A rainbow is an optical effect

A rainbow is an optical effect, a trick of the light, caused by the behaviour of light waves travelling through transparent water droplets towards an observer.

  • Sunlight and raindrops are always present when a rainbow appears but without an observer, there is nothing, because eyes are needed to produce the visual experience.
  • A rainbow isn’t an object in the sense that we understand physical things in the world around us. A rainbow is simply light caught up in raindrops.
  • A rainbow has no fixed location. Where rainbows appear depends on where the observer is standing, the position of the Sun and where rain is falling.
  • The exact paths of light through raindrops is so critical to the formation of rainbows that when two observers stand together their rainbows are produced by different sets of raindrops.

Alexander’s band

Alexander’s band (Alexander’s dark band) is an optical effect associated with rainbows. It refers to the area between primary and secondary bows that often appears to be noticeably darker to an observer than the rest of the sky.

About Alexander’s band
  • Alexander’s band is named after Alexander of Aphrodisias, an ancient Greek philosopher who commented on the effect in his writing.
  • The darker area between primary and secondary rainbows can be explained by the fact that fewer photons are directed from that area of the sky toward an observer.
  • The raindrops that form a primary rainbow all direct exiting light downwards towards an observer so away from Alexander’s band.
  • The raindrops that form a secondary bow all direct light upwards, so away from Alexander’s band, before a second internal reflection directs light downwards towards an observer.
  • Plenty of light is scattered into the area between primary and secondary rainbows but very little of it travels towards an observer.

Angle of deflection

The angle of deflection measures the angle between the original path of a ray of incident light prior to striking a raindrop and the angle of deviation which measures the degree to which the ray is bent 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 1800.
  • The angle of deflection is equal to 1800 minus the angle of deviation. So clearly the angle of deviation is always equal to 1800 minus the angle of deflection.
  • 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 all approach on a trajectory 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 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 ray changes direction as it passes through a raindrop the smaller will be the angle of deflection.
  • It is the optical properties of raindrops that determine the angle of deflection 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 deflection larger than 42.70.
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 cannot exceed an angle of deflection of more than 42.70. This corresponds with the minimum angle of deviation.
    • 42.70 is the angle of deflection that produces the appearance of red along the outside edge of a primary rainbow from the point of view of an observer.
    • 1800 – 137.60 = 42.0 4 is the maximum angle of deflection for any ray of visible light if it is to appear within a primary rainbow.
    • 1800 -139.30 =  40.70 is the angle of deflection 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 and angles of deflection between 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 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 and the law of 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 fine adjustment of the refractive index.
Viewing angle, angular distance and angle of deflection
  • The term 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.
  • The term angular distance refers to the same measurement when shown in side elevation on a diagram.
  • The angle of deflection measures the angle between the original path of a ray of incident light prior to striking a raindrop and the angle of deviation.
  • The term 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.
  • The term 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.
    • 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 with the rainbow axis.

Angle of deviation

(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 emit no light of any particular wavelength 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.

Angular distance

Angular distance is the angle between the rainbow axis and the direction in which an observer must look to see a specific colour within the arcs of a rainbow.

