Dan B, Cao

Following from Dan's alog52621, I've been looking into a Finesse model to describe the higher order mode content in  the 9 and 45 MHz sideband as we introduce differential thermal lensing with the CO2 lasers during DRMI configuration.  A -5 to 5 uD  lens is added to ITMX and ITMY  respectively in the model.  The first plot  attached below shows the ratio between the higher order modes and the fundamental mode (up to the 4th order) as functions of  the differential lens, obtained from Finesse amplitude detectors. The amplitude of each modes and their rate  of change with respect to  the change in the differential lensing can be be computed. The rate of change of ithe HG02 and HG20 amplitudes relative the HG00 are shown in the  second plot for the 9 and 45 MHz sidebands. As seen here, these rates of change are not constant and dependent on the amount of differential thermal lensing. Since the CO2 lasers have been previously optimised for DRMI locking, we assume  that we are are operating  around the  well-matched" point, around which the rates of change in HG02/HG20 amplitude are minimum at approximately 0.02-0.05%/ microDiopter change in differential lensing.

During our CO2 laser test ( alog5280), the CO2y laser power was first turned off and then doubled. to approx 2.2 W before returning to a nomnal value of approximately 0.9 W. The same procedure is repeated for  CO2Y (see the first panel below).  I estimated the amount of differential  lensing given  the change in spherical  power using the zpk  model of CO2 lensing in TCS simulation page.  Using this estimated of differential lensing,  approximate  changes in the power in HG02 and HG20 can be computed using the rates of change in spherical power from the Finesse model. The last figure shows  the CO2x, CO2y powers,  estimated change in spherical power of by CO2 zpk model and  measured ratio between the powers in HG2 and HG0  from the phase camera compared to the simulation.  The shaded regions give the simulation upper and lower limit of change in HOMs powers and are determined by the maximum and minimum rate of change in a region of  ± 2.5  mD of differential lensing. 

In summary, the model agrees well with the measurements by the phase cameras, especially  for differential lensing due to CO2y, showing 0.5%-1% change in HG20/HG02 powers during our test. There seems to be some misalignment of CO2x, as mentioned in alog52621, causing deviation from the model, which only assumes perfect differential lensing.   We must have been having the ITMX and ITMY relatively well matched at  current nominal powers of CO2x and CO2y, given the  rate of change used in the models. There are still some uncertainty regarding   the zpk models and which point of operation that we are on. We plan to  perform Bayesian inference to get a better estimation of this operating point and its corresponding rate of change in HG2 powers.