Why Does a White Rainbow Appear in Fog?
― When Droplets Are Too Small, the Seven Colours Overlap and Turn White
On an autumn morning, a large, colourless arc sometimes floats faintly in the fog. It is called a fogbow, or a "white rainbow". It works almost exactly like an ordinary rainbow. The only difference is the size of the water droplets. When droplets get too small, light starts to show its nature as a wave, and the seven colours blur together.
It is a clear autumn morning. A riverside road or a basin town is wrapped completely in white fog.
Soon the morning sun shines in from behind you, and the fog thins a little. You glance ahead, and a big arc shaped like a rainbow hangs in the fog.
But the arc has almost no colour. It is a white rainbow, like spilled milk. It isn't even raining. So why is there a rainbow? And why is it white?
A white rainbow has just two causes
A fog droplet is only about 1/50 the size of a raindrop. Light passing through such a small droplet does not go straight. It spreads a little sideways. That makes each colour band of the rainbow wide and blurry.
In a rainbow, red and violet are only about 2 degrees apart. When the bands widen, neighbouring colours overlap. When all the colours of light overlap, they make white.
So a fogbow is not a "failed rainbow". Water droplets send light back in the same way. It is simply a rainbow whose colours have mixed because the droplets are small. Let's go through it step by step.
Fog droplets are much smaller than raindrops
An ordinary rainbow forms when light bends as it enters a raindrop, bounces off the back, and bends again as it leaves. Each colour bends by a slightly different amount. So in the sky opposite the sun, the colours line up with red on the outside and violet on the inside.
Raindrops are often around 1 millimetre across. Fog droplets are said to be about 0.01 to 0.02 millimetres. By diameter, that is 1/50 to 1/100 as big.
Here the other face of light comes in. Light is a wave. When a wave meets a gap or a particle close to its own wavelength in size, it curls around behind it and spreads sideways. This is called diffraction.
The wavelength of visible light is about 0.5 thousandths of a millimetre. A raindrop is 2000 times bigger than that, so the spreading is barely noticeable. A fog droplet, though, is only a few dozen times the wavelength. Then the light leaving the droplet spreads enough to see. Try changing the droplet size with the slider in Figure 1.
Why do wide colour bands turn white?
In an ordinary rainbow, seen from the head of your shadow, red appears at about 42 degrees and violet at about 40 degrees. All seven colours are packed into a width of only about 2 degrees. Two degrees is about the width of one finger at arm's length.
With fog droplets, each colour band is said to spread by more than 1.5 degrees on each side. The bands become several times wider than the gaps between colours. Then red, yellow, green and blue light all reach the same spot at once, and your eye sees a mixed colour.
Unlike paints, light gets brighter and closer to white the more you stack it. A TV screen makes white from only tiny red, green and blue dots in the same way. So the middle of the arc turns white. Still, at the very outer edge only red arrives, and at the very inner edge only violet or blue arrives. That is why, if you look closely, a fogbow can have a faint red outer rim and a faint blue inner rim.
A fogbow is also wider than an ordinary rainbow, and it is said to make a slightly smaller circle. Because the light spreads out, its brightness is thinned too. This is why it is easy to miss.
On clear, calm nights the ground cools and morning fog forms easily. Autumn and winter, with their long nights, are the seasons when this fog is common. Riverside areas and basins ringed by mountains are especially known for morning fog. To see a fogbow, three things need to come together: (1) the sun is behind you, (2) the sun is low, as in the morning, and (3) the fog has thinned to the point where sunlight gets through. A high place that rises above the fog is an especially good spot.
When your shadow falls on fog on a mountain, you may see small coloured rings around the head of the shadow. This is a separate phenomenon called a glory, and the rings are only a few degrees across. A fogbow hangs much farther out, at about 40 degrees. Both can sometimes be seen at the same time in the same fog.
Summary
Like an ordinary rainbow, a fogbow forms when water droplets bend light and send it back. It looks white because the fog droplets are so small that light spreads sideways as a wave. The colour bands widen, overlap, and mix into white.
