Structural thermal noise in gram-scale mirror oscillators

The thermal noise associated with mechanical dissipation is a ubiquitous limitation to the sensitivity of precision experiments ranging from frequency stabilization to gravitational wave interferometry. We report on the thermal noise limits to the performance of 1 gm mirror oscillators that are part of a cavity optomechanics experiment to observe quantum radiation pressure noise. Thermal noise limits the observed cavity displacement spectrum from 80 Hz to 5 kHz. We present a calculation of the thermal noise, based on finite element analysis of the dissipation due to structural damping, and find it to be in excellent agreement with the experimental result. We conclude with the predicted thermal noise for an improved oscillator design, which should be capable of revealing the noise that arises from quantum backaction in this system.


Introduction
All mechanical systems exhibit thermally driven fluctuations. Callen and Greene give a general description of thermal noise, the fluctuation dissipation theorem, which relates the level of thermal noise at each frequency to the amount of damping at that frequency [1]. Such damping is well modeled with a complex spring constant k → k(1 + iφ( f )), where φ( f ) is known as the loss angle. When dissipation arises from the familiar velocity-dependent damping force due to, for instance, air resistance, it results in viscous damping, with ( f ) ∝ f . However, Saulson [2] proposed that in high vacuum with appropriate isolation, the damping in a mirror-suspension system (such as those used in gravitational wave detectors) would likely be structural: that is, dominated by internal friction in the suspension and mirror materials, with φ( f ) constant over the frequency band of interest. While the displacement noise due to either form of damping is identical near a mechanical resonance, away from resonance the two forms look quite different, with the amplitude spectral density of the structural form steeper by a factor of f −1/2 . The thermal noise spectrum of mechanical resonance has been well studied, but in fact it is the off-resonant behavior that reveals the underlying dissipation mechanism. This thermal noise due to mechanical dissipation provides a fundamental limitation to many precision measurement experiments, such as gravitational wave detectors [3], frequency stabilization [4,5], opto and electro-mechanics [6] and tests of deviations from Newtonian gravity via torsion pendulum. Most measurements of thermal noise exhibit non-structural damping [6,7]. The exceptions to date include measurements at very low frequencies, below 10 Hz, in reference cavities [4,5] and in torsion pendulums used for testing deviations from Newtonian gravity [8].
Here we report on an interferometric displacement measurement limited by off-resonance suspension thermal noise in the broad frequency band 80-5000 Hz, achieving a maximum displacement sensitivity of 2 × 10 −17 m √ Hz −1 at 5 kHz. Our measurement over that band matches thermal noise predictions made from a finite element model of the mirror and its suspension. Our experiment is uniquely sensitive to structural thermal noise at frequencies below a few kHz due to the mm-scale optical beam diameter, which reduces the impact of coating thermal noise and thermoelastic substrate noise [9]. This is in strong contrast to previous thermal noise measurements, where much smaller beam sizes (<0.1 mm) were used in order to magnify the effect of thermal noise [10][11][12]. Moreover, in the 80-200 Hz band our thermal noise is primarily due to above-resonance thermal noise. Our results should, therefore, be of interest both to those in the cavity optomechanics community working toward quantum noise limited experiments [6] and those in the gravitational wave community contending with structural thermal noise in the design of mirror suspensions for interferometric gravitational wave detectors [3].

