Voigt profile

Using the above definition for z , the cumulative distribution function (CDF) can be found as follows:

${\displaystyle F(x_{0};\mu ,\sigma )=\int _{-\infty }^{x_{0}}{\frac {\operatorname {Re} (w(z))}{\sigma {\sqrt {2\pi }}}}\,dx=\operatorname {Re} \left({\frac {1}{\sqrt {\pi }}}\int _{z(-\infty )}^{z(x_{0})}w(z)\,dz\right).}$

Substituting the definition of the Faddeeva function (scaled complex error function) yields for the indefinite integral:

${\displaystyle {\frac {1}{\sqrt {\pi }}}\int w(z)\,dz={\frac {1}{\sqrt {\pi }}}\int e^{-z^{2}}\left[1-\operatorname {erf} (-iz)\right]\,dz,}$

which may be solved to yield

${\displaystyle {\frac {1}{\sqrt {\pi }}}\int w(z)\,dz={\frac {\operatorname {erf} (z)}{2}}+{\frac {iz^{2}}{\pi }}\,_{2}F_{2}\left(1,1;{\frac {3}{2}},2;-z^{2}\right),}$

where ${\displaystyle {}_{2}F_{2}}$ is a hypergeometric function. In order for the function to approach zero as x approaches negative infinity (as the CDF must do), an integration constant of 1/2 must be added. This gives for the CDF of Voigt:

${\displaystyle F(x;\mu ,\sigma )=\operatorname {Re} \left[{\frac {1}{2}}+{\frac {\operatorname {erf} (z)}{2}}+{\frac {iz^{2}}{\pi }}\,_{2}F_{2}\left(1,1;{\frac {3}{2}},2;-z^{2}\right)\right].}$

If the Gaussian profile is centered at ${\displaystyle \mu _{G}}$ and the Lorentzian profile is centered at ${\displaystyle \mu _{L}}$, the convolution is centered at ${\displaystyle \mu _{V}=\mu _{G}+\mu _{L}}$ and the characteristic function is:

${\displaystyle \varphi _{f}(t;\sigma ,\gamma ,\mu _{\mathrm {G} },\mu _{\mathrm {L} })=e^{i(\mu _{\mathrm {G} }+\mu _{\mathrm {L} })t-\sigma ^{2}t^{2}/2-\gamma |t|}.}$

The probability density function is simply offset from the centered profile by ${\displaystyle \mu _{V}}$:

${\displaystyle V(x;\mu _{V},\sigma ,\gamma )={\frac {\operatorname {Re} [w(z)]}{\sigma {\sqrt {2\pi }}}},}$

where:

${\displaystyle z={\frac {x-\mu _{V}+i\gamma }{\sigma {\sqrt {2}}}}}$

The mode and median are both located at ${\displaystyle \mu _{V}}$.

A Voigt profile (here, assuming ${\displaystyle \mu _{V}=10}$, ${\displaystyle \sigma =1.3}$, and ${\displaystyle \gamma =2.5}$) and its first two partial derivatives with respect to ${\displaystyle x}$ (the first column) and the three parameters ${\displaystyle \mu _{V}}$, ${\displaystyle \sigma }$, and ${\displaystyle \gamma }$ (the second, third, and fourth column, respectively), obtained analytically and numerically.

Using the definition above for ${\displaystyle z}$ and ${\displaystyle x_{c}=x-\mu _{V}}$, the first and second derivatives can be expressed in terms of the Faddeeva function as

${\displaystyle {\begin{aligned}{\frac {\partial }{\partial x}}V(x_{c};\sigma ,\gamma )&=-{\frac {\operatorname {Re} \left[z~w(z)\right]}{\sigma ^{2}{\sqrt {\pi }}}}=-{\frac {x_{c}}{\sigma ^{2}}}{\frac {\operatorname {Re} \left[w(z)\right]}{\sigma {\sqrt {2\pi }}}}+{\frac {\gamma }{\sigma ^{2}}}{\frac {\operatorname {Im} \left[w(z)\right]}{\sigma {\sqrt {2\pi }}}}\\&={\frac {1}{\sigma ^{3}{\sqrt {2\pi }}}}\cdot \left(\gamma \cdot \operatorname {Im} \left[w(z)\right]-x_{c}\cdot \operatorname {Re} \left[w(z)\right]\right)\end{aligned}}}$

