 Branch selection by a controlled beam-radius change

Branch selection by a controlled beam-radius change resolves which of the reciprocal radius ratios is present in a centred Gaussian heterodyne overlap measurement. In reciprocal size-ratio ambiguity in centred Gaussian overlap, the efficiency is $\eta (\rho )=4{\rho }^2/(1+{\rho }^2{)}^2$, with $\rho =w_2/w_1$. Its local signed response isAn infinitesimal increase of the second radius therefore improves overlap when $\rho <1$ and worsens it when $\rho >1$, provided curvature, displacement and polarization stay fixed.

A finite one-sided change needs more care. For a known multiplier $q>1$, direct comparison givesThe threshold is below one because the finite change can overshoot the optimum. In an original example, $\rho =0.99$ is initially on the narrower-beam branch, with efficiency about $0.99990$. Increasing that radius by $10\%$ gives $\rho =1.089$ and efficiency about $0.9928$. The observed decrease alone would misclassify the initial branch if interpreted using the infinitesimal rule.

Use a symmetric beam-radius perturbation for overlap branch selection when two controlled settings are available. Compare $\eta (q\rho )$ with $\eta (\rho /q)$. Since the efficiency is an even, strictly decreasing function of $|\log \rho |$, their difference is positive for $\rho <1$, negative for $\rho >1$, and zero at matching. In the same example with $q=1.10$, the two efficiencies are about $0.9928$ and $0.9890$, correctly selecting the initially narrower beam. The perturbation must be large enough to resolve above measurement noise.

NIST's physical heterodyne setup used beam combination, polarization control and single-mode collection to establish optical matching.^[ref] A practical radius change must preserve those other mode properties; an arbitrary lens adjustment can change wavefront curvature as well and invalidate this one-variable diagnostic.