# Multi-Axis Laser Interferometer Not Affected by Installation Errors Based on Nonlinear Computation
## 1 Introduction  
With a displacement measurement resolution higher than 1 nm, the laser interferometer is used in various precision and ultra-precision measurement applications. One of the typical applications is the lithography wafer stage in semiconductor equipment, which is used to carry the silicon wafer to achieve a series of multi-degree-of-freedom (multi-DOF) precision motions. Since the positioning accuracy of the wafer stage directly affects the overlay accuracy of the lithography, a high requirement for the accuracy and bandwidth of the multi-DOF displacement measurement is put forward [1, 2, 3]. The laser interferometer is commonly used in such cases to measure displacement based on the interference of laser beams, which measure the variation in optical path length by detecting the laser intensity, thus realizing the precise displacement measurement [4, 5, 6]. According to the laser type, the interferometers can be classified as homodyne interferometers with a single-frequency laser source and heterodyne interferometers with a dual-frequency laser source. To measure the multi-DOF displacement of a precision movement stage by using a multi-axis interferometer, more lasers and other optical elements need to be adopted. Additionally, it is necessary to carefully design the number and each installation position of lasers and the multi-DOF displacement decoupling computation. For example, five lasers are used to measure the stage surface at five different points in the five-axis laser interferometer, and the 5-DOF displacement is computed from the interferometer model [7]. In addition to the measurement scheme in [7], some researchers used more lasers with redundant information to decrease the nonlinear error [8, 9, 10, 11]. Moreover, some researchers built a more accurate but more complex model of the coupling between multi-DOF displacement to improve the accuracy of the displacement computation results [12, 13, 14, 15]. However, these multi-axis laser interferometers mentioned above inevitably have measurement systematical errors, which can be summarized as two reasons. First, the interferometer model is inaccurate, which is mainly caused by installation errors. For example,the laser beams in the model are parallel, but in practice they are not, and the installation position and orientation of the lasers in the model are different from those in practice. Second, the computation of the model is inaccurate. Usually, there are many nonlinear functions such as trigonometric functions and rational fractions in the interferometer model. To simplify the computation, these nonlinear functions are approximated linearly so that the explicit expression of the multi-DOF displacement can be derived. Both of the above two reasons will result in a multi-DOF displacement measurement error, which causes the loss of accuracy. To avoid the above two factors leading to a measurement systematical error, parameters such as the laser beam angle and emitting point position should be introduced into the model as unknowns. In addition, the nonlinear measurement model should be solved directly without any linear approximation. However, in this way, the interferometers model is quite complex; that is, it contains a large number of unknowns and complex nonlinear functions, and how to solve the model accurately and quickly becomes a new problem. Although there are the Gauss-Newton method [16], Levenberg-Marquardt (LM) method, and other nonlinear equation system numerical algorithms [17; 18; 19], when it comes to practical engineering, due to the negative properties of the equations such as the singular Jacobi matrix, which are caused by the huge number of unknowns and the complex nonlinear functions, the convergence and the accuracy of the algorithm may not be ideal, which raises the necessity for a general and comprehensive analysis.