Built on Facts

Wyoming, Pt. 1

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I've been in Casper Wyoming the last few days at the Workshop of Quantum Science and Engineering, put together by Dr. Marlan Scully. The point of the workshop is to discuss the various work we're doing, exchange ideas, and all the other stuff that usually happens at these physics shindigs. We've also had a few talks celebrating 50 years of the laser and 95 years of its inventor, the Nobel laureate legend Dr. Charles Hard Townes. Dr. Scully says he's the greatest living scientist, and I have to say I think he's got a pretty good case.

Casper, Wyoming is Dr. Scully's hometown, and he makes sure we have lots of time to explore the area and do various outdoor things. For now I'm not going to talk much about the science because I haven't had time to get an outline together and organize things, so this'll be more or less just some photographic highlights.

First of all, Wyoming. Experimentalists talk about typical data, this is Typical Casper:

Here's part of one of Dr. Scully's family pieces of land, which the Audubon Society is managing as a nature preserve:

The Quantum Cowboy himself:

Your host, on Independence Rock (twice!). Ah, the old-school Oregon Trail computer game memories...

Dr. Townes in a quiet moment:

Posted by Matt Springer at 6:21 PM • 3 Comments • 0 TrackBacks

Shoving Sand

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It's a pretty nice time for quantum optics and laser physics at Texas A&M, with us moving into a brand new lab and acquiring several new laser systems. One of the systems we're moving isn't new (we've had it for a few years, and the technology is considerably older), but we're about to put it to work on a new project. It's a relatively high-power Nd:YAG laser.

The operating parameters are about 2.5 joules per pulse, with 10 pulses per second for an average power of 25 watts. Each pulse is about 8 nanoseconds long. This is a rather hefty amount of light. Though any Class 4 laser requires serious attention to safety, we're going to treat this one pretty much like a nuclear reactor when it's running.

Now as you may know, light carries momentum and can exert pressure on surfaces that it hits. This is true even in the purely classical field theory of Maxwell. But as a rule we don't notice light pressure because it's so slight. I'm a little curious as to whether it might be significant in a laser this intense. Let's run the numbers and see what happens.

First, let's review some definitions. Power is measured in watts, and it's the number of joules per second. We'll label it with a capital P. Pressure is force per area, and we'll label it with a lower-case p. Intensity is what you might colloquially think of as the brightness, and it's just the power per area, and we'll use capital I for it. The relationship between light pressure and light intensity is given by:

Where c is as usual the speed of light. Now if pressure is force per area and intensity is power per area, than the force will be given by the total power divided by c:

These are just two ways of saying the same thing. We'll start with the second one. 25 watts divided by the speed of light is: 8.34 x 10^-8 N. Very slight, as you might expect because 25 watts isn't actually much power. It's just a lot for a laser because it's kept in a very tight beam. But this is a laser and the beam is very tight. It can be focused on something very small. What if we focused the laser on a grain of sand? To simplify the math, say it's a tiny cubic crystal of sand with each side having a length of 0.25mm. Sand is frequently quartz, which has a density of about 2.6 grams per cubic centimeter. As such the grain of sand has a mass of around 40 micrograms. Force is mass*acceleration, so the acceleration the grain will be about 2.05 m/s^2. This is quite a bit, certainly enough to send it sliding across the table.

Well, unless we blow it to smithereens. Remember that 25 watts is an average. The instantaneous power during the pulse is actually about 2.5 joules / 8 nanoseconds, which is about 312 million watts. Focused onto that (0.25 mm)^2 cross section, that's an intensity of 5 x 10¹³ watts per square meter. Yikes, that's about 50 billion times more intense than direct sunlight. The pressure works out to be relatively modest - I think around 25 PSI (167 kPa) - so I'm not sure mechanical stress will break up our sand grain. However, the light that's absorbed might be enough to melt it, especially with repeated pulses. I'm not sure what the absorbance of sand is at 1063/532nm wavelengths.

