When Gaussian beam pass through a lens, the waist location is given by

(z'-f) = (z-f)M^2

Where, z' is the waist location after lens, z is waist location before lens, f is the focal length of the lens M is the magnification.

In Gaussian optics, the magnification M is given by Mr/(1+r)^(1/2), the r of Mr should be subscript is the ray optics magnification f/(z-f), the r is given by z0/(z-f), z0 is the Rayleigh length.

However, I try to use ABCD laws on q-parameter, and also geometrically and algebraically, still can't prove the waist location is given by (z'-f) = (z-f)M^2. Can you give me some idea to solve it?

**1. Homework Statement**

**2. Homework Equations**

**3. The Attempt at a Solution**