Recently, circularly polarized high harmonics (HHs) have attracted great attention from the viewpoint of their application to ultrafast spectroscopy of magnetic materials and chiral molecules. However, circularly polarized HHs from gaseous media require a complicated setup due to strong constraints upon the high-harmonic generation (HHG) mechanism. HHG from solids is a new approach to this problem. Because the mechanism of solid HHG is intrinsically different from that of gas HHG, HHs from solids can be generated under single-color circularly polarized excitation. Their polarization states have been predicted to obey simple selection rules depending on the symmetry of a crystalline solid. In this Letter, we report on an experimental demonstration of circular HHG from solids that is fully consistent with the predicted selection rules. We irradiate circularly polarized mid-infrared pulses to a thin gallium selenide crystal and measure the spectrum and polarization of the emitted HHs. The threefold rotational symmetry of the crystal causes extinction of every third-harmonic order, as well as the generation of counter-rotating circularly polarized HHs. This result paves the way toward compact circularly polarized sources covering the spectra from the infrared to the extreme ultraviolet region.
© 2017 Optical Society of America
Circularly polarized high-harmonic generation (HHG) is recognized as a promising tool for investigating femto- to attosecond dynamics in magnetic materials and chiral molecules. By using rare gas atoms as target media, circular HHG has been realized with counter-rotating two-color [1,2] or counter-rotating noncollinearly overlapped  beams. Magnetic circular dichroism measurement with these circular high harmonics (HHs) in the extreme ultraviolet (XUV) has been also demonstrated, showing a great potential for probing ultrafast spin dynamics . However, in the case of HHG in gas-phase atoms, a single-color circularly polarized driving field alone cannot produce circularly polarized HHs, requiring a complicated experimental setup. This is because of the intrinsic HHG mechanism. According to the semiclassical three-step model of atomic HHG , when an atom is exposed to intense laser fields, a valence electron is tunnel ionized, and then accelerated by the electric field. The accelerated electron wave packet recombines with the parent ion, and a high-energy photon is emitted. If a single-color circular field is applied to an atom, the trajectory of the ionized electron deviates from the parent ion and the recombination is completely suppressed . Attempts have also been made to generate circularly polarized HHs by irradiating single-color linear or elliptical pulses to anisotropic molecules, but these techniques require either alignment of molecules or fine adjustment of the driving ellipticity [6,7].
An alternative approach to circular HHG is to use solids exposed to intense laser fields. The mechanism of solid HHG has been shown to be significantly different from that of atomic HHG [8–12]. When intense laser fields are applied to solids, electrons in valence bands are excited to conduction bands via interband tunneling. The excited electrons are then accelerated in the conduction bands. Unlike continuum states in a gas atom, conduction bands in solids have a highly non-parabolic dispersion. Thus, the resulting intraband current produces HHs [8–10]. Interband polarization between multiple bands can also be considered as another source of HH radiation [11,12]. In such processes, it is not necessary for ionized electrons to return to the original position. Therefore, HHG is possible, even if one uses a single-color circular field. In fact, vacuum ultraviolet HHs have been observed by using a magnesium oxide crystal and circularly polarized infrared pulses .
The polarization state of HHs under single-color circular excitation has been predicted to obey simple selection rules originating from the rotational symmetry of an HHG medium [14,15]. For a material with -fold rotational symmetry, the selection rules are given by
Here, is the harmonic order. represents the state of circular polarization of the th harmonic, which is the same as or opposite to that of the fundamental field, respectively. By using Eq. (1), one can immediately see that the th harmonics are circularly polarized with , the th harmonics are also circularly polarized with , and the other harmonics are forbidden, as shown in Table 1. Intuitively, these rules can be interpreted as conservation of the spin angular momentum of light [16,17]. The left-hand side of Eq. (1) represents the sum of the angular spin momentum of the fundamental photons, and the first term on the right-hand side represents the spin angular momentum of the HH photon. The second term on the right-hand side can be regarded as the angular momentum transferred to the HHG medium with -fold rotational symmetry. Another interpretation of the selection rules in the temporal domain can also be provided (see Supplement 1). The selection rules have been derived theoretically by both classical tensor calculation  and quantum Floquet formulation . Experimentally, they have been verified in perturbative second- or third-harmonic generation from bulk crystals [16,17] and nanostructures [18,19]. However, the selection rules for higher-order harmonics have not been experimentally investigated with single-color circular excitation. Instead, the combination of counter-rotating fundamental and its second harmonic is used as a drive field in atomic HHG to satisfy the angular momentum conservation [1,2] as
Here, represents the number of second harmonic photons and the left-hand side is the total spin angular momentum of the fundamental photons. Equation (2) leads to the selection rule that is the same as the case of in Eq. (1).
In this Letter, we report on the first experimental observation of the selection rules for circular HHG in crystalline solids driven by a single-color circular field. We observe up to the eighth harmonic in a gallium selenide (GaSe) crystal with mid-infrared (MIR) pulses. The polarization states of these HHs are analyzed and confirmed to be circular, as predicted by the theoretical works [14,15], thereby demonstrating controllability in circularly polarized HHG from solids.
