Deployable Reflector Gain

What is modelled

A paraboloid z = r²/(4f) reflects every ray from its focus into a plane wave, because all paths have the same length. The dish unfolds from the hub outward in rings of petals, so a partly deployed dish is a smaller circular aperture on the same paraboloid. The gain of such an aperture depends on its diameter over the wavelength, on how evenly the feed lights it, and on how far the panels depart from the ideal surface.

Equations

• Path length |FP| + (zrim − zP) = f + zrim for every point P of the paraboloid.
• Gain G = ηill ηRuze (π D / λ)².
• Ruze efficiency ηRuze = exp(−(4π ε / λ)²) for rms surface error ε.
• Uniform circular aperture beam |2 J1(x)/x|² with x = π D sinθ / λ; first null at 1.22 λ/D; half-power width about (1.02 + 0.0135 T) λ/D for an edge taper of T dB.

Checks run in the page

• Equal path lengths of 24 reflected rays.
• Monte Carlo of random phase errors against the Ruze factor.
• First null of the numerically integrated aperture field.
• Surface area by integration against the closed form, and petal areas summing to the projected disc.

Honest simplifications

Partial deployment is modelled as a smaller disc on the same paraboloid with the feed fixed, which only holds for ring-by-ring deployment. There is no feed blockage, spillover or struts, the taper is a simple parabolic-on-pedestal model, and the beamwidth formula is the usual engineering approximation. The reflector is a generic design and not a particular instrument.

• Press High frequency, rough panels and watch the gain collapse as surface error approaches a tenth of a wavelength.
• Press Half deployed and see the gain drop by about 6 dB while the beam widens.
• Raise the feed taper and watch the illumination efficiency fall in the last graph.