  • Angular distance, viewing angle and angle of deflection all produce the same value measured in degrees.
  • Angular distance is a measurement on a ray-tracing diagram that represents a cross-section of a rainbow in side elevation.
  • A cross-section of a rainbow in side elevation shows the sun, observer and the highest elevation of a rainbow above the horizon arranged along the rainbow axis.
  • Think of angular distance as an angle between 40.70 and 42.40 from the centre of a rainbow that accounts for the spectrum of colours between red and violet.
  • Angular distances for different colours are constants determined by the laws of refraction and reflection.
  • The elevation of the Sun, the location of the observer and exactly where rain is falling are all variables that determine where a rainbow will appear to an observer.
  • The coloured arcs of a rainbow form the circumference of circles (discs or cones) and share a common centre.
  • The angular distance to any specific colour is the same whatever point is selected on the circumference.
  • The angular distance for any observed colour within a primary bow is between approx. 40.70 and 42.40 from its centre.
  • The angular distance for any observed colour within a secondary bow is between approx. 50.40 and 53.40 when you are looking at its centre.
  • The angular distance can be calculated for any specific colour visible within a rainbow.
  • Considered from an observer’s viewpoint, it is clear that all incident rays seen by an observer run parallel with each other as they approach a raindrop.
  • Most of the observable incident rays that strike a raindrop follow paths that place them outside the range of possible viewing angles. The unobserved rays are all deflected towards the centre of a rainbow.
Viewing angle, angular distance and angle of deflection
  • The term 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.
  • The term angular distance refers to the same measurement when shown in side elevation on a diagram.
  • The angle of deflection measures the degree to which a ray striking a raindrop is bent back on itself in the process of refraction and reflection towards an observer.
  • The term 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.
  • The term 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.
    • 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 with the rainbow axis.

Areas around rainbows

The area inside a primary rainbow

The area inside the arcs of a primary rainbow, from its centre-point out to the violet band at 40.70 appears tonally lighter or brighter than the area of sky on the outside.

  • The area inside the arcs of a primary rainbow contains:
    • Light that has been reflected off the outside surface of raindrops towards an observer. This light has not undergone refraction or dispersion so reflects white sunlight back towards an observer.
    • Light that has been randomly scattered after being intercepted and refracted by multiple droplets in succession. The result is a mixture of different wavelengths inside a primary rainbow that produces a whiter or lighter appearance to an observer.
The area outside a primary rainbow

Very little light is directed into the area outside a primary rainbow. As a result, rainbows appear to be brightly coloured and stand out against the sky beyond.

  • The outer red edge of a primary rainbow corresponds with the minimum angle of deviation of light for the raindrops that contribute to an observer’s perception of a rainbow.
  • The minimum angle of deviation is the minimum number of degrees that light must bend back on itself after it passes through a raindrop if it is to form part of a rainbow seen by an observer.
  • The minimum angle of deviation of light through a raindrop is measured between the original path of incident light before it strikes a raindrop and the path along which an observer looks towards a rainbow.
  • The area outside a primary rainbow corresponds with angles that are less than the minimum angle of deviation. Relatively few photons are directed towards an observer from this area.
  • The coloured arcs of a primary rainbow correspond with angles greater than the minimum angle of deviation.
  • The brightness of the arcs of a primary rainbow results from the fact that as light of any particular wavelength passes through a raindrop it tends to concentrate near the minimum angle of deviation (see rainbow rays).
The area between a primary and a secondary rainbow

The area between a primary and a secondary rainbow is called Alexander’s band and is tonally darker than the area inside a primary rainbow or outside a secondary rainbow:

  • As refraction and dispersion take place in raindrops that form a primary rainbow, light is directed inwards towards its centre and so away from Alexander’s band.
  • As refraction and dispersion take place in raindrops that form a secondary rainbow, light is directed outwards away from Alexander’s band.
The area inside a secondary rainbow

The inner red-coloured edge of a secondary rainbow corresponds with the minimum angle of deviation.

  • Relatively few photons are directed towards an observer from the area between primary and secondary rainbows (Alexander’s band).
The area outside a secondary rainbow

Some light is reflected or refracted into the area outside a secondary rainbow but it does not significantly lighten the sky.

