How to design and build helix antennas is described here. Also some of the theory behind it. There are tree radiation modes of helical antennas. The most used mode is the axial mode, it provides maximum radiation along the helix axis.
For axial mode, the circumference of one helix-turn is in the order of one wavelength. In normal mode, radiation goes broadside to the helix axis. It occurs when helix diameter is small in respect to the wavelength. If wavelength is small compared to the helix diameter, in conical mode, radiation will go into multiple lobes.
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Feedpoint impedance of air wound helix antennas in axial mode is 137 Ohm (almost purely resistive). Matching is required if 50 Ohm coax cable is to be connected. There are two common ways to do this matching. Either with a coax section or a stripline above a ground plane, both a quarter wavelenght long and of proper resistance.
I have found different formulas to calculate impedances of striplines over a ground plane. In the picture above, the tree most “close” are shown. I finally used formula number 3. A quarter wavelength long (58 mm) piece of copper sheet, 0.5 mm thick, 12.5 mm wide and at 5 mm distance from the reflector will do the job for a 1296 MHz helix. For 13 cm, the piece should be about 3.2 mm.
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