Measuring the Shortest Events Ever Created 
by Andrew Kerr
August 2007
Q - So you can measure very fast events. How do you do that?
Dr. Rick Trebino's lab is measuring the shortest events ever produced. They are ulrashort laser pulses, a few millionths of a billionth of a second long. They're faster than the fastest stopwatch ever created. So how do you measure them? I sat down with Rick in order to find out.
A - To measure some event in time—say, a bullet going through a card—you need a shorter event. Your eye is not that shorter event because your eye can only resolve events more than a hundred milliseconds long. But a strobe light is a shorter event that can resolve these fast events. With it, you can see what's happening to the card as the bullet goes through it. But let's say we want to know what that strobe light pulse looks like. We need a shorter event still. That shorter event might be a light detector of some sort, a camera with a shutter speed that's faster than the duration of the strobe light. But what if we want to know about the shutter speed? We need another shorter event. It goes on and on. Eventually, you get to the shortest event ever created.
What are the shortest events ever created? They are pulses of laser light—ultrashort laser pulses. It's routine to generate pulses shorter than about a picosecond, which is a decimal point, eleven zeros, and a 1. We can create pulses a few femtoseconds long. A femto is 10-15 or a decimal point, fourteen zeros, and a 1. This pulse is to one minute as a minute is to the age of the universe. Or if you prefer to think in terms of money, this pulse is to 1 second as 5 cents is to the US national debt.
Even though they sound quite exotic, there are all kinds of applications for these very short pulses - in physics, chemistry, biology, and engineering. An ophthalmologist can buy an eye surgery device that uses ultrashort pulses. Micro-machinists (people who cut stuff out on a very fine scale) can get better accuracy and more detailed work from an ultrashort pulse laser than from anything else. There are microscopes out there that require such very short pulses. There must be a thousand biologists that have multi-photon imaging set-ups in their labs generating beautiful images that they could never see otherwise.
Right now, in your computer, you've got billions of bits per second happening. Those are pulses of electrons less than a nanosecond, or a billionth of a second, long, so that's pretty short. People would like them to go faster. Anytime the internet is slow it's because these pulses aren't going fast enough. In fact, when you get movies sent to your house by the Dish Network, those are pulses of light coming to your house. Those aren't as fast as what we're considering, but we try to be a few orders of magnitude ahead of everybody, and someday, such short pulses will be used for your internet connection, and complete HD movies will arrive to your home in less than a second.
Okay, so how do you measure these femtosecond pulses of light? Well, in principle, you can't do it because you don't have a shorter event. The shortest event you have available is the event itself. But it's a start, and so the first trick we use is to use the event to measure itself.
You probably think that using an event to measure itself is pretty far out, and, if so, you're not alone. Sitting next to me on a plane once was the wife of William Shatner ("Captain Kirk"). I told her what I've told you, and when I got to this point, her face just went blank. She said, "Stop it! You're blowing my mind!" And I was thinking, "Here's the wife of Captain Kirk, whose husband fought Klingons and Romulans, went faster than the speed of light, went back in time, transported himself to planets, and she can handle that, but my work—that was just too much!
Q - It's too much for me, too! How do we do that?
A - You have a mirror that only reflects a fraction of the light. We see that everyday in our windows when some of the light gets reflected off the window and some goes through. We have mirrors that have 50% reflectivity. Therefore, these mirrors can create [from one original pulse] two pulses that are identical. You can then reflect them off other mirrors, moving one mirror around to delay one of the pulses with respect to the other. When two pulses overlap in space and time in a "nonlinear-optical crystal," they can affect each other and actually create new colors of light. For example, if two red pulses overlap in such a crystal in space and time, then out comes a blue pulse. And then you can see how much of that blue pulse you get. And if you vary the delay of one pulse, eventually they stop overlapping in time, which tells you how long the pulse is.
But it turns out that's not even good enough, because that just tells you kind of roughly how long the pulse is. What we really want to know is whether the pulse is one pulse, two pulses, or a pulse that stays constant for a while then comes down. Does it have a lot of little satellite pulses coming after it? It's all happening so fast our eyes completely fail us. So we want to know what the intensity versus time is.
That's still not everything. The pulse can change color with time. You've seen laser light shows where a beam changes color. That information is in what's called the "phase" of the pulse. Is the pulse red at the beginning, green in the middle, blue at the end? Is it red, then blue, then green? Is it blue, then green, then red? All those things can happen. Nothing is stopping it from happening. In fact, what these lasers do depends sensitively on tiny misalignments of mirrors in them. You need some kind of feedback as to what you've done. So we need to know the intensity and the phase of the pulse vs. time, that is, the intensity and the color versus time.
Even in the 1990s, methods for measuring these pulses remained fairly primitive. I came along and said, "I'm gonna figure out how to do this." I said, "Let's make a spectrogram of the pulse." A spectrogram is like a musical score. It's a plot of the color, or frequency, versus time.
This is the pulse we're trying to measure, and I've drawn it in color to indicate that it's going from red to green, and that's called "chirp." It's called chirp because, as you know, when a bird sings its song (chirps), it sings words that go from low frequency to high frequency. Since the frequency of a light wave is its color, physicists have adopted this term for light waves, too.
Q - The chirp itself is a slurring-type thing.
A - Here's our chirped pulse, which goes from red to green, and then we've got another pulse, and it's going to come through and it's going to multiply that pulse in that nonlinear medium. A pulse is going to come out, which is the product of those two.
Q - What sort of math are you using to do these kinds of calculations? Is it calculus?
A - It's calculus, it's complex numbers, and it's also the same kind of math that scientists were using to deblur the Hubble Space Telescope when it was blurry.
