Fig. 2

Dependence of WRA on the choice of quantization axis. Using the \(\hat{z}\)-axis as the quantization axis and the photons on the \(\hat{y}\hat{z}\)-plane as an example. Wave vector \(\hat{k}\) and polarization vector \(\hat{p}\) are illustrated by red and blue arrows, respectively (A). In the absence of any Lorentz-induced rotation, when the wave vector’s \(\hat{z}\)-component aligns with the quantization axis, the \(\hat{z}\)-component matches this axis in the standard frame (B) by rotating the frame about \(\hat{k} \times \hat{z}\), represented by the green arrows (along \(\hat{x}\) in A). (C and D) present the case where the direction of \(\hat{z}\)-component of wave vector is opposite to the quantization axis. Regardless of the direction of the \(\hat{z}\)-component, the polarization angle (with respect to the \(\hat{z}\)-axis) is consistent in both cases as shown in (E). However, when the system is rotated by spatial rotation or boost of the frame, \(\hat{z}\)-component direction of the wave vector, resulting in varied polarization angles in the standard frame. As an example, under a frame rotation about \(\hat{y}\)-axis, (F-J) show the polarization angle varies in the standard frame depending on the direction of \(\hat{z}\)-component of wave vector but does not equate to the frame rotation angle due to the non-commutativity. A general observation as depicted in J, is that polarization angle in the standard frame depends on the relative direction of \(\hat{z}\)-component of wave vector compared to the quantization axis (here the \(\hat{z}\)-axis), as shown in J, which could lead to asymmetry in WRA. Here, \(\hat{k^{\prime}}\) and \(\hat{p}^{\prime}\) are represent the wave vector and polarization vector after the Lorentz transformation of \(\hat{k}\) and \(\hat{p}\), respectively.