Ambulances and x-ray lasers
Why an ambulance siren changes pitch after it drives past you, and how the same effect enables the world’s most powerful x-ray lasers.
Have you ever heard of the Doppler effect? You’ve almost certainly experienced it, though you may not have noticed it. The most common example is when a car with a loud siren drives by you — like an ambulance on its way to deal with an emergency.
If you’re standing on the side of the road and an ambulance using its siren drives by you, something funny will happen. As the ambulance approaches you, you’ll hear its siren wailing at a certain pitch. But as soon as it passes and starts moving away from you, you’ll hear it wailing at a lower pitch (see for example this Youtube video).
This is the Doppler effect in action, and in this article I’ll talk about why it happens, and how this same effect enables the most powerful x-ray lasers in the world.
Understanding sound waves
Before we can talk about the Doppler effect we need to first get up to speed on some of the basic features of light and sound — both of which are waves. Broadly speaking, a wave is something that travels through some medium (air, water, the vacuum, et cetera) with a certain speed, and exhibits some kind of periodicity (it repeats after a fixed amount of time). What does that mean? To understand let’s go back to our ambulance example
The ambulance is sitting parked with its siren blaring. The siren emits a “sound wave” that starts at the ambulance and moves outward — this is represented by the rings in the picture. Waves oscillate up and down, in the picture the rings represent points where the wave is at its maximum — in between the rings it hits its minimum. The number of times it peaks in a second is called the frequency of the wave. For example, we see four peaks in the cartoon above (indicated by the rings), if the first one was emitted 5 milliseconds ago, the frequency of this wave would be 4/(5 ms) = 800 Hz.
As we noted, that wave also travels with some speed. In many places (it depends on the air temperature and pressure), the speed of sound in air is around 343 meters/second (or 767 miles per hour or 1234 kilometers per hour). Looking back at our cartoon, it means that one of those rings will travel 343 meters in one second. That means that when something makes a sound, it takes time for you to actually hear it. We don’t notice this so much in our daily life, except when lightning strikes: you see the lightning almost immediately, but you only hear the “thunder” a few seconds later.
Before I move on, I wanted to note that we also often talk about the “wavelength” of a wave — that’s how much physical space there is between two maxima (two rings, in the picture). If you know the speed and the frequency, the wavelength is speed/frequency.
With these basics, we’re ready to ask the question: what happens when the ambulance starts driving?
The Doppler effect for sound
Let’s think about what happens right after the ambulance emits its first peak. That wave starts moving outward from where it was emitted, as in the first picture. But in the mean time, the ambulance moves away from where it emitted the first peak: it gets closer to one side of the ring and further from the other! You can see this better in this cartoon:
where now we’ve got a little guy there ready to hear the siren for us. Each subsequent sound wave is emitted a little further to the right than the previous one, so on the right side the rings looks closer together while on the left they look further apart. So our friend up there, who is on the right side of the ambulance, will actually receive those rings more frequently than if the ambulance was stationary. For our friend, this means a higher pitched sound (more rings per second).
After the ambulance passes by our friend, we have instead this picture
Now that the ambulance is moving away from him, the waves that eventually reach him are observed less frequently than if the ambulance was stationary. The sound is lower pitched.
To summarize it concisely: the waves the ambulance emits always move outward from where the point that they were emitted, but the ambulance doesn’t stay in the same place that it emitted the earlier waves.
This is all kind of neat, but so far it sounds like this effect should happen whenever something simultaneously makes a sound and runs around — why don’t we notice it more often? The key is that the effect is only large if the thing is moving close to the speed of the wave: in this case, around 767 miles per hour. An ambulance driving down the street might be going about 50 miles per hour, or about 7% the speed of sound — that’s big enough to hear a small shift in its frequency.
By the way, this effect will also happen if the ambulance is stationary but the observer starts running towards or away from it. It’s the same argument: if you run towards the source of the sound, you’ll encounter the waves it emits more frequently than if you were standing still. Similarly, if you run away you’ll hear a lower frequency.
For those of you who like formulas, the frequency heard by the observer, f, is related to the frequency of the ambulance at rest, f₀, by f=vₛf₀/(vₛ ± vₐ), where vₛ and vₐ are the speeds of sound and the ambulance, and we use the plus if it’s moving away from us and the minus if it’s moving towards us.
In case it is the observer moving rather than the source, the formula becomes f=f₀(vₛ ± vₒ)/vₛ, where now vₒ is the speed of the observer, and we use the plus sign if the observer is moving towards the source and the minus sign otherwise.
You might suspect that funny things might happen if the ambulance starts moving faster than the speed of sound: that’s how we get cool effects like sonic booms (see this for example), and in the world of radiation it gives us an effect called Cherenkov radiation.
From sound to light
I promised that we’d talk about giant x-ray lasers, but first we need to talk about light. Light, just like sound, is a wave. It has a frequency and a speed, but how do these things correspond to the light we see everyday? And how do we start talking about x-rays? To do that we need to remind ourselves about the electromagnetic spectrum.
