. 2021 Mar 18;29(7):10346–10363. doi: 10.1364/OE.414405

Table 2. Third order non-rotationally symmetric aberration coefficients for infinite conjugates in an afocal refractive relay formed by two tilted thin plano-convex lenses in terms of: half field of view ( $H F O V$ ), entrance pupil radius ( $h_{0}$ ), focal length of the first lens ( $f_{1}$ ), magnification ( $m)$ and lens tilt about the local x- ( $α$ ) and y-axis ( $β$ ). Note that $B_{220_{M}}$ is a scalar.

${\overset{⇀}{A}}_{131}$	$\frac{h_{0}^{3}}{2 f_{1}^{2}} [\begin{matrix} - \frac{n_{1} \cos α_{1} \sin β_{1}}{n_{1} - 1} + \frac{m n_{2} \cos α_{2} \sin β_{2}}{n_{2} - 1} \\ \frac{n_{1} \sin α_{1}}{n_{1} - 1} - \frac{m n_{2} \sin α_{2}}{n_{2} - 1} \end{matrix}]$
${\overset{⇀}{A}}_{220 M}$	$\frac{h_{0}^{2} \tan (H F O V)}{4 f_{1}} [\begin{matrix} - \frac{(n_{1}^{2} + 1) \cos α_{1} \sin β_{1}}{n_{1} (n_{1} - 1)} + \frac{(n_{2}^{2} + 1) \cos α_{2} \sin β_{2}}{n_{2} (n_{2} - 1)} \\ \frac{(n_{1}^{2} + 1) \sin α_{1}}{n_{1} (n_{1} - 1)} - \frac{(n_{2}^{2} + 1) \sin α_{2}}{n_{2} (n_{2} - 1)} \end{matrix}]$
$B_{220_{M}}$	$\frac{h_{0}^{2}}{4 f_{1}} [\frac{(n_{1}^{2} + 1) (\cos^{2} α_{1} \sin^{2} β_{1} + \sin^{2} α_{1})}{n_{1}} - \frac{m (n_{2}^{2} + 1) (\cos^{2} α_{2} \sin^{2} β_{2} + \sin^{2} α_{2})}{n_{2}}]$
${\overset{⇀}{A}}_{222}$	$\frac{h_{0}^{2} \tan (H F O V)}{2 f_{1}} [\begin{matrix} - \frac{\cos α_{1} \sin β_{1}}{n_{1} - 1} + \frac{\cos α_{2} \sin β_{2}}{n_{2} - 1} \\ \frac{\sin α_{1}}{n_{1} - 1} - \frac{\sin α_{2}}{n_{2} - 1} \end{matrix}]$
${\overset{⇀}{B}}_{222}^{2}$	$- \frac{h_{0}^{2}}{2 f_{1}} [\begin{matrix} \sin 2 α_{1} \sin β_{1} - m \sin 2 α_{2} \sin β_{2} \\ \cos^{2} α_{1} \sin^{2} β_{1} - \sin^{2} α_{1} - m (\cos^{2} α_{2} \sin^{2} β_{2} - \sin^{2} α_{2}) \end{matrix}]$