 Mode-overlap factor in optical heterodyne detection

The 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 $\int |\mathbf{u}{|}^{2}dA=\int |\mathbf{v}{|}^{2}dA=1$. Their field overlap isThe star denotes complex conjugation. The bound is the Cauchy-Schwarz inequality. In the optical heterodyne detection current model, the beat amplitude is proportional to $|\gamma |\sqrt{P_s{P}_{\mathrm{LO}}}$; 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 $\theta$ givesThis Gaussian beam overlap under lateral displacement follows by completing the square in the overlap integral. At $d=w$ and $\theta =60^{\circ }$, $|\gamma |\approx 0.3033$, so the beat power is only $e^{-1}/4\approx 9.20\%$ 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.^[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.