= Branch selection by a controlled beam-radius change
{title2=$d\log\eta/d\log\rho=2(1-\rho^2)/(1+\rho^2)$}
\b[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 is
$$
\frac{d\log\eta}{d\log\rho}=\frac{2(1-\rho^2)}{1+\rho^2}.
$$
An 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 gives
$$
\eta(q\rho)>\eta(\rho)\quad\Longleftrightarrow\quad\rho<q^{-1/2}.
$$
The 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.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF p. 1, Heterodyne measurements]{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.
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