1. Sidelobes for Case of Uniformly-Illuminated Aperture

For a rectangular aperture antenna having a uniform amplitude distribution (or uniform weighting), the first sidelobe is −13.26 dB relative to the peak of the main beam. For such antennas the radiation pattern has a canonical form of

[math]\displaystyle{ \displaystyle\text{Radiation Pattern (in units of dB)} \propto 20\log_{10}\left|\frac{\sin X}{X}\right| }[/math]

Simple substitutions of various values of X into the canonical equation yield the following results:

For a circular aperture antenna, also having a uniform amplitude distribution, the first sidelobe level is −17.57 dB relative to the peak of the main beam. In this case, the radiation pattern has a canonical form of

[math]\displaystyle{ \displaystyle\text{Radiation Pattern (in units of dB)} \propto 10\log_{10}\left|2\cdot\frac{J_1(X)}{X}\right|^2 }[/math]

where [math]\displaystyle{ \displaystyle J_1(x) }[/math] is the Bessel function of the first kind of order 1. This is known as the Airy pattern. Simple substitutions of various values of X into the canonical equation yield the following results:

A uniform aperture distribution, as provided in the two examples above, gives the maximum possible directivity for a given aperture size, but it also produces the maximum side lobe level. Side lobe levels can be reduced by tapering the edges of the aperture distribution (changing from uniformity) at the expense of reduced directivity.

The nulls between sidelobes occur when the radiation patterns passes through the origin in the complex plane. Hence, adjacent sidelobes are generally 180° out of phase to each other.

2. Grating Lobes

A typical radiation pattern of phased arrays whose inter-element spacing is greater than half a wavelength, hence the radiation pattern has grating lobes. https://handwiki.org/wiki/index.php?curid=1178519

For discrete aperture antennas (such as phased arrays) in which the element spacing is greater than a half wavelength, the spatial aliasing effect causes some sidelobes to become substantially larger in amplitude, and approaching the level of the main lobe; these are called grating lobes, and they are identical, or nearly identical in the example shown, copies of the main beams.

Grating lobes are a special case of a sidelobe. In such a case, the sidelobes should be considered all the lobes lying between the main lobe and the first grating lobe, or between grating lobes. It is conceptually useful to distinguish between sidelobes and grating lobes because grating lobes have larger amplitudes than most, if not all, of the other side lobes. The mathematics of grating lobes is the same as of X-ray diffraction.

The animation shows the main lobe and grating lobes of a phased array in polar coordinate system. https://handwiki.org/wiki/index.php?curid=1115982