Astigmatism of an equal-strength quadrupole doublet (${\ell }_x=f^2/L-f,{\ell }_y=f^2/L+f$)

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.