 Solution

ID: past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2017/1/c/solution

Write

N_{2} = N_{th} + u

and

n = n_{ss} + w

. Keeping terms linear in the perturbations gives

\frac{d}{d t} (u w) = (- r / τ_{2} f (r - 1) / τ_{2} - 1/ (f τ_{c}) 0) (u w) .

Equivalently,

\overset{w}{¨} + \frac{r}{τ _{2}} \overset{w}{˙} + \frac{r - 1}{τ _{c} τ _{2}} w = 0.

The characteristic exponents are

s_{\pm} = - \frac{r}{2 τ _{2}} \pm i \frac{r - 1}{τ _{c} τ _{2}} - \frac{r ^{2}}{4 τ _{2}^{2}} .

Consequently the above-threshold steady state has damped oscillations precisely when

τ_{2} > \frac{r ^{2}}{4 ( r - 1 )} τ_{c}, r > 1.

Their ordinary frequency is

f_{ro} = \frac{1}{2 π} \frac{r - 1}{τ _{c} τ _{2}} - \frac{r ^{2}}{4 τ _{2}^{2}} ≃ \frac{1}{2 π} \frac{r - 1}{τ _{c} τ _{2}} .

The approximation neglects the damping correction when

r^{2} τ_{c} ≪ 4 (r - 1) τ_{2}

. The exponential envelope decays on time

2 τ_{2} / r

. The doped-volume fraction cancels out of this linear frequency once the pump is expressed through

r

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