How to construct a microwave antenna feed for linear and circular polarization ? Is the design of feed different ? How to evaluate the performance of the feed ?
The difference depends on the antenna type. Some antennas require two feeds with 90-degree phase shift, others just a single feed. Do you have an idea about your antenna type? That will help to make the answer more specific.
You can evaluate the performance of the feed in an anechoic chamber by far filed measurements. For example, you can use the antenna with circular polarization as an transmitting antenna and an antenna with linear polarization as a receiving antenna. By two measurements, you can estimate the performance of the feed with circular polarization.