Astigmatism of an equal-strength quadrupole doublet
= Astigmatism of an equal-strength quadrupole doublet
{title2=$\ell_x=f^2/L-f,\quad\ell_y=f^2/L+f$}
For an initially parallel bundle passing through opposite equal thin-quadrupole strengths separated by $L$, the two downstream focal distances differ as shown. Approximating the doublet by a common effective <focal length> $f^2/L$ discards that difference. A real common final focus requires additional incoming optical information or adjusted strengths.