence of integrals with a uniform structure, (2) a rational expansion based on fast Fourier transform (FFT) is used to approximate the Faddeeva function on the entire complex plane, (3) second-order inhomogeneous linear difference equations are derived for previous integrals and they are solved stably with Olver’s algorithm. Numerical quadrature to compute Gabor coefficients is avoided. Numerical examples show this rational-expansion-based method significantly outperforms numerical quadrature in terms of computation time while maintaining accuracy.

Impact of Faddeeva–Voigt broadening on line-shape analysis at critical points of dielectric functions

Faddeeva–Voigt broadening (FVB) couples the physical characteristics of both Lorentzian and Gaussian profiles as a combined analytic function shaping the dielectric response. Accurate extraction of the Gaussian and Lorentzian broadening contents in line-shape analysis is essential for reliable optical characterization of semiconductors and dielectrics. By adding the Gaussian-broadening width to each Lorentzian width, we investigate how FVB affects critical-point (CP) analysis. We revisit a selection of earlier work based on classical Lorentz broadening in modulation spectroscopy and spectral ellipsometry. To generalize CP analysis, we derive the FVB’s analytical representation in terms of fractional derivatives of the Faddeeva function and apply the twenty-pole Martin–Donoso–Zamudio approximation for its precise and efficient computation of the FVB of model dielectric functions and derivatives. We investigate the FVB of the electroreflectance line shape of HgCdTe for three-dimensional M0 transitions and of the photoreflectance line shape of InP excitonic E0 transitions. Furthermore, we explore how FVB affects the dielectric functions of three-dimensional excitonic and two-dimensional M0 transitions vs Tanguy’s analytical two-dimensional exciton E1 and E1+Δ1 fits of GaAs to the second-order derivatives. We use the Akaike information criterion to quantitatively estimate the goodness of fit that statistically penalizes overfitting due to extraneous parameters. By consolidating both Gaussian and Lorentzian broadenings, the FVB significantly affects the CP analysis of modulation-spectroscopy line shapes and second-order derivatives of the dielectric function.

Series form of the characteristic functions of scale mixtures of multivariate skew-normal distributions

Based on two series representations of the error function and a series representation of the Faddeeva function, the characteristic functions of scale mixtures of skew-normal distributions in univariate and multivariate cases are derived in series form instead of the previously suggested two-dimensional integral form without using contour integration. This is accomplished by applying the Lebesgue dominated convergence theorem to change the order of integral and limit. After integrating out each mixing variable, the series representation is expressed as in closed-form. Series form greatly reduces the computing times compared to the integral form, which is visualized via parallel box plots in a skew-t and a skew-slash distributions and a result of application on goodness-of-fit test.

Application of the Martin-Donoso-Zamudio multipole approximation for generalized Faddeeva/Voigt broadening of model dielectric functions

Faddeeva/Voigt broadening of dielectric functions (DFs) couples the physical properties of mutual Lorentzian and Gaussian broadening as a combined unit. This work investigates accurate computations for the general Faddeeva/Voigt broadening of complex analytic DFs and their derivatives. The paper builds on the multipole approximation for broadening profiles and exploits the reduction of the approximation into a finite sum of Lorentzian profiles. Using a two-point Padé approximant for the Faddeeva function, we computed this study's twenty-pole Martin-Donoso-Zamudio (MDZ20) approximation. First, we evaluated the accuracies of the MDZ20 approximation relative to high-precision evaluation of the Faddeeva function and its first three derivatives. Second, we applied the MDZ20 approximation to derive the general approximation formula for the Faddeeva/Voigt broadening of Kramer-Kronig-consistent complex analytic DFs. Third, we assessed the accuracy of Voigt broadening in two theoretical applications: the universal-dispersion model through piecewise polynomial elements and the three-dimensional (3D) M0 critical point and derivatives by using the MDZ20 approximation. Finally, we developed formulas for the Faddeeva/Voigt broadening of Tanguy's 3D excitons and the Tauc-Lorentz models. The Faddeeva/Voigt-broadened Tanguy model is investigated within Gilliot's dielectric model of sol-gel ZnO. The Faddeeva/Voigt-broadened Tauc-Lorentz model is applied to amorphous silicon (a-Si). Specifically, we evaluated the impact of adding a Gaussian component to each Lorentzian width on the agreement of the Voigt-broadened DF to experimental data.

