# Compensation of Frequency Drift in Frequency-Sweep Polarization-Modulation Ranging System
## 2 Principle and Method
### Influence of Thermally Induced Phase Delay and Traditional Correction Method  
In a transverse modulator configuration, the effects of temperature variation on modulation characteristics are mainly as follows: first, due to the thermo-optic effect of the crystal, the refractive index of the crystal changes with temperature. More concretely, the change in the refractive index along the direction of light transmission causes the optical path variation of light, and the change in refractive index perpendicular to the direction of light transmission results in an additional phase delay of the polarization [25]. Second, due to the elasto-optical effect of the crystal, the stress caused by the thermal expansion of the crystal also changes the refractive index and the longitudinal length of the crystal. As a result, the thermally induced phase delay can be expressed as follows:  
\[\delta_{\mathrm{T}}=\frac{2\pi}{\lambda}\Delta n_{\mathrm{f}}l+\frac{2\pi}{ \lambda}n_{\mathrm{f}}\Delta l, \tag{5}\]  
where \(\lambda\) represents wavelength of light, \(n_{\mathrm{f}}\) is the initial value of natural birefringence in the direction perpendicular to light transmission, and \(\Delta n_{\mathrm{f}}\) is the thermally induced birefringence variation. \(l\) represents the initial value of the length of the crystal, and \(\Delta l\) is the thermally induced variation of the crystal length.  
Several techniques have been applied to compensate for the thermally induced phase delay in the applications of EOM. An example of this is connecting a compensate crystal that has the same size and performance with the modulate crystal, with the optical axes of the crystals perpendicular to each other so that the phase delay in the two identical crystals are complementary [26]. Another way is to make the two crystals in series, with one half-wave plate placed in the middle of them to make the polarization plane of the polarized light rotate by \(\lambda/2\)[24]. In these methods, crystals with exactly the same parameters and their perfect assembly are the key to compensation, but that is difficult to achieve. In the FSPM ranging system, a quarter-wave plate is placed between the crystal and the target so that when light passes through the quarter-wave plate during emitting and returning, the plane of polarization can be rotated by \(\lambda/2\) to achieve the complementation [27; 28]. In this structure,the adjustment is simpler and the compensation is better. However, in the FSPM ranging system, a mismatch of polarizing devices and reflection of polarized light are inevitable and lead to the additional rotation of the polarization plane of light. Even a very small mismatch of polarization devices or any reflection in the system will result in the rotation of the polarization plane, and produce residual thermally induced phase delay.  
In our experiment, the residual thermally induced phase delay was tested first. Since thermally induced phase delay varies slowly, it can be measured through a polarization analysis system (Thorlabs, PAX5710). At the same time, the temperature in the modulator chamber was tested with a high-precision temperature controller (Wavelength, Tc_lab), with results shown in Figure 2. As is shown, with the increase of temperature, the phase delay slowly increased, accompanied by periodic oscillation, and when temperature growth slowed down, the oscillation speed also decreased.