Sadly I'm pretty sure I won't be allowed to try this experiment to find out. Frankly I think my self-preservation instinct might veto my "Holy cow, lasers!" instinct anyway. Still, it's tempting...

BLOGGY NEWS: I will be spending the next two weeks in the scenic but very small town of Casper, WY at a quantum optics / laser physics conference/school being held by our own no-kidding legendary Marlan Scully. There's a decent bit of downtime involved, and supposedly the place is quite scenic. As such I hope to be doing a bit of a travelogue, with summaries of some of the talks and pictures of the landscape. At any rate it can't possibly be any hotter than College Station is!

Posted by Matt Springer at 10:00 AM • 7 Comments • 0 TrackBacks

Sunday Function

Category: Sunday Function

Again, apologies for the hideously scanty posting. Been in the lab doing some really interesting research which will with some luck get me in a really nice journal, as well as doing the various rounds of revision on the paper for some previous research. Also putting two talks together for a conference/school. That whole wedding planning ain't doing wonders for my spare time either. But hey, these are all good things so I'm not complaining.

I can at least write up a Sunday Function for y'all. This is one we've mentioned before, but haven't actually derived. I think it's high time we did. As always, math-averse readers are encouraged to read on even without necessarily following everything in detail. The overall trajectory of the argument should still be pretty clear.

What we're doing is Stirling's approximation to the factorial function. The factorial function is the product of the integers from 1 to n, inclusive. For instance, the factorial of 4 is 1*2*3*4 = 24. Unlike most functions, this one gets a punctuation mark as its symbol. The factorial of 4 is compactly written as 4!, and that's not me being excited about the notation. Instead 4! = 24.

This is a very fast-growing function. The factorial of 50 is more than a trillion trillion trillion trillion trillion. As you'd expect, this is a pain to calculate by hand. A pocket calculator can do it without too much trouble for factorials up to about 70!, and dedicated computer algebra systems like Mathematica can come up with exact values for up to a few thousand before becoming too slow to deal with. In physics (mainly thermodynamics) we have to deal with the factorial of numbers of about the size of the Avogadro number (~10^23). Actually calculating this is absolutely, completely bananas. But we're not worried about knowing the exact value, we just need a good approximation. We can do it using the Gamma function. I'm know we've discussed it before, but I can't find the link. In any case you can take my word for it that this is exactly true, and not an approximation:

Now there's really no way to integrate that in any closed form other than just, "Hey, it's equal to n!" which isn't really helpful since we're after the number that n! represents. But we can do better if we're interested in an approximation. This approximation is due to Abraham de Moivre and James Stirling, and this particular derivation I first saw in Daniel Schroeder's thermodynamics textbook.

The plan is to recognize that the integrand looks kinda sorta like a Gaussian, so we might be able to find the best-fit Gaussian and then integrate that and see what we get. So take that expression under the integral and rewrite it using the rules for logarithms:

Where we just used the rules for changing the base of a logarithm. Still no approximations yet. Now take the stuff in the exponent (the n ln(x) - x) term and do a bit of rewriting:

Where for convenience we've defined y = x - n. The algebra to derive the above is not hard - it's about 3 lines if you remember the logarithm rules, so give it a shot if you're curious and following along at home.

Now it's time for the approximation. It's intuitively clear that the original integrand has a maximum at x = n, and you can demonstrate this via differentiation if you're so inclined. And at that point y is equal to zero, so we're justified in making a series expansion of that middle logarithm term above. This is where the approximation actually happens:

Which means that our original integrand is approximately:

But the first two are constants with respect to x (and y), so they can be pulled outside the integral, so:

Which is an easy integral to do. (You may wonder how I magically turned the 0 in the lower limit to a negative infinity, but it's justifiable because the integrand is basically zero for negative values anyway.) Performing the integral gives:

Which turns out to be a very good approximation. Keeping more terms in the series expansion would make it a better one, but this one is fine for just about every large n! approximation we need. It's very easy to evaluate for large numbers with a pocket calculator and a basic knowledge of logarithm rules. For instance, the factorial of (10^23) is about 10^(10^(10^1.38)), if I've done my math right. Which is a heck of a lot easier than doing 10^23 separate large integer multiplications.

