Rainbows in detail

To properly understand rainbows involves referring to different fields of enquiry and areas of knowledge.

• The field of optics tells us that rainbows are about the paths that light takes through different media and are the result of reflection, refraction and dispersion of light in water droplets.
• A weather forecaster might explain rainbows in meteorological terms because they depend on sunlight and only appear in the right weather conditions and times of the day.
• A hydrologist, who studies the movement and distribution of water around the planet, might refer to the water cycle and so to things like evaporation, condensation and precipitation.
• A vision scientist will need to refer to visual perception in humans and the biological mechanisms of the eye.
• An optometrist may check for colour blindness or eye disease.

Our DICTIONARY OF LIGHT COLOUR AND VISION assembles terms drawn from these different fields to explore our central interest at lightcolourvision.org which is the interconnections between these three topics.
Whenever terms that appear in the DICTIONARY are used on pages within our LIBRARY OF DIAGRAMS, a blue link appears in the text.

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.

There are three basic conditions that have to be fulfilled before an atmospheric rainbow appears:

• Bright sunlight shining through clear air.
• A curtain of falling rain in the near to middle distance.
• An observer in the right place at the right time.

The weather, season and time of day are all important if you hope to see an atmospheric rainbow.

• The best rainbows appear in the morning and evening when the Sun is strong but low in the sky.
• Northern and southern latitudes away from the equator are good for rainbows because the Sun is lower at its zenith.
• Mountains and coastal areas can create ideal conditions because as air sweeps over them, it cools, condenses and falls as rain.
• Rainbows are rare in areas with little or no rainfall such as dry, desert conditions with few clouds.
• Too much cloud is not good because it blocks direct sunlight.
• Winter is not necessarily the best season because the light is weaker and there can be excessive cloud.
• Rainbows are less common around midday because the higher the Sun is in the sky the lower the rainbow.
• If the Sun is too high, then by the time the raindrops are in the right position to form part of a rainbow they are lost in the landscape.

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.

There are several particularly noticeable things to see when looking closely at rainbows:

• The arcs of spectral colours curving across the sky with red on the outside and violet on the inside, this is a primary rainbow. The arcs appear between the angles of approx. 40.7° and 42.4° from the centre (anti-solar point) as seen from the point of view of an observer.
• There may be another rainbow, just outside the primary bow with violet on the outside and red on the inside, this is the secondary rainbow. The arcs appear between the angles of approx. 50.4° and 53.4° from its centre as seen from the point of view of an observer.
• Faint supernumerary bows often appear just inside a primary rainbow and form shimmering arcs of purples and cyan-greens. These bands appear at an angle of approx. 39° to 40° from the centre so just inside the violet arc of the primary bow.
• The remaining area inside a rainbow from its centre out to approx. 39° often appears lighter or brighter in comparison to the sky outside the rainbow. There are three main causes:
  • Light strikes multiple droplets in succession and randomly scatters in all directions.
  • Small amounts of light of all wavelengths are deflected towards the centre and combine to produce the appearance of weak white light.
  • Almost no light is deflected to the area outside a rainbow.
• When a secondary rainbow appears, the area between the two often appears to be darker in tone than any other area of the sky. This is called Alexander’s band. The effect is the result of rays being deflected away from this area as primary and secondary bows form.

To understand rainbows it is important to sort out what an observer is actually looking at.

• Rainbows only exist in the eyes of an observer.
• Every observer sees a different rainbow produces by a unique set of raindrops that happen to be in the right place at the right time.
• The individual raindrops that result in the appearance of a rainbow for one observer are always different from the raindrops that produce a rainbow for someone else.
• As an observer moves, their rainbow moves with them. Seen from a car window, the rainbow appears stationary whilst the landscape rushes past.

From an observer’s point of view

• Atmospheric rainbows appear to an observer as arcs of colour across the sky. From an aeroplane, a rainbow can appear as entire circles of colour.
• Even from the ground, it is easy to deduce that every rainbow has a centre point around which the arcs of a rainbow are arranged.
• The exact position in the sky where an atmospheric rainbow will appear can be anticipated by working out where its centre will be.
• The centre of a rainbow is always on an imaginary straight line that starts at the centre of the Sun behind you, passes through the back of your head, out through your eyes and extends in a straight line into the distance.
• The eyes of an observer are always aligned with the rainbow axis.
• To an observer, the rainbow axis appears as a point, not a line, and that imaginary point marks the centre of where every rainbow will appear.
• The idea that a rainbow has a centre corresponds with what an observer sees in real-life.
• The idea of a rainbow axis or anti-solar point corresponds with a diagrammatic view showing the scene in side elevation.

Looking for rainbows

• To work out where a rainbow might appear:
  • Turn your back on the Sun.
  • If you can see your shadow, look at the head. The axis of the rainbow runs from the Sun behind you, through your eyes and through the head of your shadow. Imagine where your eyes might be in your shadow. If a rainbow appears that point will be its centre.
  • If you can’t see your shadow, just try and imagine the line from the Sun, passing through your head and then extend it away from you till it reaches the landscape. At whatever point it touches, that will be the centre.
  • Unless you are in a plane, the centre point is always below the horizon so on the ground or in the landscape in front of you.
  • Now, with the Sun behind you spread out your arms to either side or up and down at 45⁰ from the centre point.
  • Swing them round like the blades of a windmill. That is where your primary rainbow will appear.

