# Compensation of Frequency Drift in Frequency-Sweep Polarization-Modulation Ranging System
## 2 Principle and Method  
### The Principle and Model  
The frequency-sweep polarization-modulation ranging system consists of a continuous wave laser, an isolator, a polarizing beam splitter (PBS), an EOM, a quarter wave plate (QWP), a beam expander (EXP) and the target, as shown in Figure 1.  
To illustrate the working principle of the FSPM ranging method, a Cartesian coordinate system is established. The \(z\)-axis represents the direction of wave propagation, while the \(x\)-axis is the polarization direction of linearly polarized light generated by the laser. In the system, continuous polarized light emitted by the laser travels through an isolator and a PBS tipped 45 degrees from the \(x\)-axis, so equal amplitude of polarization components in the \(x\)-axis and \(y\)-axis is achieved when light travels into the EOM. The EOM generates phase retardation between the two components in a sinusoidal manner with linearly changed frequency. The QWP has a fast axis oriented at 45 degrees to the \(x\)-axis, so the light travels to the target and returns by the same path, to then be demodulated by the EOM. The phase difference between the polarization components when light transmits through the EOM successively can be respectively described as:  
\[\begin{array}{l}\varphi_{1}=\ k\sin(\omega t),\\ \varphi_{2}=\ k\sin(\omega t+\varphi)+\delta_{\mathrm{T}},\end{array} \tag{1}\]  
where \(\omega\) is the modulation frequency, \(k\) represents the modulation depth, \(\delta_{\mathrm{T}}\) represents the additional phase delay between the polarization components, \(\varphi\) is the phase shift of the modulation signal during the round-trip time, and \(\varphi=4\pi f(t)nl/c\), where \(f(t)\) is the modulation frequency, \(c\) represents the speed  
Figure 1: Schematic diagram of the polarization-modulation ranging system. EOM: electro-optic modulator; PBS: polarizing beam splitter; QWP: quarter wave plate.  
of light in vacuum, \(n\) is the refractive index of the air, and \(l\) is the distance to be measured. Note that in the equation, \(0<k\leq 1\) for linear modulation and \(\delta_{\mathrm{T}}\) is small, typically less than \(\pi/2\). With the amplitude of \(x\) and \(y\) components normalized, the intensity of light received by the detector can be written as:  
\[\mathrm{I}\ =\ 1\ -\ \cos(\varphi_{1}-\varphi_{2})\ =\ 1\ -\ \cos\Bigl{(}2k\cos \frac{2\omega t\ +\ \varphi}{2}\sin\frac{\varphi}{2}\Bigr{)}\cdot\cos\delta_{\mathrm{T}}+\ \sin\Bigl{(}2k\cos\frac{2\omega t\ +\ \varphi}{2}\sin\frac{\varphi}{2} \Bigr{)}\cdot\cos\delta_{\mathrm{T}}. \tag{2}\]  
After simplification and filtering the high-frequency components, the equation can be rewritten as:  
\[\mathrm{I}\ =\ 1-\biggl{(}1-\frac{k^{2}}{2}+\frac{k^{2}}{2}\cos\varphi\biggr{)} \cdot\cos\delta_{\mathrm{T}}. \tag{3}\]  
Obviously, if \(\cos\delta_{\mathrm{T}}\) is constant, there exists a cosine relationship between intensity and modulation frequency. Minimums of the equations By obtaining the corresponding frequencies \(f_{x}\),\(f_{x+1}\) (k\(\in\)N) at adjacent minima, the distance can be determined by:  
\[l=\ \mathrm{N}\cdot\frac{c}{2\mathrm{n}f_{x}},\ \mathrm{in\ which}\ \mathrm{N}\ =\ \biggl{[}\ \frac{f_{x}}{f_{x+1}-f_{x}}\ \biggr{]}, \tag{4}\]  
where, N is the wavelength multiple, [ ] is the rounding operation, \(f_{x}\)/(\(f_{x+1}-f_{x}\)) is the initial value of the wavelength multiple, which comprises an integer and a fraction part, so the rounding operation is necessary. For convenience in the manuscript, The frequencies \(f_{x}\) and \(f_{x+1}\) will be defined as \"in phase\" frequency. Value of difference of adjacent in-phase frequencies i.e., \(f_{x+1}-f_{x}\) will be defined as frequency interval \(f_{\mathrm{b}}\), which is ideally a constant when distance is invariant.