Maxwell's Equations: Taking the real part of the product $\mathbf{E}\exp(j \omega t)$ in order to get the "complete field"

Question

My textbook, Laser Electronics, Third Edition, by Verdeyen, says the following in a section on Maxwell's equations:

To describe an electromagnetic wave, we need two field-intensity vectors, $\mathbf{e}$ and $\mathbf{h}$, which are related to each other by

$$\nabla \times \mathbf{h} = \mathbf{j} + \epsilon_0 \dfrac{\partial{\mathbf{e}}}{\partial{t}} + \dfrac{\partial{\mathbf{p}}}{\partial{t}} \tag{1.2.1a}$$

$$\nabla \times \mathbf{e} = -\mu_0 \dfrac{\partial{\mathbf{h}}}{\partial{t}} \tag{1.2.1b}$$

where $\mathbf{p}$ is the polarization current induced by the electric field. (A term of the form $\partial{\mathbf{m}}/\partial{t}$ can be added to (1.2.1b) but will be ignored for now.) We use lowercase letters to represent vectors that are explicit functions of time $t$ and the three spatial coordinates $x$, $y$, and $z$. Most of the time we will be talking about sinusoidal variations of the field and use the phasor representation

$$\begin{matrix} \mathbf{e}(\mathbf{r}, t) = {\rm Re} \left[ \mathbf{E}(\mathbf{r})e^{j \omega t} \right]& \mathbf{h}(\mathbf{r}, t) = {\rm Re} \left[ \mathbf{H}(\mathbf{r})e^{j \omega t} \right] \\ \mathbf{j}(\mathbf{r}, t) = {\rm Re} \left[ \mathbf{J}(\mathbf{r})e^{j \omega t} \right]& \mathbf{p}(\mathbf{r}, t) = {\rm Re} \left[ \mathbf{P}(\mathbf{r})e^{j \omega t} \right] \tag{1.2.2} \end{matrix}$$

where $Re$ is real part, $\mathbf{r} = x\mathbf{a}_x, + y\mathbf{a}_y + z\mathbf{a}_z$, $\mathbf{a}_i$, is the unit vector in the $i$th direction, and the capital letters $\mathbf{E}$ and $\mathbf{H}$ are complex vector quantities depending on space coordinates but not on time. We recognize that if we want the complete field, we must take the real part of the product $\mathbf{E}\exp(j \omega t)$.

What does the author mean by "complete field" here, and why must we take the real part of the product $\mathbf{E}\exp(j \omega t)$ in order to get it? I would greatly appreciate it if people would please take the time to explain this.

Answer 1

One of the advantages of the complex representation of the fields—as this passage is trying to emphasize—is that the normal modes of the solution are simple products of a function of the spatial position, times a function of time. However, this only true of the complexified solutions. The true, real-valued solutions are not just products—e.g. the "complete" electric field strength is $${\bf e}({\bf r},t)=\frac{1}{2}\left[{\bf E}({\bf r})e^{j\omega t}+{\bf E}^{*}({\bf r})e^{-j\omega t}\right],$$ which does not have the product form.