  • The area outside a secondary rainbow corresponds with the area inside a primary rainbow.
  • Light is widely dispersed during the formation of a secondary rainbow as a result of the second internal reflection that proceeds their observation.
  • Wider dispersion of wavelengths involved in the formation of a secondary rainbow and then the random scattering of light into the area outside a secondary rainbow produces no equivalent to the lighter or brighter look of the sky inside a primary rainbow.
Remember that:
  • White light, containing all wavelengths within the visible part of the electromagnetic spectrum in equal proportions and at equal intensities, separates into spectral colour as refraction and dispersion take place.
  • It is the small difference in the refractive index of different wavelengths of incident light that causes dispersion and separation of white light into rainbow colours.
  • As light travels through the air it is invisible to our eyes. White light is what an observer sees when all the colours that make up the visible spectrum reflect off a neutral coloured surface or particles of dust and vapour.
  • Colour is what a human observer sees when a single wavelength, a band of wavelengths or a mixture of different wavelengths strike neutral coloured surfaces or particles of dust and vapour.

Atmospheric rainbow

An atmospheric rainbow is an arc or circle of spectral colours and appears in the sky when an observer is in the presence of strong sunshine and rain.

  • Atmospheric rainbows:
    • Are caused by sunlight reflecting, refracting and dispersing inside raindrops before being seen by an observer.
    • Appear in the section of the sky directly opposite the Sun from the point of view of an observer.
    • Become visible when millions of raindrops reproduce the same optical effects.
  • Atmospheric rainbows often appear as a shower of rain is approaching, or has just passed over. The falling raindrops form a curtain on which sunlight falls.
  • To see an atmospheric rainbow, the rain must be in front of the observer and the Sun must be in the opposite direction, at their back.
  • A rainbow can form a complete circle when seen from a plane, but from the ground, an observer usually sees the upper half of the circle with the sky as a backdrop.
  • Rainbows are curved because light is reflected, refracted and dispersed symmetrically around their centre-point.
  • The centre-point of a rainbow is sometimes called the anti-solar point. ‘Anti’, because it is opposite the Sun with respect to the observer.
  • An imaginary straight line can always be drawn that passes through the Sun, the eyes of an observer and the anti-solar point – the geometric centre of a rainbow.
  • A section of a rainbow can easily disappear if anything gets in the way and forms a shadow. Hills, trees, buildings and even the shadow of an observer can cause a portion of a rainbow to vanish.
  • Not all rainbows are ‘atmospheric’. They can be produced by waterfalls, lawn sprinklers and anything else that creates a fine spray of water droplets in the right conditions.

Atmospheric rainbow summary

  • Rainbows form when sunlight encounters a curtain of rain.
  • The sunlight enters raindrops at one angle and then emerges at another.
  • The water droplets have to be in just the right place to reflect rays into an observer’s eyes.
  • Each raindrop is made of liquid water and acts as a tiny prism.
  • Raindrops break sunlight into distinct red, orange, yellow, green, blue and violet rays.
  • Rainbows can be described as being both atmospheric and optical phenomena.
Remember that:
  • If the Sun is directly behind you, rain is falling in front of you, and you look straight ahead, then you will see that the rainbow forms around a centre-point.
  • The centre-point of a rainbow is often referred to as the anti-solar point.
  • The anti-solar point, your eyes and the Sun are always in line with one another – on the same axis.
  • Anti means opposite, opposed, or at 1800. So anti-solar means a point opposite to the Sun as seen by an observer.
  • The axis of a rainbow is an imaginary line drawn between the Sun, observer and anti-solar point.
  • When sunlight and raindrops combine to make a rainbow, they can make a whole circle of light in the sky.
  • Rainbows only form a complete circle when the ground doesn’t get in the way. This only happens when you are on a plane.
  • Whenever something blocks sunlight then a shadow forms and a portion of a rainbow disappears.
  • Even if you stand on a mountain peak, the bow forms less than a circle because the mountain creates a shadow.
  • Your own shadow can get in the way of a rainbow formed by the spray from a hose or lawn sprinkler.
  • Seen from the air, the shadow of your plane is often visible at the centre of the rainbow. The further away the curtain of rain is on which the bow forms, the smaller the plane appears.
  • At ground level, the main reason rainbows don’t form a complete circle is because when droplets hit the ground they stop reflecting light so the rainbow comes to an end.

