Hilbert transform: Applications to atomic spectra

In many areas of physics, the Kramers-Kronig relations are used to extract information about the real part of the optical response of a medium from its imaginary counterpart. In this paper we discuss an alternative but mathematically equivalent approach based on the Hilbert transform. We apply the Hilbert transform to transmission spectra to find the group and refractive indices of a Cs vapor and thereby demonstrate how the Hilbert transform allows indirect measurement of the refractive index, group index, and group delay while avoiding the use of complicated experimental setups.

Figure 1

(a) Contour (blue line) used to derive the Kramers-Kronig relations, along with the pole at ${\omega }^{\prime }=\omega$ (red dot). To solve, the limits of the large semicircle are pushed to infinity, while the radius of the smaller semicircle surrounding the pole is reduced towards zero. (b) Visualization of properties of odd and even components of a causal function. The summation of odd $h_{\mathrm{o}}\left(t\right)$ and even $h_{\mathrm{e}}\left(t\right)$ components gives the causal function $h\left(t\right)$ that is zero for $t<0$.

Figure 2

Process for calculating Hilbert transform from transmission spectra. A transmission spectrum is taken and then converted to the imaginary part of the susceptibility. The imaginary susceptibility is then Hilbert transformed and converted to the refractive index. The group index is then calculated from $n\left(\omega \right)$.

Figure 3

(a) Transmission data (purple points) and theoretical fit (black line) for a Cs vapor with thickness $670\pm 10$ nm and temperature $220\pm 5{}^{\circ }\mathrm{C}$ in the vicinity of the $D1$ line. (b) Refractive index inferred from the measured transmission data using the Hilbert transform (blue points) compared to theoretical expectations (black line) from our susceptibility model. (c) Group refractive index determined from the Hilbert transformed spectra (red points) and theoretical group index calculated from modified form of [19]. Below each panel are the residuals between theory and measurements using the Hilbert transform method. The residual are within the uncertainties related to laser frequency and power fluctuations.

Figure 4

Comparison of experimental measurements and theoretical calculations of transmission line shapes for (a) the Rb $D1$ line and (b) the imaginary susceptibility ${\chi }_{\mathrm{I}}\left(\omega \right)$ for an optically thick vapor in a 2-mm cell at $T=179{}^{\circ }\mathrm{C}$. (a) Experimental (purple points) and theoretical transmissions, which appear to be in good agreement and have a maximum expected theoretical optical density of 345. (b) The full ${\chi }_{\mathrm{I}}\left(\omega \right)$ (modeled using [19], black line) cannot be determined using current detection methods (blue points).