Voigt Profile

Every real spectral line is a fingerprint smudged by two independent hands. The first is homogeneous, acting identically on every atom: a finite radiative lifetime imposes the natural linewidth of aq-optics-radiative-lifetime-natural-linewidth, collisions add pressure broadening, and — as the damped-oscillator argument of aq-optics-lorentzian-line-shape shows — both stamp the same Lorentzian shape onto the line. The second hand is inhomogeneous: different atoms move at different velocities, so each one's resonance is Doppler-shifted by a random amount, and those shifts, drawn from the Maxwell–Boltzmann distribution, scatter the line centres into a Gaussian. When both hands act at once — as they always do in a hot gas — the observed profile is neither Lorentzian nor Gaussian but the convolution of the two. Each velocity class radiates a full Lorentzian parked at its own Doppler-shifted centre, and the detector adds them up weighted by how many atoms move that fast; that sum-over-a-Gaussian-of-Lorentzians is exactly a convolution, and the result is the Voigt profile. This is not an approximation but is forced, because the total frequency offset of a photon is a sum of two independent offsets. The payoff is diagnostic. The Voigt profile interpolates smoothly between its parents — Gaussian core when Doppler wins, Lorentzian everywhere when collisions win — and because no elementary closed form exists, it is written through the Faddeeva function, tying atomic line shapes to the complex error function. Its core reports the temperature and its slow wings the density and lifetime, so one fitted curve reads out both.

Key takeaways

The Voigt profile is the convolution of a Lorentzian (homogeneous: natural plus collisional broadening) with a Gaussian (inhomogeneous: thermal Doppler broadening), describing any line where both mechanisms act at once.
It has no closed elementary form; it equals the real part of the Faddeeva function $w(z)=e^{-z^2}\mathrm{erfc}(-iz)$ , the complex error function of a complex argument.
Its narrow core follows the Gaussian, but its far wings always revert to the Lorentzian $1/\Delta\nu^2$ decay, because an algebraic tail beats a super-exponential one.
The Voigt FWHM is neither $f_L+f_G$ nor $\sqrt{f_L^2+f_G^2}$ ; the accurate combination is the Olivero–Longbothum formula, good to about $0.02\%$ .
The Voigt parameter $a=\sqrt{\ln 2}\,\gamma/\sigma_D$ (Lorentzian-to-Gaussian half-width ratio) fixes the shape: $a\ll1$ is Doppler-dominated, $a\gg1$ is pressure/lifetime-dominated.
Key idea: a Voigt profile is a Lorentzian $\otimes$ Gaussian convolution — the homogeneous Lorentzian smeared over the Gaussian spread of Doppler-shifted centres. The core reads the temperature, the wings read the density and lifetime, and both widths ride together inside the Faddeeva function.

The core insight

A Voigt profile is a Lorentzian $\otimes$ Gaussian convolution: the homogeneous Lorentzian is smeared over the Gaussian distribution of Doppler-shifted centre frequencies. The narrow core follows the Gaussian, the far wings always follow the Lorentzian, and both widths are encoded together through the Faddeeva function.

Where this lives today

Open any modern atmospheric-transmission or astrophysical line-fitting code and the Voigt profile is the atom of the calculation. HITRAN-based radiative-transfer models represent every one of millions of molecular lines as a Voigt — pressure-dependent Lorentzian width from tabulated broadening coefficients, temperature-dependent Doppler width — and lean on fast Faddeeva algorithms because a single spectral run convolves those millions of lines. In astrophysics the same profile drives the curve of growth: an absorption line is fit with a Voigt whose Gaussian $b$ -parameter yields the gas temperature plus turbulence while its damping wings yield the column density, the curve's damping branch appearing precisely because saturated cores push the measurement onto the Lorentzian wings. Fusion and combustion plasmas use it as a two-in-one probe — Gaussian FWHM for ion temperature, Lorentzian FWHM for electron density via pressure and Stark broadening — and in precision laser spectroscopy the ubiquity of the Voigt is exactly what motivates sub-Doppler methods such as saturated-absorption and two-photon spectroscopy, which dig the narrow natural Lorentzian out from under the Doppler Gaussian.

