Introduction to Rainbows

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An Introduction to Rainbows

Here are all the diagrams in our Introduction to Rainbows series

Each one can be viewed on its own page with a full explanation.
AND
Did you know that all our diagrams are FREE to download!

Take a Photo of a Rainbow
How to See a Rainbow
Rainbows Seen From the Air Form a Circle
Rainbows Seen From the Ground Form an Arc
A Rainbow is an Optical Phenomenon
Sun, Observer and Rainbow Share a Common Axis
The Lower the Sun, the Higher the Rainbow
The Higher the Sun, the Lower the Rainbow
Rainbows Appear as Bands of Spectral Colour
The Angle Between Incident and Refracted Rays
The Apparent Position of a Rainbow
Angular Distance Determines Raindrop Colour
Alexander's Band
Dispersion of White Light in a Raindrop
Reflection and Refraction in a Raindrop
The Path of a Red Ray Through a Raindrop
Parallel Light Rays Incident to a Raindrop
Rays from a Point Source Incident to a Raindrop
Non-parallel Light Rays Incident to a Raindrop
The Elevation of Raindrops Determines Their Colour
Formation of Rainbows
The Elevation of a Raindrop Determines its Colour
Rainbows and the Polarization of Light
Polarization of Light in a Raindrop
Colour Brightness and Angular Distance
Rainbow Diagrams
Rainbows as Superimposed Discs of Colour
Rainbows as Superimposed Cones of Colour

To find out more about the diagrams above . . . . read on!

About the Diagrams

There are forty pages in this section so far, and every one features a diagram and full explanation

Choose a page by clicking on one of the images above.

But don’t miss the introduction below. Here is a list of the sections that appear on this page.

IT ALL STARTS HERE

What is an atmospheric rainbow?

What is an atmospheric rainbow?

An atmospheric rainbow is an arc or circle of spectral colours that 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 observable 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 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.

Conditions for seeing a rainbow

Conditions for seeing a rainbow

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

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

See a rainbow today

See a rainbow today

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

  • 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 in the sky.
  • 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 for seeing rainbows because it blocks direct sunlight.
  • Winter is not necessarily the best season because the light is weaker and there can be too much 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 they are lost in the landscape or already on the ground.

Understanding rainbows

Understanding rainbows

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 rays take through different media and are the result of reflection, refraction and dispersion of wavelengths 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 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.

A rainbow is an optical effect

A rainbow is an optical effect

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

  • Sunlight and raindrops are always present when a rainbow appears but without an observer there is nothing, because eyes are needed to produce the visual  illusion.
  • 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 bows are produced by different sets of raindrops.

Looking closely at rainbows

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

  1. 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 anti-solar point as seen from the point of view of an observer.
  2. 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 the anti-solar point as seen from the point of view of an observer.
  3. 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 anti-solar point so just inside the violet arc of the primary bow.
  4. The area inside a rainbow from its centre out to approx. 39°  often appears lighter or brighter in comparison to the sky on the outside. The effect is produced by a mixture of scattered rays from across the visible spectrum.
  5. When a secondary rainbow appears outside a primary rainbow, the area between the two is often noticeably darker in tone than anywhere else. This is called Alexander’s band. The effect is produced by rays being directed away from this area as primary and secondary bows are formed.

Distance to a rainbow

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

  • The position of a rainbow is determined by angles rather than distance. The angles are constants determined by the physical properties of light and raindrops, not least the laws of reflection and refraction.
  • As an observer moves, the rainbow they see moves with them but the angles are preserved.
  • The shape of a rainbow can be described as a volume bounded by two cones. The vertices of both cones (the pointy ends) are within the lens of your eyes, one cone extends outwards towards the rainbow and the other extends inwards towards your retina.

Centre, axis and anti-solar point

Atmospheric rainbows always appear in the section of sky directly opposite the Sun as seen from the point of view of the observer.

  • The exact position of an atmospheric rainbow in the sky can be anticipated by imagining a line (axis) that starts at the light source behind you, passes through the back of your head, out through your eyes and extends in a straight line into the distance.
  • The centre of the rainbow is always on the rainbow axis with the primary bow forming at an angle of a bit less than 450.
  • 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.
  • The imaginary straight line that passes through the light source, the observer and the centre of a rainbow is known as the rainbow axis.
  • The point on the rainbow axis around which a rainbow appears is called the anti-solar point.
  • Every observer has a rainbow axis and anti-solar point that moves with them as they change position which means their rainbow moves too.
  • The anti-solar point is always below the horizon because the sun is above 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 and red on the inside.

