# Compensation of Frequency Drift in Frequency-Sweep Polarization-Modulation Ranging System
## 2 Principle and Method
### Reciprocating Sweeping-Based Measurement of \"in Phase\" Frequency  
The change of additional phase delay accelerates or decelerates the polarization variation when the wave is modulated across the \"in phase\" frequency, thus destroying the symmetry of the intensity curve, leading to the shift of minima. The shift can be manifested in variation of the curve slope. Through the derivation of Equation (3), the slope of the intensity curve at the ideal minimum \(\varphi_{0}\) can be written as:  
\[\mathrm{I}^{\prime}(\varphi_{0})=\lim_{\Delta\varphi\to 0}\frac{I(\varphi_{0}+ \Delta\varphi)\!-\!I(\varphi_{0}-\Delta\varphi)}{2\Delta\varphi}=\lim_{\Delta \varphi\to 0}\frac{-\Big{(}1\!-\!\frac{k^{2}}{2}+\frac{k^{2}}{2}\cos \Delta\varphi\Big{)}(\sin\delta_{0}\sin\Delta\delta)}{\Delta\varphi}, \tag{6}\]  
where \(\delta_{0}\) is the thermally induced phase delay when \(\varphi\ =\ \varphi_{0}\), \(\Delta\varphi\) is the increment on point of \(\varphi_{0}\), \(\Delta\delta\) is the increment on point of \(\delta_{0}\). According to the above equation, \(\mathrm{I}^{\prime}(\varphi_{0})\) is a result of interaction between modulation depth, frequency scanning, and phase delay variation. Large modulation depths and frequency sweep speed are beneficial for eliminating the effect of thermally induced phase delay. However, modulation depth and sweep speed are often limited owing to equipment limitations, and a larger modulation depth requires a larger microwave voltage, thus aggravating the heating problem of the modulation circuit. Interestingly, we found that Equation (6) is an odd function about \(\varphi\), so the orientation of the frequency shift can be controlled through frequency sweep directions. When modulation frequency is swept forward (i.e., from the lower frequency to the higher frequency), \(\Delta\varphi(t)>0\), otherwise, \(\Delta\varphi(t)<0\). Therefore, during the forward and backward sweep, the minimum drifts in opposite directions. Through averaging of such reciprocating sweeping, the drift errors can be diminished and frequency measurement stability can be improved. The method is further analyzed below.  
Suppose the ideal frequency under test is \(f_{0}\). \(f_{1}\) and \(f_{2}\) are frequencies very close to \(f_{0}\) and satisfy the equations \(f_{1}\!=\!f_{0}-\delta\!f\) and \(f_{2}\!=\!f_{0}+\delta\!f\). If the system is not affected by thermally induced phase delay, obviously we get the equation \(\mathrm{I}(f_{1})=\mathrm{I}(f_{2})\), then the exact value of \(f_{0}\) can be worked out through: \(f_{0}=\langle f_{1}+f_{2}\rangle/2\). However, due to thermally induced phase delay, during the forward sweeping, \(f_{1}\) and  
Figure 2: Thermally induced phase delay of round-trip polarization light in the system.  
\(f_{2}\) respectively moves \(\Delta f_{1}^{1}\) and \(\Delta f_{2}^{1}\) towards lower frequency, then in the backward sweeping, \(f_{1}\) and \(f_{2}\) respectively moves \(\Delta f_{1}^{2}\) and \(\Delta f_{2}^{2}\) towards the opposite side, as shown in Figure 3.  
Taking random measurement errors \(v\) into account, the extremum of the measurements are respectively expressed as:  
\[f_{0}^{1}=\frac{\left(f_{1}-\Delta f_{1}^{2}\pm|v_{1}^{2}|\right)+\left(f_{2} -\Delta f_{2}^{2}\pm|v_{2}^{2}|\right)}{4n}, \tag{7}\]  
Through \(n\) times of such sweeps, the frequency can be obtained by:  
\[f_{0}=\frac{2n(f_{1}+f_{2})+\sum_{i=1}^{n}\left(\Delta f_{1}^{2n}-\Delta f_{1} ^{2n-1}\right)+\sum_{i=1}^{n}\left(\Delta f_{2}^{2n}-\Delta f_{2}^{2n-1} \right)\pm\sum_{i=1}^{2n}|v_{1}^{n}|\pm\sum_{i=1}^{2n}|v_{2}^{n}|}{4n}. \tag{8}\]  
During the time of reciprocating sweeping, variation of \(\delta\) is small so items of \(\sum_{i=1}^{n}\left(\Delta f_{1}^{2n}-\Delta f_{1}^{2n-1}\right)\) and \(\sum_{i=1}^{n}\left(\Delta f_{2}^{2n}-\Delta f_{2}^{2n-1}\right)\) can be diminished to some extent, and the random measurement error of frequency can be compensated by averaging.