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I came across these interference patterns obtained from the double slit experiment online.

Interference pattern 1 Interference pattern 2

I was expecting the brightness of the fringes to drop off at some what of a regular rate from the central maximum.
When I say regular I don't necessarily mean "linear". Regular as in continually decreasing/increasing. Continually decreasing in this case. Because as far as I know, the intensity of light follows the inverse square law.
So I expect the intensity to only decrease as we move further away from the central maxima.
But there seem to be distinct bunches/packets of fringes.

In the packet of fringes in the middle the brightness drops of regularly on either side of the central maxima. But at a point the brightness drops of significantly and then rises again.
My questions are:

  1. Why is this so?
  2. Is something else at play here other than the basic two source interference?
Kyathallous
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2 Answers2

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What you are seeing is an overlay of the expected interference pattern from the two slits and the diffraction pattern produced by each individual slit.

Have a look at a single slit diffraction pattern and you will see that it has a wide central maxima flanked by secondary maxima to each side. That forms an envelope of sorts for the more closely spaced minima and maxima of the double slit interference pattern.

This page has some good images and explanations.

I tried to find a good simulation where you can play around with the slit width as well as the spacing to explore this. This is the best I could do for now.

M. Enns
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In the Fraunhofer regime, the diffraction pattern is the Fourier transform of the aperture. The aperture function of a single slit of finite width is a rectangular function and its Fourier transform is a sinc function. The aperture function of two point sources is the sum of two delta functions and its Fourier transform is a cosine function.

Therefore, two identical slits of finite width is the convolution of the two functions mentioned above. In general, there is a straightforward method to find the convolution of a given function $f(x)$ with a sum of delta functions $\sum_i \delta(x-a_i)$: For each delta function in the sum, create a copy of the given function with origin centered on it, and take the sum of the copies i.e. $\sum_i f(x-a_i)$.

The Fourier transform of the convolution is the product of the Fourier transforms of the individual functions, which is why you are seeing (the square of) a double-slit cosine pattern multiplied by the sinc function. Since the width of the slits is usually much smaller than their separation, the sinc function has a much larger scale than the cosine so it appears as an "envelope". This convolution method is an efficient way of constructing diffraction patterns of more complicated apertures out of simpler ones without the need to compute the entire Fourier transform.

Vincent Thacker
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