The Doppler effect in a bit more detail

Let’s talk about special relativity.

This is a follow-up post giving a more detailed explanation of the Doppler effect in free-electron lasers. The original post is here. If you want this to flow well, imagine replacing the text of the “The Doppler effect and the free-electron laser” in the original post with this.

Everything is relative

We know how particles emit light if they are wiggling in place somewhere, but our electron is now moving near the speed of light in a certain direction. We’re cautious to jump straight to using the Doppler formula from sound because we’re afraid of some relativistic effects we might not foresee — maybe the act of moving near the speed of light and exposing yourself to relativistic effects fundamentally changes how the system behaves. We’ll find out that that isn’t true, but let’s convince ourselves using what we know for sure.

Here, Professor Einstein come to save the day. He tells us that everything in life is relative — the following statements are both equally valid:

1. We are standing at the end of the undulator, and an electron is flying towards us at some speed as it wiggles.
2. An electron is wiggling in place, and we are flying towards it at that same speed.

That’s a big deal! We don’t know how to deal with the first situation, but the second situation is well within our grasp! We call the second situation the “quasi-rest frame” of the electron, since in that frame it’s motion is just the wiggling in place with none of the near speed of light velocity from the accelerator. For brevity I’ll just call it the rest frame. Hopefully this poorly drawn picture helps to make it clear what the two cases look like:

Two ways of thinking about us observing radiation from the fast moving electron. Case #1: the electron moves quickly towards us, we are standing still. Case #2: the electron sits and wiggles in place, we sprint towards it near the speed of light.

In the rest frame, the electron is wiggling in place at some frequency, let’s call it fᵣₑₛₜ which is not necessarily the same as the frequency it wiggles with in the original frame (case #1, often called the “lab frame”). Thus the light emitted by the electron in the rest frame has the same frequency as its wiggles: fᵣₑₛₜ. But we aren’t sitting in place observing that light — we’re sprinting at the electron at almost the speed of light. As a result, we’re going to see the light with a Doppler shift, similarly to the case of the ambulance for sound.

Since we’re running towards the source of the waves, we’re going to see them at a higher frequency. It turns out that for light and relativistic motion, that Doppler shift means that we see the light at a frequency that is the rest frame frequency multiplied by 2γ, where γ is a quantity that shows up in relativity all the time called the Lorentz factor. For an electron whose motion towards us had a speed v, the Lorentz factor is defined by γ²=1/(1-v²/c²), where c is the speed of light.

If you want some intuition about this magic value of 2γ, it turns out that in this situation (speed very close to the speed of light), we can approximately use the same Doppler formula that we had for an observer moving towards a source of sound waves, only now we have to multiply by the Lorentz factor γ because of relativity (time dilation, to be precise): the Doppler formula would have been f=f₀(vₛ + vₒ)/vₛ with f₀ the original frequency, vₛ the speed of the wave, and vₒ the speed of the observer (us), but in this case both vₛ and vₒ are roughly the speed of light c, so f=2f₀. Tacking on the γ for time dilation, we get f=2γf₀.

This is good stuff, we’re close to the final answer but we still have one mystery left to solve: what frequency does the electron wiggle with in its rest frame? To figure that out we’ll need to invoke one of the wonky features of special relativity: length contraction.

The wiggle frequency in the rest frame

What determines the frequency of the electron’s wiggles in its rest frame? In the lab frame the frequency is determined by the undulator magnet: the undulator is set up with some periodicity (meaning that we go through a pair of alternating magnets every undulator period), and the electron wiggles with that same periodicity. In particular, if the undulator period is λᵤ, and the electron beam moves basically at the speed of light c, the frequency of its wiggles in the lab frame is c/λᵤ. What about the rest frame?

Comparison of the picture of the FEL in the “lab frame” (where the electron moves, and we and the undulator stand still), and the “rest frame” (where the electron is at rest, and we and the undulator sprint towards it.

I’ve drawn the basic picture of what we’re looking at in the “lab frame” (where we and the undulator are stationary) and in the “rest frame” (where the electron is stationary) in the cartoon above. In the rest frame the electrons sit at rest waiting to be wiggled around by the undulator, which is barreling towards the poor electron at nearly the speed of light. Here we must invoke one of the strange effects from Einstein’s theory of special relativity: length contraction. In the rest frame of the electron, the undulator looks shorter than it does to us in the lab frame! Not only is the undulator as a whole shorter, but also every period is shorter. That means that instead of wiggling at the frequency c/λᵤ like it did in the lab frame, the electron in its rest frame wiggles with a frequency that is γ times higher! In other words, our fᵣₑₛₜ=γc/λᵤ.

Summary of the long answer

We’ve made a lot of conceptual hurdles in the last few sections, so let’s take a second to tie things together and craft the story of how the electron flying towards the undulator ends up emitting light, and what frequency that light has. In steps it goes something like this:

1. An electron is produced by an accelerator with a large velocity v close to the speed of light c. For later convenience, it has a Lorentz factor γ defined by γ²=1/(1-v²/c²).
2. The electron approaches the undulator which has some period λᵤ. In the electron’s rest frame, the undulator looks γ times shorter thanks to length contraction. Correspondingly, once the undulator reaches the electron in the rest frame, the electron starts wiggling with a frequency γc/λᵤ.
3. Still in the rest frame, the electron is wiggling so it starts emitting light. In the rest frame that light has a frequency of γc/λᵤ, just like the electron’s wiggles.
4. We want to see the light, so in the rest frame we start sprinting towards the electron at the same speed v and with the same Lorentz factor γ. Because we’re moving towards a source of waves, we see them with a higher frequency, in this case 2γ times the original frequency, or 2γ²c/λᵤ.

There you have it. In the lab frame, the electrons wiggle up and down with a frequency c/λᵤ, but because of the magic of the Doppler effect they emit radiation 2γ² times higher! To understand how amazing that is, you should know that electrons in our accelerator at SLAC generally have Lorentz factors around 10000. That means that the frequency of the light it emits is 200,000,000 higher than the frequency of the undulator!