### 6.2 Fabry–Pérot length sensing

In Figure 25 we have plotted the circulating power in a Fabry–Pérot cavity as a function of the laser frequency. The steep features in this plot indicate that such a cavity can be used to measure changes in the laser frequency. From the equation for the circulating power (see Equation (88)),
we can see that the actual frequency dependence is given by the term. Writing this term as
we can highlight the fact that the cavity is in fact a reference for the laser frequency in relation to the cavity length. If we know the cavity length very well, a cavity should be a good instrument to measure the frequency of a laser beam. However, if we know the laser frequency very accurately, we can use an optical cavity to measure a length. In the following we will detail the optical setup and behaviour of a cavity used for a length measurement. The same reasoning applies for frequency measurements. If we make use of the resonant power enhancement of the cavity to measure the cavity length, we can derive the sensitivity of the cavity from the differentiation of Equation (88), which gives the slope of the trace shown in Figure 25,
with as defined in Equation (103). This is plotted in Figure 33 together with the cavity power as a function of the cavity tuning. From Figure 33 we can deduce a few key features of the cavity:
• The cavity must be held as near as possible to the resonance for maximum sensitivity. This is the reason that active servo control systems play an important role in modern laser interferometers.
• If we want to use the power directly as an error signal for the length, we cannot use the cavity directly on resonance because there the optical gain is zero. A suitable error signal (i.e., a bipolar signal) can be constructed by adding an offset to the light power signal. A control system utilising this method is often called DC-lock or offset-lock. However, we show below that more elegant alternative methods for generating error signals exist.
• The differentiation of the cavity power looks like a perfect error signal for holding the cavity on resonance. A signal proportional to such differentiation can be achieved with a modulation-demodulation technique.