Angle of deflection | lightcolourvision.org

When discussing the formation of rainbows, the angle of deflection measures the angle between the initial path of a light ray before it hits a raindrop, and the angle of deviation, which measures how much the ray bends back on itself in the course of refraction and reflection towards an observer.

The angle of deflection and the angle of deviation are always directly related to one another and together add up to 180 degrees.
The angle of deflection equals 180 degrees minus the angle of deviation. So, it’s clear the angle of deviation is always equal to 180 degrees minus the angle of deflection.
In any particular case, the angle of deflection is always the same as the viewing angle because the incident rays of light that form a rainbow all follow paths that run parallel with the rainbow axis.

Any ray of light (stream of photons) travelling through empty space, unaffected by gravity, travels in a straight line forever.
When light travels from a vacuum or from one transparent medium into another, it undergoes refraction causing it to change both direction and speed.
The more a light ray changes direction when it passes through a raindrop, the smaller the angle of deflection will be.
It is the optical properties of raindrops that determine the angle of deflection of light as it exits a raindrop.
Because of the optical properties of primary rainbows, no rays of light within the visible part of the electromagnetic spectrum exit a raindrop at an angle of deflection larger than 42.4 degrees.
Because of the optical properties of secondary rainbows, no rays of light within the visible part of the electromagnetic spectrum exit a raindrop at an angle of deflection larger than 50.4 degrees.

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 cannot exceed an angle of deflection of more than 42.4 degrees. This corresponds with the minimum angle of deviation.
- 42.4⁰is the angle of deflection that results in red appearing along the outside edge of a primary rainbow from the point of view of an observer.
- 180 degrees minus 137.6 degrees equals 42.4 degrees, which is the biggest angle of deflection for any visible light ray if it is to appear within a primary rainbow.
- 180 degrees minus 139.3 degrees equals 40.7 degrees, which is the angle of deflection for a light ray that appears violet and forms the inside edge of a primary rainbow.
- Angles of deviation between 137.6 degrees and 139.3 degrees match viewing angles and angles of deflection between 42.4 degrees (red) and 40.7 degrees (violet).
- An angle of deviation of 137.6 degrees (so viewing angles of 42.4 degrees) matches the appearance of red light with a wavelength of about 720 nm.
The range of angles of deflection 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 work out the angle of deviation of white light in a raindrop.
The angle of deviation can be fine-tuned for any specific wavelength by making small adjustments to the refractive index.

- Viewing angle refers to the number of degrees through which an observer must move their eyes or turn their head to see a specific colour within the arcs of a rainbow.
- Angular distance refers to the same measurement when shown in a side elevation diagram.
- Angle of deflection measures the angle between the original path of a ray of incident light before striking a raindrop and the angle of deviation.
- Angle of deviation measures the degree to which the path of a light ray is bent back by a raindrop in the course of refraction and reflection towards an observer.
- Rainbow ray refers to the path taken by the deflected ray that produces the most intense colour experience for any particular wavelength of light passing through a raindrop.

- In any particular example of a ray 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 rays of light that form a rainbow are all approaching on a trajectory running parallel to the rainbow axis.

Each of the diagrams below can be viewed on its own page with a full explanation.

When discussing the formation of rainbows, the angle of deflection measures the angle between the initial path of a light ray before it hits a raindrop, and the angle of deviation, which measures how much the ray bends back on itself in the course of refraction and reflection towards an observer.
See this diagram for an explanation: Rainbow anatomy
The angle of deflection and the angle of deviation are always directly related to one another and together add up to 180 degrees.
The angle of deflection equals 180 degrees minus the angle of deviation. So, it’s clear the angle of deviation is always equal to 180 degrees minus the angle of deflection.
In any particular case, the angle of deflection is always the same as the viewing angle because the incident rays of light that form a rainbow all follow paths that run parallel with the rainbow axis.