I'm trying to use a 3x3 beam transfer matrix from "Generalized Beam Matrices: Gaussian Beam
Propagation in Misaligned Complex Optical Systems" by Anthony A. Tovar and Lee W. Casperson https://pdxscholar.library.pdx.edu/cgi/viewcontent.cgi?article=1068&context=ece_fac to track the modulation of a misligned Gaussian beam through a misaligned optical system. I'm not entirely sure where the input and output parameters are supposed to be on the optical axis z. For example using a thin lens, should the parameters for the complex beam parameter q and the complex misalignment parameter S be taken from the uncorrected position of the lens, or do I have to calculate an effective lens positon where the beam hits the lens like in the illustration below? The math is very unintuitive to me so maybe this z-offset is already taken care of in the ABCDGH matrices. 
z': optical axis of the initial beam, z'': optical axis of the transfomred beam $\theta_0$: angular offset of the lens from the x-axis, $x_1$: osset of the lens in x-direction, $z_0$: positons of the beam waists, $d$: offset of the beams at various positions on z. $\epsilon_0$: angular offset of the beams. For a thin lens without loss, the matrix looks like this:
$\begin{bmatrix}A & B & 0\\C & D & 0\\\ G & H & 1\end{bmatrix}$ = $\begin{bmatrix}1 & 0 & 0\\f^{-1} \cos\theta_x & 1 & 0\\\ -\beta_{air}x_0f^{-1}\cos\theta_x & 0 & 1\end{bmatrix}$, with $\beta=\frac{2\pi n}{\lambda}$.
The beam parameters being transformed are $\frac{1}{q(z)}=\frac{1}{R(z)} -\frac{i\lambda}{\pi n w(z)^2}$, and $S(z)= \beta(\frac{d(z)}{q(z)} + d(z)')$. The transfomration then look like this: $\frac{1}{q_2}=\frac{C+\frac{D}{q_1}}{A+\frac{B}{q_1}}$, $S_2=\frac{S_1}{A+\frac{B}{q_1}}+\frac{G+\frac{H}{q_1}}{A+\frac{B}{q_1}}$ according to to "Generalized Beam Matrices. III. Application to Diffraction Analysis" by Anthony A. Tovar and Lee W. Casperson https://pdxscholar.library.pdx.edu/cgi/viewcontent.cgi?article=1088&context=ece_fac
