How Far Away Do a Car's Two Headlights Merge into One?
― The Limit of Your Eye's Sharpness Is Set by the Small Size of the Pupil
When a car comes towards you along a long, straight road, its two headlights first look like a single light. For a person with 20/20 vision (a decimal acuity of 1.0), they begin to split into two at about 5 km away. What sets this distance is not how good your eyes are, but first of all the fact that "light is a wave" and that "the opening of the pupil is small".
It is dusk, on a straight road by the sea. Far ahead, you can see a single faint light.
After a while, the light stretches upwards and then splits into two dots, left and right. It was a car's headlights all along.
The two lights are only about 1.5 m apart. Yet from several kilometres away, they melt into one. It isn't that your eye is out of focus.
There are only two reasons they look like one
Light is a wave. A wave spreads out a little after passing through a narrow opening. The pupil is a small hole about 3 mm across, so every point of light becomes a small blurred circle on the retina.
The closer together the two lights seem, the more their blurs overlap. When the dip between them becomes shallow, the brain can only take them as one light. This border is the "limit of sharpness" of the eye.
So when distant headlights look like one, it is less a failing of the eye than a property of light itself, a wave. Even a perfect lens cannot beat the limit set by the size of its opening.
What happens to the blur when the pupil changes size?
Look at Figure 1. On the left is the entrance of the eye (the pupil). On the right is the brightness of the two headlights on the retina. Widen the pupil with the slider, and the blurred peaks on the right get narrower, while the dip in the middle gets deeper, so they split in two. Narrow the pupil, and the two peaks melt into a single hill.
The angle of the spread is set by the wavelength of light divided by the diameter of the pupil. Green light has a wavelength of about 0.00055 mm, and the pupil is about 3 mm. The limiting angle given by this ratio comes to about 0.77 arcminutes (one arcminute is one sixtieth of a degree). By this calculation, two lights 1.5 m apart can in principle be told apart up to about 6.7 km.
So would a wider pupil at night let us see more?
On paper, if the pupil opened to 7 mm, the limit would become more than twice as fine. But a real eye doesn't work that way. When the pupil widens, light passing through the edge of the lens is used too. The edge of the eye's lens is the most distorted part, and that blur ends up larger than the wave's own spreading.
The human eye is said to see most finely when the pupil is about 2 to 3 mm. At just this size, the blur from the wave and the spacing of the light-catching cells at the centre of the retina are thought to be roughly in balance. The opening, the lens and the cell layout: improving any one of them alone gains nothing. It is a design with very little waste.
In practice, 20/20 vision (decimal 1.0) is defined to mean "you can tell things apart down to 1 arcminute". That is a little coarser than the limit of 0.77 arcminutes, but of the same order. Using this 1 arcminute, the distance at which headlights split into two comes to about 5 km.
The Moon is about 380,000 km away. At that distance, 1 arcminute spans a width of about 110 km. So the smallest markings you can pick out on the Moon with the naked eye are about the size of the distance from Tokyo to Utsunomiya (about 100 km). This is why even big craters look like mere dots.
The second star from the end of the Big Dipper's handle has a small star right beside it. The two are about 12 arcminutes apart. That is far wider than the 1 arcminute limit, but one of them is faint and easy to miss, so it is said to have been handed down since ancient times as a star for testing eyesight.
Summary
Two distant lights look like one because light is a wave and blurs when it passes through a small hole, the pupil. The angle of the blur is set by "wavelength ÷ pupil diameter", and for the human eye it is about 1 arcminute. For headlights 1.5 m apart, about 5 km is the border of being told apart.
The limit of the eye's sharpness is set by
the wave of light and the size of a 3 mm hole.
The story of how we came to know light is a wave is in "How did we find out that light is a "wave"?". The same blur that makes rings in the sky is in "Why do small rainbow-coloured rings appear around the Moon?". And the limit on how much light the eye can pick up is in "How many particles of light does it take for the human eye to "see" something?".
- On a sheet of white paper, draw two parallel lines with a black pen, 1 mm apart.
- Stick the paper on a wall and step slowly backwards. Measure, in paces, the distance at which the two lines first look like a single grey line.
- For a person with 20/20 vision, the calculation says they merge at about 3.4 m (1 mm ÷ 1 arcminute). Compare a bright room with a dim one.
If you wear glasses or contact lenses, measure as you normally are. If you poke a tiny hole in a piece of paper (with a pin) and look through it, it is the same as making your pupil smaller. You can also check that the blur grows and the distance at which the lines merge gets shorter.
For those who want to know more ― Terms, formulas and links to textbooksWe state clearly which level each topic belongs to, from junior high science to university specialist courses
- Junior highCovered in junior high school science
- High schoolCovered in high school "physics"
- High school+Advanced high school material, or treated as a textbook sidebar
- UniversityNot taught in high school; university specialist courses (optics, vision science)
- ResearchNot taught even at university as settled fact; being studied by researchers right now
Junior highTerms: this phenomenon has a name
- Diffraction: the spreading of a wave as it bends around a narrow gap or the edge of an object.
- Resolving power: how finely the eye can tell two nearby points apart as two. It is often expressed as an angle.
- Airy disk: the bright circle, with faint rings around it, that forms when light from a point spreads out through diffraction after passing through a round hole.
