How to improve S11 of helix antenna
Now the total length of the wire I used to build the antenna is about half-wavelength
The measured S11 of the antenna looked ok but not very good
The main reason is the real part of the input impedance of the antena is a bit low
Is there anything I can do to improve it? e.g. using a longer wire like 3/4 wavelength
The challenge is mainly because there is no descent ground plane for this antenna
which is also why I tried this half-wavelength helix antena
The suggestion about other types of antennas (unbanlanced) do not require ground plane is also welcome
You can find useful matching hints in ARRL Radio Amateur's Handbook as well as in ARRL Antenna Book. Usually a variable-impedance stripline section was used. If your application is not too wideband, you can use some coaxial pad tuner, or, having measured the S11, you can design a matching device of coaxial-cable sections.
A good helical is not good without a good "ground" plane; as a directional antenna you should require a good directivity impossible without the good ground (back) plane. To reduce its size, try to use the "backfire" resonating structure after Ehrenspeck.
There is a tip how to improve S11 on a Helical Antenna. Have to increase the thickness of the conductor near the feed-point.
The wire of the first quarter of the first turn should be flat in the form of a copper strip (triangle shape the best) and the transition into a helix should be very gradual. This triangle need some tuning for getting the best VSWR.