The Photon

The photon is the smallest unit of light we can have, in much the same way that the atom is the smallest
unit of matter and the electron the smallest unit of electricity in a conducting wire. Like the electron, and
unlike the atom, the photon is a fundamental particle - it is indivisible, that is it has no smaller parts, and
appears to be a point with no spatial size. Unlike the electron, which has one unit of negative electric
charge, the photon has no electric charge - it is electrically neutral.

Let's put things into perspective. A 100 W light bulb emits about 3 x 10^20 (3 hundred million million
million) photons per second (about 90% of which are infrared and so invisible to our eyes).

On this page we shall look at some key experiments with light and photons and so illustrate some very
unusual and amazing properties of photons!

Experiment 1: Diffraction of light waves

In this experiment we shall shine a laser beam through what is called a single-slit diffraction plate -
simply an opaque piece of material with one tiny slits in it, such that light from the laser can only pass
the grating by moving through the slit. The slit is very narrow (only 0.007 millimetres in diameter). Our
laser happens to be a red laser (wavelength 650 nanometres) though any colour will do, we could even
use white light, which is made up of the full spectrum of colours, but we have chosen a monochromatic
light source to clarify our results. The laser beam also has the advantage of being coherent light,
meaning that it is made up of waves that are all in phase (moving together in synch). We also have our
waves moving in parallel and they are plane waves (straight rather than curved) so that all parts of the
wave arrive at the grating simultaneously. Our laser has a lens in front to focus the laser beam and we
ensure that the beam is focused. We position a white screen about 2 meters from the slit, to catch the
focused image of the light passing through the slit. This arrangement is shown below:

Results:

In the picture above the red lines show the positions of light wave crests (bright regions) as they move toward the screen, in-between each pair of lines is a trough (dark region). We have also indicated the pattern of light seen on the screen. The laser light passes through the slit and then spreads outward or diffracts, forming more or less semi-circular waves. When these strike the screen we see a large bright spot in the center of the image, directly opposite the slit, but surprisingly we see a series of light spots, alternating with dark regions, on either side of the main spot. These fringe spots (called interference fringes) get progressively  smaller and fainter with distance from the central spot, until they are no longer detectable.

This pattern on the screen is presented below, for clarity, with a graph above it showing the variation in light intensity across the screen that produced this pattern (note that for a light wave, a trough corresponds to a dark region):

Is this what we expected to see?

We have said that photons are discrete particles, rather like minute balls. Imagine then that instead of parallel straight waves, we had a beam of tiny spheres all traveling straight toward the grating. We know they cannot penetrate the grating except through the single slit (the material is opaque), so we might expect only those photons that travel straight through the hole to hit the screen, in this case we might expect something like that shown below, and is similar to the pattern obtained by shooting a shot-gun at a screen through a narrow lit:

We get a sharp peak of hits directly opposite the slit, but this sharply falls to zero on either side. Particles, as we ordinarily think of them, do not diffract - our shot gun pellets certainly do not diffract like waves! Our experiment suggest that photons diffract and so are behaving as waves rather than as particles.

However, if the waves spread out in circles as shown, then we might expect to see a bright spot in the center of the screen, directly opposite the slit, and we might expect this to slowly dim on either side, since far from the center the light has traveled further to reach the screen, and we know that light intensity falls with distance. Thus, we might expect something like the following:

This is something like what we would see if diffraction of waves was the only phenomenon taking place. However, this does not quite look like what we saw in our experiment. Our experiment was similar, except instead of a single broad peak, we had a narrow peak and a number of diminishing peaks on either side, with each peak or bright spot separated by a dark region. The pattern that we obtained can be understood by thinking of the waves as diffracting and interfering with one another.

