In the realm of beam shaping, diffractive optical elements (DOEs) can manipulate the laser intensity distribution by altering the laser phase through microstructures. Beam shaping algorithms play a significant role in the design of diffractive optical elements. Specifically, the most representative is the Gerchberg-Saxton (GS) algorithm proposed by R.W. Gerchberg and W.O. Saxton. It involves an iterative process of performing a Fourier transform between the input plane and output plane, while simultaneously imposing known constraints on both planes. To enhance the convergence effect of the algorithm, an input-output algorithm and the phase-mixture algorithm have been developed based on the GS algorithm. In recent years, there have been advancements in mixed-region amplitude freedom algorithms, particularly those demonstrating superior convergence effects in the signal region, as well as offset mixed-region amplitude-freedom algorithms. Global optimization algorithms, such as the feedback GSGA (Gerchberg-Saxton genetic algorithm) and the last place elimination GSGA, have also gained attraction. These algorithms are derived from the GS algorithm. However, the phase complexity of these designs is high, presenting significant challenges to the physical processing of DOEs. Furthermore, as the number of limiting conditions increases, the computational time required also escalates, especially for the feedback GSGA and last place elimination GSGA.

To optimize time efficiency and reduce phase complexity, we discover that in conventional laser applications, the use of the Hankel transform is more effective than the Fourier transform in numerical calculations when both the incident beam and target beam exhibit circular symmetry. We introduce a beam shaping algorithm, the pQDHT-GS algorithm, for a circularly symmetric beam system based on the GS algorithm. The implementation process is based on the characteristic that the Hankel transform is solely related to the radial coordinate. This is achieved by iteratively alternating between the input plane and output plane to perform the Hankel transform. We employ a quadratic surface type phase as the initial phase, select a Gaussian beam with a full width at half maximum of 2 mm as the input light source, set an iteration time of 500, and use sample numbers of 512×512 (where the sample number in the pQDHT-GS algorithm is 1×256). We then apply this algorithm and the traditional GS algorithm to shape the incident beam into a circular Airy (CA) beam, Bessel beam, and Laguerre-Gaussian (LG) beam respectively. We compare the root mean square error and energy utilization efficiency of these two algorithms. Subsequently, we set the LG beam as the target beam, adjust the iteration times to 500 and 1000, and sample numbers to 512×512 and 1024×1024 respectively, to further analyze the computational performance of both algorithms. Furthermore, we evaluate the shaping performance of the pQDHT-GS algorithm by using the CA beam as the target beam in our experimental system.

In terms of the phase distribution of DOEs, the phase calculated by the GS algorithm exhibits rotational symmetry, and it contains some high-frequency components. In contrast, the phase of DOEs computed by the pQDHT-GS algorithm displays circular symmetry, simplifying the DOEs structure and reducing processing complexity. When examining the intensity distribution at the focal plane (output plane), both algorithms exhibit superior intensity profiles. However, compared to the results obtained from the GS algorithm, the shaping beam output calculated by the pQDHT-GS algorithm is smoother (Fig. 3). In identical conditions, the pQDHT-GS algorithm achieves rapid convergence within fewer iterations and saves computational time (approximately 100 times) (Table 2). Furthermore, experimental results indicate that the shaping beam intensity distribution aligns closely with the target beam intensity distribution, with consistent light intensity curve trends. The shaping beam exhibits noticeable burrs, with a root mean square error of 0.545 and an energy utilization efficiency of 78.07%. After being filtered, these burrs are substantially reduced, and the light intensity curve distribution becomes smoother, exhibiting a root mean square error of 0.491 and an energy utilization efficiency of 78.14% (Fig. 6).

In this study, we introduce a beam-shaping algorithm based on the pQDHT proposed for the circularly symmetric beam system. This approach achieves the circular symmetry of the DOEs structure by substituting the Fourier transform in the conventional GS algorithm with the Hankel transform. We select the CA beam, the Bessel beam, and the LG beam as target beams. A numerical simulation method is employed to juxtapose and assess the performance of both the GS algorithm and the pQDHT-GS algorithm in terms of shaping outcomes. Our findings indicate that, in comparison to the GS algorithm, the pQDHT-GS algorithm converges rapidly with fewer iterations. Moreover, it refines the intensity of the output-shaped beam, ensuring that the DOEs phase exhibit a circular symmetry distribution and thereby simplifying processing. Given that the pQDHT-GS algorithm requires significantly fewer sampling points than the GS algorithm, it significantly reduces matrix operation time, leading to nearly two orders of magnitude reduced computational time. Conclusive experiments on the CA beam validate the efficacy of this method. In conclusion, the pQDHT-GS algorithm exhibits rapid and precise capabilities in circular symmetric beam shaping, holding significant implications for the design and processing of DOEs. Its potential applications extend to various areas of beam shaping, including the choice of initial phase values through the integration of global search algorithms, deep learning, neural networks, and other intelligent algorithms. Furthermore, its utility is evident in designing phase plates within large aperture lasers. Future research will further explore this area.