In the last post I made an offhand mention of wave dispersion, which is the phenomena of different wavelengths propagating a different speed. In general this does exactly what it sounds like it should. It disperses the light. If you start off with a tightly grouped bunch of runners, the pack will spread out as soon as they start running and the fast ones pull away and the slow ones get left behind.
If you're a physicist and your paycheck comes from making tightly-bunched pulses of light, this is a pain in the neck. You perform miracles of science with beautiful and tremendously expensive lasers and optical equipment and the moment that pulse interacts with glass or water is starts smearing out into as much broader and more mundane pulse. In fact the tighter your initial pulse is the greater the spread in its frequencies and thus the more it spreads out when in a dispersive material. It's a vicious cycle.
But scientists are nothing if not resourceful and so they've developed some ways to compensate for dispersion. Think about the pack of runners again. Imagine you start with the runners spread out loosely at the start of the race with the slow people in front and the fast people at the back. In that case dispersion will actually tighten the pulse. It's not perfect; eventually the pulse will start spreading out again after the fast people pass up the slow ones. But it's an improvement.
This uneven frequency spread is called chirp, because in some ways the frequency pattern resembles that of certain birds. Yesterday I spend a lot of time fighting with writing a simulation in Mathematica to demonstrate this effect for a presentation, and I figure it might as well do double duty there and on the blog too. Without further ado, the electric field of a chirped pulse in a dispersive medium:
You can see the faster waves in the back catch up and compress the pulse as the move to the front. Generally it's not possible to directly measure the electric field, instead we see effects resulting from the energy density of the light pulse. This energy density is called the intensity, and it's proportional to the square of the field. It spikes very sharply when the pulse tightens:
The speed with which the pulse distorts is exaggerated. For a pulse of a few femtoseconds duration the spatial extent of the pulse will be in the micron range. The distance required to substantially change the shape of the pulse is typically in the millimeter to centimeter range.
You can imagine all sorts of speculative uses for this kind of phenomenon. To pick one at random, by controlling the degree of chirp it could be possible to shine a medical laser through human tissue, with the intensity low enough to be harmless until the dispersion of the body brings the pulse to a compressed high intensity at a specific tumor depth. In fact a similar principle has been used in radar for years to improve the sensitivity and resolution of aircraft sensors. There's likely to be a lot more interesting work to come.
[Personal matters will likely keep me away from the web (and indeed out of state) this weekend. If so, Sunday Function will as usual be rescheduled for Monday. - Matt]
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Yer gonna trust an ex-premed with a "harmless" laser beam that propagates through tissue then chirps into an explosion? This device was not tested on animals! Swell. The training bench will be as always - warehoused demented old people.
As an interesting note, this is analogous to the approach that we use for almost all MR imaging; it's not a spatial dispersion, but the principle is the same.
Ooh ooh tell us about anomalous dispersion and ftl travel and those spoilsports who say it doesn't quite work like that!
Back in the day, I had a 16 fs FWHM pulse impinging on a decorated metal interface. We were doing time-correlated spectroscopy of surface plasmon propagation.
I concur on the wrangling necessary to keep these suckers in phase.
Matthew the Springer, your intelligence astounds me. I frequently read your blogs, but I seldom fully comprehend them. I cannot comment on any of your math or science, because I'd make a fool of myself. However, I will comment on a typo!
"You can see the faster waves in the back catch up and compress the pulse as the move to the front."
"the move" should be "they move"
I feel as though I've made some sort of valuable input to your education and to science in general with this comment. Thank you.
-Cy-fi
Clearly you need to make a post out of the news about the Cowboy's new stadium and the subject of "4.9 second punts" as a measure of height.
I suppose the formula for the shape of a chirp could be worked out from propagating a unit impulse in a dispersive medium.
1 - express a unit impulse as a sum of sine waves (ie: a fourier transform)
2 - move the waves to the right by a distance according to their frequency (and time t, obviously)
3 - add 'em back up again.
Is the velocity of a wave in a dispersive medium proportional to its frequency, or does it depend on the meduim - some having weird properties?
I thought 'chirp' came from the radio people; when it happens you really hear a chirp.
@Paul Murray: The velocity through a medium can be calculated from the complex refractive index (which will give you velocity and attenuation). Anything can happen really; velocity can increase or decrease with frequency over relatively narrow spectral regions so there is just no substitute for making measurements. If you want a general rule (which of course is probably broken in all cases) the higher frequencies will travel slower.
You were planning on linking you simulation code, right?
Paul: In fact that's not far from how I did it. I take the initial condition shape of the wave I want, fast-Fourier transform it, tack on the time-dependent phase in the dispersive medium, and inverse transform that. That gives a plottable time-dependent function, if in table form.
travc: The code is currently an unreadable hacked-together mess, with much of the meat cribbed from this tutorial anyway. But I think you're right, once I get it neater (and I will, I'm adapting it for bona-fide research) I'll release it.
In a class we did simulations with dispersion and nonlinear pulse shaping. I think you should do that next... instead of wasting time you know sleeping.