Recently, solid-state high-harmonic generation (HHG) has been observed in diverse materials, including dielectric crystals, semiconductors, two-dimensional layered materials, strongly correlated systems, and topological insulators. Topological insulators have attracted significant attention in HHG and strong field-driven dynamics due to their topologically protected surface states. These surface states exhibit spin-momentum locking, leading to unique optical responses in HHG such as even-order harmonic generation, non-integer harmonics, and anomalous dependence of harmonic intensity on driving laser ellipticity. However, the polarization characteristics of HHG in topological insulators remain systematically unexplored. We investigate the polarization properties of HHG in Bi₂Se₃ crystals driven by linearly polarized intense laser fields. By varying the crystal azimuth angle (θ), we measure the evolution of the polarization state (polarization orientation angle α and ellipticity ε) of harmonics. The experiments reveal that HHG transitions from linear to elliptical polarization with rotating orientation angles as θ changes, which is particularly pronounced in even harmonics. Theoretically, using the tight-binding approximation and semiconductor Bloch equations (SBEs), we demonstrate that the polarization states of even-order harmonics are primarily governed by the phase of complex inter-surface-state transition dipoles. These findings enhance our understanding of HHG mechanisms in topological materials and suggest new approaches for all-optical control of harmonic polarization.

We employ a linearly polarized mid-infrared femtosecond laser (central wavelength 3.8 µm, pulse duration 60 fs, peak electric field 5.2 MV/cm corresponding to an intensity of 3.6×10¹⁰ W/cm²) incident on a Bi₂Se₃ crystal at ~5° angle to produce HHG (Fig. 1). High-harmonic emission is collected via a focusing lens and detected by a grating spectrometer. The polarization states of the HHG are measured using a Stokes parameter measurement device that includes a quarter waveplate (QWP) and a polarizer. Stokes parameter analysis determines harmonic orientation angle and ellipticity [Fig. 1(c)]. Theoretical simulations combine tight-binding models with semiconductor Bloch equations, which incorporates topological surface-state electronic structures to analyze how transition dipole phases and interband Berry phases modulate polarization.

By rotating the Bi₂Se₃ crystal (azimuth angle θ), we observe modulated polarization states in even-order harmonics (e.g., H6 and H8). Figure 3 shows the θ-dependent polarization modulation of the even-order harmonics (e.g., H6 and H8). As θ increases from 0° to 30°, the orientation angle α of H6 decreases linearly from 90° to near 0°, which indicates polarization rotation from parallel to perpendicular relative to the driving field [Fig. 2(a)]. The ellipticity ε exhibits periodic oscillations, peaking at θ=15° (ε=0.313), while maintaining linear polarization (ε≈0) at high-symmetry orientations (θ=0°, 30°, 60°) [Fig. 2(b)]. This behavior stems from the threefold rotational symmetry of the Bi₂Se₃ crystal: along the high-symmetry orientations (Γ-M and Γ-K), parallel/perpendicular harmonic polarization alignment is enforced by mirror symmetry, whereas ellipticity is induced from the symmetry orientations through transition dipole phase differences (Fig. 6). Theoretical simulations, considering the strong-field-driven dynamics in topological surface states, reproduce the experimental observations [Figs. 7(a) and 7 (b)]. Interband Berry phase analysis reveals that the elliptical polarization originates from the spectral phase differences (ΔΦ≠0 or π) between orthogonal harmonic components, determined by the quantum geometric phase during electron-hole recombination. These phase differences stem from the complex transition dipole moments at the instant of electron-hole quasiparticle recombination, fundamentally linking polarization ellipticity to quantum geometric properties. Additionally, the crystal’s threefold rotational symmetry (C3v) directly governs sinusoidal θ-dependent ellipticity oscillations in harmonics, which confirms symmetry-controlled polarization states (Fig. 4). Odd-order harmonics (e.g., H5, H7) exhibit polarization characteristics fundamentally distinct from their even-order counterparts. While H5 maintains near-parallel alignment relative to the driving field (maximum deviation is 16.3°), H7 undergoes polarization plane rotation from parallel (θ=0°) to perpendicular (θ=15°), followed by realignment to parallel (θ=30°) with increasing azimuth angle [Fig. 3(a)]. These harmonics display significantly lower maximum ellipticity (ε_max=0.176) compared to even-order harmonics, with only moderate θ-dependent variations [Fig. 3(b)]. This contrast originates from their divergent generation mechanisms: odd-order harmonics predominantly stem from bulk-band optical responses, whereas even-order harmonics arise from topological surface state dynamics. This difference in generation mechanisms emphasizes the key role of band topology in controlling HHG processes. Notably, the anomalous polarization rotation observed in H7 likely originates from hybridization effects between topologically protected surface states and trivial bulk bands.

Through combined experimental and theoretical studies, we systematically characterize the polarization properties of HHG from Bi₂Se₃ and elucidate the physical mechanism governing ellipticity in even-order harmonics. Our findings reveal that the polarization states of even-order harmonics are predominantly determined by the orthogonal projection of the complex transition dipole moments at the instant of electron-hole recombination. By rotating the crystal azimuth angle, we achieve effective modulation of harmonic polarization states. This work not only advances the understanding of HHG mechanisms in topological materials but also proposes a strategy for HHG polarization control. These findings provide insights into nonlinear optical responses in topological states and establish a foundation for future all-optical harmonic polarization control.