Energy Transport and the Poynting Vector — Question 3

Question 3

A Gaussian beam requires integrating intensity over its cross section

A circular paraxial beam has time-averaged intensity $I(r)=I_0e^{-2r^2/w^2}$. Treat the local energy flow as axial and neglect diffraction over the measurement region.

1. Integrate to find total power and the power inside radius R.

2. Find the radius enclosing one-half of the total power and the fraction inside radius w.

Question 3 - Solution

A Gaussian beam requires integrating intensity over its cross section

(a) Method and calculation

$$P=\int_0^\infty I_0e^{-2r^2/w^2}2\pi r\,dr=\boxed{\pi I_0w^2/2}.$$$$\boxed{P(R)=P[1-e^{-2R^2/w^2}]}.$$

(b) Interpretation and further calculation

$$P(R_{1/2})=P/2\Rightarrow\boxed{R_{1/2}=w\sqrt{\ln2/2}=0.5887w}.$$$$\boxed{P(w)/P=1-e^{-2}=0.864665}.$$

Physical check

The beam parameter w is the intensity 1/e-squared radius, not the radius containing all of the power.