Young's Double Slit Experiment
Thomas Young, born in 1773, has been referred to as "the last man who knew everything". After his day the rate at which the body of human knowledge grew simply made it impossible for anyone after him to know everything. Of course it's a silly thing to say he knew everything, but he did have his hand in many branches of knowledge - from the natural sciences to medicine to history to Egyptology (he helped translate the Rosetta stone) to biblical studies. There's a good chance that he could have added useful comment to almost any academic conversation of his day.
Our discussion of him will be regarding his wave theory of light, which he held as his most notable accomplishment. After all, in his day, to convince the world that light was a wave also required them to abandon Newton who claimed around a hundred years earlier that light is made of particles. What we will see in later chapters is that they were both right and probably had no idea how wonderfully perplexing light really is!
Young initially (1801) split a very narrow beam of sunlight (which fortuitously is coherent if used in only narrow beams around a tenth of a millimeter or smaller across) with card stock and saw signs of interference. He presented his findings to the world with his double slit experiment later.
What he did was split a narrow beam of sunlight coming through a hole in a window shade with a piece of thin card that was a fraction of a millimeter thick. It resulted in colored fringes on the dark laboratory wall behind the card. If he obstructed the light from passing on one or the other sides, the colored fringes would disappear. He followed this presentation with the mathematics that we will now do, which ultimately convinced the world that light is a wave.
Double Slit Interference
Double Slit Math
The whole idea is that given two paths to follow to a common destination on a screen, the two light waves will travel a different distance to all points on the screen that are not directly in line with the card stock. It is worth asking why light from the sun or a laser that's traveled a long ways in a straight line will tend to curve or bend once it passes through little holes. The detail of the math will come later when we speak of diffraction. For now the fact that it does this is generally referred to as Huygen's principle. This principle says that each little slit will act like a source of spherical waves as seen in the animation above. Where the waves meet we'd have constructive interference. Where they are farthest apart (half a wavelength apart)we'd have destructive interference, or the cancelling of the wave.
It is worth noting that this principle is true of other waves as well - ocean waves coming into a harbor or sound passing through openings such as doorways. In such cases, you can think of the wave front as being made of very many little sources of outgoing spherical waves as seen in the interactive graphic below.
Huygen's Wavelets
Double Slit Math
Regarding the math, we want to be able to add two waves that meet at different points on a screen with some relative phase angle between them. If we can find the difference in path length, then we will be able to do what we've done before, and find the relative phase between the two waves by using , and add the waves using phasors. To find the path length difference requires only some simple geometry.
It can be seen below that the path length difference is approximately While in the diagram it might look like the treatment of the triangle as a right triangle is a stretch, in reality it's quite a good approximation. The problem with the diagram is one of scale. Actual double slits are a fraction of a millimeter apart while the distance to the screen is usually a good part of a meter away. Therefore the two paths are essentially parallel except to the extent that they converge to meet at a single point on the screen a relatively long distance beyond the slits.
Double Slit Geometry
Double Slit Phasor Diagram
Adding the Phasors
The math of adding the phasors for the double slit and for any number of slits is most easily done by setting up the phasors to be symmetric about one of the axes. I chose the x-axis in the diagram above. The two phasors each carry half the wave amplitude (here assumed electric field amplitude of light) and are then mirror images of one another with an angle between them. Adding them together, as seen in the diagram, leads to
What we usually wish to find is the intensity rather than the amplitude of the combined waves. The intensity is proportional to the amplitude squared, so we can write Plugging in our expression above and solving for intensity gives:
Interpretation of the Double Slit Results
Notice that in the function above that there is a delta that stands for the difference in phase between the light reaching the screen after passing through one slit versus the light reaching the screen after passing through the other slit. The path length difference is the entire reason there is a phase difference. As we look at places on the screen farther and farther from the center, the path length difference grows. Since , we can see that at certain values of we should get maxima and minima of the intensity. For instance, the cosine squared function has maxima at values of . Here, the symbol represents the set of natural numbers starting from zero, or {0,1,2,...}. So when we expect maxima of the cosine squared function. These will be the bright spots on the screen. Plugging in for delta and cleaning up the algebra, we find that
AN ASIDE ON NOTATION: I know my use of the set of natural numbers is slightly awkward. I am a physicist, so efficiency is the goal of my notation. In principle, when is written here we should understand that may take on any value such that it is an element of that set. I am not claiming that it is equal to the set itself. The alternative found in other texts where they write 1,2,3,... grows tiresome, as does mentioning notationally each time that N is an element of the set of natural numbers starting from zero.