Time-Resolved sub-Ångström Metrology by Temporal Phase Interferometry near X-Ray Resonances of Nuclei

We introduce an analytical phase-reconstruction principle that retrieves atomic scale motion via time-domain interferometry. The approach is based on a resonant interaction with high-frequency light and does not require temporal resolution on the time scale of the resonance period. It is thus applicable to hard x rays and $\gamma$ rays for measurements of extremely small spatial displacements or relative-frequency changes. Here, it is applied to retrieve the temporal phase of a 14.4 keV emission line of an ${}^{57}\mathrm{Fe}$ sample, which corresponds to a spatial translation of this sample. The small wavelength of this transition ($\lambda =0.86\AA$) allows for determining the motion of the emitter on sub-Ångström length and nanosecond timescales.

Figure 1

Spectral (a) and temporal (b) intensity of two single-line emitters separated by a frequency detuning $\mathrm{\Delta }<0$. In both panels, the orange solid line corresponds to a temporally constant phase between the two emitters, whereas the blue dashed line illustrates the case of a temporally varying phase shown in (c). For the temporally constant phase, the two emission frequencies lead to a beating in the temporal signal. The temporally varying phase leads to modulations of the temporal signal, and in turn imprints additional interference structures onto the spectrum.

Figure 2

(a),(c) Spectrogram $S(t,\mathrm{\Delta })$ as a function of time and frequency detuning with temporal phase $\phi =0$ (a) and $\phi =\phi (t)$ (c) [see Fig. 1]. The temporal phase modifies the resonance of the emitter and accounts for the asymmetry. (b),(d) The respective squared absolute Fourier-signal $|\tilde{S}(\tilde{\omega },\mathrm{\Delta }){|}^2$ of (a) and (c), respectively. Values below 14 ns (transparent) were excluded from the Fourier transform to match the experiment in Fig. 3. The temporal phase $\phi (t)$ causes a modulation of the positive oscillating term in (d), while the linear component of $\phi (t)$ in the range of 20 to 40 ns corresponds to a detuning shift in (c). For high enough detuning, the nonoscillating and oscillating terms are separated, enabling a clear reconstruction of the phase evolution.

Figure 3

(a),(c) Experimental spectrogram measured vs time and frequency detuning without (a) and with PT translation (c). Panels (b) and (d) depict the absolute square of the corresponding Fourier signals $\widetilde{|S}(\tilde{\mathrm{\omega }},\mathrm{\Delta }){|}^2$. We used spectrogram (a) to calibrate the measurements. Only signals arriving after $t_0=14\mathrm{ns}$ are recorded in order to avoid saturation and dead time caused by the prompt synchrotron pulse at $t=0\mathrm{ns}$ which we used to excite emitter and analyzer. The phase reconstructed from the indicated area $\mathrm{\Delta }\gtrsim 3.0\gamma$ in (c) is shown in Fig. 4 (green).

Figure 4

(a),(b) Temporal phase and translation of the emitter retrieved from three measurements for (a) contraction $(\mathrm{\Delta }U>0)$ and (b) extension $(\mathrm{\Delta }U<0)$ of the piezo onto which the emitter is sputtered. The green phase in (a) was retrieved from the spectrogram shown in Fig. 3. The error intervals represent the statistical error discussed in the text.