Terms and concepts

Terms and concepts

To really make sense of rainbows, an appropriate set of terms helps to explore key concepts.

In the process of writing this introduction, we have carefully selected a set of closely interrelated terms. Click any item in the list below to see examples without leaving this page.

Alternatively, every time a new term appears in the text it is highlighted in blue. Click on it, and it will open the corresponding page of the REFERENCE LIBRARY.

RAINBOWS AND COLOUR

Rainbow colour

Rainbow colour

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 we all generally agree upon, but they are not strictly defined by anything in the nature of light itself.
  • Modern portrayals of rainbows show six colours – ROYGBV. This leaves out other colours such as cyan and indigo.
  • Atmospheric rainbows contain millions of spectral colours. Measured in nanometres there are around 400 colours between red and violet, measured in picometres there are 400,000.

Bands of colour

What is an atmospheric rainbow?

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.

Why the sky is blue

Why the sky is blue

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 wavelength.

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 object results from the fact that photons travel through the vacuum of space in completely 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, there are no media to cause refraction or dispersion into different colours and no surfaces to mirror or reflect light.

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.

Five factors help to account for why the sky appears blue:

  1. The sky around the Sun is intensely white in colour because vast numbers of photons in this region make the journey from Sun to an observer in an almost straight line. An observer sees the ones that haven’t been affected by the atmosphere.
  2. In the rest 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 so many diffuse colours during the course of the day.
  3. 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 continue on their way with very few redirected towards an observer.
  4. 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.
  5. Because the relative intensity of blue is greater than violet an observer sees blue.
  6. However, there is a whole band of wavelengths that corresponds with what we simply call blue. As a result, different atmospheric conditions can fill the sky with an enormous variety of distinctly different colours at the same time.

Why the sky is sometimes red

Why the sky is sometimes red

If you understand why the sky is blue when clouds don’t get in the way then its easier to understand why it can be filled with reds and pinks at sunrise and sunset.

Five factors help to account for why the sky appears blue:

  1. The sky around the Sun is intensely white in colour because vast numbers of photons in this region make the journey from Sun to observer in an almost straight line. An observer sees the ones that haven’t been affected by the atmosphere.
  2. In the rest 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 so many diffuse colours during the course of the day.
  3. 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 continue on their way with very few redirected towards an observer.
  4. Shorter wavelengths (blue and violet) are just the right size to interact with obstacles in the atmosphere. These collision scatter light in every possible direction including towards an observer.
  5. Because the relative intensity of blue is greater than violet an observer sees blue.
  6. However, there is a whole band of wavelengths that corresponds with what we simply call blue. As a result, different atmospheric conditions can fill the sky with an enormous variety of distinctly different colours at the same time.

Why the sky turns red:

RAINBOWS AND LIGHT

Rainbows and light

Rainbows and light

Rainbows result from light encountering raindrops in the presence of an observer. The phenomenon of rainbows offers many clues as to the nature of light.

Light is a form of radiation, a type of energy that travels in the form of electromagnetic waves and can also be described as a flow of particle-like ‘wave-packets’, called photons. Radiation, electromagnetic waves and photons are all concepts that are interchangeable with the more general concept of light.

Theories of light

There are four principal theories that underpin our understanding of the physical properties of light as it relates to rainbows:

  • Wave theory – the idea that light is transmitted from luminous bodies in an undulatory wave-like motion.
  • Particle theory – the idea that the constitution and properties of light can be described in terms of the interactions of elementary particles.
  • Electromagnetic theory – the classical theory of electromagnetism that describes light as coupled electric and magnetic fields, transporting energy as it propagates through space as a wave. The energy is stored in its electric and magnetic fields and can be measured in terms of its intensity.
  • Quantum theory – explains the interactions of light with matter (atoms, molecules etc.) and describes light as consisting of discrete packets of energy,  photons. Quantum physics suggests that electromagnetic radiation behaves more like a classical wave at lower frequencies and more like a classical particle at higher frequencies, but never completely loses all the qualities of one or the other.

