Controlling both the phase and orbital angular momentum of beam waves during optical frequency conversion has become one of the most active areas of research in nonlinear optics^[1–3]. Nonlinear photonic crystals, characterized by their spatially varying second-order nonlinear coefficients and periodic modulation along the direction of beam propagation, can satisfy quasi-phase-matching conditions, thereby enabling efficient optical frequency conversion^[4,5]. Additionally, modulating the nonlinear coefficients in the plane perpendicular to the beam propagation direction allows for the manipulation of the phase distribution, or wavefront, as well as the orbital angular momentum of the beam wave^[6–12]. Initially, due to limitations in the fabrication techniques for nonlinear photonic crystals, nonlinear coefficient modulation was typically restricted to one-dimensional or two-dimensional configurations, which constrained the achievable beam wave control functionalities. However, with the advent of femtosecond laser direct-write ferroelectric domain inversion technology^[13,14], three-dimensional nonlinear photonic crystals were successfully fabricated^[15,16]. Building upon these advancements, optical frequency conversion and beam wave control have seen rapid progress^[17–21].

Existing research on nonlinear beam wave control primarily focuses on converting incident Gaussian beam waves into new frequency spatially structured beams, such as vortex beams generated via second-harmonic generation^[22–24]. Nonlinear interference techniques are commonly employed, leveraging nonlinear diffraction and nonlinear fork-shaped gratings [as illustrated in Fig. 1(a)] to transform the incident fundamental Gaussian beam (${\omega }_1$) into multiple noncollinearly propagating second-harmonic (${\omega }_2=2{\omega }_1$) vortex beams. Vortex beams, which carry orbital angular momentum, have emerged as significant carriers for optical information processing^[25], garnering considerable attention in the field. Recent studies suggest that the conversion efficiency from Gaussian beams to second-harmonic vortex beams can be significantly enhanced by utilizing periodic fork-shaped structures^[2,26]. Furthermore, the introduction of a structural focusing factor based on nonlinear fork-shaped gratings enables the generation of perfect second-harmonic vortex beams^[27–29], where the divergence of the vortex beam remains invariant regardless of the topological charge. However, the reverse process—where incident vortex beam generates a second-harmonic Gaussian beam—has not been extensively investigated. In fact, the conversion of vortex beam to Gaussian beam is equally critical. By capitalizing on the stark differences in their beam intensity distributions, it is possible to realize an all-optical switch^[30], which could provide pivotal support for future all-optical networks and optical information processing.

This study investigates the physical process by which an incident fundamental vortex beam interacts with a fork-shaped nonlinear photonic crystal to generate a Gaussian second-harmonic beam. It demonstrates the conservation of optical orbital angular momentum during the second-harmonic generation process, particularly when both the incident beam and the nonlinear optical medium possess topological charges. As illustrated in Fig. 1(b), we designed and fabricated a three-layer fork-shaped nonlinear grating structure with distinct orientations, which enables the conversion of an incident fundamental vortex beam into second-harmonic Gaussian beams propagating along different directions. Furthermore, by varying the topological charge of the incident beam, the position of the second-harmonic Gaussian beam can be correspondingly adjusted. This approach offers significant flexibility in the manipulation of photonic orbital angular momentum through nonlinear frequency conversion.

The first layer’s fork-shaped grating was obtained using the computational nonlinear holography (CGH) method^[31]. Specifically, the structure was designed by allowing a plane (or Gaussian) polarization wave to interfere noncollinearly with a second-harmonic vortex beam carrying the topological charge. The calculated interference pattern was then binarized, enabling its fabrication through ferroelectric domain inversion. In fact, this structure can be described by the formula $f(r,\varphi )=|r|\cos (\varphi )/\mathrm{\Lambda }$, where $\mathrm{\Lambda }$ is the lateral structural period of the fork-shaped nonlinear grating phase hologram^[6]. After designing the first layer of the fork-shaped grating, the second and third layers were obtained by rotating the first layer by 120° and 240°, respectively.

