Orbital angular momentum (OAM) of photons, manifested by the phase singularities¹ and unlimited number of mutually orthogonal states, has been proved to hold great potentials in optical imaging^2,3, optical manipulations⁴⁻⁶, optical communications^7,8, information multiplexing⁹, and quantum information processing¹⁰⁻¹², etc. Methods on generating OAM-carrying vortex beams and relevant applications have been developed from traditional spiral phase plates^13,14 and liquid crystal based diffractive optical elements¹⁵ to spatial light modulators^16,17, optical metasurfaces¹⁸⁻²¹, photonic crystals²², and on-chip vortex lasers^23,24. Recent studies show that optical spin–orbit interaction (SOI)^25,26 represents a novel way for OAM generations by utilizing the mutual conversions between the spin angular momentum (SAM) and OAM of photons, via structured media^15,27 or anisotropic optical crystals^28,29.

In the meantime, nonlinear optical processes in materials will definitely introduce new degrees of freedom for manipulating the light fields and have been attracting scientists’ attention. Moreover, OAM-related structured light³⁰ in nonlinear optical regime has also been widely explored³¹, such as generation of spatiotemporal OAM light in SHG process³² and toroidal light pulses³³. Early works studying the conservation law of OAM in the second harmonic generation (SHG) process³⁴ are reported soon after the proposal of OAM of light. In recent years, nonlinear photonic metasurfaces³⁵⁻³⁷ and photonic crystals^38,39 provide new platforms for generating the OAM states of photons at the second and third harmonic frequencies. By virtue of the unlimited OAM modes of photons, a myriad of classical or quantum applications have entered into the high-dimensional realm⁴⁰⁻⁴². In addition, the spatiotemporal properties of the optical vortices are experimentally demonstrated in both linear and nonlinear optical regimes^30,32,43.

One interesting topic in the OAM community is to simultaneously generate the high dimensional OAM states of light. In linear optics, many kinds of artificial media, such as diffractive optical elements¹⁰ and optical metasurfaces^12,44, have been successfully utilized to realize this target. It should be noticed that a uniaxial optical crystal can be also used to produce two kinds of OAM states of light through the SOI process. The SOI conversion efficiency is up to 50% and no alignment to the singularity point is needed²⁹ as that in the conventional diffractive optical element. Previously, we show how to generate four angular momentum states in the SHG process, in which the β-BBO crystal is pumped by the circularly polarized fundamental wave (FW)⁴⁵. In such a strategy, two angular momentum states of the FWs coming from the linear SOI process are used to generate the corresponding angular momentum states of the SHWs. Then, the SHWs experience a cascaded SOI process, leading to four angular momentum states in total. Since the complex light–matter interactions occur in a single crystal, the conversion efficiencies of two angular momentum modes of the SHWs are much stronger than the other two, making it difficult to use all the angular momentum states.

To circumvent this constraint, here we propose the concept of sequential optical spin–orbit interactions to achieve eight angular momentum states of the SHWs with nearly equal conversion efficiency. As shown in Fig. 1, the experimental configuration consists of two sequential β-BBO crystals with threefold (C₃) rotational symmetry. In the first crystal, the linear SOI process leads to two angular momentum states of the FWs with almost the same energy. By flipping the SAM states of the FWs with polarization optics before the second crystal, we are able to obtain two new angular momentum states of the FWs. After being refocused into and passing through the second BBO crystal, the four FW states generate four angular momentum states of the SHWs (see Supplementary information Section 1). Further through a sequential linear SOI process, the SHWs carry eight angular momentum states with almost the same energy.

Firstly, the generation and spin–orbit interaction process of the SHWs in one BBO crystal is revisited⁴⁶. From the symmetry selection rules in nonlinear optics⁴⁷⁻⁵⁰, when a circularly polarized FW propagates along the rotational axis of a crystal possessing m-fold rotational symmetry, the allowed orders of harmonic generation are $n=pm\pm 1$, p is an integer, and the ‘+’ or ‘–’ sign means that the circular polarization state of the harmonic wave is same as or opposite to that of the FW. In this work, the FW is propagating along the C₃ rotational axis of the BBO crystal ($m=3$), therefore the SHW ($n=2$) with opposite circular polarization to that of the FW can be generated. In this process, two FW photons with SAM of $\sigma \hslash$ are annihilated to create a second harmonic photon with SAM of $-\sigma \hslash$, where $\sigma =\pm 1$ represents the left or right circular polarization states (LCP/RCP) of the FW, and $\hslash$ is the reduced Plank’s constant. The SAM difference of $-3\sigma \hslash$ before and after the SHG process is offset by the BBO crystal. In the meantime, if the FW carry an OAM of $\mathcal{l}\hslash$ per photon, the SHW photon should carry an OAM of $2\mathcal{l}\hslash$ as a result of the conservation law of OAM³⁴. To sum up, if we denote the SAM and OAM states of photons by $(\sigma \mathrm{,}\mathcal{l})$, the SHG process in the BBO crystal can be represented as ${(\sigma \mathrm{,}\mathcal{l})}_{\text{FW}}+{(\sigma \mathrm{,}\mathcal{l})}_{\text{FW}}\to {(-\sigma \mathrm{,2}\mathcal{l})}_{\text{SHG}}$.

