Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S19 / 1 / a / Solution

Let $x_i=L/f_i$, with positive $f$ focusing in the first transverse plane. The doublet matrix is $M=Q_{2}D(L)Q_1$, with $Q_i=\left(\begin{array}{cc}1& 0\\ -1/f_i& 1\end{array}\right)$. Its lower-left entry is $C_x=-(x_1+x_2-x_1{x}_2)/L$. In the other plane both quadrupole signs reverse, giving $C_y=-(-x_1-x_2-x_1{x}_2)/L$. Thus positive effective focusing power in both planes requiresFor a parallel incident ray bundle, actual downstream focal distances are ${\ell }_x=L(1-x_1)/(x_1+x_2-x_1{x}_2)$ and ${\ell }_y=L(1+x_1)/(-x_1-x_2-x_1{x}_2)$. Requiring both foci downstream adds $|x_1|<1$. Positive effective power alone need not imply a real image after the final element. The original schematic below distinguishes these conditions.