These theories tell us things about the properties of light such as:

  • It is electromagnetic radiation, the force carrier of radiant energy.
  • Whilst it carries energy and has momentum, it has no mass.
  • It is the result of the interaction and oscillation of electric and magnetic fields.
  • It is a microscopic phenomenon that needs macroscopic metaphors such as waves and particles to describe it.
  • Once emitted at its source, it can propagate indefinitely through a vacuum in a straight line at the speed of light ( 299,792,458 metres a second) but can be deflected by gravity.
  • In any specific instance, it can be described in terms of the inter-relationship of its wavelength, frequency and energy.
  • Light slows down and is deflected as it propagates through air, water, glass and other transparent materials as photons interact with matter.

Phenomena associated with light include:

  • Absorption
  • Diffraction
  • Dispersion
  • Interference
  • Photoelectric effect
  • Polarization
  • Reflection
  • Refraction
  • Scattering
  • Transmission

Some facts about electromagnetic waves

  • An electromagnetic wave carries electromagnetic radiation.
  • Electromagnetic radiation is measured in terms of the amount of electromagnetic energy carried by an electromagnetic wave.
  • Electromagnetic waves can be imagined as synchronised oscillations of electric and magnetic fields propagating at the speed of light in a vacuum.
  • The kinetic energy carried by electromagnetic waves is often simply called radiant energy or light.
  • Electromagnetic waves are similar to other types of waves in so far as they can be measured in terms of wavelength, frequency and amplitude.
  • Other terms for amplitude are intensity and brightness.
  • Another term for the speed at which light travels is its velocity.
  • We can feel electromagnetic waves release energy when sunlight warms our skin.
  • The position of an electromagnetic wave within the electromagnetic spectrum can be identified by its frequency, wavelength or energy.

Some facts about photons

  • Photons are the fundamental building block and so the smallest unit of light.
  • Photons are the carriers of electromagnetic force and travel in harmonic waves.
  • Photons are zero mass bosons.
  • Photons have no electric charge.
  • The amount of energy a photon carries can make it behave like a wave or a particle. This is called the “wave-particle duality” of light.

Facts about the electromagnetic spectrum

  • Visible light is just one tiny part of the electromagnetic spectrum.
  • Our eyes only respond to the visible light which we see as the colours between red and violet.
  • The electromagnetic spectrum includes, in order of increasing frequency and decreasing wavelength: radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays and gamma rays.
  • The limit for long wavelengths might be the size of the universe, while it is thought that the short wavelength is in the vicinity of the Planck length (approximately 1.6 x 1035 meters).

Rainbows and rays of light

Rainbows and rays of light

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

  • The idea that light is made up of rays is so commonplace when describing and explaining rainbows that it is easily take 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 they accurately describe one of the physical properties of light. More accurate descriptions use terms like photons or waves.
  • Modelling light as ray is a way of discussing and representing the path of light through different media in a simple and easily understandable way.
  • When light rays are drawn in ray-tracing diagrams they are represented as straight lines connected at angles to illustrate how light moves and what happens when it encounters different media.
  • The nearest thing to a light ray in terms of everyday appearance is the narrowly focused beam of light produced by a laser.

Light sources for rainbows

The best light source for a rainbow is a strong point-source such as sunlight. Sunlight is ideal because it is so strong 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 defuse sunlight, it can cause too much scattering of rays before they reach raindrops, in which case 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 have the energy to reach sufficient water droplets.

Rainbows are reflections of the Sun

Tiny reflections 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 a microscopic reflected 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 if seen by an observer.
  • 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 the greatest number of photons towards an observer from the striking bands of colour within the arcs of a rainbow.
  • Raindrops in the area between a rainbow and its anti-solar point reflect light towards an observer causing it to appear lighter or brighter than the rest of the sky. Three factors, in particular, determine their appearance in this area:
    • Lower intensity: Each raindrop reflects far fewer photons in the direction of an observer once they have fallen beyond 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 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 other raindrops on its journey causing random scattering of light in all directions.
  • Six key concepts help to pick apart how and why the relative position of individual raindrops within a rainbow determines what an observer sees. Each one is dealt with separately in the sections below:
    • Viewing angle
    • Rainbow ray
    • Rainbow angle
    • Angle of deviation
    • Minimum angle of deviation
    • Peak angle of deviation

Rainbows and electromagnetic waves

To understand the formation of rainbows it is important to remember that they are composed of light waves, which is to say, electromagnetic waves.