The physical picture

Picture a hot vapour. Each atom is a tiny damped oscillator whose lifetime and collisions give it a Lorentzian emission line, sharp and with long algebraic wings. But each atom is also flying, and its line-of-sight velocity Doppler-shifts that Lorentzian by $\delta=\nu_0 v_z/c$ : fast atoms shift blue, slow atoms shift red, and the number at each speed is set by the Maxwell–Boltzmann distribution — a Gaussian. The spectrum you record is the whole crowd of shifted Lorentzians stacked up, each weighted by how populous its velocity class is. That stack is the Voigt profile. The cleanest way to feel the operation is as blurring. Convolution smears the sharp Lorentzian across the Gaussian spread of centres, like a camera's point-spread function: the blur rounds the peak and fills the core, which is why a Doppler-dominated Voigt core looks Gaussian. But blurring can never make a slow tail out of a fast one — the Gaussian dies as $e^{-\Delta\nu^2}$ while the Lorentzian dies only as $1/\Delta\nu^2$ , so far from centre the Lorentzian always wins and the wings stay Lorentzian however strong the blur.

Figure

Building a Voigt profile by convolution: each thermal velocity class radiates the same homogeneous Lorentzian (thin green) but centred at its own Doppler shift, with amplitudes set by the Maxwell distribution (dashed amber envelope, Doppler FWHM f_G ≈ 1.70 GHz). Summing the shifted Lorentzians over the Gaussian spread yields the Voigt profile (bold blue)

The single number that governs which parent dominates is the Voigt parameter $a$ , the ratio of the Lorentzian to the Gaussian half-width. When $a\to0$ Doppler rules and the line is Gaussian; when $a\to\infty$ homogeneous broadening rules and it is Lorentzian. Real lines live in between, and the fitted $a$ is the physics — separating the temperature in the Gaussian from the density and lifetime in the Lorentzian. The second figure shows the giveaway on a log scale: the Voigt clings to its Gaussian parent through the core, then peels away to trace its Lorentzian parent in the wings.

Figure

Three area-normalized profiles on a log scale for the sodium D2 line at 500 K: the Doppler Gaussian f_G ≈ 1.70 GHz, a collisional Lorentzian f_L ≈ 0.50 GHz, and their Voigt convolution with a ≈ 0.245 and f_V ≈ 1.98 GHz. The Voigt core hugs the Gaussian, but past about ±1.9 GHz the Gaussian plunges below the Lorentzian and the Voigt wings revert to the 1/Δν² tail

The core equations

The definition is a single statement: the observed line is the homogeneous shape summed over the inhomogeneous spread of centres. Because the two random frequency offsets add independently, the density of their sum is the convolution of the two densities — the Voigt profile is a convolution and nothing more.

V(\nu)=(L*G)(\nu)=\int_{-\infty}^{\infty}L(\nu')\,G(\nu-\nu')\,\mathrm{d}\nu' \qquad (1)

where: $V(\nu)$ — Voigt profile, unit area; $L(\nu')$ — Lorentzian (homogeneous) line shape, unit area; $G(\nu)$ — Gaussian (Doppler) line shape, unit area; $*$ — convolution operator. Physically: slide the Gaussian across the Lorentzian and integrate the overlap at every offset; because both inputs have unit area, so does $V$ . The homogeneous factor is the Lorentzian owned by aq-optics-lorentzian-line-shape, the far-winged shape of a damped classical oscillator (quantum-mechanically, a state of finite lifetime). Its width is the sum of the natural and collisional contributions, since both are homogeneous and add linearly.

