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

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particular frequency can be achieved by a slightly larger detuning. This shifts the
frequency of best sensitivity to a lower value.
Following the derivation of Mizuno in which the denominator function of Eq. (2.1)
is expanded to first order in frequency around the carrier frequency, the true peak
frequency and bandwidth for a given detuning and target frequency can be approxi-
mated by
where the function E(f, φ) is defined
fp � ftar + ℑ {E(ftar, φdt0)} /(4πlarm/c) (2.8)
∆f3dB � |ℜ {E(ftar, φdt0)}| /(2πlarm/c) (2.9)
E(f, φ) = 1 −
1 − r ′ secritme −i2φ
ritme −i2πfτa − Aitmr ′ seme −i2πfτa+φ
(2.10)
These approximations work best when the Q is high, that is f/∆f3 dB ≫ 1. Figure 2.5
shows the peak frequency and bandwidth as a function of the detuning phase for the
same optics in Figure 2.4.
In these cases, the peak frequency is seen to increase from DC to some maximum
frequency, at which point the trend reverses and the peak frequency drifts back to
DC. The highest peak frequency is found by taking a derivative of Eq. (2.6). 2
fpmax =
c
4πlarm
�
r
arctan
′ semTitm
�
(Ritm − AitmR ′ sem)(1 − RitmR ′ �
sem)
(2.11)
The fact that there is a peak frequency is consequence of the under-coupled nature
of the signal cavity. The phase on reflection of an under-coupled cavity starts at zero
on resonance, deviates some amount away from zero, then slowly comes back, in the
same fashion as in Figure 2.5.
It’s interesting to compare the peak frequency to the RSE bandwidth at zero
2 More precisely, taking the derivative of the argument of the arctangent. Maximizing f is the
same as maximizing tan(πfτa).