Summary

In a previous elog, I demonstrated that the RoC mismatch between ETMX and ETMY does not result in appreciable degradation in the mode overlap of the two arm modes. Koji suggested also checking the effect on the contrast defect. I'm attaching the results of this investigation (I've plotted the contrast,   rather than the contrast defect 1-C).

Details and methodology

• I used the same .kat file that I had made for the current configuration of the 40m, except that I set the reflectivities of the PRM and the SRM to 0. 
• Then, I traced the Y arm cavity mode back to the node at which the laser sits in my .kat file to determine what beam coming out of the laser would be 100% matched to the Y arm (code used to do this attached)
• I then set the beam coming out of the laser for the subsequent simulations to the value thus determined using the gauss command in finesse.
• I then varied the RoC of ETMX (I varied the sagittal and tangential RoCs simultaneously) between 50m and 70m. As per the wiki page, the spare ETMs have an RoC between 54 and 55m, while the current ETMs have an RoC of 60.26m and 59.48m for the Y and X arms respectively (I quote the values in the "ATF" column). Simultaneously, at each value of the RoC of ETMX, I swept the microscopic position of the ETMX HR surface through 2pi radians (-180 degrees to 180 degrees) using the phi functionalilty of finesse, while monitoring the power at the AS port of this configuration using a pd in finesse. This guarantees that I sweep through all the resonances. I then calculate the contrast using the above formula. I divided the parameter space into a grid of 50 points for the RoC of ETMX and 1000 points for the microscopic position of ETMX. 
• I fixed the RoC of ETMY as 57.6m in the simulations... Also, the maxtem option in the .kat file is set to 4 (i.e. higher order modes with indices m+n<=4 are accounted for...)

Result:

Attachment #1 shows the result of this scan (as mentioned earlier, I plot the contrast C and not the contrast defect 1-C, sorry for the wrong plot title but it takes ~30mins to run the simulation which is why I didn't want to do it agian). If the RoC of the spare ETMs is about 54m, the loss in contrast is about 0.5%. This is in good agreement with this technical note by Koji - it tells us to expect a contrast defect in the region of 0.5%-1% (depending on what parameter you use as the RoC of ETMY). 

Conclusion:

It doesn't seem that switching out the current ETM with one of the spare ETMs will result in dramatic degradation of the contrast defect...

Misc notes:

1. Regarding the phase command in Finesse - EricQ pointed out that the default value of this is 3, which as per the manual could give unphysical results sometimes. The flags "0" or "2" are guaranteed to yield physical results always according to the manual, so it is best to set this flag appropriately for all future Finesse simulaitons. 
2. I quickly poked around inside the cabinet near the EX table labelled "clean optics" to see if I could locate the spare ETMs. In my (non-exhaustive) search, I could not find it in any of the boxes labelled "2010 upgrade" or something to that effect. I did however find empty boxes for ETMU05 and ETMU07 which are the ETMs currently in the IFO... Does anyone know if I should look elsewhere for these?
   EDIT 17Jun2016: I have located ETMU06 and ETMU08, they are indeed in the cabinet at the X end...
3. I'm attaching a zip file with all the code used to do this simulation. The phase flag has been appropriately set in the (only) .kat file. setLaserQparam.py was used to determine what beam parameter to assign to be perfectly matched to the Y arm. modeMatchCheck_ETM.py was used to generate the contrast as a function of the RoC of ETMX.
4. With regards to the remaining checks to be done - I will post results of my investigations into the HOM scans as a function of the RoC of the ETMs and also the folding mirrors shortly...