Vectorial Phase Retrieval for Linear Characterization of Attosecond Pulses

The waveforms of attosecond pulses produced by high-harmonic generation carry information on the electronic structure and dynamics in atomic and molecular systems. Current methods for the temporal characterization of such pulses have limited sensitivity and impose significant experimental complexity. We propose a new linear and all-optical method inspired by widely used multidimensional phase retrieval algorithms. Our new scheme is based on the spectral measurement of two attosecond sources and their interference. As an example, we focus on the case of spectral polarization measurements of attosecond pulses, relying on their most fundamental property—being well confined in time. We demonstrate this method numerically by reconstructing the temporal profiles of attosecond pulses generated from aligned ${\mathrm{CO}}_2$ molecules.

Figure 1

An example for our reconstruction procedure for two artificially generated pulses with the same compact support. The blue dots, red crosses, and green squares are the roots, in the complex plane, of $|E_x(z){|}^2$,$|E_y(z){|}^2$, and $E_x(z)\overline{E_y(z)}$, respectively. A time domain plot of the two polarizations is shown in the inset.

Figure 2

(a) Two time domain polarization components of the attosecond pulse generated from ${\mathrm{CO}}_2$ molecules at an alignment of 58° (between the molecule symmetry axis and the IR polarization) and IR laser intensity of 0.07 a.u. at 800 nm. (b) $\mathrm{Err}(T)$ [defined in Eq. (8)] as a function of assumed compact supports.

Figure 3

Comparison between original (amplitude, blue; phase, black) and retrieved (amplitude, dashed red; phase, dash-dotted green) pulses, for the two polarizations of the HHG pulse generated from the ${\mathrm{CO}}_2$ molecule at an alignment of 58° (between the molecule symmetry axis and the IR polarization) and IR laser intensity of 0.07 a.u. at 800 nm. The spectral resolution of the data was $3.9\times {10}^{14}\mathrm{Hz}$. The lower figure shows the radiation in the IR polarization, and the upper shows the perpendicular polarization. The assumed pulse duration $T$ which minimizes $\mathrm{Err}(T)$ is 415 attoseconds (from $t=0$ to $t=415$), which corresponds to downsampling from $N=769$ to $N=250$.