Fig. 1. Generation of second harmonic waves (SHWs) with multiple angular momentum states through sequential optical spin–orbit interactions (SOIs) in two BBO crystals with threefold rotational symmetry. For a circularly polarized fundamental wave (FW) with angular momentum state of ${(\sigma \text{,}\mathcal{l})}_{\text{FW}}$, two states of the FWs will be generated in BBO1 through optical SOI process. Then the spin angular momentum state of the two states can be flipped by using an assembly of polarization optics (POs) after BBO1. Therefore, totally four kinds of angular momentum states of the FWs are prepared after the first crystal. The SHWs generated in BBO1 are blocked by using the long-pass filter (LPF). Then, by refocusing the FWs into the second BBO crystal, the SHWs with four corresponding angular momentum states are generated in the Direct channels. Through a sequential SOI process of the SHWs in the second BBO crystal, four new angular momentum states are obtained. It is expected that these eight kinds of angular momentum states of SHWs have nearly equal energies.

Fig. 2. The linear and nonlinear optical spin–orbit interactions (SOIs) in one BBO crystal. (a) Schematic diagram of the linear and nonlinear optical SOI processes in one BBO crystal. The BBO crystal with threefold rotational symmetry is placed at the center of a pair of 10× objective lenses to enhance the SOI effect. For a fundamental wave (FW) with angular momentum state of (σ, ℓ)_FW, the two output states of the SHWs come from the Direct and SOI channels are (−σ, 2ℓ)_SH and (σ, 2ℓ − 2σ)_SH, respectively. (b–e) Circular polarization resolved intensity profiles of the FWs (λ = 800 nm) after the BBO crystal. The incident FW is left- or right- circularly polarized (L/R) and the images for the L/R components of the output FWs are experimentally recorded. For each circular polarization combination, two images of the FWs are recorded by using a spherical lens (left panel) and a cylindrical lens (right panel). The dark core of the intensity profile imaged through the spherical lens is the typical characteristic of a vortex beam, and the value of the topological charge is revealed by the dark fringes in the one imaged via the cylindrical lens. (f–i) Circular polarization resolved intensity profiles of the second harmonic waves (λ = 400 nm) generated in the BBO crystal are also recorded. It should be noted that the SHWs measured here mainly come from the Direct FW channel, as that of the SOI-induced FW channel are very weak. The color of the SHW images is converted from the original blue to cyan for better visualization of the intensity patterns.

Fig. 3. The sequential optical spin–orbit interaction process of the fundamental waves (FWs) in the double BBO crystal system. (a) Schematic illustration of the double BBO crystal system which consists of two confocal optical sections. Each section includes a pair of 10× OLs (objective lenses) and a BBO crystal. LP (linear polarizer) and QWP (quarter-wave plate) are used to generate or analyze the circularly polarized FWs and second harmonic waves (SHWs). The combination of QWP2, LP2 and QWP3 is used to prepare the specific circular polarization component of the outgoing FWs from the first crystal (BBO1). LPF, long-pass filter, which is used to block the SHWs generated in BBO1. QWP4 and LP3 are used to analyze the circular polarization states of the SHWs generated in the second crystal (BBO2). L5, tube lens (spherical or cylindrical lens). (b–q) Intensity profiles of the FWs behind BBO2. The polarization labels above the results correspond to the sixteen combinations of the circular polarization states (L/R) of the FWs at the positions marked as POS-1 to POS-4 in (a). Compared to the results in (b) to (i), the ones in (j) to (q) are obtained under the conditions that the spin angular momentum states of the FWs between POS-2 and POS-3 are flipped from LCP to RCP, or from RCP to LCP by rotating the fast axis direction of QWP3.

Fig. 4. Angular momentum states of the second harmonic waves (SHWs) through the sequential spin–orbit interactions (SOIs) in the double BBO crystal system. The experimental setup is the same as the one shown in Fig. 3(a). (a–p) Sixteen cases correspond to full combinations of the circular polarization states of the fundamental waves (FWs) that are incident on and after BBO1, before BBO2 and that of the SHWs after BBO2 (corresponding positions are marked as POS-1 to POS-4 in Fig. 3(a)). When the angular momentum state of the input FW is LCP and Gaussian, i.e. (1, 0)_FW, the double crystal system can generate eight angular momentum states of the SHWs, as revealed in (a) to (d) and (i) to (l). The energies of the eight states are nearly the same as each other. While in (a) to (h) the spin angular momentum states of the FWs are kept unchanged, they are flipped from LCP to RCP, or from RCP to LCP in (i) to (p), which is realized by rotating the fast axis direction of QWP3. (q, r) The normalized relative power distributions of the angular momentum states of SHWs, where (q) and (r) correspond to the cases for input FW states of (1, 0)_FW and (−1, 0)_FW, respectively. SAM, spin angular momentum; OAM, orbital angular momentum.