A space solar power satellite converts sunlight to DC power, then feeds a phased-array or dish transmitter that radiates it Earthward as a coherent microwave beam. Because the transmit aperture is finite, diffraction spreads the beam — it never lands as a single point, so the receiving rectenna only captures the fraction of power that falls within its own aperture.
The standard engineering figure of merit for two circular apertures of area At (transmitter) and Ar (receiver), wavelength λ, separated by distance D, is the Goubau coupling parameter:
τ = (A_t · A_r) / (λD)²
A_t = π(D_t/2)², A_r = π(D_r/2)²
For τ ≪ 1 the beam is diffraction-spread far wider than the rectenna and most power is lost to the surrounding "spillover" ring; for τ ≳ 1 the apertures are optically well-matched and capture approaches unity. A widely used closed-form approximation for a tapered-illumination beam (after W. C. Brown's power-transmission analyses) is:
η ≈ 1 − exp(−τ)
The left panel plots the true diffraction envelope behind that spot-size number: a uniformly illuminated circular aperture of diameter Dt produces an Airy pattern on the ground,
I(r) ∝ [ 2·J₁(x) / x ]², x = π·D_t·r / (λD)
whose first dark ring sits at x = 3.8317 (the first zero of the Bessel function J₁), i.e. at radius r = 1.22·λD/Dt — exactly half of the "2.44λD/Dt" spot-diameter figure used for the ground-spot readout. (Checked numerically against the closed-form first-null value while building this page — the two formulas already agreed, so no correction was needed here.)
- Drag the beam diagram left/right to pan and up/down to zoom — the geometry is drawn at an exaggerated scale for visibility, so panning/zooming helps compare the beam cone against the rectenna at a glance.
- Dt / Dr sliders — bigger apertures on either end raise τ and shrink the spillover ring; this is why real SPS studies (e.g. NASA/DOE 1979 reference design, JAXA's ongoing work) pair a ~1 km transmitter with a multi-km ground rectenna.
- Frequency — shorter wavelength (higher GHz) tightens the beam for the same apertures, but 2.45/5.8 GHz ISM bands are favoured in practice for atmospheric transparency and licensing.
- Distance — geostationary orbit (35,786 km) keeps the satellite fixed over one ground site all day; the slider lets you see how efficiency would change at other beaming ranges.
- Power delivered — assumes a fixed 2 GW transmitted RF beam; only the free-space coupling efficiency above is modeled, not the DC→RF and RF→DC conversion losses at each end.