My Story Has a Laser... (math)

Time for more fun with real physics!

In my most recent novella, I talk about a laser in use by Luca's group in an interferometer.  Initially, I just wrote "The best measurements they had suggested that the beam might be expected to increase in size by one percent over a hundred meters, a fairly difficult feat, considering the diffraction limit at that wavelength," but damn it, I can do better than that, so let's go!

First, I'm working out of Amnon Yariv's Quantum Electronics text, third edition.

Early in the book, he discusses Gaussian beams in uniform media.

We start with the Rayleigh length:

z0 = π*w0^2*n/λ

where z0 is the Rayleigh length, w0 is the minimum waist of the beam at the focus, n is the index of refraction of the medium, and λ is the wavelength of the laser.

We'll call n 1.0 for simplicity.  A good wavelength is 632 nm, a common red laser line.  There's nothing in the story that requires the laser be a specific color, so we'll go with that.

In the story, I say that the beam is collimated with a waist of 1 cm, so we'll use that for w0.

That gives a Rayleigh length of 497.1 meters.

Now, how far would a perfect Gaussian go before its waist increases by one percent?

w^2(z) = w0^2*(1+z^2/z0^2)

This describes the waist of a beam w(z) as it propagates through space a distance z.

But what we really want is w/w0 = 1.01, so we get

1.0201 = 1 + z^2/z0^2

Plugging in our value of 497.1 meters for z0 and doing a little algebra, we arrive at

z = 70.476 meters.

So I just changed the text in the story to say "The best measurements they had suggested that the beam might be expected to increase in size by one percent over sixty meters, a fairly difficult feat, considering the diffraction limit at that wavelength," which is pretty good collimation for a free-space beam in air.  That kind of precision would require very good optics and excellent alignment of the beam in those optics to avoid aberrations making the beam larger faster.  It also implies that the beam has a very flat wave-front and is nearly ideally Gaussian.  I should probably put in there somewhere that the laser is red, to narrow it down for readers.

The point is, this particular laser is entirely normal, and not at all supernatural, so it really should accord with actual physics.

Anyway, thanks for reading! If you like hard sci-fi, this calculation was incorporated into my book What the Soul Still Fears! It's got science, the occult, and ancient cosmic horror all mixed together into a fun, creepy story!