EM-Wave

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

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

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Rainbows and the laws of refraction and reflection

RAINBOW ANATOMY

Primary rainbows

The most common 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 the 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, in your mind’s eye, extend the arc you can see to form a circle. If your shadow is visible the 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 to produce both primary and secondary bows.

Secondary rainbows

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 second 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 so spreads light over a greater area
  • 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 centre-point of the bow and the eyes of an observer.
  • The space between a primary and secondary rainbow is called Alexander’s band.

Orders of rainbows

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

  • Secondary rainbows are second-order rainbows and are produced when light is reflected twice as it passes through the interior of each raindrop.
  • Each subsequent order involves an additional reflection.
  • Higher-order bows get progressively fainter because photons escape droplets at each reflection so insufficient numbers reach an observer to trigger a visual response.
  • Each higher-order of bow gets progressively broader spreading photons more widely so reducing their brightness further.
  • Only first and second-order bows are generally visible to an observer but multi-exposure photography can be used to reveal 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.

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

Alexander's band

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

  • Alexander’s band is named after Alexander of Aphrodisias, an ancient Greek philosopher who commented on the effect in his writing.
  • This darkened area between primary and secondary rainbows can be explained by the fact that very few photons are directed toward an observer from this area.
  • The raindrops that form a primary rainbow all direct exiting light downwards towards an observer so away from Alexander’s band.
  • The raindrops that form a secondary bow all direct exiting light upwards towards an observer, so away from Alexander’s band.
  • Plenty of light is scattered into the area between primary and secondary rainbows but none of it travels towards the observer.

Supernumerary rainbows

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 edge of the primary bow and is normally the sharpest. Each subsequent supernumerary bow is a little fainter. They often look like fringes of pastel colours and can change in size, intensity and position from moment by moment.
  • Supernumerary rainbows are clearest when raindrops are small and of equal size.
  • On rare occasions, supernumerary rainbows can be seen on the outside 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 rainbows are caused by interference between light waves that contribute towards the main bow but are out of phase with one another by the time they leave a raindrop and travel towards the observer.
  • Supernumerary bows result from the wave-like nature of light. They are caused by interference between the advancing wave-fronts of different colours. 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 can cause 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 areas around a rainbow

The area inside a primary rainbow

The area inside the arc of a primary rainbow, from its centre out to the band of colour, often appears tonally lighter than the area of sky outside:

  • The maximum angle from the axis of a rainbow at which any light is refracted and dispersed by raindrops towards an observer corresponds with the outside edge of the red arc (420). There is no refraction or dispersion of colours towards an observer at angles greater than 420 for primary rainbows.
  • The area inside the arc of a primary rainbow contains light that has been reflected off the outside surface of raindrops facing an observer. This light has not undergone refraction or dispersion so reflects white light back towards an observer.
  • The area inside the arc of a rainbow contains randomly scattered refracted light as rays bounce about between droplets. This produces a mixture of different wavelengths and produces a lighter appearance to an observer.

The area outside a primary rainbow

The area outside the arc of a primary rainbow often appears tonally darker than the area of sky on the inside:

    • The radius of a rainbow is determined by the refractive index of water droplets. It is the refractive index of water that determines the apparent colour of each droplet and so also the position of the arc seen by an observer.
    • The refractive index of a medium determines how much a ray of light refracts (bends) as it passes from one medium to another.
    • In the case of a primary rainbow, refraction causes all the incident light to bend slightly inwards towards the centre of the rainbow whilst none is directed outwards.
    • In the case of primary rainbows, all refracted and dispersed light is produced by incident rays striking the top half of droplets and exiting from the bottom half towards the observer after one internal reflection.
    • In the case of secondary rainbows, all reflected, refracted and dispersed light is produced by incident rays striking the bottom half of droplets and exiting from the top half towards the observer after two internal reflections.

The area between a primary and a secondary 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 takes place in raindrops that form a primary rainbow, light is directed inwards away from Alexander’s band towards an observer.
  • As refraction and dispersion takes place in raindrops that form a secondary rainbow, light is directed outwards away from Alexander’s band towards an observer.

The area outside a secondary rainbow

The area outside a secondary rainbow is effected in a similar way to the area inside a primary rainbow:

  • Secondary rainbows are not as brightly coloured as primary rainbows because of the amount of light transmitted away in other directions.
  • Whilst light is refracted and dispersed into the area outside rather than inside a secondary rainbow it does not significantly lighten the sky. The effect is enough however to make Alexander’s band appear darker in comparison.