Junior highHigh schoolChecking with formulas: the distance at which headlights look like two
We find the angle of blur of the light passing through the pupil, then use that angle to work out the distance at which headlights 1.5 m apart can be told apart.
| In symbols | θ ＝ 1.22 × λ ÷ D 、 L ＝ s ÷ θ |
| In words | Limiting angle for telling apart ＝ 1.22 × wavelength of light ÷ pupil diameter. Distance at which they can be told apart ＝ spacing between the two lights ÷ limiting angle |
| Where the formula comes from | Light diffracted by a round hole forms a first dark ring around a bright central circle. The angle of that ring is 1.22 × λ ÷ D. The rule that "the limit of resolution is reached when the centre of one light falls on the dark ring of the other" is called the Rayleigh criterion. The second formula is a rearrangement of "arc length ＝ distance × angle", which holds when the angle is small. |
The units of the symbols are as follows.
| θ | Limiting angle for telling apart (radians) |
| λ | Wavelength of light (mm). About 0.00055 mm for green light |
| D | Pupil diameter (mm) |
| s | Spacing between the two lights (m) |
| L | Distance at which they start to look like two (m) |
| Wavelength of green light | about 0.00055 mm (the colour the eye is most sensitive to) |
| Pupil diameter in a bright place | about 3 mm |
| Spacing of the two headlights | about 1.5 m |
| 1 arcminute in radians | about 0.000291 |
| 1.22 × wavelength | 1.22 × 0.00055 ＝ 0.000671 mm |
| Divide by pupil diameter (limiting angle) | 0.000671 ÷ 3 ≒ 0.000224 radians |
| Convert to arcminutes | 0.000224 ÷ 0.000291 ≒ 0.77 arcminutes |
| Distance at which they look like two, in principle | 1.5 ÷ 0.000224 ≒ 6700 m |
| Distance for 20/20 vision (1 arcminute) | 1.5 ÷ 0.000291 ≒ 5150 m |
| Width of 1 arcminute on the Moon | 384000 × 0.000291 ≒ 112 km |
The limit set by diffraction alone is about 6.7 km, and for real 20/20 vision it is about 5 km. That the two agree in order of magnitude shows that the human eye gets close to the diffraction limit.
High schoolHigh school+Why bigger telescopes see better
High schoolIn high school physics you learn about diffraction and interference of light by slits and gratings. The wider the slit compared with the wavelength, the smaller the angle of diffraction spread. The pupil is a kind of slit, and the larger its diameter, the finer you can see, for the same reason.
High school+A bigger telescope aperture is not only for collecting more light. For a space telescope 2.4 m across, the same formula gives a limiting angle about 800 times smaller than that of the human pupil. For large telescopes on the ground, the shimmer of the air blurs the image more than diffraction does, so tricks are used to cancel it out.
UniversityFraunhofer diffraction and sampling by the retina
In Fraunhofer diffraction by a circular aperture, the brightness of the image is given by a formula using the Bessel function, and the bright central part is called the Airy disk. The coefficient 1.22 comes from the position where the first-order Bessel function first becomes zero. The retina, on the other hand, is a "sampling" device made of an array of light-detecting cells, and the upper limit of its sharpness is also set by the spacing of those cells (the sampling theorem). In the human eye, the diffraction limit and the cell spacing are said to be roughly in balance when the pupil is 2 to 3 mm.
📖 Derivations of the formula, and further reading: Airy disk (Wikipedia) ／ Fraunhofer diffraction (Wikipedia)
ResearchWhat is still not clearly understood
- How finely do the eyes of birds of prey see? The eyesight of eagles and hawks is said to be around twice that of humans, but the value changes greatly with how it is measured, and how finely they actually see in the wild is still debated.
- How much the brain "corrects". It is known that a slight offset between two lines can be detected even when it is far smaller than the limiting angle (vernier acuity). How the brain adds up the information is still being studied.
- Why short-sightedness is increasing. Separately from the limit of the eye's sharpness, short-sightedness in children is rising worldwide. Its link to time spent outdoors is being investigated, but the mechanism is not yet settled.
In other words, what this article says is also "an explanation of what is known so far". The formula for the limiting angle itself is solid, but how closely a real eye can approach it varies with the person and the conditions.
Links to textbooks (by level)
| Level | Subject / unit | Where in this article |
|---|---|---|
| Junior high | Science, Year 1: "Light and sound", structure of the eye | Roles of the pupil, lens and retina |
| High school | Physics: "Diffraction and interference of waves" | Blur of light passing through a small hole |
| High school+ | Advanced physics: "Resolving power of optical instruments" | Telescope aperture and how finely you can see |
| University | Optics, vision science | Airy disk, Rayleigh criterion, sampling by the retina |
| Research | Visual physiology, ophthalmology | Eyesight of birds of prey, vernier acuity, rise in short-sightedness |
| ― | Connections to daily life | How distant lights look, eye tests, stargazing |
- Wikipedia (Japanese), "Airy disk" (エアリーディスク)
- Wikipedia (Japanese), "Visual acuity" (視力)
- Wikipedia (Japanese), "Diffraction" (回折)
- E. Hecht, Optics (a textbook of optics; chapter on diffraction by a circular aperture and resolution)
- F. W. Campbell, D. G. Green (1965), a study of pupil size and the resolution of the eye, Journal of Physiology
※This article is a science explainer for general readers. The figures given are rough guides to help you understand how things work. How well people see varies from person to person. If you are worried about your eyesight, please have it checked by an eye doctor.