# Stabilization of Transverse Modes for a High Finesse Near-Unstable Cavity
## 5 Experimental Results  
The proposed method was experimentally confirmed in an empty FP cavity and a FP cavity with a BBO crystal inside. Here, we first demonstrated the performance of the FP cavity locked to 605.78-nm laser light in a fundamental mode. When applied with a triangle-wave voltage of a frequency 10 Hz (see Figure 5a) to the ring piezoelectric transducer, the corresponding reflection signal and the error signal are shown in Figure 5b,c, respectively. The inset of Figure 5c shows the details of the error signal, which matches well with the fast modulation mode. Note that the error signal is slightly asymmetric. One possible reason is that when scanning a high-finesse FP cavity, the ringing effects will cause the asymmetry of response profile of transmission or reflection light [33; 34], as also shown in the inset of Figure 5b. It will destroy the phase information making the error signal asymmetric especially when the modulation frequency of EOM is small. The fundamental mode is widely used in cavity locking experiments for the advantage of high light transmission efficiency and easy alignment. Figure 6 shows the typical error signal when locking the empty FP cavity to a fundamental TEM\({}_{00}\) mode,  
Figure 4: The ringdown decay dynamics recorded by an oscilloscope. The red dots are experimental results, and the blue curve is the exponential fitting result. The ringdown decay time constant \(\tau\) was measured to be 420 ns.  
and the intensity distribution of the transmission beam is shown in Figure 7a. When the FP cavity was locked to the TEM\({}_{00}\) mode at 605.78 nm, all electrical signals fluctuated within a small range. The final linewidth of the cavity was determined by the accuracy of the feedback loop. The stable locking time of the FP cavity is mainly determined by the stability of the reference laser, and a 20-min stable operation was experimentally achieved, which can be easily improved when a more stable reference laser is used.  
Figure 5: The scanning voltage (**a**), the reflection signal I\({}_{\mathrm{R}}\) (**b**) and the error signal (**c**) of the FP cavity when the voltage applied on PZT on mirror M2 is scanning. The insets in (**b**,**c**) show the asymmetry of the reflection profile and the details of the error signal, respectively.  
We also demonstrated the feasibility to lock the FP cavity in the high-order transverse modes. Figure 7b,c show the intensity distribution of the TEM\({}_{01}\) and TEM\({}_{02}\) modes output from the locked FP cavity, while Figure 8 depicts the corresponding error signals from the FP cavity in the TEM\({}_{00}\), TEM\({}_{01}\) and TEM\({}_{02}\) modes, respectively. The error signals operating in the high-order transverse modes are broader than that in the fundamental mode, and the amplitude of the error signal operating in the high-order transverse modes is smaller than that in the fundamental mode, which makes the cavity more sensitive to the environmental fluctuations/vibrations and difficult to be locked in the high-order transverse modes. The key to improve the stability of the FP cavity operating in the high-order transverse mode is to isolate the vibration in the clean room and employ a stable reference laser. The output signals of the cavity, when locking in TEM\({}_{01}\) and TEM\({}_{02}\) modes, were much like those in Figure 6 (not shown here).  
For cavity-enhanced SPDC experiments, a nonlinear crystal should be put inside the cavity. The nonlinear crystal is usually birefringent, so one should choose appropriate pump polarization and align the cavity to satisfy the phase-matching condition. Nevertheless, the locking process remains the same.  
Figure 6: The output signals from the FP cavity when it is locked to the TEM\({}_{00}\) mode of the dye laser at 605.78 nm. (**a**) corresponds to the scanning voltage, (**b**) shows the transmission intensity I\({}_{\rm{T}}\), and (**c**) is the error signal.