Reunweaving The Rainbow

by Lucas Wilkins

(* Warning — Some readers may find this massively pedantic *)

A while ago I did some research into rainbows. I’d been reading about Newton and his rings, and thinking about what people say about him and his opinion on the number of colours in a rainbow. In which way these “principal colours” were important to him is a little unclear (“because they pass into one another by insensible gradation” !?!!?!, link), but it seems to me that he understood the relation between the continua of pitch and hue and their (somewhat arbitrary) relation to the respective discrete scales rather well. Anyway, I thought I would have a go at asking how many colours there are in a rainbow, perhaps there would be a sensible answer… I gave up pretty quickly, because I noticed something else.

The first thing I did was get a prism, and have a look at what it does to the light. Well actually the first thing I did was write a computer program that simulated it, then I actually got a real prism (guess that says something about me). “Hang on, That’s not right!” I thought. Naively, I expected to see a rainbow, but I didn’t!!!!!  I saw a spectrum for sure, but I didn’t see a rainbow.  This is the sort of thing you see (of course its a (probably enhanced) photo):

A prism doing its thing

(taken by D-Kuru/Wikimedia Commons)

And a rainbow, in comparison, looks more like this:

A Rainbow

They really are quite different, In particular, notice the turquoise,  violet and multiple bands in the second one. The huge amount of blue in the first.

So I thought I would  simulate an compare them. Here are the results plotted as a line in the CIE u* v* colour space. For the prism I have used Monte Carlo (a got bored waiting hence the wibbly) rays from a slit traced onto a planar screen. I assumed Sellmeier type dispersion for BK7 glass. For the rainbow I used 0.1nm resolution Mie scattering simulations of 300micron water droplets (refractive index data from “The International Association for the Properties of Water and Steam” (link)… I love that such an institution exists). I have only plotted the part of the rainbow on the inside of Alexanders dark band.

Dispersion by A Prism in CIE u*v*

Dispersion by A Prism in CIE u*v*

A Rainbow in CIE u*v*

A Rainbow in CIE u*v*

Thing thing to note here is how the graphs look absolutely nothing like each other. The source of the difference is in the added wave interference effect (not a technical term) found in the interaction of light with the tiny little water droplets (Kat, if you’re reading this, another example of how big things are different to small things).

The majority of my colleagues I have mentioned this to have been quite surprised, philosophers and physicists alike. I started to think that actually, most people think a prism makes a rainbow – despite the fact that they look completely different. I would guess that pretty much everyone in the developed world has at some point in their lives performed the experiment in the first picture. Has no one ever noticed this? perhaps the ones who do notice the difference have it beaten out of them in the playground (or by their teacher). I hadn’t noticed, even though I knew the physics behind the two phenomena are different.

I investigated further. A search on google for prism yields a whole bunch of depictions of dispersion, many of which have been manipulated to make it look like the prism produces the correct colours for a rainbow (draw what you know, not what you see). CONSPIRACY!!!! It goes quite deep: a quote from the Wikipedia page on prisms: “A prism can be used to break light up into its constituent spectral colors (the colors of the rainbow)”. refractive index: “[...] and it is what causes a prism to divide white light into its constituent spectral colors, explains rainbows”… and so on, The myth is perpetuated…

The science behind what I’m presenting here isn’t unknown, in fact, it is very well established. I mention it only because I am fascinated by the extent of the conflation. Obviously, it doesn’t effect many peoples lives in any practical way. But on the other hand, rainbows get a disproportionate amount of attention. They’re a notable battleground in the conflict between art and science. Yet somehow there is this wide misunderstanding. Most likely it comes from (incorrectly) using rainbows as examples of dispersion in schools.

More info about atmospheric optics, All code available on request.

L

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11 Responses to “Reunweaving The Rainbow”

  1. Hey dude! Just got to reading this one :) Just have a couple of questions

    So am I correct in thinking the problem you have here is semantics? and

    Does that mean to say that you reserve ‘rainbow’ to mean a continuos spectrum of colours with continuos-equivalent spectral densities at each wavelength? probably not?

    because there’s no standard for a rainbow though right, it depends on the water droplet size mixture (there will be some sort of mental distribution, I think it’s a power law without looking), the finite diameter of the sun, aerosol content, the time of day etc. etc.

    Eventually, every rainbow is different – so the only definition I can arrive at is that it’s a separation of white light into some arrangement of colours.

    So the problem you have is the conflation of public awareness of a ‘rainbow’ from a prism and a ‘rainbow’ from a rainbow being identical? As in sound of music stuff being identical to rainbows and prisms in real life? I think as long as it helps people understand that it essentially comes from the similar dispersive mechanisms this is an example of simplification for understanding, the complexity can be added in just a couple of minutes explanation later.

