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Introduction.– Vortex optical beams carrying OAM represent a powerful tool with diverse applications, ranging from high-capacity classical and quantum communication [1]–[8] to optical manipulation [9]–[12] and optical microscopy [13], [14]. While conventional optical components, such as digital micromirror device, holograms, Q-plates, and liquid-crystal spatial light modulators, [15]–[17], have been widely employed to generate these beams, integrated photonics devices have attracted great attention. The photonic chip offers a promising path toward scalable and robust vortex beam devices [18]. Engineered nanostructures, such as metasurfaces [19]–[21], plasmonic structures [22], and gratings [23]–[25] have been utilized to transform on-chip waveguide modes or resonant whispering gallery modes (WGMs) [26] into free-space vortex beams. Significant advancements have been achieved, including the demonstration of vortex lasers [27], entangled vortex photon pairs [28], and vortex microcombs [29], [30].

Despite these progresses, existing on-chip OAM beam emitters face critical limitations that hinder their practical applications. First, they rely on distributed scatterers with certain patterns for scattering input optical field, featuring fixed, complex nanostructures that are sensitive to fabrication imperfections and lack the capability for rapid reconfiguration. Second, generating and controlling arbitrary high-dimensional OAM superposition states, which is a key resource for advanced information technologies [3], [4], [6], remains a significant challenge. Beyond emitters dependent on linear scattering processes, recent studies have demonstrated that nonlinear optics effects, such as parametric \(\chi^{(2)}\) nonlinearity and Brillouin scattering, can induce the direct radiation of on-chip guided photons into free space by a uniform photonic structures [31], [32]. These nonlinear optical radiation (NOR) processes are controlled by ancillary optical or acoustic fields, thus provides opportunities for flexible reconfiguration of the radiation fields.

In this Letter, we introduce a fundamentally new mechanism for OAM beam generation based on Brillouin NOR in a grating-free microring resonator. In our system, a microwave-driven phononic mode directly stimulate the coupling between the confined optical field to free-space radiation, thereby imprinting the net OAM of the resonant modes onto the emitted beam. This acousto-optic interaction not only eliminates the need for any fixed nanostructures but also provides complete, real-time control over the output. The OAM order can be rapidly tuned by sweeping the microwave frequency, and arbitrary, high-dimensional OAM superpositions can be synthesized by simply programming the amplitudes and phases of the driving microwave fields. Our theoretical model, validated by numerical simulations, predicts a radiation efficiency exceeding 25% in experimentally feasible microcavities [33], [34] supporting confined phonon modes for Brillouin scattering [34]–[40]. This work establishes a new interface for chip-to-free-space optical conversions, paves the way for dynamically reconfigurable and programmable generation, manipulation, and detection of vortex beams by integrated photonic devices, thereby advancing high-capacity optical interconnects and quantum information processing.

Brillouin NOR for OAM radiation.– Figure 1 (a) illustrates an on-chip microring resonator that supports both photonic and phononic WGMs, with optical band photons and microwave band phonons simultaneously circulating along the waveguides. Due to the resonator’s rotational symmetry, these modes can be decomposed as \(d_j(r) = d_j(x,y) e^{i m_j\phi}\) and \(s_n(r) = s_n(x,y) e^{i M_n\phi}\), with \(d_j(x,y)\) and \(s_n(x,y)\) being the field distribution of the photonic and phononic mode in the cross-section, and \(\phi\) being the azimuth angle. The mode index \(m_{j}\) and \(M_n\) labels the azimuthal order and relates to the cavity radius \(R\) by \(m_{j} = k_{1,j} R\) for photons and \(M_{j} = k_{2,j} R\) for phonons. \(k_{1(2),j}\) are the wave vectors of the guided photons (phonons) in the waveguide, which depend on the speed of light and sound in solids. The well-defined azimuthal phase dependence \(e^{i m\phi}\) defines the characteristic OAM. The phononic modes are excited by applying a microwave signal to interdigital transducers (IDTs), which leverages the material’s piezoelectricity to drive a specific acoustic resonance.

Our proposal considers the Brillouin interaction, which couples these distinct modes and mediates their coupling to the free-space radiation continuum. This acousto-optic coupling induces a nonlinear polarization \(P_{NL}\propto d_{j}(r) P s_{n}(r)\) with \(P\) being the photoelastic coefficient, which acts as a source and leads to NOR to free-space [Fig. 1 (a)]. The Brillouin NOR is analogous to NOR from \(\chi^{(2)}\) nonlinearities [31], and exists ubiquitously between all resonant modes and free space continuum modes. There are two different types of radiation processes: a guided photon can either absorb a phonon and radiate a higher-frequency photon into free space mode (sum-frequency radiation, SFR), or it can stimulate the emission of a phonon and radiate a lower-frequency photon (difference-frequency radiation, DFR).

