A Fabry–Pérot cavity consists of two movable mirrors that are separated by a distance , where is the distance at rest with standing for a single pass light travel time, and and are the small deviations of the mirrors’ position from the equilibrium. Each of the mirrors is described by the transfer matrix with real coefficients of reflection and transmission according to Eq. (243). As indicated on the scheme, the outer faces of the mirrors are assumed to have negative reflectivities. While the intermediate equations depend on this choice, the final results are invariant to it. The cavity is pumped from both sides by two laser sources with the same optical frequency .

The coupling equations for the ingoing and outgoing fields at each of the mirrors are absolutely the same as in Section 2.2.5. The only new thing is the free propagation of light between the mirrors that adds two more field continuity equations to the full set, describing the transformation of light in the Fabry–Pérot cavity. It is illuminating to write down input-output relations first in the time domain:

Further we use notation for the light travel time between the mirrors.The frequency domain version of the above equations and their solutions are derived in Appendix A.1. We write these I/O-relations given in Eqs. (545) in terms of complex amplitudes instead of 2 photon amplitudes, for the expressions look much more compact in that representation. However, one can simplify them even more using the single-mode approximation.

(i) in GW detection, rather high-finesse cavities are used, which implies low transmittance coefficients for the mirrors

(ii) the cavities are relatively short, so their Free Spectral Range (FSR) is much larger than the characteristic frequencies of the mirrors’ motion: and (iii) the detuning of the pump frequency from one of the cavity eigenfrequencies: is also small in comparison with the FSR: In this case, only this mode of the cavity can be taken into account, and the cavity can be treated as a single-mode lumped resonator. Note also that, while our intermediate equations below depend on whether is even or odd, the final results do not. Therefore, we assume for simplicity that is even.Expanding the numerators and denominators of Eqs. (540, 545) into Taylor series in and keeping only the first non-vanishing terms, we obtain that

and where is the cavity half-bandwidth, are the cavity left and right reflectivities and its transmittance, and is the sum variation of the cavity length.

The above optical I/O-relations are obtained in terms of the complex amplitudes. In order to transform them to two-photon quadrature notations, one needs to employ the following linear transformations:

- change frequency and rewrite the relations between the input and output operators in the form: where is an arbitrary complex-valued function of sideband frequency ;
- use the definition (57) to get the following relations for two-photon quadrature operators:

Applying transformations (289) to Eqs. (280), we rewrite the I/O-relations for a Fabry–Pérot cavity in the two-photon quadratures notations:

whereTherefore, the I/O-relations in standard form read:

with optical transfer matrix defined as: and the response to the cavity elongation defined as:Note that due to the fact that the reflectivity and the transmission matrices and satisfy the following unitarity relations:

The mechanical equations of motion of the Fabry–Pérot cavity mirrors, in spectral representation, are the following:

where are the mechanical susceptibilities of the mirrors, stand for any external classical forces that could act on the mirrors (for example, a signal force to be detected), are the radiation pressure forces acting on the mirrors, and , , , are the powers of the waves , , etc. The signs for all forces are chosen in such a way that the positive forces are oriented outwards from the cavity, increasing the corresponding mirror displacement .In the spectral representation, using the quadrature amplitudes notation, the radiation pressure forces read:

The first group, as we have already seen, describes the regular constant force; therefore, we omit it henceforth.In the single-mode approximation, the radiation pressure forces acting on both mirrors are equal to each other:

and the optical field in the cavity is sensitive only to the elongation mechanical mode described by the coordinate . Therefore, combining Eqs. (301), we obtain for this mode: where is the reduced mechanical susceptibility and is the effective external force.In the simplest and at the same time the most important particular case of free mirrors:

the reduced mechanical susceptibility and the effective external force are equal to and where is the effective mass of the elongation mechanical mode.It follows from Eqs. (291) and (304) that the radiation pressure force can be written as a sum of the random and dynamical back-action terms, similarly to the single mirror case:

with the random component equal to and the ponderomotive rigidity that readsWe introduced here the normalized optical power

with standing for the mean optical power circulating inside the cavity, and is some (in general, arbitrary) mass. Typically, it is convenient to make it equal to the reduced mass .Substitution of the force (312) into Eq. (305) gives the following final equation of motion:

Thus, the effective mechanical susceptibility for a Fabry–Pérot cavity reads:
Living Rev. Relativity 15, (2012), 5
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