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 PSL Not logged in Message ID: 1374     Entry time: Sun Oct 27 20:12:25 2013     In reply to: 1365     Reply to this: 1392
 Author: tara Type: Notes Category: optic Subject: photothermal noise in AlGaAs

I revised the calculation for photo-thermal noise in AlGaAs coatings, the photo thermal noise should not be a limiting source.

==review==

photothermal noise arises from the fluctuation in the absorbed laser power (RIN + shot noise, mostly from RIN) on the mirror. The absorbed power heats up the coatings and the mirror. The expansion coefficient and refractive coefficients  convert thermal change into phase change in the reflected beam which is the same effect as the change of the position of the mirror surface.

Farsi etal 2012, calculate the displacement noise from the effect. The methods are

• Solving heat equation to get temperature profile in the mirror.
• Use elastic equation to calculate the displacement noise due to the temperature change (thermoelastic)
• For TR, the effect is estimated from effective beta (from QWL stack) and the temperature at the surface ,as most of the TR effect comes from only the first few layers

When they solve the heat equation, the assume that all the heat is absorbed on the surface of the mirror. This assumption is ok for their case ( SiO2/Ta2O5) with Ta2O5 at the top surface, all QWL, as 74% of the power is absorbed in the first four layers (with the assumption that the absorbed power is proportional to the intensity of the beam, and all absorption in both materials are similar).

However, for AlGaAs coatings with (nH/nL) = (3.48/2.977) The E field goes in the coatings more that it does in SiO2/Ta2O5, see the previous entry. So we might want to look deeper in the calculation and make sure that photo thermal noise will not be a dominating noise source.

==calculation and a hand waving argument==

The plot below shows the intensity of the beam in AlGaAs Coatings, opt4, and the estimated intensity that decreases with exponential square A exp(-z^2/z0^2). X axis is plotted in nm (distance from surface into coatings). The thickness of opt4 is about 4500 nm. To simplify the problem, I use the exponential decay function as the heat source in the diff equation. But I have not been able to solve this differential equation yet. Finding particular solution is impossible.  So I tried to solve it numerically with newton's method, see PSL:284. But the solution does not converge. I'm trying green function approach, but i'm still in the middle of it. However, the coatings optimized for TO noise should still be working. Evans etal 2008 discuss about how the cancellation works because the thermal length is longer than the coating thickness. The calculation (TE and TR)  treat that the temperature is coherent in all the coatings ( they also do the thick coatings correction where the heat is not all coherent, and the cancellation starts to fail at several kHz). So the clue here is that the cancellation works if the heat (temperature) in the coatings change coherently.

For photothermal calculation, if we follow the assumption that all heat is absorbed at the surface (as in Farsi etal), we get the result as shown in psl:1298, where most of the effect comes from substrate TE . In reality, where heat is absorbed inside the coatings as shown in the above plot, heat distribution in the coatings will be even more coherent, and the effect from TE and TR should be able to cancel each other better. Plus, higher thermal conductivity of AlGaAs will help distribute the heat through the coatings better.

This means that  the whole coatings should see the temperature change more coherently, thus allowing the TO cancellation in the coatings to work. The assumption that heat is absorbed on the surface should put us on an upper limit of the photothermal noise.

This means that photothermal noise in the optimized coatings should be small and will not be a dominating source for the measurement.

 Attachment 2: Int_cotings.fig  12 kB  Uploaded Mon Oct 28 14:59:38 2013
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