Optical mode matching

Optical mode matching makes the spatial field of an incident beam equal to a desired optical cavity normal mode. For fundamental circular Gaussian beams, matching the complex beam parameter fixes the beam waist size and location; the optical axis, polarization and frequency must also match. A single converging optical lens connects waist Rayleigh ranges $z_A,z_B$ only when $f^2\ge z_A{z}_B$. With $s=\sqrt{f^2/(z_A{z}_B)-1}$, waist-to-lens distances obey $u=f\pm z_{A}s$, $v=f\pm z_{B}s$ with correlated signs, followed by physical distance constraints.