= Variance of a corrected heterodyne overlap estimate
{title2=$u^2(\widehat\gamma)/\gamma^2\simeq\mathbf v^TR\mathbf v$}
The \b[<variance> of a corrected heterodyne overlap estimate] quantifies uncertainty in $\widehat\gamma=B/(2\sqrt{I_1I_2})$, where $B$ is corrected peak beat current and $I_1,I_2$ are separately measured arm currents. The <consistency bound for separately measured heterodyne arm powers> requires the physical overlap not to exceed one, but measured estimates fluctuate.
For small errors, let $\mathbf e=(\delta B/B,\delta I_1/I_1,\delta I_2/I_2)^T$ have <covariance> $R$, and put $\mathbf v=(1,-1/2,-1/2)^T$. Linearization gives
$$
\frac{u^2(\widehat\gamma)}{\gamma^2}\simeq\mathbf v^TR\mathbf v
=R_{BB}+\frac{R_{11}+R_{22}}4-R_{B1}-R_{B2}+\frac{R_{12}}2.
$$
Thus common influences can cancel rather than contribute three independent uncertainty terms. This is the <covariance> form of the primary measurement-propagation rule.<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}
For an original model, all three current readings share a $2\%$ multiplicative calibration error, with additional independent $0.5\%$ errors on each. The common contribution to $\mathbf v^T\mathbf e$ is zero. The relative standard uncertainty is only $\sqrt{3/2}\times0.5\%\approx0.6124\%$, or $0.00490$ at overlap $0.80$. Treating the common calibration error as independent in all three readings would instead give about $2.525\%$ relative uncertainty.
The <common-gain cancellation in heterodyne overlap uncertainty> is exact for a single positive gain $g$: replacing $(B,I_1,I_2)$ by $(gB,gI_1,gI_2)$ leaves the ratio unchanged. A separate radio-frequency gain uncertainty affecting only $B$ does not cancel, nor does an additive dark-current offset generally have this property.
NIST's heterodyne receiver procedure corrected electrical loss and impedance effects separately from matching optical powers and modes.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF pp. 1–2, Heterodyne measurements, equations 1–4]{ref} Build the <covariance> from those actual shared and separate calibration paths. This first-order <variance> model also needs reconsideration when an arm current is close to zero or background subtraction makes its uncertainty comparable to its value.
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