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Message ID: 2250     Entry time: Fri Oct 19 17:05:10 2018
 Author: anchal Type: Summary Category: NoiseBudget Subject: Summary of present noise budget

Attached is the latest full noise budget we have. I generated it by rerunning current noise budget notebook. The aim of this post is to create a checkpoint of where we are, how the analysis was done and what we are going to do in the future.

Current Measurements:
There are 5 measurements of ASD of the beat note currently. 1 set (dark red) is from un-dated data (committed as ported from 40mSVN). Others have dates marked on them. These data sets are the spectrum of beat note frequency. No post-processing is done on these measurements and they are plotted as measured.

Coating Brownian Noise (Green):

• This is calculated using Harry et al. (2001) Eq 21 (written slightly differently here with only coating contribution)
$S_x^{\text{(cBr)}}(f) = \frac{2 k_\text{B} T}{\pi^2 f} \frac{d \phi_\text{c}}{w^2 E_\text{s}^2 E_\text{c}(1-\sigma_\text{c}^2)} \left[E_\text{c}^2 (1+\sigma_\text{s})^2 (1-2\sigma_\text{s})^2 + E_\text{s}^2 (1+\sigma_\text{c})^2(1-2\sigma_\text{c})\right]$
• Here, d: Coating Thickness  =  $4.6806 \pm 0.0004 \mu m$
w: Beam spot size on mirrors = $215.4 \pm 0.5 \mu m$
$\phi_c$: Coating Dielectric Loss Angle assuming $\phi_c = \phi_{||} = \phi_{\perp} = (2.41 \pm 0.2)\times 10^{-5}$
$E_s$: Substrate's Young modulus = $(72 \pm 1)\times 10^9 Pa$
$E_c$: Coating Young modulus =  $(100 \pm 20)\times 10^9 Pa$
$\sigma_s$: Substrate's Poisson ratio = $0.170 \pm 0.001$
$\sigma_c$: Coating Poisson Ratio = $0.311 \pm 0.062$
• Noise from above formula is in displacement. It is converted into frequency noise using the conversion factor for the cavity, $f_{conv} = \frac{c}{L \lambda} = (7.65 \pm 0.05)\times 10^{15} Hz/m$.
• It is the highest estimated source of noise in relevant frequency ranges.
• This is an oversimplified model and we are working on incorporating analysis due to Hong et al. (2013) for calculating this noise.

Coating Thermo-Optic Noise (Pink):

• This is calculated using analysis given in Evans et al. (2008). Particularly, Eq 4:
$S^{\Delta z}_{TO} = S^{\Delta T}_{TO}(\bar{\alpha_c}d - \bar{\beta}\lambda -\bar{\alpha_s}d\frac{C_c}{C_s})$
• Here, $S^{\Delta z}_{TO}$: Noise spectral density of displacement noise due to thermo-optic noises.

$S^{\Delta T}_{TO}$: Profile-weighted temperature fluctuation PSD, calculated using:
$S_T(f) = \frac{2^{3/2} k_\text{B} T^2}{\pi\kappa_\text{s}w}M(f/f_\text{T})$     with          $M(\Omega) = \Re\left[\int\limits_0^\infty\! \mathrm{d}u\, \frac{u\ \mathrm{e}^{-u^2/2}}{\left(u^2-\mathrm{i}\Omega\right)^{1/2}}\right]$
$\bar{\alpha_c}$: Effective thermoelastic coefficient for coating, calculated using Evans et a. (2008) Eq A1 and A4:
$\bar{\alpha_k} = \alpha_k \frac{1+\sigma_s}{1-\sigma_k}\left [ \frac{1+\sigma_k}{1+\sigma_s} + (1-2\sigma_s)\frac{E_k}{E_s} \right ]$        and        $\bar{\alpha_c} = \sum^{N}_{k=1} \bar{\alpha_k} \frac{d_k}{d}$
Here,  $\alpha_k$ : Thermoelastic coefficient for the k-th layer. $\alpha_{AlGaAs} = (5.24 \pm 0.52) \times 10^{-6} K^{-1}$ and $\alpha_{GaAs} = (5.97 \pm 0.6) \times 10^{-6} K^{-1}$
$d_k$ : Thickness of each layer. This is read from coatingLayers.csv which contains optimized. There are 57 layers in total.
Rest variables are the same as previously stated.
$\bar{\beta}$: Effective thermo-refractive coefficient for coating, calculated using Evans et a. (2008) Eq B3-B8. This is bit tedious calculations so I'll leave out the details.
$\lambda$ : Wavelength of beam light = 1064 nm
$C_c$: Effective heat capacity per unit volume of the coating,$C_c = \sum^N_{k=1} C_k \frac{d_k}{d}$ where $C_k$ is the heat capacity of the kth layer.
$C_{AlGaAs} = (1.698 \pm 0.001)\times 10^6 \frac{J}{Km^3}$        and       $C_{GaAs} = (1.75 \pm 0.09)\times 10^6 \frac{J}{Km^3}$
$C_s$ : Heat capacity per unit volume of substrate = $(1.6 \pm 0.1)\times 10^6 \frac{J}{Km^3}$
Rest variables are as stated before.

