40m QIL Cryo_Lab CTN SUS_Lab TCS_Lab OMC_Lab CRIME_Lab FEA ENG_Labs OptContFac Mariner WBEEShop
 PSL Not logged in Message ID: 1239     Entry time: Mon Jul 15 10:55:40 2013     In reply to: 1192
 Author: tara Type: Notes Category: NoiseBudget Subject: AlAs/GaAs layer structure optimized for TO

After a discussion with Eric and Matt, here I'll summarize about thermo optic(TO) noise calculation plus some other important noise sources.

1) goal

We aim to measure the limiting noise in AlGaAs coatings. If we order just 1/4 quarter wave stack, no optimization, the limiting noise source will be from TO noise due to high values of thermo elastic(TE) and thermo refractive(TR) coefficients of the materials. However, by optimizing the coatings structure to cancel TO noise we can:

• Probe thermo-elastic (TE) noise in SiO2 substrate at low frequency and coating Brownian noise(BR) at higher frequency
• Prove that TO cancellation can be done (according to Evans etal).

We can tell what kind of noise from the slope.  BR, TO noise or TE noise in substrate have different slopes at the interested band, see fig 1.

2) Is the calculation correct? fig1: noise budget with some fundamental noise sources. The noise budget is for AlGaAs coatings on a mirror with ROC=1m. The cap is GaAs (high index material) with 1/8 lambda thickness. See explanation below for more details.

The fundamental noise sources in our setup (1.45" cavity, 1m roc mirror, optimized AlGaAs coatings) will be:

==BR in coatings==:

• The calculation is taken from Harry2002, for half infinite mirror.
• The result is compared with Somiya&Kazuhiro2009 for finite size mirror calculation (see solid blue line and dashed cyan line). The difference is small due to our small spotsize, so using either calculation is ok for us, but Harry's calculation is less time consuming.
• The analytical result should be valid as it was verified by Numata and TNI measurements.

==BR in substrate==:

• The calculation is taken from Levin1998, with finite size correction by Liu&Thorne(LT2001).
• The loss angle for bulk fused silica is frequency dependent ~ 10-11 x f0.8(Penn2006). This loss is much lower than conservative constant loss (10^-8) (number from DCC LIGO-T0900161) from dc upto 10kHz.
• In this calculation, for constant loss of 10^-8, BR noise in substrate is still ~ a factor of 3 lower than BR in coatings.

==TE noise in substrate==:

• BGV1999 gave a result for adiabatic limit (most of the heat flow is in 1-D heat diffusion length is much smaller than beamsize, sqrt(kappa/C * 2pi*f)<<r0 )for half infinite space mirror, Liu Thorne2001 verify the result. I used comsol to simulate the noise (with adiabatic assumption) and it agreed with the analytical solution.
• However for our setup with a small spot size the assumption beaks down. Cerdonio2001, computed the noise that valid for low frequency and small beamsize which is a  case for our setup (cut off frquency ~ 10 Hz). All the factors and corrections are summarized in TNI2004 measurement and Nawrodt2012.  The calculation will be valid for our setup.

==TE and TR noise calculation:

•   The temperature fluctuation sensed by the beam is taken from BGV1999 using Langevin approach, and Mike Martin Thesis (this takes care of the fluctuation at low frequency where adiabatic assumption breaks down. The calculation assume that coating thickness << thermal diffusion length. For AlGaAs, because of its high thermal conductivity, this assumption is still hold at the bandwidth of interest.
• The thick coating calculation is given in Evans 2008.  It is important at high frequency and coatings with low thermal conductivity. This means that TE and TR effects won't be coherent in the coatings. This is not a problem for AlGaAs due to its high thermal conductivity.
• TE and TR coefficients calculations are treated coherently in Evans2008.  The cancellation only depends on coating structure. With a cap of GaAs (nH) 1/8 lambda thickness, the cancellation is very good reducing the TO noise below other noise upto a few kHz.
• The cap thickness has to be withing +/- 20Angstrom so that the TO is about a factor of2 below coating BR. G. Cole mentioned that each layer thickness varies about 0.3% or less which is about lambda/(4*n) * 0.3% = 2Angstrom. So the cancellation should be ok.

TR coefficients are calculated numerically (GWINC) and analytically (Gorodetsky2008). The results match up well (less than 1% difference), if all the parameters/ averaged values are from Evans.

In GWINC there is one correction noted as "Yamamoto thermo-refractive correction", this changes the Beta eff ~ 10% causing the cancellation to be not as good (still ok up to 1kHz). I emailed Kazuhiro Yamamoto asking him if he has anything to do with this. Otherwise all the calculations and optimization are in good shape.

 Attachment 2: alGaAsnb.fig  154 kB  Uploaded Wed Jul 17 03:03:18 2013
ELOG V3.1.3-