### Optimization of a Fabry-Perot Filter

The Fabry-Perot etalon is often used as a narrow-band filter in optics experiments. Usually, transmission, extinction ratio and bandwidth of etalon are three important parameters in application. For a typical quantum optics experiment, a transmission of 60~70 %, an extinction ratio of~ 40dB and a bandwidth of ~150 MHz should satisfy experimental needs. FPE series are designed to meet these specifications. In real application, two factors should be taken care of, they are mode matching, temperature stability / tunablity of etalon.

##### Mode Matching

The first thing need to know for mode matching is the cavity mode. A wave inside the cavity that can reproduce itself both in phase and in amplitude after around trip travel is called the mode of a cavity. The mode in cavity should satisfy two conditions:

1. Laser frequency match cavity resonance frequency : standing wave condition
2. Laser beam wavefront match the curvature of cavity reflection mirror : stable mode condition

The first one is often accomplished by tuning laser current, temperature, PZT voltage or turning cavity length directly. The second one is accomplished by a mode matching lens to adjust the wave-front to match the cavity curvature, and this process is often called mode matching process. Take a planar-convex cavity, such as our product FPE001 shown in Fig1, for example; the wavefront in side-A should be a plane and the beam waist is W0 while the curvature of wavefront in side-B is Rw. Rw should be equal to Rc, the radius of curvature of the cavity in convex side. Assume the cavity center length is d and n is the index of refraction of the cavity material. We can obtain eq.1 and calculate the value of W0.

Figure 1. Laser beam wavefront and a planar-Convex cavity.

Once the wavefront characteristic in cavity is known, the input beam of laser can be adjusted to the same characteristic. The process is called mode matching. Some of the parameters used for mode matching calculation are defined as below.

f: focal length of mode matching lens
L1: distance between input beam waist (W1) and mode matching lens
L2: distance between output beam waist (W2) and mode matching lens

L1 and L2 can be calculated from below equation derived from ABCD matrix.

Where

##### The Termal Management

The temperature stability will affect the filtering performance of etalon. The temperature drift of the etalon will cause the output power fluctuation and decrease S/N ratio. Take Fused Silica etalon with 99% reflectivity for example, ±1.84 m°K temperature will cause ±5 MHz central frequency shift. And it will introduce 4.3% output power fluctuation. So for the design or selection of the temperature controller of the etalon, it will be better to have long-term temperature stability of 3 m°K with m°K temperature adjustability.

##### Experimental Tips for Using a Fabry-Peort Etalon

There are several tips to know before using a Fabry Perot etalon.

##### How to coupling the light into a Fabry Perot etalon.

Scan the laser frequency at least half FSR of the cavity.

Adjust mirror so that the reflective light from the cavity is coincident with the incident light

With above tips, it should be easier to get the transmission light from the cavity. After that, the optimization can be done by fine tune and aliment and cavity position.

##### The transverse mode of the etalon

The transverse mode exists when the surface of cavity is not planar. The transverse mode spacing is shown in below equation. User should choose the radius of curvature carefully to avoid the signal to be filtered fall into the transverse mode the cause the low extinction ratio. To maximize the output power of a Fabry Perot etalon, it is very important to eliminate the power in high order transverse mode. If the light source doesn't have a good shape like that of an ECDL, it is suggested to use a spatial filter or a single-mode fiber to shape the beam into a TME00 mode.

Where R1 and R2 means the radius of curvature in each side.

##### Polarization of the incident beam

Due to the birefringence characteristic of etalon material, different polarization light will have different optical path in the etalon. It will result in different resonant frequency in the cavity and reduce the extinction ratio of un-wanted frequency. User should adjust the polarization of the incident beam until there is only transmission peak.