## 1 Introduction  
Optical lenses, biochips, and fuel cells, which are used as core components in various high-tech industries, are fabricated in a form that consists of a microstructure on the surface [1; 2; 3; 4; 5; 6]. In particular, a head-up display device mounted on an aircraft or a vehicle, or a head-mounted display device for realizing a VR (virtual reality) and an AR (augmented reality) requires a high-performance lens. Microstructures designed on the surface of the lens constitute one of the key methods to achieve high optical performance. Therefore, many studies have been conducted on ultra-precision machining technology to fabricate the aspherical surface or the freeform surface of a lens with a repeated fine structure [7; 8]. Recent precision machine tools have evolved to the extent where the resolution is less than or equal to 1 nm, and the development of this equipment technology has led to the study of the micropattern machining of various contents [9; 10; 11; 12; 13; 14; 15]. Currently, the turning process can fabricate micropatterns of a smaller size than those obtained using a milling process. Precision turning can produce microstructures with a simple cross-section with a single repeated shape such as triangles or rectangles at higher speeds. In addition, more complicated cross-sectional shapes can be realized when the precision turning process is combined with additional devices such as a fast tool servo (FTS) and a slow tool servo (STS) [13; 14; 16; 17]. Various materials such as aluminum, PMMA (poly methyl methacrylate), plated copper, and nickel are used as a work piece for precision machining [7; 9; 10; 12]. In particular, metals have been used as holds for the lens with repeated micropatterns on their surface [6; 9; 18; 19]. For instance, copper and nickel have excellent optical and mechanical properties, and are widely used as lens mold materials. On the other hand, a transparent PMMA is used as an optical element [12]. In ultra-precision machining, the shape accuracy of work pieces is the most important issue. There have been many experimental works that investigated the factors that cause the shape errors such as burr and deformation, and found ways to improve them. Theoretical approaches have also been proposed to predict the occurrence of shape errors [10; 11; 13; 14; 15; 20]. In recent years, studies have been undertaken to fabricate patterns of a smaller size than in the past [4; 21; 22]. Diamond tools have often been used for micropatterns ranging from tens of microns to several microns in size [4; 10; 20]. Microstructures smaller than one micrometer can be machined using a diamond tool made by a focused ion beam. In order to obtain structures smaller than one micrometer, machining studies using AFM (atomic force microscope) have also been reported [21; 22]. It was more difficult to obtain the high-dimensional accuracy, as the size of the micropattern was reduced from several tens of microns to a few microns or submicrons. The micropatterns appear to be very uneven in a paper showing examples of machining submicron-sized micropatterns using AFM [22]. However, there have been no studies to identify the causes of the shape error of microstructures of less than several microns, or to determine the proper machining process conditions to improve the shape accuracy. Today, research is still focused on whether microstructures of less than a few microns can be successfully produced by precision machining. Machining microstructures with a size of a few micrometers or less with the use of a diamond tool is still a challenging task. Making very small tools, setting more precisely the devices, and not lowering productivity should be considered together [4]. In this paper, precision machining was performed using a diamond tool to fabricate microstructures of several microns in size. In this process, two factors affecting the shape error of the microstructure were discussed.