Remember that:

• Every observer has a rainbow axis and a centre-point on that axis that moves with them as they change position. It means that their rainbow moves too.
• The centre of a secondary rainbow is always on the same axis as the primary bow and shares the same anti-solar point.
• To see a secondary rainbow look for the primary bow first – it has red on the outside. The secondary bow will be a bit larger with violet on the outside and red on the inside.

Rainbows as discs of colour

• Close consideration of why rainbows appear as arcs or circles can be explained by the idea that an observer is looking at superimposed, concentric discs of colour.
• Think in terms of each observed band of colour within a rainbow forming on the edge of a separate coloured disc.
• The area close to the circumference of each disc produces the most intense and brilliant colour.
• The intensity of each colour drops sharply away from the circumference of its disc and towards the centre.
• The observed colour of each disc corresponds with the band of wavelengths that produces it.
• The fact that we see distinct bands of colour in a rainbow is often described as an artefact of human vision.
• Each disc contributes small amounts of its own colour to the area towards the shared centre of the six concentric discs making the sky appear lighter.

When an observer looks up into the sky and sees an atmospheric rainbow they are looking at tiny images of the Sun mirrored in millions of individual raindrops. This is what produces the impression of arching bands of colour.

• It is the mirror-like surfaces on the inside of raindrops that reflect microscopic images of the Sun towards an observer.
• The images are tiny because raindrops are small, but also because the surface they reflect off is concave.
• At a micro-scale, each image of the Sun is different:
  • Each and every image is a different colour and depends on the wavelength of light each raindrop is reflecting towards an observer’s eyes at any particular moment.
  • For convenience sake, wavelength is usually measured in nanometres, but nanometres can be divided into picometres (or even smaller units). This means that an observer is looking at countless wavelengths of light and so countless colours.
  • The images range in size and shape depending on the dimensions of the droplets and turbulence in the atmosphere. The size and roundness of raindrops also affect the appearance of a rainbow as a whole.
• The millions of microscopic images of the Sun that produce the impression of a rainbow is similar to the way pixels of light produce the images we see on digital displays.

Notice that:

• If all the rays of incident light that contribute to the formation of an observer’s rainbow are traced back from each raindrop towards the Sun it transpires that they are produced by parallel rays and that each incident ray is polarized as it passes through a droplet.
• If all the rays of incident light that travel through a single raindrop as it falls are compared, it transpires that they are all parallel with the axis of the rainbow.

The exact position at which an atmospheric rainbow will appear in the sky can be anticipated by imagining a straight line that starts at the centre of the Sun behind you, passes through the back of your head, out through your eyes and extends in a straight line into the distance.

• The imaginary line that joins the Sun, observer and the centre of the rainbow is called the rainbow axis.
• The point on the rainbow axis around which a rainbow appears is called the anti-solar point. The centre of a rainbow coincides with the anti-solar point.
• Stand with the Sun on your back and look at the ground on a sunny day, the shadow of your head marks the point called the antisolar point, it is 180° away from the Sun.
• The red arc of a primary bow forms at an angle of 42.4⁰ from the rainbow axis.
• Seen from an observer’s point of view, the angle outwards from the rainbow axis to the coloured arcs is called the viewing angle.
• In diagrams, the same angle between the axis and a line extended from an observer’s eyes to the arcs of a rainbow is called the angular distance.
• With the Sun behind you, spread out your arms to either side or up and down to get a sense of where a rainbow should appear if the conditions are right.
• Unless seen from the air, the centre of a rainbow and the anti-solar point will always be below the horizon.
• The centre of a secondary rainbow is always on the same axis as the primary bow and shares the same anti-solar point.
• To see a secondary rainbow look for the primary bow first – it has red on the outside. The secondary bow will be a bit larger with violet on the outside at an angle of 53.4⁰ and red on the inside.

• 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.

Rainbow colour refers to the colours seen in rainbows and other situations where visible light separates into its component wavelengths and the corresponding hues become visible to the human eye.

• Rainbow colour (also called spectral colour) is a colour model.
• A colour model is a theory of colour that establishes terms, definitions, rules and conventions for understanding and describing colours and their relationships with one another.
• A spectral colour is a colour evoked in normal human vision by a single wavelength of visible light (or by a narrow spread of adjacent wavelengths).
• When all the spectral colours are mixed together in equal amounts and at equal intensities, they produce white light.
• In order of wavelength, the rainbow colours (ROYGBV) are red (longest visible wavelength), orange, yellow, green, blue and violet (shortest visible wavelength).
• It is the sensitivity of the human eye to this small part of the electromagnetic spectrum that results in our perception of colour.
• Whilst the visible spectrum and its spectral colours are determined by wavelength (and corresponding frequency), it is our eyes and brains that interpret these differences in electromagnetic radiation and produce colour perceptions.
• Naming rainbow colours is a matter more closely related to the relationship between perception and language than anything to do with physics or optics.
• Even commonplace colour names associated with rainbows such as yellow or blue defy easy definition. These names are concepts related to subjective impressions.
• Modern portrayals of rainbows show six colours – ROYGBV. This leaves out other colours such as cyan and indigo.
• Atmospheric rainbows actually contain millions of spectral colours. Measured in nanometres there are around 400 colours between red and violet, measured in picometres there are 400,000.

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.

Perhaps the most common of atmospheric effects, the blueness of the sky, is caused by the way sunlight is scattered by tiny particles of gas and dust as it travels through the atmosphere.

The sky is blue because more photons corresponding with blue reach an observer than any other colour.

