Physics 204: Spring 2011

Diffraction

Optical interference effects are easy to see with laser light, but first a word of caution: Our lasers are low power, but you must still be VERY CAREFUL not to look into the beam, or even to let laser light reflect from something shiny into your eye. The lasers will be fixed in place, so you will know where the beams are. Nothing reflective should be placed into the beams except the ruler in (I) below (which will also be fixed). Thus it should be easy to avoid any risk.

We’ll look at three different situations where interference has an interesting meaning. You’ll

(1) Measure the wavelength of the laser light (using a machinist’s ruler!)

(2) Measure the regular separation distance of an array of invisible scatterers

(3) Measure the diameter of your hair.

In each case we’ll have to determine the angle at which light is scattered, or occasionally the angle at which light is not scattered. The basic geometry is always the same. The unscattered light (the beam) hits a distant wall, perpendicularly, at a nice, well defined location O. The scattered light hits the same wall at a distance H, measured from O. Both rays come from the scatterer, a distance D from the wall. Therefore the scattering angle A satisfies tan(A)=H/D, or A=arctan(H/D).

I. Measuring the wavelength of light with a ruler

This sounds impossible! The wavelength of light is much too small to be measured with a ruler, isn’t it? As far as I know, this clever trick was invented by Arthur Schawlow, one of the inventors of the laser, and I myself heard him still chuckling over it years later. The trick makes use of diffraction, of course, and also the fact that on a machinist’s ruler the subdivisions are represented by little grooves, precisely scratched at very regular intervals. Thus the ruler is effectively a diffraction grating, although a very coarse one (because the spacings are still much larger than a wavelength of light). To make up for the fact that the grating is so coarse, we will make the laser beam hit the ruler at a very small “grazing” angle A0. The light that reflects specularly from the shiny ruler, as if from a mirror, “scatters” through an angle 2*A0 in the sense that we defined above. That is, the angle that you measure on the wall would be 2*A0. Dividing by 2, you can determine A0.

When you set this up in such a way that the laser light grazes not just the shiny part of the ruler but also the regular array of grooves, you see light reflected at other angles too! These are the angles at which constructive interference occurs in the contributions from the grooves. It is a good idea to determine the angles A1, A2, A3, etc. at which the scattered light leaves the ruler in order to interfere constructively on the wall. The angle you measure on the wall is A0+A1, A0+A2, A0+A3, etc., so you have to subtract A0 to find these angles. The information that constructive interference is occurring at these places indirectly tells us what the wavelength must be.

When we look at the geometry of rays that scatter from two adjacent grooves, separated by distance d, both rays coming at angle A0 to the ruler and leaving at angle An to the ruler, we see that the rays, although they originate in the same place and end at the same place, travel different distances. The difference between one path and the other is d*cos(A0)-d*cos(An). If the two rays are to represent light that interferes constructively, this difference must be an integer multiple of the wavelength of the light, n, where is the wavelength, and n labels the order of the spot (counting from the zeroth order spot of specular reflection).

Measure several angles A0, A1, A2, etc., enough to make a decent determination of . Then show that a graph of cos(An) vs n should be a straight line with slope -/d, and make such a plot, thereby determining . Estimate how far off this determination could be, given the scatter in the data. Notice that there is a systematic error associated with the distance D to the wall. Uncertainty in this distance propagates into all the angle measurements, so comment on this too. This known wavelength will now be our “ruler” to measure other things, below.

II. Measuring the regular separation distance of an array of invisible scatterers

The invisible scatters form, in effect a diffraction grating, although we cannot see it. If we hold it like a transmission grating, perpendicular to the beam, then the condition for constructive interference is just given by the usual grating equation n=d*sin(An). Measure a suitable number of An, and plot in such a way that the data lie on a line with slope d/. (You can do this even if you haven’t determined yet, and come back to interpret it later when you have a value.)

III. Measure the diameter of your hair

If you hold a single hair in the laser beam, you see a familiar diffraction pattern on the distant wall. It is just the single-slit diffraction pattern, as if the beam were going through a narrow slit in an otherwise opaque wall. This might seem rather odd, since the fact of the matter is just the reverse: there is no wall, unless you consider the hair a tiny wall, where the slit should have been. A clever observation, called Babinet’s principle, says that for the purpose of scattering, these two complementary situations give rise to the same diffraction pattern. The argument is that if you could add the light that goes past the hair, on either side, to the light that goes through a slit of the same size, which is just the light that was blocked by the hair, then you would have all the light. But all the light in the beam would just produce a single spot, with no diffraction. The diffraction patterns occur only when we remove some of the light. Thus the light scattered by the hair must be the same as the light scattered by the slit, except for a minus sign, so that if they were added together they would cancel. The intensity of the pattern is the square of the amplitude of the scattered light, so whether it is positive or negative doesn’t matter. The patterns look the same.

Realizing this, you can interpret the pattern as a single slit pattern. In this case destructive interference occurs at angles An such that n=a*sin(An), where a is the diameter of the hair. (Note: this is for any n except 0!) Measure several An, and determine a by means of an appropriate plot of the data.

How does your hair compare with your friends’ hair?