Efficient radiation requires the conservation of momentum parallel to the waveguide. As illustrated in Fig. 1 (b), this phase-matching condition is \(k_{1}\pm k_{2}=k_{r,||}\), with \(k_{1}\), \(k_{2}\), \(k_{r,||}\) being the wave vector components of the guided photon, phonon and radiated photon, respectively. Since the radiated photon must obey \(|k_{r,||}|\leq k_r\approx k_{1}/n_{\mathrm{eff}}\) (\(n_{\mathrm{eff}}\) is the mode’s effective refractive index), a stringent selection rule emerges. Of the four possible interaction configurations shown in Fig. 1 (c), two are forbidden by this condition: the DFR of counter-propagating modes and the SFR of co-propagating modes, as they would require \(|k_{r,||}|> k_r\). Consequently, only two processes lead to efficient radiation: SFR of counter-propagating modes and DFR of co-propagating modes. This imposes a requirement on the phonon wave vector: \[\left(1 - \frac{1}{n_\mathrm{eff}}\right) k_1 \lesssim k_2 \lesssim \left(1 + \frac{1}{n_\mathrm{eff}}\right) k_1. \label{eq:wavevector}\tag{1}\] This condition is critical, as it defines the necessary microwave frequency range, via the phononic dispersion \(k_{2}(\Omega)\) of the waveguide, to activate the Brillouin NOR.

The phase structure of the radiated field is a direct consequence of the interaction. The nonlinear polarization carries the combined phase of the interacting modes, \(P_{NL}\propto e^{im_j \phi}e^{\pm iM_n \phi}=e^{i(m_j\pm M_n)\phi}\), which is directly mapped onto the radiated field. By integrating the contributions from all plane waves, the far-field radiation pattern can be calculated from [41] \[\begin{align} E_{j,n}(r) \propto \alpha_{1}\alpha_{2} e^{i(m_{j}\pm M_{n})\phi} \int dr' d_{j}(r')Ps_{n}(r'), \label{eq:radiationfield} \end{align}\tag{2}\] where \(\alpha_{1,2}\) are the photon and phonon amplitudes, and the integral is over the whole cavity. Equation (2 ) shows that the resulting radiation is a vortex beam with a well-defined OAM order of \(m_{j}\pm M_{n}\). The sign depends on the radiation process (SFR or DFR). Therefore, by selecting the initial photonic mode (\(m_j\)) with a laser and the phononic mode (\(M_n\)) with a microwave signal, any desired OAM order can be generated.

OAM generation.–

To validate our theoretical model, we perform numerical simulations of the Brillouin NOR process in a microring resonator (see Supplemental Materials for geometric parameters [41]). We focus on the SFR [upper panel in Fig. 1 (c)] involving a clockwise (CW) photonic mode of order \(m\) and counter-clockwise (CCW) phononic mode of negative order \(-M\), which is predicted to generate a free-space beam with OAM order of \(\widetilde{M}=m-M\).

The calculated far-field patterns of \(\text{Re}[E]\) (\(100\,\mathrm{\mu m}\) above the microring with a radius of \(\SI{10}{\micro m}\)), shown in Fig. 2 (a) for \(\widetilde{M} = 0, \pm3, \pm5\), clearly exhibit the characteristic spiral phase fronts of vortex beams. The number of azimuthal phase cycles corresponds exactly to the OAM order \(|\widetilde{M}|\) and the handedness of the spiral matches the sign of \(\widetilde{M}\), confirming the successful transfer of OAM from cavity modes to the radiated field. The corresponding field distributions in both real space [Fig. 2 (b)] and momentum space [Fig. 2 (c)] display the distinctive annular, or "hollow" profile. The momentum is reflected by the angle \(\theta_{k}\) between \(k\) and the z axis. As expected for vortex beams, the radius of the central dark core expands with increasing OAM order \(|\widetilde{M}|\). This expansion is directly linked to the momentum conservation principle. As shown in Fig. 2 (d), the angle of maximum radiation intensity, \(\theta_{\text{max}}\), increases with \(|\widetilde{M}|\). A larger OAM order implies a greater momentum mismatch between the CW photonic and CCW phononic modes. To conserve momentum, this larger mismatch must be compensated by a larger transverse momentum component (\(k_{r,||}\)) for the radiated photon, resulting in a larger emission angle, as depicted in Fig. 1 (c).

