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Gaussian beam overlap under lateral displacement quantifies how a transverse offset reduces interference between Gaussian beams. For equal linear polarizations, matched wavefront curvature and an unlimited receiving plane, use normalized field amplitudesThe parameter $w_j$ is the radius at which the intensity has fallen to $e^{-2}$ of its central value. Completing the square in their overlap integral givesThe equal-radius case reduces to the expression in mode-overlap factor in optical heterodyne detection. The prefactor shows that centring the beams does not remove waist-size mismatch in optical heterodyne mode matching.

For an original $w_2=2w_1$ example, centred beams have $|\gamma |=0.80$ and only $64\%$ of the perfectly matched beat power at fixed collected powers. Adding displacement $d=w_1$ gives $|\gamma |=0.8e^{-1/5}\approx 0.6550$, reducing the beat-power fraction to about $42.9\%$.

An alignment tolerance follows in the equal-radius case. To retain at least $90\%$ of the matched beat power, require $e^{-d^2/w^2}\ge 0.90$, hence $d\le 0.3246w$. A chosen $100\mu \mathrm{m}$ radius permits about $32.5\mu \mathrm{m}$ displacement under these assumptions.

NIST's primary heterodyne setup used a polarizer and single-mode fibre to ensure matching rather than infer it from total power alone.^[ref] These original overlap examples omit aperture clipping, angular tilt and wavefront-curvature mismatch. They describe the detector's optical field overlap, not a universal coupling efficiency for every fibre or lens arrangement.

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.