Magnetizations and de Haas-van Alphen oscillations in massive Dirac fermions

We theoretically study magnetic field, temperature, and energy band-gap dependences of magnetizations in the Dirac fermions. We use the zeta function regularization to obtain analytical expressions of thermodynamic potential, from which the magnetization of graphene for strong field/low temperature and weak field/high temperature limits are calculated. Further, we generalize the result by considering the effects of impurity on orbital susceptibility of graphene. In particular, we show that in the presence of impurity, the susceptibility follows a scaling law which can be approximated by the Faddeeva function. In the case of the massive Dirac fermions, we show that a large band-gap gives a robust magnetization with respect to temperature and impurity. In the doped Dirac fermion, we discuss the dependence of the band-gap on the period and amplitude of the de Haas-van Alphen effect.

Computation of the Complex Error Function Using Modified Trapezoidal Rules

In this paper we propose a method for computing the Faddeeva function $w(z) := {e}^{-z^2}{erfc}(-{i}\,z)$ via truncated modified trapezoidal rule approximations to integrals on the real line. Our s...

Approximating fractional derivative of Faddeeva function, Gaussian function, and Dawson's integral

A new method for approximating fractional derivatives of Faddeeva function, Gaussian function, and Dawson's integral are presented. Unlike previous approaches, which are dominantly based on some discretization of Riemann‐Liouville integral over a finite interval using polynomial or discrete Fourier basis, we take an alternative approach that is based on expressing computation of Riemann‐Liouville definition of the fractional integral for the semi‐infinite axis in terms of a moment problem. As a result, fractional derivatives of Faddeeva function, Gaussian function, and Dawson's integral are expressed as a weighted sum of complex scaled Gaussian and Dawson's integral. Error bounds for the approximations are provided. Another feature of the proposed method compared with the previous approaches is it can be extended to approximate partial derivatives with respect to the order of the fractional derivative that can be used in PDE constraint optimization problems.

Analytical expression for the convolution of a Fano line profile with a gaussian

Asymmetric Fano line profiles are frequently encountered, e. g., in the photoionization spectra of atoms and ions. For the fitting of spectral line profiles to experimental spectra the line profiles have to be convolved with the experimental window function. The latter is often taken to be a gaussian. It is shown that the convolution can be represented by a rather simple analytic expression involving the Faddeeva function for the evaluation of which efficient and accurate numerical algorithms are available.

Optimized higher-order automatic differentiation for the Faddeeva function

Considerable research efforts have been directed at implementing the Faddeeva function w(z) and its derivatives with respect to z, but these did not consider the key computing issue of a possible dependence of z on some variable t. The general case is to differentiate the compound function w(z(t))=w∘z(t) with respect to t by applying the chain rule for a first order derivative, or Faà di Bruno’s formula for higher-order ones. Higher-order automatic differentiation (HOAD) is an efficient and accurate technique for derivative calculation along scientific computing codes. Although codes are available for w(z), a special symbolic HOAD is required to compute accurate higher-order derivatives for w∘z(t) in an efficient manner. A thorough evaluation is carried out considering a nontrivial case study in optics to support this assertion. Program summaryProgram title: HOAD_MathFunCatalogue identifier: AFAG_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AFAG_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: yesNo. of lines in distributed program, including test data, etc.: 4737No. of bytes in distributed program, including test data, etc.: 103450Distribution format: tar.gzProgramming language: Fortran 90.Computer: Non-specific.Operating system: Non-specific.RAM: 1 megabyteClassification: 4.12.External routines: Algorithm 680 [8], Mathematical functions (included)Nature of problem:General optimized higher-order automatic differentiation of mathematical functions. Complex refractive index as a case study.Solution method:Higher-order differentiation for the general second order ordinary equation defining the mathematical functions. Optimized operator overloading recurrence formula.Unusual features:Automatic differentiation, Quadratic complexity.Running time:1 second.

