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:
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
Expanding the numerators and denominators of Eqs. (540, 545) into Taylor series in and keeping only the first non-vanishing terms, we obtain that
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:
Applying transformations (289) to Eqs. (280), we rewrite the I/O-relations for a Fabry–Pérot cavity in the two-photon quadratures notations:
Therefore, the I/O-relations in standard form read:
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:
In the spectral representation, using the quadrature amplitudes notation, the radiation pressure forces read:
In the single-mode approximation, the radiation pressure forces acting on both mirrors are equal to each other:
In the simplest and at the same time the most important particular case of free mirrors:
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:
We introduced here the normalized optical power
Substitution of the force (312) into Eq. (305) gives the following final equation of motion:
Living Rev. Relativity 15, (2012), 5
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