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

What is an atmospheric rainbow?

An atmospheric rainbow is an arc or circle of spectral colour 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 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 see the upper half of the circle with the sky as a backdrop.
  • Rainbows are curved because light is reflected, refracted and dispersed symmetrically around its 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 rainbow can easily disappear if anything gets in the way, causing a shadow that blocks the path of sunlight before it reaches raindrops. Hills, 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 create a fine spray of water droplets.

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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.

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Rainbows: 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 opens the corresponding page of the REFERENCE LIBRARY.

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 understand how light and raindrops produce the rainbow effects seen by an observer.
  • In the natural world, light is not really made up of rays. More accurate descriptions use terms like photons or waves.
  • Light rays don’t exist in the sense that they accurately describe the properties of light.
  • A light ray is a way of discussing and representing the path of light through different media in a simple and easily understandable way.
  • The idea of light rays is popular because light is invisible and is otherwise a difficult and complex subject. So the study of rainbows tends to be focus on matching and equally spaced parallel rays of incident light and their subsequent paths through raindrops.
  • When light rays are drawn in 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.

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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 spherical droplets of water towards an observer.

  • An optical effect is an observable event that results from the interaction of light and matter. In the case of an atmospheric rainbow, sunlight and raindrops are always present, but without an observer, there is no event because eyes are needed to produce the effect.
  • A rainbow isn’t an object in the sense that we see and recognise physical things in the world around us that we can touch.
  • Rainbows have no fixed location. Their appearance depends on where the observer is standing, the position of the Sun and where rain is falling.

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Preconditions for seeing atmospheric rainbows

There are three basic preconditions for seeing an atmospheric rainbow:

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

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Best conditions for seeing rainbows

The right weather conditions are 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 are good for rainbows because the Sun is lower in the sky all day than nearer the equator.
  • Mountains and coastal areas can create ideal conditions for seeing rainbows because as air sweeps over them, it cools, condenses and falls as rain – especially during spring and autumn.
  • 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 for seeing rainbows because the Sun isn’t as strong 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 in the sky, then by the time the raindrops are in the right position they are lost in the landscape or already on the ground.

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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 colours are also known as spectral colours.
  • A spectral colour is a colour evoked in normal human vision by a single wavelength of visible light (or by a narrow band 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 on 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. In reality, atmospheric rainbows contain millions of spectral colours and which ones appear to be the most vivid varies depending upon conditions at the time of observation.

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Bands of colour

The fact that we see a few distinct bands of colour in rainbows, 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 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.

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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. 420 from its centre, as seen from the point of view of the observer. The inside (violet) edge forms at an angle of approx. 400.
  • To get a sense of where the centre of a rainbow might be, extend the arc you can see in your mind’s eye to form a circle. If your shadow is visible the centre is aligned with your head.
  • 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 big enough to produce both primary and secondary bows.

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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 500 to 530 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 the 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 eyes of an observer and the centre-point of the bow.
  • The space between a primary and secondary rainbow is called Alexander’s band.

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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 do not reach the observer.
  • 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

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The centre and axis of a rainbow

  • 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 a rainbow can be anticipated by imagining a line that starts at the light source behind you, passes through the back of your head, out through your eyes and then extends in a straight line into the distance. The centre of the rainbow is always on that line (at 00) with the primary bow forming at an angle of a bit less than 450. Look towards the centre then spread out your arms on either side of your head to get a sense of where it will appear.
  • An imaginary straight line that passes through the light source, the observer and the centre of a rainbow is known as its axis.
  • The point on the axis of a rainbow around which the arc forms is called the anti-solar point.
  • The centre of a secondary rainbow is always on the same axis as the primary bow.
  • To see a secondary rainbow look for the primary bow first – it has red on the outside. The secondary bow will be a little bit larger with red on the inside and violet on the outside.

