All right, I’m gonna delay the next installment of the quantum bouncing ball for a brief diversion. I have a friend who’s also a physics grad student, and he suggested that we along with a few other fellow students form a yet-to-be-named unofficial club whose raison d’etre is to get together every few weeks and do interesting Mythbusters-style science for no good scientific reason. After all, we are fortunate enough to be surrounded by expensive cutting edge equipment and brilliant people, and we should have some scientifically interesting fun.
One suggestion for something we might try was inspired by a Big Bang Theory episode, which was itself inspired by a real experiment. The idea is to measure the distance to the moon with a laser. Shoot a pulse of light at the moon, measure the time it takes to bounce back, and using the known speed of light you can calculate the distance to the moon. Since the speed of light is about 1 foot per nanosecond, measuring with nanosecond accuracy (actually pretty easy) could mean measuring that rather vast distance to within the length of a ruler. In practice other effects like atmospheric dispersion increase the uncertainty, but these can be corrected for. While the lunar surface itself is reflective enough to do the experiment unassisted, the Apollo astronauts also left behind retroreflectors which make this experiment much easier and much, much more precise. (It turns out to increase the total return of photons by about a factor of 100, and eliminates the problem of varying elevations in the topography.)
So to do this you need a laser that spits out short (ie, nanosecond or shorter) pulses with as high an energy per pulse as you can muster. You need high energy because you ain’t gettin’ much energy back from the moon. We should try to put numbers on this, to see if anything we have around the lab might work.
First, we need to see how much of the laser light will make it to the retroreflector on the moon. Lasers travel in pretty straight lines, but as anyone with a laser pointer can verify, they do spread out some with distance. Typical values for the lasers we use might be somewhere in the 1 milliradian range. The distance to the moon is roughly 400,000 km, so the portion of the lunar surface illuminated by our laser will be about (0.001)*(400,000 km) = 400 kilometers in diameter. This is an area of about 125 billion square meters. If the retroreflector is one square meter, only about 1 part in 10^11 of our emitted light even makes it to the reflector. Now the reflected light has to make it back to the earth. If we’re extremely optimistic, we might say that the reflector introduces no extra angular spread, and its reflected light is spread over a 400 km diameter of the earth’s surface. To a first approximation this just means the total there-and-back efficiency is about 1 part in (10^11)^2, or one photon in 10^22. This, my friend, is Bad.
But hey, we can use a photomultiplier tube to count individual photons, and photons are pretty tiny units of energy. Our laser sends out a heck of a lot of them. The energy of one photon of wavelength λ is:
The laser we’d likely use has a wavelength of 532 nanometers, and plugging into the equation we find that each photon has an energy of about 3.7 x 10^-19 joules. Therefore we’d need about 1000 joules per pulse to get up to the neighborhood of 10^22 photons per pulse. And we need 10^22 photons just to get one photon back per pulse on average.
We have a few compact 15 mJ/pulse q-switched Nd:YLF lasers with 1 KHz rep rates, and we even have a few not-so-compact 2 J/pulse with a 10 Hz rep rate. With a 2 second round trip time, the repetition rate isn’t so relevant since we can effectively only use one pulse per round-trip time. Even 2 joules won’t cut it unless we’re willing to do many thousands of shots worth of statistics. And that’s without even thinking about noise, which will not be inconsequential even with good filtering.
So if it’s that hard for us, how does NASA do it? First and most importantly, they use a laser with a divergence of a microradian or so. This reduces the spot diameter by a factor of a thousand compared to ours, and thus increases the intensity at the lunar surface by a factor of a million. There’s an inverse relationship between beam divergence and beam waist size, so with some beam-expanding optics this is something we could do in theory – but it’s not trivial and at that point it’s not so much a fun project to spend a few hours on as it is a major research effort, especially since a smaller illuminated lunar area means aiming is that much trickier. Also, NASA’s lasers tend to have higher pulse energies.
Thus I regret to say that lunar ranging is probably not in our club’s future. It’s also probably not something the intrepid cast of the Big Bang Theory could have done on their roof with a few shots from a small laser, given that it wasn’t passing through collimating optics (ie, a large telescope). Still, it’s not impossible and I think a dedicated and well-equipped amateur group could probably do it. If you succeed, let me know!