Energy Transport and the Poynting Vector — Question 4

Question 4

A Gaussian pulse envelope gives a finite fluence even with infinite time limits

At a detector a many-cycle vacuum pulse has cycle-averaged intensity $I(t)=I_pe^{-t^2/\tau^2}$, uniform over area A. Use $\int_{-\infty}^{\infty}e^{-u^2}du=\sqrt\pi$.

1. Find the fluence, defined as energy per unit area, and total pulse energy.

2. Find the peak carrier electric-field amplitude and the fraction of total pulse energy delivered between -tau and +tau, using erf(1)=0.842701.

Question 4 - Solution

A Gaussian pulse envelope gives a finite fluence even with infinite time limits

(a) Method and calculation

$$\mathcal F=\int_{-\infty}^{\infty}I(t)dt=I_p\tau\int_{-\infty}^{\infty}e^{-u^2}du=\boxed{I_p\tau\sqrt\pi}.$$$$\boxed{U=A\mathcal F=AI_p\tau\sqrt\pi}.$$

(b) Interpretation and further calculation

$$I_p=\tfrac12\epsilon_0cE_{0,p}^2\Rightarrow\boxed{E_{0,p}=\sqrt{2I_p/(\epsilon_0c)}}.$$$$\boxed{U_{[-\tau,\tau]}/U=\operatorname{erf}(1)=0.842701}.$$

Physical check

The envelope intensity is already averaged over the rapid carrier cycles; applying another factor of one-half would undercount the energy.