Angle of Deviation in a Raindrop

(1) The angle of deviation measures the angle between the direction of an incident ray and the direction of a refracted ray when light travels from one medium to another

(2) The angle of deviation measures the degree to which the path of light through a raindrop is altered in the course of refraction and reflection towards an observer.

About the angle of deviation (Raindrops)

• The angle of deviation is measured between the path of light incident to a raindrop and its path after it exits the raindrop back into air.
• In any particular example of light passing through a raindrop, the angle of deviation and the angle of deflection are directly related to one another and together add up to 180⁰.
• The angle of deviation is always equal to 180⁰ minus the angle of deflection. So clearly the angle of deflection is always equal to 180⁰ minus the angle of deviation.
• In any particular example, the angle of deflection is always the same as the viewing angle because the incident light that forms a rainbow, if thought of in terms of rays, is approaching on trajectories running parallel with the rainbow axis.

Remember that:

• Any ray of light (stream of photons) travelling through empty space, unaffected by gravitational forces, travels in a straight line forever.
• When light leaves  a vacuum or travels from one transparent medium into another, it undergoes refraction causing it to change both direction and speed.
• The more a ray changes direction as it passes through a raindrop the greater will be its angle of deviation.
• Amongst the optical properties of air and water, absorption, reflection, refraction, and scattering of light are the most important.
• It is the optical properties of raindrops that determine the angle of deviation of incident light as it exits a raindrop.
• It is the optical properties of raindrops that prevent any ray of visible light from exiting a primary raindrop at an angle of deviation less than 137.6⁰.

Now consider the following:

• For a single incident ray of light of a known wavelength striking a raindrop at a known angle:
  • To appear in a primary rainbow it must reach an angle of deviation of at least 137.6⁰ if it is to be visible to an observer.
  • 137.6⁰ is the angle of deviation that produces the appearance of red along the outside edge of a primary rainbow from the point of view of an observer.
  • 137.6⁰ is the minimum angle of deviation for any ray of visible light if it is to appear within a primary rainbow.
  • 139.3⁰ is the angle of deviation for a ray that appears violet along the inside edge of a primary rainbow.
  • Angles of deviation between 137.6⁰ and 139.3⁰ correspond with viewing angles between 42.4⁰ (red) and 40.7⁰ (violet).
  • For any raindrop to form part of a primary rainbow it must be between the viewing angles of 42.4⁰ (red) and 40.7⁰ (violet)
  • An angle of deviation of 137.6⁰ (so viewing angles of 42.4⁰) corresponds with the appearance of red light with a wavelength of approx. 720 nm.
• The range of angles of deviation that create the impression of colour for an observer is not related to droplet size.
• The laws of refraction (Snell’s law) and reflection can be used to calculate the angle of deviation of white light in a raindrop.
• The angle of deviation can be fine-tuned for any specific wavelength by making a small adjustment to the refractive index of water.

Minimum angle of deviation

• The optical properties of an idealised spherical raindrop mean that no light of any specific wavelength can deviate less than its minimum angle of deviation.
• The minimum angle of deviation for red light with a wavelength of approx. 720 nm is always 137.6⁰ but similar rays with other points of impact can deviate up to a maximum of 180⁰.
  
• Imagine a falling raindrop:
  • At a specific moment, the droplet is at an angle of 50⁰ from the rainbow axis as seen from the point of view of an observer. This corresponds with an angle of deviation of 130⁰ which is insufficient to be visible to an observer.
  • A moment later the droplet is at an angle of 42.4⁰ which is the viewing angle for red in a primary rainbow so the droplet becomes visible to the observer.
  • 42.4⁰ corresponds with the rainbow angle for light with a wavelength of 720 nm, so at this moment the droplet appears red at maximum intensity.
  • As the droplet continues to fall, the minimum angle of deviation for red is passed and so that colour fades just as the minimum angle of deviation for orange arrives. For a second the same droplet now appears intensely orange.
  • The sequence repeats for yellow, green, blue and then violet at which point the viewing angle drops below 40.7⁰. A moment later, it briefly produces ultra-violet light.
  • As soon as the minimum angle of deviation for violet is exceeded, increasing towards 180⁰, it no longer forms part of the arcs of colour seen by an observer, but continues to scatter light into the area between the bow and anti-solar point.

By way of summary

• Raindrops do not reflect light at an angle less than its minimum angle of deviation.
• The minimum angle of deviation for any wavelength of visible light is never less than 137.6⁰  whilst the maximum is always 180⁰.
• When the angle of deviation is 180⁰, the angles or refraction (on the entry and exit of a raindrop) = 0⁰ and the angle of reflection = 180⁰.