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* Take notes here! [#ye89c8c6]
Optickle cannot answer questions about thermal lensing effects on the ASC because it only calculates up to the 1st HG mode. We confirmed the limitations of using Optickle for such problems by comparing Optickle results with the following FINESSE calculations:
\begin{enumerate}
\item Limiting the simulation up to 1st HG mode, automatically mode-matched to FP with thermal lensing
\item Maximum HOM simulation (up to 14th HG mode), mode-matched to FP without thermal lensing
\end{enumerate}
Optickle results matched with FINESSE results in the first case, but not in the other. The thermal lens was modeled by putting a lens (Optickle function \texttt{addTelescope}) right behind the ITM. In Optickle, you can create mode-mismatch by explicitly defining waist size and position of the incident beam, but this does not change the results because Optickle uses the cavity mode for Gouy phase calculations.

Anyhow, we can address the sample problem of calculating a sensing matrix for a FP without any thermal lensing. Figure \ref{fig:optickle_WFSGouy} and Table \ref{tab:optickle_ASC_matrix} are the Optickle results. Optickle and FINESSE results matched well within 0.5\%. It is unclear why there is a sign flip for ETM signals. We have checked that angle conventions are the same for Optickle and FINESSE.

Optickle results matched with analytical calculation within 5\%, including signs, when Optickle results are multiplied by $\sqrt{2/\pi}$. This factor comes from the overlap integral between 00 mode and 1st mode. When there is an interference between normalized HG$_{00}$ mode and HG$_{10}$ mode, intensity difference between the left side and right sides on a QPD is
\begin{eqnarray}
  I_{\rm{diff}} &=& I(x>0) - I(x<0) \nonumber \\
    &=& 2 \int_{0}^{\infty} \d x \int_{-\infty}^{\infty} \d y U_{10}^{*}(x,y,z)U_{00}(x,y,z) \nonumber \\
    &=& 2 \int_{0}^{\infty} \d x \sqrt{\frac{2}{\pi w^2(z)}} \frac{2x}{w(z)} \exp{\left( -\frac{2x^2}{w^2(z)} \right)} \nonumber \\
    &=& \sqrt{\frac{2}{\pi}}.
\end{eqnarray}
Optickle (and FINESSE) ignores this factor.

The formula below was used for the analytical calculation of the WFS signal.
\begin{equation}
  P_{\rm{WFS}}^{\rm{I}} = \sqrt{\frac{8}{\pi}} P_0 J_0(\beta) J_1(\beta) (r_{\c 0}r_{\s 1}-r_{\c 1}r_{\s 0}) \left( \frac{\delta x}{w_0} \sin{\eta} - \frac{\delta \theta}{\alpha_0} \cos{\eta} \right),
\end{equation}
where $P_0$ is the incident beam power, $J_i$ is the Bessel function, $\beta$ is the modulation depth, and $r_{{\rm c/s} i}$ are the cavity reflectivity for carrier/sideband HG$_{i0}$ mode. $w_0$ and $\alpha_0$ are the waist size and the divergence angle of the cavity mode, and $\eta$ is the accumulated Gouy phase from the cavity waist to the WFS QPD.

$\delta x$ and $\delta \theta$ are translation and tilt of the cavity axis created by mirror misalignments. They can be calculated by
\begin{eqnarray}
 \delta x &=& -\frac{L(g_2\alpha_1 + g_1\alpha_2)}{2g_1g_2-g_1-g_2} , \\
 \delta \theta &=& \frac{(1-g_2)\alpha_1 - (1-g_1)\alpha_2}{1-g_1g_2} ,
\end{eqnarray}
where $L$ is the cavity length, $g_i$ are g-factors of the mirrors, and $\alpha_i$ are tilt of the mirrors ($i=1$ for ITM and $i=2$ for ETM). The sign of the mirror tilts $\alpha_i$ are defined with respect to the mirror HR surface, not the beam, just like the convention Optickle uses.


\begin{figure} 
\begin{centering} 
\includegraphics[width=100mm]{Figures/Optickle_WFSgouy.pdf}
\caption{Gouy phase dependence of the WFS signal for each mirror. Optickle results are multiplied by $\sqrt{2/\pi}$. Gouy phase swept was done by using Optickle function \texttt{setGouyPhase}.}
\label{fig:optickle_WFSGouy}
\end{centering}
\end{figure}

\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{ } & ITM & ETM \\
\hline
\multirow{3}{*}{WFSA} & Optickle & 168.52 & $-$138.88 \\
& Optickle $\times{\sqrt{2/\pi}}$ & 134.46 & $-$110.81 \\ & analytical
& 135.66 & $-$113.23 \\
\hline
\multirow{3}{*}{WFSB} & Optickle & 71.08 & $-$141.36 \\
& Optickle $\times{\sqrt{2/\pi}}$  & 56.71 & $-$112.79 \\
& analytical & 53.80 & $-$110.38 \\
\hline
\end{tabular}
\caption{Optickle and analytical calculation results for the alignment sensing matrix without thermal lensing. All values in units of W/rad.}
\label{tab:optickle_ASC_matrix}
\end{center}
\end{table}