= Separating power imbalance from mode mismatch in optical beating
{title2=$|\gamma|=I_{\rm beat,pk}/(2\sqrt{I_sI_{\rm LO}})$}
\b[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:
$$
m=|\gamma|\frac{2\sqrt r}{1+r},\qquad r=P_s/P_{\rm LO}.
$$
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 $I_{\rm LO}$. After correcting the receiver's beat-frequency response,
$$
|\gamma|=\frac{I_{\rm beat,pk}}{2\sqrt{I_sI_{\rm LO}}}.
$$
This <consistency bound for separately measured heterodyne arm powers> requires $I_{\rm beat,pk}\leq2\sqrt{I_sI_{\rm LO}}$. 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.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF pp. 1–2, Heterodyne measurements, equations 1–4]{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.
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