The final measurement obtained from the sideband sweep data taken last Thursday was of the IMC g-factor (or alternatively the MC2 radius of curvature).

For this measurement, we wanted to measure the resonant frequencies of higher-order modes in the IMC. To do this we need to observe beats between higher-order mode sidebands transmitted through the cavity with the HG00 mode carrier also transmitted through the cavity.

Due to the spatial orthogonality of higher-order modes, a single element photodiode would would usually be insensitive to this signal (hence using wavefront sensors to detect misalignment). In the absence of a broadband wavefront sensor in transmission of the IMC, we can instead break the spatial symmetry at the REFL AIR photodiode using any physical obstruction to recover the signal. In this case, we used an iris offset with respect to the beam centre to partially block the beam. In addition to this, the IMC input beam was misaligned by disengaging the WFS loops and applying offsets to the periscope PZT in order to better ring up higher-order modes.

The first attached plot shows a sweep of one whole FSR around the 45.5MHz FSR peak. Overlaid on the measured data is the output from a Finesse model of the IMC, with the IMC geometry as designed expect for the MC2 radius of curvature which was taken as 27.275m from the nebula optics page. The Finesse model allows us to quickly identify each peak in the measured transfer function.

The higher-order mode content in the model was hand-tuned to match the data, as the effects of the iris are effectively un-quantifiable.  Since for this measurement we are only interested in the higher-order mode resonance freuqencies, correct y-axis values scaling is not necessary.

Higher resolution scans were then made of 6 individual higher-order mode peaks. The second attached plot shows the results of fitting a Lorentzian function to these higher-order mode peaks. The cavity g-factors corresponding to the fitted higher-order mode difference frequency (using the mean FSR reported in 8087) were then calculated. From there, the MC2 Rc could be calculated.

The results are shown in the following table:

The "Rc_eff" column shows the effective MC2 radius of curvature experienced in either the tangential or sagittal planes. The IMC beam probes MC2 under a small angle (0.82^o - see T0900386), and as a result the Rc experienced in the tangential (xz) plane is shorter than the normal incidence Rc by a factor of cos(0.82^o), and the sagittal plane (yz) Rc is longer than the normal incidence Rc by a factor of cos(0.82^o). The difference frequency between the HG00 mode and the HG10 mode is determined by the effective Rc in the tangential plane, and the difference frequency between the HG00 mode and the HG01 mode is determined by the effective Rc in the sagittal plane. The Rc_eff, g-factor and HOM diff. freq. entries for the Nebula/design rows were calculated from the quoted Rc of 27275mm and the incident angle 0.82^o.

The Rc column shows the equivalent normal incidence Rc values. For the fitted peaks these values were calculated from the Rc_eff values and the incident angle 0.82^o. For the Nebula/design values these are just the quoted Rc from the vendor measurement.

The mean value of MC2 Rc from all 6 peak measurements is 27277mm, with standard deviation 11mm*. This can be compared with the polisher's reported value of 27275mm (from the nebula page), the manufacturing requirement of 27240±140mm (see E070079-A-D), and the manufacturing goal of 27240±30mm (see also E070079-A-D). The value of Rc from this measurement agreed with the vendor value within the error bars, and is well within the manufacturing requirements (although it is slightly outside the manufacturing goal).

* The fits for HG11 and HG20/02 are a little suspect due to the influence of other nearby resonances. In both cases this influence can be expected to lower the measured peak frequency. For the HG11 mode this has the effect of increasing the HOM diff freq. and therefore lowering the calculated Rc. For the HG20/02 mode this has the effect of lowering the HOM diff freq. and therefore increasing the calculated Rc. The mean value for the Rc when these two measurements are discounted is 27278mm, and the standard deviation is 10.5mm. It appears therefore that the nearby resonances have a roughly equal and opposite effect on the HG11 and HG20/02 Rc values, and thus don't strongly affect the final result.