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Looking closely at primary rainbows

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

  1. The arc of spectral colour that curves across the sky with red on the outside and violet on the inside. These bands appear at an angle of approx. 40° to 42° from the anti-solar point as seen from the point of view of an observer.
  2. Faint supernumerary colours sometimes appear just inside the primary rainbow and form narrow bands of purples, teals or greys. These bands appear at an angle of approx. 39° to 40° from the anti-solar point so just inside the violet arc.
  3. The lighter area inside the bow from its centre out to approx. 39°. This appearance causes the background and sky inside the bow to look brighter or lighter than the area outside the bow.
  4. A relatively darker look to the sky outside the arc of a primary rainbow. If a secondary bow is also visible, the darker area begins on the outer edge of this arc.

When a secondary rainbow appears outside a primary rainbow, the area between the two is often noticeably darker in tone than anywhere elsewhere. This is called Alexander’s band.

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 effect is of pale, narrow, shimmering arcs of colour.

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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 them in his writing.
  • The darkening of the area between primary and secondary rainbows results from all the incident rays that form the two arcs being directed away from this area during refraction and reflection.
  • The raindrops that form a primary rainbow direct light downwards, away from Alexander’s band, before exiting towards the observer. The secondary bow directs light upwards, away from Alexander’s band, before exiting towards the observer.
  • Plenty of light is scattered into the area between primary and secondary rainbows but it isn’t travelling towards the observer.

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Light sources for rainbows

The best light source for rainbows 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, and after leaving raindrops, so 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 ranges of wavelengths, so some colours are excluded.

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Invisible dimensions of rainbows

A typical atmospheric rainbow includes 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:
    • All the bands of wavelengths of the electromagnetism spectrum are composed of photons and so are all different forms of light.
    • Each band of wavelengths is associated with distinct properties.
    • The idea of bands of wavelengths has been adopted for convenience sake. The entire electromagnetic spectrum is composed of a smooth and continuous range of wavelengths.
  • Radio waves, at the end of the electromagnetic spectrum with the longest wavelengths but 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 closest band with wavelengths longer than the visible light. 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 down.
  • X-rays and gamma-rays are both forms of ionising radiation meaning that they are able to remove electrons from atoms to create ions.

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 are reflections of the Sun

Tiny reflections of the Sun mirrored in millions of individual droplets 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 in the sky and each one is produced by a water droplet from which an observer manages to catch a glimpse of a microscopic 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.
  • 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 and colour reflected by each raindrop drops off sharply.
  • Raindrops reflect the greatest number of photons from the areas that produce the striking bands of colour observed within the arc of a rainbow.
  • Raindrops in the area between the arc of a rainbow and its anti-solar point also reflect light and colour towards an observer causing it to appear lighter or brighter than the rest of the sky. Three factors, in particular, determine the appearance in this area:
    • Lower intensity: Each raindrop reflects far fewer photons in the direction of an observer once they exit 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. The mixture of a wide range of wavelengths in similar proportions produces white light.
    • 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 experiences. 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

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Viewing angle of a rainbow

The viewing angle of a rainbow is the angle between its centre-point and coloured arcs. The viewing angle is calculated from the observer’s point of view.

  • To establish where the centre of a rainbow is, imagine extending the ends of the bow until they meet and form a circle.
  • The viewing angle for a primary bow is between approx. 400 and 420 from its centre as seen from the point of view of the observer.
  • The viewing angle for a secondary bow is between approx. 500 and 530 from its centre as seen from the point of view of the observer.
  • The viewing angle can be calculated for any specific colour within a rainbow.
  • The centre of a rainbow is always on its axis. The axis is an imaginary straight line that connects the light source, observer and anti-solar point.

Remember that:

  • Sunlight contains all rainbow colours and so all the colours of the visible spectrum. As sunlight travels through empty space all the corresponding wavelengths are mixed together and this is referred to as white light. When white sunlight propagates through air, water or any other media it undergoes refraction, reflection and dispersion which scatters light in every possible direction. When thinking about rainbows only the light scattering in the direction of the observer is given serious consideration.
  • The viewing angle of a rainbow, or for a particular colour, can be calculated using the law of reflection (Snell’s law).
  • The elevation of the Sun, the location of the observer and where rain is falling are all variables that determine where a rainbow will appear, but, the viewing angle is a constant determined by the laws of refraction and reflection.