L(\nu)=\frac{\gamma/\pi}{(\nu-\nu_0)^2+\gamma^2},\qquad \gamma=\gamma_{\mathrm{nat}}+\gamma_{\mathrm{coll}} \qquad (2)

where: $\gamma$ — Lorentzian half-width at half maximum (HWHM); $\nu_0$ — line-centre frequency; $\gamma_{\mathrm{nat}}$ — natural (lifetime) width; $\gamma_{\mathrm{coll}}$ — collisional (pressure) width. The $1/(\nu-\nu_0)^2$ far-field decay is the crucial feature that survives into the Voigt wings. The inhomogeneous factor is Gaussian because the line-of-sight velocity is Maxwell–Boltzmann distributed and the non-relativistic Doppler shift maps velocity linearly to frequency. Its width is set by temperature and mass and — unlike the Lorentzian widths — may never be added linearly to $\gamma$ .

G(\nu)=\frac{1}{\sqrt{2\pi}\,\sigma}\,\exp\!\left[-\frac{(\nu-\nu_0)^2}{2\sigma^2}\right],\qquad \sigma=\frac{\nu_0}{c}\sqrt{\frac{k_BT}{m}} \qquad (3)

where: $\sigma$ — Gaussian standard deviation of frequency; $T$ — gas temperature; $m$ — atomic mass; $k_B$ — Boltzmann constant; $c$ — speed of light. Hotter gas or lighter atoms broaden the Gaussian. The convolution has no elementary closed form, but it collapses onto one special function. Derivation 1 reduces equation (1) to a single dimensionless complex integral and identifies it with the Faddeeva function $w(z)$ , whose real part is the profile.

V(\nu)=\frac{\mathrm{Re}[w(z)]}{\sqrt{2\pi}\,\sigma},\qquad z=\frac{(\nu-\nu_0)+i\gamma}{\sqrt{2}\,\sigma},\qquad w(z)=e^{-z^2}\mathrm{erfc}(-iz) \qquad (4)

where: $w(z)$ — Faddeeva function; $\mathrm{Re}$ — real part; $z$ — complex argument packing the detuning into its real part and the Lorentzian width into its imaginary part. The imaginary part of $w(z)$ is the dispersive profile, the Kramers–Kronig partner of $V$ . For tabulation and fitting the profile is written in fully dimensionless form as the Voigt function $K(x,a)$ , whose entire shape is controlled by the single parameter $a$ . This is the object every line-fitting code actually evaluates.

K(x,a)=\frac{a}{\pi}\int_{-\infty}^{\infty}\frac{e^{-t^2}}{a^2+(x-t)^2}\,\mathrm{d}t=\mathrm{Re}[w(x+ia)],\qquad a=\sqrt{\ln 2}\,\frac{\gamma}{\sigma_D} \qquad (5)

where: $x$ — reduced detuning scaled to the Gaussian half-width; $a$ — Voigt parameter, the ratio of Lorentzian to Gaussian HWHM; $t$ — scaled-velocity integration variable; $\sigma_D=\sqrt{2\ln 2}\,\sigma$ — Gaussian HWHM. Small $a$ is Gaussian-like, large $a$ Lorentzian-like: $a$ sets the shape, $\sigma$ the scale. Finally, the width you measure is neither parent width. Because the half-maximum condition of a convolution has no elementary inverse, the Voigt FWHM is captured by the empirical Olivero–Longbothum fit, accurate to about $0.02\%$ and correct in both limits.

f_V\approx 0.5346\,f_L+\sqrt{0.2166\,f_L^2+f_G^2},\qquad f_L=2\gamma,\qquad f_G=2\sqrt{2\ln 2}\,\sigma \qquad (6)

where: $f_V$ — Voigt FWHM; $f_L$ — Lorentzian FWHM; $f_G$ — Gaussian (Doppler) FWHM. Setting $f_G\to0$ gives $(0.5346+\sqrt{0.2166})f_L=f_L$ and $f_L\to0$ gives $f_G$ — the formula lands exactly on each parent.

