Visible light is a sliver of the electromagnetic spectrum less than one octave wide, and color is what your brain makes of it with just three kinds of sensor. This chapter follows light through glass, raindrops and the sky, and then into your eye.
Common mix-up: "every color is a wavelength." It isn't. Color is what three cone signals add up to in your brain. Pink, magenta, brown and white have no wavelength of their own, and the yellow on your phone screen contains no yellow light at all: it is red and green light that happen to excite your cones the same way.
Light is an electromagnetic wave, exactly the same kind of thing as the radio in the 2.4 GHz chapter. The only difference is frequency. Green light at 550 nm oscillates about 545 trillion times a second, roughly 227,000 times faster than Wi-Fi.
Your eyes respond from about 380 nm (deep violet) to about 750 nm (deep red). In frequency that is 789 THz down to 400 THz: a range of just under two to one, less than one octave if light were music. On the whole spectrum, which spans more than twenty powers of ten, that is a hairline.
Each visible photon carries between 3.26 eV (violet) and 1.65 eV (red), as the photons chapter works out. That is a sweet spot: enough energy to flip the shape of a single pigment molecule in your retina, not enough to break most chemical bonds.
Why this window and not another? Two good reasons. Water is transparent here and nearly opaque in much of the infrared and ultraviolet, and eyes evolved in water. And the Sun pours out a lot of its energy in this range.
That second point is about temperature. Anything hot glows, and the hotter it is, the shorter the wavelength where its glow peaks. A candle flame near 1,850 K looks orange; an old incandescent bulb at 2,700 K looks warm yellow-white; the Sun's surface at 5,772 K looks white. Lighting people call this color temperature, and it runs backwards from the words: "warm" light comes from the cooler source, "cool" daylight from the hotter one.
An incandescent filament at 2,700 K peaks at 1,073 nm, in the infrared. Most of what an old bulb emitted was heat you could not see, which is why it was such a poor light source. An LED labeled "2700 K" is not hot at all; the label only means its color matches that glow.
Wien's 502 nm peak is real but slippery. It is the peak of the Sun's output per nanometer of wavelength. Plot the very same light per hertz of frequency instead and the peak moves to about 339 THz, which is 883 nm, in the near infrared. Nothing physical changed; a nanometer-wide slice and a hertz-wide slice just cover different amounts of spectrum at different places.
So "the Sun peaks in green and our eyes match it" is a nice story with a hidden choice of axis. The sturdier facts are that sunlight is broad, it covers the whole visible range fairly evenly, and that is why it looks white to eyes adapted to it.
Light slows down in matter. The refractive index n says by how much: n = c / v. Water has n = 1.333, so light crosses it at about 224,900 km/s. Ordinary crown glass (the common type called BK7) has n = 1.517 and slows light to about 197,600 km/s. Diamond, n = 2.417, holds it to about 124,000 km/s.
Now send a beam into glass at a slant. One edge of each wavefront reaches the glass first and slows down while the rest is still moving at full speed. The front wheels around, like a line of marchers stepping from pavement into mud at an angle. The beam changes direction. That is refraction.
Snell's law gives the new angle exactly. A ray striking glass (n = 1.5) at 45° from the perpendicular continues at 28.1°. Light entering a denser material bends toward the perpendicular; leaving it, light bends away.
Refraction is why a pool looks shallower than it is. Looking straight down, depths appear divided by 1.333: a 2-meter deep end looks about 1.5 meters deep. It is also why a straw looks broken at the water's surface.
Going out of a dense material, something dramatic happens. Past a certain angle, Snell's law has no answer and no light escapes at all: every bit of it reflects back inside. This is total internal reflection, and the critical angle is arcsin(1/n): 48.6° for water, 41.2° for BK7 glass, just 24.4° for diamond. Diamond's tiny critical angle traps light and bounces it around inside a cut stone, which is most of its sparkle. Optical fiber uses the same trick: its core is made very slightly denser than the glass around it, so light glancing along it is trapped for kilometers.
| Material | n (at 589 nm) | Speed of light inside | Critical angle |
|---|---|---|---|
| Air | 1.0003 | 299,700 km/s | none (light leaving air into anything denser always gets in) |
| Ice | 1.31 | 228,800 km/s | 49.8° |
| Water | 1.333 | 224,900 km/s | 48.6° |
| Crown glass (BK7) | 1.517 | 197,600 km/s | 41.2° |
| Diamond | 2.417 | 124,000 km/s | 24.4° |
Between atoms, light still moves at c. What changes is the wave as a whole. The incoming field shakes the electrons in the material; each shaking electron radiates a small wave of its own (the mechanism from How a wave is born); and those little waves come out slightly behind in phase. Add the original wave and the re-radiated ones and you get a single wave whose crests advance more slowly. That combined wave is the light you measure in glass.
