## Theory  
### The Principle of FSI Using Frequency Sampling Method  
As is shown in Figure 1, the ECL has been used for five years and is connected with two interferometers. One is the auxiliary interferometer (as shown in the green box), whose output is used as the sampling signal to correct the tuning nonlinearity. The other is the measurement interferometer (as shown in the red box), whose output contains the information of measured distance. First, 10% power of the ECL goes into the auxiliary interferometer and the output is received by photo-detector 2 (PD2). Meanwhile, the remaining 90% of the power of the ECL goes into the measurement interferometer. Then, making use of a 90:10 optical splitter, 10% is used as the reference signal and 90% is used as the measurement light. Photo-detector 1 (PD1) obtains the output of the measurement interferometer. Finally, PD1 and PD2 convert the optical signals to the electric signals, and send the signals to the data acquisition card (DAQ). The He-Ne interferometer is used to verify the accuracy of our system.  
The output of the measurement interferometer can be expressed as:  
\[I_{m}(\,f\,)=\,\,A_{m}\cos(2\pi f\tau_{m})\!\!=\,\,A_{m}\cos(2\pi(f_{0}\,+\Delta f )\tau_{m}) \tag{1}\]  
Figure 1: Schematic diagram of our frequency scanning interferometry (FSI)-based system and the experimental setup.  
where \(A_{m}\) is the amplitude of the measurement interference signal, \(f_{0}\) is the initial frequency of the ECL, \(\Delta f\) is the variation of the ECL instantaneous frequency, and \(\tau_{m}\) is the time delay of the measurement interferometer.  
Similarly, the ideal output of the auxiliary interferometer can be expressed as:  
\[I_{a}(f\ )\ =\ A_{a}\cos(2\pi f\tau_{a})\ =\ A_{a}\cos(2\pi(f_{0}\ +\Delta f)\tau_{a}) \tag{2}\]  
where \(A_{a}\) is the amplitude of the auxiliary interference signal, and \(\tau_{a}\) is the time delay of the auxiliary interferometer.  
To eliminate the influence of the ECL tuning nonlinearity, taking the output of the auxiliary interferometer as the sampling signal yields:  
\[2\pi\Delta f(k)\tau_{a}=\ \pi k,\ k=\ 1,\ 2,\ \ldots,\ N \tag{3}\]  
where \(\Delta f(k)\) is the variation of instantaneous frequency, \(k\) is the sampling point index, and \(N\) is the number of the ideal sampling point. So, the output of the measurement interferometer can be rewritten as:  
\[I_{m}(k)=A_{m}\cos(2\pi f_{0}\tau_{m}+\ 2\pi\frac{\tau_{m}}{2\tau_{a}}k) \tag{4}\]  
Equation (4) by fast Fourier transform (FFT) yields:  
\[\frac{\tau_{m}}{2\tau_{a}}\ =\ \frac{P}{N} \tag{5}\]  
where \(P\) is the abscissa index of the peak point in the frequency spectrum of the measurement interferometer. When the dispersion is considered, according to [PERSON] [23], it is easy to know:  
\[\begin{array}{l}\tau_{m}\ =\ \frac{2\pi\mu_{air}L_{m}}{c}\\ \tau_{a}\ =\ \frac{\pi f_{La}}{c}(1-2\pi\alpha\beta v_{g}t)\end{array} \tag{6}\]  
where \(n_{air}\) is the refractive index of air, \(L_{m}\) is the measured distance, \(c\) is the speed of light in the vacuum, \(n_{f}\) is the refractive index of the single mode fiber, \(L_{a}\) is the length of the single mode fiber in the auxiliary interferometer, \(\beta\) is the group velocity dispersion of the single-mode optical fiber, \(\alpha\) is the tuning rate of the ECL, \(v_{g}\) is the group velocity, and \(t\) is the time.  
So, the measured distance can be described as:  
\[L_{m}=\left(1-2\pi\alpha\beta v_{g}t\right)\frac{n_{f}L_{a}P}{n_{air}N} \tag{7}\]  
The second item in the right side of Equation (7) is caused by dispersion mismatch. Consequently, according to distance offsets, the absolute distance can be corrected by Equation (7).