In the experiment, the nonlinear fork-shaped grating was fabricated using femtosecond laser direct writing of ferroelectric domain inversion^[32]. The crystal employed was a $Z$-cut ${\mathrm{Sr}}_{0.61}{\mathrm{Ba}}_{0.39}{\mathrm{Nb}}_2{\mathrm{O}}_6$ (SBN) crystal with dimensions of $4\mathrm{mm}\times 5\mathrm{mm}\times 1\mathrm{mm}$. The setup for laser direct writing is illustrated in Fig. 2. The direct writing laser had a wavelength of 750 nm, a pulse duration of 180 fs, and a repetition rate of 76 MHz (Coherent MIRA 800). The output power of the laser was controlled through a half-wave plate and a polarizer, with the polarization direction of the processing laser aligned horizontally. A microscope objective with a numerical aperture of 0.65 was used to focus the laser onto the $Z$-cut SBN crystal, which was mounted on a Thorlabs three-dimensional motorized stage capable of moving along the $x$, $y$, and $z$ axes. A LabVIEW program controlled the optical shutter and the motorized stage, enabling the creation of photoinduced ferroelectric domain structures in the SBN crystal via a point-by-point technique with approximately uniform intervals. Specifically, the laser was focused on a designated point at a specified depth for 0.5 s. During exposure, the laser focus was moved downward along the $z$ axis at a speed of 0.01 m/s for a distance of 5 µm, resulting in the inversion of ferroelectric domains at the focal point. The 5 µm scanning depth balances domain uniformity and experimental boundaries. Shallower depths may induce incomplete domain inversion (affecting nonlinear optics), while excessive scanning depths reduce available layering capacity, since the femtosecond laser focus degradation in high-refractive-index crystals limits the maximum achievable domain inversion depth. Afterward, the shutter was closed, the sample was translated to the next position, and the beam switch was activated again, repeating the process to achieve further domain inversion. This procedure was repeated until the entire fork-shaped grating structure was fabricated.

The nonlinear photonic crystal is composed of three layers of fork-shaped gratings, each with topological charge numbers $l_c=1$, 2, and 3, respectively. The depths of these layers are set to 40, 80, and 120 µm. We begin by processing the deepest third layer of the fork-shaped grating with a writing power of 588 mW. The second and first layers are then processed using writing powers of 451 and 341 mW, respectively. The lateral period of all three fork-shaped structures is 3 µm, with each layer rotated by 120° relative to the preceding one. Each domain structure layer has dimensions of 60 µm ($x$), 60 µm ($y$), and 5 µm ($z$), requiring approximately 950 inverted domains per layer. The fabrication of a single domain involves a 0.5-s laser exposure (optical shutter activation) followed by 1.1-s stage repositioning, totaling 1.6 s per domain. This translates to 25–30 min per layer (950 domains) and over an hour for three-layer structures.

The three-dimensional nonlinear photonic crystal structure, as shown in Fig. 3, was fabricated, and its morphology was captured using a Čerenkov second-harmonic microscope^[33]. At the ferroelectric domain walls, the abrupt change in the second-order nonlinear coefficient leads to a higher intensity of the Čerenkov second-harmonic signal compared to the homogeneous domains. By scanning the Čerenkov second-harmonic signals at various positions within the crystal, a three-dimensional distribution map of the ferroelectric domains is generated. The imaging results reveal that the three-layer fork-shaped structures are highly uniform, exhibiting smooth curves.

The fundamental beam for the second-harmonic generation experiment was provided by a femtosecond fiber laser (Langyan Tech, ErFemto ProH) with a pulse duration of 130 fs, a repetition rate of 80 MHz, and a wavelength of 1560 nm. This laser beam was first incident on a liquid crystal spatial beam modulator (LC-SBM), which imparted the desired topological charge, and then focused using a $5\times$ microscope objective ($\mathrm{NA}=0.1$) onto the three-dimensional nonlinear photonic crystal. The focused spot size was approximately 100 µm, slightly larger than the diameter of the designed structure’s cross section (60 µm). In the experiment, the polarization direction of the incident beam was adjusted using a combination of a half-wave plate and a polarizer, while the power of the fundamental beam was controlled. Specifically, the fundamental beam propagated along the $z$ axis of the sample, with its polarization along the $x$ axis, and generated a second-harmonic beam through the nonlinear coefficient $d_{31}$. The second-harmonic beams were then projected onto a screen placed approximately 2 mm behind the sample. A short-pass filter was positioned behind the sample to block the transmission of the fundamental beam, and a CCD camera was used to directly record the spatial profile of the emitted second-harmonic beam. The power of the second-harmonic signal was measured using a power meter (PM100D, Thorlabs).

