 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2015/s19/1/b/solution

Let

D (d) = (10 d 1)

be the drift beam transfer matrix. Start immediately before the first lens and end immediately before its counterpart in the next FODO cell. Matrices act on column vectors from right to left. With

A = 1/ f_{1}

and

B = 1/ f_{2}

, where both focal lengths are signed in the chosen plane,

M = D Q (f_{2}) D Q (f_{1}) = (1 - 2 d A - d B + d^{2} A B - A - B + d A B 2 d - d^{2} B 1 - d B) .

The two drifts account for both gaps. Direct multiplication gives

det M = 1

and

\frac{1}{2} tr M = 1 - d (A + B) + \frac{d ^{2} A B}{2} .

For the equal-strength focusing-defocusing case,

f_{1} = f > 0

and

f_{2} = - f

M = (1 - d / f - d^{2} / f^{2} - d / f^{2} 2 d + d^{2} / f 1 + d / f), tr M = 2 - \frac{d ^{2}}{f ^{2}} .

Choosing a different start within the cell gives a conjugate matrix, with the same trace and eigenvalues. If

f_{2}

is specified as a positive magnitude of a defocusing lens, its signed value in these formulas is

- f_{2}

 New to topics? Read the docs here!