# A Differential Measurement System for Surface Topography Based on a Modular Design
## 1 Introduction  
Surface topography measurement [1; 2; 3] is an important metrological tool for quality assessment in industries that include semiconductors [4; 5], micro-electromechanical system (MEMS) [6], additive manufacturing [7], and aerospace [8]. A common challenge with developing surface metrology technologies is the achievement of multi-scale measurement capability, which means high accuracy over a large measurement area.  
Optical approaches that are playing important roles in this area [9; 10], such as confocal microscopy [11; 12], coherence scanning interferometry [13], and focus variation [14], are able to meet nanometer accuracy, but the measurement range is within a few millimeters or even smaller. The data stitching technique is able to expand the measurement range, but it relies heavily on common features of the overlapping area from neighboring data sets [15]. If the overlapping area does not show distinct features, the stitching accuracy will rely heavily on the performance aspects of the motion system, such as the straightness and flatness [16].  
Therefore, high-precision 2D/3D positioning [17] is a key technical enabler for multi-scale measurement systems. In reference [18], an optical probe was mounted on a Coordinate Measuring Machine (CMM). In references [19; 20], Scanning Probe Microscopy (SPM) probes were equippedwith high-precision positioning systems, to achieve a sub-nanometer resolution in a millimeter measurement area.  
When integrating a high-accuracy probe with a motion system, there are two types of critical measurement errors that researchers have to consider. They are the motion error, introduced by the imperfect straightness of the guideway [21; 22], and the Abbe error, introduced by the system design [23; 24]. Motion errors can be minimized by using high-precision components as geometric references [25]. However, the accuracy loss of mechanical references is inevitable after heavy usage. The motion error can also be compensated by implementing differential measurement [26; 27] or by introducing additional optical measurement systems [28; 29]. The Abbe error can be minimized by employing a static coordinate system in which the position sensors track the central stage along the coordinate axis [30; 31]. In reference [30], three laser interferometers that formed a static coordinate system were used to track the position of the central stage. In reference [31], the central stages were mechanically tracked by optical encoders. However, both concepts have their drawbacks. Optical tracking is very sensitive to environmental disturbances such as temperature and airflow. Mechanical tracking systems require a very complicated assembly and adjustment process due to the over-constrained mechanism. Furthermore, the geometric accuracy of the mechanical components will significantly affect the measurement accuracy.  
In order to address the challenges discussed in the above paragraph, a modular system for surface topography measurement was developed using conventional components. The central stages were mechanically tracked by optical encoders with consideration of the cost and robustness [31]. An optimal system design was proposed in order to address the challenge of the over-constrained mechanism. A film interferometer was developed to compensate for the straightness error from the motion system.  
The motivation and scope of the presented work can be illustrated by the graphic abstract shown in Figure 1.