Bands of colour

The fact that we see a few distinct bands of colour in a rainbow, rather than a smooth and continuous gradient of hues, is sometimes described as an artefact of human colour vision.

  • We see bands of colour because the human eye distinguishes between some ranges of wavelengths of visible light better than others.
  • It is the interrelationship between light in the world around us on one hand and our eyes on the other that produces the impression of different bands of colour.
  • The visible spectrum is made up of a smooth and continuous range of wavelengths that correspond with a smooth and continuous range of hues.
  • There is no property belonging to electromagnetic radiation that causes bands of colour to appear to a human observer.

Chromatic dispersion

Chromatic dispersion means dispersion according to colour and associated wavelengths of light. Under certain conditions, chromatic dispersion causes light to separate into its component wavelengths producing a rainbow of colours for a human observer.

  • Chromatic dispersion is best demonstrated by passing a beam of light through a glass prism.
  • A familiar example of chromatic dispersion is when white light strikes raindrops and a rainbow of colours becomes visible to an observer.
  • As light first enters and then exits each raindrop, it separates into its component wavelengths which the observer sees as a band of distinct colours.
  • Chromatic dispersion can be explained in terms of the relationship between wavelength and refractive index.
  • When light propagates from one medium (such as air) to another (such as glass or water) every wavelength of light is affected to a different degree according to the refractive index of the media concerned. As a result, each wavelength changes direction by a different degree. In the case of white light, the separate wavelengths fan out with red on one side and violet on the other.
  • Remember that wavelength is a property of electromagnetic radiation, whilst colour is a feature of visual perception.

Cones of colour

Rainbows can be modelled as a set of six nested cones with the apex of each aligned with the lenses of an observer’s eyes.

  • Each cone has a different radius and each is composed of a narrow spread of wavelengths of light that determines its apparent colour. Red fills the cone with the largest radius and violet fills the smallest.
  • The cones do not have a simple 2D base. At their nearest, droplets may be within reach of an observer. At the other extreme, distant raindrops also refract and reflect light back towards an observer.
  • Modelling a rainbow as a cone that shows depth, as well as height and width, demonstrates that all the raindrops contained within one of the cones at any moment can contribute to the visual experience of an observer regardless of how far they are away.
  • Whilst modelling rainbows as discs corresponds with what an observer sees, the idea that rainbows are formed from cones of colour corresponds with a diagram showing a side elevation with the Sun, observer and rainbow arranged along the rainbow’s axis.

Discs of colour

Rainbows can be modelled as six concentric two-dimensional discs as seen from the point of view of an observer. Each disc has a different radius and contains a narrow spread of colours. The red disc has the largest radius and violet the smallest.

  • The colour of each disc is strongest and most visible near its outer edge because this is the area into which light is most concentrated from the point of view of an observer.
  • This concentration of light near the outer edge of each disc results from the path of rainbow rays.
  • The term rainbow ray describes the path that produces the most intense experience of colour for any particular wavelength of light passing through a raindrop.
  • The intensity of the colour of each disc reduces rapidly away from the rainbow angle because other rays passing through each raindrop diverge from one another and so are much less concentrated.
  • The divergence of rays of light after exiting a raindrop is often called scattering.
  • From the point of view of an observer, the six discs are superimposed upon one another and appear to be in the near to middle distance in the opposite direction to the Sun.
  • There is no property belonging to electromagnetic radiation that causes a rainbow to appear as bands or discs of colour to an observer. The fact that we do see distinct bands of colour in the arc of a rainbow is often described as an artefact of human colour vision.
  • To model rainbows as discs allows us to think of them as forming on flat 2D curtains of rain.
  • Rainbows are often modelled as discs for the same reason the Sun and Moon are represented as flat discs – because when we look into the sky, there are no visual cues about their three-dimensional form.
  • Each member of the set of discs has a different radius due to the spread of wavelengths of light it contains. This can be explained by the fact that the angle of refraction of rays of light as they enter and exit a droplet is determined by wavelength. As a result, the radius of the red disc is the largest because wavelengths corresponding with red are refracted at a larger angle (42.40) than violet (40.70).
  • From the point of view of an observer, refraction stops abruptly at 42.40 and results in a sharp boundary between the red band and the sky outside a primary rainbow.
  • The idea of rainbows being composed of discs of colour fits well with the fact that there is a relatively clear outer limit to any observed band of colour.

