Tunable Fiber Fabry-Perot Cavities with High Passive Stability

: In the following, we present details of the fabrication process for building the three FFPCs presented in the main manuscript. All components and materials are described such that the interested readers can easily reproduce these systems. Details about the measurement of the frequency noise spectral density are given for all three FFPCs. The methods for combining ﬁnite element simulations with the modeling of thermal noise of the FFPCs are described. At the end, the thermal tunability of the realized FFPCs is presented.


FFPC fabrication
Besides the fiber mirrors (see Sec. 3 main text), the important components for the demonstrated FFPCs include the glass ferrules, piezo-electric elements, glues and the electrical contact wires, which are specified below. Fig. 1. Photograph of a full slot FFPC, a compact and miniaturized high finesse resonator with wide frequency tuning and high stability, and a 1 euro cent coin (for size comparison). The FFPC slotted region allows easy access to the cavity mode volume.

Optical cavity fabrication
After the successful testing of the electrical contacts, the assembly is temporally fixed on an insulating base. Two high magnification USB-cameras (Dino-Lite digital microscope) are used to observe the ferrule bore and the central slot while inserting the fibers using translation stages. The distance between the two fiber mirrors is scanned using a piezo driven translation stage to observe the cavity resonance and thereby estimating the finesse and the coupling depths, where the coupling depth is defined as the ratio of the on-resonance versus off-resonance reflected light of a probe laser from the cavity. The dependence of these optical properties in relation to the length of the cavity is shown in Fig. 3. Although, the mirrors used in all FFPCs are expected to have the same transmission properties as mentioned in the main text, there are many factors which lead to slight differences in the optical properties (see main text Tab. 1) between the three FFPCs. These factors are , e.g., the decentration of the mirror with respect to the fiber, slightly different behaviour of the mirror transmission after annealing and possible contamination of the fiber mirrors during the assembling process.
The optical properties of the resonators also depend on the relative alignment of the two fiber mirrors. However, we observe that the cavity length adjustment using the guide provided by the ferrule maintains the resonator alignment and therefore the limitation of the achievable finesse is only due to the losses in the dielectric coating and due to the clipping of the beams on the fiber mirrors. The coupling depth, on the other hand, additionally depends on the mode matching between the fiber and the cavity mode. For the assembled FFPCs, we observe that the finesse before gluing the fibers is in most cases close to the one obtained after the mirrors are glued, Fig. 3.
Here we have used a cavity length of around 93 µm for the FFPCs. The fibers are glued using a low viscosity and vacuum compatible UV-curable glue (UV16, MasterBond ® ) as also described in the main text. The cavity length is adjusted close to the target resonance by observing the resonance peak while scanning the piezo and successively reducing the scan range. Once the UV-light is switched on, the scan is reduced to zero. We have observed that this procedure is good enough to keep the FFPC close to the target resonance while the glue hardens. For the full slot cavity (FSC) we have observed a degradation of the Finesse after gluing the mirrors (Fig. 3) which we believe is due to a contamination of the mirrors.

Laser frequency noise
The linewidth and frequency noise of the laser is measured by using a linewidth analyser from HighFinesse ® . For the unlocked laser, the measured linewidth (∆ν) is 20 kHz ≤ ∆ν ≤ 200 kHz with a typical value around 50 kHz. The measured laser frequency noise is shown in Fig. 4. These measurements confirms that the linewidth and the frequency noise of the laser are much smaller compared to the FFPC linewidth and the frequency noise of the locked cavity (see next section).

Frequency noise spectral density
To demonstrate the high passive stability of our monolithic FFPCs, we perform the analysis of the frequency noise spectral density (FNSD), S ν , of the error signal under different lock conditions (see main text Sec. 4.3). For this purpose, we measure the noise spectral density of the locked error signal using a spectrum analyser. The noise amplitude is then converted to the frequency noise of the cavity resonance by the slope of the PDH-error signal to obtain S ν . Fig. 5 shows the FNSD for three FFPCs for a sub-Hertz (blue) and a few kHz (red curve) locking bandwidths. The gray trace is the off-resonance (detection) noise limit. To gain insight about the noise distribution, we also show the integrated rms frequency noise, ∆ν rms , in the inset for the respective FFPCs.
The results of these measurements are summarized in Tab. 1. ∆ν rms is only slightly increased while locking even at sub-Hertz locking bandwidth as compared to higher locking bandwidths, testifying the high passive stability of the FFPCs. As shown in the insets of Fig. 5, ∆ν rms is increased close to the 50 Hz, due to the coupling to the electrical line frequency, and at around 1 kHz by ambient acoustic noise.. The latter is confirmed in a separate measurement using a microphone sensor.

Finite-element modeling and noise spectral density calculation
To estimate the influence of the mechanical modes of the devices on the frequency stability, we performed finite element simulations (COMSOL [1]). The obtained mode frequencies and displacement fields are used to calculate the optomechanical coupling of the mechanical modes to the optical cavity mode. The resulting frequency noise spectral density of the optical mode is then retrieved via the fluctuation-dissipation theorem.