In outer space, the Sun forms a blinding disk of white light set against a completely black sky. The only other light is produced by stars and planets (etc.) that appear as precise white dots against a black background. The sharpness of each of these distant objects results from the fact that photons travel through the vacuum of space in straight lines from their source to an observer’s eyes. In the absence of gas and dust, there is nothing to scatter or diffuse light into different colours and no surfaces for it to mirror or reflect off.

All of this changes when sunlight enters the atmosphere. Here, the majority of photons do not travel in straight lines because the air is formed of gases, vapours and dust and each and every particle represents a tiny obstacle that refracts and reflects light. Each time a photon encounters an obstacle both its speed and direction of travel change resulting in dispersion and scattering. The outcome is that, from horizon to horizon, the sky is full of light travelling in every possible direction and it reaches an observer from every corner.

The following factors help to account for why blue photons reach an observer in the greatest numbers:

• The sky around the Sun is intensely white in colour because vast numbers of photons of all wavelengths make the journey from Sun to an observer in an almost straight line.
• In every other area of the sky, light has to bend towards an observer if they are to see colour. It is this scattering of light that fills the sky with diffuse light throughout the day.
• Longer wavelengths of light (red, yellow, orange and green) are too big to be affected by tiny molecules of dust and water in the atmosphere so scatter the least so few are redirected towards an observer.
• Shorter wavelengths (blue and violet) are just the right size to interact with obstacles in the atmosphere. These collisions scatter light in every possible direction including towards an observer.
• Because blue is relative intense compared with violet in normal conditions and in the absence of the longer wavelengths the sky appears blue.
• However, there is a whole band of wavelengths corresponding with what we simply call blue. As a result, different atmospheric conditions fill the sky with an enormous variety of distinctly different blues during the course of the day.

If we understand why the sky is usually blue it’s easier to understand why it can be filled with reds and pinks at sunrise and sunset.
 

Let’s review why the sky is blue

• In most weather conditions, the Sun and the area around it appear intensely white to an observer because vast numbers of photons of every wavelength make the journey from Sun to their eyes in an almost straight line.
• The Sun, and the area around it, appears white because it contains a mixture of all wavelengths of light (white light).
• In every other area of the sky, sunlight is striking billions of particles that make up the atmosphere and scattering in every possible direction.
• If it were not for this scattering (deflection of light in all directions), the sky would be as black as night. In reality, an observer is bathed in light arriving from every direction and the sky, as a result, appears to be full of diffuse light.
• Not all wavelengths of light behave in the same way when scattered by the small particles that make up the atmosphere.
• Longer wavelengths of light (red, yellow, orange and green) are too big to be affected by tiny molecules of dust and water so scatter the least.
• Shorter wavelengths (blue and violet) are just the right size and are affected by reflection, refraction and scattering as they strike successions of particles. It is these collisions that direct light in every possible direction including towards an observer.
• Because human eyes are more sensitive to blue than violet, in most atmospheric conditions, and in the absence of the longer wavelengths, the sky appears blue.
• A wide band of wavelengths corresponds with what we often describe as blue. As a result, the sky is filled with an enormous variety of distinctly different blues during the course of every day.

Why the sky is sometimes red

• A red sky suggests an atmosphere loaded with dust or moisture and that the Sun is near the horizon.
• In the morning and evening, photons must travel much further through the atmosphere than at mid-day.
• Assuming the air above our heads is around 20 km, the total distance light travels increases fivefold to around 500 km when the Sun is on the horizon.
• Remember that:
  • Longer wavelengths of light (red, yellow, orange and green) are too big to be affected by tiny molecules of dust and water so scatter the least.
  • Shorter wavelengths (blue and violet) are just the right size and are affected by reflection, refraction and scattering as they strike successions of particles.
• In the right weather conditions, light travelling horizontally through the atmosphere undergoes so much scattering that no yellow, green, blue or violet wavelengths remain.
• In these conditions, the light that reaches us, illuminating the sky and clouds and reflecting off every surface around us, is composed of wavelengths that bath the world in red and orange.

A ray of light (light ray or just ray) is a common term when talking about how and why rainbows appear.

• 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.
• Modelling light as rays is a way to discuss and represent the path of light through different media in a simple and easily understandable way.
• When light rays are drawn in a ray-tracing diagram they are represented as straight lines connected at angles to illustrate how light moves and what happens when it encounters different situations and conditions.
• More accurate descriptions of light use terms such as photons or electromagnetic waves.
• Don’t forget that:
  • The incident rays of light that contribute to a rainbow seen by an observer are those that approach raindrops parallel with the rainbow axis.
  • To understand why incident rays are always parallel with the rainbow axis we need to think in terms of what the observer sees. See: 2.7 Observer’s point of view.

The best light source for a rainbow is a strong point source such as sunlight. Sunlight is ideal because it is so intense and contains all the wavelengths that make up the visible spectrum.

• A human observer with binocular vision (two eyes) has a 120⁰ field of view from side to side. In clear conditions, the Sun can be considered to be a point-source filling just 0.5⁰ of their horizontal field of view.
• A wide range of visible wavelengths of light is needed to produce all the rainbow colours. The Sun produces a continuous range of wavelengths across the entire visible spectrum.
• When atmospheric conditions like cloud or fog cause too much diffusion of sunlight before it strikes a curtain of rain, no bow is formed.
• Artificial light sources such as LED’s, incandescent light bulbs, fluorescent lights and halogen lamps all make poor light sources because they emit too narrow a range of wavelengths and don’t emit sufficient energy.

Tiny images of the Sun mirrored in millions of individual raindrops create the impression of bands of colour arching across the sky when an observer sees an atmospheric rainbow.

