Twisting attosecond pulse trains by amplitude-polarization IR pulses

According to the definition of amplitude-polarized (AP) electric fields (see main text) and assuming a Gaussian-shaped envelope, the explicit expressions for the components of $\bm{E}(t)$, when its polarization direction is at an angle of $45^{\circ}$, are:

	$\displaystyle E_{x}(t)$	$\displaystyle=$	$\displaystyle E_{0}\;e^{-2\ln(2)\left(\frac{t+\tau/2}{\Delta t}\right)^{2}}\;\mathrm{cos}(\omega_{0}(t+\tau/2)+\phi)$		(1a)
	$\displaystyle E_{y}(t)$	$\displaystyle=$	$\displaystyle E_{0}\;e^{-2\ln(2)\left(\frac{t-\tau/2}{\Delta t}\right)^{2}}\;\mathrm{cos}(\omega_{0}(t-\tau/2)+\phi).$		(1b)

In the central temporal region, i.e. around $t=0$, the peak field amplitude ($\left|\bm{E}(t)\right|_{\mathrm{max}}=\left(\sqrt{E_{x}(t)^{2}+E_{y}(t)^{2}}\right)_{\mathrm{max}}$) for such an AP pulse results

$\displaystyle\left|\bm{E}(0)\right|_{\mathrm{max}}=\sqrt{2}E_{0}e^{-2\ln(2)\left(\frac{\tau/2}{\Delta t}\right)^{2}}.$

(2)

Therefore, when $\tau=\Delta t$, we have

$\displaystyle\left|\bm{E}(0)\right|_{\mathrm{max}}=\sqrt{2}E_{0}e^{-\frac{1}{2}\ln(2)}=E_{0}.$

(3)

In this section, we describe how the angle $\alpha$ between two adjacent “petals” in the Lissajous curves of the driving AP field, varies as a function of the temporal delay $\tau$. The importance of this angle lies in the fact that it represents the angular difference between the polarization directions of two adjacent attosecond pulses in the twisted APT, near the center of the temporal region.

Supplementary Figure 1 shows the Lissajous curve of the electric field of an AP pulse whose temporal FWHM is $\Delta t=2$ opt. cycles. In this figure, $\alpha$ corresponds to the angular separation between the black and blue dashed lines. The blue dashed line intersects the position of the maximum field amplitude at 45^∘ (global phase $\phi=0$) while the black dashed line intersects the position of the adjacent maximum. Mathematically, we can find the angle $\alpha$ after evaluating the field ${\bm{E}}(t)$ at $t=1/2$ opt. cycle, which corresponds to the temporal position of the petal indicated by the black dashed line in the figure, and taking the angular distance to the angle $\frac{\pi}{4}$, which gives the position of the central petal at $t=0$ :

$\displaystyle\alpha(t=1/2,\tau,\Delta t)$

$\displaystyle=$

$\displaystyle\mathrm{tan}^{-1}\left(\frac{E_{x}(1/2)}{E_{y}(1/2)}\;\right)-\frac{\pi}{4}\;=\;\mathrm{tan}^{-1}\left(\frac{e^{-2\ln(2)\left(\frac{1/2-\tau/2}{\Delta t}\right)^{2}}}{e^{-2\ln(2)\left(\frac{1/2+\tau/2}{\Delta t}\right)^{2}}}\;\right)-\frac{\pi}{4}.$

(4)

It is worth mentioning that, for ultra-short pulses in the sub-two-cycle regime, the temporal position at $t=1/2$ opt. cycle is not well described by $T_{0}\equiv 2\pi/\omega_{0}$ but by the principal period $T_{P}\equiv 2\pi/\omega_{P}$, where $\omega_{P}$ refers to the principal frequency (see Refs. [neyra2021principal, PhysRevResearch.4.033254] for definition and in-depth analysis of these concepts). It is this period (frequency) that makes it possible to correctly locate the position of the maxima field amplitude for such ultra-short pulses. Also, it should be noted that the angle $\alpha$ results independent of $\tau$ when this is an integer multiple of $T_{0}$. Here, the AP pulse has right or left polarization (see Main Text for more details).

From Eq. (4) we can extract the linear approximation of $\alpha(\tau,\Delta t)$, that can be expressed as $\alpha_{1}(\tau,\Delta t)=\frac{ln(2)\tau}{\Delta t^{2}}$. In Supplementary Figure 2 we show the angle $\alpha$ as a function of the delay $\tau$ for each of the two AP pulses considered in this work. Red (blue) circles represent the case of FWHM $\Delta t=2$ ($\Delta t=5$) opt. cycles, while the corresponding linear approximation can be seen in red (blue) triangles. As can be seen, $\alpha$ exhibits a linear-like behavior as a function of the delay $\tau$ in the case of $\Delta t=5$ opt. cycles. However, for $\Delta t=2$ opt. cycles the linearity holds, approximately, until $\tau\approx 2$ opt. cycles.

From the results of the time-frequency analysis using the Gabor transform (see the Methods Section in the Main Text), here we calculate the time-resolved recollision angle of ionized electrons [murakami2013high]. This quantity can be derived as

$$\Theta(\omega,t)=\mathrm{tan}^{-1}\left(\sqrt{\frac{G_{y}(\omega,t)}{G_{x}(\omega,t)}}\;\right),$$

(5)

where we restrict ourselves to the first quadrant, i.e. $\Theta(\omega,t)>0$. This angle determines the angle at which the electron recombines with its parent ion.

The complete wavelet analysis can be seen in Supplementary Figure 3, where we show the results for an AP pulse with a temporal FWHM of $\Delta t=5$ opt. cycles and a global phase $\phi=0$. In this figure, panels (a) and (b) depict the color maps of the Gabor transforms for the associated HHG spectrum, $G_{x}(\omega,t)$ and $G_{y}(\omega,t)$, while the color map in panel (c) indicates the corresponding value of $\Theta(\omega,t)$. It is worth mentioning that values of $\Theta(\omega,t)$ above, approximately, the 28th harmonic order, have no physical meaning since the emission probability is negligible (they are numerical artifacts that come from the computation of the expression in Eq. (5)). From panel (c), we can observe how is the variation of the emission angle between the different bursts of XUV radiation, which indicates the polarization angle of the different attosecond pulses in the APT. In agreement with what is seen in the Main Text, this angle goes from 0^∘ to 90^∘, indicating the rotation of the polarization of the AP pulse in the $x-y$ plane.

Supplementary Figure 4 shows the analysis analogous to the one performed in SM Figure 3, but for the case of an AP pulse with $\Delta t=2$ opt. cycles. In panel (c), it can be seen the variation of the emission angle between the different bursts of XUV radiation, or attosecond pulses, which, as expected, is greater than in the previous case.

Finally, in Supplementary Figure 5, we show the variation of the emission angle for a single burst which is generated, approximately, between 0 opt. cycles and 0.5 opt. cycles. From this figure, it can be observed the angle of the recombination for the “short” and “long” trajectories in a single burst, or in a half laser cycle. Between the 16th and the 20th harmonic order, the “short” trajectories recombine with an angle of 40^∘, while the “long” trajectories do so at an angle of 50^∘. Above the 20th harmonic order (cutoff region), the angle of recombination of both trajectories converges, approximately, to 45^∘.