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viewing-angle

Rainbow ray

The term rainbow ray describes the path along which light is most concentrated as it is refracted and reflected through a raindrop towards the observer. Whilst the rainbow ray produces the most vivid, pure colours, rays on other nearby paths produce more diffuse and weaker colours.

  • When an observer sees a rainbow they are seeing light that has been bent back on itself by refraction and reflection as it passes through raindrops.
  • Rainbow rays are produced near the minimum angle of deviation from their original path prior to striking a raindrop.
  • A rainbow ray is a ray that has the smallest angle of deviation of all the rays incident upon a raindrop.
  • The notion of light rays and rainbow rays is useful when considering the path of light through different media in a simple and easily understandable way. But light is not really made up of rays. More accurate descriptions use terms such as photons or waves.

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rainbow-ray

Rainbow angle

The term rainbow angle refers to the peak angle of deviation of rainbow rays as they bend back on themselves towards an observer in the course of refraction and reflection. The peak angle is dependent on the angle of incidence and the initial point of impact as a light ray strikes a droplet.

  • The rainbow angle is measured at the point where the path of an incidence ray intersects the path of the same ray once it exits a raindrop towards the observer.
  • To find the point at which they intersect, the path of the incident ray has to be extended in a straight line beyond its point of impact on the droplet, whilst the path after it exits the droplet has to be traced back in a straight line until they cross. The intersection point is always on the far side of a droplet from the light source and observer.
  • The rainbow angle, for any ray  contributing directly to the arcs of a primary rainbow, is always between approx. 400 and 420.
  • Because the angle of incidence follows the axis of a rainbow the viewing angle and the rainbow angle are always the same.
  • To calculate the corresponding angle of deviation for any rainbow angle subtract it from 1800. eg. 410 – 1800  = 1390.

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rainbow-angle

Angle of deviation

The angle of deviation (sometimes called 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.

  • A ray of light travelling through empty space, unaffected by gravitational forces, will travel in a straight line for ever. When a ray encounters another medium it deviates from its previous path. 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 any ray to appear in a primary rainbow it must reach a minimum angle of deviation of approx. 1380.
    • 1380 is the minimum angle of deviation for a ray with a wavelength that will appear red to the observer whilst 1400 is the minimum angle of deviation for a ray that will appear violet.
    • Angles between 1380 and 1400 correspond with viewing angles between 420 (red) and 400 (violet).
    • An angle of deviation of 1380 corresponds with a refractive index of 1.33 for red light with a wavelength of approx. 720 nanometres and travelling from air into water.
      • The intensity of light produced by this ray at angles less than 1380 is near zero.
      • The intensity of light produced by this ray at angles above 1380 decreases steadily because this is its peak angle of deviation.
      • Angles above 1380 direct light from this ray towards the centre of the bow.
  • Using Snell’s law and the law of reflection we can work out by how much a ray deviates after it first hits a raindrop.

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angle-deviation

Minimum and maximum angles 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 trace any particular colour round the arc of a rainbow you are 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

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minimum-angle-deviation

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

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peak-deviation

Rainbows and the laws of refraction and reflection

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refraction-reflection

Water droplets

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.

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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.

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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.

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Areas around the arc of 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.

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Rainbows as discs or cones of colour

Let’s think of 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 band of colour. 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 band 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.

Now let’s think of 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 band 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 arc 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.

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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.
  • 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.

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Rainbows from other substances

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.

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Rainbows, reflection and refraction

The colours of the rainbow are a result of refraction splitting the light into its constituent components, just as happens when light shines through a prism. The white light that arrives from the Sun is a combination of electromagnetic waves with varying frequencies. You see white when this mix of frequencies hits your eye at the same time, but when your eye catches an individual wave on its own, you perceive a particular colour.

