Bright circularly polarized soft X-ray high harmonics for X-ray magnetic circular dichroism

Abstract

We demonstrate, to our knowledge, the first bright circularly polarized high-harmonic beams in the soft X-ray region of the electromagnetic spectrum, and use them to implement X-ray magnetic circular dichroism measurements in a tabletop-scale setup. Using counterrotating circularly polarized laser fields at 1.3 and 0.79 µm, we generate circularly polarized harmonics with photon energies exceeding 160 eV. The harmonic spectra emerge as a sequence of closely spaced pairs of left and right circularly polarized peaks, with energies determined by conservation of energy and spin angular momentum. We explain the single-atom and macroscopic physics by identifying the dominant electron quantum trajectories and optimal phase-matching conditions. The first advanced phase-matched propagation simulations for circularly polarized harmonics reveal the influence of the finite phase-matching temporal window on the spectrum, as well as the unique polarization-shaped attosecond pulse train. Finally, we use, to our knowledge, the first tabletop X-ray magnetic circular dichroism measurements at the N4,5 absorption edges of Gd to validate the high degree of circularity, brightness, and stability of this light source. These results demonstrate the feasibility of manipulating the polarization, spectrum, and temporal shape of high harmonics in the soft X-ray region by manipulating the driving laser waveform.

Keywords: X-rays; high harmonics generation; magnetic material; phase matching; ultrafast light science.

Fig. 1.

Experimental scheme. Bright, circularly polarized, soft X-ray beams were generated by focusing 0.79- and 1.3-µm counterrotating circularly polarized laser fields into a gas-filled waveguide; they are then used for XMCD measurements at the N_4,5 absorption edges of Gd as well as the M_2,3 absorption edge of Fe from an out-of-plane magnetized Gd/Fe multilayer sample. (Left Inset) Combined field of the two drivers.

Fig. 2.

Circularly polarized EUV and soft X-ray HHG. Experimental HHG spectra generated from Ar (A), Ne (B), and He (A and C) driven by counterrotating 0.79- and 1.3-µm laser fields. All spectra show a peak-pair structure, located at positions predicted by energy and spin angular momentum conservation (circles). The separation within each pair is ω₁ – ω₂, and different pairs are separated by ω₁ + ω_2.

Fig. 3.

Simulations of circularly polarized soft X-ray HHG from He. (A) Dominant electron trajectories for counterrotating lasers that result in the emission of a 100-eV photon. (B) Laser wavelengths of 1.3 and 0.78 µm generate an eightfold symmetric field, where the ionization (circles) and recombination (triangles) times for the trajectories in A are indicated. (C) Simulated HHG spectra after macroscopic propagation for counterrotating laser drivers with perfect circularity (${\mathit{ϵ}}_1=-1,{\mathit{ϵ}}_2=1$, cyan) and slight ellipticity (${\mathit{ϵ}}_1=-0.985,{\mathit{ϵ}}_2=0.985$, magenta). Additional peaks appear when a slight ellipticity is introduced. (D) Attosecond pulse trains [E_x(t), E_y(t), and E_total(t)] obtained by performing a Fourier transform of the magenta spectrum in C shows a short phase-matching temporal window limits bright HHG to 5-attosecond bursts, with 2.6 linearly polarized bursts per 1.3-µm cycle.

Fig. 4.

EUV and X-ray magnetic circular dichroism of Fe and Gd. (A and B) HHG spectra around the Fe M_2,3 and Gd N_4,5 edges, transmitted through a Gd/Fe multilayer as the magnetization direction is parallel (red) and antiparallel (blue) to the HHG propagation direction. Gray curves, transmission of Fe and Gd. (C and D) XMCD asymmetry of Fe and Gd, with opposite signs for left (green) and right (red) circularly polarized HHG demonstrating opposite circularity of adjacent harmonics. (E and F) Extracted MO absorption coefficients at the Fe M_2,3 and the Gd N_4,5 edges (after correcting for ellipticity) agree well with literature values (41, 43, 44) (see SI Text, XMCD Shows the Brightness and Stability of Soft X-ray Circular HHG, for details).

Fig. S1.

Combined electric field of counterrotating, bichromatic laser fields: (A) 0.78 μm and 1.3 μm; (B) 0.79 μm and 1.3 μm, over 10 cycles of the 1.3-μm field; (C) 0.79 μm and 1.3 μm, over 20 cycles of the 1.3-μm field.

Fig. S2.

