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
Figure 24: Typical optical layout of a two-mirror cavity, also called a Fabry–Pérot interferometer.
Two mirrors form the Fabry–Pérot interferometer, a laser beam is injected through one of the
mirrors and the reflected and transmitted light can be detected by photo detectors.
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
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: Power enhancement in a two-mirror cavity as a function of the laser-light frequency.
The peaks marks the resonances of the cavity, i.e., modes of operation in which the injected light is
resonantly enhanced. The frequency distance between two peaks is called free-spectral range (FSR).
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.
Figure 26: This figure compares the fields reflected by, transmitted by and circulating in a
Fabry–Pérot cavity for the three different cases: over-coupled, under-coupled and impedance
matched cavity (in all cases and the round-trip loss is 1%). The traces show the
phase and amplitude of the electric field as a function of laser frequency detuning.
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
Figure 27: Power transmitted and circulating in a two mirror cavity with input power 1 W. The
mirror transmissions are set such that and the reflectivities of both mirrors are set
as . The cavity is undercoupled for , impedance matched at
and overcoupled for . The transmission is maximised in the impedance-matched case and
falls similarly for over or undercoupled settings. However, the circulating power (and any resonance
performance of the cavity) is much larger in the overcoupled case.
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
- 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