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.

Figure 1. Gaussian-beam envelope through a thin lens by the ABCD law, with the transverse spot shown at the input waist and at the focus; the focused waist follows lambda f / (pi w0). Drag the object or the lens. Method: complex q-parameter propagated by ray-transfer matrices (free space, thin lens).

input waist w0 (um)200

focal length f (mm)120

object z0 (mm)60

lens position (mm)250

wavelength (nm)1064

WHAT TO TRY

Drag the object or the lens: the q-parameter transforms by the lens ray-transfer matrix, the beam refocuses to a new waist w0-prime, and the spot-size readout shows how much tighter the focus is.
Shorten the focal length f: a stronger lens drives a smaller, closer focus but a faster-diverging beam beyond it. There is no free lunch, tighter waist means larger far-field angle.
Change the input waist or wavelength: a bigger input beam focuses to a smaller spot (the focusing analogue of a larger aperture), and longer wavelengths focus more loosely.