# Diffraction: Definition, Equation & Examples

Instructor: Aaron Miller

Aaron teaches physics and holds a doctorate in physics.

## Diffraction: A Test For Waves

Ripples on water's surface are classified as waves. That's because we can see the water's surface move up and down in a repeating pattern of troughs and crests as the ripples travel out from their source (the source is whatever hit the water's surface to initiate the wave). The characteristic 'up-and-down' wave motion is not easy to see or feel all the time. Many different phenomena we experience every day have a fundamental wave nature, but our senses are unable to detect the characteristic up-and-down motion, called oscillation.

Take sound, for example. We are told that sounds are waves that propagate through air from a source (like a speaker) to our ears. However, unlike a water wave, we cannot sense individual troughs or crests arriving at our eardrums. So how do we know sound is a wave? The answer is technically and mathematically quite complex, but the bottom line is that waves behave in a unique yet predictable way when they encounter the edge of a solid object.

Unlike the translational motion of, say, a ball or a bullet, traveling waves naturally 'bend' or 'spread' around corners. This effect is known as wave diffraction. You can hear what's happening around the corner of your house because sound waves diffract toward your ear as they pass the corner. If sound were not a wave, it would not reach your ear unless you were in the direct path of the sound's source. Observing diffraction effects is the definitive test of whether a physical phenomenon is fundamentally wave-like.

## Diffraction is Different For Different Wavelengths

Scientists have learned to use diffraction to their advantage. Diffraction phenomena allow scientists to deduce--from simple measurements--the wave characteristic of principle scientific interest: wavelength. Wavelength is the physical distance between consecutive troughs (or crests) in a traveling wave. For sound waves, the wavelength correlates with the sound's pitch. In light waves, it corresponds to the color we perceive.

However, wavelength is notoriously difficult to measure accurately because:

1. A traveling wave never stops moving.
2. Traveling waves can move very fast.
3. The individual oscillations of most waves are not accessible to our human senses (for example, we cannot see infrared light).

Diffraction in certain controlled scenarios produces predictable wave intensity patterns whose characteristics depend on wavelength. You can see this in Figure 1, which depicts a multi-wavelength argon laser reflecting from a diffraction mirror and separating into beams of its constituent wavelengths.

In the following two sections, we will discuss two simple experimental setups involving diffraction that scientists utilize to make measurements: single-slit diffraction and diffraction gratings.

### Single-Slit Diffraction

Single-slit diffraction is the simplest experimental setup where diffraction effects can be observed. In this setup, waves are sent toward a solid material that has one rectangular opening, referred to as the 'slit' or 'aperture.' The material blocks the wave, except in the region where the slit is cut. Waves passing through the slit diffract, spreading out as they pass by the edges of the slit. On a viewing screen behind the slit, they form a characteristic intensity pattern (called a diffraction pattern) of alternating bright (high intensity) and dark (low intensity) spots.

What makes this experimental setup useful? If the waves passing through the slit are characterized by a single wavelength (i.e., they are monochromatic), and if they all have the same phase (i.e., they are coherent), then the size of the diffraction pattern on the viewing screen can be related mathematically to the wavelength by a simple formula, which we will discuss in detail below. This type of experiment is often done using a laser light as the source, since it is monochromatic, coherent and used in many modern applications.

So how exactly does the wavelength of the wave relate to the size of the pattern? Unfortunately, there is not a formula that directly tells us, 'The size of the pattern is X for waves with wavelength Y.' Instead, there is a mathematical relation that lets you calculate the locations of the dark spots in the diffraction pattern. If the distance between neighboring dark spots is larger, then the size of the pattern is larger. In its most common form, the formula that relates the angular position theta of a dark spot (measured with respect to the horizontal line perpendicular to the center of the slit, see Figure 2 above) and the wavelength lambda is:

where w is the width of the slit, and the integer variable m

labels a particular dark spot you would like to locate. The absolute value of m is sometimes called the order of the intensity minimum. In Figure 3, the diffraction pattern's intensity (or brightness) is represented graphically with the order labels displayed along the x-axis.

The angle theta can be related to the distance y along the viewing screen using trigonometry, if the perpendicular distance from the slit to the viewing screen L is known. One such trigonometric relationship is:

in which theta is calculated from Equation 1. Finally, all of these results are only valid when the viewing screen is far from the slit, compared to the width in the slit, because the derivation of Equation 1 relies on the approximation that L is much bigger than w.

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