• Rainbows are formed from tiny indistinguishable dots of light and each one is produced by a water droplet from which an observer manages to catch a glimpse of an image of the Sun.
• It is the precise position of each individual raindrop in the sky that determines:
  • Whether or not it is in the range of possible positions that will enable it to reflect an image of the Sun towards the observer.
  • The exact spectral colour that it will produce at any moment and over the passage of time as it falls.
• The precise position of each raindrop changes over time as it falls, causing its colour to change from red through to violet. Prior to reflecting red, each raindrop is invisible to an observer. After reflecting violet the amount of light reflected by each raindrop drops off sharply.
• Raindrops reflect and refract the greatest concentration of photons towards an observer from the intense bands of colour within the arcs of a rainbow.
• Raindrops inside the coloured arcs, in the area between the anti-solar point and the inside edge of the violet bow, direct light towards an observer causing this area to appear lighter or brighter than the rest of the sky.  Factors that determine the appearance of this area include:
  • Lower intensity: Each raindrop reflects far fewer photons in the direction of an observer once they have fallen below the violet band of a rainbow.
  • Reduced saturation: The saturation of each rainbow colour reduces sharply as raindrops leave the violet band because they mix with other droplets that are reflecting other colours.
  • Any situation where an observer is exposed to a mixture of a wide range of wavelengths in similar proportions produces the impression of white rather than a specific colour.
  • Scattering: Light reflected by a raindrop in the direction of an observer may encounter a series of other raindrops on its journey causing random scattering of light in other directions.

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

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.

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.

The most common atmospheric rainbow is a primary bow.

•  Primary rainbows appear when sunlight is refracted as it enters raindrops, reflects once off the opposite interior surface, is refracted again as it escapes back into the air, and then travels towards an observer.
• The colours in a primary rainbow are always arranged with red on the outside of the bow and violet on the inside.
• The outside (red) edge of a primary rainbow forms an angle of approx. 42.4⁰ from its centre, as seen from the point of view of the observer. The inside (violet) edge forms at an angle of approx. 40.7⁰.
• To get a sense of where the centre of a rainbow might be, imagine extending the curve of a rainbow to form a circle.
• If your shadow is visible as you look at a rainbow its centre is aligned with your head.
• A primary rainbow is only visible when the altitude of the sun is less than 42.4°.
• Primary bows appear much brighter than secondary bows and so are easier to see.
• The curtain of rain on which sunlight falls is not always large enough or in the right place to produce both primary and secondary bows.

A secondary rainbow appears when sunlight is refracted as it enters raindrops, reflects twice off the inside surface, is refracted again as it escapes back into the air, and then travels towards an observer.

• A secondary rainbow always appears alongside a primary rainbow and forms a larger arc with the colours reversed.
• A secondary rainbow has violet on the outside and red on the inside of the bow.
• When both primary and secondary bows are visible they are often referred to as a double rainbow.
• A secondary rainbow forms at an angle of between approx. 50.4⁰ to 53.4⁰ to its centre as seen from the point of view of the observer.
• A secondary bow is never as bright as a primary bow because:
  • Light is lost during the second reflection as a proportion escapes through the surface back into the air.
  • A secondary bow is broader than a primary bow because the second reflection allows dispersing wavelengths to spread more widely.

Remember that:

• The axis of a rainbow is an imaginary line passing through the light source, the eyes of an observer and the centre-point of the bow.
• The space between a primary and secondary rainbow is called Alexander’s band.

Primary rainbows are sometimes referred to as first-order bows. First-order rainbows are produced when light is reflected once as it passes through the interior of each raindrop.

Secondary rainbows are second-order bows. Second-order bows are produced when light is reflected twice as it passes through the interior of each raindrop.

• Each subsequent order of rainbows involves an additional reflection inside raindrops.
• Higher-order bows get progressively fainter because photons escape droplets after the final reflection. As a result, insufficient light reaches an observer to trigger a visual response.
• Each higher-order of bow gets progressively broader spreading photons more widely and reducing their brightness further.
• Only first and second-order bows are generally visible to an observer but multi-exposure photography can be used to capture them.
• Different orders of rainbows don’t appear in a simple sequence in the sky.
• First, second, fifth and sixth-order bows all share the same anti-solar point.
• Zero, third and fourth-order bows are all centred on the Sun and appear as circles of colour around it.

https://www.atoptics.co.uk/rainbows/orders.htm

Supernumerary rainbows are faint bows that appear just inside a primary rainbow. Several supernumerary rainbows can appear at the same time with a small gap between each one.

• The word supernumerary means additional to the usual number. The first supernumerary rainbow forms near the violet edge of the primary bow and is the sharpest. Each subsequent supernumerary bow is a little fainter.
• Supernumerary bows often look like fringes of pastel colours and can change in size, intensity and position from moment to moment.
• Supernumerary rainbows are clearest when raindrops are small and of equal size.
• On rare occasions, supernumerary rainbows can be seen on the outside of a secondary rainbow.
• Supernumerary rainbows are produced by water droplets with a diameter of around 1 mm or less. The smaller the droplets, the broader the supernumerary bands become, and the less saturated are their colours.
• Supernumerary bows result from the wave-like nature of light and are caused by interference between the waves that contribute towards the main bow. In some places, the waves amplify each other, and in others, they cancel each other out.
• The theory is that rays of a similar wavelength have slightly different distances to travel through misshapen droplets affected by turbulence, and this causes them to get slightly out of phase with one another. When rays are in phase, they reinforce one another, but when they get out of phase they produce an interference pattern that appears inside the primary bow.

