# A Novel 2D Micro-Displacement Measurement Method Based on the Elliptical Paraboloid
## 1 Introduction  
The measurement of micro angle and displacement plays an important role in science and technology [1; 2]. Optical angle or displacement apparatuses have been applied in many fields, such as in the calibration of machine tools and the measurement of multi-degrees-of-freedom systems [3]. Under certain conditions, the principles of angle measurement are also applicable to displacement measurement. Conventionally, the three main optical methods for angle measurement with high resolution and accuracy are: interferometers, internal-reflection and autocollimators [4; 5; 6; 7; 8]. An interferometer utilizes phase information to measure angular and linear displacement with high signal-to-noise ratio and high resolution [9]. The internal-reflection effect can make a wide range of devices very stable and compact with high resolution [10]. An autocollimator is customarily used for measurement of straightness, flatness and angular position [11]. As mentioned, the principles of interferometry and internal-reflection have been transformed into the displacement measurement field [12; 13; 14; 15]. However, seldom has the autocollimator principle been studied in the field of displacement measurement. Compared with the other two methods, displacement measurement based on the autocollimator principle has important application value, especially in two-dimensional displacementmeasurement, which is worth researching. In reality, an interferometer is usually large in size with a high cost in many applications [16]. Moreover, the displacement measurement based on internal-reflection is limited by the measurement range, the geometric accuracy of the reflection prisms, and material inhomogeneity [17]. For example, [PERSON] developed a two-axis position measurement using an optical slope sensor based on an autocollimator with a resolution of 0.1 \(\upmu\)m, with a low cost and a small size [18]. In addition, with the development of optical surface processing technology, the application of an optical freeform surface has become more and more extensive [19]. For instance, [PERSON] realized the nanometer scale measurement of two-dimensional displacement based on an optical freeform surface with a double sinusoid [20; 21]. The space wavelength of the sinusoid was 150 \(\upmu\)m and the amplitude was 100 nm. However, a nonlinear error is generated by using the superimposed sinusoid freeform surface as the reference element of the displacement measurement. Therefore, it is necessary to design a novel optical freeform surface element to realize linear transformation of angle and displacement for micro-displacement measurement, which we refer to as the elliptical paraboloid.  
In this paper, a novel two-dimensional micro-displacement measurement method based on the elliptical paraboloid is proposed. The elliptical paraboloid device is the key to realizing the transformation of angle to displacement. The basic optical structure of an autocollimator is slightly improved to accommodate the measurement of the curved surface. The experimental results show that the principle of an autocollimator is applicable to micro-displacement and linear measurement, and that two-dimensional displacement can be measured simultaneously with an accuracy within \(\pm\) 0.3 \(\upmu\)m. Although some aspects need to be improved, this measurement has a good range of application, such as the detection of geometric errors of CNC machine tools.