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Lognormal bias of a heterodyne overlap estimate is the upward mean shift that remains when a zero-mean Gaussian logarithmic error is exponentiated. It does not undo common-gain cancellation in heterodyne overlap uncertainty: a perfectly shared gain cancels in every reading, while residual mismatched gains can still fluctuate.

Write the normalized positive estimate as $\hat{\gamma }=\gamma e^d$, with fixed true overlap magnitude $\gamma$ and $d\sim N({\mu }_d,v_d)$. The moment-generating function of the normal distribution givesThese are exact lognormal distribution identities under the stated error model. When ${\mu }_d=0$, the logarithmic estimate is centered correctly and the median is $\gamma$, but the arithmetic mean exceeds it by relative amount $e^{v_d/2}-1$.

For an original deliberately visible example, $\gamma =0.8$ and logarithmic standard deviation 0.2 give $v_d=0.04$ and mean about 0.81616: a 2.02% mean shift. The values are chosen statistical parameters, not uncertainties reported for a receiver. An estimate can also exceed the physical consistency bound for separately measured heterodyne arm powers; the noisy readout is not itself a physical overlap above one.

If ${\mu }_d$ and $v_d$ are known, multiplying the estimate by $e^{-{\mu }_d-v_d/2}$ corrects its mean in this model. Uncertain variance in heterodyne lognormal bias correction matters when those parameters are estimated: plugging in a noisy variance, especially from related data, is not automatically an exact correction.

Averaging logarithms has a different interpretation from averaging overlaps. For $n$ independent zero-mean log-errors of variance $v_d$, the geometric mean is $\gamma e^{\overline{d}}$, whose arithmetic expectation is $\gamma e^{v_d/(2n)}$. A gain component shared across the entire run does not acquire the independent-shot $1/n$ variance reduction.

The physical calibration paths determine the logarithmic covariance. NIST's heterodyne apparatus controlled the optical modes while separately correcting electrical loss and impedance mismatch.^[ref] A first-order uncertainty calculation alone is not this exact distributional model; nonlinear effects and correlated inputs must be assessed explicitly.^[ref] State the assumed distribution and the intended summary rather than treating median, geometric mean and arithmetic mean as interchangeable.

Separating power imbalance from mode mismatch in optical beating requires more information than a single modulation-depth reading. In the optical heterodyne modulation depth model, two effects multiply:The power ratio controls one term, while $\gamma$ describes the mode-overlap factor in optical heterodyne detection. A low beat depth can result from either effect or both.

Measure each arm separately with the same detector and collection geometry, subtracting dark current. For sufficiently close optical frequencies with the same detector responsivity, let the separate currents be $I_s$ and $I_{\mathrm{LO}}$. After correcting the receiver's beat-frequency response,This consistency bound for separately measured heterodyne arm powers requires $I_{\mathrm{beat},\mathrm{pk}}\le 2\sqrt{I_s{I}_{\mathrm{LO}}}$. An apparent value above unity indicates inconsistent calibration, background treatment, conditions or model assumptions.

For an original example, separate currents are $0.40$ and $0.10\mathrm{mA}$. Their power-imbalance limit is $m_{\max }=0.80$. A corrected beat depth $m=0.56$ means peak beat current $0.28\mathrm{mA}$ and overlap $|\gamma |=0.70$. If the electrical chain instead attenuates the beat amplitude by $0.75$ relative to DC and that effect is ignored, the apparent depth becomes $0.42$ and inferred overlap $0.525$.

NIST's physical receiver experiment matched the two optical modes and received photocurrent contributions; it separately corrected electrical loss and impedance mismatch when inferring receiver response and modulation.^[ref] Holding alignment fixed while changing one arm's power tests whether the corrected overlap remains constant. Source drift, changed polarization or aperture clipping during the separate-arm measurements can spoil that inference.

The variance of a corrected heterodyne overlap estimate quantifies uncertainty in $\hat{\gamma }=B/(2\sqrt{I_1{I}_2})$, 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 givesThus common influences can cancel rather than contribute three independent uncertainty terms. This is the covariance form of the primary measurement-propagation rule.^[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 $\sqrt{3/2}\times 0.5\%\approx 0.6124\%$, 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 $(gB,g{I}_1,g{I}_2)$ 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.^[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.