Frequency does not change at a boundary; it can't, because each crest arriving must produce a crest leaving. Wavelength shrinks instead, to λ/n. A 600 nm orange beam is 450 nm long in water, yet it still looks orange to a diver, because the eye responds to frequency, not to wavelength in the medium.
Snell's law also falls out of Fermat's principle: light takes the path of least time. Like a lifeguard who runs further along the beach before swimming the shorter distance, the ray trades a longer path in the fast medium for a shorter one in the slow medium, and the best trade-off is exactly n₁ sin θ₁ = n₂ sin θ₂.
The refractive index is not one number. It depends on wavelength. For BK7 glass it is about 1.531 at 400 nm (violet), 1.517 at 589 nm (yellow) and 1.513 at 700 nm (red). The difference is barely one percent, but it is enough to split white light.
The reason is resonance. Electrons in glass have natural frequencies in the ultraviolet. The closer the light's frequency is to that resonance, the harder the electrons respond, the bigger the phase lag, and the more the light slows. Violet is closest to the ultraviolet, so violet slows most and bends most. This is called normal dispersion.
A prism makes the effect large by refracting twice, at two tilted faces that both bend the light the same way. A 60° BK7 prism at its best angle turns yellow light through 38.6°, violet through 39.9° and red through 38.3°. That 1.6° fan is the spectrum Isaac Newton cast on his wall in 1666, when he showed the colors were in sunlight all along rather than added by the glass.
Rainbows are the same physics in water. A raindrop is a tiny sphere: sunlight refracts going in, reflects once off the back, and refracts again coming out. Rays entering at different heights exit at different angles, but they pile up near one limiting angle, and that pile-up is the bright bow. For red light (n ≈ 1.331) it sits at about 42° from the point directly opposite the Sun; for violet (n ≈ 1.343) at about 40.6°.
That makes a rainbow a circle of 42° radius centered on the shadow of your head. Red is on the outside, violet inside. You only see it with the Sun behind you and lower than 42° in the sky, which is why summer rainbows come in the morning and late afternoon. Every observer sees their own bow, made by a different set of drops.
Two internal reflections make a fainter secondary bow at about 50–53°, with its colors reversed: red inside, violet outside. Inside the primary the sky is brighter, because drops send light to every angle smaller than the bow. Between the two bows almost no single- or double-bounce light reaches you, so the sky there is darker. That stripe is Alexander's dark band, named after Alexander of Aphrodisias, who described it around 200 AD.
Follow a ray that hits a drop at angle of incidence i and refracts to angle r, where sin i = n sin r. After one internal reflection it has been turned through a total deviation D = 180° + 2i − 4r. As i varies, D has a minimum, and near a minimum many neighboring rays leave in almost the same direction. Setting dD/di = 0 gives cos² i = (n² − 1)/3.
For n = 1.333: i = 59.4°, r = 40.2°, D = 137.9°, so the light comes back at 180° − 137.9° = 42.1° from the anti-solar point. With two reflections, cos² i = (n² − 1)/8 and the bow lands near 51°.
Look closely just inside a bright primary bow and you may see faint pastel fringes: supernumerary bows. Ray optics can't explain them. They are interference between rays that leave at the same angle along different paths, and in 1803 Thomas Young used them as evidence that light is a wave.
Drag anywhere on the bench to aim the beam (or use the angle slider). Switch between white light and a single wavelength. The ray trace uses Snell's law at every face and real BK7 indices; if a ray can't get out, you'll see total internal reflection.
Sunlight crossing the atmosphere hits nitrogen and oxygen molecules. Each is about a thousand times smaller than a wavelength of light, so the light's electric field simply shakes the molecule's electrons back and forth. A shaking charge is a tiny antenna, and it re-radiates in all directions. That is scattering.
