## 1 Introduction  
In the era of rapid development of technology, the uniformity of film thickness is very important issue for large-area coatings. In particular, the error of optical coating thickness has achieved the nanometer level. Because considerable production and large-area manufacturing have become relatively important to reduce costs. How to control the film thickness in the evaporation process to achieve the theoretical design value is very critical. The uniformity of the film thickness is affected by the placement position between the evaporation source and the substrate, the vacuum value inside the chamber, and the emission characteristics of different materials which the different factors will have an impact [1]. These factors include substrate holder geometry, vacuum pressure, system temperature (i.e., substrate and material flux temperature), and angle configuration of deposition material and substrate [2]. In 2018, [PERSON] et al. [3] proposed a novel spatial-distribution function of diffusion theory, which replaced [PERSON]'s law of cosine and empirical rule. The thin film deposition of electron beam evaporation on the substrate was investigated. The thickness distribution obtained the accurate prediction of the deposition thickness of titanium and aluminum film confirms that this method has developed to improve the uniformity of the thin films. The most widely used method is to rotate the substrate holder to increase the uniform distribution of the film thickness, and also use a mask to obtained better film uniformity. But the disadvantages will cause material waste. Therefore, by optimizing the mask geometry can be obtained the best uniform distribution and the least loss of materials. In 2000, [PERSON] et al. [4] designed a mask to be applied to large-area substrates of different geometric shapes and obtained a good film thickness distribution. In 2018, [PERSON] et al. [5] used simulation modeling to compare different complex geometric structures. A new program for modelling and simulating thin film uniformity for physical vapor deposition was developed by using MathCAD. A mask for optimizing thin film thickness distribution designed using the program was shown to improve thickness uniformity from \(\pm\)4% to \(\pm\)0.56%. The advantage of electron beam evaporation is that the quality of the deposited film has higher purity, evaporation rate fast, and better control of film thickness [6]. For example, relevant applications include mirrors for gravitational wave detection [7], photovoltaic semiconductor industry, anti-reflection films [8], etc. Over the past decades, many methods have been developed to improve film thickness uniformity. The most widespread method is to achieve thickness uniformity through the substrate holder revolution or planetary rotation [9]. At the same time, the use of a modified coating mask can not only improve the film thickness uniformity, but also achieve the specified thickness distribution [10]. Basically, the modification of the coating mask requires time-consuming trial-and-error methods. In recent years, the application of physical deposition theory in the study of film thickness distribution and theoretical mask design has attracted more and more attention. The shape modification of the coating mask can improve the efficiency of material utilization. For the formation of thin films using other different techniques, electrochemical synthesis is a promising method for preparing thin films with a given thickness and uniform phase composition [11]. The dependence of the structure, phase composition and thickness of the electrodeposited films was obtained. Due to different preparation methods, the substrates have different microstructures and surface properties. [PERSON] et al. [12] reported the effect of the structure and surface properties of the substrate on the composition gradient of NiFe films. Some results of the preparation and investigation of other substituted oxides promising for practical applications are reported in the literature. [PERSON] et al.