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Gaussian beam overlap under lateral displacement quantifies how a transverse offset reduces interference between Gaussian beams. For equal linear polarizations, matched wavefront curvature and an unlimited receiving plane, use normalized field amplitudesThe parameter $w_j$ is the radius at which the intensity has fallen to $e^{-2}$ of its central value. Completing the square in their overlap integral givesThe equal-radius case reduces to the expression in mode-overlap factor in optical heterodyne detection. The prefactor shows that centring the beams does not remove waist-size mismatch in optical heterodyne mode matching.

For an original $w_2=2w_1$ example, centred beams have $|\gamma |=0.80$ and only $64\%$ of the perfectly matched beat power at fixed collected powers. Adding displacement $d=w_1$ gives $|\gamma |=0.8e^{-1/5}\approx 0.6550$, reducing the beat-power fraction to about $42.9\%$.

An alignment tolerance follows in the equal-radius case. To retain at least $90\%$ of the matched beat power, require $e^{-d^2/w^2}\ge 0.90$, hence $d\le 0.3246w$. A chosen $100\mu \mathrm{m}$ radius permits about $32.5\mu \mathrm{m}$ displacement under these assumptions.

NIST's primary heterodyne setup used a polarizer and single-mode fibre to ensure matching rather than infer it from total power alone.^[ref] These original overlap examples omit aperture clipping, angular tilt and wavefront-curvature mismatch. They describe the detector's optical field overlap, not a universal coupling efficiency for every fibre or lens arrangement.

Optical heterodyne detection combines a signal field with a reference field at a different frequency, producing an electrical beat within a receiver's bandwidth. The reference is a local oscillator. A photodetector responds to the squared sum of the fields; averaging over optical carrier cycles removes the fast sum-frequency term but can retain the difference-frequency interference.

For constant powers $P_s,P_{\mathrm{LO}}$ incident on a linear detector of responsivity $R$, a useful model isThe mode-overlap factor in optical heterodyne detection satisfies $|\gamma |\le 1$ and includes spatial and polarization matching. The beat frequency is $|{\nu }_s-{\nu }_{\mathrm{LO}}|$, while its phase carries a relative optical wave phase. Orthogonal polarizations can give $\gamma =0$ and no beat despite nonzero power from both sources.

For an original chosen $P_s=1\mathrm{nW}$, $P_{\mathrm{LO}}=1\mathrm{mW}$, $R=0.50A/W$ and $|\gamma |=1$, the beat's peak current is $1.0\mu \mathrm{A}$. The signal alone produces only $0.50\mathrm{nA}$ of current, but the mixed DC level is about $0.50\mathrm{mA}$. A chosen frequency difference of $10\mathrm{MHz}$ is measurable electronically even though each carrier has an optical frequency. Increasing the reference also increases its noise and can overload the receiver; beat amplitude is not an unlimited sensitivity gain.

NIST's physical receiver-characterization experiment combined an external-cavity semiconductor laser and a Nd:YAG laser near $1319\mathrm{nm}$. Variable attenuators, a beam splitter, a polarizer and single-mode fibre controlled the powers and matched the modes before the receiver. Sweeping the optical frequency difference generated a tunable test modulation.^[ref]

The receiver must pass the beat frequency and preserve the intended overlap. A single real beat frequency supplies the magnitude of the frequency difference; identifying which optical frequency is higher needs additional information. At zero difference, the corresponding fixed-phase measurement is optical homodyne detection.

The optical heterodyne modulation depth measures the fractional oscillation of the combined optical power. For a stable sinusoidal beat, defineUsing the optical heterodyne detection field-mixing model giveswhere $\gamma$ is the mode-overlap factor in optical heterodyne detection. The inequality $2\sqrt{P_s{P}_{\mathrm{LO}}}\le P_s+P_{\mathrm{LO}}$ shows that $m\le |\gamma |\le 1$. 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.^[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 $P_s+P_{\mathrm{LO}}+P_b$ without contributing the intended coherent beat. Background correction, receiver response and unsaturated readout are needed before interpreting an electrical amplitude ratio as optical overlap.

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.