In this explainer, we will learn how to describe the interference patterns produced by waves that are diffracted by passing through gaps and traveling different distances.
Light waves from a light source travel away from the source in all directions. A barrier containing a small gap can be placed near a light source. The gap is sometimes called an aperture or a slit. Only light waves traveling through the gap pass to the other side of the gap, as shown in the following figure.
The figure, which shows the waves that pass through the gap, is in fact not fully correct, however. A more correct representation of the wave fronts of these waves is shown in the figure below.
We can see in this figure that these wave fronts become longer as they move away from the gap. Waves that pass through the gap therefore change direction.
It is important to understand that the change in the direction of the light waves is neither due to the waves being reflected from the boundary nor is it due to the the light waves being refracted.
Light waves can change direction due to reflection and also due to refraction. However, the phenomenon being described in this explainer is a separate phenomenon from reflection and to refraction.
In both reflection and refraction, light waves must be incident at a boundary between regions containing substances with different refractive indices. In this case, though, the same substance occupies the space on either side of the gap and also within the gap. No change in refractive index occurs.
The process of light waves changing direction without the waves being incident at a boundary between regions with different refractive indices is called diffraction.
Diffraction does not only occur at gaps. Diffraction also occurs when light waves travel parallel to a surface and reach the end of the surface. This is represented in the following figure.
The amount of direction change of the waves can vary when light waves are diffracted. The amount of direction change is called the diffraction angle. The diffraction angle for light waves passing through a gap is shown in the following figure by the angle .
For light waves diffracted by passing through a gap in a barrier, the diffraction angle depends on the width of the gap and on the wavelength of the light. The angle of diffraction must be less than 90 degrees.
Let us now look at an example concerning diffraction.
Example 1: Defining Diffraction
Which of the following is a correct definition of diffraction?
- Diffraction is the change in direction of a wave that passes close to an object and changes its direction by an angle smaller than 90 degrees.
- Diffraction is the change in direction of a wave that passes from one medium into another with a different density.
- Diffraction is the change in wavelength of a wave that passes through an aperture.
- Diffraction is the change in direction of a wave that passes close to an object and changes its direction by an angle greater than 90 degrees.
- Diffraction is the change in speed of a wave that passes through an aperture.
Answer
Light waves changing direction when passing through an aperture is an example of diffraction.
Diffraction will occur if the medium that the wave travels in on either side of the aperture and within the aperture is the same. This means that the density of the medium in which a wave moves is not part of a correct definition of diffraction.
The speed at which a light wave moves depends on the refractive index of the medium in which the light travels. Light can diffract while traveling in a medium with a constant refractive index. This means that a change in the speed of a light wave is not part of a correct definition of diffraction.
The wavelength of a light wave of a given frequency depends on the refractive index of the medium in which the light travels. Light can diffract while traveling in a medium with a constant refractive index. This means that a change in the wavelength of a light wave is not part of a correct definition of diffraction.
The only options that are not ruled out are that diffraction is the change in the direction of a wave that passes close to an object. The angle of the change in the direction of the wave is less than 90 degrees in one option and more than 90 degrees in the other option.
Diffraction of light waves cannot reverse the direction in which the waves travel. The angle of diffraction of diffracted light cannot be more than 90 degrees, and, in fact, it must be less than 90 degrees.
There is a relationship between the diffraction angle for diffracted light passing through a gap, the wavelength of the light, and the width of the gap.
Relationship: The Wavelength of Light Diffracted by an Angle and the Width of a Gap That the Light Passes Through
The smaller the width of a gap, the greater the diffraction angle of light passing through it.
The following figure shows the diffraction of light waves through three gaps of different widths. Light with the same wavelength, , passes through each gap.
We can see that the greatest diffraction angle occurs for the narrowest gap. However, it is important to understand that when light passes through a gap narrower than the wavelength of the light, although the light is diffracted, this does not result in the production of an observable diffraction pattern of the kind that will be described later in this explainer.
In the diagram, we can see that the wave fronts of the wave that passes through the widest gap have a noticeably different shape to light passing through the other gaps.
We can also see that the wave fronts of the light that passes through the widest gap are wider just after passing through the gap than those of light passing through the other gaps.
Light that passes through the gap of width diffracts with a greater angle than light that passes through the widest gap, however.
The following figure shows what would happen to the wave fronts of the waves passing through the widest gap and through the gap of width after they had both traveled some distance from the gaps.
We see that the spreading of the wave fronts from the gap of width is greater than those from the wider gap beyond a certain distance from the gaps, shown by the green line in the following figure.
We can see that because the slope of the white line is greater than the slope of the black line, for horizontal distances from the slit greater than that to the green line, the spreading of wave fronts is greater for the slit of width .
