# A Novel 2D Micro-Displacement Measurement Method Based on the Elliptical Paraboloid
## 2 System and Principle
### Measurement Principle  
The basic element of the measurement system is the elliptical paraboloid, which is shown in Figure 2. The elliptical paraboloid was micromachined by an ultraprecision diamond tool using brass (H62) as the original material. Through the feedback loop between precise machining and inspection, the elliptical paraboloid was processed successfully. Its processing quality was detected by 3D Surface Measuring System (LuphoScan, Taylor Hobson, UK) and the result shows that the RMS is 0.021 \(\upmu\)m. It takes 3 h of skilled labor to produce such a part. Of course, the machining time depends on the size and quantity of the elliptical paraboloid. Here, its mouth diameter is 10 mm. In reality, it can be scaled up to a large volume (mouth diameter 50 mm). Nevertheless, the measurement range will limit its unnecessary volume. A Cartesian coordinate system is established in the vertex of the elliptical paraboloid. It is assumed that the equation of the elliptical paraboloid is  
\[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}=2z\big{(}a^{2}=0.1547\big{)} \tag{1}\]  
where \(a\) is the parameter of the elliptical paraboloid. The derivatives of Equation (1) in the direction of x and y are calculated as follows, respectively:  
\[\tan\alpha_{x}=\frac{dz}{dx}=\frac{x}{a^{2}} \tag{2}\]  
\[\tan\alpha_{y}=\frac{dz}{dy}=\frac{y}{a^{2}} \tag{3}\]  
where \(\alpha_{x}\) is the angle between the tangent of the point on the elliptical paraboloid and the \(x\) direction, and \(\alpha_{y}\) is the angle between the tangent of the point on the elliptical paraboloid and the \(y\) direction. According to the above equations, the relationship between the displacement on the elliptical paraboloid and the tangent of the angle is linear.  
The measurement system based on the elliptical paraboloid utilizes the optical autocollimator principle to obtain a 2D micro-displacement variation, as shown in Figure 3. The measurement system is similar to the typical optical autocollimator system except that the plane mirror is replaced by an elliptical paraboloid. The measuring beam propagating along the horizontal direction is reflected by the elliptical paraboloid, which is received by a CCD detector after being focused by lens 2 with focal length \(f\). While the measuring beam moves along the \(x\) direction, the reflected beam converges on another point on the CCD detector. Supposing that the angle between the t  
Figure 1: Schematic of the measurement system.  
point and the vertical direction is \(\alpha\) and the displacement of the light spot on the CCD detector is \(x^{\prime}\), the following equation can be obtained:  
\[\tan 2\alpha=\frac{x^{\prime}}{f} \tag{4}\]  
Generally speaking, the angle \(\alpha\) is especially small. According to Equation (2) and (4), the following result can be obtained, which shows the relationship between the displacement of the measuring beam in the x direction and the displacement of the light spot on the CCD detector:  
\[x=\frac{x^{\prime}}{2f}a^{2} \tag{5}\]  
Similarly, the relationship between the displacement of the measuring beam in the y direction and the displacement of the light spot on the CCD detector can be easily written as follows:  
\[y=\frac{y^{\prime}}{2f}a^{2} \tag{6}\]  
From Equations (5) and (6), the linear relationship between the displacement of the laser beam and the light spot can be utilized to measure micro-displacement.