Feasibility of near-unstable cavities for future gravitational wave detectors

Near-unstable cavities have been proposed as an enabling technology for future gravitational wave detectors, as their compact structure and large beam spots can reduce the coating thermal noise of the interferometer. We present a tabletop experiment investigating the behavior of an optical cavity as it is parametrically pushed to geometrical instability. We report on the observed degeneracies of the cavity’s eigenmodes as the cavity becomes unstable and the resonance conditions become hyper-sensitive to mirror surface imperfections. A simple model of the cavity and precise measurements of the resonant frequencies allow us to characterize the stability of the cavity and give an estimate of the mirror astigmatism. The significance of these results for gravitational wave detectors is discussed, and avenues for further research are suggested.

Figure 1

Stability diagram of a Fabry-Perot cavity. All colored regions bounded by the curves $g_{1,2}=0$, and $g_1{g}_2=1$ correspond to stable configurations. The colored cavity configurations correspond to the regions of the diagram with matching color. Cavities close to the edges are less stable.

Figure 2

The separation factor of the first-order mode ${\delta }_1$ as a function of the cavity $g$ factor in different gravitational wave detectors. The dots represent the separation factors in their current configurations. They all use mode separation factors above 100, except from the AEI 10 m prototype with ${\delta }_1\approx 20$ when pushed to its maximum design stability. Our tabletop cavity, with a finesse of around 2000, has separation factor similar to the prototype at its targeted stability.

Figure 3

Simulation showing the resonant frequencies of the fundamental mode and the first- and second-order spatial modes in the tabletop cavity in its initial stable configuration. Offsets in resonant frequencies can be measured and used to extract information about the cavity stability. The resonances of all modes have periodicity over the cavity’s free spectral range. In the near-unstable plane-concave cavity the even-order modes become coresonant with the fundamental mode, and the odd-order modes become antiresonant.

Figure 4

Schematic of the near-unstable cavity setup. The blue and red lines represent the beams from the 1064 nm control and probe lasers, respectively. They pass through phase modulators (EOM) that generate the sidebands used for control purposes. The beams are combined at the central beam splitter (BS) and then coupled into the cavity through two mode matching lenses. The beams have orthogonal polarizations upon combination to ensure that they don’t interfere. The control laser is locked to the cavity via a Pound-Drever-Hall loop (PDH). A fraction of the PDH signal is fed back to the probe laser to reduce common mode frequency noise. The probe laser is then locked to the control laser with a tunable frequency offset via an offset phase locking loop. The frequency offset is controlled by the source of a network analyzer. The probe laser is then used to study the cavity. A camera and a photodetector (PD) are placed at the cavity’s transmission port to observe the resonant mode. The subscripts of R and T stand for reflection and transmission ports of the cavity, respectively. An alignment sensing system based on wavefront sensing is used to help maintain accurate alignment of the cavity.

Figure 5

A cavity scan measurement showing the resonant frequencies the fundamental mode, and the second- and fourth-order spatial modes. The resonances of modes of the same order but orthogonal spatial profile appear separate due to the imperfect surface of the spherical mirror. The shapes of the modes as they are scanned are shown in the bottom. The separation can be reduced by increasing the stress of the screw holding the spherical mirror, thus compensating the surface deformation. The shapes of the compensated second- and fourth-order modes are shown in the right, which appear as circularly symmetric Laguerre-Gauss modes.

Figure 6

Resonances measured by scanning the frequency of the probe laser when pushing the cavity to near-unstable region. We track resonant frequencies of the fundamental mode, the HG02 and HG20 modes, which are plotted in black, blue and red, respectively. L5 is the position where the translation stage gives a reference reading of roughly 999 mm. Legends from L1 to L18 display relative length changes from this value. The y-axis represents the cavity length deviation from L5 for each measurement. Peaks on the right are resonant frequencies of the fundamental mode, which give a very good measurement of the free spectral range of the cavity. The two peaks on the left at first represent resonances of the separated HG20 and HG02 modes. Mode separation frequencies between second-order modes and the fundamental mode become smaller and smaller. But beyond a certain point, it seems that second-order modes begin to move back away from the fundamental mode. We believe that what we tracked are no longer the pure second-order modes; indeed, the measured beam profiles appear to have an increasing fraction of higher-order mode content. It could be that the cavity has in these cases become unstable, but these modes are near-resonant in the cavity.

Figure 7

Resonant frequencies of the ${\mathrm{HG}}_{00}$ (at one FSR), ${\mathrm{HG}}_{20}$, and ${\mathrm{HG}}_{02}$ modes. The measurement of the FSR allows us to obtain the cavity length for every point. Fits of the cavity length to the frequencies of the ${\mathrm{HG}}_{20}$ and ${\mathrm{HG}}_{02}$ modes yield two values of the mirror RoC that differ by $145\mu \mathrm{m}$, indicating astigmatism. Only the filled points are used in the fits, as the data start deviating from the expected behavior for Gaussian beams as it gradually becomes unstable. The measured resonances of the ${\mathrm{HG}}_{20}$ and ${\mathrm{HG}}_{02}$ modes start to deviate from the fitting at L7 and L10, respectively, where the deviation is 1% in frequency. The cavity $g$ factor is derived as an average property by using the average RoC of the mirror. The $g$ factor suggests that the cavity becomes unstable at approximately L15, which agrees well with observations.

Figure 8

Spatial profile of the transmitted beam at each step of the translation stage of the end mirror, the horizontal position of each image depicts the frequency of each resonance as seen in Fig. 7. Together with our fitting results shown in Fig. 7, we regard these points below as turning points. L7: ${\mathrm{HG}}_{02}$ starts to deviate from the predicted behavior and shows a gradual clockwise rotation in its profile. L10: ${\mathrm{HG}}_{20}$ starts to deviate and shows a clear rotation. L13: ${\mathrm{HG}}_{02}$ arrives at its stability edge, however, the cavity is still stable for ${\mathrm{HG}}_{20}$. L15: ${\mathrm{HG}}_{20}$ arrives at its stability edge and the cavity becomes completely unstable. From L13, the spatial profile of the fundamental mode goes from a typical round-spot shape to a two-spot shape, as it overlaps with the second-order mode. Modes we tracked after this point no longer appear to belong to Gaussian modes if the cavity is unstable.