Diffraction

What is diffraction?

When starting out thinking about lasers, a lot of people think that the laser light just goes in a perfectly straight line forever. But no, physics is not as permissive as that. Light is a kind of wave, a wave made up of electric and magnetic fields but a wave nonetheless. And when waves encounter an obstacle, they can bend around the obstacle. Think of ripples in a still pond encountering a reed poking up through the water - the reed doesn’t cast a perfect shadow behind it. No, those ripples bend around and fill in the space behind the reed as well. In a similar way, if the ripples hit a wall with a hole in it, the ripples that go through the hole won’t march on in a perfectly straight column. Rather, they spread out as soon as they go through the hole. This bending and spreading out is called diffraction. Diffraction is a fundamental property of any wave in nature.

Waves diffracting through an opening.

The smaller the wavelength of a wave in comparison to the size of the obstruction or opening, or the width of the beam, the less the wave diffracts. Since visible light has a very small wavelength, this is why light seems to travel in straight lines. It takes very narrow objects, such as hairs or thin scratches, or very long distances, to get noticeable diffraction. This is easy to demonstrate with a laser pointer in a darkened room. Shine the beam at a wall. Now put a hair in the beam. You will see a streak appear on the wall through the laser dot perpendicular to the hair. The streak is the light diffracting around the hair.

A uniformly illuminated laser beam profile through a circular aperture (left) and the far-field diffraction pattern from this aperture (right).
The beam profile of a laser beam encountering a thin horizontal obstruction (left) and the far-field diffraction pattern from this aperture showing the resulting vertical streak of light(right).

Spot Size

For a laser, we want to shine our beam on something. But as soon as the laser gets out of the machine that is making it and guiding it, it diffracts around the aperture that emits the beam into the wider world. This diffraction limits the amount by which the laser can be focused, or even by which it can move straight. This means that if you try to focus the beam down to a tiny point at a distant target, the beam might spread out to a much larger spot by the time it gets there. The amount the beam expands depends on the ratio of the initial width of the beam to the wavelength of the beam. In fact, we can determine the smallest possible spot size into which you can focus the beam: if we denote the diameter of the mirror, lens, or other opening in the laser or focusing element as $D$, the wavelength of the light as $\lambda$, and the distance to the target as $R$, then the diameter of the smallest spot, $S$, is given by

$S=1.22\,R\,{\frac {\lambda }{D}}$.

This is called the Rayleigh critereon.

Note that to focus well, which gives you long range and maximizes your damage and armor penetration, you want to have your focal aperture be as large of diameter as you can get it! This is entirely the opposite of the intuitive idea that you want to shoot out a straight, narrow beam out of a small hole; because the beam is focused, not straight and wide apertures let you focus better. Just like a bigger diameter lens on your camera lets you take higher resolution pictures from a long way away, a bigger diameter lens or mirror on your laser gun lets your beam have a higher resolution (smaller spot size) when projected far away.

Diffraction also encourages using shorter wavelengths so you can focus damaging spots at longer ranges. Practically, you will probably not want to consider any wavelength longer than short-wave infrared for causing physical damage to a target. Historically, some long-wave and mid-wave infrared lasers were explored for laser weapons, but these are largely obsolete now. Microwave beams are used for counter-electronics weapons, but not for burning your target up because they simply can’t focus well enough. For space combat, you will want to use as short of a wavelength as is practically achievable by your technology. (In air, very short wavelengths have other problems.)

Beam Shape

A focused, diffraction limited beam can be approximated as converging on a focal point at range at $R$ with a beam width

$W=\left|D\,\left(1-{\frac {x}{R}}\right)\right|$

at any given distance $x$ along the beam as long as $W>S$. Once $W<S$, the beam will propagate in a nearly collimated line for a distance

$Z={\frac {2\,S\,R}{D}}={\frac {2\,S^{2}}{1.22\,\lambda }}$

In practice, you will probably want to focus a distance $Z/2$ behind your target, in order to get the beam going straight for as long as possible once it encounters the target. When you take pictures, $Z$ is called the depth of field, reflecting the limitations on the camera’s ability to focus at all distances. Likewise, we will refer to this distance for laser weapons as depth of focus for the same reason. (Note that in the literature, the region where the beam is nearly parallel is the beam waist, our spot size $S$ is closely related to the waist radius $\omega _{0}$ and our depth of focus is closely related to the Rayleigh range $z_{R}$.)

Beam Quality

The equation given for the diffraction-limited spot size assumes that you have perfect beam quality. This means you need to be lasing in the best cavity modes of your laser. But sometimes other, unwanted modes creep in, and the more these modes contribute the worse your ability to focus becomes. These other modes diffract worse than the best modes. So good quality lasers end up being able to shoot much farther than poor quality lasers.

Often, lasers that are good quality at low power become worse quality as you try to shove more power through them. The extra power heats and distorts the laser generator, leading to lasing on multiple less favorable modes. Some lasers, such as free electron lasers and fiber lasers, are largely free of this issue. But other lasers like slab solid state lasers and especially diode lasers suffer from this.

Interference and Phased Arrays

If you took an introductory class in physics during college, you might still remember the infamous double slit interference pattern. If you have two apertures that remain in phase with each other (maybe a laser beam is being split in two and sent through both of them), and if the spots from the beams from both apertures overlap, then the beams will interfere with each other. For only two beams, you get a stripe pattern, but with multiple beams you can get more complicated patterns of spots.

Diffraction, it turns out, is just a special kind of interference. It is the pattern you get when you consider the waves of each different part of the aperture interfering with each other.

So here's an example of how we can use diffraction and interference to our benefit. here is a picture that shows a hexagonal aperture on the left and the diffraction pattern in produces at the target on the right (not to scale):

Credit

Author: Luke Campbell