Circularly polarized waves

Consider a time-harmonic E field, given as:

\[\mathbf{E}(\mathbf{r},t) = \text{Re}\Big[\mathbf{E}(\mathbf{r})\text{e}^{j\omega t}\Big]\]

For a plane wave, the \(\mathbf{E}\) field phasor is

\[\mathbf{E}(\mathbf{r}) = \hat{n}~E_{0}~\text{e}^{-j~\mathbf{k}\cdot \mathbf{r}}\]

Now, if we consider a plane wave with \(\mathbf{E}\) field of amplitude \(E_{0}\), pointing in a direction \(45^{circ}\) with respect to the \(x\)-axis and traveling in the \(+z\) direction, the \(\mathbf{E}\) field phasor is

\[\mathbf{E}(\mathbf{r}) = \Bigg(\frac{\hat{x}+\hat{y}}{\sqrt{2}}\Bigg)~E_{0}~\text{e}^{-j~k~z}\]