[Koji Gautam]
With Gautam's help, I ran a coating design code for an HR mirror with the standard quarter-wave design. The design used here has 17 pairs of lambda/4 layers of SiO_{2} and Ta_{2}O_{5} (=34 layers) with the fused silica as the substrate to realize the transmission of tens of ppm. At the AOI (angle of incidence) of 4 deg (=nominal angle for the aLIGO OMC), there is no significant change in the reflectivity (transmissivity). With 95% of the case, the phase difference at the AOI of 4 deg is smaller than 0.02 deg for given 1% fluctuation (normal distribution) of the layer design and the refractive indeces of the materials. Considering the number of the OMC mirrors (i.e. 4), the total phase shift between P and S pols is less than 0.08 deg. This makes P and S resonances matched well within 1/10 of the cavity resonant width (360/F=0.9deg, F: Finesse=400).
Of course, we don't know how much layer-thickness fluctuation we actually have. Therefore, we should check the actual cavity resonance center of the OMC cavity for the polarizations.
Attachment 1 shows the complex reflectivity of the mirror for P and S pols between AOIs of 0 deg and 45 deg. Below 30 deg there is no significant difference. (We need to look at the transmission and the phase difference)
Attachment 2 shows the power transmissivity of the mirror for P and S pols between AOIs of 0 deg and 45 deg. For the purpose to check the robustness of the reflectivity, random fluctuations (normal distribution, sigma = 1%) were applied to the thicknesses of each layer, and the refractive indices of Silica and Tantala. The blue and red bands show the regions that the 90% of the samples fell in for P and S pols, respectively. There are median curves on the plot, but they are not well visible as they match with the ideal case. This figure indicates that the model coating well represents the mirror with the transmissivity better than 70ppm.
Attachment 3 shows the phase difference of the mirror complex reflectivity for P and S pols between AOIs of 0deg and 45deg. In the ideal case, the phase difference at the AOI of 4deg is 1x10^{-5} deg. The Monte-Carlo test shows that the range of the phase for 90% of the case fell into the range between 5x10^{-4} deg and 0.02 deg. The median was turned to be 5x10^{-3} deg.
Attachment 4 shows the histogram of the phase difference at the AOI of 4deg. The phase difference tends to concentrate at the side of the smaller angle. |