# Stabilization of Transverse Modes for a High Finesse Near-Unstable Cavity
## 4 Experimental Setup  
In general, a high-finesse FP cavity is used to lock a dye laser, and therefore to get a narrow linewidth laser source, but not the reverse. The main reason is that it is difficult to lock two spatially separated high-finesse cavities resonating simultaneously. Nevertheless, the dye laser has the advantage of broad wavelength tuning range, which is usually not reachable by other narrow-band laser sources such as semiconductor diode lasers. In some specific cases, one may need to lock a cavity to resonate at a specific wavelength, which may not be easily available from other laser sources. For example, the rare-earth ions play an important role in quantum storage, and the operating wavelength is determined by the transitions between the energy levels of the rare-earth ions, which may locate in the visible spectral range. One such a typical example is the electromagnetically induced transparency effect (EIT) in Pr\({}^{3+}\)-doped Y\({}_{2}\)SiO\({}_{5}\) crystal, which has many potential applications in quantum storage and quantum repeater. The operating wavelength is at 605.78 nm determined by the hyperfine energy-level transitions \({}^{3}\)H\({}_{4}\ \leftrightarrow\ ^{1}\)D\({}_{2}\) of Pr\({}^{3+}\) ions. In this case, it is necessary and very convenient to utilize the dye laser as the narrow-band light source. Figure 2: The simulation results of error signals for three different transverse cavity modes. By misaligning the mirror M1 in Figure 1 by \(3\times 10^{-4}\) rad and moving the mirror M1 along the optical axis, three different transverse cavity modes can be observed. The blue, red, and green curves correspond to the error signals of TEM\({}_{00}\), TEM\({}_{01}\) and TEM\({}_{02}\) modes, respectively. Figure 3 shows our experimental setup. The laser we used was a Matisse 2 DX dye laser (Spectra-Physics), and the wavelength was tuned to 605.78 nm. The linewidth of the output laser was about 100 kHz. Firstly, the laser beam passed through an electro-optic phase modulator (PM-C-BB, LINOS), which was driven by a function generator with a frequency of 5 MHz. Then it was launched into a home-made FP cavity that consists of a plane mirror M1 and a concave mirror M2, as shown in Figure 1. All these two mirrors were mounted on the invar cavity to isolate vibrations and to minimize the thermal-induced expansion/contraction. A specially designed tiny motorized pitch and yaw platform with a temperature-controlled unit was installed between the two mirrors, which could used to hold the crystal for OPO or other experiments. It ensures that the phase-matching condition can be achieved by tilting the crystal slightly. In Figure 1, we marked each side of mirrors M1 and M2 with a letter. The side a of M1 was coated with antireflection coating for 302.89 nm and 605.78 nm laser light. Both the side b of M1 and the side c of M2 were coated with antireflection coating for 302.89 nm light but highly reflection coating for 605.78 nm light, respectively. The side d of M2 was coated with antireflection coating for both 302.89 nm light and 605.78 nm light. We measured the finesse at 605.78 nm employing the cavity ringdown spectroscopy [31; 32]. An accousto-optical modulator AOM (MT110-A1.5-VIS, AA Opto Electronic) was used to switch the input light on and off. Figure 4 shows the ringdown decay dynamics with a decay time constant \(\tau\) of 420 ns. The measured finesse \(F\) of the FP cavity was 7330, corresponding to a reflection coefficient of 99.957%. The optical losses of the mirrors were estimated to be about 270 ppm. A ring-shaped piezoelectric transducer with a travel distance of 4 \(\mu\)m was mounted on the side d of mirror M2 to adjust the cavity length. The cavity length \(L\) was set to be 54 mm. The free spectral range \(\Delta v_{FSR}\) and the linewidth \(\delta v\) of the FP cavity can be calculated using Equations (3) and (4), respectively. \[\Delta v_{FSR}=\frac{c}{2n_{0}L}\,, \tag{3}\]  
Figure 3: The experimental setup of locking an FP cavity.