Theory
The fluctuation dissipation theorem provides a unified way of understanding thermal noise. It relates the power spectral density S x of fluctuations in a generalized coordinate x to its dissipation into a coupled thermal reservoir at temperature T as: where k B is Boltzmann's constant, and Y ( f ) is the generalized mechanical admittance at frequency f . The admittance acquires a real part when the system has some loss. It is computed from the complex velocity response v( f ) of the coordinate to an oscillating driving force F( f ): For systems with a single degree of freedom, the admittance can be calculated exactly from its equation of motion, and then equation (1) yields the exact thermal noise spectrum. For multi-mode or continuous systems, one may often obtain a good approximation to the measured thermal noise by summing the thermal noise power spectral density due to each mode, but only if the normal modes are reasonably orthogonal (a condition dependent on the modal Qs and frequencies) and sufficiently many modes are included in the sum. This approximation may be written as: where m n is the modal mass of the nth mode with frequency f n and loss angle φ n . Typically at least the lowest 100 modes must be included to obtain good estimates for the mirror substrate thermal noise in a gravitational wave interferometer [13]. The extent to which each normal mode couples to the generalized coordinate x of interest is quantified by its change x n under maximal expansion of that mode, via its modal mass where U is the stored energy in the mode. For a system made of several materials, the loss angle φ n of mode n is the sum over the loss angles θ i of the materials weighted by α n,i , the fraction of energy stored in that material at maximal expansion: Based on the fluctuation dissipation theorem, Levin derived a 'direct approach' to calculate the exact thermal noise spectrum of mirror position as measured by an incident laser beam, as in an interferometer [14]. One calculates the mechanical admittance by applying a generalized force F, oscillating at frequency f , that drives only the generalized coordinate x. As measured by a laser, the mirror position is x(t) = d 2 r w( r )y( r , t), where w(r ) is the intensity weight of the beam at radius r , and y( r , t) is the deformation of the surface at position r in thex-direction at time t. For a Gaussian beam, we have w(r ) = e −(r/r 0 ) 2 /(πr 2 0 ), where r 0 is the radius at which the intensity has dropped to 1/e 2 . After finding the admittance, the power spectral density of the thermal noise is computed with equation (1). This formulation is particularly amenable to numerical calculation with finite element simulations.
In our analysis, we use Levin's direct approach to calculate predicted thermal noise levels, but the normal mode method provides a simple way to interpret those results qualitatively. For instance, thermal noise due to a particular mode often dominates in a particular frequency band, and knowledge of which mode is the culprit may suggest a route to mitigating it. Further, in a multi-material system, the thermal noise from the lossiest (highest φ) material may often completely overwhelm the thermal noise from lower loss components (see equation (4)) over certain frequency bands, even if the amount of lossy material is minuscule in comparison. We also observed that computing thermal noise via the normal modes approach with finite element models is about an order of magnitude faster than the Levin approach, and with the inclusion of the lowest ∼100 modes, agreed at the ∼30% level.

Experiment
A schematic of the experimental setup is shown in figure 1. The optomechanical system consists of two Fabry-Perot cavities with movable mirrors, embedded in the arms of a Michelson interferometer. The Michelson enables a sensitive readout of the differential displacement of the two cavities. A 1 gm mirror oscillator forms the end mirror of each cavity. The primary goal of the experiment is to observe the quantum backaction of the differential displacement measurement on the 1 gm mirrors. Details of the experimental design can be found in [15]. Central to the present work are the 1 gm mirror oscillators, which are suspended from fused silica fibers, as shown in figure 2. The fibers are bonded to the mirror with Vac-Seal epoxy (Tra-Con), see table 1. Each fiber is 40 mm in length, and tapers from a maximum diameter of 3 mm to a minimum of ∼150 µm. A thicker region, called the 'ear', is located at the bottom, and is about 10 mm long and 1.5 mm in diameter. The shape of the ear was informed by the need for a steep spatial gradient at the transition between the small diameter fiber and the large diameter ear, which allows for better decoupling of ear and mirror motion. See figure 3, for a close-up of the ear region.
The purpose of the tapering fibers is to minimize the thermal noise of the fundamental mode of oscillation (which occurs at about 10 Hz), by ensuring that it stores energy mostly in the pristine fused silica near the middle of the fiber, and not in the lossy epoxy at either end. Quality factors of 10 6 or greater are measured for the 10 Hz mode, suggesting that the thermal noise of this mode is far better than what is required to observe the quantum backaction. However, another thermal noise floor limits the measured displacement noise spectrum in the 80-5000 Hz band, as shown in figure 4. In the next section, we present the results of finite element simulations confirming that the ear-mirror epoxy bond is the origin of the observed thermal noise.

