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Common-gain cancellation in heterodyne overlap uncertainty is the exact invariance of the normalized beat amplitudewhen the corrected peak beat current $B$ and both separately measured arm currents $I_1,I_2$ are multiplied by the same factor $g>0$. The factor occurs once in the numerator and once in the denominator. This is scale invariance: the ratio is a homogeneous function of degree zero. The cancellation holds for every realization of a random common factor, without requiring it to be independent of the currents.

More generally, take $B,I_1,I_2>0$ and let their residual multiplicative measurement calibration factors be $g_B,g_1,g_2$. The observed normalization satisfiesThus the gain-matching condition for heterodyne normalization is $g_B^2=g_1{g}_2$; the three factors need not all be equal. For example, $g_B=1$, $g_1=1.04$ and $g_2=1/1.04$ leave the ratio unchanged. The denominator depends on the geometric mean of its two factors.

Writing the natural logarithms of the factors as a random vector $\mathbf{G}$ makes the remaining uncertainty particularly clear. With $\mathbf{v}=(1,-1/2,-1/2{)}^T$,This identity is exact for the logarithmic error. The relative variance of the original ratio is only approximately equal to it when the errors are small, as in the variance of a corrected heterodyne overlap estimate. Keeping the covariance terms follows the general measurement-propagation rule.^[ref]

For an original imperfect-sharing model, set $\log g_j={\alpha }_{j}Z+{ϵ}_j$. Assume $Z,{ϵ}_B,{ϵ}_1,{ϵ}_2$ are zero-mean independent random variables. The coefficient of the shared fluctuation is $c={\alpha }_B-({\alpha }_1+{\alpha }_2)/2$, soTake ${\alpha }_B=1$, ${\alpha }_1={\alpha }_2=0.95$, with standard deviations $0.03$ for $Z$, $0.01$ for ${ϵ}_B$ and $0.004$ for each arm's separate error. Then $c=0.05$ and $v_d=0.00011025$, giving logarithmic standard uncertainty $0.0105$, approximately $1.05\%$ relative uncertainty. A large shared fluctuation can leave a small residual, but incomplete matching prevents exact cancellation.

If the log-errors additionally have a normal distribution, then $d\sim N(0,v_d)$ and the normalized readout has a lognormal distribution. Its exact expected value is $\gamma e^{v_d/2}$, not $\gamma$. At $\gamma =0.75$, this example gives mean $0.7500413$ and standard deviation $0.0078757$. The lognormal bias of a heterodyne overlap estimate comes from exponentiating the remaining error; the perfectly shared factor still cancels.

The physical calibration paths determine which model applies. NIST's receiver experiment combined two lasers through a beam splitter, polarizer and single-mode optical fiber, while separately correcting electrical loss and impedance mismatch.^[ref] An electronic gain change confined to the beat-frequency path need not appear in the arm-current measurements. Likewise, adding a current offset is not multiplication by a common factor. Background correction and the sharing of calibration factors must therefore be established before claiming cancellation.

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.