# 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

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