# Which direction does half-wave plate rotate field?

It is well known that if you have linearly polarized light impinging on a half-wave plate with an angle $\theta$ = 0 between the fast axis and the field polarization direction, then the output light will have the same polarization as the input. Now if you rotate the half-wave plate, such that $\theta$ is now non-zero,the output polarization of the beam will be along 2$\theta$.

However, it is not clear which direction the field should rotate. For example, if I rotate a half-wave plate $\theta$ clockwise (from the perspective of the propagating beam), then the field can be rotated either clockwise or counterclockwise to end up at 2$\theta$ relative to its initial state. Or is there always a fixed relationship between the two rotation directions (that of the plate and that of the field)?

According to a quick test I performed using a half-wave plate and a polarizer, the field rotates in the same direction as the plate. However, I am not sure if this generalizes to all half-wave plates, nor am I aware of an argument supporting that speculation.

## 1 Answer

After some thought and discussion with a friend, the answer seems blatantly obvious:

By definition, if the polarization is at an angle $\theta$ to the fast axis of the half-wave plate at the input, then at the output it will be rotated by 2$\theta$, AKA it will appear as a mirror image on the other side of the fast axis. Thus, since the polarization is always flipped about the fast axis, its rotation should follow that of the waveplate, in agreement with my experiment.