A Differential Phase-Modulated Interferometer with Rotational Error Compensation for Precision Displacement Measurement  
Liping Yan  
Precision Measurement Laboratory, Zhejiang Sci-Tech University, Hangzhou 310018, China; [EMAIL_ADDRESS] (L.Y.); [EMAIL_ADDRESS] (YY.); [EMAIL_ADDRESS] (B.C.)  
Yucen Yan  
Department of Electrical Engineering, University of California, Berkeley, CA 94720, USA; [EMAIL_ADDRESS] (L.Y.); zhenby@astro  
-fringe tilt and interval change analysis [11]. For differential sensing technology, it is mainly divided into differential power sensing (DPS) technology and differential wavefront sensing (DWS) technology. The two technologies both adopt the four-quadrant detector (QD) as the detector, but the difference is that DPS technology uses QDs to obtain the power unbalanced signals, and DWS technology uses QDs to obtain the phase unbalanced signals. For DPS technology, the DPS signal is used for angle measurement, and the average phase of four interference signals is used for displacement measurement, but this will be limited by the performance of the detector itself [12]. A Mach-Zehnder interferometer is developed by combining DPS and DWS; it uses the DWS signal to measure a large rotation range and the DPS signal to provide high measurement accuracy [13]. In our previous study, we proposed a modulated laser homodyne interferometer using an integrated four-photodetector to obtain a DWS signal to measure the displacement and rotational angle simultaneously [16]. An electro-optic phase modulator (EOM) was applied in the reference arm as a phase modulator. Although a high measurement sensitivity was achieved, the unbalanced environmental disturbances on the reference and measuring arms leads to an increase in measurement uncertainty.  
To minimize the impact of unbalanced environmental disturbances, a series of differential interferometers have been developed [17; 18; 19; 20]. For example, a differential interferometer was designed to eliminate the influence of environmental fluctuations by constructing a measuring interferometer and a reference interferometer that share the same reference arm [17]. Displacement can be measured by detecting the phase change in the interference signals of two interferometers. However, using PZT as the phase modulator, its modulation speed limits the displacement measurement speed. An all-fiber differential interferometer was proposed to measure displacement, using two circulators as a reference arm and a measuring arm that do not affect each other [18]. Furthermore, a compact, double-pass differential plane interferometer with coaxial four parallel and coplanar beams in the reference and measuring arms was presented [19]. The interferometer has a non-linearity of less than 0.1 nm but is limited by complex fabrication processes and long-term temperature sensitivity.  
In this paper, a differential phase-modulated interferometer (DPMI) with rotational error compensation for precision displacement measurement is proposed. In DPMI, by constructing the reference and measuring interferometers with a common reference arm, the unbalanced disturbance between the reference and measuring arms of the measuring interferometer is converted into a common-mode disturbance between the two interferometers, and the common-mode disturbance can be minimized by subtracting the displacement results of the two interferometers. Using DWS technology, an integrated 2 \(\times\) 2 array photodetector is applied in the measuring interferometer for rotational angle compensation. With the proposed design, the displacement without unbalanced disturbance and rotational angle errors can be measured. The optical configuration is presented in Section 2, the measurement principle is described in Section 3, and experiments are carried out to verify the feasibility of DPMI in Section 4.