= Reciprocal size-ratio ambiguity in centred Gaussian overlap
{title2=$\eta(\rho)=\eta(1/\rho)$}
\b[Reciprocal size-ratio ambiguity in centred Gaussian overlap] means that one beat-efficiency measurement cannot distinguish a beam that is too wide from one that is reciprocally too narrow. Under the matched-curvature, matched-polarization model of <waist-size mismatch in optical heterodyne mode matching>,
$$
\eta(\rho)=\frac{4\rho^2}{(1+\rho^2)^2}=\eta(1/\rho),\qquad\rho=w_2/w_1>0.
$$
Solving for the ratio at $0<\eta\leq1$ gives two roots,
$$
\rho_\pm=\frac{1\pm\sqrt{1-\eta}}{\sqrt\eta},\qquad \rho_+\rho_-=1.
$$
For an original measured efficiency $\eta=0.64$, the ratios are $1/2$ and $2$. With a known reference radius $100\,\mathrm{\mu m}$, the other beam could have radius $50$ or $200\,\mathrm{\mu m}$. This is an exact two-branch ambiguity, even for a noiseless measurement.
Put $x=\log\rho$. Then $\eta=1/\cosh^2x$, an <even function> of the signed logarithmic mismatch. Close to matching, $\eta=1-x^2+O(x^4)$, so the first-order response to a small signed mismatch vanishes. Beat efficiency alone is both sign-ambiguous and weakly sensitive near its maximum.
A <branch selection by a controlled beam-radius change> can distinguish the two possibilities if other optical properties stay fixed. Increasing $w_2$ by $10\%$ changes the ratio $0.5$ to $0.55$, increasing efficiency to about $0.7132$; it changes $2$ to $2.2$, decreasing efficiency to about $0.5677$. This is a conditional diagnostic: a lens adjustment that also changes curvature or displacement need not follow this one-parameter curve.
NIST used matched optical modes and polarization in the physical heterodyne calibration setup.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF p. 1, Heterodyne measurements]{ref} Independent beam profiling at the receiving plane supplies the missing radius information directly; power balancing alone does not resolve this ambiguity.
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