A key advantage of our approach is its rapid reconfigurability. The OAM order is determined by the selected phononic mode, which is directly controlled by the microwave drive frequency. Because the phononic free spectral range in such cavities is typically in the MHz range, switching between adjacent OAM orders requires only a small MHz-level frequency shift. This allows for high-speed OAM reconfiguration solely through microwave control, eliminating the need for the complex, static nanostructures [24] and slow thermo-optic modulators [25] used in conventional devices.

The phase-matching condition in Eq. 1 also sets the operational frequency range and an upper limit on the achievable OAM order. Figure 2 (e) shows the relationship between the generated OAM order \(\widetilde{M}=m-M\) and the required phonon frequency for resonators of different radii. For low-order OAM generation, the required microwave frequencies are around half of the Brillouin scattering frequencies in materials. Here we use the parameters of lithium niobate for calculation, and the 4.5 GHz frequency is consistent with half of the scattering frequencies at 1550 nm wavelength [42]. For a microring resonator of radius \(R\), the maximum order of OAM beam at wavelength \(\lambda\) is fundamentally bounded by the ratio of the cavity circumference to the optical wavelength \[|\widetilde{M}|_{max} = \frac{2\pi R}{\lambda}. \label{max95order}\tag{3}\] For a typical \(50\,\mathrm{\mu m}\)-radius microring cavity, it allows for the generation of OAM beams with orders exceeding 200 at \(1550\) nm, demonstrating the potential for creating highly complex structured light.

Universal control of OAM superpositions.– A crucial requirement for advanced applications in coherent information processing is the ability to generate arbitrary OAM superposition states, \(|\Psi\rangle=\sum c_j |\widetilde{M}_j\rangle\), at a single optical frequency. For conventional on-chip devices, creating the complex, spatially-varying phase profiles required for such states is exceptionally challenging. In our Brillouin NOR scheme, however, the direct mapping between microwave frequency and OAM order provides an elegant and powerful solution.

The principle is illustrated in Fig. 3. To generate a superposition of OAM states \(\{\widetilde{M}_j\}\), we simultaneously excite a single photonic mode (\(m\)) and multiple phononic modes (one for each target OAM order, \(\{-M_j\}\)). This is achieved by driving the IDTs with a multi-tone microwave signal, where each frequency tone \(\Omega_j=\Omega_0+j\Delta\) excites a specific phononic mode \(M_j\). The complex amplitude of each OAM component in the final superposition is directly controlled by the amplitude \(\varepsilon_j\) and phase \(\theta_j\) of the corresponding microwave tone.

To ensure all radiated OAM components are frequency-degenerate, we must compensate for the different phonon frequencies. This is accomplished by preparing a corresponding multi-tone optical field to pump the photonic mode, where each optical tone \(\omega_j=\omega_0-j\Delta\) is set to satisfy the SFR condition \(\omega_j+\Omega_j=\omega_{out}\). The resulting field, before spectral filtering, is a sum of OAM states, each having its own optical frequency, and can be wirtten as \[|\Psi\rangle = \frac{1}{\mathcal{N}_0}\sum_{j}\varepsilon_{j}e^{i\theta_{j}}|\widetilde{M}_j\rangle_{\omega_{out}}+ \frac{1}{\mathcal{N}_1}\sum_{j}\sum_{j}\varepsilon_{j}e^{i\theta_{j}}|\widetilde{M}_j\rangle_{\omega_{out}-\Delta}+...,\] with \(\mathcal{N}\) being the normalization factor and \(\widetilde{M}_j = m-M_j\). By then using an optical filter or wavelength-division multiplexer to select the single frequency \(\omega_{out}\), we project the system into the desired pure OAM superposition state \(|\Psi\rangle = \frac{1}{\mathcal{N}} \sum_{j} \varepsilon_{j}e^{i\theta_{j}}|\widetilde{M}_j\rangle\).

The number of distinct OAM states that can be coherently superposed is ultimately limited by the need for all optical tones to address the same photonic resonance. This sets the achievable dimension of the superposition to be on the order of \(\kappa/\Delta\), where \(\kappa\) is the linewidth of the optical cavity. For typical cavities with radius of tens micrometers, this allows for the creation of superpositions with hundreds of dimensions.

Radiation efficiency.– In conventional on-chip OAM generators, the radiation efficiency is a static parameter determined by the fixed geometry of a scattering grating. In contrast, the efficiency of our grating-free Brillouin NOR scheme is a dynamic quantity that depends on the intracavity photon and phonon populations. This allows the efficiency to be dramatically enhanced by using high-quality (\(Q\)) factor microcavities, which confine both optical and acoustic energy in a small volume [43].

