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.

 ==Implication to AdvLIGO coatings==

As noted in SK2009, the estimated values for half infinite and finite size analyses are about the same (~2.5% difference) (I have not verified this). Then, the result from GWINC using Harry2002 formula is still accurate.

==note/comments==

• The calculation in SK2009 uses an overall loss angle of the coatings, while calculation in Harry2002 separates the elastic energy in two directions,parallel and perpendicular to the surface, and also loss angles in the associated directions.  I use the perpendicular average under the assumption that most energy/deformation occurs in that direction.
• The result matches the measurement quite well. This reassures us that other noises introduced by the setup (i.e. noise in optical bonding/ noise from supporting structure/ thermoelastic/ brownian noise in the spacer) are not higher than coating thermal noise.