and

${\displaystyle {\begin{aligned}{\frac {\partial ^{2}}{\left(\partial x\right)^{2}}}V(x_{c};\sigma ,\gamma )&={\frac {x_{c}^{2}-\gamma ^{2}-\sigma ^{2}}{\sigma ^{4}}}{\frac {\operatorname {Re} \left[w(z)\right]}{\sigma {\sqrt {2\pi }}}}-{\frac {2x_{c}\gamma }{\sigma ^{4}}}{\frac {\operatorname {Im} \left[w(z)\right]}{\sigma {\sqrt {2\pi }}}}+{\frac {\gamma }{\sigma ^{4}}}{\frac {1}{\pi }}\\&=-{\frac {1}{\sigma ^{5}{\sqrt {2\pi }}}}\cdot \left(\gamma \cdot \left(2x_{c}\cdot \operatorname {Im} \left[w(z)\right]-\sigma \cdot {\sqrt {\frac {2}{\pi }}}\right)+\left(\gamma ^{2}+\sigma ^{2}-x_{c}^{2}\right)\cdot \operatorname {Re} \left[w(z)\right]\right),\end{aligned}}}$

respectively.

Often, one or multiple Voigt profiles and/or their respective derivatives need to be fitted to a measured signal by means of non-linear least squares, e.g., in spectroscopy. Then, further partial derivatives can be utilised to accelerate computations. Instead of approximating the Jacobian matrix with respect to the parameters ${\displaystyle \mu _{V}}$, ${\displaystyle \sigma }$, and ${\displaystyle \gamma }$ with the aid of finite differences, the corresponding analytical expressions can be applied. With ${\displaystyle \operatorname {Re} \left[w(z)\right]=\Re _{w}}$ and ${\displaystyle \operatorname {Im} \left[w(z)\right]=\Im _{w}}$, these are given by:

${\displaystyle {\begin{aligned}{\frac {\partial V}{\partial \mu _{V}}}=-{\frac {\partial V}{\partial x}}={\frac {1}{\sigma ^{3}{\sqrt {2\pi }}}}\cdot \left(x_{c}\cdot \Re _{w}-\gamma \cdot \Im _{w}\right)\end{aligned}}}$

${\displaystyle {\begin{aligned}{\frac {\partial V}{\partial \sigma }}={\frac {1}{\sigma ^{4}{\sqrt {2\pi }}}}\cdot \left(\left(x_{c}^{2}-\gamma ^{2}-\sigma ^{2}\right)\cdot \Re _{w}-2x_{c}\gamma \cdot \Im _{w}+\gamma \sigma \cdot {\sqrt {\frac {2}{\pi }}}\right)\end{aligned}}}$

${\displaystyle {\begin{aligned}{\frac {\partial V}{\partial \gamma }}=-{\frac {1}{\sigma ^{3}{\sqrt {2\pi }}}}\cdot \left(\sigma \cdot {\sqrt {\frac {2}{\pi }}}-x_{c}\cdot \Im _{w}-\gamma \cdot \Re _{w}\right)\end{aligned}}}$

for the original voigt profile ${\displaystyle V}$;

${\displaystyle {\begin{aligned}{\frac {\partial V'}{\partial \mu _{V}}}=-{\frac {\partial V'}{\partial x}}=-{\frac {\partial ^{2}V}{\left(\partial x\right)^{2}}}={\frac {1}{\sigma ^{5}{\sqrt {2\pi }}}}\cdot \left(\gamma \cdot \left(2x_{c}\cdot \Im _{w}-\sigma \cdot {\sqrt {\frac {2}{\pi }}}\right)+\left(\gamma ^{2}+\sigma ^{2}-x_{c}^{2}\right)\cdot \Re _{w}\right)\end{aligned}}}$

${\displaystyle {\begin{aligned}{\frac {\partial V'}{\partial \sigma }}={\frac {3}{\sigma ^{6}{\sqrt {2\pi }}}}\cdot \left(-\gamma \sigma x_{c}\cdot {\frac {2{\sqrt {2}}}{3{\sqrt {\pi }}}}+\left(x_{c}^{2}-{\frac {\gamma ^{2}}{3}}-\sigma ^{2}\right)\cdot \gamma \cdot \Im _{w}+\left(\gamma ^{2}+\sigma ^{2}-{\frac {x_{c}^{2}}{3}}\right)\cdot x_{c}\cdot \Re _{w}\right)\end{aligned}}}$