What’s so important about the electromagnetic spectrum? It tells us that the light we see with our eyes is just one example of a broader thing called an “electromagnetic wave”. Blue light is a wave with a higher frequency than red light, for example. As you go to higher frequencies than we can see with our eyes, you reach ultraviolet light (like the kind that gives you a sunburn, here’s a cool video shot with a camera sensitive to ultraviolet), and past that you reach x-rays (like the kind that lets the doctor see your bones).
If it helps to have some numbers to associate with, blue light has a frequency of around 750×10¹² oscillations per second — that’s 750 followed by 12 zeros! Red light on the other hand has a frequency that’s more like 400×10¹² oscillations per second. A typical frequency for x-rays on the other hand is closer to 250×10¹⁵ oscillations per second, almost 1,000 times higher.
Just like sound, light also has a speed. The speed of light is much much larger than the speed of sound, it’s about 300,000,000 meters/second, I won’t bother converting that to miles per hour or kilometers per hour, needless to say it’s unimaginably large for us in our daily lives.
So we know now that light is a wave. But to fully understand how the Doppler effect plays a role in x-ray lasers, we need to understand how we make those waves in the first place! We need to know what our ambulance is this time.
It turns out that light is produced whenever a charged particle is accelerated, like in this cartoon:
In this cartoon, an electron (a small negatively charged particle) is wiggling up and down. “Wiggling” is just an easy way of saying that it is accelerating back and forth, so it emits waves of light — when a particle emits electromagnetic waves we say that it is “radiating” or “producing radiation”. As in the ambulance example, the rings show where the wave it emits reaches its maximum, and the number of rings emitted in one second is the frequency of the wave. In this case that wave is travelling away from the electron at the speed of light.
The frequency of the light that it emits is the same as the frequency of the electron’s wiggles. For example, if the electron wiggles three million times in one second, the light it produces will have a frequency of 3 million oscillations per second. For reference, that frequency corresponds to radiowaves like the ones you use to listen to the radio in your car (if you still do that nowadays)!
[Note: doing something three million times in one second may sound like a lot to us, but electrons are much lighter and smaller and generally operate on faster timescales.]
Now we know two basic things:
- All light, from x-rays to radio waves and beyond, can be described as a wave traveling through space with some frequency.
- When a charged particle wiggles up and down in place, it emits light with the same frequency as its wiggles.
Armed with these facts and our sonic intuition about the Doppler effect, let’s finally talk about one of my favorite topics: the free-electron laser.
What is a free-electron laser?
I’m going to start by describing what a free-electron laser is, then we’ll use the information we learned earlier to understand how it works.
I need to give a disclaimer here before I get yelled at by someone in my field: technically we’re really only going to understand how “undulator radiation” works: the free-electron laser is a whole other beast that I’ll talk about in more detail in another post. For simplicity (and because they’re so cool), I’m going to keep saying “free-electron laser” even though this isn’t completely accurate: you can think of this as just the first step in your journey to understand free-electron lasers.
Let’s start with the facts: free-electron lasers are light sources driven by a particle accelerator. They are arguably the most versatile sources of light on the planet: whereas most light sources can only work in specific frequency ranges, free-electron lasers can in principle be tuned through the entire electromagnetic spectrum if you’ve got the right tools.
These days some of their most exciting renditions are as big sources of x-rays. I plan to make another post in the near future about why we like making big x-ray sources, but I’ll give you a hint here. We use light to look at things everyday: our eyes use visible light to figure out what’s going on in the world around us. And light is used in science for tons of purposes: think about how impactful the microscopic has been to biology and medicine, or how impactful the telescope has been to astrophysics. Both of these devices let us see the world at different scales using light.
Microscopes let us see small things, like bacteria, but they have limits — they can’t resolve anything much smaller than about one micrometer (0.000001 meters). If microscopes gave us a glimpse of the micrometer world, x-ray sources like the x-ray free-electron laser let us see the world on the Angstrom scale, that’s 0.0000000001 meters, 10,000 times smaller than a micron! At that point we’re talking about the size of single atoms. With precision like that we can study atoms at a fundamental level, watch chemical reactions as they occur, and figure out the structure of the proteins that make our life saving drugs work.
So how do free-electron lasers work, and why do we need an accelerator to drive them? Well, based on what we learned in the last section you can probably guess that it’s going to work by taking the beam of particles from the accelerator and wiggling it up and down — if you do that, it’ll spit out light! To do the wiggling, we use a device called an “undulator magnet” (sometimes you’ll hear it aptly called a “wiggler” — technically, a “wiggler” is a very strong “undulator”). You can see how this might look in this picture, where an electron traveling to the right near the speed of light enters an undulator:
First of all, as the poorly drawn picture tries to convey, an undulator is a series of alternating magnets (N meaning a “north” pole, S meaning a “south” pole). When a particle flies through these magnets, it starts wiggling in response. As we now know well, that wiggling prompts it to start making light. But what is the frequency of the light?