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.

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.

Rainbows can be formed by droplets of liquids other than water, or even by a cloud of solid transparent microspheres. The table below shows the viewing angles for primary rainbows produced by a number of different media.
 

Substance	Refractive index	Viewing angle
Water	1.33	42.5
Kerosene	1.39	34.5
Carbon tetrachloride	1.46	26.7
Benzene	1.50	22.8
Plate glass	1.52	21.1
Other glass	1.47 to 1.61	25.7 to 14.2

Primary rainbow viewing angles for various media

• Materials with an index of refraction of 2.00 or more do not produce primary rainbows.
• Diamonds, for example, do not produce primary rainbows because their index of refraction is 2.42. However, if a diamond is ground into microspheres, it can produce secondary and higher-order rainbows.

Data from https://www.basic-physics.com/rainbows-figuring-their-angles/

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.

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.

An idealized raindrop forms a sphere. These are the ones that are favoured when drawing diagrams of both raindrops and rainbows because they suggest that when light, air and water droplets interact they produce predictable and replicable outcomes.

• In real-life, full-size raindrops don’t form perfect spheres because they are composed of water which is fluid and held together solely by surface tension.
• In normal atmospheric conditions, the air a raindrop moves through is itself in constant motion, and, even at a cubic metre scale or smaller, is composed of areas at slightly different temperatures and pressure.
• As a result of turbulence, a raindrop is rarely in free-fall because it is buffeted by the air around it, accelerating or slowing as conditions change from moment to moment.
• The more spherical raindrops are, the better defined is the rainbow they produce because each droplet affects incoming sunlight in a consistent way. The result is stronger colours and more defined arcs.

Real-life raindrops

• Raindrops start to form high in the atmosphere around tiny particles called condensation nuclei — these can be composed of particles of dust and smoke or fragments of airborne salt left over when seawater evaporates.
• Raindrops form around condensation nuclei as water vapour cools producing clouds of microscopic droplets each of which is held together by surface tension and starts off roughly spherical.
• Surface tension is the tendency of liquids to shrink to the minimum surface area possible as their molecules cohere to one another.
• At water-air interfaces, the surface tension that holds water molecules together results from the fact that they are attracted to one another rather than to the nitrogen, oxygen, argon or carbon dioxide molecules also present in the atmosphere.
• As clouds of water droplets begin to form, they are between 0.0001 and 0.005 centimetres in diameter.
• As soon as droplets form they start to collide with one another. As larger droplets bump into other smaller droplets they increase in size — this is called coalescence.
• Once droplets are big and heavy enough they begin to fall and continue to grow. Droplets can be thought to be raindrops once they reach 0.5mm in diameter.
• Sometimes, gusts of wind (updraughts) force raindrops back into the clouds and coalescence starts over.
• As full-size raindrops fall they lose some of their roundness, the bottom flattens out because of wind resistance whilst the top remains rounded.
• Large raindrops are the least stable, so once a raindrop is over 4 millimetres it may break apart to form smaller more regularly shaped drops.
• In general terms, raindrops are different sizes for two primary reasons,  initial differences in particle (condensation nuclei) size and different rates of coalescence.
• As raindrops near the ground, the biggest are the ones that bump into and coalesce with the most neighbours.

Raindrops, incident light and primary rainbows

Let’s look at the rays of incident light that contribute to a primary rainbow.

• All rays of light that contribute to a primary rainbow strike the surface of each raindrop three times. Once as they enter a droplet and undergo refraction, again as they reflect off the rear interior surface and once more as they undergo refraction for the second time and exit in the direction of the observer.
• Whilst some photons are following paths that will produce a primary rainbow there are many other possibilities for every photon and the vast majority go off in other directions.
• Incident rays of light that form the curved apex of a primary rainbow strike the upper half of raindrops in line with their vertical axis. These rays initially deviate downwards during refraction and internal reflection towards an observer.
  • Rays bend downwards (and slow down) as they enter a droplet and are refracted towards the normal.
  • Rays then reflect off the interior surface on the far side of a droplet and are directed downwards again.
  • When they strike the surface a third time, they are refracted away from the normal (and speed up) as they exit in the direction of the observer.
  • In some cases, this final step is an upward bend and so reduces the overall angle of deviation relative to their source.
• Incident rays of light that form the curved sides of a primary rainbow strike the side of a raindrop in line with their horizontal axis. These rays initially deviate inwards during refraction and internal reflection towards an observer.
• Incident rays of light striking the lower half of raindrops are initially directed upwards and away from the observer.

Raindrops, incident light and secondary rainbows

Now let’s look at the rays of incident light that contribute to a secondary rainbow.

• All rays of light that contribute to a secondary rainbow strike the surface of each raindrop four times. Once as they enter a droplet and undergo refraction, twice as they reflect off the interior surface and once more as they undergo refraction for the second time and exit in the direction of the observer.
• Incident rays of light that form the curved apex of a secondary rainbow strike the lower half of raindrops in line with their vertical axis. These rays initially deviate vertically upwards during refraction and internal reflection.
  • Rays bend upwards (and slow down) as they enter each droplet and are refracted towards the normal.
  • Rays then reflect twice off the interior surface on the far side of the droplet. After the second strike, they are directed downwards towards the observer.
  • Finally, at the fourth strike, they refract away from the normal (and speed up) as they exit.
• Incident rays of light that form the curved sides of a secondary rainbow strike the side of a raindrop in line with their horizontal axis. These rays deviate inwards during refraction and internal reflection towards an observer.
• Incident rays of light striking the upper half of raindrops at the apex of a rainbow during the formation of a secondary rainbow are initially directed downward and away from the observer.