All right, that's all for now. See y'all around!

Posted by Matt Springer at 10:00 AM • 6 Comments • 0 TrackBacks

Sunday Function

Category: Sunday Function

Happy birthday to the United States! It's one of the younger countries on the planet and yet has still managed to have one of the oldest continuous systems of government. Not too shabby. Here's hoping our current wobbles get straightened out and our next few Independence Days occur under more pleasant conditions.

I'm not sure what function might fit very well in the context of this holiday, but I'll make sort of a stretch and do a function about the Pythagorean theorem. The Pythagorean theorem is a very ancient discovery with numerous different methods of proof. One of them was discovered by James Garfield, who would later become a president of the United States. I don't think many of our 44 presidents have made independent nontrivial mathematical discoveries, so good for him. Here's a figure from his proof:

The Pythagorean theorem states that for the sides of a right triangle A, B, and C, the identity A^2 + B^2 = C^2 will be satisfied. You may be familiar with the Scarecrow's statement of the theorem in the Wizard of Oz, though he actually butchers it somewhat and what he says isn't actually true.

For certain side lengths, you'll find that the lengths can be specified with pure integers. For example, the triple {3, 4, 5} is a so-called Pythagorean triple because 3^2 + 4+2 = 5^2. Go ahead, try it out. There's an infinite number of these triples, and they can be generated with our Sunday Function:

Given any two integers m and n (with m > n), and this formula will generate three numbers a, b, and c which are Pythagorean triples. For instance, with n = 1776 and m = 2010, you'll get the triple {885924, 7139520, 7194276}.

Ok, enough math for the day. Now go cook some burgers, celebrate your freedoms, and try not to lose any fingers to fireworks!

Posted by Matt Springer at 5:20 PM • 12 Comments • 0 TrackBacks

You Probably Shouldn't Buy This Laser

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The laser pointer, much beloved of PowerPoint lecturers, cat owners, amateur scientists, and middle school boys at movie theaters, is actually a pretty amazing device. There's quite a bit you can do with a relatively cheap laser, and they're just plain fun. They're also relatively safe. The red pointers are usually Class 2 and the green ones are usualy class 3R. Class 2 lasers are very difficult to hurt yourself with, and while Class 3R lasers can cause eye injury, in general brief exposures are unlikely to cause permanant damage.

There's two higher classifications for lasers: 3B and 4. Class 3B lasers are officially in the "not a toy" category, with even very brief direct eye exposure having the potential to do serious and permanant eye damage. For the higher-power end of that class, it's not merely potential but essentially certain. Class 4 lasers cause permanant eye damage on direct exposure, and have the potential to cause burn injuries especially when focused. Because Class 4 is the highest class, there's no labling difference between relatively low-power Class 4 lasers that you barely feel as a tingle even if you stick your hand in them and huge CO2 lasers that can slice through steel.

As an example of the former type of Class 4 laser, one of the lasers in my lab emits a stream of roughly 35 femtosecond long pulses of near-infrared 800nm light. The pulses have an energy of about 1mJ and are emitted at a rate of 1 kHz. As a result, the average power of the laser is 1 watt, which puts it solidly in Class 4 territory. The infrared beam in invisible, so while a green 1 watt laser would be so blazingly bright as to leave no doubt as to its hazard the IR beam doesn't give such an obvious warning. At 1 watt it's just barely intense enough to singe something that's black and flammable, but if it's focused it's considerably nastier. Focused or unfocused, it's instant eye damage and we have to take serious precautions - among many other things, the laser and all the optical components the beam interacts with are bolted to a 1-ton optical table. A permanent blind spot would a rather steep price to pay for shoddy safety, so we practice good laboratory safety standards.

So I'm not so sure I like this: a 1 watt violet diode laser being marketed as a lightsaber.

Now in fairness the actual company selling these isn't marketing along these lines, and they do spend a pretty decent amount of time spelling out the danger of these things. But be assured, this is serious business.