Remember that:

    • White light, containing all wavelengths within the visible part of the electromagnetic spectrum, 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 the separation of white light into rainbow colours.
    • When all the different wavelengths of the visible spectrum are mixed together they produce white light.
    • White light is the name given to visible light that contains all wavelengths of the visible spectrum in equal proportions and at equal intensities.
    • 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 strike a white or neutral coloured surface.
    • Colour is what a human observer sees when a single wavelength, a band of wavelengths or a mixture of different wavelengths strike a white or neutral coloured surface.

The invisible dimensions of rainbows

A typical atmospheric rainbow includes six bands of colour between red and 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 are composed of photons and so each produces a different kind of light.
    • Each band of wavelengths has 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.
  • 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 that 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.
  • 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.

Other types of rainbows

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.

Primary rainbow viewing angles for various media

Substance Index of refraction Viewing angle for a primary rainbow (in degrees)
Water 1.33 42.5
Kerosene 1.39 34.5
Carbon tetracloride – used in paints, adhesives and degreasers 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
  • 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.

Rainbows: Figuring Their Angles

Fogbows, dewbows, moonbows and more

There are many optical effects similar to rainbows.

  • A fogbow is a similar phenomenon to a rainbow. As its name suggests, they are associated with fog rather than rain. Because of the very small size of water droplets that cause fog a fogbow has only very weak colours.
  • A dewbow can form where dewdrops reflect and disperse sunlight. Dewbows 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 moonbow is produced by moonlight rather than sunlight but appears for the same reasons. Moonbows are often too faint to excite the colour receptors (cone cells) but sometimes appear in photographs taken at night with a long exposure.
  • Twinned rainbows are produced when two rain showers with different sized raindrops overlap one another. Both raindows have red on the outside and violet on the inside. The two bows often intersect at one end.
  • A reflection rainbow is produced when light reflects off large lakes or the ocean before striking a curtain of rain. The conditions must be ideal with no wind so that the reflecting water acts like 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 which 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.

RAINDROPS

Raindrops

An idealised raindrop in free-fall and not buffeted by the wind forms a sphere. The more perfect the sphere, the better the rainbow it produces because each droplet affects incoming sunlight in a consistent way. Raindrops in real conditions don’t form perfect spheres.

Let’s think about the real-life of a raindrop.

  • Water molecules collect around dust and smoke particles high in the atmosphere and begin to form clouds. Raindrops start off roughly spherical in shape because of surface tension.
  • Surface tension is stronger on small drops which helps them to maintain their shape. But as raindrops fall they collide with others and increase in size.
  • As larger raindrops begin to fall they lose some of their rounded shape. They become flattened on the bottom and with a curved top because the airflow on the bottom is greater than on the top.
  • Once the size of a raindrop gets too large, it will break apart to form more small, spherical drops.
  • The size of raindrops is important. When all the droplets are the same size, they produce rainbows with vivid bands of colour. If the droplets are too large (over 3 to 4 mm) then air resistance affects their shape and causes colours to blur. If the droplets are too small they float and form mist or fog which makes for faint, fuzzy rainbows.
  • The spatial distribution of raindrops is important. If a curtain of rain is crossing an observer’s entire field of view, the rainbow may appear continuous from end to end. Patches of rain produce fragments of rainbows.
  • Where and when rain falls is the result of endless changes in atmospheric conditions as air and clouds are blown across the landscape, so a long arc may appear suddenly or shrink down to nothing in seconds.
  • The temporal distribution of raindrops is important. A rainbow that at one moment looks almost close enough to touch 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.

Raindrops and incident light

Primary rainbows

Let’s look at rays of incident light that contribute to a primary rainbow whilst ignoring the other directions they can be refracted, reflected or transmitted.

  • 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 again and exit in the direction of the observer.
  • Incident rays of light striking the upper half of raindrops at the apex of a primary rainbow initially deviate vertically downwards during refraction and internal reflection towards an observer. Rays bend downwards (and slow down) as they enter each droplet and are refracted towards the normal. They then reflect off the interior surface on the far side of the droplet and are directed downwards again. When they strike the surface a third time, they are refracted away from the normal (speed up) and exit in the direction of the observer. In some cases, this final step is an upward bend that reduces the overall angle of deviation relative to their source.
  • Incident rays of light striking the outer half of raindrops at the sides of a primary rainbow are affected in a similar way but their path is along the horizontal axis of individual droplets and of the rainbow’s arc.
  • Incident rays of light striking the lower half of raindrops and following a similar path to those described about but are directed upwards and away from the observer.