Derivation AVoigt profile as the real part of the Faddeeva function

The goal is to prove that the two-width Lorentzian–Gaussian convolution — an integral that looks intractable — collapses onto a single special function evaluated at one complex point. We write the convolution explicitly, rescale both widths into dimensionless variables so only their ratio survives, recognize the resulting integral as the standard representation of the Faddeeva function, and restore the physical variables. The reward is practical: it turns an expensive numerical convolution into one fast evaluation of $w(z)$ , and makes the two limiting cases fall out of the known behaviour of $w$ .

Step 1. Write the convolution with the Lorentzian centred at the Doppler-shifted frequency $\nu_0+\delta$ and weighted by the Gaussian in $\delta$ : V(\nu)=\int_{-\infty}^{\infty}\frac{\gamma/\pi}{(\nu-\nu_0-\delta)^2+\gamma^2}\,\frac{e^{-\delta^2/2\sigma^2}}{\sqrt{2\pi}\,\sigma}\,\mathrm{d}\delta. Why: this is just equation (1) with the two shapes substituted, and it is the honest statement that each velocity class contributes one shifted Lorentzian.

Step 2. Nondimensionalize with $u=\delta/(\sqrt{2}\sigma)$ , $y=\gamma/(\sqrt{2}\sigma)$ , and $x=(\nu-\nu_0)/(\sqrt{2}\sigma)$ , collapsing both physical widths into the single ratio $y$ and rescaling detuning to $x$ . Why: once the overall width is scaled out, only the ratio $\gamma/\sigma$ can matter, and this makes it explicit.

Step 3. Collect every prefactor into the leading amplitude to reach V=\frac{1}{\sqrt{2\pi}\,\sigma}\,\frac{y}{\pi}\int_{-\infty}^{\infty}\frac{e^{-u^2}}{(x-u)^2+y^2}\,\mathrm{d}u. Why: the scale of the line rides entirely in $1/(\sqrt{2\pi}\sigma)$ , leaving a pure dimensionless integral whose value depends only on $x$ and $y$ .

Step 4. Recognize the dimensionless integral as the standard representation $\frac{y}{\pi}\int e^{-u^2}/[(x-u)^2+y^2]\,\mathrm{d}u=\mathrm{Re}[w(x+iy)]$ of the real part of $w(z)=e^{-z^2}\mathrm{erfc}(-iz)$ . Why: this is exactly the integral that defines the Faddeeva function; identifying it is the whole point of the rescaling and is what removes the need to convolve numerically.

Step 5. Restore variables through $z=x+iy=[(\nu-\nu_0)+i\gamma]/(\sqrt{2}\sigma)$ to land on $V(\nu)=\mathrm{Re}[w(z)]/(\sqrt{2\pi}\sigma)$ , equation (4). Why: this is the working formula, with detuning and Lorentzian width entering as the real and imaginary parts of one complex number. Check: $\gamma\to0$ makes $z$ real and $w(x)\to e^{-x^2}$ , a Gaussian; $\sigma\to0$ drives $z$ up the imaginary axis and the large-imaginary asymptotics of $w$ rebuild the Lorentzian.

Go deeper · Derivation B · 1 minThe far wings are always Lorentzian

Derivation BThe far wings are always Lorentzian

The goal is to establish the most useful diagnostic fact about the Voigt profile: however strongly Doppler broadening dominates the core, the far wings decay as the Lorentzian $1/\Delta\nu^2$ and carry the homogeneous width $\gamma$ alone — which makes wing fitting a clean, thermometer-free measurement of density and lifetime. We examine the convolution at detunings far larger than the Gaussian width, expand the Lorentzian about line centre, and integrate term by term against the Gaussian moments, where the super-exponential Gaussian cannot compete.

Step 1. For $|\nu-\nu_0|\gg\sigma$ , the Gaussian weight $G(\delta)$ is non-negligible only for $|\delta|\lesssim$ a few $\sigma$ , tiny compared with the detuning, so $\delta$ is a small perturbation to the Lorentzian denominator. Why: the Gaussian has fallen to essentially zero long before $\delta$ reaches the detuning, so only small shifts contribute in the wing.

