Knowledge Bank

For large ratios of the center frequency, $f0$, to the full linewidth at one-half peak height, $H(=2\beta 1)$, the phase function associated with a Lorentz absorption coefficient is $$ \displaystyle{{-\left (\frac {f-f_{0}}{\beta_{1}}\right)}\over {1+\left (\frac {f-f_{0}}{\beta_{1}}\right )^{2}}}$$ for a peak amplitude of unity. Under the same conditions a Gauss absorption coefficient has the phase function - $2\Pi D f-f0\beta 2$ where D(x) is Dawson’s integral and $\beta 2\approx 0.6H$. The phase maximum of 0.61 radian (for a peak Gauss attenuation of 1.0 neper) is larger than the phase maximum of 0.5 radian associated with a Lorentz absorption coefficient with a peak attenuation of 1.0 neper. Also, the width of the anomalous region is larger for the Gauss case (1.1 H) than for the Lorentz case (1-0 H). The phase function associated with a Voigt absorption coefficient has a profile that is located between the profiles of the two above cases and for large ratios $f0H$ (peak attenuation of unity) the phase function is $$ \frac {- 1} {{e}(\beta{1}/\beta_{2}) erfc (\beta_{1}/\beta_{2})} Im\ \ \omega (\frac {f-f_{0}} {\beta_{2}} + j \frac {\beta_{1}} {\beta_{2}})$$ where $\beta 1$ and $\beta 2$ are Lorentz and Gauss shape parameters, respectively, and $\omega (z)$ is the error function for complex arguments. For intermediate values for $f0H$ the phase functions are more complicated and depend upon the exact definitions of the absorption coefficients.