 Reciprocal size-ratio ambiguity in centred Gaussian overlap

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,Solving for the ratio at $0<\eta \le 1$ gives two roots,For an original measured efficiency $\eta =0.64$, the ratios are $1/2$ and $2$. With a known reference radius $100\mu \mathrm{m}$, the other beam could have radius $50$ or $200\mu \mathrm{m}$. This is an exact two-branch ambiguity, even for a noiseless measurement.

Put $x=\log \rho$. Then $\eta =1/{\cosh }^{2}x$, 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.^[ref] Independent beam profiling at the receiving plane supplies the missing radius information directly; power balancing alone does not resolve this ambiguity.