## PHY450H1S. Relativistic Electrodynamics Tutorial 4 (TA: Simon Freedman). Waveguides: confined EM waves.

Posted by peeterjoot on March 14, 2011

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# Motivation

While this isn’t part of the course, the topic of waveguides is one of so many applications that it is worth a mention, and that will be done in this tutorial.

We will setup our system with a waveguide (conducting surface that confines the radiation) oriented in the direction. The shape can be arbitrary

PICTURE: cross section of wacky shape.

## At the surface of a conductor.

At the surface of the conductor (I presume this means the interior surface where there is no charge or current enclosed) we have

If we are talking about the exterior surface, do we need to make any other assumptions (perfect conductors, or constant potentials)?

## Wave equations.

For electric and magnetic fields in vacuum, we can show easily that these, like the potentials, separately satisfy the wave equation

Taking curls of the Maxwell curl equations above we have

but we have for vector

which gives us a pair of wave equations

We still have the original constraints of Maxwell’s equations to deal with, but we are free now to pick the complex exponentials as fundamental solutions, as our starting point

With and this is

For the vacuum case, with monochromatic light, we treated the amplitudes as constants. Let’s see what happens if we relax this assumption, and allow for spatial dependence (but no time dependence) of and . For the LHS of the electric field curl equation we have

Similarly for the divergence we have

This provides constraints on the amplitudes

Applying the wave equation operator to our phasor we get

So the momentum space equivalents of the wave equations are

Observe that if , then these amplitudes are harmonic functions (solutions to the Laplacian equation). However, it doesn’t appear that we require such a light like relation for the four vector .

# Back to the tutorial notes.

In class we went straight to an assumed solution of the form

where . Our Laplacian was also written as the sum of components in the propagation and perpendicular directions

With no dependence in the amplitudes we have

# Separation into components.

It was left as an exercise to separate out our Maxwell equations, so that our field components and in the propagation direction, and components in the perpendicular direction are separated

We can do something similar for . This allows for a split of 1.14 into and perpendicular components

So we see that once we have a solution for and (by solving the wave equation above for those components), the components for the fields in terms of those components can be found. Alternately, if one solves for the perpendicular components of the fields, these propagation components are available immediately with only differentiation.

In the case where the perpendicular components are taken as given

we can express the remaining ones strictly in terms of the perpendicular fields

Is it at all helpful to expand the double cross products?

This gives us

but that doesn’t seem particularly useful for completely solving the system? It appears fairly messy to try to solve for and given the propagation direction fields. I wonder if there is a simplification available that I am missing?

# Solving the momentum space wave equations.

Back to the class notes. We proceeded to solve for and from the wave equations by separation of variables. We wish to solve equations of the form

Write , so that we have

One solution is sinusoidal

The example in the tutorial now switched to a rectangular waveguide, still oriented with the propagation direction down the z-axis, but with lengths and along the and axis respectively.

Writing , and , we have

We were also provided with some definitions

\begin{definition}TE (Transverse Electric)

.

\end{definition}

\begin{definition}

TM (Transverse Magnetic)

.

\end{definition}

\begin{definition}

TM (Transverse Electromagnetic)

.

\end{definition}

\begin{claim}TEM do not existing in a hollow waveguide.

\end{claim}

Why: I had in my notes

and then

In retrospect I fail to see how these are connected? What happened to the term in the curl equation above?

It was argued that we have on the boundary.

So for the TE case, where , we have from the separation of variables argument

No sines because

The quantity

is called the mode. Note that since an ampere loop requires since there is no current.

Writing

Since we have purely imaginary, and the term

represents the die off.

is the smallest.

Note that the convention is that the in is the bigger of the two indexes, so .

The phase velocity

However, energy is transmitted with the group velocity, the ratio of the Poynting vector and energy density

(This can be shown).

Since

We see that the energy is transmitted at less than the speed of light as expected.

# Final remarks.

I’d started converting my handwritten scrawl for this tutorial into an attempt at working through these ideas with enough detail that they self contained, but gave up part way. This appears to me to be too big of a sub-discipline to give it justice in one hours class. As is, it is enough to at least get an concept of some of the ideas involved. I think were I to learn this for real, I’d need a good text as a reference (or the time to attempt to blunder through the ideas in much much more detail).

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