What actually happens is due to the effects of having a slit of definite width (albeit narrow) which causes light from one one of the slit to travel a different distance to light from the center of the slit to reach the same point of the screen, resulting in interference. If we could have an infinitely narrow slit that still let light through, then we would only see the effects of diffraction, as shown above. However, because the slit must in practice have a certain width, this means that light rays or waves passing through the slit near to the edges has further to travel to reach the screen, as we said already, but this results in rays of light passing through the near the edges of the slit being out of phase with those traveling through the slit center. This way of thinking requires some equations, but there is an alternative way of putting it - waves actually spread out from each corner of the slit, and thus overlap with one another as they spread out, as shown below:

This is a more realistic way of looking at things. Remember that when waves overlap they add together in a way such that if their crests coincide then the resultant wave becomes stronger (brighter) - a process called constructive interference; whereas if a trough of one wave overlaps with the crest of the other wave then the two waves cancel out by destructive interference (and so we have a dark region). If we draw lines joining the regions where crests overlap to give the bright regions then we expect these to coincide with the bright spots on our screen, as shown below they do:

Above: a more realistic model in which waves diffract from each corner of the slit and interfere with one another. (Note I drew the slit wider here for clarity). Actually we can imagine a series of wave sources lining the slit, not just one at each corner.

If we observed the light waves in transit to the screen (e.g. by using smoke) then what would we actually see? The actual wave crests will follow contours that are tangent to the various wavelets radiating from our multiple sources inside the slit. This process of imagining multiple sources for diffracting waves, each generating its own set of wavelets, and then using these to construct the overall wave fronts that are actually observed is called Huygen's principle, and is a handy trick for dealing with diffraction. The actual result will look something like that shown below, in which the effect is similar to what we started with originally, but with troughs (dark regions) interspersed along the roughly semi-circular wave fronts, producing the pattern of light and dark bands that we observe. The black lines show these troughs and do indeed point to the dark lines on the screen.

Conclusion:

Thus our pattern of light and dark lines on the screen was due to two wave phenomena: diffraction and interference. This confirms the wave-like behavior of light, contrary to the idea of light being made up of particles called photons.

Experiment 2: Double-slit diffraction of light

This is a version of Young's classic double-slit experiment. We use an identical set-up as for the single-slit experiment (experiment 2 above) but we have two slits in our diffraction plate. The slits are both as narrow as before, and are only 0.4 millimetres apart. What we see is shown below:

Here we have again assumed that each slit acts as a single source of waves, giving us two wave patterns that overlap and interfere, producing a series of bright and dark lines on the screen again. If we draw black lines through the regions where the crests overlap and interfere constructively to give the bright red spots, as before, but this time the pattern is more complicated, each bright spot is broken up by a series of dark lines. The graph shows the pattern of light intensity on the screen, with the dotted line being identical to the pattern observed with a single slit.

Explanation:

If we were only dealing with interference then we would have a series of peaks (bright spots) formed by constructive interference alternating with a series of dark lines (destructive interference) as shown below:

However if we superimpose the effects of diffraction from a single slit, which assumed a series of sources of diffracting waves along the slit, then we get the pattern observed:

Pattern for two-slit interference.

+

Pattern for single-slit diffraction.

=

Pattern for two-slit diffraction.

Pattern for single versus two-slit diffraction compared:

We could draw four wave sources (two for each slit, just as for the single-slit experiment) and show that the bright and dark spots are predicted correctly, but such a diagram gets rather complicated and confusing!

Using single photons:

The interesting thing about the double slit experiment is that it adds further proof to the wavelike nature of light - if a photon is a discrete particle then it should travel through one slit or the other, but not both. Let us put this to the test, after all, suppose photons behave as waves only when they get together in large groups. This is a very interesting twist in the tale! We could reduce the intensity of our laser until it emits only one photon per second. This is possible since we know the wavelength of our laser (650 nm, red light) and a simple equation predicts the energy of a photon at this wavelength. Thus, we can set our laser to emit on average one photon per second.  Now, we have to use a special screen that records a hit every time a photon strikes it and then build up the pattern, photon by photon, perhaps on a computer screen. What we find is that we still get the same pattern of two-slit diffraction with interference - the photon has interfered with itself, meaning that it has travelled through both slights at once, and so is a wave.

