Calculation of ionization in direct-frequency comb spectroscopy

Direct-frequency comb spectroscopy (DFCS) is currently the highest-resolution absolute frequency spectroscopic technique known. In general, one does DFCS by scanning the repetition rate, $f_{\mathrm{rep}}$, of a comb laser and measuring fluorescence from the excited states of the species under study. The technique has already been successfully characterized by a theoretical model that starts with the optical Bloch equations and, with a few simplifying assumptions converts them into linear coupled iterative equations. In the present work we build on that successful model to predict the characteristics of the ion yield from photoionization by the comb laser, as a function of $f_{\mathrm{rep}}$. We show that the ion spectrum yields the same atomic structure as the fluorescence spectra, but with greater efficiency.

Figure 1

Simplified energy level diagram for ${}^{87}$Rb. Individual states are grouped into manifolds. The continuum is modeled by manifolds 4 and 5. See text for details.

Figure 2

Plot of the relative populations in the $5p$, $5d,$ and Rb${}^{+}$ manifolds as functions of the comb laser repetition frequency, minus a fixed reference frequency, $f_{\mathrm{ref}}=75557551$ Hz. The $5p$ population is plotted as round points connected by a line (green online) and uses the linear scale on the far right; the $5d$ is plotted as a solid line without points (red online) and uses the log scale on the left; and the Rb${}^{+}$ population is plotted as crosses connected by a line (blue online) and also uses the log scale on the left. All of the peaks in the $5d$ and ion curves have been identified as resulting from two-photon transitions from the $5s\to 5d$ manifolds. A few selected peaks are labeled as (1) $5s_{1/2},F=1\to 5d_{5/2},F=2$; (2) $5s_{1/2},F=2\to 5d_{5/2},F=2$; (3) $5s_{1/2},F=1\to 5d_{5/2},F=1$; (4) $5s_{1/2},F=2\to 5d_{5/2},F=4$.

Figure 3

Computation of $5d$ (solid line without points, red online) and ionization (solid line with crosses, blue online) manifolds under the artificial initial condition that the initial population of $5p$ states (solid line with circular points, green online) varies sinusoidally with $f_{\mathrm{rep}}$. The ionization population largely follows the $5d$ population. However, the lack of $5d$ structure in the ionization population is an indication of two-photon ionization of states in the $5p$ manifold (without being resonant with an intermediate $5d$ state).

Figure 4

Detail of $5p$ population for low- and high-intensity comb laser. The peak near 7 Hz is from the $5s_{1/2},F=2\to 5p_{3/2},F=2$ transition and clearly shows the effects of optical pumping. The peak near 15 Hz is from the $5s_{1/2},F=2\to 5p_{3/2},F=3$ transition, for which no optical pumping is expected.

Figure 5

Ground-state hyperfine populations versus $f_{\mathrm{rep}}$ when a comb tooth is resonant with the $5s_{1/2},F=2\to 5p_{3/2},F=2$ transition. (a) is for low comb laser intensity and does not exhibit strong optical pumping. In (b) the comb laser intensity is high and strong optical pumping from $5s_{1/2},F=2$ to $5s_{1/2},F=1$ is clearly seen. (c) and (d) are the same intensity conditions as (a) and (b), respectively, but $f_0$ has been adjusted such that a second comb tooth is resonant with the $5s_{1/2},F=2\to 5p_{3/2},F=2$ transition. The optical pumping is minimal, as expected.

Figure 6

The same as Fig. 2, but for 50 pulses in the train. The peaks are slightly broader and ringing is seen in the populations. The ringing is actually due to the Fourier transform of a rectangular pulse, superimposing a sinc function on the comb teeth, and is especially evident on the $5d$ curve.