Voigt lineshape function as a solution of the parabolic partial differential equation

The goal of this paper is to show that the Voigt function may be found as a solution of a parabolic partial differential equation, like the heat conduction equation or other diffusion equations. A square of the Gaussian half-width of the Voigt function plays the role of ‘time’ and initial conditions are determined by a Lorentz function. Some questions concerning the practical application of the numerical grid methods for the calculation of the Voigt function are discussed. It is shown, that in some cases the offered calculation algorithm can be both faster and more accurate than other known algorithms.     

The exact expression of the Voigt profile function

We solve in exact and closed-form a classical problem of mathematical physics of great interest in spectroscopy: the convolution of a Gaussian and a Lorentzian distribution that define the Voigt profile function, V(x). The solution is based in three steps: a power series development following the integral expression for V(x), the ordinary differential equation (ODE) satisfied by that expansion, and the corresponding solution of the ODE. This work converts in obsolete all graphical, numerical and semi-analytical approximations published previously. All results are clearly expressed in terms of the complementary error function Φ_c(a)=1-erf(a), where is, basically, the relation between Lorentzian and Gaussian widths.     

Voigt线形函数二阶导数研究

在强展宽条件下, 光谱信号二阶导数相互的交叠影响较小, 是一种潜在的反演光谱信息的手段. 本文研究了光谱Voigt线形函数的二阶导数, 得出了其二阶导数全域积分为0的性质, 计算了二阶导数最小值与偶数高阶导数最大值和最小值的解析结果, 并通过数值计算与曲线拟合得出了其极大值位置与零点位置的比例与洛仑兹-多普勒半宽比的关系, 为强展宽下由光谱二阶导数准确反演光谱信息提供了理论基础. 关键词： Voigt函数 二阶导数最小值 零点位置     

Rapid and accurate calculation of the Voigt function

Two general techniques significantly improve both the accuracy and speed of spectral line shapes that use only the complex Voigt function. Such line shape functions include the Voigt line profile, which is closely related to the real part of this function. The first technique is a new algorithm, which trades a small amount of RAM (at most 1.49 megabytes) for a considerable gain in speed and accuracy. The accuracy is one part in 10⁶ of the function itself. In addition, the derivatives of the function with respect to x and y are returned with an accuracy of 0.5%. The algorithm is up to nine times faster than the Drayson or Humlí?ek algorithms and about two orders of magnitude more accurate. A second independent improvement is a set of simple criteria to decide when an evaluation is required and when a value of zero can be assumed. This section alone can reduce the computation time by three orders of magnitude or more and it can be applied to algorithms using any Voigt approximation.     

A highly accurate Voigt function algorithm

A complex Voigt lineshape algorithm is presented whose maximum relative error over the complex plane is less than 1 × 10^-8. The algorithm consists of series, rational approximations and Gauss-Hermite integrations which makes it suitable as a general purpose software module for a wide variety of uses, including a Voigt function standard.     

Voigt线型函数及其最大值的研究

研究了多普勒和洛伦兹线型函数卷积形式的Voigt线型函数, 给出了它的最大值.结果表明, Voigt线型函数是关于中心频率的对称函数, Voigt线型函数的最大值由多普勒和洛伦兹线型函数的半宽度决定, 与中心频率无关, 且比洛伦兹和多普勒线型函数的最大值都小.提出了利用Voigt线型函数最大值和半宽度获得多普勒线型函数和洛伦兹线型函数的方法, 并利用Monte Carlo方法进行了验证. 关键词： Voigt线型函数 半宽度 最大值 傅里叶变换     

Rapidly convergent series for high-accuracy calculation of the Voigt function

A rapidly convergent series, based on Fourier expansion of the exponential multiplier, is presented for highly accurate approximation of the Voigt function (VF). The corresponding algorithm enables the rapid calculation, required for its implementation as a subprogram in an interpolation approach. The numerical analysis of this VF approximation suggests that it may be more accurate than 10⁻⁹ in the Humlí?ek regions 3 and 4.     

An efficient method for evaluation of the complex probability function: The Voigt function and its derivatives

An efficient method is developed to evaluate the function w(z)=e^-z2(1+(2i/√π)∫^z₀e^t2dt) for the complex argument z = x + iy. The real part of w(z) is the Voigt function describing spectral line profiles; the imaginary part can be used to compute derivatives of the spectral line shapes, which are useful, e.g. in least-squares fitting procedures. As an example of the method a simple and fast FORTRAN subroutine is listed in the Appendix from which w(z) in the entire y ? 0 half-plane can be calculated, the maximum relative error being less than 2 × 10^-6 and 5 × 10^-6 for the real and imaginary parts, respectively.     

