As one type of non-diffracting beam, the Bessel beam can be applied in areas such as super-resolution microscopy, optical trapping, and metrology. Moreover, the infinite orthogonal modes of the Bessel beam make it a promising candidate for future high-capacity optical communication and quantum information processing. Therefore, researchers have made great efforts to develop on-chip vortex beam emitters, where the orbital quantum number can be introduced as a new degree of freedom for information transmission. The quality of the emitters can be assessed by factors such as mode purity, efficiency, high-order mode realization, bandwidth, and integrability. Among the previous works on on-chip vortex beam emitters, waveguide structures have difficulty producing high-order vortex beams, thus limiting the development of high-capacity orbital angular momentum (OAM) communication. Vortex beams generated by resonators typically exhibit high purity and efficiency, while their narrow bandwidths make it challenging to achieve wavelength and frequency division multiplexing. Additionally, on-chip holographic gratings, affected by diffraction effects, yield vortex beams with lower efficiency. Therefore, challenges persist in generating high-quality on-chip vortex beams. In this work, we propose an inverse design of a Bessel-Gaussian beam emitter based on an adjoint optimization method. After hundreds of iterations, we obtain the optimized structure with outstanding performance, which is confirmed through finite difference time domain (FDTD) simulations. The final on-chip emitter achieves a correlation of over 86% at a communication wavelength of 1550 nm. In addition, we analyze the stability of the on-chip emitter under fabrication errors, including over-etching, under-etching, and variations in the operating wavelength. This proposed on-chip emitter is expected to play an important role in optical communication and optical computing.

The design is conducted on a Silicon on Insulator (SOI) platform (Fig. 1), which takes advantage of the higher refractive index contrast of SOI for compact device dimensions and stable mode propagation. The structure consists of a pair of SOI waveguides and a square area of 5 μm×5 μm for inverse design, denoted by dashed lines. This design area includes two materials: silicon and silica. The operating wavelength is set to 1550 nm, and a second-order Bessel-Gaussian beam with a waist of 1.5 μm and a transverse wave number of 4×10⁴ rad/m is chosen as the emitted light. The waveguide’s cross-section has a width of 0.5 μm and a height of 0.22 μm. These parameters ensure that the waveguides can only support the fundamental transverse electric mode, namely TE₀ mode, thereby preventing interference from other modes. Next, we describe the process of generating the Bessel-Gaussian beam. The second-order Bessel-Gaussian beam is used as the source incident in the design area, and the interaction between light and materials in the design area causes the beam to couple from free space to the waveguides. The desired result is a total transformation of the Bessel-Gaussian beam into the fundamental TE mode. Two monitors are placed on the cross-sections of the two waveguides, respectively, which are used to calculate the transmittance of the fundamental TE mode. The figure of merit (FOM) in the optimization algorithm of inverse design is set as the sum of the normalized transmittance of the fundamental TE mode in both waveguides. Thus, the optimization target is to maximize FOM, with the FOM close to 1 indicating a high conversion efficiency from the Bessel-Gaussian beam to the two fundamental TE modes. Considering reciprocity, when we excite the fundamental TE mode simultaneously in both waveguides, the emitter can generate a Bessel-Gaussian beam above the design area. The entire optimization process consists of the following steps (Fig. 2). 1) Determine the initial structure of the design area, which is divided into 41 pixel×41 pixel. Each pixel is a small rectangular dielectric with dimensions of 125 nm in length and width, 220 nm in height. The dielectric constant ε(x′) of each pixel ranges from εSiO2 to εSi. 2) After the forward and adjoint simulation, the FOM will be calculated and the algorithm determines if the target is met. If not, calculate the gradient of the FOM and use the L-BFGSB method, based on the gradient, to update the dielectric constant distribution in the design area for the next iteration. 3) If the target FOM is met or convergence is achieved, discrete dielectric distribution in the design area will be binarized, which results in a region composed solely of silicon and silica. 4) Apply design for manufacturing (DFM) constraints to the optimized structure, removing parts that are smaller than the minimum manufacturable size. 5) Test the stability of this device and export the layout file.

After 349 iterations, the FOM converges to 0.7, which corresponds to a 70% conversion efficiency from the Bessel-Gaussian beam to the fundamental TE mode of the two waveguides (Fig. 3). To verify whether the structure meets the requirement of emitting a Bessel-Gaussian beam, we perform a normalized overlap integral with the theoretical second-order Bessel-Gaussian beam to calculate the correlation coefficient. The result of the overlap integral between the two electric fields is 0.8649, which demonstrates that after optimization, this beam is highly similar to the second-order Bessel-Gaussian beam (Fig. 3). Additionally, we analyze the effect of etching errors on the performance of the on-chip Bessel-Gaussian beam emitter. We also consider the deviations from the central wavelength, which may affect the propagating mode and the quality of the emitted Bessel-Gaussian beam (Fig. 4). In the cases of over-etching or under-etching, blue or red shifts occur in the correlation peaks, respectively, with the correlation at the central wavelength dropping to around 0.64. Despite the peak shifts, the device under all three etching conditions can maintain a correlation above 0.64 within a bandwidth of 30 nm.

Vortex beams have become a hot topic in the fields of optical computing and optical communication in recent years. The Bessel-Gaussian beam emitter is expected to emerge as a key research direction in the future, due to its compact size, feasibility for high-order orbital quantum numbers, and large bandwidth. The dual-port Bessel-Gaussian beam emitter proposed in this work leverages the reciprocity characteristics of the device and uses maximum conversion efficiency from the vortex beam to the dual-port TE₀ mode as the optimization goal for inverse design, which is innovative. Furthermore, a gradient descent algorithm is employed based on the adjoint method. The advantage of this machine learning algorithm is that the gradient of the FOM can be calculated with just two simulations per iteration, which greatly reduces the time and computing resources required for reverse design. Lastly, the inverse design structure optimized in this work achieves a second-order Bessel-Gaussian beam correlation of over 86%. The correlation of the outgoing beam remains above 0.64, from an under-etching of 7 nm to an over-etching of 9 nm, and is maintained above 0.64 within a 30 nm wavelength bandwidth.