Evidence of depolarization and ellipticity of high harmonics driven by ultrashort bichromatic circularly polarized fields

High harmonics generated by counter-rotating laser fields at the fundamental and second harmonic frequencies have raised important interest as a table-top source of circularly polarized ultrashort extreme-ultraviolet light. However, this emission has not yet been fully characterized: in particular it was assumed to be fully polarized, leading to an uncertainty on the effective harmonic ellipticity. Here we show, through simulations, that ultrashort driving fields and ultrafast medium ionization lead to a breaking of the dynamical symmetry of the interaction, and consequently to deviations from perfectly circular and fully polarized harmonics, already at the single-atom level. We perform the complete experimental characterization of the polarization state of high harmonics generated along that scheme, giving direct access to the ellipticity absolute value and sign, as well as the degree of polarization of individual harmonic orders. This study allows defining optimal generation conditions of fully circularly polarized harmonics for advanced studies of ultrafast dichroisms.

( , , ) s s s normalized Stokes parameters [2], it writes in the dipole F  from a synchrotron-based experiment used here as a reference [6]. The comparison of the two led to  I  LF ion fragment angular distribution [3,4]: The () I  polar dependence (integrated over  ) provides the  asymmetry parameter. If  is nonzero, the Fourier analysis of the () I  azimuthal dependence (integrated over ( , , ) s s s Stokes parameters is determined, one can express in an equivalent way the polarization state of the ionizing light in terms of the orientation  and signed ellipticity  of the polarization ellipse, and the degree of polarization P [2].

Supplementary Note 2: Molecular polarimetry measurements on harmonics H15 and H19
In this section, we present additional results obtained with the MP method, characterizing harmonics H15 -which starts building up when one of the ω and 2ω driving fields is not perfectly circularly polarized -and H19 which has a weak contribution in the APT. As displayed in Supplementary Fig. 3, the measured 3D MFPAD and product show that H19 behaves the same way as H16, as predicted for harmonics of the same group (here 3q+1) [7]. Besides, when not fully suppressed, H15 (3q) exhibits the same helicity as H16 (3q+1). This is in agreement with results reported previously for temporally overlapping pulses [7][8][9].

Supplementary Note 3: Calibration of the effects of the toroidal mirror on the XUV polarization
The polarization state of the XUV light is modified by the reflection on the gold-coated toroidal mirror (angle of incidence 78.5°, f = 60 cm) used to focus the beam into the COLTRIMS interaction region, where it induces photoionization (PI) of a gas phase target. It is thus necessary to fully characterize this reflection in order to recover the polarization state of the high harmonics composing the XUV attosecond pulse train right after generation. The XUV polarization state is described by the Stokes vectors XUV,PI measured by molecular polarimetry [2,10], and XUV,HHG the sought-for polarization state of the HHG emission. The gold mirror is described by a Mueller matrix Mirror [11] where and are the reflectivities in intensity of the s-and p-polarized components of the light, respectively, and is the dephasing between these components induced by a reflection on the mirror. To determine these parameters for each harmonic, we used molecular polarimetry to measure the ( 1 , 2 , 3 ) normalized Stokes vector of the reflected light originating from known, controlled, linearly polarized high harmonics generated in SF6 by a linearly polarized IR field for various polarization angles α in the [-90°, 90°] range, where α is defined relative to the mirror reference axis.
For this calibration study, (i) the 3 parameter was measured using dissociative photoionization (DPI) of the NO molecule with 45° sampling, providing also 1 and 2 , and (ii) the 1 and 2 parameters were additionally measured with 10° sampling by analyzing the photoelectron angular distribution in the laboratory frame for PI of helium, whose higher count rate allowed faster acquisitions. The values of the = / and parameters characterizing the mirror for harmonics H15 to H21 were successively derived by a fit of the α-dependence of the 1 (R) and 3 ( ) quantities, respectively, according to the relations: We illustrate the quality of the fits in Supplementary Figs. 4 and 5 which display respectively the evolution of the difference between the measured 1 parameter for the actual mirror and that of a perfect mirror 1 = 1 ( = 1), and the measured 3 as a function of the initial polarization direction α for harmonics 17, 19 and 21. The values of the R and δ parameters extracted from the corresponding fits are summarized in Supplementary Table 1 and compared with those of an ideal gold mirror [11]. The values of the dephasing are found of lower magnitude (by less than 10°) than the theoretical values. This difference might originate from a thin organic deposit at the surface of the gold mirror, modifying its refractive index. 8 This transmission function of the toroidal mirror is used throughout the work presented to derive the HHG reduced Stokes parameters from the values measured after the mirror using molecular polarimetry. For the even harmonics, such as H16, which are not generated with linearly polarized light, the parameters were extrapolated between those for H15 and H17.

