A Light-Hearted Introduction to Diffraction

Waves have amplitude and phase. Amplitude is the height of the wave and determines how much energy it carries. Depending on who you talk to, phase has different meanings. I would say that phase encompasses where a wave is in space and time.

All waves—ocean waves, light, sound, electrons—diffract. Diffraction is when waves interact with something close to the size of their wavelength. When this happens, the waves make beautiful patterns created by constructive and destructive interference—the crests of waves boosting or annihilating each other depending on their relative phase, their relative positions in space and time.

Purple wave = red + blue wave. Since the two dashed waves have the same amplitude, they can completely annihilate each other. When the two dashed waves line up nicely, the purple wave is twice as tall. Credit: Me. And Mathematica 11.

To be precise about this, the path length difference between two waves created by the diffracting object will change the phase relative to each other, selecting some waves to be amplified and some waves to be destroyed. I remember the term diffraction by thinking of the path length –diff-erence necessary to create it.

path length difference -> constructive and destructive interference -> diffraction

One of the more intuitive examples is water waves flowing through small gaps or around stones. The waves start to fan out circularly, making a very distinct repeating pattern. Diffraction allows you to extract information from either direction: you can guess the wave pattern from the obstacle or you can guess the obstacle from the wave pattern. Assuming, of course, that you have all of the wave information–the amplitude and the phase.

Two waves, alike in dignity… Diffract. There is a clear periodicity to the wave fronts that hit the edge of the animation. In a double slit experiment with light, you would see stripes of light instead of the hazy smear you might expect if you were expecting light to always be made of particles. Credit: Me and Mathematica 11.

Electromagnetic waves (think visible light, x-rays) also diffract, but with much, much smaller gaps and much, much smaller “stones”. Diffraction gratings are optical components with carefully tailored periodic gaps that are close to the wavelengths of visible light. They’re used in instruments called monochromators (mono = one, chrome = color) to filter out all light except for a single wavelength. You can also buy them online for $1.70 (not including shipping or discounts on bulk orders), although these aren’t the fancy ones in monochromators.

Each gap is a micron thick, or 1000 nanometers, close to the wavelengths of the visible light spectrum. As light flows through the diffraction grating the spacing selects single colors, producing rainbows when you hold this in front of a lamp or the sun. Can you afford not to buy this?

CD’s also behave like diffraction gratings, which is why you can see rainbow streaks when you look at them from certain angles.

In the natural world, some insects and birds have small scales on their wings and bodies that act like diffraction gratings, producing unbelievably vibrant colors that would be quite hard to achieve with natural pigments (cheaters!). The epitome of this technique is an African plant called Pollia Condensanata. This plant entices birds to eat it despite having low nutritional content because of its bright blue color, which arises from helical oriented stacks of fibers that resemble diffraction gratings.

This tricky plant has no blue pigment. Read the original publication in PNAS here. 

Similarly, x-rays interact with gaps even smaller than visible light. Crystals are repeating layers of atoms with spaces between each plane that are so close together that even visible light, with wavelengths of 100’s of nanometers, is too large to diffract. Luckily, the wavelengths of x-rays can be as small as 1/100th of a nanometer. X-ray diffraction can determine the atomic structure of crystals by shooting x-rays at a sample and varying the angle between the sample and the x-ray beam. Similar to a diffraction grating with visible light, some special angles will produce a very strong signal of additive x-ray wavelengths while others will completely cancel out the x-ray wavelengths. Similar to the pond metaphor, we can detect the x-rays flowing through the gaps between crystal planes and figure out the structure of the crystal without directly seeing it ourselves.

An example of a single crystal x-ray diffraction (XRD) pattern. For powder samples, there are essentially a ton of single crystals facing many different directions, which causes each diffraction spot to be spread out into a ring to account for the many angles. This image is from here. 

Electrons have an even smaller wavelength than x-rays and can similarly be used to determine the atomic structure of crystals in a Transmission Electron Microscope (TEM). A electron diffraction pattern looks similar to an x-ray diffraction pattern, a grid of points. Essentially, the spacings of the crystal (or other periodic structure, like proteins) the electrons are passing through cancel out so many directions of the electron wave that you are left with a pattern of dots. Each dot corresponds to the specific crystal plane that produced it, with higher order planes being rarer, hence producing fainter dots. Planes with “lighter” elements (think C vs. Au) will also produce fainter dots. If they were rocks, C is a smaller rock than Au. I like to imagine that TEM electron beam diffraction is essentially dropping an electric pond on an array of atomic stones.

Examples of electron diffraction patterns taken from this paper. The larger label in the top right is the zone axis, or the direction the beam is going through, while the smaller labels indicate the crystal planes that produced each diffraction spot.

Electrons and x-rays can provide similar information about crystalline materials but have different properties that might make one more favorable than the other. I might talk about that another time.