It's not as dramatic as the message board hype; this isn't the Arson-O-Matic and you couldn't blind stadiums of people by waving it around. The beam divergence is listed on their chart:

Extrapolating, the beam is reduced to the same intensity as a 5mW laser at the aperture at a distance of about 22 meters. Outside this range the laser is not exactly safe but it's unlikely to cause instant damage.

But this is sort of faint praise - a 20+ meter danger radius is Bad News for people who're buying this without the intention of being adequately safe. Which I suspect is lots of 'em.

Plenty of people online have suggested tighter regulation of these things. I'm not exactly the government's biggest fan, but I'm not sure this is the sort of thing that ought to be available by mail order on the internet. On the other hand these lasers are integral components of an increasing number of perfectly innocuous devices like Blu-Ray players and projectors. This laser is itself scavenged from a projector, so any person with a screwdriver could get one of these even if they weren't being marketed as lasers.

So I'm not taking sides. I'd just like to advise that if you're interested in one of these, be aware that it's very, very dangerous and if you're a laser hobbyist you ought to treat this in the same way a skydiving hobbyist treats a parachute. I'd really prefer you not touch Class 4 lasers unless you have dedicated safety equipment and training as well as plenty of experience on Class 3R and 3B lasers.

Posted by Matt Springer at 10:05 AM • 6 Comments • 0 TrackBacks

Quantifying the "Ouch" of heading in soccer

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I'm not normally much of a soccer fan, but the World Cup doesn't happen every day and it's pretty interesting to see all the excitement and high level of play. I personally think the rules need a little tweaking to reduce the tendency toward 0-0 and 1-1 ties, but I suppose the sport couldn't have so many billions of fans without doing something right.

In honor of the World Cup, let's do a quick example of just how tough the game can be. In soccer of course the players are generally prohibited from touching the ball with their hands - hence, football in most of the world. But in fact the players can and do use anything but the arms to manipulate the ball during play. If you watch the game, you'll notice the players using their heads to hit the ball in mid-air when kicking is not possible.

This can be a pretty traumatic thing - as in many contact sports the cumulative effect on the brain can be serious in certain cases. Let's do the math and calculate roughly just how much of a whack heading the ball can produce.

In 1-d accelerated motion, the position as a function of time is given by:

We'd like to find a, the acceleration experienced by the ball. We don't know x, the distance over which the ball decelerates against the player's head, but I think we can estimate that it's about equal to the radius of the ball, which happens to be about 35 11 centimeters. [I originally used a wrong value here, which I've noted via strikethrough in the rest of the text.] We also don't know the time t over which the ball decelerates. But we can estimate that too - since we know that the ball has to decelerate from its initial speed to a stop in the most dramatic case, we can use the equation relating velocity, acceleration, and time:

Substitute that back into the first equation and we can find the acceleration without directly using the time:

I've played kind of sloppy here and dropped initial velocity and position terms in the above equations with the understanding that really we're really working with the change in velocity and position. If you're currently in a physics class learning this for real, you might want to grab a pen and paper to verify to yourself that this is legit.

Right off the bat we see the acceleration is proportional to the velocity squared, so heading a faster ball will pack a much harder wallop. According to Google, 70 miles per hour is pretty typical of a hard kick. Plugging in that figure and the 35 11 cm value for d (mind the unit conversions!) we get an acceleration of 1399 4451 m/s^2. To find the force which the head applies to the ball to produce this acceleration, we just multiply by the 420 gram mass of the ball. I get 587 1869 newtons, or about 132 420 pounds force.

Which is quite a bit to have applied to your head, even for a few milliseconds. Now this is in some sense a worst-case scenario. Most of the time players are not heading high-speed balls directly opposite the original direction of motion. But it does give an idea of the possible hazards of the sport and why sports medicine will always be a booming business.