To quantify this, we define the efficiency \(\eta\) as the ratio of the radiation optical power to the input optical power. Figure 4 (a) shows the calculated efficiency as a function of the photonic (\(Q_1\)) and phononic (\(Q_2\)) quality factors, assuming critical coupling and fixed input optical power \(P_{\text{photon}}=1\,\text{mW}\) and microwave power \(P_{\text{phonon}}=100\,\mu \text{W}\). As expected from Eq. 2 , the efficiency initially is proportional to the product \(Q_{1}Q_{2}\) [Fig. 4 (a)], since higher Q leads to larger intracavity field amplitudes (\(\alpha_{1,2}\)) and thus stronger nonlinear radiation. However, as the Q increases further, the radiation itself opens a significant nonlinear loss channel for the cavity. This radiation loss begins to dominate the intrinsic and coupling losses, effectively moving the cavity into an under-coupled regime and causing the overall efficiency to saturate and then decrease, shown by the upper right corner of Fig. 4 (a). Remarkably, our model shows a radiation efficiency over \(25\%\) with realistic device parameters (phononic \(Q\approx5000\), photonic \(Q\approx10^6\)). In our calculations, the photoelastic coefficient are set to be the same as lithium niobate and the high radiation efficiency can also be achieved in integrated photonic platforms such as silica, silicon and gallium nitride.

To isolate the intrinsic radiative properties of the cavity from the pump powers, we define a pump-independent radiation rate \(\kappa_{2}\) of the cavity as \[\kappa_{2} = \frac{P_{\mathrm{rad}}/\hbar \omega }{|\alpha_1\alpha_2|^2}, \label{eq:disspationrate}\tag{4}\] which is determined solely by the geometry and material of the cavity. This rate is also derived in Ref. [41]. As shown in Fig. 4 (b), \(\kappa_{2}\) is nearly constant for all allowed OAM orders but is inversely proportional to the ring radius \(R\). Theoretical derivations indicate the scaling \(\kappa_{2}\propto R^{-1}\). This reveals a fundamental trade-off: smaller rings are more efficient radiators, but larger rings can support higher OAM orders, as shown in Fig. 2 (e).

The interplay between this intrinsic radiation rate \(\kappa_{2}\) and the waveguide-cavity coupling conditions can be captured in a compact expression for the efficiency [41] \[\eta = \frac{4 \kappa_{\text{nl}} \kappa_{\text{ex}}}{(\kappa_{\text{nl}} + \kappa_{\text{ex}} + \kappa_\text{int})^2}, \label{equ95P95kappa}\tag{5}\] where \(\kappa_{\text{nl}} = \kappa_{2} |\alpha_2|^2\) is the phonon-number-dependent nonlinear decay rate due to Brillouin NOR, while \(\kappa_\text{ex}\) and \(\kappa_\text{int}\) are the standard linear external coupling and intrinsic loss rates of the photon mode. For critically coupled microcavity (\(\kappa_{\text{ex}}=\kappa_{\text{int}}\)), \(\eta\) gets its maximum \(50\%\) at \(\kappa_{\text{nl}} = 2\kappa_\text{int}\). In the strong microwave drive regime, \(\eta\) decreases with \(\alpha_2\), where over-coupling condition \(\kappa_{\text{ex}}>\kappa_{\text{int}}\) should be utilized to increase the radiation efficiency. This model provides a clear roadmap for optimizing the device performance by engineering the cavity coupling and microwave drive power.

Discussion and conclusion.– In conclusion, we have introduced and validated Brillouin NOR for generating structured light. This approach leverages microwave-driven acoustic waves as a reconfigurable, dynamic grating within a simple microring resonator. It eliminates the reliance on fixed, nanofabricated structures and unlocks a suite of powerful capabilities. We have shown that the OAM of the emitted beam can be rapidly tuned by the microwave frequency. More profoundly, we have demonstrated that arbitrary, high-dimensional OAM superposition states can be synthesized at a single wavelength by simply programming the spectrum of the microwave drive. With a predicted efficiency exceeding 25% in standard photonic platforms, this method is not only versatile but also highly practical. Our work fundamentally shifts the paradigm of chip-to-free-space interfaces from static, lithographically-defined optics to dynamic, all-electronically controlled acousto-optic systems. This opens exciting new avenues for advanced applications in quantum communication, optical trapping, and sensing that require programmable, on-demand structured light. Furthermore, the reversal symmetry of this process suggests that our device could function not only as an emitter but also as a compact, mode-selective detector, enabling analysis of complex optical wavefronts.

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