# Single slit diffraction: Why aren't the interference patterns circles?

A picture of the single slit experiment

We were all showed this pictures when started learning about the experiment.

I assumed the picture was just a simplified 2D model from the 3D world, and to get what actually happened, just rotate the everything on the picture around the white axis and result in this pattern (sorry for the terrible drawing): However, experiments have shown that the patterns does indeed appear on 1 single line. Example:

My question is, as light traverse through space in every direction, why isn't the interference pattern look more like the second picture, instead of the third one, which appears like light only travel in one plane.

• If you use a pinhole aperature you do get the concentric-rings pattern. And that works just fine with a laser as a source, so it is easy to do in the modern classroom. Historically though, getting enough light could be a problem which made slits preferable. Feb 18, 2019 at 17:16

So the reason that you don't get something perfectly round is that you have a slit geometry: the light passes through a rectangle of size $$(\delta x, \delta y)$$ but where $$\delta x < \lambda$$ while $$\delta y \gg \lambda.$$

As for why one might prefer this in the classroom, there are a couple reasons but maybe the simplest is that it is relatively easy to devise a mechanism that will let you slowly "narrow" or "open" the slits, while changing the size of a pinhole would really require constructing a custom aperture or disassembling a camera to use that camera's aperture. Either way you'll want a gearing setup to move the thing since we're talking about slits that are a fraction of a hair's breadth, but it almost seems like if one were strapped for resources one might be able to make a single-slit interference experiment out of, say, Lego bricks if needed.

If you can narrow the space that the light is going through, you can establish an essential point: that "going in one coherent direction" as a laser does is actually an interference phenomenon requiring a beam width some wavelengths wide. As you narrow the sides of the rectangle, at first the image appears to get narrower as all the light that would have gone sideways is being blocked by the sides; but then as you keep narrowing them, the image starts to get wider as you get within the size of a wavelength. This is the phenomenon that you are seeing, the light beam spreads out along the $$x$$-axis precisely because that was the axis that it was "squished" along, it has a sharp $$y$$-spread because it has many wavelengths of space to collimate itself along that axis, but it has a broad $$x$$-spread because it was not given so many wavelengths and thus it had to revert towards what a point source of light would do -- go in every direction equally.

One can also see these sidebands emerge, which gives some nice discussion of how to view them as a superposition of point sources along the slit's $$x$$-axis. And one can talk about the interpretation for individual photons as a sort of Heisenberg uncertainty: by making the $$x$$-location well-specified one can say that we made the $$x$$-momentum poorly-specified in inverse proportion: and how this can be seen as a general property of a Fourier transform, that to have a well-specified wavelength in some direction one requires a domain many wavelengths long in that direction. So this prepares students for quantum mechanics. Those aspects are much nicer if we can discuss how the $$x$$- and $$y$$-axes are more or less independent in the analysis.

And, of course, one can prepare to discuss the case of the two-slit experiment after the one-slit experiment, though I think you might have a harder time making that one from Lego bricks, heh.

A circular aperture produces the Airy disk (George Airy, 1801-1892), so indeed it is very 19th Century:

Note that this not the same as rotating the single slit pattern:

$$I(x) \propto (\frac {\sin x}x)^2$$

The circular pattern is:

$$I(r) \propto (\frac {J_1(r)} r)^2$$

where $$J_1(r)$$ is a Bessel function.

Of course Bessel functions are solutions to the wave equation with cylindrical symmetry, while (co)sines are solutions with rectangular symmetry.

If you rotate the system to produce a three dimensional system the slit becomes a circular hole and the diffraction pattern does consist of circles.

However if you stick to a slit in three dimensions then the pattern is a series of vertical lines which is not obvious when a slit is illuminated by a laser beam of small cross sectional area. Think of the pattern produced by a diffraction grating when using a spectrometer where the spectrum (diffraction/interference pattern) is a series of lines.