= Optical heterodyne modulation depth
{title2=$m$}
The \b[optical heterodyne modulation depth] measures the fractional oscillation of the combined optical power. For a stable sinusoidal beat, define
$$
m=\frac{P_{\max}-P_{\min}}{P_{\max}+P_{\min}}.
$$
Using the <optical heterodyne detection> field-mixing model gives
$$
m=\frac{2|\gamma|\sqrt{P_sP_{\rm LO}}}{P_s+P_{\rm LO}}
=\frac{2|\gamma|\sqrt r}{1+r},\qquad r=P_s/P_{\rm LO}>0,
$$
where $\gamma$ is the <mode-overlap factor in optical heterodyne detection>. The inequality $2\sqrt{P_sP_{\rm LO}}\leq P_s+P_{\rm LO}$ shows that $m\leq|\gamma|\leq1$. Full modulation requires both equal powers and complete overlap.
An original example shows an inference ambiguity. A measured $m=0.60$ is compatible with equal powers and $|\gamma|=0.60$. It is also compatible with perfect overlap and power ratio $r=1/9$ or $9$. The beat depth alone therefore does not identify mode mismatch. <Separating power imbalance from mode mismatch in optical beating> requires additional information, such as measuring the two arms separately under the same collection conditions.
NIST's receiver-characterization experiment matched both polarization and spatial mode and equalized the received photocurrent contributions to obtain full modulation. Its calibrated transfer-standard relation also explicitly uses modulation depth.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF p. 1, Heterodyne measurements; p. 2, equation 4 and definition of modulation depth]{ref}
The measured electrical modulation can differ from the optical one if the receiver attenuates the beat frequency relative to its DC response. An additive background $P_b$ also changes the observed denominator to $P_s+P_{\rm LO}+P_b$ without contributing the intended coherent beat. Background correction, receiver response and unsaturated readout are needed before interpreting an electrical amplitude ratio as optical overlap.
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