Control of attosecond light polarization in two-color bicircular fields

We develop theoretically and confirm both numerically and experimentally a comprehensive analytical model which describes the propensity rules in the emission of circularly polarized high harmonics by systems driven by two-color counter-rotating fields, a fundamental and its second harmonic. We identify and confirm the three propensity rules responsible for the contrast between the $3N+1$ and $3N+2$ harmonic lines in the HHG spectra of noble gas atoms. We demonstrate how these rules depend on the laser parameters and how they can be used in the experiment to shape the polarization properties of the emitted attosecond pulses.

Figure 1

Numerical TDSE HHG spectrum of (a) helium and (b) neon. The inset in (a) shows the $y$ vs $x$ components of the driving electric field. For helium, the fundamental and second harmonic drivers have field strengths of $F_{\omega }=F_{2\omega }=0.056$ a.u. and duration of 12 fs. For neon, the field strengths of the drivers are $F_{\omega }=F_{2\omega }=0.073$ a.u. and their duration is 12 fs. The red lines indicate the ${\widehat{\mathbf{e}}}_{+}$ spherical component, which corresponds to light rotating with the fundamental driver (counterclockwise), while the blue line indicates the ${\widehat{\mathbf{e}}}_{-}$ spherical component, which corresponds to light rotating with the second harmonic (clockwise). Angular momentum conservation imposes that the $3N+1$ harmonic lines rotate with the fundamental field, the $3N+2$ lines rotate with the second harmonic, and the $3N$ lines are forbidden. The inset in (b) shows the cutoff harmonics for which the blue lines start to dominate in the spectrum of neon.

Figure 2

Schematic of the optical setup. The BBO (beta-barium-borate) crystal was used for the generation of the second harmonic of the fundamental beam, MCP denoted the detector for the XUV radiation, and BS is the beam splitter.

Figure 3

High harmonics generated in neon for different peak intensities of the fundamental $I_{\omega }$ and second harmonic $I_{2\omega }$ circularly polarized fields. Intensities (in units of ${10}^{14}{\mathrm{W}/\mathrm{cm}}^2$) are estimated from the pulse energy, focal diameter, and the cutoff position: (a) $I_{\omega }\sim 4$, $I_{2\omega }\sim 3.3$; (b) $I_{\omega }\sim 2.6$, $I_{2\omega }\sim 3.3$; (c) $I_{\omega }\sim 2.5$, $I_{2\omega }\sim 2.0$; (d) $I_{\omega }\sim 1.6$, $I_{2\omega }\sim 2.0$. The ratios of the intensities of the adjacent harmonics with orders $3N+1$ and $3N+2$ are marked in the figures near to the corresponding harmonic pairs.

Figure 4

High harmonics generated in neon for different peak intensities of the fundamental $I_{\omega }$ and second harmonic $I_{2\omega }$ fields. Intensities, in units of ${10}^{14}{\mathrm{W}/\mathrm{cm}}^2$, were estimated from the pulse energy, focal diameter, and the cutoff position: (a) $I_{\omega }\sim 1.2$, $I_{2\omega }\sim 0.7$; (b) $I_{\omega }\sim 1.2$, $I_{2\omega }\sim 2.8$; (c) $I_{\omega }\sim 1.7$, $I_{2\omega }\sim 0.9$; (d) $I_{\omega }\sim 1.7$, $I_{2\omega }\sim 2.0$. The ratios of the intensities of the adjacent harmonics with orders $3N+1$ and $3N+2$ are marked in the figure near the corresponding harmonic pairs.

Figure 5

Top: permitted dipole transitions in recombination to (a) $s$, (b) $p+$, and (c) $p-$ states. Thick arrows indicate those favored by the Fano-Bethe propensity rule. Center: recombination matrix element [Eq. (16)], in arbitrary units, for final (d) $s$, (e) $p+$, and (f) $p-$ states. Bottom: numerical TDSE high harmonic spectra when an electron is emitted from (g) $s$, (h) $p+$, and (i) $p-$ states. Colored lines indicate the helicity of the emitted light: red for ${\widehat{\mathbf{e}}}_{+}$ (counterclockwise) and blue for ${\widehat{\mathbf{e}}}_{+}$ (clockwise).

