Lasers and Laser Light — Question 4

Question 4

Integrating a Gaussian profile determines the power captured by an aperture

A circularly symmetric laser beam has $I(r)=I_0e^{-2r^2/w^2}$. A centered circular aperture has radius a. Assume the aperture is thin and calculate incident transmitted power before subsequent diffraction.

1. Integrate the total beam power and the fraction within a.

2. Find a in units of w required to pass 95 percent of the power.

Question 4 - Solution

Integrating a Gaussian profile determines the power captured by an aperture

(a) Method and calculation

$$P=2\pi I_0\int_0^\infty re^{-2r^2/w^2}dr=\boxed{\pi I_0w^2/2}.$$$$P(a)=\frac{\pi I_0w^2}{2}(1-e^{-2a^2/w^2})\Rightarrow\boxed{P(a)/P=1-e^{-2a^2/w^2}}.$$

(b) Interpretation and further calculation

$$.95=1-e^{-2a^2/w^2}\Rightarrow2a^2/w^2=\ln20.$$$$\boxed{a/w=\sqrt{\ln20/2}=1.224}.$$

Physical check

A Gaussian has no hard edge. Passing 95 percent requires an aperture radius larger than w, which encloses about 86.5 percent of the power.