A tiny antenna radiates much more strongly at high frequency: the power goes as frequency to the fourth power, which is one over wavelength to the fourth. This is Rayleigh scattering. Blue at 450 nm is scattered (700/450)⁴ ≈ 5.9 times more than red at 700 nm. Violet at 400 nm is scattered 9.4 times more.
So look anywhere away from the Sun and what reaches you is sunlight that air has redirected, heavily weighted to short wavelengths. That is the blue sky. On a clear day it's a real effect of every molecule above you; the air overhead scatters away about 20% of 450 nm light but only about 4% of 700 nm light.
Why blue and not violet, if violet scatters more? Sunlight contains less violet than blue to begin with, your eyes are far less sensitive to it, and the scattered mix also carries plenty of green. That blend lands on your cones as a sky blue, slightly washed with white.
Sunset is the same physics from the other side. When the Sun sits on the horizon, its light crosses about 38 times as much air as when it is overhead. Over that path, 450 nm light is cut to about 0.02% of its starting strength while about a quarter of the 700 nm light survives. The blue has been scattered away to make someone else's sky, and what's left is orange and red. Dust, smoke and volcanic aerosols add even more scattering, which is why sunsets after big wildfires or eruptions are so vivid.
Clouds break the rule because their droplets are big: typically 10 to 20 µm across, dozens of wavelengths. Particles that size scatter all visible wavelengths about equally (Mie scattering, heading toward plain geometric reflection and refraction), so clouds look white. Thick clouds look gray only because less light makes it through them. Mars shows the flip side: fine dust gives it a butterscotch daytime sky, and its sunsets look blue.
For a whole column of clean air, the Rayleigh optical depth is roughly τ ≈ 0.0088 λ−4.05 with λ in micrometers (the exponent is a little more than 4 because air's refractive index also rises toward the blue). That gives τ ≈ 0.22 at 450 nm and τ ≈ 0.037 at 700 nm, so a vertical beam keeps e−τ = 80% and 96%. Multiply τ by the air mass, about 38 at the horizon, and the blue drops to e−8.5 ≈ 0.0002.
Scattered light also has a direction preference: Rayleigh scattering goes as 1 + cos²θ, strongest straight ahead and straight back, weakest at 90° from the Sun. And at 90° it is strongly polarized, which the last section uses.
One correction the simple model misses: at twilight the sky overhead stays blue partly because ozone absorbs orange and yellow light (the Chappuis bands). Without ozone, the zenith at dusk would look grayer.
Slide the Sun from overhead to the horizon. The panorama is computed from a single-scattering Rayleigh model: a 5,772 K sunlight spectrum, the λ⁻⁴ law, and the real air mass for each direction. It ignores dust, ozone and light scattered more than once, so treat colors as approximate. Near sunset the real zenith stays bluer than this model shows, thanks to ozone (see “the numbers behind the sky” above).
Your retina holds about 6 million cone cells and roughly 100 million rods. Rods are sensitive enough to work by starlight, but there is only one kind, peaking near 498 nm. One kind of sensor can only report "how much," never "which," which is why color drains away at night.
Cones come in three kinds, named for the wavelengths they favor: S (short) peaking near 420 nm, M (medium) near 534 nm, L (long) near 564 nm. M and L overlap heavily. Every cone also has the same blind spot as a rod: it reports only how strongly it was excited, not by what wavelength. A dim light at its peak and a bright one off its peak can look identical to a single cone.
So your entire experience of color comes down to three numbers per point in your view: how hard the S, M and L cones are firing. That's a huge compression. A light spectrum has a value at every wavelength; your eye throws all but three numbers away. Researchers estimate people can still tell apart roughly a million colors.
The compression has a famous consequence. Many different spectra produce the same three numbers, and when they do they look identical. Such look-alike pairs are called metamers. Pure 580 nm light and a mix of red and green light can excite your L and M cones in exactly the same ratio; you see both as the same yellow.
That is the entire trick of every screen. A display pixel has a red, a green and a blue emitter, roughly near 610–630, 530 and 460 nm. By setting just three brightnesses it can produce the cone triple for almost any color you need. No screen can hit every color, though: pure spectral colors sit outside what three real primaries can reach, which is why a laser's green looks more intense than anything your monitor can show.
