Norway Rainbow & René Descartes

Rainbows over Troms�ysund, Norway imaged by Per Ivar Somby (more images). ©Per Ivar Somby, shown with permission.

The primary rainbow is formed by rays reflected once inside raindrops. How much a ray is deflected depends on where it strikes the drop. The ray diagram shows paths for a variety of rays. It is of the type first computed by Ren� Descartes and published in 1637 in his 'Discourse on the Method of Rightly Conducting the Reason, and Searching for Truth in the Sciences'. His description was the first quantitative explanation of the rainbow and one of the first uses of the methods later to be known as mathematical physics.

Rays passing on a line closest to the drop centre are sharply reflected backwards and emerge considerably deflected. As the ray distance from the centre (called the impact parameter) increases its deflection gets less. At about 86% of the radius a minimum deflection is reached. This ray is shown coloured white. At larger impact parameters the deflection increases again.

The minimum deflection ray corresponds to the rainbow rim. Many rays cluster at similar deflections and this makes the rainbow rim bright. In fact, classical ray theory predicts an infinite light intensity at the minimum deflection angle. This is a characteristic of 'caustics' and the rainbow is an example of one.

Rays deflected through greater angles fill the inside of the rainbow caustic with light.

The diagram is for a single colour. Red light is refracted less than blue and its minimum deviation angle is less. The red caustic is therefore on the outside of the primary bow.

Rays leave out of the front of the drop. More than 90% of sunlight does so. These 'zero order' rays have no minimum deviation. There is no caustic and no rainbow. Instead they make a bright glow around the sun and help prevent us seeing bows from three and four reflections inside raindrops.