# Misaligned lens in analytic ray transfer matrix to produce coma aberration

i want to do analytic ray tracing by using ABCD matrices and retrieve the wavefront at the detector via Zernike coefficients.

$\begin{pmatrix}x_{out} \\ \theta_{out} \end{pmatrix} = \begin{pmatrix}A & B \\ C & D \end{pmatrix} \begin{pmatrix}x_{in} \\ \theta_{in} \end{pmatrix}$

where $x$ and $\theta$ are position and angle.

A simple setup would be to misalign a lens in an optical system (tilt/decenter). Apparently this induces a coma-like aberration as seen in the wikipedia article for coma.

When i use extended ABCD matrices from literature (Siegmann 3x3 or Shaomin 4x4), tilting a lens would have no effect on the wavefront. However, this is not true in reality.

I found two papers (Link1, Link2) that just add the offset $\triangle \theta$ and $\triangle x$ to the tilt and decenter, respectively.

The only effects i can reproduce are tip/tilt and defocus. However, i cannot produce the effects stated in the paper.

I know this is really specific but is anyone able to help me out here?

Thus the transfer matrix can only "see" linearized behavior. All lenses described by a $2\times 2$ transfer matrix are therefore perfect: rays converge perfectly to their focusses when described by the transfer matrix. A $4\times 4$ transfer matrix, tracing the $(x,\,y)$ positions and $(\theta_x,\,\theta_y)$ angles of rays, can account for an astigmatic lens: different focusses for the meridional and sagittal planes. By definition, it cannot "see" higher order aberrations, so astigmatism is the only aberration that be modelled.