Waves with frequencies between around 670 and 780 THz are perceived as shades of violet. On the other end of the spectrum are waves with frequencies between around 400 and 480 THz, which are perceived as shades of red. All the other colours come from frequencies in between these two bands. Electromagnetic waves of other frequencies can’t be perceived at all by the human eye.

GraphFigure 1: A ray of light being refracted, reflected and then refracted again.

When a ray of sunlight hits a spherical water droplet some of it will be reflected by the surface of the droplet, but some of it will enter it. As it enters, the light ray will be bent, or refracted. It’s the same phenomenon you see when you stick a straw in a water glass. The ray then continues until it hits the back of the droplet. Some of the light will exit, but some of it will be reflected back, leaving the droplet on the other side and being refracted again in the process. See figure 1.

Refraction is a result of a ray of light being slowed down as it passes from one medium to another. For a very crude analogy think of pushing a shopping trolley from the road onto grass at an angle: it will change direction because the side of the trolley that hits the grass first will be slowed down first.

Different coloured light being refractedFigure 2: Light of different frequencies is refracted by different amounts.

When light from the Sun travels through a vacuum (and to a very good approximation through air) all frequencies travel at the same speed $c$, roughly 300,000 km per second. As the ray of light passes into water, the frequency, and therefore colour, remains the same. However, its speed will change by an amount that depends on the frequency. This is because the atomic structure of water interacts differently with waves of different frequencies. A measure of the slowing-down of light with frequency $f$ is given by the refractive index $n_{f,w}.$ Its value depends not only on the frequency $f$ but also on the medium the light is entering (in this case water, as indicated by the subscript $w$). The index is defined as

$n_{f,w}=\frac{\mbox{Speed of light in a vacuum}}{\mbox{Speed of light with frequency $f$ in water}}.$

The refractive index barely changes as the frequency $f$ varies: $n_{f,w}$ is around 1.34 for the violet end of the spectrum and around 1.33 for the red end. But this small variation is enough to split sunlight into the beautiful spectrum of colours we see in a rainbow. (The refractive index also varies slightly with temperature, but we can ignore this here.)

Snell's lawFigure 3: The diagram shows the cross-section of the water droplet containing the incident ray, the refracted ray and the normal. The angles α and β are related by Snell’s law.

Just how much the light of different frequencies is bent when entering the droplet is described by Snell’s law. The law says that the refracted ray of light lies in the plane formed by the incident ray and the normal at the point of incidence – the normal is the line that passes through the point where the ray hits the droplet and is perpendicular to the surface of the droplet. Since we’re assuming the droplet to be spherical, the normal in this case is just the extended radius of the droplet, connecting its centre to the point of incidence.

Snell’s law also tells us that the angle by which a ray is refracted is given by this equation:

$\displaystyle \frac{\sin {\alpha }}{\sin {\beta }}=\frac{n_{f,w}}{n_{f, a}}.$

Here $\alpha $ and $\beta $ are the angles shown in figure 3 and $n_{f,a}$ and $n_{f,w}$ are the refractive indices of air and water, respectively, for light with frequency $f$. As air is very similar to a vacuum, the refractive index $n_{f,a}$ is very nearly equal to 1 for all frequencies. Thus, if a light ray hits the droplet so that $\alpha =45^\circ $, then red light with a refractive index 1.33 has

\[ \beta =\arcsin {\frac{\sin {45^\circ }}{1.33} = 32.12^\circ .} \]

(The result is rounded to two decimal places.) Violet light with a refractive index of 1.34 has

\[ \beta =\arcsin {\frac{\sin {45^\circ }}{1.34}=31.85^\circ .} \]

It’s these different refraction angles for the different frequencies of light that gives a rainbow its colours.

https://plus.maths.org/content/rainbows

Rainbow ray

 

By staring hard at figure 4 you can convince yourself that the deviation $D_ f(\alpha )$ is given by the formula

\[ D_ f(\alpha )=(\alpha -\beta )+(180^\circ -2\beta )+(\alpha -\beta )=180^\circ +2\alpha -4\beta . \]

Angle of deviationFigure 4: Working out the angle of deviation.

