= Waist-size mismatch in optical heterodyne mode matching
{title2=$\eta=4\rho^2/(1+\rho^2)^2$}
\b[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> gives
$$
|\gamma|=\frac{2\rho}{1+\rho^2},\qquad
\eta=|\gamma|^2=\frac{4\rho^2}{(1+\rho^2)^2}.
$$
Here $\eta$ 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 $\eta\geq0.90$. Solving the quadratic inequality gives
$$
\frac{1-\sqrt{1-\eta_0}}{\sqrt{\eta_0}}
\leq\rho\leq
\frac{1+\sqrt{1-\eta_0}}{\sqrt{\eta_0}},\qquad \eta_0=0.90.
$$
The allowed ratio is approximately $0.7208$ to $1.3874$. With one radius $100\,\mathrm{\mu m}$, the other must be about $72.1$–$138.7\,\mathrm{\mu m}$ in this model. A $140\,\mathrm{\mu m}$ 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.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF p. 1, Heterodyne measurements]{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.
Back to article page
