You are surrounded by lenses. There is one at the front of every camera and phone, one in a pair of reading glasses, one in a magnifying glass, one in a projector — and there is a living one inside each of your own eyes, right now, focusing these words onto the back of your eyeball. A lens is just a carefully curved piece of glass or plastic, and it does one job: it refracts light — bends every ray by exactly the right amount — to gather the light from a scene and rebuild it as an image.

You already know that light bends when it changes speed crossing into glass, and that you see a thing when light from it reaches your eye. A lens takes that single idea — refraction — and puts it to work on a grand scale. This page is about how it does it, and how to predict exactly where the picture will appear, how big it will be, and whether it will be the right way up.

Two kinds of lens

Lenses come in two families, and everything follows from their shape.

• A converging (or convex) lens is fat in the middle and thin at the edges. When parallel rays of light pass through it, they are all bent inwards and cross at a single point. It brings light together. This is the lens in a camera, a magnifying glass, and your eye.
• A diverging (or concave) lens is thin in the middle and fat at the edges. Parallel rays are bent outwards and spread apart, as if they had fanned out from a point behind the lens. It spreads light out. This is the lens used to correct short sight.

Almost all the useful image-making — cameras, projectors, magnifiers, the eye — is done by the converging lens, so that is the one we will study in detail.

The principal focus and the focal length

Point a converging lens at something very far away — the Sun, a distant window — so that the light arriving is a bundle of parallel rays, all travelling the same way. The lens bends every one of them inwards, and they all cross at a single sharp point on the far side. That point is the principal focus (or focal point), F.

The distance from the middle of the lens out to that point is the focal length, f. It is the single most important number describing a lens: a fatter, more strongly curved lens bends light harder, so it has a short focal length; a gently curved lens bends light only a little, giving a long focal length. Every lens actually has two focal points, one on each side, the same distance out — because light can pass through either way.

Hold this picture in your head, because it gives us the two special rays we need to find any image: a ray that comes in parallel to the axis leaves through F, and — run backwards — a ray that comes in through F leaves parallel.

The three special rays

To find the image of an object, we do not need to trace the millions of rays leaving it. We need just two — and any two will do — because every ray from one point on the object ends up at the matching point on the image. We use rays whose paths we can draw without calculation. From the top of the object we send out these three standard rays:

• Ray 1 — parallel, then through the focus. A ray travelling parallel to the principal axis. After the lens it bends to pass straight through the far principal focus F.
• Ray 2 — straight through the centre. A ray aimed at the very centre of the lens carries on undeviated — dead straight, no bending — because the two glass faces are parallel there, like a thin flat window.
• Ray 3 — through the focus, then parallel. A ray that passes through the near focus F on its way in leaves the lens travelling parallel to the axis. (This is Ray 1 run backwards.)

Wherever these rays cross on the far side is where the top of the image sits. Draw the image as an arrow standing on the axis directly below (or above) that crossing point, and you are done. Below you can build exactly this diagram and watch it change.

Build the ray diagram yourself

Here is a converging lens (the double-headed arrow), its two focal points F and the points 2F twice as far out, and a glowing object arrow on the left. Drag the object distance slider to slide the object in and out, and watch the three rays redraw and the image form where they cross. Read the banner at the top: it tells you whether the image is real or virtual, which way up it is, and how big.

Start far out (object beyond 2F) and the image is small, inverted and real. Slide the object in past 2F and the image swells; at exactly 2F it matches the object. Push the object inside the focal point and something dramatic happens: the rays on the far side stop meeting, the real image vanishes, and instead a large upright image appears back on the object's own side — traced by the dashed lines. That is the magnifying glass.

How the image changes as the object moves

The behaviour you just watched is worth learning as four clear cases. For a converging lens of focal length f:

• Object far beyond 2F: the image is real, inverted and smaller than the object, formed just outside F on the far side. This is a camera photographing a landscape — and it is what your eye does with the whole room.
• Object at 2F: the image is real, inverted and exactly the same size, formed at 2F on the far side.
• Object between F and 2F: the image is real, inverted and larger, formed beyond 2F. This is a projector throwing a small slide onto a big screen (which is why the slide goes in upside-down).
• Object inside F: now the refracted rays diverge and never meet. The image is virtual, upright and larger, on the same side as the object. This is a magnifying glass.

Notice the pattern: as the object comes towards the lens, the image moves away and grows — right up until the object crosses the focal point, where the real image shoots off to infinity and then reappears as a magnified virtual one.

Real images and virtual images

The single deepest idea on this page is the difference between the two kinds of image.

A real image is formed where light rays actually cross and meet. Because real light energy is genuinely arriving at that place, you can put a screen there and catch the picture on it — the film in a camera, the retina in your eye, the wall a projector shines on. A real image from a single converging lens is always inverted (upside-down).

A virtual image is formed where the rays only appear to come from — they never truly meet there, they just seem to, when your eye traces the diverging rays backwards along straight lines. No light energy actually gathers at that spot, so you cannot catch a virtual image on a screen. The magnified picture you see in a magnifying glass, and your own reflection in a mirror, are both virtual — and both the right way up.

Magnification: how much bigger?

"The image is bigger" is vague. We make it a number — the magnification, m — by comparing the height of the image with the height of the object.

Worked example 1 — a projector. A slide is 3\ \text{cm} tall and its image on the screen is 150\ \text{cm} tall. The magnification is

The picture is 50 times taller than the slide (and, being a real image, upside-down — which is exactly why slides are loaded into a projector inverted).

Worked example 2 — a magnifying glass. A ladybird 8\ \text{mm} long is viewed through a magnifier with magnification m = 4. Rearranging the formula, the image height is

Worked example 3 — from distances. An object sits 10\ \text{cm} from a lens and its real image forms 25\ \text{cm} away on the other side. Then

so a 2\ \text{cm} object would give a 2 \times 2.5 = 5\ \text{cm} image.

The eye and the camera: the same trick

A camera and an eye are the same machine. Each has a converging lens at the front and a light-sensitive screen at the back — the digital sensor in a camera, the retina in an eye. Both point at a distant scene, so the object is far beyond 2F, and both therefore form a small, inverted, real image on that screen. A camera focuses by moving its lens forwards and backwards to place the sharp image exactly on the sensor; your eye does something even cleverer — tiny muscles squeeze the soft lens fatter or thinner, changing its focal length so the image always lands on your retina, whether you are reading a book or gazing at a hill.