${\displaystyle {\begin{aligned}{\frac {\partial V'}{\partial \gamma }}={\frac {1}{\sigma ^{5}{\sqrt {2\pi }}}}\cdot \left(x_{c}\cdot \left(\sigma \cdot {\sqrt {\frac {2}{\pi }}}-2\gamma \cdot \Re _{w}\right)+\left(\gamma ^{2}+\sigma ^{2}-x_{c}^{2}\right)\cdot \Im _{w}\right)\end{aligned}}}$

for the first order partial derivative ${\displaystyle V'={\frac {\partial V}{\partial x}}}$; and

${\displaystyle {\begin{aligned}{\frac {\partial V''}{\partial \mu _{V}}}=-{\frac {\partial V''}{\partial x}}=-{\frac {\partial ^{3}V}{\left(\partial x\right)^{3}}}=-{\frac {3}{\sigma ^{7}{\sqrt {2\pi }}}}\cdot \left(\left(x_{c}^{2}-{\frac {\gamma ^{2}}{3}}-\sigma ^{2}\right)\cdot \gamma \cdot \Im _{w}+\left(\gamma ^{2}+\sigma ^{2}-{\frac {x_{c}^{2}}{3}}\right)\cdot x_{c}\cdot \Re _{w}-\gamma \sigma x_{c}\cdot {\frac {2{\sqrt {2}}}{3{\sqrt {\pi }}}}\right)\end{aligned}}}$

${\displaystyle {\begin{aligned}&{\frac {\partial V''}{\partial \sigma }}=-{\frac {1}{\sigma ^{8}{\sqrt {2\pi }}}}\cdot \\&\left(\left(-3\gamma x_{c}\sigma ^{2}+\gamma x_{c}^{3}-\gamma ^{3}x_{c}\right)\cdot 4\cdot \Im _{w}+\left(\left(2x_{c}^{2}-2\gamma ^{2}-\sigma ^{2}\right)\cdot 3\sigma ^{2}+6\gamma ^{2}x_{c}^{2}-x_{c}^{4}-\gamma ^{4}\right)\cdot \Re _{w}+\left(\gamma ^{2}+5\sigma ^{2}-3x_{c}^{2}\right)\cdot \gamma \sigma \cdot {\sqrt {\frac {2}{\pi }}}\right)\end{aligned}}}$

${\displaystyle {\begin{aligned}{\frac {\partial V''}{\partial \gamma }}=-{\frac {3}{\sigma ^{7}{\sqrt {2\pi }}}}\cdot \left(\left(\gamma ^{2}+\sigma ^{2}-{\frac {x_{c}^{2}}{3}}\right)\cdot x_{c}\cdot \Im _{w}+\left({\frac {\gamma ^{2}}{3}}+\sigma ^{2}-x_{c}^{2}\right)\cdot \gamma \cdot \Re _{w}+\left(x_{c}^{2}-\gamma ^{2}-2\sigma ^{2}\right)\cdot \sigma \cdot {\frac {\sqrt {2}}{3{\sqrt {\pi }}}}\right)\end{aligned}}}$

for the second order partial derivative ${\displaystyle V''={\frac {\partial ^{2}V}{\left(\partial x\right)^{2}}}}$. Since ${\displaystyle \mu _{V}}$ and ${\displaystyle \gamma }$ play a relatively similar role in the calculation of ${\displaystyle z}$, their respective partial derivatives also look quite similar in terms of their structure, although they result in totally different derivative profiles. Indeed, the partial derivatives with respect to ${\displaystyle \sigma }$ and ${\displaystyle \gamma }$ show more similarity since both are width parameters. All these derivatives involve only simple operations (multiplications and additions) because the computationally expensive ${\displaystyle \Re _{w}}$ and ${\displaystyle \Im _{w}}$ are readily obtained when computing ${\displaystyle w\left(z\right)}$. Such a reuse of previous calculations allows for a derivation at minimum costs. This is not the case for finite difference gradient approximation as it requires the evaluation of ${\displaystyle w\left(z\right)}$ for each gradient respectively.

${\displaystyle V(x;\sigma ,\gamma )=H(a,u)/({\sqrt {2}}{\sqrt {\pi }}\sigma ),}$

with

${\displaystyle a=\gamma /({\sqrt {2}}\sigma )}$

and

${\displaystyle u=x/({\sqrt {2}}\sigma ).}$

The Tepper-García function, named after German-Mexican Astrophysicist Thor Tepper-García, is a combination of an exponential function and rational functions that approximates the line broadening function ${\displaystyle H(a,u)}$ over a wide range of its parameters.[2] It is obtained from a truncated power series expansion of the exact line broadening function.