Step 2. Expand the Lorentzian for small $\delta$ and drop the negligible $\gamma^2$ term: L(\nu-\nu_0-\delta)\approx\frac{\gamma/\pi}{(\nu-\nu_0)^2}\left[1+\frac{2\delta}{\nu-\nu_0}+\cdots\right]. Why: far from centre the Lorentzian is a smooth slowly varying function of $\delta$ , so a Taylor expansion in $\delta/(\nu-\nu_0)$ converges rapidly.

Step 3. Integrate term by term against the Gaussian, using $\int G\,\mathrm{d}\delta=1$ and $\int\delta\,G\,\mathrm{d}\delta=0$ , so the leading correction vanishes and V(\nu)\;\xrightarrow[|\nu-\nu_0|\gg\sigma]{}\;\frac{\gamma}{\pi\,(\nu-\nu_0)^2}. Why: the zeroth moment is unity and the first is zero by symmetry, so only the bare Lorentzian tail survives; the Gaussian, dying as $e^{-\Delta\nu^2}$ , adds nothing. Check: the exact asymptotic $\mathrm{Re}[w(z)]\sim y/(\sqrt{\pi}|z|^2)$ reproduces the same law — why wing fitting isolates $\gamma$ independently of $T$ .

Limiting cases

Case	Result	Meaning
Doppler-dominated, $a\to0$	$V(\nu)\to G(\nu)=\dfrac{1}{\sqrt{2\pi}\,\sigma}\,e^{-(\nu-\nu_0)^2/2\sigma^2}$	Pure Gaussian: width set by thermal motion — low-pressure hot gases, stellar photospheres, vapour cells.
Homogeneous-dominated, $a\to\infty$	$V(\nu)\to L(\nu)=\dfrac{\gamma/\pi}{(\nu-\nu_0)^2+\gamma^2}$	Pure Lorentzian: natural plus collisional broadening win — high-pressure gases and dense plasmas.
Far wings, any $a>0$	$V(\nu)\simeq\dfrac{\gamma/\pi}{(\nu-\nu_0)^2}\;\;(\|\nu-\nu_0\|\gg\sigma_D,\gamma)$	The Lorentzian $1/\Delta\nu^2$ tail always wins; wing fitting isolates $\gamma$ with no reference to $T$ .
Peak, small $a$	$K(0,a)=e^{a^2}\mathrm{erfc}(a)\approx1-\dfrac{2a}{\sqrt{\pi}}+\cdots$	The Lorentzian lowers and rounds the peak relative to an equal-area Gaussian; peak height probes $a$ .

Common trapThe Voigt width is the sum of the Lorentzian and Gaussian widths

It is not. The Voigt FWHM is neither the linear sum $f_L+f_G$ (an overestimate) nor the quadrature sum $\sqrt{f_L^2+f_G^2}$ (an underestimate) — both are only crude limiting guesses. The correct value is the Olivero–Longbothum combination $f_V\approx0.5346\,f_L+\sqrt{0.2166\,f_L^2+f_G^2}$ , which lands between them and reduces exactly to $f_L$ or $f_G$ in the two limits. A related trap: the homogeneous widths (natural and collisional) do add linearly, because Lorentzians convolve to a wider Lorentzian by adding widths — but you may never add a Lorentzian width to a Gaussian width by any rule; those combine only through the convolution.

Worked examples

1 · Doppler width and Voigt parameter of a real sodium line (confidence builder)

Sodium vapour in a 500 K cell shows the D2 line at $\lambda=589$ nm. The excited state has lifetime $\tau=16.2$ ns. Find the Doppler FWHM, the natural Lorentzian FWHM, and the Voigt parameter $a$ , and say what regime the line is in. Take $m_{\mathrm{Na}}=22.99\,u$ with $u=1.6605\times10^{-27}$ kg.

