= Mode-overlap factor in optical heterodyne detection
The \b[mode-overlap factor in optical heterodyne detection] measures how effectively the two optical fields interfere over the receiver. Write their normalized transverse vector profiles as $\mathbf u$ and $\mathbf v$, with $\int|\mathbf u|^2dA=\int|\mathbf v|^2dA=1$. Their field overlap is
$$
\gamma=\int\mathbf u^*(\mathbf r)\cdot\mathbf v(\mathbf r)\,dA,
\qquad |\gamma|\leq1.
$$
The star denotes <complex conjugation>. The bound is the <Cauchy-Schwarz inequality>. In the <optical heterodyne detection> current model, the beat amplitude is proportional to $|\gamma|\sqrt{P_sP_{\rm LO}}$; its electrical power is proportional to $|\gamma|^2$ at fixed optical powers and receiver conditions.
An original example uses equal circular <Gaussian beams> whose field amplitude varies as $e^{-r^2/w^2}$. Over an unlimited collecting plane, displacing one beam laterally by $d$ and rotating its linear <polarization> through angle $\theta$ gives
$$
|\gamma|=e^{-d^2/(2w^2)}|\cos\theta|.
$$
This <Gaussian beam overlap under lateral displacement> follows by completing the square in the overlap integral. At $d=w$ and $\theta=60^{\circ}$, $|\gamma|\approx0.3033$, so the beat power is only $e^{-1}/4\approx9.20\%$ of the perfectly matched value. The two collected optical powers can nevertheless remain unchanged.
The NIST heterodyne receiver experiment used a polarizer and single-mode fibre to match both polarization and spatial mode, making its modulation depth calculable.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF p. 1, Heterodyne measurements, beam combining and mode matching]{ref} Changing alignment or polarization while monitoring each arm separately tests this distinction between lost optical power and lost interference.
A finite detector aperture changes the collecting integral and the powers used for normalization. Wavefront curvature and phase variation can also reduce overlap. Fluctuating relative phase introduces an additional coherence/averaging effect; a spatial overlap number alone does not characterize every loss of a measured beat.
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