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 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) 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:
with optical transfer matrix defined as: and the response to the cavity elongation 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:
whereIn 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 coordinateIn 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:
We introduced here the normalized optical power
with standing for the mean optical power circulating inside the cavity, and Substitution of the force (312) into Eq. (305
) gives the following final equation of motion:
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Living Rev. Relativity 15, (2012), 5
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