Posted by Matt Springer at 10:19 AM • 17 Comments • 0 TrackBacks

Power from the Earth's Magnetic Field

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On a web forum I frequent, a person asked if it would be possible to extract energy from the Earth's magnetic field. He was told no - static magnetic fields can't transfer energy. For all practical purposes this is true, but in fact we also know that the earth's magnetic field isn't static. It changes from day to day and from year to year - and even second to second. The changes are small over small timescales, but in fact the magnetic poles do drift around and the solar wind does perturb the fields and so forth.

Wikipedia gives a reference saying that typical local variations in the magnetic field at the surface are of the order of 1 nanotesla per second. This is pretty small compared to the total field of perhaps 50 microtesla, but it is measurable with sensitive equipment. Can we extract that energy and free ourselves from reliance on coal and oil? We expect the answer is "no" because otherwise someone would have done it, but we can crunch the numbers to make sure. First, Faraday's law:

Looks bad, but it's not. In this simple physical situation the calculation above will only involve multiplication.

The right hand side says "Make a closed shape out of a bent wire. A square, a circle, a heart, whatever. Now hold it in place and look at the local magnetic field as it passes through the loop. Take that total magnetic flux and look at the rate at which it's changing with time."

The left hand side is just the total potential difference in volts that each electron gains after making one circuit of that loop. It's that number which will tell us something about how useful this might be as a power source.

Magnetic flux is just the magnetic field multiplied by the area of the loop, assuming the field is perpendicular to and uniform within the loop. We're interested in the time rate of change of this flux, and let's say we have a circle with a diameter of 1 meter. The rate of change of the flux is thus (1 nanotesla/second)*(3.14 meters^2).

Which is 3.14x10^-9 volts. Three one-billionths of a volt per square meter of flux-collecting surface. If you tried very hard you might be able to finagle some useful energy out of such a small potential, perhaps with very long superconducting solenoids. But it would be less cost-effective than pretty much any other form of renewable energy by many orders of magnitude.

Still, it was worth a try!

Posted by Matt Springer at 10:34 AM • 17 Comments • 0 TrackBacks

Laser Safety and the Vuvuzela

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As you might expect for a guy who does experimental optical physics, I get to spend a tremendous amount of time in labs with some fairly snazzy lasers. Most of them are fairly specialized pieces of equipment that aren't really designed simply to dump huge amounts of power in industrial applications. As far as danger goes, they're not going to come to life and murder you in your sleep. But still, we have open beamlines of infrared lasers with average powers on the order of 1-4 watts. Unfocused they usually won't do much to exposed skin other than make you uncomfortable (for instance, like a Christmas light pressed against the skin for the 4 watt laser). Focused they'd have no trouble poking a very clean little pinhole in you. Unfortunately your eye is a focusing lens, and direct eye exposure would be instant permanent damage. And the infrared beam is invisible to the naked eye, so that's an extra challenge.

Of course we have pretty robust safety procedures and I've never heard of any injuries in our AMO group. We also have a wide assortment of surprisingly expensive safety glasses that are designed to filter out the wavelength of particular lasers. For lasers with a continuous beam this is not so hard since all the light is emitted at pretty much exactly one frequency. For lasers that generate ultrashort pulses the light spans a wider portion of the spectrum and the glasses have to remove more bandwidth.

This particular laser is a 15 watt Nd:YLF laser we use to pump an ultrafast amplifier. While it's possible in theory to pop the lid and get a good picture of its innards, it's not a great idea to shut down and open up expensive equipment for no research-related reason. So here's a picture of some green light diffusely reflecting from the gap between the pump laser and the amplifier:

And here's the same thing with some safety glasses held in front of the camera:

Though the images are not great, you can see that the green light is totally blocked. You can still see most of the world pretty well because the glasses still let in light with longer wavelengths than green. (For the curious, these block everything from about 540nm to around 160nm. This protects against frequency doubled Nd lasers and excimer lasers, though our lab only uses the former.)

