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: 2023     Entry time: Thu Dec 21 08:58:39 2017
 Author: Craig Type: DailyProgress Category: scatter Subject: Scatter Shoulder Fit

I've been learning about scattering from here and here.  It seems most scattering equations are left in an arbitrary form of $x_{\text{scatter}}(t)$, and simply measure seismic noise then use it as $x_{\text{scatter}}(t)$ in the following FFT:

$S_f(\omega) = \int_{-\infty}^{\infty} \sin\left(\dfrac{4 \pi}{\lambda} x_{\text{scatter}}(t)\right) e^{i \, \omega \, t} dt$

If the sine wave were perfectly sinusoidal, say $x_{\text{scatter}}(t) = x_0\,\Omega \, t$, our FFT would yield delta functions at $\omega = \pm \Omega$.  However, scattering is rarely so clean.

If $x_{\text{scatter}}(t) \ll \lambda$ where lambda is the laser wavelength, then our sine wave is approximately linear, and we get a clean spectrum at frequencies above the seismic noise.

When $x_{\text{scatter}}(t) \approx \lambda$ , we get "upconversion" of scatter noise, i.e. the higher order modes of the sine wave start to matter, and this extends the scatter shelf into higher frequencies.

I fit a scattering shelf of the functional form $S_{\text{Hz}}(f) = A \, e^{-\pi \,\Gamma \,f}$, where $A$ is a scatter coupling coefficient in units of hertz, and $\Gamma$ is a half width half maximum (HWHM) of an underlying Lorentzian $L(t) = \dfrac{1}{\pi} \, \dfrac{\Gamma}{\left(t - t_0\right)^2+ \Gamma^2}$.

I found $A = 34.0 \, \text{Hz}$ and $\Gamma = 0.04 \, \text{s}$.

We can think of the HWHM as a function of overall scattering displacement and velocity: $\Gamma \approx \dfrac{x_{\text{scatter}}}{v_{\text{scatter}}}$

If $x_{\text{scatter}}(t) \approx \lambda$, this gives $v_{\text{scatter}} \approx 27 \, \dfrac{\mu \text{m}}{\text{s}}$

 Attachment 1: PLLNoisebudget_20171221_111006.pdf  175 kB
ELOG V3.1.3-