TCS noise calculations

The heating profile of the AXICON can be fitted with a finction of this kind:
I(r)=2r^2/w1^2*4*(w2^2-r^2)^2/w2^4*exp(-2r^2/w1^2)
with w1=0.099 and w2=0.0048 for our present telescope (see attached plot).



Considering that r^2=x^2+y^2, the above function can be rewritten as:
I(x,y)=8/(w1^2*w2^4)*(w2^4*(x^2+y^2)-2w2^2*(x^2+y^2)^2+(x^2+y^2)^3)*exp(-2(x^2+y^2)/w1^2)
As stated in LIGO-T060224-00-D, to compute the flexural noise it is necessary to consider the first derivative of I(x,y) in x and y. After some algebra (if the calculations are right), I get:
dI/dy=16y/(w1^2*w2^4)*exp(-2(x^2+y^2)/w1^2)*(w2^4-2w2^2*(x^2+y^2)*(2+w2^2/w1^2)+(x^2+y^2)^2*(3+4w2^2/w1^2)-2/w1^2*(x^2+y^2)^3)
but considering that w2^2/w1^2=2.5e-3, we can approximate the above expression to:
dI/dy=16y/(w1^2*w2^4)*exp(-2(x^2+y^2)/w1^2)*(w2^4-4w2^2*(x^2+y^2)+3*(x^2+y^2)^2-2/w1^2*(x^2+y^2)^3)

Using dI/dy found in the previous topic, it is possible to evaluate the RIN requirements in case of direct application of the CO₂ beam on the TM.

Applying 10W of CO₂ on the HR face requires a RIN better than 5*10^-9/Hz^1/2 @ 50Hz (see figure below)



While if CO₂ beam is applied on both faces of the TM (2.5W on HR and 7.5W on AR faces), the RIN requirement is relaxed to 2*10^-8/Hz^1/2 @ 50 Hz (see figure below)