 Solution

ID: past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2021/section-b/b4/c/solution

P_{+} = u_{+} u_{+}^{†} = \frac{1}{2} (1 - i i 1), P_{-} = u_{-} u_{-}^{†} = \frac{1}{2} (1 i - i 1) .

J = λ_{+} P_{+} + λ_{-} P_{-} = \frac{1}{2} (λ_{+} + λ_{-} - i (λ_{+} - λ_{-}) i (λ_{+} - λ_{-}) λ_{+} + λ_{-}) .

This simply substitutes

a = (λ_{+} + λ_{-}) /2

and

b = (λ_{+} - λ_{-}) /2

and makes each eigenmode's wave amplitude transmission explicit.

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