Secondary rainbows

Now let’s look at rays of incident light that contribute to a secondary rainbow whilst ignoring the other directions they can be refracted, reflected or transmitted.

  • 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 again and exit in the direction of the observer.
  • Incident rays of light striking the lower half of raindrops at the apex of a secondary rainbow 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. They then reflect off the interior surface on the far side of the droplet and reflect upwards. When they strike the surface a third time, they reflect again but this time they turn downwards. Finally, at the four strike, they are refracted away from the normal (speed up) and exit at a downwards angle towards the observer. This final step reduces the overall angle of deviation relative to their source.
  • Incident rays of light striking the inner half of raindrops at the sides of a secondary rainbow are affected in a similar way but their path is along the horizontal axis of individual droplets and of the rainbow’s arc.
  • Incident rays of light striking the upper half of raindrops and following a similar path to those described above are directed downward and away from the observer.

Remember that:

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

Reflections off the surface of raindrops

Not all incident light striking a raindrop crosses the boundary into the watery interior of a droplet. Some light is reflected off the surface facing the observer.

  • Incident light reflected off the surface facing an observer undergoes neither refraction nor dispersion.
  • Because the outside surface of a raindrop is convex it reflects white light in every possible direction.
  • In the same way that raindrops form the coloured arc of a primary rainbow, raindrops anywhere within a cone centred on the eye of an observer, and with the circumference of its base extending to an angle of 420 from its axis, can reflect white light from the Sun towards an observer.
  • White light reflected towards an observer off the outside of raindrops helps to account for the sky within a rainbow appearing brighter and lighter than the area of sky outside.

Raindrops and the polarization of light

Polarization of an electromagnetic wave refers to a situation in which the rotation of all the coupled electric and magnetic fields is restricted to a single plane from the point of view of an observer. It is the electric field that aligns with the plane. This phenomenon is known as plane polarization. Plane polarization filters out all the waves where the electric field is not orientated with the plane.

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 be pushed inside. 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 are used in sunglasses. The polarizing plane is orientated horizontally and cuts out glare by blocking vertically aligned waves. In the case of a rainbow, it is the position of each raindrop on the arc of the bow, with respect to the observer, that determines the angle of the polarizing plane.

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 the observer.

The inside surface of each raindrop provides a highly reflective mirrored surface that produces a specular image of the Sun.

The rear hemisphere of a raindrop forms a concave mirror in which an observer sees a reflection of the Sun.

The image of the Sun in each and every raindrop is reflected back towards 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 in both horizontal and 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 sunlight scatters off in other directions.

Given that an ideal raindrop forms a perfect sphere, it is not the orientation of the droplet that is important here, it is a question of where rays of light strike the hemispherical mirror on the horizontal and vertical planes. Only rays that strike at the point where the horizontal and vertical planes intersect reflect towards the observer. Rays that strike to the left or right miss the observer completely. Rays that strike above or below the centre-point on the vertical axis

The position of a raindrop in the sky along with the effects of reflection, refraction and dispersion all determine which raindrops contribute to an observer’s rainbow.

The correct alignment of a raindrop involves the vertical axis of the hemispherical mirror being at exactly 900 with respect to a plane the includes your eyes, the centre of the droplet, and, the electric field of the electromagnetic waves from the Sun. To see a primary rainbow the hemisphere has to be titled upward by 410 on its horizontal axis to allow for the effects of refraction.

When an observer sees a rainbow the light is 96% polarized by raindrops.

Raindrops wave-fronts and interference patterns

RAINBOW GEOMETRY

Centre of a rainbow

  • The idea of a rainbow axis imagines a line crossing a diagram on which the Sun, observer and rainbow can be positioned.
  • The idea of an anti-solar point is imagined from an observer’s point of view. Because the observer is looking along the axis it appears as a point, the anti-solar point.

Axis of a rainbow

  • The idea of a rainbow axis imagines a line crossing a diagram on which the Sun, observer and rainbow can be positioned.
  • The idea of an anti-solar point is imagined from an observer’s point of view. Because the observer is looking along the axis it appears as a point, the anti-solar point.