Step 1. Line-centre frequency: $\nu_0=c/\lambda=(2.998\times10^8)/(589\times10^{-9})=\mathbf{5.09\times10^{14}\ Hz}$ . The one-dimensional thermal speed $\sqrt{k_BT/m}=\sqrt{(1.381\times10^{-23})(500)/(3.819\times10^{-26})}=425$ m/s is a useful sanity check.

Step 2. Doppler width from equation (3): $\sigma=(\nu_0/c)\sqrt{k_BT/m}=(1.697\times10^{6})(425)=7.22\times10^{8}$ Hz, so the Gaussian FWHM is $f_G=2\sqrt{2\ln 2}\,\sigma=2.3548\times0.722\ \mathrm{GHz}=\mathbf{1.70\ GHz}$ — the width a low-pressure sodium cell shows.

Step 3. Natural Lorentzian width: $f_L=1/(2\pi\tau)=1/(2\pi\times16.2\times10^{-9})=\mathbf{9.8\ MHz}$ , so $\gamma=4.9$ MHz. With $\sigma_D=f_G/2=850$ MHz the Voigt parameter is $a=\sqrt{\ln 2}\,\gamma/\sigma_D=0.8326\times(4.9/850)=\mathbf{0.0048}$ .

Step 4. Interpretation: $a\approx0.005\ll1$ , so the line is overwhelmingly Doppler-dominated — the Gaussian is roughly $170\times$ wider than the natural Lorentzian. The Olivero–Longbothum FWHM is $f_V\approx0.5346(9.8)+\sqrt{0.2166(9.8)^2+1700^2}=1704$ MHz, only $5$ MHz above the pure Doppler value. The observed width is essentially the $1.70$ GHz Doppler value with the $9.8$ MHz natural linewidth buried under it — which is exactly why measuring the lifetime demands sub-Doppler techniques.

2 · Adding collisional broadening: combining widths correctly

Now fill the same cell with a buffer gas until collisions broaden the homogeneous width to $f_L=0.50$ GHz. With the Doppler width unchanged at $f_G=1.70$ GHz, find the Voigt parameter and the true Voigt FWHM, and compare with the two wrong shortcuts.

Step 1. Homogeneous widths add linearly, so $f_L=0.50$ GHz is the full Lorentzian FWHM and $\gamma=0.25$ GHz. The Voigt parameter climbs to $a=\sqrt{\ln 2}\,\gamma/\sigma_D=0.8326\times(0.25/0.85)=\mathbf{0.245}$ — a genuine intermediate Voigt.

Step 2. Olivero–Longbothum: $f_V\approx0.5346(0.50)+\sqrt{0.2166(0.50)^2+(1.70)^2}=0.267+1.716=\mathbf{1.98\ GHz}$ .

Step 3. The wrong shortcuts bracket it: the linear sum $f_L+f_G=2.20$ GHz is too wide by $11\%$ and the quadrature sum $\sqrt{0.50^2+1.70^2}=1.77$ GHz too narrow by $11\%$ . The correct Voigt FWHM, $1.98$ GHz, sits between the two naive estimates — neither is acceptable at percent-level spectroscopy.

3 · The peak-height depression as a diagnostic

Using the same $a=0.245$ line, find how far the Voigt peak sits below an equal-area pure Gaussian.

Step 1. For fixed area the dimensionless peak is $K(0,a)=\mathrm{Re}[w(ia)]=e^{a^2}\mathrm{erfc}(a)$ , normalized so a pure Gaussian gives $K(0,0)=1$ . At $a=0.245$ : $e^{0.0600}=1.0619$ and $\mathrm{erfc}(0.245)=0.729$ , so $K(0,0.245)=\mathbf{0.774}$ . The Voigt peak stands at only $77\%$ of the equal-area Gaussian peak — the Lorentzian pulls the peak down $23\%$ and pushes that area into the wings — and since the profile is area-normalized this ratio depends only on $a$ , so the peak-to-area ratio reads out $a$ directly.