When the FW is focused into the BBO crystal with C₃ rotational symmetry, both the FW and the SHW experience the SOI processes, where the SAM of light is flipped and its difference is imparted into the OAM of light. Such a process can be briefly represented as $(\sigma \mathrm{,}\mathcal{l})\to (-\sigma \mathrm{,}\mathcal{l}+2\sigma )$, from which one can easily see the conservation of the total angular momentum of light. Due to the SOI effect, the angular momentum state ${(\sigma \mathrm{,}\mathcal{l})}_{\text{FW}}$ of the input FW will be converted to two states, ${(\sigma \mathrm{,}\mathcal{l})}_{\text{FW}}$ and ${(-\sigma \mathrm{,}\mathcal{l}+2\sigma )}_{\text{FW}}$, which then generate the SHWs with angular momentum states of ${(-\sigma \mathrm{,2}\mathcal{l})}_{\text{SHG}}$ and ${(\sigma \mathrm{,2}\mathcal{l}+4\sigma )}_{\text{SHG}}$ in the BBO crystal. The generated SHWs in the same BBO crystal also experience a SOI process, forming four angular momentum states, namely ${(-\sigma \mathrm{,2}\mathcal{l})}_{\text{SHG}}$, ${(\sigma \mathrm{,2}\mathcal{l}-2\sigma )}_{\text{SHG}}$, ${(\sigma \mathrm{,2}\mathcal{l}+4\sigma )}_{\text{SHG}}$, and ${(-\sigma \mathrm{,2}\mathcal{l}+6\sigma )}_{\text{SHG}}$. For convenience, we denote these four angular momentum states as s₁ to s₄ in the following discussions. However, the last two angular momentum states s₃ and s₄ of the SHWs, which are expressed as ${(\sigma \mathrm{,2}\mathcal{l}+4\sigma )}_{\text{SHG}}$ and ${(-\sigma \mathrm{,2}\mathcal{l}+6\sigma )}_{\text{SHG}}$ respectively, have quite low efficiency.

Figure 2 shows the results of the single crystal case, in which the SHW from the SOI-induced FW is very weak⁴⁵. As shown in Fig. 2(a), we mainly consider the SHWs with angular momentum states of ${(-\sigma \mathrm{,2}\mathcal{l})}_{\text{SHG}}$, ${(\sigma \mathrm{,2}\mathcal{l}-2\sigma )}_{\text{SHG}}$. Then, we carry out the nonlinear optical experiments to verify our assumption. In the experiment, a femtosecond laser (pulse duration: ~140 fs, repetition frequency: 80 MHz) is used to pump the BBO crystal (see Methods and Supplementary information Section 2). The crystal is coated with 400 nm/800 nm anti-reflection layers on its two facets, therefore the preferred fundamental wavelength is set to be 800 nm. We focus on the case of a circularly polarized Gaussian FW with angular momentum state of ${(\pm \mathrm{1,0})}_{\text{FW}}$. The FW is focused into the BBO crystal by using an objective lens with numerical aperture of $NA=0.25$. Both the FW and the SHW behind the crystal are collected by the second objective lens with the same NA. When the input FW state is ${(\mathrm{1,0})}_{\text{FW}}$, the measured intensity profiles of the output FWs with LCP and RCP states are shown in Fig. 2(b, c), which are captured by using a spherical lens (left panel) and a cylindrical lens (right panel), respectively. While through the spherical lens images we are able to observe the doughnut shaped intensity profile of the vortex beam, the cylindrical lens images can be used to distinguish the sign and value of the OAM states of light⁵¹. From Fig. 2(b, c), we can see that the two angular momentum states ${(\mathrm{1,0})}_{\text{FW}}$ and ${(-\mathrm{1,2})}_{\text{FW}}$ of the output FW have comparable intensities. Similar results corresponding to input FW with angular momentum state ${(-\mathrm{1,0})}_{\text{FW}}$ are shown in Fig. 2(d, e), in which the output angular momentum states are ${(-\mathrm{1,0})}_{\text{FW}}$ and ${(\mathrm{1,}-2)}_{\text{FW}}$, respectively.

