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Single-slit Diffraction: Interference Pattern & Equations

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  • 0:01 What Is Single-Slit…
  • 1:15 Single-Slit…
  • 2:22 Equation
  • 3:20 Calculation Example
  • 4:23 Lesson Summary
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Lesson Transcript
Instructor: David Wood

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

After watching this lesson, you will be able to explain what single-slit diffraction is, describe and picture the pattern that it creates on a distant screen and use the equation of single-slit diffraction to solve problems. A short quiz will follow.

What is Single-Slit Diffraction?

Diffraction is when waves like light or sound spread out as they move around an object or through a slit. Single-slit diffraction is a specific example of diffraction where certain conditions are met: monochromatic (single-color) light is used, the size of the slit is small compared to the distance to the screen and light is shined through a small, thin slit similar in size to the wavelength of the wave going through it. Diffraction, in general, only happens in a noticeable way when the size of the slit or obstacle is comparable to the wavelength of the wave. Sound has a large wavelength, which is why you can hear a car alarm from right around the corner, annoying everyone in the neighborhood, and why you have to cover your ears when your sister screams from downstairs. But light has a tiny wavelength, and so it doesn't diffract enough around large objects.

When you place a screen at a significant distance from a single slit with light shining through it, you get an interference pattern on the screen. But how can a single beam of light interfere with itself? Well, this happens because of Huygen's Principle, which is discussed in more detail in another lesson. But basically, Huygen realized that a single wave of light can be thought of as being made up of lots of tiny wavelets, and so those tiny waves can interfere with each other and produce a pattern of dark and light areas on a screen.

Single-Slit Diffraction Pattern

When you look at the screen, there is a pattern characteristic of a single slit that looks something like this:

Single-slit diffraction pattern
single slit diffraction pattern

It contains a large, wide, bright maxima in the center and alternating bars of dark and light on each side. The pattern is a little different for a larger gap (which is not a slit), and a double slit and a diffraction grating, all of which are discussed in other lessons. A circular aperture pattern is similar, but circular instead of containing bars. But why does a diffraction (or interference) pattern appear at all?

As a wave passes through the single slit, the distance to a particular part of the screen is slightly different from one end of the slit to the other. Because of this, the wavelet that begins from one end of the slit won't arrive at the same time as the other. This means that a peak of one wavelet might hit the screen at the same time as a trough of another wavelet. This creates a dark area on the screen, and is called destructive interference. Alternatively, two troughs could hit the screen together, or two peaks. Either of these cases creates a bright area on the screen, and this is called constructive interference.

Equation

Since this is physics, we of course have to come up with some algebra to describe the pattern. That's just what we do. We want to be able to predict where on the screen there will be a light area and where there will be a dark area. The derivation of the equation is rather involved. The diagram above might help, but just to explain the basic concept, you look at the difference in distance to a particular point on the screen from the two sides of the slit. If that so-called 'path difference' is equal to a whole number of wavelengths, you'll get destructive interference (a dark spot), and if it's equal to a factor of a half of a wavelength, then you'll get constructive interference (a light spot).

Equation for single-slit diffraction
single slit diffraction equation

So the equation comes out like this, where D is the slit width measured in meters, lambda is the light's wavelength also measured in meters, theta is the angle relative to the original direction of the light in degrees, and m is the order of the minimum - that's just a number to say whether you're looking at the first minimum (1), second minimum (2) and so-forth.

Calculation Example

Okay, now let's go through an example of how to USE the equation. Let's say you shine some light of wavelength 6.5 * 10^-7 meters through a slit of width 1.6 * 10^-6 meters. That light's about the wavelength of your average laser pointer, and you're asked to calculate the angle of the second minima from the center of the screen.

First of all, let's note down what we know. We know that the wavelength of light is 6.5 * 10^-7 meters, so that's lambda. And we know that the slit width (D) is 1.6 * 10^-6 meters. And we're asked for the second minima, so m is equal to 2. Do some algebraic rearrangement, and we get sine of the angle is equal to 2(6.5 * 10^-7) / (1.6 x 10^-6). To get rid of the sine part, we have to take the inverse sine of each side of the equation. And then if you type it all into a calculator you get 54.3 degrees. And that's it, we're done!

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