Single-Slit Diffraction

If we use only a single slit, is there diffraction? If the slit is not wide compared to a wavelength of light, then we can approximate its behavior by using only a single set of Huygens ripples. There are no other sets of ripples to add to it, so there are no constructive or destructive interference effects, and no maxima or minima. The result will be a uniform spherical wave of light spreading out in all directions, like what we would expect from a tiny lightbulb. We could call this a diffraction pattern, but it is a completely featureless one, and it could not be used, for instance, to determine the wavelength of the light, as other diffraction patterns could.

(a) Single-slit diffraction of water waves. (PSSC Physics.)

All of this, however, assumes that the slit is narrow compared to a wavelength of light. If, on the other hand, the slit is broader, there will indeed be interference among the sets of ripples spreading out from various points along the opening. Figure (a) shows an example with water waves, and figure (b) with light.

(b) Single-slit diffraction of red light. Note the double width of the central maximum. (Photo by the author.)

(c) A pretty good simulation of the single-slit pattern of figure (a), made by using three motors to produce overlapping ripples from three neighboring points in the water. (PSSC Physics)

Self-Check	How does the wavelength of the waves compare with the width of the slit in figure (a)?
Answer	Judging by the distance from one bright wave crest to the next, the wavelength appears to be about 2/3 or 3/4 as great as the width of the slit.

We will not go into the details of the analysis of single-slit diffraction, but let us see how its properties can be related to the general things we've learned about diffraction. We know based on scaling arguments that the angular sizes of features in the diffraction pattern must be related to the wavelength and the width, a, of the slit by some relationship of the form

This is indeed true, and for instance the angle between the maximum of the central fringe and the maximum of the next fringe on one side equals 1.5λ/ a. Scaling arguments will never produce factors such as the 1.5, but they tell us that the answer must involve λ/a, so all the familiar qualitative facts are true. For instance, shorter-wavelength light will produce a more closely spaced diffraction pattern.

An important scientific example of single-slit diffraction is in telescopes. Images of individual stars, as in the figure above, are a good way to examine diffraction effects, because all stars except the sun are so far away that no telescope, even at the highest magnification, can image their disks or surface features. Thus any features of a star's image must be due purely to optical effects such as diffraction. A prominent cross appears around the brightest star, and dimmer ones surround the dimmer stars. Something like this is seen in most telescope photos, and indicates that inside the tube of the telescope there were two perpendicular struts or supports. Light diffracted around these struts. You might think that diffraction could be eliminated entirely by getting rid of all obstructions in the tube, but the circles around the stars are diffraction effects arising from single-slit diffraction at the mouth of the telescope's tube! (Actually we have not even talked about diffraction through a circular slit, but the idea is the same.) Since the angular sizes of the diffracted images depend on λ/a, the only way to improve the resolution of the images is to increase the diameter, a, of the tube. This is one of the main reasons (in addition to light-gathering power) why the best telescopes must be very large in diameter.

Self-Check	What would this imply about radio telescopes as compared with visible-light telescopes?
Answer	Since the wavelengths of radio waves are thousands of times longer, diffraction causes the resolution of a radio telescope to be thousands of times worse, all other things being equal.

Discussion Questions

A	Why is it optically impossible for bacteria to evolve eyes that use visible light to form images?

1. A diffraction pattern formed by a real double slit. The width of each slit is much bigger than the wavelength of the light. This is a real photo. 2. This idealized pattern is not likely to occur in real life. To get it, you would need each slit to be comparable in size to the wavelength of the light, but that's not usually possible. This is not a real photo. 3. A real photo of a single-slit diffraction pattern caused by a slit whose width is the same as the widths of the slits used to make the top pattern. (Photos by the author.)

Double-slit diffraction is easier to understand conceptually than singleslit diffraction, but if you do a double-slit diffraction experiment, in real life, you are likely to encounter a complicated pattern like pattern 1 in the figure above, rather than the simpler one, 2, you were expecting. This is because the slits are not narrower than the wavelength of the light being used. We really have two different distances in our pair of slits: d, the distance between the slits, and w, the width of each slit. Remember that smaller distances on the object the light diffracts around correspond to larger features of the diffraction pattern. The pattern 1 thus has two spacings in it: a short spacing corresponding to the large distance d, and a long spacing that relates to the small dimension w.