5.1 The two-mirror cavity: a Fabry–Pérot interferometer

We have computed the field amplitudes in a linear two-mirror cavity, also called Fabry–Pérot interferometer, in Section 2.2. In order to understand the features of this optical instrument it is of interest to have a closer look at the power circulation in the cavity. A typical optical layout is shown in Figure 24: two parallel mirrors form the Fabry–Pérot cavity. A laser beam is injected through the first mirror (at normal incidence).

The behaviour of the (ideal) cavity is determined by the length of the cavity , the wavelength of the laser and the reflectivity and transmittance of the mirrors. Assuming an input power of , we obtain

with , , and , as defined in Section 1.4. Similarly we could compute the transmission of the optical system as the input-output ratio of the field amplitudes. For example,
is the frequency-dependent transfer function of the cavity in transmission (the frequency dependency is hidden inside the ).

Figure 25 shows a plot of the circulating light power over the laser frequency. The maximum power is reached when the cosine function in the denominator becomes equal to one, i.e., at with an integer. This is called the cavity resonance. The lowest power values are reached at anti-resonance when . We can also rewrite

with FSR being the free-spectral range of the cavity as shown in Figure 25. Thus, it becomes clear that resonance is reached for laser frequencies
where is an integer.

Another characteristic parameter of a cavity is its linewidth, usually given as full width at half maximum (FWHM) or its pole frequency, . In order to compute the linewidth we have to ask at which frequency the circulating power becomes half the maximum:

This results in the following expression for the full linewidth:
The ratio of the linewidth and the free spectral range is called the finesse of a cavity:
In the case of high finesse, i.e., and are close to we can use the fact that the argument of the function is small and make the approximation

The behaviour of a two mirror cavity depends on the length of the cavity (with respect to the frequency of the laser) and on the reflectivities of the mirrors. Regarding the mirror parameters one distinguishes three cases:

• when the cavity is called undercoupled
• when the cavity is called impedance matched
• when the cavity is called overcoupled

The differences between these three cases can seem subtle mathematically but have a strong impact on the application of cavities in laser systems. One of the main differences is the phase evolution of the light fields, which is shown in Figure 26. The circulating power shows that the resonance effect is better used in over-coupled cavities; this is illustrated in Figure 27, which shows the transmitted and circulating power for the three different cases. Only in the impedance-matched case can the cavity transmit (on resonance) all the incident power. Given the same total transmission , the overcoupled case allows for the largest circulating power and thus a stronger ‘resonance effect’ of the cavity, for example, when the cavity is used as a mode filter. Hence, most commonly used cavities are impedance matched or overcoupled.