Vortex beams, as specialized laser beams carrying orbital angular momentum (OAM), are fundamentally characterized by their helical wavefront structure [1]. This unique property endows vortex beams with broad application prospects in various fields such as optical communication [2], optical remote sensing [3], optical tweezers [4], etc. Conventional vortex beams primarily include Laguerre-Gaussian vortex beams [1], Bessel-Gaussian vortex beams (BGVBs) [5], and perfect vortex beams [6] carrying the helical phase of exp(ilθ), as well as asymmetric power-exponent vortex beams with the phase of exp[i2πl(θ/2π)n] [7]. Here, l denotes the topological charge, θ represents the azimuthal angle, and n is the power exponent. It is noteworthy that the topological charge l in all the aforementioned vortex beams remains constant integer or fractional value. In 2023, a breakthrough by Zhang et al. [8] demonstrated a customized vortex beam, generated by substituting the conventional helical phase exp(ilθ) with a trigonometric phase exp(ilθsinθ). This design creates a continuously tunable non-uniform OAM spectrum varying from −l to + l within a single transverse beam profile and provides unprecedented flexibility for particle manipulation and introduces an additional degree of freedom for OAM mode multiplexing. Very recently, Chen et al. [9] further extended trigonometric vortex beams to grafted perfect vector vortex beams, thereby enhancing their encryption dimensionality and multiplexing capability.

In various applications of vortex beams, accurate recognition of their OAM modes is indispensable. Taking optical communication as an example, the topological charge of a vortex beam theoretically possesses an infinite value, which could provide OAM-based multiplexing systems with unbounded channel capacity. In practice, however, this capacity is constrained by the detection precision of OAM modes. Consequently, developing characterization techniques capable of precise and effective OAM measurements has become a key prerequisite for promoting the broad application of vortex beams and remains one of the hot topics of current research. The interference method is one of the earliest and most intuitive OAM recognition techniques. Its principle is to interfere the vortex beam under test with a reference beam, and directly determine the magnitude and sign of the topological charge based on the characteristics of the resulting interference fringes. For example, double-slit interference can convert the helical phase difference of a vortex beam into a transverse shift of the interference fringes [10], while Mach–Zehnder interferometer combined with rotatable Dove prisms further enhances the detection capability and resolution for higher-order topological charges [11]. Alternatively, the diffraction method relies on the modulation of the vortex beam's wavefront by optical elements to achieve OAM recognition. When a vortex beam passes through an annular aperture, it produces a characteristic array of bright spots that correlates with its topological charge [12]. Similarly, after passing through a cylindrical lens [13] or a tilted lens [14], multiple bright spots corresponding to different OAM modes appear in an orderly arrangement near the focal plane. To achieve parallel recognition of multiple OAM modes, the geometric coordinate transformation method has developed [15]. This approach employs diffractive optical elements—such as spatial light modulators (SLMs)—to map the OAM of vortex beams to distinct transverse positions in a Cartesian coordinate system, thus achieving spatial separation of OAM modes. However, the above OAM recognition techniques directly rely on the physical characteristics of the vortex beam, which not only requires the detection environment to be almost undisturbed, but also faces inherent limitations for further improvement of detection precision and accuracy.

In recent years, owing to advances in computational power and the development of machine learning theories and algorithms, neural networks have been widely applied across numerous fields for their remarkable learning and prediction capabilities. Against this backdrop, a new interdisciplinary direction—machine-learning-assisted high-precision detection—has emerged [16]. The fundamental unit of a neural network is the perceptron, whose structure mimics the behavior of biological neurons. A perceptron consists of multiple inputs and a single output, and its computation proceeds as follows: a linear operation y = ax + b is first performed (where a represents the weight and b denotes the bias), followed by a nonlinear transformation applied through an activation function. This nonlinearity enhances the network's ability to model complex tasks. By interconnecting a large number of such neurons according to specific topologies, a deep neural network is formed. However, deep neural networks typically have high computational complexity when processing high-dimensional features. For this purpose, researchers have developed various specialized network architectures to address issues in different application domains. Among them, the convolutional neural networks (CNNs) are particularly well-suited for image-related tasks, and their typical structures include convolutional layers, pooling layers, and fully connected layers. The convolutional layer is responsible for extracting local features, the pooling layer (e.g., max pooling or average pooling) reduces dimensionality and highlights salient features, and the fully connected layer ultimately maps the features to the target labels. Theoretically, employing CNNs enables the extraction of more subtle features from light field images, thereby creating the potential for high-precision and high-accuracy detection of OAM modes across various scenarios. For instance, Xu et al. [17] employed asymmetric Bessel beams with non-diffracting characteristics as OAM carriers and constructed a CNN for OAM mode recognition of these beams at the source plane and various propagation distances in free space. The results show that the recognition accuracy of this method is stable at over 90%, verifying the good adaptability and robustness of the proposed network to changes in propagation distance. In practical scenarios, vortex beams often encounter atmospheric turbulence during propagation, which significantly increases the difficulty of OAM detection. To address this, Wang et al. [18] designed a six-layer CNN model consisting of two convolutional layers, two pooling layers, and two fully connected layers. The results indicate that even after long-distance transmission under strong atmospheric turbulence, the model can still maintain an accuracy of over 96.25% in recognizing multiplexed OAM modes. In addition, for ocean turbulence, Li et al. [19] utilized a customized CNN model to perform OAM recognition on Laguerre-Gaussian vortex beams under strong ocean turbulence, achieving an accuracy of 98.67% for dual-OAM mode recognition. The above methods all use CNN to directly detect OAM modes of vortex beams. In fact, it can also be detected through indirect methods. For example, Wang et al. [20] placed a tilted lens after a vortex beam propagating through turbulence, converting the distorted wavefront into a light field with distinct fringe patterns. Finally, a CNN model was used to achieved an accuracy of over 89% under strong turbulence conditions. Similar indirect detection methods involve introducing the cross-spectral density distribution, which carries richer feature information in low-coherence scenarios [21]. This method can solve the problem of intensity information loss caused by diffraction effects in low-coherence fractional-order vortex beams, thereby improving the accuracy of CNN-based OAM mode recognition.

Although deep learning has been applied to OAM recognition, as mentioned earlier, its application scenarios are still relatively limited, mainly involving source fields, free-space propagation, and turbulent propagation. In practical vortex-beam-based optical communication systems, however, optical system incorporating lens are indispensable as receivers [22] and perform mode sorting [23]. Note that the focal plane of the lens corresponds to the Fourier plane of the optical field, where different OAM modes produce distinct, OAM-dependent intensity patterns that enhance mode separability and enable efficient feature extraction by CNN. Moreover, these focal-plane intensity distributions are relatively insensitive to propagation distance and alignment errors, thereby improving the robustness of the recognition system under practical experimental conditions. Given that OAM is conserved during the weak focusing through a lens, deep learning can be employed to accurately identify OAM modes at any transverse plane during this focusing process.

To address the abovementioned concerns, this work will investigate the generation, focusing, and propagation characteristics of cosine-phase BGVBs. Using the CNN GoogLeNet [24], high-precision and high-accurate recognition of OAM modes are performed on focused cosine-phase BGVBs, with accuracy and precision reaching 0.05 and 97.27%, respectively.