Distance to, size and duration of rainbows

Distance to a rainbow

Rainbows are formed from the millions of individual raindrops that happen to be in exactly the right place at the right time, so it is difficult to be precise about how far away a rainbow is.

  • Because a rainbow is a trick of the light rather than a solid material object set in the landscape it has no fixed position and is at no fixed distance from an observer. Instead, rainbows move as the Sun and the observer move or as curtains of rain cross the landscape.
  • Because a rainbow is composed of light reflecting off and refracting in millions of individual raindrops it might be fair to say that the distance to a rainbow is the distance to the location of the greatest concentration of raindrops diverting photons towards an observer.
  • An observer cannot easily estimate the distance to a raindrop or a curtain of rain along their line of sight but the position of clouds or objects in the landscape can help to determine where rain is falling.
  • The position of a rainbow is primarily determined by angles. The angles are constants and result from the physical properties of light and water droplets, not least the laws of reflection and refraction.
  • As an observer moves, the rainbow they see moves with them and the angles are preserved.
Size of a rainbow
  • Just as the visual impression of the size of the moon depends on how near it is to the horizon, the apparent diameter of a rainbow is also affected by other features in the landscape.
Duration of a rainbow
  • A rainbow may be visible for minutes on end before receding slowly into the distance. In other situations, a rainbow may appear one moment and be gone the next.

Electromagnetic waves

The form and composition of rainbows are often explained in terms of electromagnetic waves.

EM-Wave

Electromagnetic waves consist of coupled oscillating electric and magnetic fields orientated at 900 to one another. (Credit: https://creativecommons.org/licenses/by-sa/4.0)

  • Electromagnetic waves can be imagined as oscillating electric (E) and magnetic (B) fields arranged at right angles to each other.
  • In the diagram above, the coupled electric and magnetic fields follow the y-axis and z-axis and propagate along the x-axis.
  • This arrangement is known as a transverse wave which means the oscillations are perpendicular to the direction of travel.
  • By convention, the electric field is shown in diagrams aligned with the vertical plane and the magnetic field with the horizontal plane.
  • In normal atmospheric conditions the geometric orientation of the coupled y-axis and z axis is random, so the coupled fields EB may be rotated to any angle.

Fog bows, dew bows and more

There are many optical effects similar to rainbows.

  • A fog bow is a similar phenomenon to a rainbow. As its name suggests, it is associated with fog rather than rain. Because of the very small size of water droplets that cause fog, a fog bow has only very weak colours.
  • A dew bow can form where dewdrops reflect and disperse sunlight. Dew bows can sometimes be seen on fields in the early morning when the temperature drops below the dew point during the night, moisture in the air condenses, falls to the ground, and covers cobwebs.
  • A moon bow is produced by moonlight rather than sunlight but appears for the same reasons. Moon bows are often too faint to excite the colour receptors (cone cells) of a human eye but can appear in photographs taken at night with a long exposure.
  • A twinned rainbow is produced when two rain showers with different sized raindrops overlap one another. Each rainbow has red on the outside and violet on the inside. The two bows often intersect at one end.
  • A reflection rainbow is produced when strong sunlight reflects off a large lake or the ocean before striking a curtain of rain. The conditions must be ideal if the reflecting water is to act as a mirror. A reflected rainbow appears to be similar to a primary bow but has a higher arc. Don’t get confused between a reflection rainbow that appears in the sky and a rainbow reflected in water.
  • A glory is a circle of bright white light that appears around the anti-solar point.
  • A halo is a circle of bright multicoloured light caused by ice crystals that appears around the Sun or the Moon.
  • A monochrome rainbow only occurs when the Sun is on the horizon. When an observer sees a sunrise or sunset, light is travelling horizontally through the atmosphere for several hundred kilometres. In the process, atmospheric conditions cause all but the longest wavelengths to scatter so the Sun appears to be a diffuse orange/red oval. Because all other wavelengths are absent from a monochrome rainbow, the whole scene may appear to be tinged with a fire-like glow.