Suppose we are not happy that a photon can do this and decide to place sensors on each slit to record a photon as it passes through - now surely the photon must travel through only one slit or the other and not both? Indeed it does! It now behaves like a particle, rather like a shot-gun pellet and the pattern we get has changed to the one shown below:

We have the pattern we would expect for firing a shot-gun through a double slit - our photons have turned from waves into particles!! If we don't measure which slit they pass through, they behave as if they are waves passing through both, but just when we watch which slit they go through we can't actually detect them passing through both slits at once, because just when we watch them closely they turn into particles and only go through one slit or the other - they can not be caught in the act of passing through both slits at once, even though they clearly do when we are not looking!

Actually, what really happens is a little more subtle and depends on how accurately we measure the position of the photons. If they are only measured to be anywhere within the slit then they will behave as waves confined to a single slit and we will see the single-slit diffraction pattern generated by each slit without interference between the slits. However, the more accurately we measure thier position the more particle-like they become!

Tricky chaps these photons! What is really happening is that the photons behave as waves until we measure their position accurately and then the wave collapses into a narrow pulse (called a wave packet) which behaves as a single discrete particle. This is a simple fact - when we measure a photon's position we force it to interact with matter (atoms) in our detector the photon becomes a particle because that is what happens when light interacts with matter - it changes from a wave to a particle. Photons are so small that we can not measure them without changing them in some way. This dual behavior is called wave-particle duality. Actually photons collapse into particles when they hit our screen, but by then they have already behaved like a wave in passing through both slits and so their wave-like behavior becomes detectable. Similar, photons become particles when they strike the retina in an eye.

Conclusion:

Once again the two slit diffraction experiment illustrates the wave-like nature of light and of photons. However, when we accurately measure the position of a photon then the wave collapse into a particle. Photons can sometimes behave as waves and sometimes as particles, depending upon the conditions imposed upon them, this is called wave-particle duality.

Mathematical Treatment

A full mathematical treatment of single and double-slit diffraction is rather involved. A number of geometric tricks exist to predict the positions of maxima or minima. The results of this analysis will be presented below.

The position of the minima (dark spots) in single-slit diffraction are given by:

Where n is the diffraction order: n = 1 corresponds to the first or innermost pair of minima on either side of the central bright spot (central maximum):

Note that essentially the same formula gives the positions of the maxima in double-slit diffraction (and also for multiple-slit diffraction) where this time, n = 0 labels the central maximum and n > 0 labels the pair of maxima on either side and w now represents the center-to-center distance of an adjacent pair of slits rather than the width of a single slit.

A more involved and comprehensive mathematical treatment makes use the mathematical formulations of diffraction derived by Kirchoff (and equivalently by Young) and expressed as the Fresnel-Kirchoff diffraction equation. This equation is quite unwieldy and assumptions are made in order to simplify it. The relevant assumption here is Fraunhofer diffraction which assumes that the incident waves are plane waves. Spherical waves emanating from a source become plane waves at an 'infinite' distance from the source. In fact, Fraunhofer diffraction assumes that the slit is far from the source and that the screen is far from the slit. In the lab this assumption can be made valid by using a diverging lens to convert the diverging incident spherical wave-fronts into plane waves before they arrive at the screen (and a lens to focus the waves leaving teh slit onto the screen). The result is given below for a single slit:

A plot of the intensity function for a wide slit is shown below:

Note that u, in units of pi radians, is given along the horizontal axis and intensity, I, on the vertical axis.

As expected, a narrower slit spreads out the pattern until the interference effect can not be seen (once the angle from the center-line to the screen exceeds 180^o on a flat screen, then no more of the pattern can be seen) and the pattern is pure diffraction, i.e. spreading out of the waves behind the aperture. The pattern predicted when the slit width is equal to the wavelength of the incident light is shown below:

Similarly, for double-slit Fraunhofer diffraction we get the following results:

So now the intensity depends on both the width of each slit (the two slits are assumed to be of equal width a) and the center-to-center distance between the two adjacent slits, b. Note that we have taken the central maximum to be of unit intensity, but the intensity can be scaled according to the units used.

Above: the double-slit intensity pattern is shown in mauve; the red envelope shows the equivalent pattern for single-slit diffraction.

Article updated: 22/6/2020

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