\begin{align}\mathbf{S} & = \mathbf{E} \times \mathbf{H}\\  & = \mathrm{Re}\left(\mathbf{\widetilde E}\right) \times \mathrm{Re}\left(\mathbf{\widetilde H} \right)\\  & = \mathrm{Re}\left(\mathbf{E_c} e^{j\omega t}\right) \times \mathrm{Re}\left(\mathbf{H_c} e^{j\omega t}\right)\\  & = \frac{1}{2}\left(\mathbf{E_c} e^{j\omega t} + \mathbf{E_c}^* e^{-j\omega t}\right) \times \frac{1}{2}\left(\mathbf{H_c} e^{j\omega t} + \mathbf{H_c}^* e^{-j\omega t}\right)\\  & = \frac{1}{4}\left(\mathbf{E_c} \times \mathbf{H_c}^* + \mathbf{E_c}^* \times \mathbf{H_c} + \mathbf{E_c} \times \mathbf{H_c} e^{2j\omega t} + \mathbf{E_c}^* \times \mathbf{H_c}^* e^{-2j\omega t}\right)\\  & = \frac{1}{4}\left(\mathbf{E_c} \times \mathbf{H_c}^* + \left(\mathbf{E_c} \times \mathbf{H_c}^*\right)^* + \mathbf{E_c} \times \mathbf{H_c} e^{2j\omega t} + \left(\mathbf{E_c} \times \mathbf{H_c} e^{2j\omega t}\right)^*\right)\\  & = \frac{1}{2}\mathrm{Re}\left(\mathbf{E_c} \times \mathbf{H_c}^*\right) + \frac{1}{2}\mathrm{Re}\left(\mathbf{E_c} \times \mathbf{H_c} e^{2j\omega t}\right). \end{align}