• Same as before, the calculated noise is displacement noise and is converted into frequency noise.
• This noise has been reduced significantly by making thermoelastic and thermorefractive noise cancel each other Its further analysis is at low priority as it is well below other noise sources.

Substrate Brownian Noise (Yellow):

• This is calculated using Cole et al. (2013) Eq.1.:
$S_x^{\text{(sBr)}}(f) = \frac{2 k_\text{B} T}{\pi^{3/2} f} \frac{1-\sigma_\text{s}^2}{w E_\text{s}}\phi_\text{s}$
• Here all variables are as stated before.
• Note: This is less by a factor of 2 from expression in the paper because the expressions in papers are for 2 mirrors together. We in our calculation take into account the number of mirrors separately.
• This noise source is the third highest source of noise in relevant frequency ranges. It is hard to make any change other than changing substrate material itself to reduce this noise source.
• This is a much better-understood source of noise and there are no proposed changes in its analysis for now.

Substrate Thermo Elastic Noise (Sky Blue):

• This is calculated using Somiya et al (2010) Eq 3 and 8: (Analytical expression for the theory by Cerdonio et al. (2001))
$S_x^{\text{(subTE)}}(f) = \frac{4 k_\text{B} T^2}{\pi^{1/2}} \frac{\alpha_\text{s}^2 (1+\sigma_\text{s})^2 w}{\kappa_\text{s}} J(f/f_\text{T})$       where    $J(\Omega) = -\operatorname{Re}\left\{\frac{\mathrm{e}^{\mathrm{i}\Omega/2}}{\Omega^2} (1 - \mathrm{i}\Omega)\, (\operatorname{Erfcom}\!{\left[\frac{\Omega^{1/2}(1+\mathrm{i})}{2}\right]})\right\} + \frac{1}{\Omega^2} - \frac{1}{(\pi\Omega^3)^{1/2}}$
• Here, $\kappa_s$ : Substrate Heat Conductivity = $1.38 \pm 0.2 W/(K m)$
$f_\text{T}$: Thermal relaxation frequency calculated by Cerdonio et al. (2001) Eq 9, $f_T = \frac{1}{2\pi}\frac{\kappa_s}{C w^2}$

• Note: There is a difference of $\sqrt{2}$ factor from Somiya and Cerdonio's expression because w in Somiya et al. and r0 are related as $r_0 = w/\sqrt{2}$ as explained in Black et al. PRL 93, 241101 (2004).
• This noise source is the highest noise source up to 90 Hz and is second to Coating Brownian Noise in the above frequencies.
• There no proposed changes in the analysis for this source for now.