We see also that near the gap, the shapes of the wave fronts are quite different, but as the distance from the gap increases, more the wave fronts of the waves from both slits approximate those of plane waves.
When light travels a distance from a gap much greater than the width of the gap, the width of the gap is negligible compared to the distance traveled from the gap by the light. The angle of diffraction can be approximated as being measured from the center of the gap when this is the case, as shown in the following figure.
When considering diffraction in this explainer, we will only consider light at horizontal distances from a gap greater than those corresponding to the distance to the green line in the preceding figures.
Let us now look at an example involving diffraction angles for light passing through different gaps.
Example 2: Comparing the Diffraction of Light Waves that Pass Through Different Gaps
Light passes through gaps in a screen, A, B, and C, as shown in the diagram. The light passing through each gap falls onto the screen perpendicularly. The wavelength of the light through each of the gaps is the same but the gaps are of different widths. The light passing through each gap is diffracted. After moving a distance past the gaps, the wave fronts of the waves one wavelength apart are shown; is much greater than the wavelength, , of the waves. The lengths of the leading wave fronts , , and vary depending on the size of the gap that the light passed through.
- Which of the gaps is the nearest in width to the wavelength of the light passing through it?
- Which of the gaps is the furthest in width from the wavelength of the light passing through it?
Answer
Part 1
The magnitudes of the lengths , , and show the magnitude of the spreading out of the wave fronts of the light that has passed through the gaps A, B, and C.
The question states that the distance is much greater than the wavelength , so diffraction angles can be taken as being from the center of a gap.
The greatest spreading out of wave fronts occurs for the greatest diffraction angle, as shown in the following figure.
The greatest diffraction angle occurs for a gap of width equal to the wavelength of the light passing through it. The magnitude of is greatest, so gap A must be closest in width to the wavelength of the light.
Part 2
The magnitudes of and are both less than that of , so gaps B and C must have widths less close to that of the wavelength of the light. The smallest diffraction angle occurs for gap B. Gap B must be furthest in width from the wavelength of the light.
Light from a single light source can be diffracted from two gaps that are so close together that the diffracted light from each gap passes through the same space. The wave fronts of the light from each gap overlap each other when this happens. This is shown in the following figure.
Where the wave fronts from each gap overlap, light waves interfere. Light waves that have traveled equal distances interfere constructively. This is shown in the following figure.
The blue and red lines start at the light source. We can see that the total length of the two blue lines equals the total length of the two red lines. This means that at the point on the overlapping wave fronts where the blue and red lines meet, light waves from the light source have traveled the same distance from the source.
When the light waves have traveled equal distances from the light source, the difference between the distance that the waves have traveled is zero.
Waves do not need to have traveled equal distances to interfere constructively, they also interfere constructively if the distance that they have traveled from their source is different by an integer number of their wavelengths.
Any integer number of wavelengths difference results in constructive interference, so interference is constructive when the number of wavelengths difference between the distance traveled by the waves has a value of where is an integer.
Light waves from the gaps can also interfere destructively. This happens when the number of wavelengths difference between the distance traveled by the waves has a value of where is an integer.
Let us now look at an example concerning interference of diffracted light waves.
Example 3: Identifying Constructive and Destructive Interference in Diffracted Light Waves
The diagram shows the wave fronts of two waves that have been diffracted through equally narrow gaps. Both waves have the same speed, wavelength, frequency, and initial displacement as each other.
- How many wavelengths of this light is the left-hand gap from point A?
- How many wavelengths of this light is the right-hand gap from point A?
- Is the interference between the two light waves at point A constructive or destructive?
- How many wavelengths of this light is the left-hand gap from point B?
- How many wavelengths of this light is the right-hand gap from point B?
- Is the interference between the two light waves at point B constructive or destructive?
- How many wavelengths of this light is the left-hand gap from point C?
- How many wavelengths of this light is the right-hand gap from point C?
- Is the interference between the two light waves at point C constructive or destructive?
- How many wavelengths of this light is the left-hand gap from point D?
- How many wavelengths of this light is the right-hand gap from point D?
- Is the interference between the two light waves at point D constructive or destructive?
Answer
There are three parts of this question for each of the points A, B, C, and D. For each point, the three questions are the same. The questions are as follows: How many wavelengths has light traveled from the left gap? How many wavelengths has light traveled from the right gap? Is the interference constructive or destructive?
Let us consider the diagram shown.
All points that intersect an orange wave front have traveled an integer number of wavelengths from the left gap. The wave front nearest the gap is 1 wave front from the gap, the next wave front is 2 wave fronts from the gap, and so on.
The same applies to the blue wave fronts, except that for these wave fronts, the distances are a number of wavelengths from the right gap.
The interference at a point is constructive if the number of wavelengths at a distance from both the left gap and right gap are integers.