COMPUTATIONAL ASPECTS OF 2D-QUASI-PERIODIC-GREEN-FUNCTION COMPUTATIONS FOR SCATTERING BY DIELECTRIC OBJECTS VIA SURFACE INTEGRAL EQUATIONS

We describe a surface integral-equation (SIE) method suitable for computation of electromagnetic fields scattered by 2D-periodic high-permittivity and plasmonic scatterers. The method makes use of fast evaluation of the 2D-quasi-periodic Green function (2D-QPGF) and its gradient using a tabulation technique in combination with tri-linear interpolation. In particular we present a very efficient technique to create the look-up tables for the 2D-QPGF and its gradient where we use to our advantage that it is very effective to simultaneously compute the QPGF and its gradient, and to simultaneously compute these values for the case in which the role of source and observation point are interchanged. We use the Ewald representation of the 2D-QPGF and its gradient to construct the tables with pre-computed values. Usually the expressions for the Ewald representation of the 2D-QPGF and its gradient are presented in terms of the complex complementary error function but here we give the expressions in terms of the Faddeeva function enabling efficient use of the dedicated algorithms to compute the Faddeeva function. Expressions are given for both lossy and lossless medium parameters and it is shown that the expression for the lossless case can be evaluated twice as fast as the expression for the lossy case. Two case studies are presented to validate the proposed method and to show that the time required for computing the method of moments (MoM) integrals that require evaluation of the 2D-QPGF becomes comparable to the time required for computing the MoM integrals that require evaluation of the aperiodic Green function. Reliable prediction of the interaction of electromagnetic fields with periodic structures plays a key role in numerous applications whether it is in the radio-frequency (RF) regime or in the optical regime. In the RF regime, periodic structures are for instance used to model the scattering properties of large arrays and frequency-selective surfaces. In the optical regime, an application of great importance is the scattering of light by diffraction gratings. To enable accurate and efficient computation of fields scattered by periodic structures, considerable effort has been invested over the years in the development of a range of numerical methods. For many of these methods the performance is excellent as long as the periodically repeated scattering objects have low contrast, but for high permittivity and plasmonic scatterers the problem of obtaining sufficiently accurate results in acceptable computation times arises often. In the RF regime, metals can mostly be modeled by PEC structures but in the optical regime they exhibit plasmonic behavior, see (1) and (2), hence in the optical regime the problem of dealing with plasmonic scatterers is especially important. While the very successful Finite Element Method (FEM) and Volume Integral-Equation (VIE) method apply a volumetric discretization, the Surface Integral Equation (SIE) formulation only uses

A depth-dependent formula for shallow water propagation.

In shallow water propagation, the sound field depends on the proximity of the receiver to the sea surface, the seabed, the source depth, and the complementary source depth. While normal mode theory can predict this depth dependence, it can be computationally intensive. In this work, an analytical solution is derived in terms of the Faddeeva function by converting a normal mode sum into an integral based on a hypothetical continuum of modes. For a Pekeris waveguide, this approach provides accurate depth dependent propagation results (especially for the surface decoupling) without requiring complex calculation methods for eigenvalues and corresponding eigenfunctions.

On the atomic line profiles in high pressure plasmas

In a previous contribution to this journal [H. P. Stormberg, J. Appl. Phys. 51(4), 1963 (1980)], Stormberg presented an analytical expression for the convolution of Lorentz and Levy line profiles, which models atomic radiative transitions in high pressure plasmas. Unfortunately, the derivations are flawed with errors and the final expression, while correct, is accompanied by misguiding comments about the meaning of the symbols used therein, in particular the “complex error function.” In this paper, we discuss the broadening mechanisms that give rise to Stormberg's model and present a correct derivation of his final result. We will also provide an alternative expression, based on the Faddeeva function, which has decisive computational advantages and emphasizes the real-valuedness of the result. The MATLAB/Octave scripts of our implementation have been made available on the publisher's website for future reference.

