Objective In fields such as laser processing, surface ablation, and medical testing, flat-topped beams with uniform energy distribution are more valuable than Gaussian beams directly output by lasers. Diffractive optical elements (DOEs) are widely used as beam-shaping devices with the advantages of simple structure, high design degree of freedom, accessible mass production, and wide range of material selection. Differing from traditional refractive devices, DOEs use surface relief microstructures to adjust the amplitude or phase of the wavefront, thereby tuning the distribution of the output beam. DOEs have high requirements for the wavelength, waist width, eccentricity, beam quality, and other conditions of the input beam. When these requirements are not met, the output light spot will deviate from the design results, which affects actual use. Unlike other tolerance constraints, the effect of beam quality on DOEs cannot be directly calculated from the diffraction propagation of coherent light fields, and the effect of input beam coherence must be considered. However, few studies have discussed the influence of beam quality on DOEs. In this work, we study the output results of Gaussian shell-model (GSM) beams with different beam qualities passing through the DOE and propose a new method for designing DOEs with GSM beam shaping. Methods In this work, an improved G-S algorithm, namely, the symmetric iterative Fourier transform algorithm (SIFTA), is used to design flat-topped DOEs. By introducing a signal window, this algorithm can obtain a highly uniform output light spot while ensuring output efficiency. With the DOE designed by SIFTA as a standard, the influence of beam quality is studied. The multimode laser can be approximately described by a GSM beam. The transmission of the GSM beam through a DOE is typically described by using a cross-spectral density (CSD) function, which involves a 4-order Fourier transform and results in significant computational complexity. In order to simplify the calculation, the mode decomposition of CSD is used to study the output light spot of the DOE under GSM beams. The coherent-mode representation uses coherent Hermite-Gaussian modes to express the CSD and requires only a limited number of modes to obtain accurate solutions. Similarly, the random-mode representation uses random modes that conform to statistical relationships to represent the CSD and reduce computational complexity. We also use variable substitution to simplify CSD transmission and find that the DOE output can be directly separated into a convolution of coherent and noncoherent parts. Results and Discussions By using the mode decomposition method, the output light spot distributions of GSM beams with different beam qualities passing through the DOE are calculated (Fig. 3). It can be seen that as the M-2 factor increases, the size of the flat-topped area of the output light spot gradually decreases until it deteriorates to a Gaussian-type light spot, and the flat-topped shaping effect of the DOE fails. The flat-topped DOE has a high requirement for beam quality. When the M-2 factor increases to 1. 5, the output result already has significant deformation. The same conclusion can be obtained using the convolutional representation of the GSM beams (Fig. 4). Only when the width of the convolution kernel is much smaller than the spot size D of the DOE will the convolution result approach the designed output light spot. Therefore, for a given DOE, the applicable maximum M-2 factor is related to the output spot size D and the input beam size w(0), and they can be expressed as M-2 = [1 + (pi Dw(0)/alpha lambda f)(2)](1/2). Specifically, a is the proportional coefficient between the output spot size and the maximum convolution kernel width; lambda is the wavelength; f is the focal length. In addition, the convolution property of DOE output is similar to the image blurring effect caused by the point spread function (PSF) in optical imaging systems. Therefore, methods in photolithography systems can be applied to the design of DOEs with GSM beam shaping. A modified coherent output target pattern is obtained by directly adding serifs to the original target shape, and then a DOE is designed using traditional coherent algorithms. Figure 5 shows an example of designing a DOE with GSM beam shaping using the proximity correction method. The output light spot of the designed DOE under the GSM beam can well match the target pattern. Moreover, the DOE with GSM beam shaping effectively improves the coherent noise phenomenon. Conclusions In this work, the SIFTA is used to design an 8-order flat-topped DOE with a square light spot. Using the methods of coherent-mode representation and random-mode representation, the output results of GSM beams with different beam qualities passing through the DOE are studied. It is found that increasing beam quality will lead to a reduction in the size of the flat-topped area of the output light spot and make the DOE ineffective. It is shown that the output light spot is a convolution of coherent and noncoherent parts, and the convolution contribution of noncoherent parts is the cause of light spot degradation. The relationship among the applicable maximum M-2 factor of flat-topped DOEs, output spot size, and input beam size is given, which provides a basis for laser selection in practical application. A design method of DOEs with GSM beam shaping is presented, which is expected to achieve the application of DOEs in lasers with low beam quality.
