Voigt Profile

Voigt Profile

The Voigt profile is the convolution of a Gaussian (Doppler-broadened) and a Lorentzian (collision/pressure-broadened) line shape. It describes the absorption coefficient of essentially every real spectral line above the natural-linewidth floor.

The Convolution

Let $\phi_D(\nu)$ be the Doppler profile and $\phi_L(\nu)$ the Lorentzian profile. The Voigt line shape is

$$\phi_V(\nu) = \int_{-\infty}^{\infty} \phi_D(\nu') \phi_L(\nu - \nu')\, d\nu'$$

Closed form involves the Faddeeva function $w(z)$. Most practical implementations use the Humlíček algorithm or a rational approximation; plasma-solvers::voigt_profile(x, a) returns the normalized profile evaluated at offset $x$ (in Doppler widths) and damping parameter $a$.

The a Parameter

The Voigt parameter $a$ is the ratio of Lorentzian to Doppler half-widths:

$$a = \frac{\Gamma_{\text{coll}}}{2 \cdot \Delta\nu_D \cdot \sqrt{\ln 2}}$$

It controls the crossover between regimes:

• $a \ll 1$: profile is essentially Doppler at line center; wings decay as a Lorentzian far out.
• $a \sim 1$: full crossover, neither pure limit applies.
• $a \gg 1$: profile is essentially Lorentzian everywhere.

Where Voigt Lives in This Wiki

The Voigt profile enters Holstein radiation trapping through the escape kernel $K(\vec{r}, \vec{r}\,')$ — the wing photons that carry escape budget in optically thick vapor are the same wings the Voigt profile parameterizes. Wrong $a$ → wrong escape factor → wrong trapped lifetime.

In cooperative emission, the line shape governs the spectral overlap that contributes to the cooperation factor $\eta$ in Dicke superradiance. Doppler-dominated profiles (low $a$) at saturated Na vapor give $\eta \sim 0.1$–$0.3$; pressure-broadened Hg lamps with $a \gtrsim 1$ have $\eta$ closer to unity, all else equal.

## See also - 02-holstein-radiation-trapping - 03-dicke-superradiance - Resonance Radiation and Excited Atoms, Mitchell & Zemansky (1934)