## 1 Introduction  
With the development of industrial automation and intelligence, the popularization and demand of multi-axis computer numerical control (CNC) equipment in industrial production are increasing. Therefore, the multi-degree-of-freedom (MDOF) measurement technology is a key basic technology. For instance, in a MDOF CNC machine tool, the current common measurement scheme is to equip each motion axis with a separate single-axis measuring device, including linear grating ruler, linear photoelectric encoder, and so forth [1; 2; 3]. The installation of these measuring devices is usually stacked along the motion axis. This brings about the abbe error problem in the assembly process, which will have an adverse effect on the measurement accuracy [4; 5].  
In order to eliminate the abbe error in multi-axis measurement and improve the integration of the measurement system, measurement devices with MDOF measurement capabilities have been paid more attention. Among them, the technique schemes of two-dimensional grating rulers and the MDOF grating rulers are of great application potential [6; 7; 8; 9]. These MDOF grating rulers, generally with usage of a two-axis planar scale grating, are able to measure not only the translational motions but the small order rotational error motions, and it can be also applied to six-DOF measurement. Compared with the traditional single-axis measuring devices, the MDOF grating ruler has a higher level integration and the abbe error could be avoided Thus, the reliability and precision of the multi-axis precision stage and measurement system can be improved effectively [10; 11; 12].  
As illustrated in these prototype MDOF grating rulers, grating diffraction beams interferometry is the key technology, which means that the interference is generated by using four beams of \(\pm 1\) order diffracted light of a two-dimensional grating so that to obtain the displacement distance by detecting the interference signal [6; 7; 8]. Compared with the one-dimensional interferometric grating ruler, the two-dimensional grating ruler is equivalent to integrating two sets of one-dimensional grating rulers whose working direction is orthogonal to each other, however this makes complexity of the two-dimensional grating ruler light path increase greatly. The complexity is mainly manifested in the need of the interference of four beams both from the reference grating and the scale grating parallel or perpendicular to the incident light in the system in order to produce an ideal interference signal. This means that the above eight diffracted beams need to be finely deflected, which determines the quality of the interference signals. In previous research, there are three proposed methods, collimation lens represented by Reference [6], separate prisms represented by Reference [5], and a home fabricated diffraction device by combining four separate one-axis line gratings in a glass substrate [13]. These methods meet the demand of the light path design and can be manufactured under the lab conditions. However, due to the complex structure and alignment, loss of signal intensity and the low sensitivity, these solutions are difficult to meet the condition of mass production.  
Therefore, in this paper, focusing on the demand of mass production and engineering prototype, facing these challenges in current technologies, we design a new lens module for this kind of grating rulers, a single prism called a quadrangular frustum pyramid, enabling simultaneously deflecting the four diffraction beams without any power loss and easy alignment. To test the lens module, a grating ruler system is designed, constructed and experimental verification are introduced in this manuscript.