Alexander’s band

• The fact that light deviates downwards when it strikes the upper half of droplets that contribute to a primary rainbow and deviates upwards when it strikes the lower half of droplets that contribute to secondary bows accounts for the darker area between the two known as Alexander’s band.

An important optical effect that explains how raindrops produce rainbows is refraction.

Refraction refers to the way that electromagnetic radiation (light) changes speed and direction as it travels across the interface between one transparent medium and another.

• As light travels from a fast medium such as air to a slow medium such as water it bends toward the normal and slows down.
• As light passes from a slower medium such as water to a faster medium such as air it bends away from the normal and speeds up.
• In a diagram illustrating optical phenomena like refraction or reflection in a raindrop, the normal is a line drawn from the surface of a raindrop to its centre.
• The speed at which light travels through a given medium is expressed by its refractive index (also called the index of refraction).
• If we want to know in which direction light will bend at the boundary between transparent media we need to know:
  • Which is the faster, less optically dense (rare) medium with the smaller refractive index.
  • Which is the slower, more optically dense medium with the higher refractive index.
• The degree to which refraction causes light to change direction is dealt with by Snell’s law.
• Snell’s law considers the relationship between the angle of incidence, the angle of refraction and the refractive indices (plural of index) of the media on both sides of the boundary. If three of the four variables are known, then Snell’s law can calculate the fourth.

More about refraction in a raindrop

• Light rays (streams of photons) undergo refraction twice when they encounter a raindrop, once as they enter, then again as they leave.
• Once inside a raindrop, a given photon may reflect off the inside surface of a raindrop several times, but on each refraction, some light crosses the boundary back and undergoes refraction as it escapes into the surrounding air.
• Some photons never escape, instead, they are absorbed when they strike electrons within a raindrop, releasing heat that can causes evaporation.

Not all incident light striking a raindrop crosses the boundary into the watery interior of a droplet. Some of the incident light is reflected off the surface and a small proportion of that travels towards the observer.

• Incident light reflected off the surface facing an observer undergoes neither refraction nor dispersion.
• Because the outside surface of a raindrop forms a shiny convex mirror, reflected light diverges in every possible direction depending on its initial point of impact.
• Just as raindrops form the coloured arc of a primary rainbow, they can also reflect white light towards an observer.
• White light reflected towards an observer off the outside of raindrops helps to account for why the sky on the inside of a rainbow (between its centre and coloured arcs) appears brighter and lighter than the area of sky outside.

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.

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.

Polarization of electromagnetic waves refers to the geometrical orientation of their oscillations.

Polarization restricts the orientation of the oscillations of the electric field of electromagnetic waves to a single plane from the point of view of an observer. This phenomenon is known as plane polarization.

• Plane polarization filters out all the waves where the electric field is not orientated with the plane from the point of view of an observer.
• To visualize plane polarization, imagine trying to push a large sheet of card through a window fitted with close-fitting vertical bars.
• Only by aligning the card with the slots between the bars can it pass through. Align the card at any other angle and its path is blocked.
• Now substitute the alignment of the electric field of an electromagnetic wave for the sheet of card, and plane polarization for the bars on the window.
• Polarizing lenses used in sunglasses rely on plane polarization. The polarizing plane is orientated horizontally and cuts out glare by blocking vertically aligned waves.
• Plane polarization is one of the optical effects that account for the appearance of rainbows.
• It is the position of each raindrop on the arc of a rainbow, with respect to the observer, that determines the angle of the polarizing plane.
• Rainbows are typically 96% polarized.

Let’s take this one step at a time

• Rainbows form in the presence of sunlight, raindrops and an observer, and involve a combination of refraction, reflection and chromatic dispersion.
• It is during reflection off the back of a droplet that light becomes polarized with respect to an observer.
• The rear hemisphere of a raindrop forms a concave mirror in which an observer sees a tiny reflection of the Sun.
• As a rainbow forms, an image of the Sun forms in each and every raindrop and the ones in exactly the right place at the right time become visible to the observer.
• The light reflected towards an observer is polarized on a plane bisecting each droplet and at a tangent to the arc of the rainbow.
• The rear hemisphere of a raindrop is best thought of as the half of the raindrop opposite the observer and with the Sun at its centre.
• Now recall that to see yourself in a normal flat mirrored surface it has to be aligned perpendicular to your eyes. Get it right and you see yourself right in the middle. If it’s not perpendicular, then you see your image off-centre because the mirror is not aligned with your eyes on either the horizontal or vertical planes.
• The Sun appears right in the centre of every raindrop from the point of view of an observer only if it is in exactly the right position in the sky at the right time. In all other cases, the light is scattered in other directions.
• Only rays that strike at the point where the horizontal and vertical planes intersect are reflected towards the observer. Rays that strike to the left or right or above/below the centre-point miss the observer.
• The correct alignment of a raindrop involves the vertical axis of the rear hemisphere being at exactly 90⁰ with respect to the observer. In the case of a primary rainbow, the horizontal axis is titled downwards by approx. 20.5⁰.

Wavefronts

Parallel electromagnetic waves with a common point of origin, the same frequency and phase, and propagating through the same medium, produce an advancing wavefront perpendicular to their direction of travel.

• Lasers that form a pencil of light made of parallel rays produce waves with flat wavefronts.
• An electromagnetic wave with a  flat wavefront is called a plane wave.

