side lobes relation too dipoles antennae
The choice of substrate material is very critical. The evidence of sidelobe reduction can be observed with materials having Er > 9.
Moreover, it also depend how you mount the antenna on the EBG surface. In a simulation, one can easily model the antenna on the EBG surface without any supporting structure just to simplify the simulation. However in practice for example a planar dipole antenna will have a balun incorporated, the matching impedance you have calculated in freespace will be distorted by the surface impedance of the EBG structure. As a result, you will find the resonant frequency is shifted slightly. So, one has to carry out a parametric study to design the dimensions of the antenna with EBG structure, not based on freespace calculation alone! You will find that your antenna dimension is usually slightly smaller than the dimension in freespace, which is another benefit after all.
One thing note is that after all that lengthy simulation and effort, you will find the reduction in the back radiation is not breath-takingly great. So, if you are funded by a company to do this research, they will tell you that you additional cost for having the EBG in place is not financially worthwhile when only a slight improvement in the radiation characteristics is observed.
Mushroom-like structure is primarily the best in its class as a EBG ground plane, solely because it provides quite a wide reflection-phase response. This ensure very little tuning of the antenna itself. The disadvantage is the unit-cell size is large which results in the overall surface area of the EBG structure large too. Bare in mind EBG ground plane works best with infinite number of cells, but in practice, the applications limit this factor to say 5 cells across each side.
There are reduced sized elements proposed to overcome the large cell size of the mushroom-like structure, but these elements have narrow bandgap and is very challenging to match the operating frequency of the antenna.
In short, the mushroom-like structure I have worked on have certain reduced the sidelobe radiation in the range of 2-5 dB with a dipole antenna matched by a bazooka balun. The performance is compare with the same antenna on copper ground plane. Beware! If your ground plane is small, the dipole is acting as a feed and the ground plane resonates and behave like a dish antenna. This may cause a misconception to some people who were expecting the dipole radiation performance to be poor when placed very close to a metal surface.
Designing EBG surface is easy. But design it to work well with an antenna is difficult because there are so many parameters to consider. A successful designer should understand all the physics involve in the making. Good luck!