 Gaussian beam overlap under lateral displacement

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.