Graphical interpretation of complex electric fields

Question

Before anything, I have read similar questions here but still something doesn't click perfectly, so I'll try to describe so.

My doubt is essentially that I have always thought $\vec E$ in the context of wave as the usual $\vec E=E_0 \cos(kz-\omega t)\vec u_x$ which in my mind is represented as a line as the linear polarisation shows below:

So now my teacher really shocked me with now having vector fields of these types:

1. $$\vec E=\vec E_0 e^{i(kz-\omega t)}$$ with $\vec E_0\in \mathbb{R^3}$ (the components of the $\vec E_0$ are real) or also this type:

2. $$\vec E=\vec E_0 \cos(kz-\omega t)$$ with now $\vec E_0$ having complex components like $\vec E_0=(0,a,bi)$.

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I have lately been wondering how would you even plot these, in the second type for example with complex numbers in the $\vec E_0$ I would've thought since complex vectors can't be plotted over $\mathbb{R^2}$ so that they couldn't be plotted, can they?

And then for the first case, I have read that there can be a complex number that accounts for a difference of phases $$\vec E=\vec E_0 e^{i(kz-\omega t+\delta)}=\vec E_0 e^{i\delta} e^{i(kz-\omega t)}$$ which is the $\delta$ apparently the one responsible for describing a circled polarisation or linear polarisation and the $\vec E'=\vec E_0 e^{i\delta}$ is the one called the phasor.

In conclusion. I think my problem is that I have a lots of ideas picked up from different classes but can't seem to connect them all and can't also seem to visualise those ideas. Mainly could be because of notation or of bad complex background but if someone could shine a light it would help

Answer 1

This is a notation we use because it makes it much much easier to work with the fields.

We just remember that we have to take a real part of the complex quantities to get a real field value. So the field is conventionally taken as the real part of what you show. $\Re \{e^{i(kz−\omega t)}\}$ = $\cos(kz−\omega t)$. When there is a phase change $\delta$, there will be both sinusoidal and cosinusoidal components (usually called in-phase and quadrature components): $\Re \{{e^{i(kz−\omega t +\delta)}}\} = \cos(\delta) \cos(kz−\omega t) - \sin(\delta) \sin(kz−\omega t) $ (apologies if I have a sign wrong in that equation).

For the general field, you will have both the $\cos(kz−\omega t)$ and $\sin(kz−\omega t)$ components, and it's much easier to describe them as complex quantities than to keep track of both separately. This is especially true when you need to take derivatives, and the derivative is just a multiplication. e.g. $d/dt(e^{i(kz−\omega t+δ)}) = -i\omega e^{i(kz−ωt+δ)}$. Otherwise $d/dt$ changes $\sin(kz−\omega t)$ to $-\omega \cos(kz−\omega t)$, and vice-versa (with a sign change).

(I can't figure out how to get the exponentials to show nicely like yours, but hopefully it's clear enough.)

Answer 2

I will not return to the well-known interest of complex representation for scalar quantities and I will insist in a little more detail on the representation of vector quantities.

I start from the example of the electric field of a progressive sinusoidal plane wave in vacuum propagating along $z$. Its most general form will be :

$\vec{E}=E_{0x}cos(\omega t -k z+\phi_x) \vec{e_x}+E_{0y}cos(\omega t -k z+\phi_y) \vec{e_y}$.

The quantities $E_{0x}$, $E_{0y}$, $\phi_x$ and $\phi_y$ determine the polarization of the electric field.

This real field is naturally the real part of the complex quantity : $\underline{\vec{E}}=E_{0x}e^{(\omega t -k z+\phi_x)} \vec{e_x}+E_{0y }e^{(\omega t -k z+\phi_y)} \vec{e_y}$

Which we can also be written : $\underline{\vec{E}}=(E_{0x}e^{j\ \phi_x} \vec{e_x}+E_{0y}e^{j\phi_y} \vec{e_y})e^{j(\omega t -k z)}$

So, the complex amplitude vector $\underline{\vec{E_0}}=E_{0x}e^{j\ \phi_x} \vec{e_x}+E_{0y}e^{j\phi_y} \vec{e_y}=\underline{E_{0x}} \vec{e_x}+\underline{E_{0y}} \vec{e_y}$ appear naturally.

The notation becomes very compact again, almost as simple as with a scalar quantity: $\underline{\vec{E}}=\underline{\vec{E_0}}e^{j(\omega t -k z)}$

But the price to pay is that the amplitude vector has complex components and should not be imagined as having a real direction!

These complex vectors must be handled with great care. For example, we naturally define a scalar product which is an extension of the scalar product for real vectors but which in the end is no longer a true scalar product. for example, the vector $\underline{\vec{E_0}}=\vec{e_x}+j \vec{e_y}$ has a scalar square which is zero while the vector is non-zero!

The simple case of rectilinear polarization corresponds to the situation for which the complex vector has a real direction. For this, the two complex components must be in a real relationship: $\underline{E_{0y}} =\alpha \underline{E_{0x}}$ with $\alpha$ real. We then have: $\underline{\vec{E_0}}=\underline{E_{0x}}(\vec{e_x}+ \alpha \vec{e_y})=\underline{E_{0x}} \vec{u}$ with $\vec{u}$ a real vector. The complex electric field is now written $\underline{\vec{E}}=\underline{E_{0x}} \vec{u}e^{j(\omega t -k z)}=E_{0x}\vec{u}e^{j(\omega t -k z+\phi_x)}$ and his real part is simply $\vec{E}=E_{0x}cos(\omega t -k z+\phi_x)\vec{u}$ which effectively corresponds to a rectilinearly polarized field.

But this is not general. For example, it is easy to verify that an amplitude of the form $\underline{\vec{E_0}}=\underline{E_{0x}}(\vec{e_x}+ j \vec{e_y})$ corresponds to circular polarization.

There is still much to say about these complex vectors. In particular, we may be dealing with complex wave vectors $\underline{\vec{k}}$ whose interpretation also requires a little thought. For example, the relation $\underline{\vec{B}}=\frac{1}{\omega}(\underline{\vec{k}} \times \underline{\vec{E}})$ does not mean that the real vectors $\vec{B}$ and $\vec{E}$ are orthogonal.

Thess complex vectors are called bivectors. A simple discussion of this topic can be found in Gibbs's very famous article on vector analysis : Gibbs. Scientific papers. Vector analysis. p 84

Hope it can help and sorry for my poor english.