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The science
The parabola
A dish of focal length f has the profile y = x²/(4f). Every ray arriving parallel to the axis reflects through the focus, and all of them travel the same total path (distance to a line perpendicular to the axis plus distance to the focus), so the signals add in phase there. Depth at the rim is D²/(16 f).
Beam width and gain
A uniformly lit circular aperture of diameter D has the Airy pattern [2 J1(x)/x]² with x = π D sinθ / λ. The first null is at 1.22 λ/D and the half-power width is about 1.02 λ/D. Ideal gain is G = η (π D/λ)² with aperture efficiency η.
Surface errors
Random surface error with rms σ costs exp[-(4πσ/λ)²] (Ruze formula): a dish that is fine at long wavelengths loses gain quickly once σ approaches λ/16.
Off-axis sources
A source tilted by θ moves the focal spot sideways by about f θ and spreads it (coma), which is why a feed far from the axis degrades the image. The graph measures the spread with an exact ray trace.
Checks
(1) Path-length equality of 41 rays through the parabola. (2) The first Airy null found by a numerical scan against 1.22 λ/D. (3) Directivity from numerically integrating the pattern against (π D/λ)².
Simplifications
Uniform illumination (no feed taper or blockage), one cross-section, geometric optics for the focus. Illustrative dish, no real instrument is modelled.
What to try
- Raise the frequency until the surface error eats the gain.
- Tilt the source and watch the focal spot smear.
- Double the diameter: gain rises by 6 dB and the beam halves.