## 4 Performance Tests
### Length Measurement in Lateral Direction  
Because of the diameter error of the ball itself, the roundness deviation of the ball and the deflection of the ball stylus at triggering point, the equivalent diameter of the probe ball in the measuring direction should be known in advance. Here, the equivalent diameter of the probe in the measuring direction is calibrated firstly in the measurement direction by a reference gauge block with high accuracy. Then, the calibrated value is subtracted from the obtained data of the sample.  
In the estimation of the equivalent diameter of the probe ball, six points on one measuring face of the reference gauge block (its calibrated value is 10.000045 mm) were measured by the developed micro CMM firstly. Using the coordinates of the six points, a reference plane was calculated by the least squares method. Then, one point on the opposite measuring face of the reference gauge block was measured. The distance between this point and the reference plane is the sum of the dimension of the gauge block and the equivalent diameter of the probe ball. Figure 10 shows the results of a 15 times repeated measurement (The horizontal coordinate is the order of measurement). Neglecting the dimensional error of the reference gauge block, the equivalent diameter of the probe ball in the \(X\) direction was calculated as \(r_{x}\) = 11.001268 mm \(-\) 10.000045 mm = 1.001223 mm. The standard deviation (\(\sigma\)) of the repeatability is about 26 nm.  
Figure 9: Step height measurement schematic diagram between two gauge blocks. 1: Optical flat; 2: Calibrated gauge block 1; 3: Calibrated gauge 2.  
After the equivalent diameter of the probe in the measurement direction had been calibrated, the reference gauge block was replaced by the sample gauge block in the same location and same direction. After the equivalent diameter of the probe ball was subtracted from the measured value, the measured dimension of the sample gauge block could be obtained. Figure 11 shows 10 times measurement results of a sample gauge block with calibrated length of 40.000039 mm. The measured dimension is about 40.000011 mm on average and the standard deviation is about 47 nm.  
Another test was done in the \(Y\) direction with the same setup as in the \(X\) direction. Figure 12 shows the probe diameter calibration results. The 10 times average equivalent diameter of the probe in \(Y\) direction is 0.998294 mm. The standard deviation (\(\sigma\)) of the repeatability is about 47 nm. Figure 13 shows 12 times measurement results of the same sample gauge block. The measured dimension is about 40.000054 mm on average and the standard deviation is about 45 nm.  
Figure 11: The measuring length of a gauge block with the nominal length of 40 mm.  
Figure 10: The repeated measuring results of sample gauge block.  
In the lateral length measurement, the measurement errors are mainly from the interferometer (26 nm, \(k=2\)), the thermal expansion error (24 nm, \(k=2\)), the Abbe error (30 nm, \(k=2\)) in lateral direction, the Abbe error in \(Z\) direction (61 nm, \(k=2\)), the repeatability error of the probe (30 nm, \(k=2\)), the error from reflection mirror (50 nm, \(k=2\)), the non-orthogonal error (40 nm, \(k=2\)). Neglecting other error sources, the measurement uncertainty is about 106 nm (\(k=2\)).  
Although the equivalent diameters of the probe ball in different directions shown in Figures 10 and 12 are different, but the measured dimensions of the sample gauge block, shown in Figures 11 and 13 have no obvious difference (about 43 nm).