Re: microwave, antenna
It sounds like you are trying to match a microstrip patch antenna. It is practical to get it into the -10 to -15 dB range for s11. The weasel word answer is that it is possible to get much lower at any specific frequency.
Try looking at the feedline and stub length to see if that changes your match. Try moving them relative to the aperture or changing their dimensions. You might also modify the aperture dimensions slightly. What I would look for is a part of the antenna that modifys the reflection and experiment some to see which way to change the dimension.
If you have a simulator, use that. Otherwise, a roll of copper tape and an exacto knife are the traditional tuniing tools..
Be sure to keep an eye on the patterns and gain. It is easy to get a good match while trashing the antenna characteristics. Don't just tune for low s11.