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 PSL Not logged in Message ID: 1194     Entry time: Tue Jun 11 16:46:52 2013     In reply to: 1193
 Author: tara Type: Notes Category: NoiseBudget Subject: noisebudget for 8" SiO2/Ta2O5 cavity

note about the calculation for coating Brownian noise in a finite size mirror .

==Coatings Parameters==

Young's modulus, Poisson's ratio, and loss angle are taken from the volume averaged value of the coatings (Yavg = d/ ( d1/Y1 + d2/Y2) , sigma avg = 1/2 (sigma1+sigma2 ). These are used for "perpendicular" direction in Harry2002 formula.

Loss angles

• SiO2 loss angle  = 1e-4
• Ta2O5 loss angle = 2.3e-4
• coatings loss = 1.32e-4

Young's moduli

• SiO2 Young's modulus = 72e9  Pa
• Ta2O5                         =140e9  Pa
• Coatings Young's modulus = 93e9 Pa

Coatings structure

• 1/2 lambda cap of SiO2
• 26 layers
• 300 ppm transmission

==calculation codes==

• I got the file for finding zeroes of the bessel function from Matlab exchange.
• The code for calculating Br noise is attached below.
• For the finite size bdy condition, the solutions include all the besselj function of all orders (m=1 to inf). I used m from 1 to 55 in the calculation since it converged quite fast after that.
• For the integration to calculate all the elastic energy, I used Riemann sum, with stepsize of ~0.15 um. The result does not change much (less than 3%) if I go from 0.8*a to a where a is the radius of the mirror. This is important to note because our coatings do not cover the whole surface of the mirror. There is an annulus edge with ~3mm width for optical contact area. The result means that the elastic energy is still localized in the spot area.