From Snell’s law, we know that we can substitute

\[ \beta = \arcsin {\frac{\sin {\alpha }}{n_{f,w}}} \]

in the above expression. (We are taking the refractive index of air to be 1 here.)

Figure 5 shows the graph of $D_ f(\alpha )$, taking the refractive index $n_{f,w}=1.33$ for a particular shade of red. Notice that it has a minimum at a value $\alpha _ m$ somewhere in the region of $60^\circ $.

GraphFigure 5: The graph of Df(α).

This minimum angle $\alpha _ m$ is what gives us the rainbow. Figure 6 shows a 2D cross-section of the droplet containing a bunch of rays for our refractive index $n_{f,w}=1.33$. The ray entering at the minimum angle $\alpha _ m$ in this cross-section is shown in red. It is called the rainbow ray. The rays that hit the droplet near the rainbow ray (with an angle close to $\alpha _ m$) cluster close to it during their passage through the droplet and when they emerge. So if your eye happens to catch the rainbow ray from this droplet after it’s emerged, you will see a whole bunch of other rays too, making the light that comes from our droplet particularly intense. Since all the clustering rays are of the same colour, our particular shade of red which has $n_{f,w}=1.33$, the droplet lights up red in the sky.

Rainbow rayFigure 6: The rainbow ray is shown in red. A cluster of rays emerges from the droplet near the rainbow ray, while rays that emerge elsewhere are more spaced out.

The fact that emerging red rays cluster near the rainbow ray is a consequence of $\alpha _ m$ being the minimum of the function $D_ f(\alpha )$. You can see this in figure 7. Take an interval $I_1$ centred on the minimum and an interval $I_2$ of the same width centred elsewhere. The range of deviation angles for values in $I_1$ (given by the interval $J_1$) is much smaller than the range of deviation angles for values in $I_2$ (given by the interval $J_2$). Thus, rays that hit the droplet with angles $\alpha $ in $I_1$ stick closer together than rays with angles $\alpha $ in $I_2$.

If you don’t believe the picture, here is a proof.

GraphFigure 7: The interval J1 is smaller than the interval J2.

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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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 in sunglasses work in this way. The polarizing plane is orientated horizontally and cuts out glare by blocking all but horizontally 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.

Remember first that every raindrop in a rainbow reflects an image of the Sun towards an observer. Now imagine that the image of the Sun in each each droplet is composed of polarized light waves and that in every case, the light reflected towards an observer is polarized on a plane bisecting each droplet and at a tangent to the arc of the rainbow.

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 rear hemisphere of a raindrop is best thought of the half of the raindrop opposite the observer and with the Sun at its centre. The Sun and the observer are always on the principal axis.

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.

your reflection in a concave hemispherical mirror. Align a raindrop perfectly and a reflection of the Sun appears right in the centre from the point of view of an observer. Get it wrong and the reflection of the Sun misses you completely.

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 observed 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 polarise the light seen by an observer.

  • The diagram shows each raindrop in cross section. Each one has been cut in half right through the centre.
  • The cross cut produces either a vertical or horizontal plane.
  • The incident light rays enter, are reflected and exit each droplet in line with the surface formed by the plane.
  • The sequence in each case is that incident light strikes the droplet and is refracted, it then
  • through the is why the diagram shows half of each raindrop as a hemi-spherical cross-section. The cross-section in this diagram forms a plane that includes the light source, the centre of the raindrop and the eye of the observer. It is positioned at the apex of the observed rainbow.

 

 

In the field of optics

The process for a primary rainbow is as follows:

The front half of every raindrop that contributes to an observer’s rainbow refracts rays of incident light as they cross the boundary from air to water and then again as they exit.

The back half acts as a semi-hemispherical concave reflector

If In the case of a primary rainbow, incident light enters the raindrop above halfway.

