Upper Limit to the Transverse to Longitudinal Motion Coupling of a Waveguide Mirror

Waveguide mirrors possess nano-structured surfaces which can potentially provide a significant reduction in thermal noise over conventional dielectric mirrors. To avoid introducing additional phase noise from motion of the mirror transverse to the reflected light, however, they must possess a mechanism to suppress the phase effects associated with the incident light translating across the nano-structured surface. It has been shown that with carefully chosen parameters this additional phase noise can be suppressed. We present an experimental measurement of the coupling of transverse to longitudinal displacements in such a waveguide mirror designed for 1064 nm light. We place an upper limit on the level of measured transverse to longitudinal coupling of one part in seventeen thousand with 95% confidence, representing a significant improvement over a previously measured grating mirror.


Introduction
Major upgrades to the worldwide network of gravitational wave detectors are currently under way. New designs for the Advanced LIGO [1], Advanced Virgo [2], KAGRA [3] and GEO-HF [4] detectors will provide unmatched ability to detect gravitational waves in the audio spectrum. At their most sensitive frequencies, these detectors are expected to be limited primarily by Brownian thermal noise arising from the reflective coatings on the detectors' test masses [5][6][7][8]. In order to help mitigate this limitation beyond the next generation of detectors, efforts are under way to develop mirror coatings with lower thermal noise [9,10].
In the case of Advanced LIGO, each end test mass (ETM) consists of a substrate with 19 pairs of sub-wavelength coatings which produce a transmission of 5 ppm for 1064 nm light [11]. Each layer within this stack contributes to the overall thermal noise [7,8]. The approach taken by Levin to calculate mechanical loss [5] shows that loss from the front surface of a mirror contributes more than from an equivalent volume in the substrate. Additionally, typical coating materials tend to exhibit mechanical loss orders of magnitude higher than typical substrate materials [7,8]. For these reasons particular attention is being given to the reduction of coating thermal noise to improve the sensitivity of future detectors.
One strategy, to be applied for example in KAGRA, is to cool the mirrors to cryogenic temperatures. While this can potentially reduce the thermal noise of the mirrors [12], the application of cryogenic mirrors requires new infrastructure, different choices of mirror substrate and coating materials and poses the challenge of heat extraction from the mirror without spoiling its seismic isolation and thermal noise performance. Efforts in the application of cryogenics are also under way to identify suitable substrate and coating materials for ET-LF, the low frequency interferometer as part of the proposed Einstein Telescope [13][14][15][16].
Apart from using different coating materials [17,18] or different beam shapes [19][20][21] such as with LG33 modes [22], another potential approach is to utilise waveguide mirrors (WGMs) [23][24][25][26]. These mirrors can possess high reflectivity at a wavelength determined by its structure. In contrast to conventional dielectric mirrors, a mirror possessing a waveguide coating can exhibit high reflectivity without requiring multiple stacks [27]. The waveguide coating instead presents the incident light with a periodic grating structure of high refractive index material n H on top of a substrate with low refractive index n L (see Figure  1). Light is forced into a single reflective diffraction order, the 0 th . In transmission, only the 0 th and 1 st diffraction orders are allowed as long as the condition in Equation 1 for p, the grating period, and λ, the light's wavelength in vacuum, is fulfilled [23]. The light diffracted into the 1 st order undergoes total internal reflection at the substrate boundary where it excites resonant waveguide modes. Light leaving the waveguide then contains a π phase shift with respect to the 0 th order transmitted light, causing destructive interference so that most of the incident light is reflected [28].
A recent set of calculations by Heinert et al. [29] showed that a suitably optimised WGM can provide a reduction in coating thermal noise amplitude waveguide (n H ) substrate (n L ) destructive interference constructive interference grating structure Figure 1: Propagation of light within a waveguide mirror. The grating and waveguide layers have refractive index n H , and sit atop a substrate of refractive index n L . Blue arrows represent incident light and red arrows represent reflected light. In realisations of waveguide mirrors such as this, a thin etch-stop layer is placed between the grating and waveguide layers to assist fabrication [26].
of a factor of 10 at cryogenic temperature compared to mirrors employed in Advanced LIGO.
Previous efforts to demonstrate grating structures as alternatives to dielectric mirrors have identified phase noise in the light reflected from the grating not otherwise present in dielectric mirrors [30,31]. The effect arises from transverse motion of grating mirrors with respect to the incident light. Incident light at angle of incidence α is reflected into the m th diffraction order, exiting at angle β m (see Figure 2). The change in path length δl between the reflected and incident light is then: where ζ a and ζ b represent the relative optical path length of each depicted ray. The noise this adds to the reflected light can be enough to offset the improvement in coating thermal noise, as witnessed in a study involving 2 nd order Littrow gratings [32]. Although WGMs possess gratings, the resonant waveguide structure can, with suitably chosen parameters, mitigate this extra coupling, as shown in simulations by Brown et al. [33]. This paper presents an experimental measurement of the coupling level.

