Dispersion corrections to the Gaussian profile describing the Doppler broadening of spectral lines

A dispersionally corrected Gaussian profile describing Doppler-broadened spectral line shapes is presented. Proposed corrections include the frequency dependence of the Doppler shifting caused by dispersion as well as by light frequency variation over the whole spectral line shape. It is shown that the frequency dependence of the Doppler shifting can have a non-negligible influence on the line-shape model and can affect the line shape even at the relative level of ${10}^{-5}$. Moreover, this effect also influences the determination of the line position at the level of kilohertz. Finally, the impact of the presented results on the Doppler width thermometry and precise molecular spectroscopy for fundamental studies is emphasized.

Figure 1

(a) Real and (b) imaginary parts of the complex Gaussian profile ${\mathcal{I}}_{\mathrm{GP}}\left(\omega \right)$ with the area normalized to unity. The remaining panels show normalized differences between dispersionally corrected Gaussian profiles including effects described in the text—(c, d) $k=\omega /c$, (e, f) $k={\omega }_{0}n/c$, (g, h) $k={\omega }_0[n+n^{\prime }\left(\omega \right)]/c$, (i, j) $k={\omega }_0{n}_a\left(\omega \right)/c$, (k, l) $k=\omega n_a\left(\omega \right)/c$—and an ordinary Gaussian profile.

Figure 2

Dependence of the relative shift $\mathrm{\Delta }{\omega }_0/{\omega }_D$ of the (a) real and (b) imaginary parts of the dispersionally corrected Gaussian profile with respect to the ordinary Gaussian profile on the dimensionless quantities $\beta =cNS/\left(2k_0{\omega }_D\right)$ and ${\omega }_D/{\omega }_0$. Results were obtained for $n-1=4\times {10}^{-8}$. Dashed black lines mark the zero values of the shift. The $\mathrm{\Delta }{\omega }_0/{\omega }_D$ values for the investigated ${\mathrm{CO}}_2$ transition, equal to $-1.4\times {10}^{-5}$ ($-1.1$ kHz) and $-2.0\times {10}^{-5}$ ($-1.6$ kHz) for the real and imaginary parts, respectively, are denoted by white stars.

Figure 3

(a) Real part of the complex Gaussian profile ${\mathcal{I}}_{\mathrm{GP}}\left(\omega \right)$ with the area normalized to unity. (b) Normalized differences among the corrected Gaussian profile including only $k=\omega /c$, the relativistic emissive Gaussian profile [40], the absorptive Gaussian profile [41], and an ordinary Gaussian profile.