In the experiment, we use MIR pulses (50 μJ, 3.5 μm, 70 fs, 300 Hz) from a home-built KTA-based optical parametric amplifier pumped by Ti:sapphire chirped pulse amplifiers to generate HH radiation. Figure 1(a) shows a schematic of the experimental setup. The MIR pulses are filtered with two wire grid polarizers (extinction ratio at 3.5 μm is more than ) to adjust their pulse energy and to ensure linear polarization. The MIR pulses are then sent to a zero-order quarter-wave plate ( waves retardance over the whole MIR spectrum) and the polarization state is converted to either circular or linear polarization. These pulses are loosely focused with a lens () into a 30-μm-thick, -cut GaSe crystal (EKSMA optics). We use -type GaSe with a hexagonal layered structure with threefold rotational symmetry [i.e., in Eq. (1)]. The lattice structure of a GaSe monolayer is depicted in Fig. 1(b). The propagation axis is perpendicular to the layer plane. The maximum pulse energy at the sample is measured to be 24.3 μJ with a spot size of 370 μm ( diameter) by the knife-edge method, corresponding to electric field amplitudes in the crystal of 11.5 MV/cm for linear polarization and 8.1 MV/cm for circular polarization. The emitted HHs in the visible and infrared regions are measured using optical multichannel analyzers with a silicon array detector (USB4000, Ocean Optics) and an InGaAs array detector (NIRQuest512-2.5, Ocean Optics), respectively. In addition to the total HH spectra, we measure the polarization states of the HHs by using polarization analyzers that are optionally inserted in front of the fiber. When measuring the ellipticity of the HHs, a wire grid polarizer (extinction ratio is more than over the whole HH spectrum) is inserted and rotated to measure angular-dependent transmission. When measuring the rotational direction of the circular HHs, a quarter-wave Fresnel rhomb retarder and a wire grid polarizer are inserted to distinguish the left-circularly polarized (LCP) and right-circularly polarized (RCP) components of the HHs. We use a Fresnel rhomb because of its spectrally flat retardance ( waves over the whole HH spectrum). The fast axis of the Fresnel rhomb is fixed while the polarizer is rotated by with respect to the fast axis of the Fresnel rhomb. When the polarization is LCP or RCP, the transmission becomes maximum at or , respectively.
Figure 2 shows the measured HH spectra under linear excitation (blue curve) and circular excitation (red curve). The MIR pulse energy is set to be 23.0 μJ, yielding peak amplitudes of 11.2 MV/cm and 7.9 MV/cm for linear and circular excitation, respectively. In the case of linear excitation, the polarization direction of the MIR pulses is set along the axis in Fig. 1(b). Multi-octave HHs above the bandgap of GaSe (1.98 eV) are recorded for both linear and circular excitation. In the case of circular excitation, clear extinction of the third and sixth harmonics is observed while the yields of the other harmonics are comparable to linear excitation. This result is fully consistent with the expected selection rules. As for the ninth harmonic, weak radiation is observed for linear excitation, but not for circular excitation. This is also consistent with the selection rules but its experimental validity is not certain with the current signal-to-noise ratio.
In the next step, we analyze the polarization of the HHs under circular excitation with the same experimental conditions. Figure 3 represents the result of the ellipticity measurement. Every harmonic shows a substantially flat transmission with respect to the rotation of the polarizer, which clearly indicates the generation of circularly polarized HHs. We fit these results with a cosine function and obtain ellipticities higher than 0.9 for all harmonics. Deviation from perfect circular polarization might be due to a slight misalignment and imperfect polarization components in the setup. We also measure the rotational directions of the circular polarization. Figure 4 shows the measured LCP and RCP components of the HH spectrum. Here, the polarization of the fundamental beam is set to be RCP. The th harmonics are predominantly LCP (, 2, 3), whereas the th harmonics are RCP (, 2). For the second to seventh harmonics, the major polarization component is more than two orders of magnitude larger than the minor polarization component, and for the eighth harmonic the RCP component is below the noise level. These observations agree well with the selection rules.
Finally, we measure the pump intensity dependence of the total HH yield for a circularly polarized fundamental beam, as shown in Fig. 5. When the MIR intensity is low, the intensity dependence is well fitted by the perturbative scaling, where the th HH yield scales as the th power of the peak electric field (dotted lines in Fig. 4). However, when the MIR intensity is increased, the HH yields begin to saturate. This observation indicates the nonperturbative nature of solid HHG as shown in previous experiments with linear excitation [8,9]. Our obtained results show that the experimental condition employed in the above measurements (Figs. 2–4) is evidently in the nonperturbative regime. The observation of the selection rules, even in this regime, assures intrinsic coherence in the HHG process, thus implying the scalability of the selection rules to shorter-wavelength regions and higher orders.
The observed selection rules are, in principle, applicable to crystals not only with threefold rotational symmetry, such as GaSe, but also with other rotational symmetries, as shown in Table 1. Therefore, our observation leads to general and versatile techniques for producing circularly polarized HH fields in crystalline solids. By employing a short-wavelength driver and a wide-bandgap material, circularly polarized HHG in the XUV is possible, offering a compact, all-solid-state light source for magnetic circular dichroism or photoelectron circular dichroism measurement.
In conclusion, we experimentally demonstrate selection rules for HHG in solids irradiated by circularly polarized MIR pulses. By using a GaSe crystal with a threefold rotational symmetry, we observe extinction of the th harmonics and generation of the counter-rotating th harmonics. The robustness of the selection rules is also confirmed by measuring the intensity dependence of HHG. These results pave the way toward compact, circularly polarized XUV attosecond light sources for various applications.
Ministry of Education, Culture, Sports, Science and Technology (MEXT); Japan Society for the Promotion of Science (JSPS) (JP17H04816); Advanced Leading Graduate Course for Photon Science (ALPS).
See Supplement 1 for supporting content.
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