Experiment
In order to quantify the level of transverse coupling, a WGM has been produced in collaboration with Friedrich-Schiller University Jena, Germany (see Table 1 for its properties). It is designed for light of wavelength 1064 nm, and consists of an etched grating structure on top of a waveguide layer, both tantala (Ta 2 O 5 ), on a silica substrate. The WGM was used as the input coupler of a Fabry-Pérot cavity, held on resonance via the Pound-Drever-Hall (PDH) technique [34], providing sufficiently large signals in response to changes in the phase of the longitudinal light field within the cavity.
There are two mechanisms in which grating mirrors can couple transverse motion into longitudinal phase changes (see Figure 3). The first is through transverse motion of the grating, which can in principle be minimised with appropriate suspension design. The second mechanism is the coupling of changes in the opposite cavity mirror's alignment into the spot position on the grating mirror. This effect is of particular importance to gravitational wave observatories, where longer arm lengths can increase its prevalence. For this reason the second mechanism is considered in more detail in this work.

Cavity Length Signals
As a non-zero transverse coupling by the WGM produces a phase shift on the reflected light, the effect can be witnessed as a cavity length change. Additional cavity length changes are produced, however, by the rotation of the ETM, due to two geometric effects present for all types of mirror (see Figure 4). The first effect, a, is due to the position offset of the beam from the centre of the mirror's surface. An offset of the beam horizontal to the centre by a distance x a will receive an additional longitudinal shift of:  for small angles. The second effect, b, is due to the depth d of the mirror, proportional to the rotation angle θ. For the change in position of the centre of the mirror for this rotation, x b , we have: and therefore: The total longitudinal effect δl E caused by the rotation of the ETM is then: In measuring the cavity length, we obtain the signal due to the combination of mirror longitudinal effects from Equation 6 and any WGM coupling. Considering the ETM's rotation, dimensions and mass, it is then possible to infer the WGM's coupling level. The phase effect associated with transverse coupling is expected to be independent of spot position, whereas the phase effect associated with the ETM's rotation has a sign flip about the centre of rotation. It is therefore expected that a spot position will exist, offset from the ETM's centre of rotation, for which there is a cavity length signal minimum. This is the result of the WGM's transverse coupling and the length change due to ETM rotation having opposite phase (see Figure 5).
By measuring the position of the spot on the ETM's surface, and applying (6), it is possible to calculate the change in cavity length attributed to its rotation. The position of the minimum allows the transverse coupling level of the WGM to be determined.
Examples of possible coupling levels smaller than (blue), larger than (red) and roughly equivalent to (green) the ETM's longitudinal effect are shown in Figure 5. For coupling levels of similar magnitude to the ETM's effect, the minimum signal is witnessed to move away from the centre of the ETM's rotation.

The Glasgow 10 m Prototype
The Glasgow 10 m prototype provides a testing bed in which a WGM's transverse to longitudinal coupling can be quantified. The prototype is housed in a Class 1000 clean room and consists of an input bench at atmospheric pressure and a vacuum envelope with pumps able to reach pressures of order 1 × 10 −5 mBar. The envelope consists of nine 1 m diameter steel tanks, each connected by steel tubes, arranged into two 10 m arms, with a shorter arm for input optics between the two.
For the experiment, the laser light was passed through a single-mode fibre to provide spatial filtering and an electro-optic modulator (EOM) to impose RF sidebands on the light to facilitate PDH control. The light was then coupled into the vacuum system via a periscope. The configuration can be seen in Figure  6.
In tanks 4 and 5 were sets of two triple suspension chains based on the GEO-600 design [35]. A viewport present to the rear of tank 5, and to the side of tank 1, allowed for light to exit the vacuum envelope for the purposes of sensing and control.
For the purposes of this experiment, 1064 nm, fundamental mode laser light of power 2 W was used, of which approximately 150 mW entered the cavity.
The WGM was attached to an aluminium block of mass 2.7 kg and suspended from tank 4's cascaded (triple) pendulum, forming the cavity's ITM. A silica test mass, also 2.7 kg, with a 40 ppm transmission coating, was used as the ETM, suspended from a similar triple pendulum in tank 5. On the rear surface of the ETM were three magnets for the purpose of actuation. With optimal alignment the mirrors formed an overcoupled cavity with finesse 155.
A three-stage reaction chain was placed behind the triple pendulum of the ETM, for the purposes of providing actuation via coils acting on the three magnets. The upper and intermediate stages were identical to those of the chain carrying the ETM, but-for the purposes of another experiment, not reported here-the lower stage was split into two parts separately suspended from the intermediate stage. The part closer to the ETM consisted of a 1.8 kg aluminium mass that carried the actuation coils. The distant part was a 0.9 kg mass needed to balance the suspension. The suspension design parameters are shown in Table 2. To minimise unintended tilt actuation of the ETM from the   The cavity is held on resonance by the frequency stabilisation servo which feeds back to the light's frequency via the EOM (high frequencies) and the laser crystal's PZT mid frequencies) and temperature (low frequencies).

