To further motivate the determination of the surface roughness, one might take a look at theoretical predictions correlating the surface roughness with the absorbed energy.^{12} As soon as multiple scattering takes place, the laser absorption is raised significantly, which in turn decreases the ablation threshold $\Phi thr$.^{5}^{,}^{6} The (cold) absorption on highly reflecting materials as, e.g., aluminum can be approximated linearly by the effective number of reflections.^{12} For a given surface correlation length and a Gaussian height distribution, this results in a linear correlation to surface roughness $Sa$, easily doubling the absorbed laser intensity.^{12} With the above knowledge and a few assumptions, a linear approximation can be deduced to indicate the amount of thrust noise $\sigma [F]$ introduced. ($\sigma [A]$ depicts the standard deviation of $A$, $E[A]$ its mean or expected value.) First, we may assume that the average surface roughness $E[Sa]$ and the surface correlation length $\tau s$ are of the same size. $\tau s$ represents the characteristic width of the autocorrelation function of the surface profile.^{12} Keeping $\tau s$ constant for all, $Sa$ introduces a linear shift of the absorbed fluence $\Phi abs(Sa/\tau s)$ depending on the surface roughness $Sa$. This again translates the fluence-dependent coupling coefficient $cm[\Phi eff(Sa)]$. The coupling coefficient depicts the ratio of generated thrust to the average laser power $Plaser$ used. Second, using the Sinko model,^{13} a realistic intermediate slope of $cm(\Phi )$ half way to its maximizing fluence $\Phi max$ ($2\Phi thr\u2248(\Phi max/2)$) can be deduced dividing $cm(\Phi max)$ by $\Phi max$. Finally, adding the relation from Bergström et al.^{12} for aluminum, $\sigma [\Phi abs]\u22483\Phi (\sigma [Sa]/E[Sa])$, leads to the following approximation: Display Formula
$\sigma [F]\u2248\sigma {cm[\Phi eff(Sa)]}\xb7Plaser\u2248cm(\Phi max)\Phi max\xb7Plaser\xb7\sigma [\Phi eff(Sa)]\u2248Fthrust\Phi max\xb73\Phi max2\xb7\sigma [Sa]E[Sa],$(1)
Display Formula$\sigma [F]\u2248Fthrust\xb7\sigma [Sa]E[Sa].$(2)
The approximation states that under relevant process parameters one may obtain a relative thrust noise proportional to the relative deviation in surface roughness. The surface roughness variations may be due to roughness variations in time as well as surface positions. In the latter case, an additional source of error has to be considered. The surface might have a large scale roughness or rather unevenness. Since the thrust vector typically points perpendicular to the local surface, this leads to a misaligned thrust vector, which primarily will generate thrust noise perpendicular to the desired thrust direction. Similar to the previous calculation, a simple geometric consideration leads to Display Formula$\sigma [Fperpendicular]\u2248Fthrust\xb7\sigma [\gamma ]$(3)
depending on the uncertainty in the slope $\sigma [\gamma ]$ of the surface in rad. To connect these approximations with, e.g., the LISA requirements of a power spectral density [$PSD(f)$] of $(0.01\u2009\u2009\mu N2)/Hz$, one has to know the frequency space over which the noise will be distributed. For the sake of the argumentation, we will assume now that white noise (WN) is generated with a constant $PSD(f)=WN2$ starting from 0 Hz up to $fmax$. In this case, the flowing relation has to be considered^{14}Display Formula$\u222b0fmaxdfWN2=\sigma 2[F]=>WN2\xb7fmax=\sigma 2[F]=>WN=\sigma [F]fmax,$(4)
with $fmax=100\u2009\u2009Hz$, $\sigma [F]$ may not exceed $1\u2009\u2009\mu N$, at which a $20\u2009\u2009\mu N$ thrust allows for a maximum relative error of 5%. Bearing these considerations in mind, we will focus on the surface roughness as a primary indicator for problematic constellations.