In its most computationally efficient form, the Tepper-García function can be expressed as

${\displaystyle T(a,u)=R-\left(a/{\sqrt {\pi }}P\right)~\left[R^{2}~(4P^{2}+7P+4+Q)-Q-1\right]\,,}$

where ${\displaystyle P\equiv u^{2}}$, ${\displaystyle Q\equiv 3/(2P)}$, and ${\displaystyle R\equiv e^{-P}}$.

Thus the line broadening function can be viewed, to first order, as a pure Gaussian function plus a correction factor that depends linearly on the microscopic properties of the absorbing medium (encoded in ${\displaystyle a}$); however, as a result of the early truncation in the series expansion, the error in the approximation is still of order ${\displaystyle a}$, i.e. ${\displaystyle H(a,u)\approx T(a,u)+{\mathcal {O}}(a)}$. This approximation has a relative accuracy of

${\displaystyle \epsilon \equiv {\frac {\vert H(a,u)-T(a,u)\vert }{H(a,u)}}\lesssim 10^{-4}}$

over the full wavelength range of ${\displaystyle H(a,u)}$, provided that ${\displaystyle a\lesssim 10^{-4}}$. In addition to its high accuracy, the function ${\displaystyle T(a,u)}$ is easy to implement as well as computationally fast. It is widely used in the field of quasar absorption line analysis.[3]

The pseudo-Voigt profile (or pseudo-Voigt function) is an approximation of the Voigt profile V(x) using a linear combination of a Gaussian curve G(x) and a Lorentzian curve L(x) instead of their convolution.

The pseudo-Voigt function is often used for calculations of experimental spectral line shapes.

The mathematical definition of the normalized pseudo-Voigt profile is given by

${\displaystyle V_{p}(x,f)=\eta \cdot L(x,f)+(1-\eta )\cdot G(x,f)}$ with ${\displaystyle 0<\eta <1}$.

${\displaystyle \eta }$ is a function of full width at half maximum (FWHM) parameter.

There are several possible choices for the ${\displaystyle \eta }$ parameter.[4][5][6][7] A simple formula, accurate to 1%, is[8][9]

${\displaystyle \eta =1.36603(f_{L}/f)-0.47719(f_{L}/f)^{2}+0.11116(f_{L}/f)^{3},}$

where now, ${\displaystyle \eta }$ is a function of Lorentz (${\displaystyle f_{L}}$), Gaussian (${\displaystyle f_{G}}$) and total (${\displaystyle f}$) Full width at half maximum (FWHM) parameters. The total FWHM (${\displaystyle f}$) parameter is described by:

${\displaystyle f=[f_{G}^{5}+2.69269f_{G}^{4}f_{L}+2.42843f_{G}^{3}f_{L}^{2}+4.47163f_{G}^{2}f_{L}^{3}+0.07842f_{G}f_{L}^{4}+f_{L}^{5}]^{1/5}.}$

The full width at half maximum (FWHM) of the Voigt profile can be found from the widths of the associated Gaussian and Lorentzian widths. The FWHM of the Gaussian profile is

${\displaystyle f_{\mathrm {G} }=2\sigma {\sqrt {2\ln(2)}}.}$

The FWHM of the Lorentzian profile is

${\displaystyle f_{\mathrm {L} }=2\gamma .}$

An approximate relation (accurate to within about 1.2%) between the widths of the Voigt, Gaussian, and Lorentzian profiles is:[10]

${\displaystyle f_{\mathrm {V} }\approx f_{\mathrm {L} }/2+{\sqrt {f_{\mathrm {L} }^{2}/4+f_{\mathrm {G} }^{2}}}.}$

By construction, this expression is exact for a pure Gaussian or Lorentzian.

A better approximation with an accuracy of 0.02% is given by [11] (originally found by Kielkopf[12])

${\displaystyle f_{\mathrm {V} }\approx 0.5346f_{\mathrm {L} }+{\sqrt {0.2166f_{\mathrm {L} }^{2}+f_{\mathrm {G} }^{2}}}.}$

Again, this expression is exact for a pure Gaussian or Lorentzian. In the same publication,[11] a slightly more precise (within 0.012%), yet significantly more complicated expression can be found.