So what's this have to do with the now-infamous vuvuzela? If you've been living under a rock lately, you may have missed watching or at least seeing clips of the World Cup being held in South Africa. Apparently it's tradition to bring this instrument to the games. It sounds something like an oversized kazoo and in the aggregate makes the stadium sound like the world's biggest beehive. But in analogy to the laser, it just so happens that the sound produced by the vuvuzela is generated in a relatively narrow frequency range - mostly at 233 Hz, 466 Hz, 932 Hz, and 1864 Hz. Block those frequencies electronically and you remove that buzzing while preserving most of the rest of the noise of the games. Various news organizations have reported that some people are doing just that. Here's the web site of the guy most of the news articles mention. It's in German, but if you scroll to the bottom the before/after sound samples are easy to find. It's not quite perfect. The vuvuzela is not quite a perfectly narrowband instrument and some of those removed frequencies contain significant parts of the spectrum of the human voice which leaves vocal audio sounding a tiny bit tinny. But it's really a remarkable improvement nonetheless.

Maybe not quite as important to health as a good pair of laser safety glasses, but certainly an application of band-stop filtering that's likely to be useful to more people.

Posted by Matt Springer at 10:00 AM • 11 Comments • 0 TrackBacks

Sunday Function

Category: Sunday Function

A while back I mentioned the St. Petersburg paradox. It's a hypothetical gambling scenario where you win money based on the outcome of a coin toss. If you get your first tails on the first throw, you get $1. If you get one head before your first tails, you get $2 dollars. If you get two heads before tails, you get $4. If three heads before tails, you get $8, and so on doubling each time.

How much should you be willing to pay to play this game? If you work out the mathematical expected value of the game, it turns out to be infinite. Play this game enough times and it doesn't matter how much it costs to play, you will certainly come out ahead in the long run. It might be the very long run, because so much of the value of the game is the one in a bazilion chance to win a bazilion dollars. But formally, any casino that offers this game at any price will eventually go broke.

There's a lot of discussion of this result and why so many people wouldn't be willing to pay much to play it despite its mathematical value. For most people (including me), it boils down to the fact that while we might jump at the chance to bet a dollar on a 2% chance at winning $100, we'd hesitate to bet ten thousand dollars on a 2% chance at winning a million dollars. The math is the same, but what happens to us after the very likely loss is not.

But real life doesn't quite fit this mathematical abstraction anyway. Any casino that offers this bet will only have finite resources. Therefore the bet they're offering is to toss the coin either until tails comes up or until the casino goes broke. And that changes the math quite a bit. I'll follow the argument on the linked Wikipedia article, since it's a nice summary and it gets me out of typesetting it myself because ScienceBlogs still doesn't have LaTeX.

Since your winnings double for each head, the maximum number of times you can flip heads before the casino goes broke is L = 1 + Floor[log(W)], where W is the casino's total assets and log is the base 2 logarithm. The floor function is just a function that leaves whole numbers unchanged and rounds everything else down to the nearest whole number. For instance, Floor[2.9] = 2.

So we calculate the expected winnings thusly:

The first line is pretty easy. It's the probability of tossing k heads times the winnings you'll get if you toss that many heads. It'll be whichever is smaller: 2^k dollars or the casino's total assets because you just bankrupted them.

The second line splits the sum for each of those possibilties. The first term is the pre-bankrupting winnings and the second is the post-bankrupting winnings. The third line calculates those sums. Remember L is a function of W, the casino's total holdings.

We can plot this. Notice I'm using a log-linear plot, so look carefully at the x-axis. The graph is giving the expected winnings on the y-axis compared to the casino's holdings:

This is pretty instructive. If your friend offers you the bet but he'll only pay up to $100, then you shouldn't be willing to pay more than about $4.28 if you want to come out ahead. But even if the maximum winnings are a million dollars, it's still only worth about $10.95. And even if you were offered maximum winnings of trillions of dollars you'd be justified in paying much more than a Jackson. The logarithmic function is very slowly growing, and so that infinite expected value only comes into play because of the infinite possible winnings. Such a situation will not obtain in Vegas, so you shouldn't go betting your house to play the game. After all, they have statisticians too.

Posted by Matt Springer at 6:55 PM • 7 Comments • 0 TrackBacks