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Rainbows and Snell’s law

Why does a rainbow form a perfect arc

Colour cones

So you see a red dot in the sky for every droplet from which your eye manages to catch an outgoing red rainbow ray. To see where in the sky these droplets are relative to you, let’s first work out the exact value of $\alpha _ m$. Solving $\frac{d D_ f(\alpha )}{d\alpha }=0$ to find the minimum gives

\[ \alpha _ m=\arccos {\sqrt{\frac{n_{f,w}^2-1}{3}}}. \]

(See here for the details).

Substituting $n_{f,w}=1.33$ (for our particular shade of red) gives $\alpha _ m=59.58^\circ $ and $D_ f(\alpha _ m)=137.48^\circ $.

Now if an emerging rainbow ray from a droplet meets your eye, then this means that the emerging ray makes an angle $r_ f=180^\circ -137.48^\circ =42.52^\circ $ with the line $L$ shown in figure 8. It’s the line you get from extending the ray of sunlight that would pass straight through your eye if your head wasn’t in the way. (Remember we’re assuming that the rays from the Sun are parallel.) Let’s call $r_ f$ the rainbow angle. Of course it depends on the frequency $f$ and therefore on colour.

GraphFigure 8: The deviated rainbow ray from your droplet makes a 42.52 degree angle with the line L.

If you take all lines that emanate from your eye and make a $42.52^\circ $ angle with $L$, what you get is a cone (see figure 9). The droplets that light up red for you all lie on this cone: if they didn’t, you wouldn’t be catching their red rainbow ray. But when you look along the surface of a cone from its vertex, as your eye is doing, all you see is a circle. You can try this by rolling a sheet of paper up into a cone shape and peering through the little hole at the tip. The rainbow comes from droplets that lie on the cone at different distances from your eye, some can be near and others can be far. But your eye can’t distinguish the distances, all it sees is red light mingling together to form a circular arc that appears to be located somewhere in the distance. The reason why you don’t see the full circle is that the Earth gets in the way. Unless you’re above the water droplets, for example when looking down from a plane, in which case you can see a beautiful circular rainbow.

GraphFigure 9: The droplets you see light up in the sky lie on the surface of a cone.

The same reasoning goes for all the other colours of the spectrum: they appear as circular arcs. But the varying indices of refraction give a different rainbow angle for each colour. For example, violet light with $n_{f,w}=1.34$ gives $\alpha _ m=59.0^\circ $ and $D_ f(\alpha _ m)=138.93^\circ $, so the rainbow angle in this case is $41.07^\circ $. Thus, the rainbow appears as a nested sequence of circles of colours in order of their refractive indices, or, equivalently, in order of their frequencies: red at the top all the way through to violet at the bottom.

This explanation also shows why you only ever see a rainbow when you’re standing with your back to the Sun: that’s the only way you can catch rainbow rays coming from droplets. It also explains why the sky appears much brighter below the rainbow than above. Since the vast majority of rays leaving a droplet do so above the rainbow ray (see figure 6), you won’t catch any rays from droplets that are “above” the rainbow (that is, outside the cones for the various colours). So you won’t see any reflected light from these droplets. However, your eye does catch reflected light from droplets “below” the rainbow (droplets that lie inside the cones) and it’s this light that makes the sky below the rainbow appear brighter. It appears as white light because non-rainbow rays for different colours, coming to your eye from different droplets, are mixed together.

The rainbow geometry also shows that any rainbow you see is yours and yours alone: whatever a person standing next to you might see, it’ll come from a different set of water droplets and therefore it’ll be a different rainbow.

Sometimes, if you are lucky, you might see a second, slightly fainter rainbow above the main one. The secondary rainbow is a result of light rays being reflected twice within the water droplets. The rainbow angles for the various colours are around 51 degrees in this case, which is why the secondary rainbow is seen higher in the sky. The double reflection also means that the colours of the secondary rainbow appear in reverse order, with violet at the bottom and red at the top. Here’s the original sketch by René Descartes, who first explained the shape of the rainbow, showing both the primary and secondary rainbow. The double reflection corresponding to the secondary rainbow is traced in red.
https://plus.maths.org/content/rainbows

Rainbows and light waves

Rainbows wave-fronts and interference patterns