Geometric raindrops

An idealised raindrop forms a geometrically perfect sphere. Although such a form is one in a million in real-life,  simplified geometrical raindrops help to make sense of rainbows and reveal general rules governing why they appear.

The insights that can be gained from exploring the geometry of raindrops apply to every rainbow, whilst the rainbows we come across in everyday life demonstrate that each individual case is unique.

Don’t forget that the idea of light rays is also a way to simplify the behaviour of light:

  • The idea that light is made up of rays is so commonplace when describing and explaining rainbows that it is easily taken for granted.
  • The idea of light rays is useful when trying to model how light and raindrops produce the rainbow effects seen by an observer.
  • Light rays don’t exist in the sense that the term accurately describes a physical property of light. More accurate descriptions use terms like photons or waves.
Basics of raindrop geometry
  • A line drawing of a spherical raindrop is the starting point for exploring how raindrops produce rainbows.
  • The easiest way to represent a raindrop is as a cross-section that cuts it in half through the middle.
  • A dot or small circle can be used to mark the centre whilst the larger circle marks the circumference.
  • Marking the centre makes it easy to add lines that show the radius and diameter.
  • Marking the centre also makes it easy to add lines that are normal to the circumference.
  • A normal (or the normal) refers to a line drawn perpendicular to and intersecting another line, plane or surface.
  • A normal is used in a diagram to connect the centre with a point where a ray strikes the circumference.
  • The diameter of a circle is a line that passes through its centre and is drawn from the circumference on one side to the other.
  • The radius of a circle is a line from the centre to any point on the circumference.
  • The horizontal axis of a raindrop is a line drawn through its centre and parallel to incident light. The vertical axis intersects the horizontal at 900 and also passes through the centre point.
  • The angle at which incident light strikes the surface of a raindrop can be calculated by drawing a line that shows where an incident ray strikes a droplet and then drawing the normal. The angle of incidence is measured between them.
  • The path of light as it strikes the surface and changes direction as it is refracted at the boundary between air and water can be calculated using the Law of Refraction (Snell’s law).
  • When light is refracted as it enters a droplet it bends towards the normal.
  • The law of reflection can be used to calculate the change of direction each time light reflects off the inside surface of the raindrop.
  • When light exits a raindrop the angle of refraction is the same as when it entered but this time bends away from the normal.

Impact parameter

The term impact parameter refers to a scale used on a ray-tracing diagram to measure the point at which incident rays strike the surface of a raindrop. Rays are given a value between 0.0 and 1.0 depending upon their point of impact.

  • For a primary rainbow, all the incident rays of interest strike a raindrop between its horizontal axis (0.0 on the impact parameter scale) and the upper-most point (1.0 on the impact parameter scale). In the second case, the ray grazes the surface at 900 to the normal and continues on its course without deviation.
  • For a secondary rainbow, all the incident rays of interest strike a raindrop between its horizontal axis (0.0 on the impact parameter scale) and the lowest point (1.0 on the impact parameter scale). In the second case, the ray grazes the surface at 900 to the normal and continues on its course without deviation.
  • An impact parameter is useful because it allows the relationship between equidistant incident rays, the angle at which they strike the surface and their angle of refraction to be plotted.

Invisible dimensions of rainbows

A typical atmospheric rainbow includes six bands of colour from red to violet but there are other bands of light present that don’t produce the experience of colour for human observers.

