# Choosing Arm Cavity ROCs

by
Robert Ward
—
last modified
2010-06-07 13:57

Some plots showing how range can depend on choice of arm cavity mirror radii of curvature. The main factors considered are the change in thermal noise due to spot size and clipping losses due to spot size. The upshot is that with the current choice, a 2% deviation from spec will be a disaster.

## Choosing the ROCs

Several competing factors must enter the decision. These include mirror thermal noise, cavity stability, clipping losses as beams become larger, and tolerance to manufacturer errors in mirror production. Figure 3 shows the consequences of errors in polishing, for the easily illustrated case of a symmetric cavity. Increasing the radius of curvature decreases the spot size on the mirror, and thus reduces detector sensitivity. Conversely, decreasing the radius of curvature increases sensitivity, until the spot size becomes too large for the mirror and clipping losses become important.

Manufacturer errors are likely to be at the 1% level, but may be larger, and so we choose mirror radii of curvature 3.4% away from instability. This specific choice is made to minimize the chance of a coincidental higher-order mode resonance, while also maintaining as large a beam size as is reasonable given manufacturing constraints. The average radii is then 1551, and concurrently minimizing the thermal noise contribution and the clipping losses yields mirror radii of 1420 for the ITM and 1683 for the ETM.

The plots below have all been made with GWINC. They thus include mirror thermal noise (due to spot size) and the quantum noise of the light. The maximum power buildup is limited by clipping losses, according the simple approximation that any light which falls outside the coating diameter of 34 cm is lost.

These plots do NOT include effects due to differential errors on the RoCs (or other surface figure errors). Such un-considered effects may include:

- Mode-mismatching (leading to a contrast defect, decreased mode matching to the OMC, and the resulting loss of signal amplitude).
- Loss imbalance (leading to laser noise couplings, problems with setting an acceptable homodyne quadrature).
- Probably lots more.

## Symmetric ROCs.

For the two arm cavity mirrors having identical radii, we can clearly see the effect of ROC on sensitivity.

### Asymmetric ROCs (baseline)

For asymmetric RoCs, we center our attention around ITM RoC = 1416 and ETM RoC=1646 from the baseline design. The boxes show 1% and 2% errors on the RoCs, and the text indicates the resulting inspiral ranges.

### Asymmetric ROCs (proposal)

For asymmetric RoCs, we center our attention around ITM RoC = 1420 and ETM RoC=1683. The boxes show 1% and 2% errors on the RoCs, and the text indicates the resulting inspiral ranges.

### History/Experience

For Virgo+, the specification on the end mirror RoCs was 3450 +/- 100m, which is a 3% error. After coating, the central 150mm of the NE mirror was 3300, which is 4.3% out of spec. See VIR-0334A-10 for the full tables of measurements.