Gaussian Beam: ABCD Propagation

A Gaussian beam is fully described by one complex number, the beam parameter $q$, with $1/q = 1/R - i\lambda/\pi w^2$. Any paraxial element acts on it by its ray-transfer matrix: $q\to(Aq+B)/(Cq+D)$. Free space of length $z$ gives $q\to q+z$ (so the spot follows $w(z)=w_0\sqrt{1+(z/z_R)^2}$, $z_R=\pi w_0^2/\lambda$), and a thin lens $f$ refocuses it. A collimated beam through a lens forms a new waist $w_0'\approx\lambda f/\pi w_0$ near the focal plane, the diffraction limit of focusing. Drag the lens along the bench and watch the envelope and the focused waist respond.