P010010-00-R - LIGO - California Institute of Technology

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where it’s been assumed that the power in the signal sidebands is negligible. The
signal of interest is the second part, and the signal sideband fields can be expressed
in terms of Eq. (2.1) and Eq. (2.14), where it has been assumed that the mirror
reflectivity retm ≈ 1.
E+f = iklarmh |Earm|tcc(+f) (2.17)
E−f = iklarmh |Earm|tcc(−f) (2.18)
Substitution into the second term of Eq. (2.16) and some algebra gives a measured
signal voltage as
Vsignal = Zimp
ηe
hν0
� �
2klarmhℜ �H(f)e−i2πft (2.19)
where Zimp is the transimpedance of the photodiode in amps to volts, η is the quantum
efficiency of the photodiode in number of electrons to number of photons, e is the
electric charge, hν0 is the energy of a carrier photon. The transfer function � H(f) is
defined as
�H(f) = |Earm||E0|(itcc(+f)) ∗ e iφ0 + (itcc(−f))e −iφ0 (2.20)
where φ0 is the phase of the carrier field E0.
If the phase of the carrier can be chosen arbitrarily, it can be seen that the output
at a particular frequency can be maximized by choosing the carrier phase such that
both terms have the same overall phase. This can be found to be 2φ0 = arg(tcc(+f))+
arg(tcc(−f)) + π. This carrier phase defines a maximum transfer function
| � Hmax(f)| = |Earm||E0|(|tcc(−f)| + |tcc(+f)|) (2.21)
If tcc(+f) ∗ = −tcc(−f), then this holds true for all frequencies. This condition is
met at resonance (RSE or dual-recycling), but fails for all detunings. Hence, all real
transfer functions will be less than or equal to Eq. (2.21). The transfer function
�Hmax(f) defines a theoretically ideal transfer function which, in general, is not be
reached at all frequencies.