Screens add light: red plus green gives yellow, all three give white. Paints and inks do the opposite; they subtract. Cyan ink absorbs red, magenta absorbs green, yellow absorbs blue. Print cyan over yellow and only green survives both, so you see green. Printers use cyan, magenta and yellow, plus black (the K in CMYK) because three real inks never quite make a clean black.
Now the strangest part. Light that strongly excites both L and S cones, but not M, has no wavelength of its own. Any single wavelength long enough to excite L and short enough to excite S would land in the middle of the spectrum and excite M most of all. Your brain names that combination anyway: magenta, purple, pink. These are non-spectral colors, the ones that join red back to violet and close the spectrum into a color wheel. White, gray and brown are not on the rainbow either.
About 8% of men of northern European ancestry and about 0.5% of women have a red-green color deficiency, usually because their L or M pigment is shifted toward the other, or missing. The genes for both sit on the X chromosome, which is why men are affected far more often.
Since 1931 color science has used the CIE color-matching functions, x̄, ȳ and z̄: three curves, mathematically related to the cones, that turn any spectrum into three numbers X, Y, Z. The instruments on this page compute displayed colors from a published multi-Gaussian fit to those curves (Wyman, Sloan and Shirley, 2013). The cone curves drawn below are simple bell-shaped approximations with the right peaks; real cone curves are lopsided, with long tails.
The mantis shrimp is often said to see superb color with 12 or more receptor types. Experiments suggest the opposite: it distinguishes colors worse than we do, apparently trading fine discrimination for speed.
Pick a single wavelength, or switch to the screen mode and mix red, green and blue. Watch the three cone responses, the color they produce, and whether any single wavelength could have produced it. Cone curves are Gaussian approximations.
The electric field in a light wave points sideways to the direction of travel (that's one of the consequences of Maxwell's equations). The direction it points is the polarization. Sunlight and lamplight are a jumble of all directions at once: unpolarized.
A polarizing filter passes only one direction. An ideal one passes half of unpolarized light. Light that is already polarized passes in proportion to cos²θ, where θ is the angle between its polarization and the filter: at 45° half gets through, at 90° none. Stack two filters at right angles and you get black.
Reflection polarizes light. Glare off a lake or a wet road is mostly polarized horizontally, and at one angle, Brewster's angle, the reflected light is completely polarized. For water that is 53.1° from the perpendicular; for glass, 56.3°. Polarized sunglasses are simply filters that block horizontal polarization, which is why they kill glare and let anglers see into the water.
The blue sky is polarized too. Rayleigh-scattered light coming from 90° away from the Sun can be more than 70% polarized on a clear day. Rotate polarized sunglasses while looking at that part of the sky and it darkens and brightens. Bees and many other insects navigate by this pattern; whether Vikings did the same with crystals called sunstones is still debated.
Every LCD is built on polarization. Light from the backlight passes a polarizer, then a thin layer of liquid crystal that rotates its polarization by an amount set by a voltage, then a second polarizer. Each subpixel is a light valve. That's why an LCD can go dark when you tilt your head wearing polarized sunglasses.
Reflected light is radiated by electrons in the surface, shaken by the light that entered. At Brewster's angle the refracted ray and the would-be reflected ray are exactly 90° apart. Electrons shaken in the plane of incidence would then have to radiate straight along their own direction of motion, and an oscillating charge radiates nothing along its axis. So that polarization simply doesn't reflect, and only the perpendicular polarization bounces off.
The same zero along the axis is why sky light at 90° from the Sun is polarized: molecules shaken by sunlight can't send light in the direction they are shaken. Circular polarization, where the field direction rotates as the wave moves, is what 3D cinema glasses and many OLED phone screens use.
| Source | Color temperature | Wien peak | Looks |
|---|---|---|---|
| Candle flame | ≈ 1,850 K | 1,566 nm (infrared) | Deep orange |
| Incandescent bulb | ≈ 2,700 K | 1,073 nm (infrared) | Warm yellow-white |
| Halogen lamp | ≈ 3,200 K | 906 nm (infrared) | Neutral warm white |
| Sun's surface | 5,772 K | 502 nm | White, from space |
| D65 "daylight" standard | 6,500 K | 446 nm | Cool white; the white point of sRGB screens |
| Clear blue sky | 10,000 K and up | below 290 nm (ultraviolet) | Blue; a scattering color, matched to a temperature only by convention |