[Note: another disclaimer here for accelerator-oriented readers. We’re making simplified arguments here to convey the basic physics, I may make later posts with more detailed discussions, but for now bear with me for the sake of of understanding.]
The Doppler effect and the free-electron laser
The short answer is what you might guess from the fact that I‘ve written this post: this problem looks just like the ambulance problem in some sense. A source of waves — this time an electron in an undulator — is simultaneously moving very quickly, so we expect to find that the frequency it wants to make is shifted by the Doppler effect. Let’s go through it in steps.
A warning to the reader: I’m going to use some simple formulas in this section but I’ll try to convey what’s going on qualitatively at all times. It’s hard to appreciate the magnitude of this effect without plugging some numbers into formulas.
If the electron was not wiggling, it would emit light at the same frequency as its wiggles. What is that frequency? It’s set by the undulator: the electron will go through one full wiggle after a pair of NS-SN magnets. In particular, if one pair of alternating magnets has a length 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.
Just like in the ambulance problem, because the electron is moving to the right towards us, we’re going to see the waves with a higher frequency. Now let’s apply our Doppler formula from the earlier section on sound. If something would emit waves with speed vₛ and frequency f₀ if it were at rest, but instead it moves at speed vₐ, the frequency we perceive is f=vₛf₀/(vₛ-vₐ). We use the minus sign here since the electron is moving towards us.
Now what are these different factors? vₛ is the speed of light: c. f₀ is the frequency of the undulator, c/λᵤ. The final thing to specify is the speed of the electron vₐ. We’re interested in using electrons that have been accelerated very close to the speed of light (we’ll see why very shortly), typically we write its speed like this: vₐ≃c(1–1/(2γ²)). γ is something that pops up in special relativity all the time called the Lorentz factor. In accelerators that drive x-ray free-electron lasers, like the one at SLAC, γ is often 10,000 or even higher! So when we say “moving close to the speed of light”, we mean it!
What does that mean for our observed frequency? If we plug things into our formula, we find that the frequency of the light that we observe is 2γ²c/λᵤ — that’s 2γ² times higher than the undulator frequency, which for typical parameters is 200,000,000 times higher! That’s a crazy upconversion, but before we talk about that let’s summarize what we’ve learned and put it in context.
[Note on this section: for those of you who have some training in special relativity, you might be uncomfortable naively applying the Doppler formula for sound to this problem — that is understandable, I was too. But amazingly, it works. I’ve made another post with a more detailed explanation using relativity finding the same answer. The really right way to figure it out is to use something called “four-vectors” or to solve Maxwell’s equations, maybe I’ll do that some other day.]
Conclusions
Now that we’ve indulged ourselves with formulas, let’s come back to the big picture. When an electron wiggles with some frequency, it normally emits light with that same frequency. But if that electron is moving in some direction close to the speed of light while it wiggles, it experiences a huge Doppler effect. The frequency of the light it emits is 2γ² times higher than its wiggle frequency, where γ for electrons in accelerators can be in the tens of thousands.
[By the way, when electrons moving near the speed of light wiggle and produce Doppler-shifted light, we call it synchrotron radiation. But don’t let the fancy new name scare you: it’s just the same old Doppler effect.]
This is extremely powerful: we can generally build undulators with few centimeter periods, and a ton of work is done to even push down to millimeters. If we didn’t know better, we might think that was disastrous because that frequency of wiggles is nowhere close to x-rays. But the Doppler effect saves us: with a centimeter scale undulator period and an electron beam moving very close to the speed of light, we can make x-rays with a wavelength of Angstroms (again that’s 0.0000000001 meters)! For some context, an angstrom is roughly the radius of the smallest atom — Hydrogen.
Amazingly, this really is exactly the same effect that makes an ambulance’s siren sound different when it’s moving towards or away from you. It’s just the good old Doppler effect. What seems like magic is really just a consequence of something emitting a wave while it moves very close to the speed of that wave.
The other amazing thing is that you might think that this simple argument wouldn’t get us very close to the real answer, but it does. The full answer is that the frequency of the light we see is given by 2γ²c(1+K²)/λᵤ, where K is a number proportional to the field of the undulator that’s typically between 0.1 and 5. That’s not so bad for a few simple arguments.
Thanks for bearing with me this far. I hope that at least some part of this taught you something new: even the basic Doppler effect for sound is interesting enough in its own right. The fact that the same basic effects lie at the heart of the world’s most versatile and powerful x-ray lasers never ceases to amaze me. If that makes you more curious about the world of accelerators and free-electron lasers, stick around!
P.S. Thanks to Janet Zhong and Rebecca Hao for help proof-reading this first substantive post :).
