= Gaussian beam overlap under lateral displacement
{c}
{title2=$|\gamma(d)|$}
\b[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 amplitudes
$$
u_j(\mathbf r)=\sqrt{\frac{2}{\pi w_j^2}}\,
e^{-|\mathbf r-\mathbf d_j|^2/w_j^2}.
$$
The 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 gives
$$
|\gamma|=\frac{2w_1w_2}{w_1^2+w_2^2}
\exp\!\left[-\frac{d^2}{w_1^2+w_2^2}\right],
\qquad d=|\mathbf d_1-\mathbf d_2|.
$$
The 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 $|\gamma|=0.8e^{-1/5}\approx0.6550$, 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 $e^{-d^2/w^2}\geq0.90$, hence $d\leq0.3246w$. A chosen $100\,\mathrm{\mu m}$ radius permits about $32.5\,\mathrm{\mu m}$ 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.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF p. 1, Heterodyne measurements, spatial/polarization matching]{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.
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