For this antenna you can use RG58 (VF=0.66) cable. You can also choose to take the better Aircom plus/Aircom premium (VF=0.85) but this makes the antenna longer.
1.0000: 299.8 in a vacuum (maximum speed)
0.9997: 299.7 in air
0.96: in bare copper wire
0.95: in ladder line (lead-in cable)
0.80-0.85: in good quality coax
0.66: cheap coax
5.887 GHz example
The top segment (1/4 wavelength * VF 1.0):
1/4 wavelength * 0.961 = 12.22 mm
1/4 wavelength * 0.66 = 8.40 mm
The center segments (1/2 wavelength * VF 0.66):
Use foam to support the elements which are in line with the top segment.
The bottom segment:
Bazooka balun (or sleeve balun):
1/4 wavelength * 0.96 = 12.22 mm + 5.5mm = 17.72mm
15 mm diameter pipe
4 mm shielding
11/2 = 5.5mm bridging
Why use it? Normally a transmission line is unbalanced. When making an antenna we would like balanced operation (to make the rest of the dipole function properly). There is some theory decribed here.
To make it, use a 15 mm piece of copper tube as a template. The pipe is not used in the actual antenna.
Measure the width of the copper foil and add some extra length to connect it to the braid of the coax cable. Create a folding line to bend it more easy.
Wrap a piece of copper foil around the copper pipe and slide it of the template. Discard the copper tube.