1 The Subtlety of Rainbows Dr Andrew French A. French Page 1

Summary When I worked in the radar systems simulation & modelling group of BAE Systems we often constructed mathematical models of physical processes (such as reflection of microwaves off aeroplanes, rain, sea…) in order to run accurate computer simulations which allowed us to minimise the amount of (expensive) tests using real radar equipment. The example here of a mathematical model of a rainbow aims to help us to understand how simple principles of physics can be used to predict observable quantities such as the shape, angular width and order of colours. All models are built upon assumptions - essentially what set of scientific laws can we apply to the situation. A. French Page 2

Descartes theory of the rainbow ‘des cartes postal’ ! A. French Page 3

Descartes theory of the rainbow - assumptions Light is internally reflected off the interior of spherical raindrops. The wavelength of light is much smaller than the dimensions of the raindrop, so interference effects can be ignored. The mathematical relation between the angle that light is bent by the raindrops and the angle of incidence to the raindrop, has an ‘extremum’.* This results in the focussing of light of particular wavelengths into particular angles. Incident light Spherical raindrop   Light from rain cloud   Maximum 0 degrees 90 degrees *max or min. A. French Page 4

Schematic of a rainbow ‘Anti solar direction’   Sun Parallel rays of incident sunlight Rain cloud Rainbow Horizon Observer See next page for close up! A. French Page 5

Internal reflection of sunlight by a spherical raindrop Close up of rainbow showing deflection of incident rays of sunlight by internal reflection within a spherical raindrop of radius a   a  a a     Spherical raindrop Incident ray Deflected ray  A. French Page 6

  a  a     Spherical raindrop Incident ray Deflected ray  A Internal reflection of sunlight by a spherical raindrop a A. French Page 7

  a  a     Spherical raindrop Incident ray Deflected ray  A B Internal reflection of sunlight by a spherical raindrop a A. French Page 8

  a  a     Spherical raindrop Incident ray Deflected ray  A B C Internal reflection of sunlight by a spherical raindrop a A. French Page 9

  a  a     Spherical raindrop Incident ray Deflected ray  A B C Internal reflection of sunlight by a spherical raindrop a A. French Page 10

  air water Light ray Air-water interface Normal to interface Snell’s Law of refraction A. French Page 11

Snell’s Law of refraction Refractive index   air water Light ray Air-water interface Normal to interface A. French Page 12

Refractive index Refractive index is the ratio of the speed of light in a medium to that of the speed of light in a vacuum For air n ~1 For water n ~1.4 Note n often varies with the frequency of light A. French Page 13

Snell’s law again!   n = 1,  n >1,  n So for light entering a medium of higher refractive index, light is always bent towards the normal to the interface A. French Page 14

Formula for rain cloud light elevation Incident light Spherical raindrop   Light from rain cloud   Maximum 0 degrees 90 degrees A. French Page 15

Visible light and refractive indices Remember Frequency Wavelength Speed nm=10 -9 metres In water A. French Page 16

Frequency (or wavelength) variation of refractive index n of water A. French Page 17

Minimum deviation and light focussing Graph produced using MATLAB Gradient of  Angle of rainbow A. French Page 18

Rainbow structure! So a rainbow is observed at a mean angle of about 41.7 o with an angular width of about 1.6 o. Moving outwards in angle, colours go from violet (higher frequency, smaller wavelength) to red (lower frequency, larger wavelength) A. French Page 19

Rainbow structure! Rain cloud Rainbow Horizon 41.7 o Light from rain cloud 1.6 o A. French Page 20

See a circular rainbow when flying since rain cloud is illuminated above and below aircraft Sun Incident sunlight Rainbow Rain cloud A. French Page 21