## 2 Measurement Method and Systems  
### Measurement Method  
The principle of the proposed microsphere measurement system is demonstrated in Figure 1. In this system, a measured microsphere made of ruby is clamped with a clamping fixture, and the fixture is assembled on a micro stage (Micro stage-3). The line profile of the microsphere in the maximum cross section is scanned simultaneously and differently with two scanning probes (scanning probe-1 and scanning probe-2). During the measurement, the microsphere moves along the Y direction (as shown in Figure 1) via micro stage-3, and the two probes move along the X direction separately and oppositely driven by micro stage-1 and micro stage-2. After one of the cross-sectional profiles is scanned, the microsphere is rotated by the rotary stage with a predetermined degree, such as 30\({}^{\circ}\), and a new profile of the cross section is scanned. Two CCD cameras are installed on the top (This one is not shown in Figure 1) and front of the microsphere to confirm the visual alignment. The schematic of the measurement process is shown in Figure 2, where a maximal cross-sectional profile of the microsphere is scanned. Figure 2a illustrates the initial position of the measurement  
Figure 1: Diagram of the measurement system. process. After one two-point measurement is completed, probes move away from the microsphere in the X direction. Afterward, the microsphere moves down to a new position along the Y direction, and two probes move toward the microsphere to perform a new two-point measurement (Figure 2b). Figure 2c provides the complete result of a maximal cross-sectional profile of the microsphere. After the measurement of one cross-sectional profile is completed, the microsphere is rotated by rotary stage, and another cross-sectional profile can be scanned. By the same way, several cross sections of the microsphere can be measured. The dimensions of the microsphere can be fitted after integrating the data of the multi-sections. In the process of measuring a cross-sectional profile, the yaw error is the main factor affecting the measurement accuracy. In our method, after the measurement of one cross-sectional profile is completed, the microsphere is turned with the angle of 180 degrees, and the cross-sectional profile at the opposite position is measured [13]. By measuring the two opposite cross-sectional profiles, the yaw error can be eliminated. Figure 3a shows the measurement principle of a cross-sectional profile, \(F_{L}(i)\) is the measurement data of scanning probe-1, \(F_{R}(i)\) is the measurement data of scanning probe-2, \(F_{0}(i)\) is the assumed true value of the chord length, \(f_{L}(i)\) and \(f_{R}(i)\) are the morphological errors at left and right, \(f_{c}(i)\) is the yaw error, \(i\) is the index of measurement positions. After the measurement of the profile circle shown in Figure 3a is completed, the microsphere is turned with the angle of 180 degrees, and the opposite cross-sectional profile can be measured. The measurement principle of the opposite profile is shown in Figure 3b, \(F_{L^{\prime}}(i)\) is the measurement data of scanning probe-1, \(F_{R^{\prime}}(i)\) is the measurement data of scanning probe-2, \(F_{0}{{}^{\prime}}(i)\) is the assumed true value of the chord length of the opposite circle. For the two opposite circles, the measurement results of the two probes can be expressed as:  
\[F_{L}(i)=-\frac{1}{2}F_{0}(i)-f_{L}(i)+f_{c}(i) \tag{1}\]  
\[F_{R}(i)=\frac{1}{2}F_{0}(i)+f_{R}(i)+f_{c}(i) \tag{2}\]  
\[F_{L}{{}^{\prime}}(i)=-\frac{1}{2}F_{0}{{}^{\prime}}(i)-f_{R}(i)+f_{c}(i) \tag{3}\]  
\[F_{R}{{}^{\prime}}(i)=\frac{1}{2}F_{0}{{}^{\prime}}(i)+f_{L}(i)+f_{c}(i) \tag{4}\]  
Sum the four equations, the morphological errors \(f_{L}(i)\) and \(f_{R}(i)\) can be automated eliminated, and the yaw error \(f_{c}(i)\) can be separated as the following equation.