Application of the low-finesse $\gamma$-ray frequency comb for high-resolution spectroscopy

High-finesse frequency combs (HFC) with large ratio of the frequency spacing to the width of the spectral components have demonstrated remarkable results in many applications such as precision spectroscopy and metrology. We found that low-finesse frequency combs having very small ratio of the frequency spacing to the width of the spectral components are more sensitive to the exact resonance with absorber than HFC. Our method is based on time domain measurements reviling oscillations of the radiation intensity after passing through an optically thick absorber. Fourier analysis of the oscillations allows to reconstruct the spectral content of the comb. If the central component of the incident comb is in exact resonance with the single line absorber, the contribution of the first sideband frequency to oscillations is exactly zero. We demonstrated this technique with $\gamma$-photon absorption by Mössbauer nuclei providing the spectral resolution beyond the natural broadening.

Figure 1

(a) Power spectrum of the source radiation field, seen by the vibrated absorber (normalized to the peak intensity). (b) The absorption spectrum of the field, normalized to unity. The vibration frequency is $\mathrm{\Omega }={\mathrm{\Gamma }}_S/5$. The modulation index $a$ is 0 (solid line), 2 (dotted line), 3 (dashed line), and 6 (dash-dotted line) in (a). The modulation index $a$ is 0 (solid line) and 6 (dotted line) in (b). The optical thickness of the absorber is $T_a=\alpha \left(0\right)l=5.2$. The linewidth of the absorber ${\mathrm{\Gamma }}_A$ and spectral components of the source ${\mathrm{\Gamma }}_S$ are equal to ${\mathrm{\Gamma }}_0$, which is defined by the lifetime of the excited state nucleus (141 ns).

Figure 2

Oscillations of the intensity of high-finesse frequency comb, transmitted through the absorber. Phase modulation frequency is $\mathrm{\Omega }=10$ MHz. Intensity oscillations are shown by solid lines for the detuning from resonance (a) $\mathrm{\Delta }=-200$ and (b) 200 kHz. Dotted line shows the level of ${\mathtt{I}}_0\left(\mathrm{\Delta }\right)$. The value of the parameters are given in the text.

Figure 3

The dependence of the first-harmonic amplitude $D_1$ of the intensity oscillations on the modulation index $a$. The modulation frequency and resonant detuning are $\mathrm{\Omega }=\mathrm{\Delta }=200$ kHz.

Figure 4

Time dependence of the photon counts $N\left(t\right)$ for different values of the resonant detuning. The number of counts (in relative units) are normalized to the mean value at exact resonance. The value of the detuning in units of ${\mathrm{\Gamma }}_0$ is indicated in each panel. The dots are experimental points and solid line is a theoretical fitting.

Figure 5

(Left) Comparison of the dependence of $D_1/D_2$ on $\mathrm{\Delta }$ for LFC and HFC. Experimental data for LFC are shown by dots. (Right) Fourier content of the intensity oscillations for $\mathrm{\Delta }=0.3{\mathrm{\Gamma }}_0$. Other parameters are defined in the text. Dots correspond to the data, obtained from the Fourier analysis of the experimentally observed intensity oscillations. Solid line is the analytical approximation by the set of Lorentzians.