Reducing cavity length noise for a better squeezing source

Detuning of the OPO cavity a small amount away from its ideal resonance point results in small shifts in the phase of the out-coupled fundamental squeezed quadrature. If the simplifying assumptions applied for OPO cavity equations of§3.4.4 are relaxed to include cavity detuning, an expression for quadrature rotation as a function of static detuning may be formed. Solving equations 3.124a-b and computing the shift to first order about the linear region of pump phase dependence for variance ( b=⇡/2), the squeezing angle

as a function of cavity detuning is approximated as [145]:

d✓SQZ d = ✓ 1 2 dV /d dV /d b ◆ b=⇡/2, L=0 = 1 b total + 1 a total(1 +x2) . (6.2)

where, again,x is the normalised non-linear coupling fraction of threshold (x= 1 1/pg, where g is the classical parametric gain) and a,b _{are the cavity fundamental and pump}

field decay rates of the OPO as defined in equation 3.79.

use co-propagating RF o↵set control sidebands, that references the phase of the OPO pump light, to provide an error signal to feed back and stabilise the relative phase with the measurement device. Similar such schemes have been implemented in injection experiments for GW detectors [66, 123] (see also, [64, 132, 151] for further developments). However, residual phase fluctuations still exist. Di↵erences in intra-cavity phase sampled by RF control sidebands in the OPO relative to its on-resonance field, lead to lock-point errors driven by cavity length noise. Additionally, e↵ects in the interferometer such as alignment o↵sets in the presence of higher order modes, allow the conversion of alignment jitter into locking-point o↵sets2. Thus the relative motion between the OPO and the interferometer fields as well as residual length noise of the OPO coupled from direct mechanical disturbances (such as air flow and direct mechanical couplings in the laboratory) are not completely cancelled by locking control loops. A squeezed light source mounted in an inherently quiet environment, such as a suspended stage in vacuum, o↵ers a way to isolate a squeezing subsystem from noise coupled from the surrounding environment.

Cavity length noise within an OPO arises from mechanical couplings that enter the mirror assembly and mounts. Although this detuning is largely sensed and suppressed using a Pound-Drever-Hall lock [156], some residual length perturbations remain. Because the cavity’s mechanical vibration frequencies are well below its optical line-width, their evo- lution is adiabatic with respect to the cavity’s optical dynamics and the static detuning model is sufficient to approximate the translation of cavity length noise into phase noise. A small shift in round trip length Ldetunes the fundamental field by a=! L/L¯, where

¯

Lis the on-resonance cavity length and! is the fundamental laser frequency. Likewise the pump harmonic field detuning is altered by b = 2 a. Substituting this length induced

detuning into equation 6.2, the shift to first order for the squeezing angle as a function of cavity length is approximated as

✓SQZ(t) = ✓ 1 b total + 1 a total(1 +x2) ◆ ! L(t) ¯ L . (6.3)

From this expression we may make predictions of the expected intrinsic phase noise of a doubly resonant OPO source with knowledge of the cavity length noise.

For the parameters selected for the initial LIGO-H1 squeezing experiment, the OPO generated 90 mrad per nanometer RMS length noise [144]. Cavity length fluctuations above the coherent control loop bandwidth translate directly in this manner contributing directly to RMS phase noise. The squeezing phase control loop bandwidth in the LIGO-H1 implementation was limited to 10 kHz due to potential instabilities induced by the arm cavity dispersive response that phase shifts audio-band fluctuation components close to each free spectral range [145]. Furthermore, those length fluctuations within the coherent control loop bandwidth were only half suppressed. The intra-cavity phase sampled by control sidebands, widely o↵set from OPO resonance by tens of MHz, experience a di↵erential phase shift for the detuned cavity that di↵ered from the on-resonance squeezed fields. The wide detuning of coherent sidebands is necessary to prevent contamination of the OPO with coherent light at the fundamental frequency; seeding of the OPO can open

2_{See [144] and references therein for a detailed treatment squeezing quadrature fluctuations in GW}

up potential audio-band noise couplings from a range of classical noise sources (see [101], chapter 5). Thus a large portion of the length noise induced quadrature jitter is still present even with coherent control implemented in gravitational wave detectors. Cavity length noise is therefore a source of degradation not wholly cancellable by feedback control because of bandwidth or lock point limitations.

The squeezing preparation optics for the LIGO-H1 test were mounted at some height on a table grouted to the floor but only contained in a box primarily designed for dust exclusion and stray beam containment, providing only limited isolation from air currents and acoustic disturbance [27]. For the cavity length fluctuations measured at the time, the estimated RMS quadrature fluctuations below 100 kHz were 24.6_±3 mrad [144]. These factors notwithstanding, even under ideal conditions the phase noise is expected to be too high for either 6 dB or 10 dB levels of injection when operated in air. These bow-tie cavities were constructed with mirrors mounted in standard ‘o↵-the-shelf’ Newport ‘Suprema’ low drift 1/2” inch adjustable mounts bolted to a single piece aluminium block [157]. Such a construction provides a relatively long term stable cavity alignment but is susceptible to thermally induced cavity length changes and has limited rigidity compared to non-adjustable fixed mirror mounting constructions or monolithically bonded reference style cavities. The GEO600 hemilithic squeezer had a more compact linear cavity assembly with the coupling mirror and crystal contained within a permanently bolted (non-adjustable) assembly with a short 22.5 mm air gap [155]. This is a better de- sign, in terms of length noise, but has poor intrinsic backscatter isolation (see next section).

An alternative design approach, the one adopted in this thesis work, was to build the OPO cavity in a monolithic construction similar to the output mode cleaner (OMC) cavities installed in the Advanced LIGO Michelson output chain. These OMC cavities consisted of rectangular optical fused-silica prisms and 1/2” (12.7 mm) mirrors UV-epoxy bonded with PZTs to mounting blocks that were in turn epoxy bonded to a large 450⇥150 mm fused-silica breadboard. The permanently bonded mirror assembly formed a 1.132 m (round trip) cavity with an ultra low thermal expansion and a length noise better than 1_⇥10 15_m/p_{Hzat 100 Hz and above when mounted with seismic and acoustic isolation}

[158]. Length noise scaling of monolithic type cavities are known to scale as 1/pf below 100 Hz [159, 160], which leads to an expected upper bound for total RMS length noise of order 10 12 m [146]. An OPO cavity constructed in a similar manner and housed in an isolated vacuum environment is expected to yield similar length noise performance. The selection criteria for cavity parameters are outlined over the course of the following chapter. However, taking the parameters summarised in table 6.1 and substituting them into equation 6.3 we arrive at a estimated length noise to phase noise transfer of 82.9_±0.7 mrad/nm length RMS when operating the OPO at 75% of threshold power. This fraction of threshold is sufficient to reach the 6 dB e↵ective squeezing goal at 22% loss (the lower bound of presently attainable total loss). Combining this with the estimated cavity RMS length noise of a similarly construction to the OMC cavity, we should expect 0.08 mrad of length noise induced quadrature jitter intrinsic to the squeezer. For this reason we selected our primary candidate design as an all glass construction for our cavity, selecting the same low noise models of PZTs and a similar glass breadboard and prism/mirror design.