Rainbows as cones of colour

A rainbow can be thought of as being composed of a set of six concentric cones, as seen from the point of view of an observer. Each cone has a different radius and each is filled with a narrow spread of wavelengths of light that determine its apparent colour . Red fills the cone with the largest radius and violet fills the smallest.

  • To model rainbows in three dimensions allows us to think of their coloured arcs as forming within six 3D cones each of which reaches from the eye of an observer at its apex. The cones do not have a simple 2D base. At their nearest, droplets may be within reach  of an observer. At the other extreme are distant raindrop that are barely able to refract light back to an observer.
  • So, a 3D model of a rainbow accurately explains the fact that all raindrops contribute to the visual experience regardless of how far they are away from the observer.

Rainbows as discs of colour

Rainbows can be thought of 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 the outer edge because this is the area into which refraction and dispersion concentrates the most light. This concentration of colours is called the rainbow angle.
  • The apparent colour of each disc fades rapidly away from the rainbow angle because the density of rays drops towards the centre.
  • From the point of view of an observer the discs are superimposed upon one another and appears to be in the near to middle distance, in the opposite direction to the Sun, with the sky beyond as a backdrop.
  • 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 shape in three-dimensions.
  • 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 (420) than violet (400).
  • From the point of view of an observer, refraction stops abruptly at 420 and results in a sharp boundary between the red band and the sky outside the 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.
    • Reflection off the front face of raindrops and light being refracted multiple times in different raindrops causes scattering of light across the face of each disk.

Viewing angle

The viewing angle of a rainbow is the angle between its anti-solar point and the coloured arcs of its bows, measured from the observer’s viewpoint.

  • To establish where the centre of a 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 and is always below the horizon.
  • To locate the viewing angle as you look at a rainbow, trace two lines towards you, one from the anti-solar point and the other from the outer edge of the rainbow. The viewing angle is between those two lines, at their vertex, which is always within the lens of your eye.
  • The coloured arcs of a rainbow form the circumference of arcs and circles with centres at the anti-solar point.
  • The viewing angle is the same whatever point is selected on the circumference. This is directly related to the fact that rainbows appear as arches of colour.
  • The viewing angle for a primary bow is between approx. 40.70 and 42.40 from its centre.
  • The viewing angle for a secondary bow is between approx. 50.40 and 53.40 when you are looking at 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 the imaginary straight line that connects the light source, observer and anti-solar point.
  • Most incident rays striking a raindrop will follow paths that place them outside the viewing angle. These rays pass by an observer and play no part in the observer’s perceptions of colour.
  • A carefully measured ray-tracing diagram can establish whether or not the path of a ray will fall within the viewing angle of an observer.
  • The viewing angle for all rainbows is a constant 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.

Rainbow ray

The term rainbow ray describes the path that produces the most intense colour experience for any particular wavelength of light passing through a raindrop.

  • When an observer sees a rainbow they are seeing light, of all wavelengths of the visible spectrum, bent back on itself by refraction and reflection as it passes through raindrops.
  • Each intense area of colour an observer sees within the arcs of a rainbow is produced by rainbow rays within a multitude of raindrops. All other visible rays are drowned out by the intensity of the colour produced by rainbow rays.
  • When plotting the path of incident rays through a raindrop to identify rainbow rays, parallel incident rays provide the most useful information.
  • Parallel incident rays emerge from a raindrop at many angles. Those that deviate the least bunch together and it is amongst these that the rainbow ray is to be found.
  • A rainbow ray is the source of the most intense appearance of colour for any particular wavelength.
  • At any particular wavelength, the rainbow ray follows a path corresponding with the minimum angle of deviation.
  • 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.

Rainbow angle

The rainbow angle measures the angle at which light from a rainbow ray is reflected back towards an observer as it passes through a raindrop.

  • Remember that when discussing or calculating the paths and angles at which light travels through raindrops only incident rays that are parallel to one another are usually taken into consideration.
  • Because the angle of incidence of all rays drawn on a ray-tracing diagrams are shown as being parallel to the axis of a rainbow, the resulting geometry reveals that the viewing angle and the rainbow angle are always the same.
  • The rainbow angle and the angle of deviation together add up to1800 on a ray-tracing diagram.
  • The 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 made 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 rainbow angle lies between the two.
  • The 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.

Impact parameter

The term impact parameter refers to the angle at which a ray of light incident to a raindrop strikes its surface.