If the distances from the two gaps at a point is different by half a wavelength, then the interference at that point is destructive.
We can now answer the questions for each point.
Parts 1 to 3
Point A is 2 wavelengths from either gap. The distances traveled by the waves are equal. The interference at point A is therefore constructive.
Parts 4 to 6
Point B is 4 wavelengths from either gap. The distances traveled by the waves are equal. The interference at point B is therefore constructive
Parts 7 to 9
Point C is 3 wavelengths from the left gap. Point C is halfway between the third and fourth wave front from the right gap. Point C is then 3.5 wavelengths from the right gap. The difference in the wavelengths traveled by waves from the two gaps at point C is
The interference at point C is therefore destructive.
Parts 10 to 12
Point D is 3 wavelengths from the left gap and 2 wavelengths from the right gap. The difference in the wavelengths traveled by waves from the two gaps at point D is
The interference at point D is therefore constructive.
At a point where two light waves interfere constructively, the amplitude of the light waves is the sum of the amplitudes of the two light waves. Where light waves interfere destructively, the amplitude is zero.
Let us consider the wave fronts of light diffracted from two nearby gaps. This is shown in the following figure.
We can see that for some diffraction angles, the interference is constructive, and for others, the interference is destructive.
We see that parallel to the gaps, at the point halfway between them, interference is constructive.
We can also see that the angles for constructive and destructive interference alternate symmetrically on either side of the gap.
If the light diffracted from two nearby gaps is incident on a screen perpendicular to the gaps, it produces a pattern of alternating light and dark regions. This is shown in the following figure.
We see that the brightest part of the pattern is at the center. On either side of the center, there are symmetrically alternating bright and dark regions called fringes. The brightness of the middle of a bright fringe decreases with distance from the center of the pattern.
It was mentioned earlier in this explainer that light passing through a gap narrower than the wavelength of the light would not produce a diffraction pattern.
We recall that the diffraction angle for light diffracted by a gap narrower than the wavelength of the light was very great. The diffraction angle would in fact be sufficiently great that the entire pattern would consist of a single bright region. Only when multiple bright regions that are separated by dark regions are observed is this considered a diffraction pattern.
Let us now look at an example involving a diffraction pattern.
Example 4: Identifying a Pattern Produced by Light Diffracted through Nearby Gaps
A light source emits light that passes through two narrow slits and then falls onto a screen, as shown in the diagram. Which of the four patterns on the screens A, B, C, and D most correctly shows the pattern that would be produced on the screen by the light diffracted through the slits?
Answer
Light passing through the slits will form a pattern of alternating light and dark fringes. This is not present in pattern B, so we can eliminate it.
The brightest part of the pattern produced will be at the center of the pattern. In pattern C, the center of the pattern is a dark fringe. We can eliminate pattern C.
It is not clear whether the center of pattern D is bright or dark. Pattern D is very asymmetrical. The pattern produced would be symmetrical. We can eliminate pattern D.
Pattern A has a bright fringe at its center. On either side of the center, there are symmetrically alternating bright and dark fringes. Pattern A is the pattern that would be produced.
There is nothing special about the number of gaps that light diffracts from being equal to two. Similar patterns are produced by greater numbers of gaps than two. Also, a similar pattern is produced by just one gap.
Let us look at an example involving the patterns produced by light diffracting through different numbers of gaps.
Example 5: Relating the Pattern Produced by Diffraction of Light Passing through Gaps to the Number of Gaps
Which of the following statements most correctly describes the relationship between the number of nearby gaps that waves pass through and the diffraction pattern produced by the waves?
- A diffraction pattern is produced by any number of gaps.
- A diffraction pattern is only produced if there is one gap.
- A diffraction pattern is only produced if there are two gaps.
- A diffraction pattern is only produced by a number of gaps equal to the wavelength of the waves.
Answer
It is familiar from many examples that light diffracted through two nearby gaps produces a pattern. The pattern is produced by interference of light passing through the gaps. If more gaps are added, light diffracted through the added gaps will interfere with light diffracting through the originally existing gaps. This means that a pattern will be produced if the number of gaps is greater than two.
The only option that is consistent with this fact is that a pattern will be produced by any number of gaps. It is important to recognize that this includes the number one. A diffraction pattern will be produced by light passing through a single gap.
Let us now summarize what has been learned in this explainer.
Key Points
- The process of light waves changing direction without the waves being incident at a boundary between regions with different refractive indices is called diffraction.
- Diffraction occurs when light waves pass through a gap or travel parallel to a surface and reach the end of the surface.
- The angle of diffraction of light waves is maximum for a gap with a width equal to the wavelength of the light waves.
- Diffraction of light waves produces both constructive and destructive interference.
- Diffracted light incident on a screen perpendicular to gaps that diffract the light produces a pattern of bright and dark fringes.
