Young's double slit experiment is a classic demonstration of the wave-like nature of light and one of the most important experiments in the history of physics. It was conducted by the English physicist Thomas Young in 1801 and provided strong evidence in favor of the wave theory of light, which had been proposed by Huygens in the 17th century. The experiment involves shining a beam of light through two narrow slits and observing the interference pattern that appears on a screen behind the slits.

One of the more prominent aspects of waves is their ability to ** diffract** off opaque materials. Diffraction can be thought of as the “bending” of the wave around obstacles. The most common explanation for this odd phenomenon in light waves is called

**Huygens principle**which treats the

**wavefront**of a light beam as made of many point-sources of light which evolve to produce the next phase of the wave front. According to Huygens principle, these two slits (if narrow enough) behave as two coherent point sources of light.

This behavior of light is best demonstrated with Thomas Young’s 1800s **double slit** experiment which greatly improved our understanding of light behavior. In the double slit experiment, a wave reaches an obstacle with two narrow openings. According to Huygens principle, these two slits (if narrow enough) behave as two coherent point sources of light.

The waves coming from these slits __interfere__ and produce what we call an **interference pattern.**

Geometry of the Double Slit Experiment

In the __post on interference__ we have seen that the intensity depends on the phase difference, which in turn depends on the optical path difference between two waves. We consider a double slit geometry in order to produce a general dependence of the path difference on the physical parameters of the system - the width between the two slits, the distance of the interference pattern from the screen and the wavelength of the incident beam.

In this scheme, the path difference of two rays coming out of the two slits is given by the length BD which is approximately

By considering a screen which is far from the slits (L>>d), we can relate the angle theta to the point on the screen, given by a coordinate (y).

Combining those two equations, we can the path difference for a give y point as follows

We know that the intensity is maximal (constructive interference) for integer multiples of wavelengths (again - consult the post on __interference__ if you're not sure why) which leaves us with the famous result

Where m is some integer. Those are the points on the screen on which constructive interference occur, and which to us are seen as the point of maximum brightness.

In between are the points of destructive interference - the patches of darkness as seen in the figure above. These patches of darkness are essentially the ultimate proof that light is a wave, since there is no reason to think that particle-like objects would be able to produce this pattern in which certain areas remain vacant of them. It is easy to see that the distance between two successive bright spots is constant

This equation allows us to calculate (given a certain wavelength of light) the effects of the screen distance (L) and the slit-width (d) on the density of the interference pattern. As one takes the screen further away from the slits (makes L larger) the distance between peaks increases linearly. On the other hand, decreasing the width between the slits causes the pattern to spread more.

Franhaufer Diffraction

The Our analysis thus far is only able to account for points for miximal (constructive) or minimal (destructive interference). Maybe the most rigorous way of getting a more general result which accounts for the intensity on every point on the screen is calculating the **Franhaufer integral** which makes direct use of the Huygens principle which is the underlying explanation for diffraction. We might get into what is the Franhaufer integral and how it useful in other cases in later post in the future, but for now we present here the result for the double slit intensity

where we introduce the sinc function which is defined as follows

We also introduce the a new parameter - **b** which is the width of each slit. Our initial discussion assumed that the slits we infinitely thin and that the sources of light are points, but in practice the slits have finite width and each acts as a single slit on its own. This is why the double slit pattern **contains** the __single-slit__ pattern.

In the graph we can see the result of the Franhaufer diffraction integral in blue, which consists of the fast oscillations due to the interference from the two slits, but also slow oscillations (the dotted black line) which result from the single slit diffraction due to each slit independently.

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