A short detour here will prove useful. We will consider the case of a three-mirror coupled cavity, such as the one depicted in figure 3.7. Of particular interest is the frequency response of such a cavity, considered in three distinct but interesting cases. In order to keep this discussion relevant to gravitational wave detectors, we will limit ourselves to the coupling of a long (arm) cavity with a short (recycling) cavity, where in each case the long (length L) cavity is held at a length such as to be resonant with the laser carrier field (the primary optical frequency), and the mirror reflectivities are ordered such thatrr< ri< re. We will continue to use a subscript c for the long cavity, while introducing a subscript r for the short cavity (theris for ‘recycling,’ discussed in section3.9).
L l
Figure 3.7: A three mirror coupled cavity
Replacing re withrc in equation (3.38), we can write down the field in the recycling cavity,
Er=Ein tr
1−rrrce2iφ, (3.63)
whereφis the one-way phase in the recycling cavity. We note here that for the case of the overcoupled long cavity,rc will have opposite sign tori.
The three cases, which encapsulate possible resonant conditions for the carrier in the short cavity, are discussed in the following sections.
3.8.1 Antiresonant Short Cavity
As depicted in figure3.7, the recycling cavity is antiresonant whenφ= 0 (because of our convention that the back surface of a mirror has negative reflectivity). Because on resonance for the arm cavity rc is opposite in sign to ri (thus rc is positive), this actually represents the configuration with maximum power buildup in the recycling cavity, and thus also in the arm cavity (since the power in the short cavity is effectively the input power to the arm cavity). Thus, an anti-resonant recycling cavity means the coupled system has a greater finesse than the arm cavity, with a consequently lower bandwidth. This is the situation employed in the technique known as power recycling, described in section3.9.1, and also in signal recycling, described in section3.9.2.
With the help of equation (3.63) and [49], we can write down the frequency response of this coupled system with the updated definitions of the field amplitude gain,
gr= tr
1 +rrrc
, (3.64)
the coupled cavity pole,
ωcc=1 +rrr0
1 +rr
ωc, (3.65)
and defining
scc=iωa
ωcc, (3.66)
we have the field in the recycling cavity:
Ecc=Eingr
1 +sc
1 +scc
. (3.67)
Note that these represent the transfer function from outside the recycling mirror to just inside the mirror. For an anti-resonant short cavity, scc< sc, and this bandwidth reduction means that the coupled cavity acts as a low-pass filter on the light in excess of the filtering provided by the arm cavity alone, as shown in figure3.8(a). The presence ofsc in the numerator represents a zero in this transfer function at the arm cavity pole frequency; physically, sidebands above this frequency are
no armer resonant in the arm cavity, and thus fall out of resonance in the recycling cavity as well since they do not experience a phase flip on reflection from the arm cavity. For a source field in the arm cavity, there is no zero at the arm cavity pole frequency; it would instead appear at the pole frequency of the short cavity.
3.8.2 Resonant Short Cavity
In this case, the finesse of the coupled system is lowered, and so the bandwidth is higher than that of the arm cavity alone (cf. figure 3.8(a)). This is because, when viewed from the arm cavity, the reflectivity of the recycling cavity is lower thanri, which lowers the overall finesse. This situation is useful in the technique known as resonant sideband extraction, described in section3.10.
3.8.3 Detuned Short Cavity
A detuned cavity is neither resonant nor anti-resonant for the carrier, which implies that the system is resonant for some other frequency of light. Cavity detunings are typically specified as a carrier optical phaseφ, which specifies how far the cavity is from a carrier resonance. The specific details of the system (mirror reflectivities, cavity lengths) can then be used to determine the frequency at which the system will resonate; since that frequency will typically be shifted from the carrier by an audio frequency, such a system can be characterized as having an audio frequency optical resonance.
What this means is that the response of the system can be tuned to have a maximal response to a certain audio frequency.
Figure3.8(b)depicts a typical profile for a resonant system, with the frequency axis shifted such that the origin (the carrier frequency) is not at the center of the resonance. This represents the situation under discussion. When a phase modulation is produced by test mass motion atωa, a pair of sidebands at±ωa appear. One of these sidebands will be closer to resonance than the other, and
100 101 102 103 104 105
−40
−30
−20
−10 0 10 20 30 40 50 60
f (Hz)
Mag (dB)
100 101 102 103 104 105
−40
−30
−20
−10 0 10 20 30 40 50 60
f (Hz)
Mag (dB)
resonant short cavity anti−resonant short cavity no short cavity
(a) A short additional cavity can increase or decrease the system bandwidth, compared to no additional cavity.
−2000 −1000 0 1000 2000
100 101 102
sideband amplitude (a.u.)
f (Hz)
−2000 −1000 0 1000 2000
100 101 102
sideband amplitude (a.u.)
f (Hz)
ωa ωa
USB LSB
(b) Detuning the short cavity affects upper and lower signal sideband amplitudes differ- entially.
Figure 3.8: Effect of adding a third mirror with a short cavity on the bandwidth of the coupled system.
the total response of the system as a function ofωa will have a resonant peak in the response. We thus encounter the first delicate issue with detuning: sideband imbalance. In general, upper and lower signal sidebands will be imbalanced, and the RF sidebands used as local oscillators can also be imbalanced. One particular effect of this RF sideband imbalance is a reduction of cancellation efficiency for some noises on the sidebands, due to the differing amplitudes on the upper and lower sidebands.
There is another delicacy that we have neglected up to this point. We have been fortunate in being primarily concerned with cavities on resonance or anti-resonance, where dPdl = 0; this has allowed us to ignore the effects of radiation pressure, which in cavities detuned from resonance can affect the dynamics in significant ways.
The force due to radiation pressure is due to the momentum change of photons upon reflection from a mirror; it is given by the average number of photons per second striking a mass, times the momentum change experienced by each:
F =N×2~ω c = 2P
c , (3.68)
where P is the average power. If we takexto be the position of one of the cavity mirrors, we can define an effective spring factor,
dF dx =2
c dP
dx, (3.69)
which is the optical rigidity of a light beam in a cavity, and is frequency dependent. We can show heuristically why this matters by doing a rough estimate of this rigidity for a cavity with parameters relevant to LIGO: length 4 km, with 100 kW of circulating power, when held at the point of half- max-power (the maximum rigidity), for a system with a finesse of∼18000. The DC radiation force is∼600 µN, but the effective spring constant at DC is dFdx ∼2c
PF
λ →107N/m. For comparison, the Young’s modulus of diamond is 1220 GPa and so this light beam is about as stiff as an equivalent length of diamond rod with diameter 20 cm. If we attach a 10 kg mass, the system would have a resonant frequency of about 160 Hz. This cannot be ignored.
These two concerns, the sideband imbalance and the radiation pressure, are also related: at a given audio frequency ωa, the audio sidebands beat with the carrier field, resulting in a power fluctuation atωa. This power fluctuation pushes on the mirror, creating further phase modulation atωa. The optical fields and mirror positions are thus linked at all audio frequencies, and the upper and lower audio sidebands must be calculated together to correctly determine the radiation pressure force on the mirror. This complicates the derivation of frequency responses of coupled systems, because now, instead of just considering the fields in the interferometer at a single frequency, we must simultaneously consider two frequencies (±ωa)andthe mirrors; the resulting algebra becomes inconvenient enough that, for analytical work, it is simpler to work in the two-photon formalism
developed for quantum optics by Caves and Schumaker [53, 54]. This will be discussed more in section3.11, and more details can also be found in appendixD.