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 24View Image: two parallel mirrors form the Fabry–Pérot cavity. A laser beam is injected through the first mirror (at normal incidence).
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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 L, the wavelength of the laser λ and the reflectivity and transmittance of the mirrors. Assuming an input power of |a0|2 = 1, we obtain

P = |a |2 = ------------T1------------, (88 ) 1 1 1 + R1R2 − 2r1r2cos(2kL )
with k = 2π∕λ, P, 2 T = t and 2 R = r, 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,
a2-= --−-t1t2exp-(− ikL-)- (89 ) a0 1 − r1r2exp (− i2kL )
is the frequency-dependent transfer function of the cavity in transmission (the frequency dependency is hidden inside the k = 2πf ∕c).
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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 25View Image shows a plot of the circulating light power P1 over the laser frequency. The maximum power is reached when the cosine function in the denominator becomes equal to one, i.e., at kL = N π with N an integer. This is called the cavity resonance. The lowest power values are reached at anti-resonance when kL = (N + 1∕2)π. We can also rewrite

2L 2L 2πf 2kL = ω c--= 2 πf-c- = FSR-, (90 )
with FSR being the free-spectral range of the cavity as shown in Figure 25View Image. Thus, it becomes clear that resonance is reached for laser frequencies
fr = N ⋅ FSR, (91 )
where N is an integer.
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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 T1 + T2 = 0.2 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, fp. In order to compute the linewidth we have to ask at which frequency the circulating power becomes half the maximum:

2 ! 1 2 |a1(fp)| = 2|a1,max| . (92 )
This results in the following expression for the full linewidth:
( ) FWHM = 2fp = 2FSR--arcsin 1-−√-r1r2 . (93 ) π 2 r1r2
The ratio of the linewidth and the free spectral range is called the finesse of a cavity:
-FSR---- -------π--------- F = FWHM = ( 1−r1r2) . (94 ) 2arcsin 2√r1r2-
In the case of high finesse, i.e., r1 and r2 are close to 1 we can use the fact that the argument of the arcsin function is small and make the approximation
√ ---- F ≈ π--r1r2-≈ ---π----. (95 ) 1 − r1r2 1 − r1r2
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Figure 27: Power transmitted and circulating in a two mirror cavity with input power 1 W. The mirror transmissions are set such that T1 + T2 = 0.8 and the reflectivities of both mirrors are set as R = 1 − T. The cavity is undercoupled for T1 < 0.4, impedance matched at T1 = T2 = 0.4 and overcoupled for T1 > 0.4. 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 cases5:

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 26View Image. The circulating power shows that the resonance effect is better used in over-coupled cavities; this is illustrated in Figure 27View Image, 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 T + T 1 2, 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.

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