Message ID: 1192
Entry time: Thu Jun 6 22:28:46 2013
In reply to: 1180
Reply to this: 1193 1239

Author:

tara

Type:

Notes

Category:

NoiseBudget

Subject:

AlAs/GaAs layer structure optimized for TO

Here is an outline for TO calculation. I tried to summarize it and make it as simple to follow as possible.

Use Levin's direct approach to calculate thermal fluctuations seen by the beam.

Apply power injection at the coating surface, with proper boundary condition, take coating into account. (Evans2008 see thick coating correction, Somiya2009)

To calculate the loss due to the dissipated heat, we need to solve heat equation. The loss associated with the injected heat is proportional to (gradient of temperature)^{2}

The calculation for gradient of temperature has to be calculated in both longitudinal and transverse direction, as thermal length is comparable to the beam size [Cerdonio 2001]. Other papers usually approximate grad T = dT/dZ, which is 1-D treatment [Evans2008, Somiya2009]. The effect from Heat flow in transverse direction shows up at low frequency, where the noise level becomes lower.

When solve heat diffusion equation, apply boundary condition for finite size mirror (somiya2009).

Once we have thermal fluctuations, S_{T}, we convert it to displacement noise with TE and TR coefficients. Sx = S_{T} *(TE + TR)

TE and TR coefficients can be calculated from the layer structure. The cancellation will occur only at lower frequency where temp fluctuations in coatings are uniform. At higher frequency the effect from TE and TR will sum up in quadrature (if heat equation is solved in coatings), see thick coat correction section in Evans2008.

This means that for TO optimized coatings, we have to make sure that TE and TR coefficients are comparable for maximum cancellation. The calculation for TE and TR are quite well defined, [Fejer2004, Evans2008, Gorodetsky2008]. This part is independent from temperature fluctuation calculation outlined above. So we can choose the optimized design and then calculate the total TO noise level later. The proposed optimization can be found in psl:1183. (Here is the result for 1/8 cap of nH).

Note:

Basically most of the calculations outlined above are done in Somiya2009, except transverse heat flow. If we consider transverse heat flow in coatings and substrate, the result will be valid at low frequency as well.

The decision for G Cole etal to use substrate parameters in temperature fluctuations as suggested by Rana seems to be ok, since their calculation also include the thick coat correction (Evans2008), it means that temperature fluctuations in coatings are taken into account. However, the cutoff frequency might be off a bit, since the equation for transverse flow is only in substrate (BGV1999, cerdonio2001). I think the real cutoff frequency should be higher because kappa is larger in the coatings, and transverse heat flow becomes more significant at higher frequency.

Somiya paper also include Brownian noise in Coatings with finite size substrate/coatings (see fig2) which is not done in Harry etal 2002. Finite size effect increases the noise level by a lot, I think this might explain why the beat result we measured from 8" cavities is a bit higher than the estimated noise using the result from Harry etal. I'll check that later.

I'm not quite sure about The TO calculation in Somiya. The injected heat from TO and TE are added independently, however, the result is similar to that of Evans (with half infinite limit). I'm checking it.