Headed down the BCN main line to the Black Country Boat Festival last weekend, I was delighted to catch a really vivid rainbow in the few minutes during (and after) a brief shower in between bright spells. You don't 'chase' rainbows; they follow you. And what you see is mainly luck.
But more significantly, this time, the occasionally-seen 'second rainbow' was visible, something which I've never managed to catch on camera before.
You can see spots of rain in the picture, caught by the camera's flash.
I also photographed the complete arc, using several pictures from right to left, and could in theory 'stitch' these together for a panorama, except they wouldn't produce an interesting image. The scene on the left wasn't all that interesting.
There's no technical mystery about the rainbow, especially with the Wikipedia entry on rainbows to hand.
Your head is always in between the sun and the center of a rainbow circle, the sun is at your back.
A normal rainbow has a lighter area of sky under it. White is the reflected color under the rainbow. Dark, or no added reflected light is over the top of the rainbow, over the red. Yellow light is in the middle of a rainbow.
A warm magenta is barely visible on top of the red on a strong rainbow, it's stronger at the cool bottom of the rainbow. Also, yellow will appear again under the magenta on a strong rainbow.
If you make your own rainbow, with a small spray of water, you don't see the green; white will take its place. This happens in the middle band.
Red is on the outside of the main rainbow and on the inside of the second rainbow, the second rainbow is separated by 15 degrees and would have a darker band between them.
The third rainbow's color progression is the same as the main rainbow, ad infinitum.
He doesn't blind you with optics and tell you how it occurs; he just tells you what you'll see. This is to optics as JK Rowling is to magic.