Thermal noise analysis
The finite element method is an approach to solving for continuum dynamics in various physical scenarios by discretizing the geometry. We perform finite element simulations with the COMSOL Multiphysics package, using its structural mechanics module. With that software we model the three-dimensional geometry and then implement the Levin approach [14]. We apply a Gaussian load to the mirror face, and solve for the steady-state mirror response to the load oscillating at a set of frequencies { f }, yielding the complex admittance Y ({ f }) directly. See table 1 for the material parameters used in the simulations. Using experimental data for different quantities of glue we infer the loss angle, φ epoxy = 0.035, of the epoxy from the amplitude spectra density. This value results in excellent agreement between our experimental data and the output from the model. We take this result as confirmation that our displacement noise floor is dominated in the 80-5000 Hz band by structural thermal noise. The 80-200 Hz band is in the above-resonance 'wing' of the 80 Hz pitch mode. If all the pitch modal energy were in the very low loss fused silica, then we would expect to measure Q = 1/φ fusedsilica ∼ 3 × 10 6 . However, we measure only ∼5 × 10 5 , indicating both that pitch thermal noise is setting our noise floor in that band, and that epoxy loss is dominating that noise. In the 200-5000 Hz band the spectrum shows the characteristic f −1/2 below-resonance thermal noise spectrum of a structurally damped oscillator, which is well reproduced by simulations. The modes responsible for this noise are the bending modes of the ear-mirror joint, with resonant frequencies starting around 30 kHz, outside the band of our measurement. See figure 3, for an example of one such ear-mirror bending mode.

Thermal noise mitigation
Various schemes have been presented to reduce the level of thermal noise in an interferometer [17][18][19][20], relying either on carefully tuned passive cancelations or multiple measurements to disentangle the thermal noise from displacement signal. Our modeling results suggest several simpler geometric ways to lower our noise floor.
In the 80-200 Hz band, the measured spectrum is dominated by the thermal noise from the pitch mode of the oscillator. Introducing a vertical offset of the incident laser beam [14],  shifting it ≈1 mm from the center of the mirror, places the beam near the node of the pitch mode. This offset should decrease the sensed pitch thermal noise (by increasing the modal mass), as depicted in figure 4.
Above 200 Hz, the thermal noise of the mirror-ear bending modes takes over. As the loss angle φ epoxy is five orders of magnitude larger than φ fusedsilica (for ∼150 µm fibers [16]), it is the epoxy that dominates both the pitch and the mirror-ear bending thermal noise. To minimize the effect of this thermal noise in our measurement band, we studied alternative mirror-ear geometries to maximize the strength of the ear-mirror bond per amount of epoxy. By maximizing this quantity, we redistribute the thermal noise of the mirror-ear modes to higher frequencies, out of the measurement band. This quantity of merit is optimized in the case of plane-on-plane contact, and we propose a design that incorporates ears with square cross-section, and 'chopped' mirrors with flattened regions on the circular edge. Figure 5 shows a model of this square-ears/chopped-mirror geometry, and figure 6 shows the thermal noise for variations on that geometry predicted from our modeling. We expect these square ears to be the most expedient path forward, and are pursuing this design for the next iteration of the end mirror suspension. This chopped geometry is also compatible with other adhesives such as silicate bonding [21].

Conclusion
We have experimentally demonstrated and theoretically analyzed the structural thermal noiselimited performance of a 1 gm mirror oscillator, finding excellent agreement between the model and the measurement. We have also proposed some avenues to reduce the thermal noise floor in this experiment. The observation of off-resonance thermal noise has importance for a broad class of experiments in cavity optomechanics and gravitational wave interferometry. Accurate modeling of these complex structures depends on detailed knowledge of their geometry and material properties, but lets us better understand and optimize their mechanical design. The optimizations we have presented should permit our experiment to enter the quantum backactionlimited regime.