 Waist-size mismatch in optical heterodyne mode matching

Waist-size mismatch in optical heterodyne mode matching reduces interference even when the beam centres coincide. For two circular Gaussian fields with matching polarization and wavefront curvature, define their radius ratio $\rho =w_2/w_1>0$. The centred overlap from Gaussian beam overlap under lateral displacement givesHere $\eta$ is the beat-power fraction relative to a fully matched receiver at the same collected arm powers. Matching total power alone does not set $\rho$.

An original tolerance calculation asks for $\eta \ge 0.90$. Solving the quadratic inequality givesThe allowed ratio is approximately $0.7208$ to $1.3874$. With one radius $100\mu \mathrm{m}$, the other must be about $72.1$–$138.7\mu \mathrm{m}$ in this model. A $140\mu \mathrm{m}$ radius gives only about $89.5\%$ beat power.

The interval has reciprocal endpoints because replacing $\rho$ by $1/\rho$ leaves the overlap unchanged. This reciprocal size-ratio ambiguity in centred Gaussian overlap means the beat magnitude alone cannot tell which beam is wider. Independent beam profiling or a known reference radius resolves that ambiguity.

NIST's heterodyne metrology setup used single-mode fibre and polarization matching to establish a calculable modulation rather than infer matching from collected power alone.^[ref] The radius test above omits displacement, curvature mismatch, clipping and angular misalignment. Radii must be compared at the same receiving plane; equal spot sizes there do not by themselves establish the same full Gaussian optical mode.