A couple options to just get a feeling whether it's circular or not.
After your first antenna pattern, rotate your antenna 45 degrees and plot the antenna pattern again. Rotate your 90 degrees (from the original position) and plot the antenna pattern a second time. If all the plots look similar, then it's probably circular polarization, where the plots diverge it's not circular. An antenna with one or two dB difference for these antenna patterns is pretty good. For wide angle antennas, even 5 dB difference in the patterns is often acceptable.
Or, a quick measurement on a Vector Network Analyzer, measure S21 from your antenna to a dipole, calibrate on this initial measurement to get a flat line on your analyzer. Then rotate the dipole and see how much the S21 changes relative to your flat reference line. You can repeat that at various angles relative to your antenna boresight by positioning the dipole off axis. Use some absorber around the test area, or keep the two antenna fairly close to one another (a few inches) for strong coupling so that bounces off the operator and other hardware don't hurt the measurement too much.
The most accurate mathematical result is to take two antenna patterns, one with the test chamber transmit antenna vertical, one with the transmit antenna horizontal. You need phase data in the measurement to calculate circular polarization antenna pattern for both right hand and left hand circular.
What frequency are you working at, just curious.
kch