Photographing a double rainbow (Shallow Thoughts)

Photographing a double rainbow

The wonderful summer thunderstorm season here seems to have died down. But while it lasted, we had some spectacular double rainbows. And I kept feeling frustrated when I took the SLR outside only to find that my 18-55mm kit lens was nowhere near wide enough to capture it. I could try stitching it together as a panorama, but panoramas of rainbows turn out to be quite difficult -- there are no clean edges in the photo to tell you where to join one image to the next, and automated programs like Hugin won't even try.

There are plenty of other beautiful vistas here too -- cloudscapes, mesas, stars. Clearly, it was time to invest in a wide-angle lens. But how wide would it need to be to capture a double rainbow?

All over the web you can find out that a rainbow has a radius of 42 degrees, so you need a lens that covers 84 degrees to get the whole thing.

But what about a double rainbow? My web searches came to naught. Lots of pages talk about double rainbows, but Google wasn't finding anything that would tell me the angle.

I eventually gave up on the web and went to my physical bookshelf, where Color and Light in Nature gave me a nice table of primary and secondary rainbow angles of various wavelengths of light. It turns out that 42 degrees everybody quotes is for light of 600 nm wavelength, a blue-green or cyan color. At that wavelength, the primary angle is 42.0° and the secondary angle is 51.0°.

Armed with that information, I went back to Google and searched for double rainbow 51 OR 102 angle and found a nice Slate article on a Double rainbow and lightning photo. The photo in the article, while lovely (lightning and a double rainbow in the South Dakota badlands), only shows a tiny piece of the rainbow, not the whole one I'm hoping to capture; but the article does mention the 51-degree angle.

Okay, so 51°×2 captures both bows in cyan light. But what about other wavelengths? A typical eye can see from about 400 nm (deep purple) to about 760 nm (deep red). From the table in the book:

Wavelength	Primary	Secondary
400	40.5°	53.7°
600	42.0°	51.0°
700	42.4°	50.3°

Notice that while the primary angles get smaller with shorter wavelengths, the secondary angles go the other way. That makes sense if you remember that the outer rainbow has its colors reversed from the inner one: red is on the outside of the primary bow, but the inside of the secondary one.

So if I want to photograph a complete double rainbow in one shot, I need a lens that can cover at least 108 degrees.

What focal length lens does that translate to? Howard's Astronomical Adventures has a nice focal length calculator. If I look up my Rebel XSi on Wikipedia to find out that other countries call it a 450D, and plug that in to the calculator, then try various focal lengths (the calculator offers a chart but it didn't work for me), it turns out that I need an 8mm lens, which will give me an 108° 26‘ 46" field of view -- just about right.

So that's what I ordered -- a Rokinon 8mm fisheye. And it turns out to be far wider than I need -- apparently the actual field of view in fisheyes varies widely from lens to lens, and this one claims to have a 180° field. So the focal length calculator isn't all that useful. At any rate, this lens is plenty wide enough to capture those double rainbows, as you can see.

About those books

By the way, that book I linked to earlier is apparently out of print and has become ridiculously expensive. Another excellent book on atmospheric phenomena is Light and Color in the Outdoors by Marcel Minnaert (I actually have his earlier version, titled The Nature of Light and Color in the Open Air). Minnaert doesn't give the useful table of frequencies and angles, but he has lots of other fun and useful information on rainbows and related phenomena, including detailed instructions for making rainbows indoors if you want to measure angles or other quantities yourself.