A quadrupole doublet in a beamline consists of a thin focusing quadrupole (focal…
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Nuclear Physics · Linear algebra and vector spaces
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Problem
A quadrupole doublet in a beamline consists of a thin focusing quadrupole (focal length +f) and a thin defocusing quadrupole of the same magnitude (focal length −f), separated by a field-free drift of length d, all in the horizontal plane. Treat the lenses as thin (all bending at a point) and work in the paraxial approximation, ignoring fringe fields. Take f=1.8m and d=0.9m.
((a)) Write the thin-lens matrices for focal lengths +f and −f and the drift matrix for length d. Multiply them, in the order the beam meets them, for the case focusing-then-drift-then-defocusing, to find the doublet's transfer matrix M.
((b)) Show that the off-diagonal element of M gives an effective focal length F=f2/d for the whole doublet, and evaluate F. Then repeat the multiplication for the reversed order (defocusing lens first) and show F is unchanged.
((c)) In one sentence, explain why the identical hardware also focuses in the vertical plane, where the two quadrupoles' roles are interchanged.