Description of phasor form of Maxwell's equations in linear media

Question

I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Chapter 1.3 FIELDS IN MEDIA AND BOUNDARY CONDITIONS says the following:

If linear media are assumed ($\epsilon$, $\mu$ not depending on $\overline{E}$ or $\overline{H}$), then Maxwell's equations can be written in phasor form as $$\nabla \times \overline{E} = - j \omega \mu \overline{H} - \overline{M}, \tag{1.27a}$$ $$\nabla \times \overline{H} = j \omega \epsilon \overline{E} + \overline{J}, \tag{1.27b}$$ $$\nabla \cdot \overline{D} = \rho, \tag{1.27c}$$ $$\nabla \cdot \overline{B} = 0. \tag{1.27d}$$ The constitutive relations are $$\overline{D} = \epsilon \overline{E}, \tag{1.28a}$$ $$\overline{B} = \mu \overline{H}, \tag{1.28b}$$ where $\epsilon$ and $\mu$ may be complex and may be tensors. Note that relations like (1.28a) and (1.28b) generally cannot be written in time domain form, even for linear media, because of the possible phase shift between $\overline{D}$ and $\overline{E}$, or $\overline{B}$ and $\overline{H}$. The phasor representation accounts for this phase shift by the complex form of $\epsilon$ and $\mu$.

$\epsilon$ is the complex permittivity of the medium, and $\mu$ is the complex permeability of the medium.

I don't understand this part:

Note that relations like (1.28a) and (1.28b) generally cannot be written in time domain form, even for linear media, because of the possible phase shift between $\overline{D}$ and $\overline{E}$, or $\overline{B}$ and $\overline{H}$. The phasor representation accounts for this phase shift by the complex form of $\epsilon$ and $\mu$.

What exactly is this saying / what exactly does this mean?

Answer 1

Pozar is talking about a wide bandwidth case in which the frequency dependence of the dielectric permittivity $\epsilon = \epsilon(\omega)$ and/or magnetic permeability $\mu = \mu(\omega)$ cannot be ignored. While the equations Eq 1.27, 1.28 are still true for pure sinusoidal excitation as written it is not true in the time domain. For that you have to calculate the (inverse) Fourier Transform of $\epsilon = \epsilon(\omega)$, say $\tilde \epsilon = \tilde \epsilon(t)=\int_{-\infty}^{+\infty} \epsilon(\omega) e^{\mathfrak {j} \omega t}d\omega/(2\pi)$ and then you get the time-domain convolution $$ \tilde D (t)= \tilde \epsilon(t) \otimes \tilde E (t)$$ where $\tilde E (t) = \bar E (t)\int_{-\infty}^{+\infty} \epsilon(\omega) e^{\mathfrak {j} \omega t}d\omega/(2\pi) $