 Solution

ID: past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2019/1/c/ii/solution

At the end of the pulse, the photon density returns to negligible size. Put

u = N_{f}^{*} / N_{i}^{*}

and

r = N_{i}^{*} / N_{th}^{*}

. The result of i gives

0 = ln u + r (1 - u) .

Using

x = β η = 1 - u

, the nonzero-pulse solution obeys

r x = - ln (1 - x), x = 1 - e^{- r x}, x = β η .

For

r > 1

, select

0 < x < 1

, corresponding to depletion below threshold. The root

x = 0

is also an algebraic solution for every

r

but represents no macroscopic pulse. A positive nonzero root exists only above threshold.

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