It is mathematically more convenient to deal with complex exponentials rather than sines and cosines in many circumstances, especially when it comes to combining functions or taking derivatives. The only thing we really need to recall about complex exponentials is the fact that
$latex \displaystyle e^{i x} = \cos{x} + i \sin{x} $
from which a host of other useful formulae may be derived. If we want to rewrite our generalized plane wave as a complex exponential, we can write
$latex \displaystyle \vec{E} = Re(A \hat{a}_p e^{i(k_x x + k_y y + k_z z – \omega t + \phi)} ) $
where we’ve taken the real part of the complex exponential, although many physics texts prefer the following:
$latex \displaystyle \vec{E} = \frac{1}{2} (A \hat{a}_p e^{i(k_x x + k_y y + k_z z – \omega t + \phi)} + cc.) $
where ‘cc.’ is the complex conjugate of the preceding expression. If we introduce the position vector $latex \vec{r} = x \hat{a}_x + y \hat{a}_y + z \hat{a}_z$, which points from the origin to any observation point in Cartesian space, we can shorten our complex exponential expression to
$latex \displaystyle \vec{E} = Re{(A \hat{a}_p e^{i(\vec{k} \cdot \vec{r} – \omega t + \phi)} )} $
When dealing with a problem in which all fields are oscillating at the same frequency $latex \omega$, it’s typical to write the fields as phasors, which means not bothering to write down the $latex Re$ nor the $latex \omega t$ – they are implied. This eases mathematical manipulations greatly. The phasor electric field of the plane wave, then, is
$latex \displaystyle \vec{E}_{phasor} = A \hat{a}_p e^{i(\vec{k} \cdot \vec{r} + \phi)} $
The ‘starting phase’ $latex e^{i \phi}$ can be folded into the amplitude $latex A$ to make a complex amplitude $latex A’ = A e^{i \phi}$, further simplifying our expression:
$latex \displaystyle \vec{E}_{phasor} = A’ \hat{a}_p e^{i \vec{k} \cdot \vec{r}} $
This is all very convenient, but we should always bear firmly in mind that in the real world, an electric field can’t be complex – it represents the force applied to a charge, after all, and what is an imaginary force? So if we’re playing with phasor electric fields (which are complex) and we want to return to the real-world electric field that would be measured by an instrument (the ‘time-domain field’), we must multiply the phasor by $latex e^{-i \omega t}$ and take the real part of the whole thing.