Supplementary Note 4: Breaking the dynamical symmetry with short pulses
In the main text Fig. 1, we discuss the deviations from circularity and depolarization calculated for high harmonics generated in helium by 5 -cycle FWHM (≈ 13.3 fs) circularly-polarized driving pulses at I=I2=2 x 10 14 W/cm². The same calculations have been performed for 10 -cycle FWHM (≈ 26.6 fs) pulses at the same intensities. The corresponding ellipticity and degree of polarization are displayed in Supplementary Fig. 6. Harmonics 22 to 41 possess a high degree of polarization although not equal to 1 ( ≈ 0.9). This value is of the order of the degree of polarization found in the main text Fig. 1e for orders lower than 33 that are still narrower than the +/-0.25 order bandwidth. With 10 -cycle pulses, the harmonics' ellipticity is higher (in absolute value) compared to the 5 -cycle case. However, deviations from circularity due to the fast-varying envelope are still evidenced with | | ≈ 0.8 − 0.9 for the 3q+1 orders and | | ≈ 0.7 − 0.8 for the 3q+2, even for these 26.6-fs driving pulses. Given the slow evolution with increasing pulse duration, we expect that significant effects may still occur for much longer pulse durations (for which the computing time becomes prohibitive).
Considering the observed decrease of harmonic ellipticity when the pulses' durations shorten, we anticipate that harmonics generated with few-cycle pulses will strongly deviate from the perfect circularity. Obviously, in the extreme case where the HHG process is limited to 1/3 of  cycle, the emitted attosecond pulse is linearly polarized [12]. , and ε the field ellipticity, i = 1,2. is the carrier-envelope phase, acting on both 1 and 2 fields, whereas is a phenomenological phase term applied only on the IR field for simulating a dephasing between the two fields induced, e.g., by instabilities in the Mach-Zehnder interferometer used to split the beams. Figure 7: Influence of the relative phase on the total field. Lissajous curves of the bicircular field for two − 2 relative phases IR in the case of (ε  = +0.9, ε 2 = -1) driving fields with trapezoidal envelopes.

Supplementary
In the case of constant pulse envelope and perfect circularity of the drivers, a change in the − 2 relative phase IR (caused by, e.g., an instability of the two-color interferometer) simply rotates the total field rosette shape. The threefold dynamical symmetry therefore remains. When one of the drivers is elliptical, and a fortiori for short pulse envelopes, the dynamical symmetry is broken. The bicircular field evolution in the polarization plane then strongly depends on the − 2 relative phase IR . Supplementary Fig. 7 shows the Lissajous curves of the total field between = 0 and = 8 (end of the constant envelope region), for the trapezoidal envelopes used in the calculations of main text Figs. 2 and 3. The construction of the total field over the 2-cycle turn-on is significantly different for the two relative phases, which has strong consequences when ionization confines efficient harmonic emission to the first cycles, as is the case in argon at 1.2 x 10 14 W/cm².
The variations with IR of the ellipticity and degree of polarization for H16 and H17 generated in argon by such trapezoidal pulses are displayed in Supplementary Fig. 8. The ellipticity of both harmonics is strongly affected by a variation of IR. The degree of polarization of H17 is more sensitive than the one of H16, as generally observed all along this study. In experiments where this relative phase is not controlled, long acquisition times result in an averaging over all the phase values. The consequence is a strong decrease of the degree of polarization of H17 to 61%, that of H16 remaining high at 93% (see Supplementary Table 2). Figure 8: Influence of the -2 relative phase on the harmonic polarization. Degree of polarization (left) and ellipticity (right) of H16 and H17 calculated over the harmonic spectral width (+/-0.25 order) generated in argon by trapezoidal bichromatic counter rotating (ε  = +0.9, ε 2 = -1) pulses at an intensity I0=I=I2= 1.2 x 10 14 W/cm² as a function of the relative phase between the two pulses (see Supplementary Eq. 9). the variation of the H17/H16 ratio among the different effects mentioned above. Finally, we stress that discussing the ellipticity of the XUV emission only based on the relative spectral intensity of the components co-rotating or counter-rotating with the  field (as shown in main text Fig. 1d) may be misleading. Indeed, the degree of circular polarization, calculated from the intensities of the left-and right-circular components of the light, writes:

Supplementary
On the other hand, the ellipticity writes: If the light is fully polarized, Supplementary Eqs. (13) and (14) show that a 2-order-of-magnitude difference between the intensities of the left-and right-circular components of the light only gives = 0.8, which is far from full circularity. A 4-order-of-magnitude difference is necessary to achieve = 0.98.