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I am reading the wikipedia page for parabolic antennas, and have a question about the below quote:

In order to achieve narrow beamwidths, the parabolic reflector must be much larger than the wavelength of the radio waves used

Why is this so? The geometric explanation of these antennas doesn't really explain it---from what I understand, the paraboloid has a unique property:

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The lengths of $FP_1Q_1 = FP_2Q_2 = FP_3Q_3$.

So if we emit light at the point $F$, and coat a paraboloid in a mirror-like material, it should send the light in a straight beam (following the $FP_1Q_1$, $FP_2Q_2$, $FP_3Q_3$, etc lines) parallel to the line $VF$, no matter the wavelength.

So why do the dishes need to be bigger than their wavelength?

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The law of reflection (that incoming and outgoing wavefronts are at equal angles to the normal to the mirror surface) is a consequence of wave interference. A wave approaches the mirror. Each atom on the surface of the mirror absorbs and immediately re-emits the wave as a point source, with the same phase and wavelength as that point receives. At different points along the mirror, they are at different distances from the source, and so the phase of the emitted spherical waves is different. The outgoing spherical waves interfere, and if the mirror is large compared to the wavelength, the result is an outgoing wavefront of the appropriate shape.

The law of reflection implicitly assumes that the wavelength is negligible compared to the size of the mirror. If the wavelength is significantly larger than the mirror, then all points on the mirror re-emit spherical waves with essentially the same phase, and the result is a spherical outgoing wave. The mirror is 'too small for the wavelength to resolve' and acts as a point scatterer.

When the mirror and the wavelength are roughly the same, you get intermediate behaviour, where the reflection is roughly concentrated in the expected direction, but is spread out. This is called 'diffraction'. The ability of an optical system to resolve fine details is determined by the diffraction limit (the Rayleigh criterion) that depends on the aperture of the instrument and the wavelength of the light (or any other type of wave).

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I would only add to @NulliusinVerba's answer that you are showing a ray diagram to represent wave propagation. Rays themselves are short wavelength approximate concepts to represent and simplify the more precise wave propagation phenomena including refraction and scattering, where the length scale of being "short" is to be understood relative to the dominant geometric scales that the "ray" encounters and interacts with. Here the primary scale is that of the mirror, but even in this idealized case what the ray model hides is what happens in the focus, the so-called primary radiator, that is in fact not point like, it usually is a "horn".

If you want to know how precise the mirror must be then the surface errors relative to the operating wavelength come into play and the ray model using reflection and scattering quickly becomes useless just as the scattering effect of the spars holding the radiator in the focus cannot be analyzed properly with a ray model because spars' thickness is usually of the order of the wavelength at RF or microwaves.

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