Light loss when passing a Gaussian beam through a circular aperture
Consider the light loss when a Gaussian beam passes through a circular aperture as shown in the figure below.

The intensity at a distance r from the central axis of a Gaussian beam is expressed as follows:

Here, w is the beam waist radius and i 0 =i(0).
The amount of light V(α) after leaving an aperture of radius α can be calculated using the "shell integration method" as follows:

In addition, the amount of light V0 before it leaves the aperture is equal to the result of changing α →∞ in the above equation, and is given by:

Therefore, the ratio of the amount of light before and after passing through the aperture is expressed as follows:

In particular, when α = w, the value is:

In this way, when the aperture is determined based on the beam waist radius, approximately 13.5 % of the light amount is lost.