What separated the colours of a rainbow was the size of the water droplets.
In front of droplets that are too small, light cannot keep its seven colours apart.
Why an ordinary rainbow appears at a fixed angle opposite the sun is explained in detail in Why does a rainbow always appear opposite the sun? Another example of light waves overlapping to make colours is in Why do soap bubbles look rainbow-coloured? For what morning fog actually is, see also Why is the grass wet in the morning?
- On a sunny morning or evening, stand with your back to the sun. Spray water ahead of you with a hose shower, and an ordinary rainbow appears.
- Next, spray in the same direction with a spray bottle (one that gives the finest mist). Notice that the finer the droplets, the paler the colours and the wider the band.
- Compare the two rainbows by how well the colours separate and how wide the band is. Photograph them with your phone and put them side by side to see the difference clearly.
Spray-bottle droplets are often bigger than fog droplets, so the bow may not turn fully white. Even so, you should be able to confirm that the colours look paler and blurrier than with the hose. Do this where the water won't land on anyone.
For those who want to know more ― terms, formulas and links to textbooksEach part is labelled with its level, from middle-school science to university courses
- Middle schoolCovered in middle-school science
- High schoolCovered in high-school physics
- High school+Advanced high-school material, or textbook sidebars
- UniversityUniversity courses not taught in high school (optics, atmospheric optics)
- ResearchTopics researchers are still investigating, not yet taught as settled
Middle schoolTerms: this phenomenon has names
- Fogbow (kiri-niji) / white rainbow (hakkō): a whitish rainbow formed opposite the sun by tiny water droplets in fog or cloud.
- Refraction and dispersion of light: light bends as it enters and leaves water. Because the amount of bending differs slightly by colour, the colours separate.
- Diffraction: a wave curls around and spreads behind a particle or gap close to its own wavelength in size. It happens with sound and water waves too.
Middle schoolHigh schoolCheck with a formula: how much wider do the colour bands get with fog droplets?
We calculate the angle by which light spreads behind a droplet, using the simplest estimate. We work it out for both a fog droplet and a raindrop, and compare it with the gap between rainbow colours.
| In symbols | θ ≒ λ ÷ d |
| In words | Spreading angle of light (radians) ≒ wavelength of light ÷ droplet diameter |
| Where it comes from | It comes from the diffraction of waves. In the direction where the path difference between waves passing the two edges of the droplet is about one wavelength, the waves begin to cancel. So the smaller the droplet and the longer the wavelength, the larger the spreading angle. |
| θ | Spreading angle of the light (in radians; 1 radian is about 57.3 degrees) |
| λ | Wavelength of the light (in micrometres; 1 micrometre is 1/1000 of a millimetre) |
| d | Diameter of the water droplet (in micrometres) |
| Wavelength of green light (roughly the middle of visible light) | about 0.55 micrometres |
| Fog droplet diameter (a typical value) | about 20 micrometres (0.02 mm) |
| Raindrop diameter (a typical value) | about 1000 micrometres (1 mm) |
| Angle between red and violet in an ordinary rainbow | about 2 degrees (red about 42, violet about 40) |
| 1 radian in degrees | about 57.3 degrees |
| Spread for a fog droplet (radians) | 0.55 ÷ 20 ＝ 0.0275 |
| Convert to degrees | 0.0275 × 57.3 ≒ 1.58 |
| Spread for a raindrop (radians) | 0.55 ÷ 1000 ＝ 0.00055 |
| Convert to degrees | 0.00055 × 57.3 ≒ 0.032 |
| Gap between neighbouring colours of the seven (divided into 6 gaps) | 2 ÷ 6 ≒ 0.33 |
| How many times the colour gap is the fog droplet's spread? | 1.58 ÷ 0.33 ≒ 4.8 |
With fog droplets, each colour band spreads by about 1.6 degrees on each side. That is about 5 times the gap to the neighbouring colour. Light of one colour reaches the place of the colour five steps away, so the colours mix and turn white. With raindrops the spread is only about 0.03 degrees, around 1/10 of the colour gap, so the colours stay clearly separate.
