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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}}

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