As shown in Fig. 2(f–i), the intensity profiles of the SHWs are obtained by filtering out the FW after the BBO crystal. For the FW with angular momentum state of ${(\mathrm{1,0})}_{\text{FW}}$, two SHW states of ${(-\mathrm{1,0})}_{\text{SHG}}$ and ${(\mathrm{1,}-2)}_{\text{SHG}}$ can be well observed (Fig. 2(f, g)), corresponding to states s₁ and s₂ in the SHG process. While s₁ is the direct harmonic generation from the input FW, s₂ is the linear SOI-induced state from s₁. The states s₃ and s₄ generated from the SOI-induced FW of ${(-\mathrm{1,2})}_{\text{FW}}$, are very weak and difficult to be measured. When the angular momentum state of the FW is switched to ${(-\mathrm{1,0})}_{\text{FW}}$, similar phenomena can be found in Fig. 2(h, i). These results are consistent with the previous ones discussed in ref.⁴⁵. It should be noted that the SHW at state s₂ has almost the same energy as that of s₁ (see Supplementary information Section 3).

Based on the above analysis, we propose to achieve the uniform energy distributions of the angular momentum states of the SHWs by using the sequential optical SOI processes in two BBO crystals. The first crystal and an assembly of polarization optics are utilized to prepare the four FW states with a near 50% SOI conversion efficiency, and the second crystal provides the channel for generating the SHWs as well as the SOI process. Figure 3(a) shows the experimental setup used to generate the eight SHW states with the sequential SOI processes. For a circularly polarized FW with Gaussian beam profile, the LCP and RCP components of the FWs with specific orbital angular momentum states are first prepared with BBO1, and then refocused into BBO2. To avoid the complex interferences of the SHWs generated from the two crystals, the SHWs generated from BBO1 are filtered out by using a long-pass filter.

The angular momentum states of the FWs after BBO2 are recorded and shown in Fig. 3(b–q) (see Supplementary information Section 4). For each circularly polarized FW state leaving from BBO1, as shown in Fig. 2(b–e), it experiences the linear SOI process and is partially converted to the one with opposite circular polarization. The circular polarization states of the FWs at the input, after BBO1, before and after BBO2 are labelled in Fig. 3(b–q). For the case of input FW state ${(\mathrm{1,0})}_{\text{FW}}$, the one focused into BBO2 is ${(\mathrm{1,0})}_{\text{FW}}$ when the ‘L_FW-L_FW’ polarization combination is chosen after BBO1 (Fig. 3(b, c)). The FW continues to experience a linear SOI process in BBO2, leading to two FW states, namely ${(\mathrm{1,0})}_{\text{FW}}$ in Fig. 3(b) and ${(-\mathrm{1,2})}_{\text{FW}}$ in Fig. 3(c). When the circular analyzer (QWP2 and LP2) after BBO1 is switched to the RCP condition (‘L_FW-R_FW’), we can extract the SOI-induced FW state, which is ${(-\mathrm{1,2})}_{\text{FW}}$. This state remains unchanged in BBO2, as shown in Fig. 3(d), or is converted back to an OAM-free state of ${(\mathrm{1,0})}_{\text{FW}}$ through one more linear SOI process (Fig. 3(e)). It should be noted that although the spherical lens image in Fig. 3(e) exhibits a ring-like pattern, the cylindrical lens image clearly shows that the FW carries no OAM. In addition, the experimental setup in Fig. 3(a) offers a convenient way to control the SAM state of the FWs after BBO1 by controlling the fast axis direction of QWP3. If the fast axis of QWP3 is rotated by an angle of ${90}^{\text{o}}$, the SAM state of the FW before BBO2 can be flipped from LCP to RCP (‘L_FW-L_FW-R_FW’ as in Fig. 3(j–k)), or from RCP to LCP (‘L_FW-R_FW-L_FW’ as in Fig. 3(l–m)), therefore leading to four new angular momentum states of the FWs after BBO2, thanks to the SOI process of the FWs in BBO2. Similar results can also be observed with the input FW state of ${(-\mathrm{1,0})}_{\text{FW}}$, as shown in Fig. 3(f–i) and Fig. 3(n–q).

