Laser rifle recoil?

There's a really snazzy physics blog run out of Cornell called The Virtuosi. Sort of like this blog, they're big fans of looking at interesting scenarios through the equations of physics. Not long ago, they had an interesting post looking at whether or not laser guns would have recoil.

I'm not going to duplicate the steps of the problem here, as you should go read it on their excellent site. I'll just state the conclusions: if the laser pulse has energy roughly comparable to a bullet, the momentum imparted to the gun will be tiny - on the order of 10^6 times smaller than that even of the gently-recoiling .22 LR. But the emission of the light happens over a very short period of time, perhaps a few nanoseconds in the case of the "typical" movie laser with a beam of visibly short length.* Force is change in momentum divided by time (say, a few nanoseconds), so carrying out this calculation gives a force of around 15 N, though only for those few nanoseconds. This is macroscopic, and perhaps comparable to the force on the butt end of a .22 rifle. But on the other hand, the rifle is applying this force for a few hundred milliseconds, not for the factor of a million shorter time that the laser is firing. Can this still be felt?

There's some discussion in the comments on this. I suspect the answer is no, and that the total impulse is the parameter of interest in a case of very short application of a force - not least of which is that the body of a pistol may not be able to propagate a sound wave of ~30ns duration. GHz ultrasound just doesn't propagate well; the propagating pressure wave carrying momentum from the laser cavity to the shoulder stock is likely going to widen in temporal duration into something much longer, meaning we're back to a small force over a longer time.

How good are humans at detecting this sort of high-force, low-duration impulse? One rough comparison might be our response to sound. Sound is after all a sequence of short-duration pressure waves. Normally we can't feel it, but high-intensity sounds such as thundering bass at a rock concert can sometimes be felt in the skin. Let's try to draw a rough comparison between the recoil and an equivalent sound wave.

We know from The Virtuosi's work that the recoil impulse is about 4.5e-7 kg m/s. Let's say the laser gun has a mass of 5 kilograms. Dividing the impulse by the mass of the gun gives a recoil velocity of 9e-8 m/s. Since kinetic energy is equal to:

We have a recoil energy of about 2e-14 J. This is pretty tiny, but then again human ears if not human skin can detect some pretty tiny sound energies. Now in a context like this the pressure of a sound wave is given by

Where I is the intensity in watts per square meter and v is the speed of sound in our material. We don't know what the intensity is, but we can estimate it by saying that it's equal to the energy (2e-14 J) divided by the time of recoil (probably milliseconds, not micro/nanoseconds because of the difficulty of propagating very short acoustic pulses in bulk media - let's say 1ms to be generous), divided by the area of the shoulder stock (say, 25 cm^2). Taking the speed of sound in the shoulder stock to be the 3300 m/s value typical of wood gives us a sound pressure level of about 2.4e-12 pascals. The threshold of hearing is about 20 micropascals, which is much higher. Even assuming the sound pulse really is only 1ns long doesn't quire reach the threshold of hearing. If the pulse couldn't even be heard it probably can't be felt.

Now this is a pretty rough calculation with a lot of estimates and assumptions. Nonetheless I'd expect the recoil of a bullet-energy pulsed laser would not be detectable to the person firing.

*Of course the fact that you can see it moving means it's not a light pulse, but leave that aside.