Point sources emitting electromagnetic waves in all directions, at same frequency and phase, and propagating through the same medium, produce spherical wavefronts tangental to their origin.

Diffraction

Diffraction of electromagnetic radiation refers to various phenomena that occur when a light wave encounters an obstacle or opening.

• Diffraction describes the way light waves bend around the edges of an obstacle into regions that would otherwise be in shadow.
• An object or aperture that causes diffraction is treated as being the location of a secondary source of wave propagation.
• Diffraction causes a propagating wave to produce a distinctive pattern when it subsequently strikes a surface.
• Diffraction produces a circular pattern of concentric bands when a narrow beam of light passes through a small circular aperture.

• In classical physics, the diffraction of electromagnetic waves is described by treating each point in a propagating wavefront as an individual spherical wavelet.
• As each wavelet encounters the edge of an obstacle it bends independently of every other. However, interference between wavelets alters the angle to which they bend and the distance they must travel before striking a surface.
• The explanations that best describe the process of diffraction belong to Wave Theory and are the result of two centuries of study in the field of optics.

• In modern quantum mechanics, diffusion is explained by referring to the wave function and probability distribution of each photon of light when it encounters the corner of an obstacle or the edge of an aperture.
• A wave function is a mathematical description concerning the probable distribution of outcomes of every possible measurement of a photon’s behaviour.

In real-life, full-size raindrops don’t form perfect spheres because they are composed of water which is fluid and are only held together by surface tension.

• In normal atmospheric conditions, the air a raindrop moves through is itself in constant motion and even at a cubic metre scale or smaller, it is composed of areas at different airflows, temperatures and pressure.
• As a result of turbulence, a raindrop is rarely in free-fall because it is buffeted by the air around it, accelerating or slowing as conditions change from moment to moment.
• Raindrops start to form high in the atmosphere around tiny particles called condensation nuclei — these can be composed of little pieces of salt left over after seawater evaporates, or particles of dust or smoke.
• Raindrops form around condensation nuclei as water vapour cools producing clouds of tiny droplets that start off roughly spherical.
• Surface tension is the tendency of liquids to shrink to the minimum possible surface area.
• At water-air interfaces, the surface tension that holds water molecules together results from them being attracted to one another more than to the nitrogen, oxygen, argon or carbon dioxide molecules that make up our atmosphere.
• As clouds of water droplets begin to form, they are between 0.0001 and 0.005 centimetres in diameter.
• As soon as droplets form they start to encounter more vapour and collide with one another. As larger droplets bump into other smaller droplets they increase in size — this is called coalescence.
• Once they are big and heavy enough they begin to fall and continue to grow. Droplets can be thought to be raindrops once they reach 0.5mm in diameter.
• Sometimes, gusts of wind (updraught) force raindrops back into the clouds and coalescence starts over.
• As full-size raindrops fall they lose some of their rounded shape. The bottom becomes flattened due to wind resistance whilst the top remains rounded.
• Large raindrops are the least stable, so once a raindrop is over 4 millimetres it may break apart to form smaller more regularly shaped drops.
• In general terms, raindrops are different sizes for two primary reasons,  initial differences in particle (condensation nuclei) size and different rates of coalescence.
• As raindrops near the ground, the biggest are the ones that bumped into and coalesced with the most droplets.

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.

The viewing angle of a rainbow is the angle between a line extended from an observer’s eyes to a bow’s centre point and a second line extended out towards the coloured arcs.

• In all cases, viewing angle, angular distance and angle of deflection all produce the same value measured in degrees.

Viewing angle and rainbows

• Viewing angle refers to the number of degrees through which an observer must move their eyes or turn their head.
• On the vertical plane, the viewing angle is a measure of how far an observer must raise their eyes or head to look from the centre of a rainbow out to the coloured arcs.
• On the horizontal plane, the viewing angle is a measure of how far an observer must look from the centre out to the side to see the coloured arcs.

Viewing angle and raindrops

• The idea of a viewing angle for a specific raindrop within a rainbow is nonsense really because they are too small to see. However, the viewing angle for a specific raindrop can be derived from the angle of deflection.
• 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.
• Of all the rays deflected towards an observer by a single raindrop, there is always one that produces the most intense impression of colour for an observer at any specific moment. It is often called the rainbow ray.
• 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.
• A ray-tracing diagram can calculate which of the rays of a specific wavelength, exiting a raindrop is the rainbow ray.
• If an observer could watch a single raindrop as it falls, they would see its viewing angle decrease and its colour change from red, through intermediate colours, to violet. With each change of viewing angle, colour and wavelength the exact trajectory of the rainbow ray must be recalculated.

Find the viewing angle

• To find the viewing angle as you look at a rainbow, trace two lines away from your eyes, one to the centre of the rainbow, and the other to any point on the coloured arcs. The viewing angle is between those two lines, which intersect within the lenses of your eyes.
• If you are not sure where the centre of the rainbow is, imagine extending the ends of the bow until they meet and form a circle. The centre (the anti-solar point) is right in the middle.
• For atmospheric rainbows seen from the ground, the anti-solar point is always below the horizon.
• The coloured arcs of a rainbow form the circumference of circles (discs or cones) and share centres at their anti-solar point.
• The viewing angle is the same whatever point is selected on the circumference of the circular arcs of the rainbow visible above the horizon.
• The viewing angle for a primary bow is between approx. 40.7⁰ and 42.4⁰ from its centre. The exact angle depends on which rainbow colour is selected.
• The viewing angle for a secondary bow is at an angle of between approx. 50.4⁰ and 53.4⁰ when you are looking outwards from its centre.
• The viewing angle can be calculated for any specific colour within a rainbow.
• The centre of a rainbow is always on its axis. The rainbow axis is an imaginary straight line that connects the light source, observer and anti-solar point.
• 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.
• The viewing angles for all rainbow 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.

