Observation of Rotational Doppler Shift for Harmonic Generation in Solids

Harmonic generation controlled by electric field and nonlinear media symmetries has attracted considerable attention. It can be applied to generate coherent circularly-polarized beams in the extreme-ultraviolet regime and measure chiral molecules [1] . The underlying principles of this harmonic generation can be explained by considering the Spin Angular Momentum (SAM) conservation between the electric field and the nonlinear media. For example, when the nonlinear media with the three-fold rotational symmetry is irradiated by the circularly polarized incident beam, the counter-rotating circularly polarized second harmonic is generated as shown in Fig. (a). The incident beam photons and the generated harmonic photon have 2ℏ and -ℏ SAMs, respectively. In this process, the media can receive 3ℏ SAMs as a torque, not excited by the SAM and remaining the ground state. As a result, the SAM conservation between the incident photons, the harmonic photon, and the nonlinear media is satisfied. This SAM transfer can be explained using the analogy with the Umklapp scattering of phonons with the translational symmetry [2] . Generally, the media with the n -fold rotational symmetry can receive SAMs with multiple n .