# Vibration Compensation of the Frequency-Scanning-Interferometry-Based Absolute Ranging System
## 2 Theory Background  
### FSI-Based Absolute Ranging System  
The schematic diagram of the FSI-based absolute ranging system is shown in Figure 1. The light emitted by the tunable laser is first split using an 80:20 optical splitter, and then 80% of the optical power goes through the measuring optical path, and the remaining 20% goes through the reference optical path, as shown in Figure 1. After that, the frequency-swept light going through the measuring optical path is split by another 80:20 optical splitter such that 20% of the optical power goes through the 50:50 optical coupler as a local oscillator light and the remaining 80% of the optical power goes through a circulator as a measuring light. The local oscillator light can be expressed as:  
\[I_{LO}(t)=A_{L}\cos[2\pi(f_{0}t+\frac{1}{2}\alpha_{0}t^{2})+\varphi_{0}],0 \leq t\leq T_{m}/2, \tag{1}\]  
where \(A_{L}\) is the amplitude of the local oscillator light, \(f_{0}\) is the center frequency of the frequency-swept light, \(\alpha_{0}\) is the modulation speed of the laser, \(\varphi_{0}\) is the initial phase, and \(T_{m}\) is the period of frequency modulation. Subsequently, the measuring light is incident on the target mirror through the circulator and collimating lens and returned to the circulator through the target mirror reflection path. After that, the measuring light goes through the 50:50 optical coupler and merges with the local oscillator light. Thus, the measuring light can be expressed as:  
\[I_{M}(t-\tau)=\eta A_{L}\cos\{2\pi[f_{0}(t-\tau)+\frac{1}{2}\alpha_{0}(t-\tau )^{2}]+\varphi_{0}\},\tau\leq t\leq\tau+T_{m}/2, \tag{2}\]  
where \(\eta\) is the signal attenuation coefficient, and \(\tau\) is the time delay corresponding to the distance to be measured. So, the measuring light and the local oscillator light converge and interfere in the optical coupler, and the beat signal detected by the photodetector 1 from the measuring optical path can be written as:  
\[I=\frac{1}{2}\eta{A_{L}}^{2}\cos[2\pi(f_{0}\tau-\frac{1}{2}\alpha_{0}\tau^{2} +\alpha_{0}t\tau)],\tau\leq t\leq T_{m}/2. \tag{3}\]  
Since \(\tau^{2}\) term in Equation (3) is extremely small, it can be ignored.  
According to Equation (3), the frequency of the beat signal is a fixed value equal to \(\alpha_{0}\tau\) when the target is stationary. In that case, the distance is calculated by:  
\[R=\frac{\tau c}{2}=\frac{f_{beat}c}{2\alpha_{0}}, \tag{4}\]where \(f_{beat}\) is the frequency of the beat signal, and \(c\) is the light speed in the vacuum.  
Unfortunately, the frequency of the laser sweeping-signal is not completely linearly modulated, which leads to a serious broadening of the beat signal spectrum and a significant measurement error [21; 22; 23; 24; 25]. To reduce the influence of nonlinear tuning, the beat signal of the reference optical path is used as a clock signal to resample the beat signal of the measuring optical path. After equispaced-phase resampling, the beat signal of the measuring optical path is given by:  
\[I=\frac{1}{2}\eta{A_{L}}^{2}\cos[2\pi(f_{0}\tau-\frac{1}{2}\alpha_{0}\tau^{2} +\frac{\tau}{2\tau_{ref}}k)],k=0,\;1,\;2,\;\cdots, \tag{5}\]  
where \(\tau_{ref}\) is the time delay corresponding to the OPD of the reference optical path, and \(\tau_{ref}\) is generally taken as more than twice of \(\tau\) to satisfy the sampling theorem.