  • The path of parallel rays through a raindrop is determined first of all by their different angles of incidence – the angle at which they impact the surface when they first strike the surface.
  • The range of possible angles of impact for parallel rays can be represented on an impact parameter scale, drawn on a ray-tracing diagram and dividing the relevant portion of possible impact points into equal fractions between 0 and 1.
    • 0 on the scale refers to the point on the surface of a droplet at which the angle of incidence and angle of reflection are both at 900. In this case, the incident ray is aligned with the normal, aligned with the centre of the droplet and reflects back on itself.
    • For a primary rainbow  1 on the scale refers to the uppermost point on the vertical axis of the droplet. This ray strikes the surface at 900 to the normal so skims the very topmost point without deflection.
    • For a secondary rainbow 1 on the scale refers to the lowest point on the vertical axis of the droplet. This ray strikes the surface at 900 to the normal so skims the lowest point without deflection.

Angle of deviation

The angle of deviation (sometimes referred to as the angle of deflection) measures the degree to which the path of a light ray is bent by a raindrop in the course of its refraction and reflection towards an observer. The angle of deviation and rainbow angle are directly related to one another, together they always add up to 1800.

  • Remember that:
    • Any ray of light (stream of photons) travelling through empty space, unaffected by gravitational forces, will travel in a straight line forever.
    • When light travels from a vacuum or one medium into another, it deviates from its previous path (and changes speed).
    • The more a ray changes direction the greater the angle of deviation.
    • A ray reflected directly back on itself has an angle of deviation of 1800.
  • Now consider the following closely related facts for a single ray of a known wavelength striking a raindrop at a known angle:
    • For the ray to appear in a primary rainbow it must reach a minimum angle of deviation of at least 137.60.
    • 137.60 is the angle of deviation that produces the appearance of red along the outside edge of a rainbow.
    • 139.30 is the angle of deviation for a ray that will appear violet along the outside edge of a rainbow.
    • Angles of deviation between 137.60 and 139.30 correspond with viewing angles between 42.40 (red) and 40.70 (violet).
    • An angle of deviation of 137.60 corresponds with red light (wavelength of approx. 720 nm) travelling from air through a raindrop.
    • The optical properties of light and water prevent any ray from exiting a droplet at an angle of deviation less than 137.60 from the point of view of an observer.
    • The intensity of light produced by this ray at angles of deviation above 137.60 decreases steadily. These rays are directed inwards towards the centre of the rainbow. Some affect the colour of
    • Angles above 137.60 direct light from this ray towards the centre of the bow.
  • The point at which an incident ray must strike a droplet to produce an angle of deviation that creates the impression of colour for an observer depends on the size of the droplet.
  • Using Snell’s law and the law of reflection we can work out by how much any ray will deviate when it strikes a raindrop.

Minimum angle of deviation

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

  • If you see any specific colour around the arc of a rainbow you are looking at multitudes of raindrops each of which is at exactly the same angle from its centre. Because they are all the same colour they share the same rainbow angle and the same angle of deviation.   following a path marked by its minimum angle of deviation on the outside and its peak angle of deviation.
  • Different colours have different minimum angles of deviation as they pass through raindrops because the refractive index of water must be adjusted according to each wavelength.
  • The angle of deviation increases, with decreasing angles of incidence, until the angle of minimum deviation is reached.
  • Once the maximum angle of deviation angle has been reached the angle of incidence decreases.
  • The curve of the arcs of colour within a rainbow follow the minimum angle of deviation as it is reproduced in every drop of rain that directs light towards the observer.
  • In the right conditions, the minimum angle of deviation describes a circle around the anti-solar point.

Maximum deviation occurs when the angle of incidence to the surface of a raindrop is 90 degrees. In this case an incident ray “grazes” along the surface at the shallowest of angles. The maximum deviation for the emergent light ray causes it to graze along the surface after leaving the prism at the same angle.

  • The angle of incidence and angle of emergence for any ray striking a raindrop are always the same

Peak angle of deviation

The peak angle of deviation describes the point at which a ray of light produces appears most intense colour for an observer.

The minimum angle of deviation for a ray of light passing through a raindrop produces a concentration of scattered rays that accounts for both the location and the position of colours within the arc of a primary rainbow.

  • If a ray of light strikes a raindrop at a right angle, it is either transmitted directly through its centre without deviation (an angle of deviation of 00)  or reflects back along it original path.
  • In the case of a primary rainbow, rays that strike above