Bandwidth and Community Expectations

Derek Lowe has posted an article about X-ray lasers in chemistry, which amused me because of the following bit:

Enter the femtosecond X-ray laser. A laser will put out the cleanest X-ray beam that anyone's ever seen, a completely coherent one at an exact (and short) wavelength which should give wonderful reflection data.

This is funny to somebody in my end of the science business, because we usually think of femtosecond lasers as have an extremely broad spectrum, not an "exact wavelength." It's a striking example of something I see all the time with chemists-- what chemists think of as "narrow," atomic physicists think of as "broad-band."

Who's right? Everybody...

The key question here is how to build up a short-duration pulse. If you want a beam of light that has a short duration in time, the only way to get that is through adding together a large number of different frequencies. You can think of this as a consequence of the energy-time uncertainty principle:

Δ E Δ t ≥ h/4 π

(though it almost makes more sense to saw that energy-time uncertainty is a consequence of the need to add frequencies to get short pulses). The uncertainty in time multiplied by the uncertainty in energy has to be greater than or equal to some minimum value, meaning that if you want a small time uncertainty, you need to accept a large energy uncertainty. Energy is proportional to frequency, so a large energy uncertainty means a large frequency uncertainty.

To make a 20fs laser pulse, as in one of Derek's examples, you need a bandwidth of roughly 5 x 10¹³ Hertz-- that is, the frequency uncertainty has to be at least that big, so the laser frequency is known only to +/- 5 x 10¹³ Hz. That's a huge number, roughly equal to the absolute frequency of far-infrared light.

Of course, if you're talking about a femtosecond X-ray laser, you're dealing with a much larger base frequency-- 2.9x10¹⁸ Hz for the 12 keV laser in Derek's example. The uncertainty is a small fraction of the total-- about 1.7x10^-5, or not quite 20ppm. That's pretty darn good, by everyday standards.

In my part of the scientific world, though, that's an absolutely terrible laser. The frequency of visible light (taking 500 nm as the center of the range) is around 6x10¹⁴ Hz, so the same fractional uncertainty would correspond to a laser with a bandwidth of 10¹⁰ Hz, or 10 GHz. That's roughly ten thousand times larger than the minimum frequency resolution I need for my experiments. I'd prefer my laser to have a frequency spread something like a million times smaller than that.

That's why most precision spectroscopy experiments use continuous-wave lasers-- the frequency spread you get from leaving your laser on all the time can be extremely small, and is usually limited by technical issues, not quantum uncertainty. Fairly ordinary diode-laser systems, the sort you can make in an undergrad laboratory, have a bandwidth of less than a megahertz (1MHz = 10⁶ Hz), and the features we measure in our advanced spectroscopy labs are separated by only a few hundred MHz, and the natural width of those features is on the order of 1MHz. If I can get a laser to scan over 2 GHz, I call that a good day in the lab.

Chemists, on the other hand, generally show spectroscopic data with a wavelength scale measured in tens or hundreds of nanometers. If they see a feature narrower than 10¹⁴Hz, they gush about how narrow it is. I always have to make sure that my students explain the scale they're talking about when presenting things to the summer colloquium series, because otherwise, the chemists assume we're talking about lasers that span half of the visible spectrum.

So, whether a femtosecond laser is a single-frequency source or a broad-band source really depends on what the expectations of your particular research community are. By the standards of chemistry, it's incredibly narrow, but for laser spectroscopy types, it's comically broad.

(Which is not to say that femtosecond lasers are not interesting-- on the contrary, this is a really exciting field. The exciting thing is not the spectral purity of the lasers, though, but the fact that the pulses are so short-- you can use them to stroboscopically follow chemical reactions or electron dynamics in real time, and that's way cool.)