③ In everyday terms, 1.6 degrees is about the width of one finger at arm's length. The seven colours of a rainbow are packed into one finger's width, and each colour then smears out by one finger's width. That is a fogbow.
High schoolHigh school+The rainbow angle and the diffraction spread
High schoolThe rainbow angle can be found with the law of refraction (Snell's law) for light reflected once inside a droplet. As you vary where the light enters, there is a turning point where the outgoing direction barely changes. Light piles up there, forming a bright arc. Water's refractive index is slightly different for red (about 1.331) and violet (about 1.343), so the angle shifts with colour.
High school+The formula above, θ ≒ λ ÷ d, has the same form as single-slit diffraction (the dark-line condition) taught in high school. For a round particle the coefficient changes slightly, to about 1.22. For a rainbow, though, the light leaves not from the whole droplet but from a part near its edge. So the real spread is said to be larger than this estimate. The calculation here is only meant to give a sense of scale.
UniversityAiry theory and Mie scattering
The first wave-based treatment of the rainbow is said to be the Airy theory (Airy's theory of the rainbow) by the 19th-century astronomer Airy. It considers the shape of the wavefront leaving the droplet and expresses the brightness distribution of the rainbow with the "Airy function". The theory shows that the smaller the droplet, the wider the rainbow band, and the more the primary bow shifts slightly inward (to a smaller angle). This also explains why a fogbow's circle is a little smaller. More accurately, Mie scattering (Mie theory), which solves the scattering of light by a spherical particle directly from the equations of electromagnetic waves, is said to reproduce even the whiteness of the fogbow and the faint supernumerary bows that appear inside it.
📖 Derivations and further reading: Fogbow - Wikipedia (Japanese) ／ Mie scattering - Wikipedia (Japanese)
ResearchWhat is still not well understood
- Measuring cloud droplet size from rainbows. The brightness and polarization pattern of a rainbow formed on top of a cloud (a cloudbow) change with droplet size. Researchers are working on estimating cloud droplet size by measuring this from satellites and aircraft. How accurately it can be read in real clouds, where droplets are not all the same size, remains a challenge.
- How fog droplets grow. The number and size of fog droplets change with the amount of dust and pollution in the air and with how fast it cools. Forecasting when fog forms and when it clears is still considered a hard problem.
- Droplets that are not round. When fog droplets are about to freeze, or ice particles mix in, the way light scatters changes. How rainbows and halos look in such mixed clouds has not been fully reproduced even by calculation.
So the content of this article, too, is "an explanation of what is known so far". The whiteness of a fogbow itself is well explained, but efforts to read the contents of clouds and fog from that whiteness are still under way.
Links to textbooks (by level)
| Level | Subject / unit | Where in this article |
|---|---|---|
| Middle school | Science, Year 1: "Refraction of light", "Colours of light" | How rainbows form, why colours separate |
| High school | Physics: "Properties of waves", "Diffraction of light" | Why smaller droplets spread light more, the formula θ ≒ λ ÷ d |
| High school+ | Physics (advanced): "Diffraction by a single slit and a circular aperture" | The 1.22 coefficient, limits of the estimate |
| University | Optics, atmospheric optics | Airy theory, Mie scattering |
| Research | Atmospheric physics, satellite observation of clouds | Estimating cloud droplet size from rainbows |
| ― | Link to daily life | Look for one in autumn and winter morning fog, with the sun at your back |
- Fogbow - Wikipedia (Japanese)
- Diffraction - Wikipedia (Japanese)
- Fog - Wikipedia (Japanese)
- Raymond L. Lee Jr., Alistair B. Fraser, "The Rainbow Bridge: Rainbows in Art, Myth, and Science", Pennsylvania State University Press, 2001
- M. Minnaert, "Light and Color in the Outdoors", Springer, 1993
*This article is a popular-science explanation for general readers. The figures given are rough guides to help you understand the mechanism. The size of fog droplets and raindrops, and the angles of rainbows, vary with conditions. When moving through fog, take great care, as visibility is poor.