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.

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 the Sun, an observer and a rainbow in side elevation.
• Think of angular distance as an angle between the centre of a rainbow and its coloured arcs with red at 42.4⁰ and violet at 40.7⁰.
• 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 in a primary bow is between 42.4⁰ and violet at 40.7⁰.
• The angular distance for any observed colour in a secondary bow is between 53.4⁰ and 50.4⁰ from 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.

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.

• Rainbows are composed of rainbow rays.
• Rainbow rays are responsible for an observer’s perception of a rainbow.
• Rainbow rays are rays of light of a single wavelength that have their origin in individual raindrops. They can be explained in terms of their angular distance from the rainbow axis at the moment they contribute to an observer’s view of a rainbow.
• Rainbow rays are ephemeral. They are not individually observable but more a way of conceptualizing the fact that at a specific moment and in a specific position a raindrop will transmit one spectral colour towards an observer before falling further, perhaps to reappear in a different position and another colour.
• Individual rainbow rays produce the intense appearance of each of the different spectral colours that together constitute the phenomenon of rainbows.
• Rainbows are composed of millions of rainbow rays and each one has its origin within a single raindrop.
• A rainbow ray is a ray of a single wavelength that for a second is responsible for a bright flash of its corresponding colour as a result of being in exactly the right place at the right time.
• Rainbow rays are always located amongst the rays that deviate the least as they pass through a raindrop and bunch together around the minimum angle of deviation.
• The millions of microscopic images of the Sun that produce the impression of a rainbow function in a similar way to the pixels that produce the images we see on digital displays.
• Rainbow rays tend to out-shine all other sources of light in the sky (other than the Sun) and account for the brilliance and imposing appearance of rainbows.
• Because raindrops polarize light at a tangent to the circumference of a rainbow, the path of rainbow rays dissects raindrops exactly in half.
• So:
  • Individual rainbow rays account for the appearance of spectral colours of a single wavelength within the arcs of a rainbow.
  • Bands of colour within a rainbow are composed of rainbow rays that together transmit narrow spreads of wavelengths towards an observer.
  • The overall appearance of a rainbow as a singular phenomenon can be accounted for by optical and geometric rules that determine the passage of light through raindrops and in the process account for rainbow rays.
• Remember: the notion of light rays and rainbow rays are useful when considering the path of light through different media in a simple and easily understandable way. But in the real world, light is not really made up of rays. More accurate descriptions use terms such as photons or electromagnetic waves.

The term rainbow angle is often paired with rainbow ray to measure the angle at which light is deflected back towards an observer as it passes through a raindrop.

• At lightcolourvision.org the term rainbow angle is avoided but is treated as being synonymous with the angle of deflection.
• The angle of deflection (rainbow angle) is measured at the point where the path of an incidence ray and the path of the same ray after it exits a raindrop towards the observer can be shown to intersect.
• To make the incident and exiting ray intersect in a ray-tracing diagram the incident ray is extended forwards in a straight line beyond the raindrop. The ray exiting the droplet towards the observer is then extended backwards until both intersect. The angle of deflection (rainbow angle) lies between the two.
• The angle of deflection (rainbow angle), for any ray that is contributing directly to the arcs of a primary rainbow, is always between approx. 40.7⁰ and 42.4⁰.

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 rays 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.

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.

The minimum angle of deviation of a ray of light of any specific wavelength passing through a raindrop is the smallest angle to which it must change course before it becomes visible within the arcs of a rainbow to an observer.

• 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 deviates from its original path (and changes speed).
• The more a ray changes direction the greater its angle of deviation.
• A ray reflected directly back on itself has an angle of deviation of 180⁰ – the maximum possible angle of deviation.
• It is the optical properties of air and raindrops that determines the angle of deviation of any ray of incident light.
• It is the optical properties of raindrops that prevent any ray of visible light within the visible spectrum from exiting a raindrop towards an observer at an angle of deviation less than 137.6⁰.
• The angle of deviation and the angle of deflection are directly related to one another and together always add up to 180⁰.
• The angle of deviation and the viewing angle are always the same.

More about the minimum angle of deviation

• The optical properties of an idealised spherical raindrop mean that no light of any particular wavelength can deviate at less than its minimum angle of deviation.
• The minimum angle of deviation of visible light depends on its wavelength.
• The minimum angle of deviation for red light with a wavelength of approx. 720 nm is 137.6⁰ but similar rays of the same wavelength but with other impact parameters can deviate up to a maximum of 180⁰.
• Different colours have different minimum angles of deviation because the refractive index of water changes with wavelength.

Impact parameter and minimum angle of deviation

• To form a primary rainbow, incident light must strike each raindrop above its horizontal axis.
• Rays of incident light of a single wavelength strike a raindrop at every possible point on the side of a raindrop facing the Sun.
• Only those that strike above the horizontal axis contribute to a primary rainbow.
• Points of impact of incident light striking a droplet can be measured on an impact parameter scale.
• It is the point of impact of rays of incident light of the same wavelength that is the variable factor that determines their subsequently different paths.
• Rays that strike nearest the horizontal axis, so with a value near 0.0 on an impact parameter scale have the largest angles of deviation.
• Rays that strike farthest away from the horizontal axis (near the topmost point on an impact parameter scale and so near 1.0) also have a large angle of deviation.