A new procedure for obtaining the Voigt function dependent upon the complex error function

This work proposes a new method for obtaining the differential equation of the Voigt function and, from this equation, expressing the Voigt function as dependent upon the complex error function. In addition, the integral expression of the successive derivatives of the Voigt function is given, and from this a method is generalized which permits the representation, also, of other functions depending on the complex error function. This enables us to simplify other functions which are the convolution of a Gaussian function with rational polynomial functions. Moreover, the relationship between the Lorentzian (w_L), Gaussian (w_G) and Voigt (w_V) widths at half maximum for the function is given, which is of great interest in diverse branches of physics, such as plasma spectroscopy, astrophysics, nuclear magnetic resonance, etc.     

Calculation of optical depths from an integral of the Voigt function

The optical depth along a vertical path in an atmosphere in hydrostatic equilibrium can be calculated from an integral of the Voigt function for the case where the absorption is due to spectral lines. Series expansions are presented that allow rapid evaluation of this integral over all values of the independent variables, frequency and pressure.     

On path lengths of integration in the computation of the Voigt function

Path lengths are discussed for the computation of the Voigt function in a stellar atmosphere using numerical integration and quadrature.     

Note on the paper by T. Andersen entitled “the differential equations of the Voigt functions” (JQSRT 19, 169 (1978))

Solutions in terms of quadratures are presented for two differential equations defining the Voigt function.     

Stochastic differential equations

In chapter I stochastic differential equations are defined and classified, and their occurrence in physics is reviewed. In chapter II it is shown for linear equation show a differential equation for the averaged solution is obtained by expanding in ατ_c, where α measures the size of the fluctuations and τ_c their autocorrelation time. This result is the underlying reason for the existence of “renormalized transport coefficients”. In chapter III the same treatment is adapted to nonlinear equations. In chapter IV an alternative treatment is described, applicable only in a special case, but not confined to small ατ_c.The emphasis is on physical usefulness rather than mathematical rigor. Throughout the text applications are given at the points where they appeared to serve best as illustrations of the method. The list of references is not complete, but hopefully representative of the literature.     

A comparison of fast codes for the evaluation of the Voigt profile function

Several fast codes for the evaluation of individual Voigt profile functions are compared for accuracy and speed. An apparent error in one of these codes is discussed, along with a suggested correction. A general comment regarding the use of such codes in line-by-line radiative transfer programs is made.     

Optimized implementations of rational approximations for the Voigt and complex error function

Rational functions are frequently used as efficient yet accurate numerical approximations for real and complex valued functions. For the complex error function w(x+iy), whose real part is the Voigt function K(x,y), code optimizations of rational approximations are investigated. An assessment of requirements for atmospheric radiative transfer modeling indicates a y range over many orders of magnitude and accuracy better than 10⁻⁴. Following a brief survey of complex error function algorithms in general and rational function approximations in particular the problems associated with subdivisions of the x, y plane (i.e., conditional branches in the code) are discussed and practical aspects of Fortran and Python implementations are considered. Benchmark tests of a variety of algorithms demonstrate that programming language, compiler choice, and implementation details influence computational speed and there is no unique ranking of algorithms. A new implementation, based on subdivision of the upper half-plane in only two regions, combining Weideman's rational approximation for small |x|+y<15 and Humlicek's rational approximation otherwise is shown to be efficient and accurate for all x, y.     

The Gibbs measures and partial differential equations

We investigate the connections of the Gibbs measures, which appear in Euclidean Field Theory, and the corresponding partial differential equations of Classical Euclidean Field Theory.     

Calculation of the apparatus function for the case of a generalized Voigt contour

The Rayleigh-Bracewell method of finite differences is applied to cases where the apparatus function is either entirely Lorentzian or results from the superposition of Lorentzian and Gaussian components. A firstorder correction is calculated, and order of magnitude estimates are given for the remaining terms in the series expansion representing the correction. The method is employed to correct self-reversed contours of spectral lines.     

Determination of nonlinear optical properties using the Voigt function: Stochastic considerations

In this work, the absorptive and dispersive profiles of a molecular system have been calculated through a methodology based on the averaging of the coherence obtained by the resolution of the optical stochastic Bloch equations. To this aim, a generalized Lorentzian approximation to the Voigt function has been used as a probability distribution, which allows a more generalized analysis of the interactions between the solvent and the molecule. This has been modeled with the parameters of the standard organic dye Green Malaquite, which exhibits a nonlinear behavior under the interaction with a high intensity electric field.