= Consistency bound for separately measured heterodyne arm powers
{title2=$B\leq2\sqrt{I_1I_2}$}
The \b[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:
$$
B\leq2\sqrt{I_1I_2}.
$$
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 $|J|^2\leq I_1I_2$ and the beat amplitude is $B=2|J|$. Equivalently, their <Gram matrix>
$$
G=\begin{pmatrix}I_1&J\\J^*&I_2\end{pmatrix}
$$
is 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 $\widehat\gamma=B/(2\sqrt{I_1I_2})=1.02$. 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 $u(\widehat\gamma)\approx0.0125$. 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>.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3]{ref}
NIST's receiver experiment separately addressed optical mode/polarization matching, electrical losses and impedance mismatch.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF pp. 1–2, Heterodyne measurements, equations 1–4]{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.
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