The commonly quoted lambda/4 criterion is not really a limit on the flare angle itself; it is a limit on the resulting path-length or surface-position error caused by a construction error.
The reason comes from wave phase. A path-length error of lambda/4 corresponds to a phase error of:
Delta phi = 360 deg x (Delta L / lambda)
If Delta L = lambda/4, then:
Delta phi = 90 deg
For a Horn of Plenty horn, the aperture field is formed by waves arriving from different parts of the horn. If construction errors cause some regions of the aperture to be displaced by more than about lambda/4 relative to others, the aperture illumination becomes noticeably distorted.
The consequences are:
- Gain decreases.
- Beam shape changes.
- Sidelobes increase.
- Aperture efficiency falls.
The lambda/4 rule is therefore a practical engineering tolerance that keeps performance degradation reasonably small.
For the hydrogen line:
lambda = c / f
where c=speed of light and f=frequency being measured by the antenna = frequency of hydrogen.
At 1420.405 MHz:
lambda = 299792458 / 1420405000
lambda approximately 0.211 m
Therefore:
lambda/4 approximately 0.0528 m
or about 52.8 mm.
So if a flare-angle error causes the horn edge to end up more than about 5 cm from its intended position, performance will begin to suffer noticeably.
A useful way to think about it is that the flare-angle error itself is not what matters. What matters is how far the resulting surface is displaced from where it should be.
For example:
- A 1 degree error on a very long horn may move the aperture edge by more than lambda/4 and therefore be unacceptable.
- The same 1 degree error on a much smaller horn may produce only a few millimetres of displacement and be insignificant.
This is why antenna engineers usually specify allowable surface or phase errors rather than allowable angular errors.
For amateur SETI and hydrogen-line horns, lambda/4 is a reasonable upper-limit construction tolerance. If you can achieve lambda/8 (about 26 mm at 1420 MHz) or better, the horn will more closely match the design gain and beam pattern.
The guideline of lambda/4 is acceptable, provided it is interpreted correctly: the important quantity is the resulting path-length or surface-position error being less than about lambda/4, not the flare-angle error by itself.