Euler’s spiral

When talking diffraction, one of the more amusing curves is the curve showing the intensity of light near the edge of a shadow. It is shown below.

Fig 30-9

Light becomes more intense as we move away from the edge, then it overshoots (so it is brighter than further away), then the intensity wobbles and oscillates, to finally ‘settle’ at the intensity of the light elsewhere.

How do we get a curve like that? We get it through another amusing curve: the Cornu spiral (which was re-named as the Euler spiral for some reason I don’t understand), which we’ve encountered also when adding probability amplitudes. Let me first depict the ‘real’ situation below: we have an opaque object AB, so no light goes through AB itself. However, the light that goes past it, casts a shadow on a screen, which is denoted as QPR here. And so the curve above shows the intensity of the light near the edge of that shadow.

Fig 30-7

The first weird thing to note is what I said about diffraction of light through a slit (or a hole – in somewhat less respectful language) in my previous post: the diffraction patterns can be explained if we assume that there are sources distributed, with uniform density, across the open holes. This is a deep mystery, which I’ll attempt to explain later. As for now, I can only state what Feynman has to say about it: “Of course, actually there are no sources at the holes. In fact, that is the only place that there are certainly no sources. Nevertheless, we get the correct diffraction pattern by considering the holes to be the only places where there are sources.”

So we do the same here. We assume that we have a series of closely spaced ‘antennas’, or sources, starting from B, up to D, E, C and all the way up to infinity, and so we need to add the contributions – or the waves – from these sources to calculate the intensity at all of the points on the screen. Let’s start with the (random) point P. P defines the inflection point D: we’ll say the phase there is zero (because we can, of course, choose our point in time so as to make it zero). So we’ll associate the contribution from D with a tiny vector (an infinitesimal vector) with angle zero. That is shown below: it’s the ‘flat’ (horizontal) vector pointing straight east at the very center of this so-called Cornu spiral.

Fig 30-8

Now, in the neighborhood of D, i.e. just below or above point D, the phase difference will be very small, because the distance from those points near D to P will not differ much from the distance between D and P (i.e. the distance DP). However, as h increases, the phase difference will become larger and larger, it will not increase linearly with h but, because of the geometry involved, the path difference – and, hence, the phase difference (remember – from the previous post – that the phase difference was the product of the wave number and the difference in distance) will increase proportionally with the square of h. In fact, using similar triangles once again, we can easily show that this path difference EF can be approximated by EF ≈ h2/s. However, don’t lose sleep if you wouldn’t manage to figure that out. 🙂

The point to note is that, when you look at that spiral above, the angle of each vector that we’re adding, increases more and more, so that’s why we get a spiral, and not a polygon in a circle, such as the one we encountered in our previous post: the phase differences there were linearly proportional and, hence, each vector added a constant angle to the previous one. Likewise, if we go down from D, to the edge B, the angles will decrease. Of course, if we’re adding contributions to get the amplitude or intensity for point P, we will not get any contributions from points below B. The last (or, I should say, the first) contribution that we get is denoted by the vector BP on that spiral curve, so if we want to get the total contribution, then we have to start adding vectors from there. [Don’t worry: you’ll understand why the other vectors, ‘down south’, are there in a few minutes.]

So we start from BP and go all the way… Well… You see that, once, we’re ‘up north’, in the center of the upper-most spiral, we’re not adding much anymore, because the additional vectors are just sharply changing direction and going round and round and round. In short, most of the contribution to the amplitude of the resultant vector BP∞ is given by points near D. Now, we have chosen point P randomly, and you can easily see from that Cornu spiral that the amplitude, or the intensity rather (which is the square of the amplitude) of that vector BP∞, increases initially, to reach some maximum, depending upon where P is located above B, but then it falls and oscillates indeed, producing the curve with which we started this post.

OK. […] So what else do we have here? Well… That Cornu spiral also shows how we should add arrows to get the intensity at point Q. We’d be adding arrows in the upper-most spiral only and, hence, we would not get much of a total contribution as a result. That’s what marked by vector BQ. On the other hand, if we’d be adding contributions to calculate the intensity at a point much higher than P, i.e. R, then we’d be using pretty much all of the arrows, down from the spiral ‘south’ all the way up to the spiral ‘north’. So that’s BR obviously and, as you can see, most of the contribution comes, once again, from points near D, so that’s the points near the edge. [So now you know why we have an infinite number of arrows in both directions: we need to be able to calculate the intensity from any point on the screen really, below or above P.]

OK. What else? Well… Nothing. This is it really − for the moment that is. Just note that we’re not adding probability amplitudes here (unlike what we did a couple of months ago). We’re adding vectors representing something real here: electric field vectors. [As for how ‘real’ they are: I’ll entertain you about that later. :-)]

This was rather short, isn’t it? I hope you liked it because… Well… What will follow is actually much more boring, because it involves a lot more formulas. However, these formulas will help us get where we want to get, and that is to understand – somehow, if only from a classical perspective – why that empty space acts like an array of electromagnetic radiation sources.

Indeed, when everything is said and done, that’s the deep mystery of light really. Really really deep. 

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