PDH Shot Noise (Orange):

• This is calculates using (in $W^2/Hz$):
$S_P^\text{(PDHshot)} = 2h\nu P_0 \left[J_0(\Gamma)^2 (1-\eta) +3 J_1(\Gamma)^2\right]$
• Here, $\Gamma$: PDH modulation index of FSS = $0.2 \pm 0.005 \, rad$
$\eta$: Cavity Visibility = $0.35 \pm 0.05$
$P_0$: Incident power on FSS EOM = $3 \pm 0.2 \, mW$
• This is then converted into frequency noise spectral density by using PDH Slope (Also accounting for cavity pole):
$S_f^\text{(PDHshot)} = S_f^\text{(PDHshot)} \left(\frac{1 + \frac{f}{f_p}}{\Gamma'} \right)^2$
• Here, $f_p$ : Cavity Pole = $136 \pm 9 kHz$
$\Gamma^'$ :  PDH slope = $8.85 \pm 0.86 mW/Hz$
• This is a fairly low noise source and is a straightforward calculation. No updates required here other than remeasuring $\Gamma$ and $\eta$ .

PLL Oscillation Noise (Grey):

• This is directly measured and the data on the attached graph is from 2012.
• This data should be taken again and if possible should be included in automated scripts.

• This is calculated with what is referred to as "Tara's Magic Number":
$S^{f}_{PLL} = (f\times0.0207\times5.04\times 10^{-5})^2$
• I have no idea how this formula emerged. It would be best to measure the noise from PLL again and dissect and measure all independent noise sources again.

Seismic Noise (Black):

• This was measured last in 2011 October with a seismometer on the table.
• This doesn't affect much our region of interest but should be updated as we are 7 years in the future now.

Photothermal Noise (Brown):

• This is the coupling mechanism of intensity noise into frequency noise.
• To calculate this, following conversion from incident power noise to photothermal noise is used:
$S^{photoThermal}_f(f) = |H(f)|^2 P_{abs}^2 S_{RIN}$
• Here,  $H(f)$ :  Photothermal Transfer function in (m/W)
$P_{abs}$: Power absorbed by mirror calculated by $P_{abs} = \alpha_c \frac{P_0 \mathcal{F}}{\pi}$
Here, $\alpha_c$ : Coating absorptivity = $(6 \pm 1)\times 10^{-6}$
$P_0$: Incident Power
$\mathcal{F}$: Cavity finesse = $15000 \pm 1000$
$S_{RIN}$: Relative Intensity Noise Spectral Density
• Relative Intensity Noise is measured directly. For the plot attached, RIN was measured only on the south path and north path's RIN was assumed same.
• Photothermal Transfer Function is calculated due to Farsi et al. (2012) Eq. A51:
$H(f) = H_\text{c}(f) + H_\text{s}(f) + H_\text{tr}(f)$
• Here, $H_\text{c}(f)$ : is contribution through the coating, calculated using Eq. A44 (Details excluded)
$H_\text{s}(f)$: is contribution through the substrate, calculated using Eq. A45 (Details excluded)
$H_\text{tr}(f)$: is contribution through the thermorefractive process, calculated using Eq. A49 (Details excluded)
• This noise, it turns out is quite low in the relevant frequency range. No further updates are required in the analysis.
• The RIN needs to be measured again though for both the paths.

Residual NPRO Noise (Green):

• This calculation requires free-running frequency noise of NPRO. It is assumed to be following which is taken from Wilke et al. Opt. Lett. 25, 14 1019-1021 (2000):
$\sqrt{S_{\nu}(f)} = (10^4 \text{ Hz/Hz}^{-1/2})\times(1 \text{ Hz}/f)$ .
• Open Loop Transfer Function of FSS was measured in 2004 (see PSL:1504). The measurements are fitted with model transfer functions to get zeros and poles which are then used to estimate OLTF at frequency points of interest.
• The assumed NPRO free-running noise is then suppressed by these OLTFs (north and south) and added in quadrature.
• We need to measure the OLTFs again and if possible, measure the true free-running frequency noise of the NPROs.

### Near future plan:

We should be able to finish aligning South Path with the newly installed PMC by end of this week. Then if everything works fine, we should start taking beat note measurements every week and focus on updating old data in the estimate plots.

 Attachment 1: 20181019_154643noiseBudget.pdf  923 kB  Uploaded Mon Oct 22 17:27:25 2018
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