Quantitative Evaluation of the Raman Spectra of Carbon Layers

The Raman effect has been widely used for years for carbon layers analysis for qualitative analysis of samples as well as quantitative. Two main lines of attitude to the quantitative analysis of the Raman spectra have been de¬ve¬lo¬ped. The first one is applied mainly in forensic investigations when one has pre¬pa¬red comparative samples which consist of known substances with known concentration. This is the ba¬se for comparison of spectra from unknown sample. A tool used in these in¬ves¬ti¬ga¬tions is the Voigt profile based on Faddeeva function, allowing a very precise com¬pa¬rison of spectra with elimination of individuality of me¬a¬s¬u¬re¬ment equipment. This at¬ti¬tu¬de seems not to be the best solution when one has not comparative samples and the sam¬p¬le un¬der test is unique – the situations often meet in scientific investigation. Using Voigt profiles may cause a loss of in¬for¬ma¬tion car¬¬ried with the spec¬trum, especially related with small quantities of nanotubes or ful¬le¬renes generated in the process of imposing diamond layers in plas¬ma arc. Another function for approximation shape of the Raman peak based on Breit-Figner-Fano curve is frequently used but difficulties arise when iden¬ti¬fy¬ing physical meaning of parameters and subtracting idle pedestal from the ex¬pe¬ri¬men¬tal spec¬t¬rum. A different attitude based on simple Gaussian peak approximation is pre¬sen¬ted as better solution. Use of this procedure as a metrological tool has been dis¬cus¬sed leading to the conclusion that engaging more advanced and so¬p¬hi¬s¬ti¬ca¬ted me¬thods does not provide automatically better results.

A Note on the Error Function

A new exact representation of the error function of real arguments justifies an accurate and simple analytical approximation. Two of the most widely used functions in physical sciences are the error function erf(x) and its related complimentary error function erfc(x). These functions occur extensively in problems relating to diffusion, heat conduction, and probability. When the argument x is real, rational approximations for these functions provide a high accuracy. In addition, the Faddeeva function, which is a variant of the error function for the complex argument z = x + iy, is used extensively in nuclear physics and spectroscopy.

Rapid approximation to the Voigt/Faddeeva function and its derivatives

Existing algorithms for the calculation of the Voigt/Faddeeva function are adapted to produce a method which is more efficient for atmospheric line-by-line calculations. The approach allows greater computational speed to be obtained if the maximum relative error criteria can be relaxed. A supplementary algorithm to calculate the partial derivatives of the Voigt function simultaneously with the Faddeeva function is also specified.

Evaluation of Molecular Integrals in a Mixed Gaussian and Plane-Wave Basis by Means of the Faddeeva Function and Its Derivatives

We present formulas for two-electron free-free exchange integrals and integrals with three Gaussians and one plane-wave function that are required in calculations of electron scattering by polyatomic molecules. The formulas of integrals with p- and d-type Gaussians were obtained by differentiation of the fundamental integrals (sk'|sk) and (ss|sk) that contain s-type Gaussians only and that may be evaluated by means of the Faddeeva function w. Explicit formulas are given for the two types of integrals in the spd Gaussian basis.

Flexural vibration of an infinite wedge

A new solution to the problem of flexural vibration of an infinite wedge excited by a point harmonic load is obtained via a superposition of the geometric and diffraction fields, for simply supported and roller boundary conditions along the edges. The diffraction integrals are evaluated both in closed form in terms of the Faddeeva function and also as an asymptotic series using the steepest descent method. An eigensolution is also derived by expanding eigenfunctions in the angular direction and propagation terms in the radial direction. These two representations of the solution are compared numerically for accuracy and spatial distribution in different frequency regimes. The relative contribution of the geometric and diffraction fields to the overall solution is examined. This result is then compared to the second-order Helmholtz wave equation solution in order to demonstrate the near-field effects arising due to the fourth-order biharmonic formulation. The authors’ motivation for obtaining the solution for a wedge-shaped infinite plate is that it is an essential step toward devising numerical models for the vibration of arbitrary polygonal plates. By combining the contribution from diffraction at the vertices of a polygonal plate with the geometrical solution obtained by the method of images, the authors expect to obtain better predictions at mid and high frequencies than is possible by other available methods.