Test mass suspensions
Triple pendulum, resonant frequency ∼ 1 Hz, final stage suspension length 280 mm [35] Steering optics suspensions Double pendulum, resonant frequency ∼ 1 Hz Voice coil resistance 10 Ω  voice coils, their DC levels were tuned by viewing the light transmitted through the mirror using a CCD camera positioned to view the viewport on tank 5.

Measuring Cavity Length Changes
A resonant photodiode was placed at the viewport on tank 1, where it could view the light reflected from the cavity. By using PDH demodulation, the signal from this photodiode provided an error signal for the cavity length. This signal was fed back via a circuit to control the laser's wavelength and the EOM to maintain cavity resonance.

Measurements and Simulations
To provide a rotation signal to the ETM, a sinusoidal signal V S = 9 V was produced using a signal generator at a frequency of 70 Hz. This frequency was chosen to be higher than the suspensions' resonant frequencies but low enough to provide an adequate signal-to-noise ratio. The signal was split into two, V L and V R , and one path's signal was inverted. Each signal was fed to a separate coil on the reaction mass to rotate the ETM. The piezoelectric transducer (PZT) within the laser had a response of 1.35 MHz/ V, corresponding to a cavity length change of approximately 135 nm/ V. This provided a length calibration for the cavity error signal.
By recording the PZT feedback during actuation of the ETM at f = 70 Hz for a period of 120 s, with coil driving signals of equal magnitude and sign, it was possible to calibrate V S in terms of cavity length change. The calibrated length signal, along with the ETM's mass m, could be entered into Equation 7 to obtain the combined force applied to the ETM per volt driving signal from the coils.
Four spot positions corresponding to x a were chosen across the surface of the mirror. The mirrors were aligned to produce a fundamental mode resonance at each spot position. For each position, the sinusoidal signals V L = −V R = V S were applied to the ETM's coils and the PDH length signal was measured for a period of 300 s. For each position an additional measurement was taken with V L = V S ± 0.1 V and V R = −V S for a period of 60 s, to provide an extra signal calibration check.

Obtaining Length Changes due to Waveguide Mirror Coupling
The knowledge of the ETM's coils' distance from the centre, x c , the ETM's moment of inertia, I, the coil driving frequency, f , and the force calibration from Equation 7, allowed the rotation angle to be obtained using Equation 8. A simulation using the frequency-domain tool FINESSE [36] was carried out to calibrate the effect of ETM rotation on the WGM, taking Gaussian optics into account. This indicated the rotation to transverse coupling to be a factor of 10, which is similar to the ray optical case. This calibration then allowed the spot movement on the WGM to be obtained. Combining this value with the residual length signal from the cavity, it was then possible to calculate the WGM's transverse coupling level.

Analysis of the Coupling Level
A model was built to simulate the cavity length change due to the ETM and WGM. The WGM's coupling factor was assumed to be frequency independent. The ETM's two effects were subtracted from the simulated longitudinal signals, and the effect of beam smearing was considered due to residual suspension displacement and the integration time of the measurements. The beam smearing was modelled with the assumption that the motion of the spot on the WGM followed a Gaussian distribution about a fixed point. A Markov-Chain Monte-Carlo algorithm as part of a Bayesian analysis was applied to the model to marginalise over three parameters: coupling level, beam smearing standard deviation and scaling. Scaling is a constant factor multiplied with each spot position's amplitude, used to calibrate the cavity's PDH signal into changes in length. The Bayesian analysis assumes the spot positions yielding a possible cavity length signal follow a distribution resembling Gaussian noise. The parameters governing this Gaussian noise were determined from the error in the measured spot positions (assumed to be ±1 mm from visual inspection) and the misalignment between coils and magnets on the ETM suspension. Misaligned coils and magnets could lead to a drop in the expected force coupling, leading to a change in the centre of rotation of the ETM. From measurements it was found that the effect of any possible misalignment during the experiment could only account for a drop in force of 0.11 %, contributing a negligible error (0.03 mm) to the results.
From the parameter marginalisation it was possible to produce a probability density distribution for the coupling level, as shown in Figure 7. The coupling level predicted from the distribution is bounded between 1:52500 and 1:8700 with 95% confidence. Cavity length signals produced by the model for these bounds are shown in Figure 8.

Results and Future Work
The final results are shown in Figures 7 and 8. The upper coupling level bound, 1:8700, shown in Figure 7, represents a significant improvement over the coupling levels of previous grating designs such as the 2 nd order Littrow grating measured in [32], where the coupling factor was of order 1:100. The limit to the precision to which the coupling can be determined is due to the method used to estimate the position of the end test mass. A future experiment might seek to utilise a more sensitive means of position measurement, such as through the use of local readout.
The authors would like to thank members of the LIGO Scientific Collaboration for fruitful discussions. The Glasgow authors are grateful for the support from the Science and Technologies Facility Council (STFC) under grant number ST/L000946/1. The Jena authors are grateful for the support from the Deutsche Forschungsgemeinschaft under project Sonderforschungsbereich Transregio 7.