I calculated brownian noise in AlAs/GaAs coatings, brownian noise and thermoelastic noise in fused silica substrate for different beam sizes. From the plot, we can see that a smaller spotsize might be better for us.

     This is a quick study to see the how spotsize on a mirror affects Brownian noise and thermoelastic noise in coatings and substrate. The radii of the beam (where the beam intensity drops by 1/e^2) used in the calculation are 91, 182, 364 um. Loss in coatings is 10^-5, loss in substrate is 10^-7. Note for 1.45" cavity with 0.5m RoC mirrors, the beam radius is 182 um.

Reminders:

• The plot is shown in displacement noise, not frequency noise from cavity.
• The psd (m^2/Hz) of coating Brownian noise is proportional to 1/w^2 (w is the beam radius)
• The psd of substrate Brownian is proportional to 1/w
• The psd of substrate thermoelastic is proportional to 1/w^3 ,at high frequency where adiabatic assumption valids. But at low frequency, when heat diffusive flow rate is comparable to the beam radii, TE noise is reduced from that of adiabatic assumption.

The Brownian noise in the coatings is more comparable to TE noise in substrate with smaller beam size although the crossing between the two noises are at higher frequency. So it should be able to see the total noise from both effects. However, to get smaller beamsize, we probably have to use even shorter cavities, or smaller RoC mirrros. So it might not be practical for us.. Nevertheless, going to smaller beam size should  be a good idea.

Note:

• for 1.45" long cavity, no choices of RoC give w = 92 um,
• for mirror with RoC = 0.5m, cavity length of 0.1 inch(2.5 mm) gives w = 92 um
• I think I made a mistake in the proposal since Brownian noise in substrate was higher than coatings' noise. I double checked it for this calculation and Brownian noise in substrate is always lower than coating brownian.