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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.

Inversion of a symmetric Gaussian overlap log-ratio recovers a radius ratio from two positive overlap-efficiency measurements made at known symmetric radius settings. It applies to the one-parameter model of symmetric beam-radius perturbation for overlap branch selection, with one shared response scale and fixed remaining mode properties.

Let $h=\log q>0$, $D_h=\log (V_{+}/V_{-})$ and $u=\log \rho$, where $\rho =w_2/w_1$. Since the symmetric log-overlap ratio satisfies $D_h=-4\mathrm{artanh}(\tanh h\tanh u)$, defineThenA finite positive radius ratio exists exactly when $|z|<1$, equivalently $|D_h|<4h$. The strictly decreasing forward relation makes the solution unique inside this interval. Equality corresponds only to a limiting zero or infinite radius ratio; an out-of-range noisy measurement has no finite solution in this ideal model.

For an original example, $q=2$ and $D_h=\log (289/64)$ give $\tanh h=0.6$ and $z=-0.6$, hence $\rho =1/2$. If the known first radius is $100\mu \mathrm{m}$, the inferred second radius is $50\mu \mathrm{m}$. The nuisance scale is unnecessary for that inversion. Once $\rho$ is obtained, one can reconstruct it as $A=V_{+}{\cosh }^2(u+h)$, but two fitted readings alone do not validate the Gaussian optical model.

The inverse sensitivity isFor small errors, relative radius uncertainty is approximately $u(\rho )/\rho \simeq |du/d{D}_h|u(D_h)$. At the worked point the derivative is $-17/30$; stipulated $u(D_h)=0.02$ gives about 1.13% relative radius uncertainty, before including uncertainty in the setting $h$ or the reference radius.

Near the allowed interval's edges, the inverse becomes highly sensitive. Boundary-sensitive Gaussian overlap inversion needs an uncertainty model that respects the nonlinear transformation; clipping an out-of-range value to a convenient radius does not supply the missing information. The GUM distinguishes local uncertainty propagation from cases requiring higher-order treatment.^[ref]

The apparatus must implement a radius change without unintended displacement, polarization or wavefront changes. NIST's physical heterodyne receiver experiment used mode and polarization control and separate electrical corrections.^[ref] Those controls motivate checks on the inversion assumptions; the paper is not a report of the particular symmetric-radius example derived here.