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The initial question was, if at 1GHz circular polarization, can we say the electrical field rotates at the rate = 10^9 rotations/s?
So in your view, at 1GHz circular polarization, how many (full) rotations would be during one second?
Yes, the field rotates at the same rate with the frequency.
Let us start with simpler case, linear polarization. In linear polarization the field changes its direction at the rate of the frequency driven.
Imagine a vector oscillating in +x and -x directions.
In circular polarization, you add a vector oscillating in +y and -y directions with 90 degree phase shift. Addition of these vectors will give a rotating vector, it rotates at the same rate.
The rotation rate should be the instantaneous frequency of the signal. The modulation does not complicate it. It is not expected to see a extraordinary effect on the polarization in a simple medium (homogeneous, non-dispersive, isotropic).
However, the polarization may undergo interesting effects in an anisotropic medium. It can be changed from RHCP to LHCP for example. The rate should still be the same.
The rotation rate is nothing but the frequency of the signal itself just like the case for linear polarization.
If it is changing in 1 GHz and 1.001 GHz, yes, the rotation rate will vary between those frequencies.