It also uses what's called the Fundamental Theorem of Algebra, which you may have learned about. The Fundamental Theorem of Algebra says if you have a polynomial, like x2 + 2x + 1, you can factor it into the product of factors, here (x+1)2. But what happens if you have a polynomial with two variables in it—not just x, but x and y? So you could imagine x2 + 2xy + y2. You can factor that. That's (x + y)2. But what if you have some weird polynomial with two variables? If you get a polynomial with two variables, like say x2 + xy + y2, most of the time you won't be able to factor it (x2 + 2xy + y2 is a really unusual case). When you learn in high school that you can factor a polynomial, your teacher never tells you about what happens if you have polynomials with two variables, and no one ever thinks to ask. One of the reasons why your teacher doesn't tell you about it and why your teacher was probably really glad you didn't ask is that there isn't a Fundamental Theorem of Algebra for polynomials with two variables. It doesn't work. Most polynomials with two variables can't be factored!
Now, the [graph of] intensity versus time is a curve, and the [graph of] color versus time is also a curve—it's just a plot vs. one variable, time. And what we've done is made spectrograms that are plots of intensity vs. time and frequency. We took this problem, which was a one-dimensional problem and made it into a two-dimensional problem. Now, it turns out that our problem has a connection to the Fundamental Theorem of Algebra for polynomials. Previous methods that didn't involve spectrograms made plots with only one variable (time), and they were mathematically equivalent to the Fundamental Theorem of Algebra for polynomials of one variable. Our spectrograms, with two variables, correspond to the Fundamental Theorem of Algebra for polynomials of two variables. It turns out that the existence of the 1-D Fundamental Theorem of Algebra makes it impossible to extract out information about the pulse from previous measurements using only one variable. And the nonexistence of the 2-D Fundamental Theorem of Algebra actually allows us to measure pulses from spectrograms. It's the failure of it that makes it work!
Q - Algebra's just kind of romantic; it's an elegant thing. I think algebra teachers would be thrilled to learn that there are still real-life applications of it in cutting-edge research.
A - For us to do anything, algebra is involved, trigonometry is involved—everything I learned in high school math class is involved. All that stuff that you thought was boring that you were never going to use because you weren't using it in your daily life in high school becomes really interesting and relevant when you want to do something really interesting with it.
Q - Is most of your research theoretical at first, and then you build something afterward to test it?
A - We'll usually think of something in our heads, we'll have a discussion, something will come out of it, and the math will look good, so then we'll go try it. Then we'll go look at the results, and we'll go, "Ehh, not quite the way we want it." We use every trick that we can—we do calculations that could be algebra, calculus, or just arithmetic. We do computer simulations (we let the computer figure it out for us), and we go in the lab and we try it. We do everything we can, and all that stuff is important.
Everyone in my group is a theorist and an experimentalist at the same time. Sometimes in some fields in physics and science you have just theorists and just experimentalists. The theory is so hard that you have to be a full-time theorist to be able to do it. In some cases the experiments are so hard you just have to be a full-time experimentalist to do it. The theorist collaborates with the experimentalist.
In what we're doing, neither is that hard. The theory keeps you connected to what should happen, makes you understand stuff. Doing some experiments keeps you connected to reality. Sometimes, as a theorist, you could go off and consider some interesting effect that you've discovered that you know could happen. But there might be other effects that are much bigger than it that you sort of neglect because you're so interested in this one effect. Getting in the lab shows you what those other effects are; it keeps you grounded in reality. But the theory is also important because you really want to understand what it is that you're doing.
Q - Digging back to your own high school days—were you more fond of mathematics, were you more fond of physics, or were you into something else entirely and stumbled into this field?
A - I liked math. But in grade school I found word problems difficult. As a result of that, I was intrigued by them. I hated them, but I was also intrigued because I thought that I should be able to do them. When I became a physicist I realized that my whole life is word problems. And it wasn't just solving word problems; it was also making them up and then solving them. I'm drowning in word problems now, but that's really what science is all about. Even life.
I just installed lamps in my kitchen. We decided to do six of them in a spiral configuration. A handyman helping me with it said, "OK, how do we do it?" I was realizing, "Okay, we need to solve two equations in two unknowns to get it just right." Otherwise we'd just be poking holes in the ceiling everywhere. I also did a little geometrical calculation for him based on the idea that a hexagon is six equilateral triangles. That allowed us to determine the distance between most of the pairs of lights, which was x, and the distance between other pairs was 2x. We placed all the lights at the vertices of a hexagon. Then the question was, "What about some other distances?" Well those ended up being the square root of 3x/2, and a little trigonometry came in handy there. The lights look really nice now, and we have exactly the numbers of holes in the ceiling as lights.
Q - Proof that when people say "I won't use trig after graduation," that's really only because they're not seeing where they might apply it.
A - If you say you won't use trigonometry, I suppose you wouldn't, but then you'd not accomplish what you might be able to if you did use it.
Q - I read on your site that when you began doing laser work as a graduate student the pulses were much longer than they are today.
A - When I was a grad student, the shortest pulses were just shorter than a nanosecond—a billionth of a second—long, but that's really long to us now, and many applications that are now possible weren't then. If people then had said that a nanosecond is short enough, then we wouldn't have shorter pulses or all those cool applications now.
Q - Yeah, that's an eternity! (laughing)
A - In my grad school career, I think I used about a minute of laser light. In the course of their whole graduate careers, my grad students will probably use no more than about a second of laser light. But that's still a lot of pulses! Whatever the case, we'll certainly know them a lot better now than ever before.