## 2 Proposed Differential Quadrature Fabry-Perot Interferometer  
Based on the pending patent (application number: 108142443), the optical structure of the proposed differential quadrature Fabry-Perot interferometer is extended and realized experimentally. It enables the arrangement with two measurement types simultaneously for determining the linear displacement. This structure contains a laser source, an optical cavity, and an optical element and signal sensing module (Figure 5a). The laser beam is incident to the optical cavity module from the laser source module and the optical element and signal sensing module. Then the laser beam is divided into multiple beams and output from the cavity, and PDs placed in the optical element and signal sensing module will inspect the interferometric signals. The laser source module is composed of the high stabilized He-Ne laser and the optical isolator. The purpose of placing the optical isolator is to avoid the laser beam to reflect back from other optical components. The optical cavity module contains two measurement types depending on the selection of different measurement mirrors (Figure 5b). One is the PM type whose optical cavity is composed of two PMs, and the other is the corner cube retro-reflector (CCR) type whose optical cavity consists of a PM and a CCR. In the CCR type, due to its folded measurement structure, the optical resolution can be improved by a factor of two compared to the type of the PM [19]. In addition, owing to CCR bearing large tilt angles of the measurement mirror, this type is more suitable for the displacement measurement within the large range. Due to the multi-beam interferometric measurement structure, the incident beam travels forward and backward in the cavity and is divided into numerous reflected beams (order number of beam: 1 st, 2 nd, 3 rd... Nth). The Nth beam will pass through the quarter-wave plate twice more than the N-1 th beam, so the polarization direction of each laser beam will be orthogonally converted. Then they will be transmitted back to the optical element and signal sensing module. In the optical element and signal sensing module, it contains a quarter-wave plate, two BS, two PBS, and four PDs. According to the proposed optical structure, the interferometric signal can be acquired by PDs for further displacement measurement. The linearly polarized laser beam with a polarization direction angle of -\(\pi\)/4 with respect to the \(y\)-axis will be acquired after passing through the optical isolator, and it is converted to the right-handed circularly polarized beam by the first quarter-wave plate, whose fast axis is along the \(y\)-axis. Then the laser beam will be incident to the BS\({}_{1}\) and the optical cavity. In this cavity, the beam is transmitted to the second quarter-wave plate whose fast axis is at a zero angle with respect to the \(y\)-axis multiple times. Numerous reflected laser beams output by the optical cavity are sequentially converted into the orthogonally polarized beams (Figure 5c). The backward laser beam is split into two beams by BS\({}_{2}\), and then each beam is transmitted to the PBS\({}_{1}\) and the PBS\({}_{2}\), respectively. The optical axis of the PBS\({}_{2}\) is rotated around the incident beam by \(\pi\)/4 relative to that of PBS\({}_{1}\). In the optical arrangement, two pairs of the interferometric signal with a phase difference of \(\pi\) can be obtained. And by subtracting each pair signal, the orthogonal signal for displacement measurement will be determined.