Figure 4 shows the angular momentum states of the SHWs observed after BBO2 for FW states of ${(\mathrm{1,0})}_{\text{FW}}$ and ${(-\mathrm{1,0})}_{\text{FW}}$. From the knowledge learnt in Fig. 2, we will focus on the efficient generation of SHWs from the Direct FW channel. In the case of ${(\mathrm{1,0})}_{\text{FW}}$, the four angular momentum states (Fig. 3(b, d, j, l)) of the Direct FWs prepared with BBO1 have almost the same energy. The corresponding four Direct-generated SHW states (Fig. 4(a, c, i, k)) in BBO2 have similar nonlinear conversion efficiencies. In the meantime, another four angular momentum states (Fig. 4(b, d, j, l)) of the SHWs are generated through the linear SOI process of the SHWs in BBO2. From the spherical and cylindrical lens images shown in Fig. 4(a–d), one can easily distinguish the angular momentum states of the SHWs, which are ${(-\mathrm{1,0})}_{\text{SHG}}$, ${(\mathrm{1,}-2)}_{\text{SHG}}$, ${(\mathrm{1,4})}_{\text{SHG}}$ and ${(-\mathrm{1,6})}_{\text{SHG}}$, corresponding to the s₁ to s₄ states discussed in previous section. It should be noted that all the generated angular momentum states here originate from the Direct FWs. More importantly, the intensities of these four states are almost equally distributed, with a standard deviation of ~1.2% as evaluated from the SHW spectra recorded by using a spectrometer (see Supplementary information Section 5). These results verify that the proposed sequential SOI process can greatly improve the efficiencies of the SHW states which are very weak in the single crystal scheme⁴⁵. By flipping the SAM states of the FWs before BBO2, as we did in Fig. 3(j–m), the Direct generated (Fig. 4(i, k)) and SOI-induced (Fig. 4(j, l)) SHWs in BBO2 carry another four angular momentum states, namely ${(\mathrm{1,0})}_{\text{SHG}}$, ${(-\mathrm{1,2})}_{\text{SHG}}$, ${(\mathrm{1,2})}_{\text{SHG}}$ and ${(-\mathrm{1,4})}_{\text{SHG}}$, corresponding to ${(\sigma \mathrm{,2}\mathcal{l})}_{\text{SHG}}$, ${(-\sigma \mathrm{,2}\mathcal{l}+2\sigma )}_{\text{SHG}}$, ${(\sigma \mathrm{,2}\mathcal{l}+2\sigma )}_{\text{SHG}}$, and ${(-\sigma \mathrm{,2}\mathcal{l}+4\sigma )}_{\text{SHG}}$, respectively. These states are also experimentally observed with uniform intensity distribution. Similar phenomena can be found in Fig. 4(e–h) and Fig. 4(m–p) when the angular momentum state of the FW is switched to ${(-\mathrm{1,0})}_{\text{FW}}$.

Since the four kinds of angular momentum states of FWs prepared before BBO2 have covered all the situations in this sequential crystal system, the angular momentum states of the generated SHWs are limited to eight types. It can be regarded as a “foldable” process, which means that the type of the angular momentum states of SHWs as well as their energy equality will not be affected by the number of sequential crystals. However, the amplitude and phase of the generated SHWs in each crystal may change with the number of the sequential crystals, which will affect the total efficiency. This method can be extended to the field of high-order harmonic generations. According to the symmetry selection rules of harmonic generation in nonlinear optics⁵⁰, under the incidence of FW with angular momentum state of ${(\sigma \mathrm{,}\mathcal{l})}_{\text{FW}}$, the allowed harmonic orders in the c-cut BBO crystal are $n=3p\pm 1$. The angular momentum states of n^th-order harmonic waves are ${(+\sigma \mathrm{,}n\mathcal{l})}_{\text{HHG}}$ and ${(-\sigma \mathrm{,}n\mathcal{l})}_{\text{HHG}}$, corresponding to the cases where the circular polarization state of the harmonic wave is the same as or opposite to that of the FW. In the sequential process, the prepared angular momentum states of the FWs through the SOI process in BBO1 are ${(\sigma \mathrm{,}\mathcal{l})}_{\text{FW}}$ and ${(-\sigma \mathrm{,}\mathcal{l}+2\sigma )}_{\text{FW}}$. In the case where the circular polarization state of the harmonic wave is the same as that of the FW, the generated angular momentum states of n^th-order harmonic waves in BBO2 are ${(+\sigma \mathrm{,}n\mathcal{l})}_{\text{HHG}}$ and ${(-\sigma \mathrm{,}n\mathcal{l}+2n\sigma )}_{\text{HHG}}$. Through the SOI process in BBO2, two new states of n^th-order harmonic waves are generated, which are ${(-\sigma \mathrm{,}n\mathcal{l}+2\sigma )}_{\text{HHG}}$ and ${(+\sigma \mathrm{,}n\mathcal{l}+2n\sigma -2\sigma )}_{\text{HHG}}$. Similarly, in the case where the circular polarization state of the harmonic wave is opposite to that of the FW, four kinds of the generated angular momentum states of n^th-order harmonic waves in BBO2 are ${(-\sigma \mathrm{,}n\mathcal{l})}_{\text{HHG}}$, ${(+\sigma \mathrm{,}n\mathcal{l}+2n\sigma )}_{\text{HHG}}$, ${(+\sigma \mathrm{,}n\mathcal{l}-2\sigma )}_{\text{HHG}}$ and ${(-\sigma \mathrm{,}n\mathcal{l}+2n\sigma +2\sigma )}_{\text{HHG}}$ in total. In addition, if the SAM flipping of the FW states prepared after BBO1 is considered, four new states of n^th-order harmonic waves will be generated.

