## 1 Introduction  
As a universal length standard for calibrating instruments, step gauges are widely used in the precision calibration and error compensation of high-precision instruments, such as the Coordinate Measuring Machine (CMM), numerical control machine tool, optical instruments and so on.  
Moreover, the International Bureau of Metrology (BIPM) has also listed the step gauge as a key length comparison, making it an important component in measuring the metrological capacity of a national geometric measurement calibration laboratory [1; 2; 3; 4]. Many ambitious countries have established their own step gauge calibration devices [5; 6; 7].  
On the basis of Leitz CMM, the National Metrology Institute of Japan (NMIJ) realize their step gauge calibration by use of the large displacement measurement of a laser interference measuring system, and the micro displacement of the probe of the CMM, thereby obtaining the dimension values of the measured step gauge. The measurement uncertainty of this device is U= (0.06 + 0.22L) \(\upmu\)m [8].  
Based on the Moore M48 CMM, the National Institute of Standards and Technology (NIST) has developed a step gauge calibration device using a specially designed induction type probe that provides a short-term repeatability of 7 nm [9].  
The above two calibration systems are based on the measurement platform of CMM, but the Abbe error, pitch and yaw errors are considered only as the error sources of measurement uncertainty, rather than being directly measured and compensated.  
The National Institute of Metrology of Finland (VTT MIKES) has constructed a step gauge calibration system on a vibration isolated stone table with a 1D movable guide rail, rather than CMM, to eliminate mechanical disturbances [10]. The measurement uncertainty of the system is U = Q[0.090; 0.14L] \(\upmu\)m. This system also considers the pitch and yaw errors as the error sources of measurement uncertainty, and so does not directly measure them.  
At Physikalisch-Technische Bundesanstalt (PTB), the Nanometer Comparator is upgraded with a tactile sensor system to add the capability to measure step gauges. The 1\(\sigma\) repeatability of the device is better than 7 nm [11]. The system adopts the precise displacement measurement method based on the principle of vacuum interference, which has high precision, but is relatively complex.  
Based on the analysis of the advantages and disadvantages of the above-mentioned instruments, this paper proposes a new computational model of step gauge calibration based on the Synthesis Technology of Multi-Path Laser Interferometers (SMLI), in which three of four interferometers are used for distance measurement, and the fourth channel is used for the wavelength compensation of the laser [12; 13; 14], thereby establishing our own step gauge calibration device on the platform of Leitz CMM. By adding four laser interferometers in the measurement platform, the spatial position relationship model of three out of four lasers is constructed, and the synthetic optical path of these lasers is translated to the measurement line of the measured step gauge, which can effectively eliminate the influence of the Abbe error of the CMM on the measurement result, and compensate the errors of pitch and yaw. Based on the presented calculation model of the closed-loop vector principle, the error caused by pitch and yaw can be corrected to the measured length in real time by measuring attitude change with an auto-collimator to obtain a better calibration accuracy.  
This paper will firstly introduce the composition and structure of the developed step gauge measurement system by in detail. Then, the principle and implementation of the measurement method are be addressed systematically. Finally, the experimental results are given and discussed to provide a conclusion.