Lasers and Laser Light — Question 10

Question 10

Two unequal pulse trains can share average power but differ greatly in peak power

A laser emits Gaussian power pulses $P(t)=P_{\mathrm{pk}}\exp[-4\ln2\,t^2/\Delta t^2]$ at repetition rate f, with negligible overlap. Use $\int_{-\infty}^{\infty}e^{-at^2}dt=\sqrt{\pi/a}$. Let average power be 2.00 W, $f=80.0\,\mathrm{MHz}$, and $\Delta t=200\,\mathrm{fs}$.

1. Derive the pulse energy in terms of peak power and find the pulse energy numerically.

2. Find the peak power and how it changes if f is halved at fixed average power and pulse duration.

Question 10 - Solution

Two unequal pulse trains can share average power but differ greatly in peak power

(a) Method and calculation

$$U=P_{\mathrm{pk}}\Delta t\sqrt{\frac{\pi}{4\ln2}},\quad\bar P=fU.$$$$\boxed{U=\bar P/f=25.0\,\mathrm{nJ}}.$$

(b) Interpretation and further calculation

$$\boxed{P_{\mathrm{pk}}=\frac{U}{\Delta t}\sqrt{\frac{4\ln2}{\pi}}=1.174\times10^5\,\mathrm W}.$$$$P_{\mathrm{pk}}\propto f^{-1}\Rightarrow\boxed{P_{\mathrm{pk}}\prime=2P_{\mathrm{pk}}}.$$

Physical check

The Gaussian pulse-shape factor is retained; dividing energy by FWHM alone overestimates the peak. The pulse spacing is much longer than its duration, consistent with negligible overlap.