Imaginary components of Poynting vector in rectangular waveguide

Question

Consider a rectangular wave guide, as in the example of Jackson - Classical Electrodynamics

The Poynting vector turns out to be $$\vec{S}=\epsilon c^2 \vec{E_0} \times \vec{B_0} e^{2i(kct-\alpha z)}$$

Where (that's the important part): $$\vec{E_0} \times \vec{B_0}=\hat{i}\bigg(iC^2\lambda^2\epsilon_r\mu_r \frac{k}{c}\frac{m \pi}{a} \mathrm{sin}\frac{m\pi }{a}x \mathrm{cos}\frac{m\pi }{a}x \mathrm{sin^2}\frac{n\pi }{b}y\bigg)+\hat{j}\bigg(iC^2\lambda^2\epsilon_r\mu_r \frac{k}{c}\frac{m \pi}{b} \mathrm{sin^2}\frac{m\pi }{a}x \mathrm{sin}\frac{n\pi }{b}x \mathrm{cos}\frac{n\pi }{b}y\bigg)-\hat{k}\bigg\{C^2\alpha\epsilon_r\mu_r \frac{k}{c}\bigg[(\frac{m \pi}{a})^2 \mathrm{cos^2}\frac{m\pi }{a}x \mathrm{sin^2}\frac{n\pi }{b}y+(\frac{m \pi}{b})^2 \mathrm{sin^2}\frac{m\pi }{a}x \mathrm{cos^2}\frac{n\pi }{b}y\bigg]\bigg\}$$

With $m$ and $n$ integers that determine also $\lambda^2$, which is the eigenvalue.

I would like to understand the meaning of imaginary trasversal components of $S$. As stated in Jackson:

The transverse component of $S$ represents reactive energy flow and does not contribute to the time-average flux of energy.

I do understand the meaning of reactive energy, as an analogy with AC circuits, where power has indeed an imaginary reactive component.

But I do not see why mathematically the flux of energy from these components should be zero only because of the fact that they are imaginary. Here are some point I would like to clarify:

• We are talking about $x$ and $y$ components of a vector, hence clearly the flux through a surface perpendicular to $z$ axis is zero, so what "flux" is to be consider in order to have a quantity that is not trivially zero only for the orientation of the discussed components?
• Mathematically I can see that $i$ in front of $x$ and $y$ components as $e^{i \frac{\pi}{2}}$, hence as a shift in the phase, which has not much to do with the fact that the flux vanishes, apparently. Moreover, the time dependence of the components is the same for all of them and does not average to zero over time (the complex exponential gives a $\sin^2$ or $\cos^2$ which averages to $\frac{1}{2}$). So how does the flux vanishes mathematically?
• Is the "flux" here to be interpreted as the "average" on the surface of $S$ (except for a factor equal to the area itself)?

Plotting for $m=n=1$ and $a=10$ and $b=5$ the function that gives the (normalized) first component : $$S_x(x,y)=\mathrm{sin}\frac{\pi }{10}x \mathrm{cos}\frac{\pi }{10}x \mathrm{sin^2}\frac{\pi }{5}y$$

It looks clear that this component should "average to zero over the surface" (there is a positive and a negative part), but is this "average" to be done mathematically with a flux?

If so, again, the main question is : which flux (or average, over space or time) should one consider in order to see why the imaginary components do not contribute to energy propagation and how this flux vanishes?

Answer 1

Keep in mind that the physical electric and magnetic fields are real-valued. They are often expressed as complex functions for computational convenience, but then the physical interpretation of what is happening becomes complicated. A mathematically consistent way to define the electric and magnetic fields as complex-valued fields is with the aid of phasors. A phasor electric field is a complex field, which, when multiplied by $\exp(i\omega t)$, the real part would give the physical electric field $$ {\bf E}({\bf x},t) = {\rm Re}\{ \exp(i\omega t) {\cal E}({\bf x})\} . $$ Similar for the magnetic field. This obviously requires that the electric field is monochromatic. In case the system under investigation is linear one can always separate any electric field into monochromatic parts and express it in terms of a phasor field. Note also that the time dependence dropped out.

In terms of the phasors, one now finds that the Poynting vector is given by (ignoring some proportionality constants) $$ {\cal S} = {\cal E}\times{\cal H}^* . $$ One can see that the effect of the complex conjugation is actually to produce an averaging over one period of oscillation in time.

Why is this averaging incorporated into the definition of the Poynting vector? It is because the oscillation of the field over one period may involve an exchange in energy/power in the field that does not contribute to the propagation of the field. This exchange contributes to what is called the reactive field, often found in antenna theory. The averaging removes this ineffectual exchange, leaving only that part that should contribute to the propagating part of the flux.

Back to the more general way of using complex fields. One can now try to understand what the imaginary parts of these expressions for the flux mean. As you rightly pointed out, the $i$ represents a phase shift. This implies a relative phase shift between different parts that would then destructively interfere when one computes the propagating part of the flux.

So the imaginary part of the Poynting vector in the transverse direction inside a waveguide indicates power that oscillates back and forth without contributing to the propagation of the field along the waveguide.