  • It is useful to remember that:
    • Each band of wavelengths within the electromagnetism spectrum (taken as a whole) is composed of photons that produce different kinds of light.
    • Remember that light can be used to mean visible light but can also be used to refer to other areas of the electromagnetism spectrum invisible to the human eye.
    • Each band of wavelengths represents a different form of radiant energy with distinct properties.
    • The idea of bands of wavelengths is adopted for convenience sake and is a widely understood convention. The entire electromagnetic spectrum is, in practice, composed of a smooth and continuous range of wavelengths (frequencies, energies).
  • Radio waves, at the end of the electromagnetic spectrum with the longest wavelengths and the least energy, can penetrate the Earth’s atmosphere and reach the ground but are invisible to human eyes.
  • Microwaves have shorter wavelengths than radio waves, can penetrate the Earth’s atmosphere and reach the ground but are invisible to human eyes.
  • Longer microwaves (waves with similar lengths to radio waves) pass through the Earth’s atmosphere more easily than the shorter wavelengths nearer the visible parts of spectrum.
  • Infra-red is the band closest to visible light but has longer wavelengths. Infra-red radiation can penetrate Earth’s atmosphere but is absorbed by water and carbon dioxide. Infra-red light doesn’t register as a colour to the human eye.
  • The human eye responds more strongly to some bands of visible light between red and violet than others.
  • Ultra-violet light contains shorter wavelengths than visible light, can penetrate Earth’s atmosphere but is absorbed by ozone. Ultra-violet light doesn’t register as a colour to the human eye.
  • Radio, microwaves, infra-red, ultra-violet are all types of non-ionizing radiation, meaning they don’t have enough energy to knock electrons off atoms. Some cause more damage to living cells than others.
  • The Earth’s atmosphere is opaque to both X-rays or gamma-rays from the ionosphere downwards.
  • X-rays and gamma-rays are both forms of ionising radiation. This means that they are able to remove electrons from atoms to create ions. Ionising radiation can damage living cells.
Remember that:
  • All forms of electromagnetic radiation can be thought of in terms of waves and particles.
  • All forms of light from radio waves to gamma-rays can be thought to propagate as streams of photons.
  • The exact spread of colours seen in a rainbow depends on the complex of wavelengths emitted by the light source and which of those reach an observer.

Laws of refraction and reflection

The path of light through a raindrop is a key factor in determining whether it will direct light towards an observer and contribute to their perception of a rainbow. This can be broken down as follows:

  • The impact parameter is a measure of the direction from which rays of incident light approach a raindrop and the point at which they strike the surface.
  • When using a ray-tracing diagram to map the path of rays through a raindrop, an impact parameter scale is used to select which incident rays are of interest.
  • An impact parameter scale is aligned with parallel incident rays and divides the relevant part of the surface of a droplet into equal parts.
  • Using a scale with steps between zero and one, 0 is aligned with the ray that passes through the centre of a droplet and 1 with the ray that grazes the surface without refraction or reflection.
Remember that:
  • Primary rainbows form when incident light strikes raindrops above their horizontal axis reflecting once off the inside before exiting towards an observer.
  • Incident light that strikes raindrops below their horizontal axis and reflects once on the inside before exiting, directs light upwards away from an observer.
  • Secondary rainbows form when incident light strikes raindrops below their horizontal axis reflecting twice off the inside before exiting downwards.
  • The Law of reflection deals with the angles of incidence and reflection when light strikes and bounces back off a surface and can be used for calculations relating to the curved surfaces of a raindrop.
  • Remember that the law of reflection states that the angle of incidence always equals the angle of reflection for a mirror-like (specular) surface.
  • The Law of Refraction (Snell’s law) deals with the changes in the speed and direction of incident light as it crosses the boundaries between air and a raindrop and then between a raindrop and the surrounding air.
  • An equation can be derived from Snell’s law that deals with the relationship between the angle of incidence and the angle of refraction of light with reference to the refractive indices of both media.