Physics of the portrait

The Voigt profile sews two nineteenth-century strands of spectroscopy — the Doppler smearing of moving atoms and the damping wings of radiating oscillators — into one curve, and the seam bears the name of the man who tied it.

Origins · 1 minWoldemar Voigt and how this was discovered

Woldemar Voigt

1850–1919 · line-shape theory and crystal optics

Woldemar Voigt spent his career in Göttingen turning the phenomenology of light in matter into clean mathematics, and much of the vocabulary is still his: he coined the word tensor in its modern physical sense, systematized crystal physics in his Lehrbuch der Kristallphysik (1910), and gave his name to the Voigt effect in magneto-optics and to Voigt notation in elasticity. Around 1912, working on spectral-line shapes, he recognized that a real line in a hot gas must carry both the damping wings of the radiating oscillators — the Lorentzian that Lorentz had derived — and the Doppler smearing of thermal motion, and that the observed profile is the convolution of the two. He had also, in 1887, written coordinate transformations preserving the wave equation that anticipated the Lorentz transformation, a near-miss on special relativity by two decades. His convolution profile resisted tabulation for years, until astrophysical curve-of-growth work in the 1940s and the identification with the complex error function in the 1960s turned it into the workhorse of every spectroscopy laboratory today.

History

1889

— Michelson attributes the residual width of low-pressure spectral lines to the Doppler effect of thermal motion

1906

— Lorentz derives the Lorentzian shape from the damped, collision-interrupted classical oscillator

1912

— Woldemar Voigt combines Doppler and damping broadening into the convolution profile that bears his name

1967

— Armstrong expresses the Voigt function through the complex error (Faddeeva) function with efficient algorithms

1977

— Olivero and Longbothum publish the accurate empirical formula for the Voigt FWHM

Going further

Texts. Foot, Atomic Physics (2005), Ch. 8 · Demtröder, Laser Spectroscopy, Vol. 1, 5th ed., Ch. 3 · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 2 · Saleh and Teich, Fundamentals of Photonics, 3rd ed., Ch. 13 · Hecht, Optics, 5th ed., Ch. 7 · Olivero and Longbothum, J. Quant. Spectrosc. Radiat. Transfer 17, 233 (1977).

Provenance. The convolution definition and the Doppler width $\sigma=(\nu_0/c)\sqrt{k_BT/m}$ follow Foot, Atomic Physics Ch. 8 and Demtröder, Laser Spectroscopy Ch. 3; the Lorentzian factor is recalled by name from aq-optics-lorentzian-line-shape. The Faddeeva-function form and $K(x,a)=\mathrm{Re}[w(x+ia)]$ are standard (Armstrong 1967; Abramowitz and Stegun). A verifier should confirm: $f_G=2\sqrt{2\ln2}\,\sigma$ and $f_L=2\gamma$ ; the Olivero–Longbothum coefficients $0.5346$ and $0.2166$ and their reduction to $f_L$ and $f_G$ ; the far-wing $1/\Delta\nu^2$ law; and the worked numbers $\nu_0=5.09\times10^{14}$ Hz, $f_G=1.70$ GHz, natural $f_L=9.8$ MHz, $a=0.0048$ , and (with $f_L=0.50$ GHz) $a=0.245$ , $f_V=1.98$ GHz, $K(0,0.245)=0.774$ .

Prerequisites. Lorentzian Line Shape · Radiative Lifetime and Natural Linewidth · Resonant Absorption · Einstein A and B Coefficients. Leads to. Oscillator Strength · Resonant Absorption.

Ask the tutor: ask me to fit a real Voigt line, walk through the Faddeeva evaluation, or derive why the wings stay Lorentzian while the core turns Gaussian.

Why is the Voigt profile a convolution rather than the product of a Lorentzian and a Gaussian?

