# Stabilization of Transverse Modes for a High Finesse Near-Unstable Cavity
## 3 PDH Technique in Cavity Stabilization for Different Transverse Cavity Modes  
Due to mechanical vibration of mirror mounts, temperature drift, air flows, and frequency shift of the incident laser, the optical cavity suffers a frequency drift and line broadening. There are various methods to lock the cavity, for example, the side of fringe stabilization technique [21], the PDH stabilization technique [22; 23], the Hansch-Couilaud technique [24], and the tilt-locking technique [25]. The side of fringe stabilization technique can only lock the cavity to the side of transmission or reflection peak, whereas it is not the optimal position of the cavity. The Hansch-Couilaud technique requires a nonlinear crystal cut at the Brewster angle or putting a polarizer in the middle of the cavity to alter the polarization of the incident light, which may reduce the finesse of the cavity. The tilt-locking technique takes the advantage of the interference among different spatial modes and needs split detectors with a high position precision [26]. Compared to the above mentioned techniques, the PDH stabilization technique is stable, easy to operate, and capable of operating continuously for a long time.  
While most PDH stabilization experiments are focused on locking the fundamental cavity mode, high-order transverse cavity modes may be useful for some specific cases, for example, using the higher-order Laguerre-Gauss modes to reduce the contribution of coating Brownian noise for gravitational wave detection [27; 28; 29; 30]. For type-I SPDC, the down-converted photons are distributed on a down conversion ring. If locking the cavity in the fundamental mode, the two entangled photons may leave the cavity collinearly and only one light spot can be observed. On the other hand, if locking the cavity in a high-order Hermite-Gaussian mode, e.g., the TEM\({}_{01}\) mode, the generation rate of entangled photons can be enhanced by the cavity in two different directions. In this way, the enhanced SPDC occurs in a specific down conversion angle. For different cavity modes TEM\({}_{\text{mnd}}\)(m = 0, 1,..., n = 0, 1,..., q = 0, 1,...) with m and n being the horizontal and vertical orders of the transverse mode and q being the order of the longitudinal mode, respectively, the resonant frequency of each mode is related to not only the longitudinal mode number but also the transverse mode numbers [20]  
\[v_{mnd}=\frac{c}{2L}\left(q+\frac{1}{\pi}(m+n+1)\cos^{-1}(g_{1}g_{2})\right). \tag{2}\]  
We simulated the error signals of the TEM\({}_{00}\), TEM\({}_{01}\), and TEM\({}_{02}\) cavity modes using the Finesse software, which has been used in Advanced LIGO and GEO600 to calculate the transmission matrix of different optical components. Figure 2 shows the simulation results for different transverse cavity modes. The high-order transverse cavity modes were generated through the simulation of the PDH stabilization technique when deliberately misaligning one of the mirrors (M1 in Figure 1) by \(3\times 10^{-4}\) rad. The modulation frequency of electro-optic phase modulator (EOM) is 5 MHz, and the cavity parameters are the same as the practical values we used in the following experiments. By scanning the cavity length, three different transverse cavity modes were demonstrated in the simulation. The simulated error signals showed that it is possible to lock this near-unstable cavity in different transverse modes based on the PDH stabilization technique.