When the fundamental beam is incident along the $z$ axis to the central position of the fork-shaped nonlinear grating, and the three-layer fork-shaped structure is simultaneously illuminated, three pairs of second-harmonic output signals are generated, as shown in Fig. 4. To analyze the output of the second-harmonic beam in terms of its direction and shape, we consider the case shown in Fig. 4(a). In this scenario, the incident fundamental vortex beam carries a topological charge of $l_{\mathrm{FB}}=-0.5$. According to the conservation of orbital angular momentum during the second-harmonic process^[6,34], the topological charge of the second-harmonic beam satisfies the following condition: $$l_{\mathrm{SH}}=2l_{\mathrm{FB}}+m{l}_c,$$(1)where $l_{\mathrm{SH}}$ represents the topological charge carried by the second-harmonic beam, $l_{\mathrm{FB}}$ denotes the topological charge of the fundamental beam (for the case where the incident fundamental beam is a Gaussian beam, $l_{\mathrm{FB}}=0$), $m$ represents the order of nonlinear diffraction for the second-harmonic, with particular focus on the cases where $m=\pm 1$, and $l_c$ refers to the topological charge of the fork-shaped grating structure itself. According to Eq. (1), when $l_{\mathrm{FB}}=-0.5$, a second-harmonic Gaussian beam ($l_{\mathrm{SH}}=0$) is generated along the $m=1$ diffraction order of a fork-shaped nonlinear grating with $l_c=1$. In the experiment, the fork-shaped grating with $l_c=1$ is vertically oriented, so the corresponding second-harmonic Gaussian beam appears at the horizontal 3 o’clock position. When the topological charge of the incident vortex beam changes to $l_{\mathrm{FB}}=-1$, Eq. (1) predicts that a fork-shaped nonlinear grating with $l_c=2$ will output a second-harmonic Gaussian beam along the $m=1$ diffraction order. The grating with $l_c=2$ has been rotated 120° from the vertical orientation, and consequently, the observed second-harmonic Gaussian beam is also rotated counterclockwise by 120° compared to its horizontal position in Fig. 4(a), as shown in Fig. 4(b). Similarly, when $l_{\mathrm{FB}}=-1.5$, the second-harmonic Gaussian beam undergoes another 120° rotation, as depicted in Fig. 4(c). Figs. 4(d)–4(f) present the simulated far-field patterns of the second-harmonic beams obtained through the Fourier transform, showing excellent agreement between the experimental results and theoretical simulations.

When the topological charge of the incident fundamental beam changes the sign, i.e., for $l_{\mathrm{FB}}=0.5$, 1, and 1.5, the second-harmonic Gaussian beam appears at the $-1$ diffraction order, corresponding to $m=-1$ in Eq. (1). As a result, the position of the second-harmonic Gaussian beam is flipped 180° compared to the case where $l_{\mathrm{FB}}$ is negative, as demonstrated in the experimental results [Figs. 4(g)–4(i)] and theoretical simulations [Figs. 4(j)–4(l)].

From the experimental results and analysis above, it is evident that the conservation of optical angular momentum in laser frequency conversion processes can be leveraged for all-optical control. For example, we can monitor the output optical field of the second-harmonic signal, using the position of the second-harmonic Gaussian beam as an indicator for the opening and closing of the optical channel. Additionally, by analyzing the position of the second-harmonic Gaussian beam, we can infer the topological charge of the incident beam signal. The multilayer fork-shaped grating significantly enhances the accuracy of this inference.

We also investigated the relationship between the second-harmonic output power and the input power of the fundamental beam, with the experimental results shown in Fig. 5. The scattered points in the figure represent the experimental measurements, while the curve denotes the second-order fit. It is evident that the second-harmonic power exhibits a quadratic relationship with the fundamental beam power.

We have designed and fabricated a three-dimensional nonlinear photonic crystal to enable the control of optical orbital angular momentum during nonlinear frequency-conversion processes. Experimental results demonstrate that fork-shaped nonlinear gratings with different orientations can convert the incident vortex beam into second-harmonic Gaussian beams with distinct spatial distributions. Moreover, altering the topological charge of the incident vortex beam causes a corresponding change in the emission direction of the second-harmonic Gaussian beam. These findings are significant for advancing the understanding of optical orbital angular momentum and for the realization of all-optical switching based on orbital angular momentum.