Logic of inverse relation between flaring and beamwidth in piramid antenna

[yefj]

Hello,in piramid horn antenna there is an issue when we flaring wider in some direction then the beam width in that direction is getting narroer.
As shown in the formulas bellow,
Why is that?
Is there some physical explanantion for that?
Thanks.

 

[superkuh]

The sky pattern of an aperture is the Fourier transform of the current distribution. The longer the span in space of the aperture the smaller the fourier representation in the sky as antenna pattern. A longer effective aperture always means a smaller beamwidth. It's easier to understand this coming from fourier optics than from the radio side. Look up Fourier transforms from lenses turning wavefront angles into positions. And if you want a rigorous derivation then chapter 15 of Interferometry and Synthesis in Radio Astronomy on the Van Cittert-Zernike Theorem. The below videos give good general summaries as well.

Generally this holds for horns until they are >>> wavelength in aperture and phase errors re: middle versus edges start to hurt the gain.