COMSOL simulations
For an overview of the basic modes and their coupling, a symmetrized geometry is simulated first, see Fig. 6 Fig. 6 (b). These modes occur in all simulated geometries and correspond to a bending and stretching mode of the devices. From the displacement fields u(x, y, z) as retrieved in the simulations the effective mass m eff of each mechanical mode [2] for all geometries is calculated as with V the full simulation volume and ρ(x, y, z) the local density at position (x, y, z). The zero point motion x zpm of each of the modes is then calculated using x zpm = /2m eff Ω m , with Ω m the simulated angular eigenfrequency of the respective mechanical mode. The vacuum optomechanical coupling rates g 0 are then extracted by where G = −2πν 0 /L cavity is the optomechanical coupling to the optical mode at frequency ν 0 and L cavity the cavity length (here fixed at 95 µm). The factor u z,mirror,i /max V (|u|) scales the displacement along the cavity axis of each of the mirrors i to the full maximum displacement of the mode, since the calculated x zpm refers to the maximum displacement position [2].

Noise spectral density modeling
The optomechanical coupling strengths and eigenfrequencies of the mechanical modes as retrieved from the simulations are used to calculate the expected frequency noise of the optical mode due to the thermal excitation of the mechanical resonances. This is achieved by applying the fluctuation-dissipation theorem to obtain the spectral density of displacement fluctuations of the mechanical modes of the devices. These are then coupled to the optical mode frequency by the respective optomechanical coupling strengths [3]. The resulting single-sided frequency noise spectral density S ν ( f ) of the optical mode from a single mechanical mode at Ω m is thereby given as Where the noise frequency is f = Ω/2π, k B the Boltzmann constant, T the temperature of the device and Γ m /2π the linewidth of the mechanical resonance. The rms frequency fluctuation up to a certain cutoff frequency ν co is then calculated as Alternatively, the contributions of the mechanical modes can be used to obtain the rms frequency fluctuations by j 2g 2 0, j n j /2π [3], with g 0, j being the vacuum optomechanical coupling strength and n j the expectation value of the mechanical mode occupancy (k B T/ Ω m, j for k B T Ω m, j ) of the jth mode. Figure 6 (c) and (d) show the resulting modeled frequency noise spectral densities of the optical mode and the corresponding integrated rms fluctuation for the three designs. Since the finite-element simulations did not take damping effects into account, the linewidth of the mechanical modes was arbitrarily fixed at Γ m /2π = 1 kHz for all modes in the preliminary analysis. Note that the full (ν co → ∞) rms frequency fluctuation does not depend on Γ m .
For the final combination of measured frequency noise spectral densities and simulations, the frequencies and linewidths of the mechanical modes of the exemplarily chosen full slot FFPC were fitted in the measured spectrum. The fitted noise peaks were then attributed to mechanical modes in the simulations at close-by frequencies. To get a final estimate of the noise caused by the thermal fluctuations, S ν ( f ) was calculated using the fitted mode frequencies and linewidths together with the vacuum optomechanical coupling strengths of the attributed mechanical modes from the simulations. Thereby slight shifts of the mechanical mode frequencies caused by deviations from idealized geometry are captured. Also, the corresponding linewidths are extracted without the necessity to include mechanical damping in the finite-element simulations.

Effect of structure asymmetries
In order to get a more realistic prediction of the expected frequency noise, a detailed model without the symmetry simplification and including some more properties of the final devices and some common imperfections was set up exemplarily for the full slot FFPC geometry, see Fig. 7. The introduced asymmetry causes more modes to couple to the optical resonance, e.g. sideward bending modes, whose coupling would otherwise be prevented by the symmetry. The longer fiber parts and the included silver glue pads at the bottom of the piezo cause the mechanical mode frequencies to slightly shift. More importantly, also the optomechanical coupling of the mechanical modes changes since the effective mass and the displacement fields are altered. For the model presented here, slightly smaller coupling strengths and thereby also smaller rms noise was retrieved from the more realistic model. Some pronounced noise peaks in the spectrum slightly below 100 kHz appear, which can also be found in the measured noise spectra. The final combination of measurement and simulation as presented in the main text used the more detailed model with the parameters (δ longitudinal , δ lateral , θ tilt ) = (1.9 mm, 0.05 mm, 0.2°) and a height of 0.5 mm for the glue pads on the bottom of the piezo (see Fig. 7). Fig. 7. Effect of asymmetries on the modeled spectra and rms noise. (a) shows an exaggerated sketch of the asymmetries and imperfections that were included in the finite-element simulations. To get a more realistic estimate of the spectrum the parameters were chosen as (δ longitudinal , δ lateral , θ tilt ) = (1.9 mm, 0.15 mm, 0.3°) here. In addition the length of the fiber pieces outside the ferrule was increased to 5 mm and the conductive glue points were included as silver pads at the bottom of the device (glue height: 0.7 mm, width fitting to the piezo-element). The comparison of the resulting spectra (exemplary Γ m, j /2π = 1 kHz) is plotted in (b). The asymmetry leads to a number of weakly coupled additional resonances as well as to a small reduction of the optomechanical coupling strengths, which results in a smaller estimate for the rms fluctuations in (c). Aside the integrated spectra also the separated contribution of each individual mode j to the rms noise 2g 2 0, j n j /2π for both the original, simplified symmetric and for the more realistic asymmetric model are indicated.

Thermal tuning
In addition to the fast piezo tuning as described in the main text, thermal tuning can also be used to change the cavity length. Fig. 8 shows the change in the cavity resonance frequency as the ambient temperature of single slot FFPC is varied. Here the FFPC is placed inside a thermally isolated box which has a Peltier element and a sensor to change and stabilize the temperature to a desired value. From the slope of the curve, the measured temperature tunability is 8 GHz/K.