In summary, we propose a strategy to generate eight angular momentum states of SHWs through a sequential optical SOI in two uniaxial BBO crystals which have threefold rotational symmetry. With one input angular momentum state of the fundamental wave, one can prepare four kinds of FWs through linear optical SOI in the first BBO crystal and the manipulation of the SAM with a quarter-wave plate. Then, the FWs will be used to generate four angular momentum states of the SHWs in the second BBO crystal. As the SHWs also experience the linear optical SOI in the same crystal with a conversion efficiency close to 50%, we are able to generate eight angular momentum states of the SHWs with the double crystal system. These results are theoretically predicted and successfully verified in the experiment. The intensities of the eight SHW angular momentum states are uniformly distributed. It should be noted that the generated angular momentum states of SHWs by this approach are fixed. To realize flexible manipulation on the angular momentum states, it can be combined with conventional diffractive optical elements or spatial light modulators. The sequential optical SOI in the double crystal system can be also used to improve the energy distributions of the photon-pairs with desired angular momentum states of light in the spontaneous down conversion⁵², and similar concept can be extended to four-wave mixing and THz wave generation processes. This strategy is also applicative for other uniaxial nonlinear crystals. With the strong ability to prepare multiple angular momentum states, the proposed method may find more applications in optical communication, optical information processing and optical storage, etc.

The sequential optical spin–orbit interaction process in the double BBO crystal system is investigated by using a home-made nonlinear optical system. The fundamental waves (FWs) with a wavelength 800 nm are from a femtosecond laser (repetition rate 80 MHz, pulse duration ~ 140 fs), whose polarization states are controlled by using a linear polarizer and a quarter-wave plate. After passing through an objective lens ($NA=0.25$), the focused FWs are normally incident into the first BBO crystal (size: 6 mm × 6 mm, thickness 5 mm). The FWs and the generated second harmonic waves (SHWs) behind the first BBO crystal, which are collected by an objective lens ($NA=0.25$) and circular polarization resolved by using a quarter-wave plate and a linear polarizer, are projected onto a camera by a tube lens (spherical or cylindrical lens). By removing the flip mirror (see Supplementary information Fig. S2), the polarization controlled FWs behind the first BBO crystal are subsequently focused into the second BBO crystal (size: 6 mm × 6 mm, thickness 5 mm) by an objective lens ($NA=0.25$). The SHWs are collected by an objective lens ($NA=0.25$) and resolved by using the polarization optics, which consist of a quarter-wave plate and a linear polarizer. Finally, the FWs and SHWs behind the second BBO crystal are projected by a tube lens (spherical or cylindrical lens) and captured by using a camera. To evaluate the energy distributions of different angular momentum states of linear and nonlinear optical waves, the quantitative characterization is also conducted. The power of the FWs is measured by using a power meter. The SHWs are captured by using an Andor spectrometer (SR193i), then by integrating the SHG spectra the relative power of the SHWs are obtained. It should be noted that the selection of objective lens with numerical aperture $NA=0.25$ and thick β-BBO crystals is critical for the experiments, which has been investigated in ref.⁴⁵ and proved to be important for obtaining the angular momentum states of FWs with almost equal energy. In addition, the SAM states and the symmetry of intensity distribution of the generated SHWs are very sensitive to the angle between the incident light and the optical axis of the crystal, therefore, all the optical elements are finely adjusted to make the FWs normally incident into the BBO crystal.