perforated plastic for dielectric quarterwave matching transformer

Lets say i had an horn antenna, and wanted to launch a wave into a material with a dielectric constant of around 2.3.

Could i use a quarter wave matching transformer (a slab of dielectric material) made of Polyethelene, but with lots of small holes uniformly drilled into it so that the effective dielectric constant of the polyethelene was made to be √(1 * 2.3) = 1.52? Polyethelene is around a er=2.2-2.3, but if i drilled enough air holes in it, would it act like it had a dielectric constant of the desired 1.52???

If so, what would be the formula? Something like just a volume ratio:

εeff = 1 + 1.25*(Volume remaining of plastic/total volume of slab)

all air (zero plastic volume remaining) would be εeff = 1, and all plastic (no holes drilled) would be εeff = 1 + 1.25 = 2.25?

or would a sandwich be better? Maybe plastic sheet, RHO cell, Plastic sheet, etc?

I think you should measure the permittivity of the perforated material at the frequency of interest, and then use the material in an ordinary way.
The perforation size and distance should be much smaller than the wavelength, otherwise the analysis becomes quite complex.

Instead of a quarter wave transformer, a set of pyramidal pieces providing a smooth transition between free space and the dielectric material (like pyramidal absorbers but made of polyethylene or similar material) perhaps could be best suited.
It would need more space and material, but it would have wider bandwidth.

Z

not a bad idea, but I do not have a lot of room for length in this project.