Additional channels allowed by imperfect circularity of the drivers. (A) Magenta curve: experimental circular HHG spectrum from He. Circles: channels when zero wrong photons are absorbed. Squares: channels when one wrong photon is absorbed. Crosses: channels when two wrong photons are absorbed. (B) With perfectly circularly polarized driving beams, zero wrong photons are absorbed, and only the l = n ± 1 channels are allowed. (C–G) With slightly elliptically polarized laser drivers, one (C and D) and two (E and F) wrong photons can be absorbed, (C and E) l = n ± 1; (D and F) l = n ± 3; (G) l = n ± 5, where n and l are the number of photons of 0.79 µm and 1.3 µm used to generate a circularly polarized HHG photon. The side peaks of the HHG spectrum match well with the predicted positions of these new channels, validating this analysis.

Fig. S3.

Ellipticity of circularly polarized HHG. (A and B) Polarization analysis using the simple photon model for He phased-matched at low gas pressure (510 torr, EUV region) (A) and high gas pressure (970 torr, soft X-ray region) (B). Magenta curves: experimental HHG spectra. Blue dashed curves: simulated HHG spectra using the simple photon model. Brown curves: circularity [(I_RCP – I_LCP)/(I_RCP + I_LCP)] of the HHG, which decreases from ∼1 in the EUV region (A) to ∼0.6 in the soft X-ray region (B). (C and D) Polarization analysis of the simulated spectra presented in Fig. 3C when the drivers are slightly elliptical (C) and perfect circularly polarized (D). Red (blue) curves: RCP (LCP) components of the HHG spectra in Fig. 3C. Green dots: circularity of the harmonics calculated from (I_RCP – I_LCP)/(I_RCP + I_LCP).

Fig. S4.

Macroscopic simulations for circular HHG from He. Simulated HHG spectrum from He at pressures of 100 torr (A) and 300 torr (C), as well as the corresponding time-frequency analysis (B and D) for counterrotating driving wavelengths of 0.79 µm and 1.3 µm and ellipticities ${\mathit{ϵ}}_1=-0.985,{\mathit{ϵ}}_2=0.985$. When the pressure is increased, the phase-matching window is shortened, so the number of attosecond pulses emitted is reduced and, consequently, the harmonic bandwidth in C is broadened. The time-frequency analysis is performed for the LCP harmonics. RCP harmonics exhibit very similar structure. From the strong peaks in B and D, it is clear that short trajectories make the dominant contributions.

Fig. S5.

Single-atom HHG from He. (A) (Upper) HHG photon energy as a function of the electron travel time τ (in units of the period T = 2π/ω₀, ω₀ = 2πc/λ₀, where λ₀ is the smallest common multiple of the driving laser wavelengths, i.e., λ₀ = 3.9 µm). The electron travel time refers to the total time the electron spends in the continuum. (Lower) exp(ImS)/|τ|^3/2, which is proportional to the probability of the HHG process, as a function of the electron travel time, where S is the action. (B) Dominant trajectories for photon energies 100 eV and 160 eV. (C) Three example trajectories corresponding to a photon energy of 100 eV in the three maximum value regions at the bottom of A, which have travel times of 0.1104 T (blue, dominant), 0.3294 T (green), and 0.6997 T (red). (D) Ionization (circles) and recombination (triangles) time are denoted on the combined driving field corresponding to the three trajectories in C.

Fig. S6.

Flux characterization for EUV and soft X-ray circular HHG. Experimental circular HHG flux from Ar (A), Ne (B), and He (C and D). Note that the driving bichromatic laser field intensity used in C is higher than in D, which leads to a higher flux in C than in D. However, because the phase-matching window in C is shorter than in D, the harmonics of the former spectrum are broader and merge stronger than in the latter case.

Fig. S7.

The 4-min XMCD of Gd. (A) HHG spectra around the Gd N_4,5 edges, transmitted through a Gd/Fe multilayer as the magnetization direction is parallel (solid) and antiparallel (dashed) to the HHG propagation direction. The gray curve shows the Gd absorption edge. (B) XMCD asymmetry of Gd obtained by taking a 2-min single spectrum when the magnetization is parallel/antiparallel to the HHG propagation direction (14).

Fig. S8.

Absorption coefficient of Gd. The absorption coefficient we measured (blue, unaveraged single-spectrum data) agrees very well with the absorption coefficient from ref. (purple) and Prieto and colleagues (43, 44) after appropriate scaling (brown, yellow).