Because the two mechanisms shift a photon's frequency by two statistically independent random amounts — an intrinsic homogeneous offset and a Doppler offset from the atom's velocity — and the density of a sum of independent variables is the convolution of their densities, never their product. Physically, each velocity class radiates a complete Lorentzian centred at its own Doppler-shifted frequency, and the detector sums those Lorentzians weighted by a Gaussian population weight — summing a shape over a distribution of shifted copies is the very definition of convolution.

If there is no closed form, how does software actually compute a Voigt profile?

It evaluates the Faddeeva function $w(z)=e^{-z^2}\mathrm{erfc}(-iz)$ at the single complex point $z=[(\nu-\nu_0)+i\gamma]/(\sqrt{2}\sigma)$ and takes the real part, as in equation (4). Derivation 1 shows this is exactly equivalent to the convolution, so no numerical integration over velocity is needed. Optimized rational and series approximations for $w(z)$ — the Humlíček and related algorithms — reach machine precision in a handful of operations. That speed matters: an atmospheric radiative-transfer run needs a Voigt value at millions of line-and-frequency combinations, where one $w(z)$ call versus a full convolution is the difference between finishing and not.

Why do the far wings stay Lorentzian even when Doppler broadening dominates the core?

Because the Gaussian dies far faster than the Lorentzian. At large detuning the Gaussian weight $e^{-\Delta\nu^2/2\sigma^2}$ has collapsed to essentially nothing, while the Lorentzian falls off only algebraically as $1/\Delta\nu^2$ . Derivation 3 makes this quantitative: expanding the convolution in the wing and using the Gaussian moments (area one, first moment zero) leaves exactly $V\to\gamma/[\pi(\nu-\nu_0)^2]$ , the bare Lorentzian tail. So however Gaussian the core looks, the wings revert to the homogeneous shape — which is why wing fitting measures $\gamma$ (density and lifetime) with no reference to the temperature that sets the core.

What is the Voigt parameter a, and how do I estimate it quickly?

The Voigt parameter $a=\sqrt{\ln 2}\,\gamma/\sigma_D$ is the ratio of the Lorentzian to the Gaussian half-width, and it alone controls the shape of the normalized profile — everything else is scale. Since $\gamma=f_L/2$ and $\sigma_D=f_G/2$ , you can read it straight from the FWHMs as $a\approx0.833\,f_L/f_G$ . If $a\ll1$ the line is Doppler-dominated and nearly Gaussian; if $a\gg1$ it is pressure- or lifetime-dominated and nearly Lorentzian; $a\sim1$ is a genuine two-parameter Voigt. A vapour cell often has $a$ of a few thousandths, a dense plasma above one.

Do the natural and collisional widths add, and can I add them to the Doppler width the same way?

The natural and collisional widths do add — linearly. Both are homogeneous, both produce a Lorentzian, and convolving two Lorentzians gives a wider Lorentzian whose HWHM is the sum, so $\gamma=\gamma_{\mathrm{nat}}+\gamma_{\mathrm{coll}}$ . But you must never add that combined Lorentzian width to the Gaussian Doppler width by any linear or quadrature rule: a Lorentzian convolved with a Gaussian is a Voigt, whose width follows the Olivero–Longbothum formula. So collapse the homogeneous contributions into one $\gamma$ and the inhomogeneous ones into one $\sigma$ , then combine those two only through the convolution.

How does a Voigt fit separate temperature from density in one measurement?

The two live in different parts of the same curve. The Gaussian Doppler width $f_G\propto\sqrt{T/m}$ sets the core, so the core width reports the temperature (plus any turbulence folded into the astrophysical $b$ -parameter). The Lorentzian width $\gamma=\gamma_{\mathrm{nat}}+\gamma_{\mathrm{coll}}$ sets the wings, and since $\gamma_{\mathrm{coll}}$ tracks the collision rate, the wings report the density and — through $\gamma_{\mathrm{nat}}$ — the lifetime. Fitting a single Voigt therefore returns $f_G$ and $f_L$ at once: a thermometer from the core, a densitometer from the wings — a two-parameter diagnostic disguised as one line.