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The consistency bound for separately measured heterodyne arm powers is the requirement that the corrected peak beat current not exceed the maximum permitted by the two separately measured arm currents:Here $I_1,I_2$ are positive, dark-corrected currents and $B$ is the beat amplitude after correcting the electrical frequency response. The same collection geometry, detector responsivity and stable illumination must apply to all three measurements, as in separating power imbalance from mode mismatch in optical beating.

The Cauchy-Schwarz inequality gives a short physical proof. Write the collected optical fields as vectors $u,v$ in the detector-weighted inner product, with squared norms proportional to $I_1,I_2$. Their cross term $J$ obeys $|J{|}^2\le I_1{I}_2$ and the beat amplitude is $B=2|J|$. Equivalently, their Gram matrixis a positive semidefinite matrix, so its determinant cannot be negative. This is an overlap constraint rather than an empirical correction factor.

Measurement uncertainty matters when interpreting an apparent violation. In an original example, $I_1=I_2=0.100\mathrm{mA}$ and $B=0.204\mathrm{mA}$ give $\hat{\gamma }=B/(2\sqrt{I_1{I}_2})=1.02$. Suppose the three readings have independent $1\%$ relative standard uncertainty. From $\delta \log \gamma =\delta \log B-(\delta \log I_1+\delta \log I_2)/2$, the relative uncertainty is approximately $\sqrt{1+1/4+1/4}\%=1.225\%$, giving $u(\hat{\gamma })\approx 0.0125$. The excess above unity is only about $1.6$ standard uncertainties; it does not establish a physical overlap above one. Correlated calibration influences change the variance of a corrected heterodyne overlap estimate.^[ref]

NIST's receiver experiment separately addressed optical mode/polarization matching, electrical losses and impedance mismatch.^[ref] Interleaving the two arm-current measurements with the beat measurement helps detect source drift. Dark subtraction, detector nonlinearity or a changed collection aperture can also produce inconsistent inputs to the bound.

The optical heterodyne modulation depth measures the fractional oscillation of the combined optical power. For a stable sinusoidal beat, defineUsing the optical heterodyne detection field-mixing model giveswhere $\gamma$ is the mode-overlap factor in optical heterodyne detection. The inequality $2\sqrt{P_s{P}_{\mathrm{LO}}}\le P_s+P_{\mathrm{LO}}$ shows that $m\le |\gamma |\le 1$. Full modulation requires both equal powers and complete overlap.

An original example shows an inference ambiguity. A measured $m=0.60$ is compatible with equal powers and $|\gamma |=0.60$. It is also compatible with perfect overlap and power ratio $r=1/9$ or $9$. The beat depth alone therefore does not identify mode mismatch. Separating power imbalance from mode mismatch in optical beating requires additional information, such as measuring the two arms separately under the same collection conditions.

NIST's receiver-characterization experiment matched both polarization and spatial mode and equalized the received photocurrent contributions to obtain full modulation. Its calibrated transfer-standard relation also explicitly uses modulation depth.^[ref]

The measured electrical modulation can differ from the optical one if the receiver attenuates the beat frequency relative to its DC response. An additive background $P_b$ also changes the observed denominator to $P_s+P_{\mathrm{LO}}+P_b$ without contributing the intended coherent beat. Background correction, receiver response and unsaturated readout are needed before interpreting an electrical amplitude ratio as optical overlap.