Figure 6

(a) SFA-calculated ionization factor, in arbitrary units, of electrons emitted from the $p+$ (magenta) and $p-$ (green) orbital of neon. (b) SFA-calculated square of the product of the ionization factor and recombination matrix element in the spectrum of neon separated in the two helical components (in arbitrary units), including (dashed lines) and neglecting (solid lines) the interference term in Eq. (17). (c) Spectrum of neon calculated solving the TDSE and separated in the two helical components [same as Fig. 1].

Figure 7

Control of helicity of attosecond pulses generated in helium, using the bicircular driving field. Each of the two columns represents a different intensity ratio between the fundamental and the second harmonic: $I_{\omega }/I_{2\omega }=1$ for panels (a),(c) and $I_{\omega }/I_{2\omega }=2.25$ for panels (b),(d). Top row: SFA-calculated recombination matrix element for helium, as a function of harmonic order, in arbitrary units. Center row: TDSE-calculated spectrum of helium in linear (top) and logarithmic (bottom) scale. Bottom row: attosecond pulse trains (APT) generated by taking an energy window from harmonic 9 to harmonic 42. The degree of circularity $e$ of the generated APTs is given (see text for details) and electric-field strengths are expressed in arbitrary units. The pulse parameters used in both the SFA and TDSE calculations are the same.

Figure 8

Top row: SFA-calculated matrix elements for recombination to the (a) $p+$ and (b) $p-$ orbitals of neon in arbitrary units. Bottom row: (c) SFA-calculated ionization factor of electrons emitted from the $p+$ (magenta) and $p-$ (green) orbitals of neon and (d) SFA-calculated product of the ionization and recombination dipoles in the spectrum of neon, in arbitrary units. For all panels, solid red (blue) lines indicate emission of light with ${\widehat{\mathbf{e}}}_{+}$ (${\widehat{\mathbf{e}}}_{-}$) helicity using a driver intensity ratio of $I_{\omega }/I_{2\omega }=0.5$. The dashed red (blue) lines indicate emission of light with ${\widehat{\mathbf{e}}}_{+}$ (${\widehat{\mathbf{e}}}_{-}$) helicity using a driver intensity ratio of $I_{\omega }/I_{2\omega }=1$.

Figure 9

Same as Fig. 8, but solid lines now indicate results using a driver intensity ratio of $I_{\omega }/I_{2\omega }=2$.

Figure 10

Control of helicity of attosecond pulses generated in neon, using the bicircular driving field. TDSE-calculated HHG spectra in neon (left panels, with linear scale in the top and logarithmic scale in the bottom) and APT obtained from the spectra (right panels) for three different driver intensity ratios: (a), (e) $I_{\omega }/I_{2\omega }=0.5$, (b),(f) $I_{\omega }/I_{2\omega }=1$, and (c), (g), (h) $I_{\omega }/I_{2\omega }=2$. The APTs in panels (e), (f), (g) were obtained by inverse Fourier transform of the plateau harmonics in panels (a), (b), (c), respectively, while the APT in panel (h) was obtained from the cutoff harmonics of panel (c). Electric-field strengths are expressed in arbitrary units. Panel (d) shows the ratio between the peak intensity of $H_{3N+1}$ and $H_{3N+2}$ for an $I_{\omega }/I_{2\omega }$ ratio of 0.5 (blue diamonds), 1 (green crosses), and 2 (red squares). A factor of 0.1 and 0.5 has been applied to $H_{13}/H_{14}$ and $H_{16}/H_{17}$, respectively, in the case of $I_{\omega }/I_{2\omega }=2$. The black line indicates equal intensity of $H_{3N+1}$ and $H_{3N+2}$ harmonics.