    Just a couple of other things that you’re probably aware of:

    The bulb used in the picture from wikipedia has a very different spectrum to the spectrum of the sun so the spectral content will be different and

    the rainbow in real life is occurring on a background of Raleigh scattering – spectral densities at those blue wavelengths for the rainbow are reduced through scattering so appearances are going to be deceiving.

    Hahaha, you did warn that it might be a bit pedantic, I think this one is a little bit for me :)

    I hope you are doing well, I still tell people about the letter you sent with Mischa! That and the cockcopter. Also, the ninjas came in very handy in the defence. What research are you up to at the moment? Do you have anything I can read? Behrens.paul@gmail.com is the address if you have time to send something.

    • So am I correct in thinking the problem you have here is semantics?

      Not at all

      and

      Does that mean to say that you reserve ‘rainbow’ to mean a continuos spectrum of colours with continuos-equivalent spectral densities at each wavelength? probably not?

      Hell no!

      because there’s no standard for a rainbow though right, it depends on the water droplet size mixture (there will be some sort of mental distribution, I think it’s a power law without looking), the finite diameter of the sun, aerosol content, the time of day etc. etc.

      indeed, it’s fucked: http://www.philiplaven.com/p2.html

      Eventually, every rainbow is different – so the only definition I can arrive at is that it’s a separation of white light into some arrangement of colours.

      How about – “those colourful things that you sometimes see opposite the sun after it rains?”
      Also, the difference between different rainbows or different white light dispersions is far smaller than between the two classes.

      So the problem you have is the conflation of public awareness of a ‘rainbow’ from a prism and a ‘rainbow’ from a rainbow being identical? As in sound of music stuff being identical to rainbows and prisms in real life? I think as long as it helps people understand that it essentially comes from the similar dispersive mechanisms this is an example of simplification for understanding, the complexity can be added in just a couple of minutes explanation later.

      Superficially thats my problem, but it’s just an example of a much bigger problem. One which I will explain soon I reckon. The interesting thing though, is that every scientist I have spoken to about it thought they were qualitatively the same, which they certainly are not. The amount of prisms I seen dawn with a pink is quite remarkable too.

      Just a couple of other things that you’re probably aware of:

      The bulb used in the picture from wikipedia has a very different spectrum to the spectrum of the sun so the spectral content will be different and

      Sure, it’s one of the reasons why I did calculations instead of using the images. That and the problem of the difference between cameras spectral sensitivity and our eyes spectral sensitivity – and the obvious photoshopping.

      the rainbow in real life is occurring on a background of Raleigh scattering – spectral densities at those blue wavelengths for the rainbow are reduced through scattering so appearances are going to be deceiving.

      The choice of colour space goes a long way to reducing this – but not all the way I admit.

      • I think I used actual recordings of sunlight for the calculations, It may be blackbody though. I seem to have forgotten to add it to the post, I’ll try and remember to look at the calculation when I don’t have to do it remotely.

      • “How about – “those colourful things that you sometimes see opposite the sun after it rains?””

        Right, yes, that is not white light separation that is blackbody then subtracting atmospheric absorption and adding in all the other effects we just mentioned dispersion.

        “Also, the difference between different rainbows or different white light dispersions is far smaller than between the two classes.”

        Sure, that seems reasonable – although I’ve not seen anything on this.

        “Superficially thats my problem, but it’s just an example of a much bigger problem. One which I will explain soon I reckon. The interesting thing though, is that every scientist I have spoken to about it thought they were qualitatively the same, which they certainly are not. The amount of prisms I seen dawn with a pink is quite remarkable too.”

        So that doesn’t explain to me what you are trying to say :( Clearly they are both dispersive – so what is your beef? that they are different types of dispersion and that the conflation of the two is a mistake? Or are you more annoyed about the adding of colours in a prism spectrum to make it match up with rainbows we see in the sky? You’re being very coy about explaining something soon and not mentioning what the bigger problem is so I can get a handle on what you mean!

      • Sure, that seems reasonable – although I’ve not seen anything on this.

        The two graphs are pretty standard. The rainbow will always be a medium loop and a sqiggle, the prism a horse shoe shape.

        Clearly they are both dispersive

        Well, the rainbow differs in having interference effects – I don’t think there’s any “need” for dispersion at all (I think I ran a simulation with constant refractive index and it was very similar, it was over a year ago now). I have some code, will run it – takes a while though due to the numerical instability of the all known Mie scattering solutions.

        You’re being very coy about explaining something soon and not mentioning what the bigger problem is so I can get a handle on what you mean!

        Yes, I am, I will make a post pretty soon though :)

    • I will send you an email soon, personal comments on this blog feel wrong!

  2. Ha! I thought “red and yellow and pink and green and blah blah blah blah” was in the sound of music. Turns out it wasn’t – goes to show how many times I’ve watched that film.

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