A lens bends light by refracting it at each of its two surfaces

1. A lens is a shaped piece of glass or clear plastic with at least one curved surface.
2. Light is refracted as it crosses the boundary from air into the glass and refracted again as it leaves, so a lens steers light by refraction and not by reflection.
3. In a thin lens the glass is so much thinner than the distances the rays travel that you can treat all of the bending as happening at one line drawn straight down the middle of the lens.
4. The middle of that line is the centre of the lens, and every distance you quote for a lens is measured from it.
5. Rays described as parallel to the principal axis run alongside that line and stay parallel to one another until they reach the glass.
   1. A ray travelling exactly along the principal axis strikes both surfaces head on, so it passes straight through without changing direction.

A converging lens brings a parallel beam together at a single point

1. A converging lens is fatter in the middle than at its rim, which is why it is also called a convex lens.
2. Every ray of a beam that is parallel to the principal axis is refracted towards the axis as it passes through the lens.
   1. A ray entering near the rim meets a steeply sloping surface and is bent through a large angle, a ray entering close to the axis meets an almost flat surface and is barely bent at all, and the amounts are graded so that all of the rays cross the axis in the same place.
3. That place is the principal focus, and real light energy arrives there, so the spot is genuinely bright and can be caught on a screen.
4. Hold a hand lens in bright sunlight and you see this as a tiny dazzling dot on a sheet of paper, hot enough to scorch it.

A diverging lens spreads a parallel beam out, so nothing meets on the far side

1. A diverging lens is thinner in the middle than at its rim and is also called a concave lens.
2. Each ray of a beam parallel to the principal axis is refracted away from the axis, so the beam leaves the lens wider than it arrived.
3. The outgoing rays are diverging, so they never cross, and there is nowhere behind the lens where a screen picks up a bright dot.
4. Extend the outgoing rays backwards as straight dashed lines and they all meet at one point, on the same side of the lens as the incoming beam.
5. That point is the principal focus of a diverging lens, and it is called virtual because no light actually passes through it and the rays only appear to diverge from it.
6. Fit a diverging lens in front of a torch and its beam is spread over a wider area instead of being concentrated.

The principal focus belongs to the lens and always sits on the principal axis

1. The principal focus is also called the focal point, and on diagrams it is labelled $F$.
2. For a converging lens, $F$ lies on the far side of the lens from the incoming parallel beam, and the light really does travel through it.
3. For a diverging lens, $F$ lies on the same side as the incoming beam, because only the backward extensions of the rays reach it.
4. The position of $F$ is fixed by the shape and the material of the lens, so it does not shift when you move a lamp or an object about in front of the lens.
   1. Nothing has to be placed at the principal focus for it to exist, since it is defined purely by what the lens does to a parallel beam.

A fatter lens bends light more, so its focal length is shorter

1. Focal length is given the symbol $f$, and because it is a distance you quote it in units such as $cm$ or $m$.
2. Start the measurement at the centre of the lens, not at the front or back face of the glass, and finish it at the principal focus.
3. A strongly curved, fat lens refracts each ray through a large angle, so the rays cross close to the lens and $f$ is short.
4. A gently curved, nearly flat lens changes each ray's direction only slightly, so the rays run a long way before they cross and $f$ is long.
   1. A chunky hand lens may have a focal length of about $5cm$, while a nearly flat lens of the same width may have one of $50cm$ or more.
5. A diverging lens has a focal length in exactly the same way, measured from the centre of the lens to its virtual principal focus, and one with more steeply curved surfaces again has a shorter $f$.

Light from a distant object arrives almost parallel and focuses on the focal plane

1. Light spreads out from every point of an object, but by the time it has crossed several metres to reach a small lens the rays that get there are very nearly parallel.
2. A converging lens therefore brings light from a distant window or streetlamp to a sharp focus almost exactly one focal length behind it.
3. Light from a distant object off to one side arrives as a parallel beam that is tilted to the principal axis, and it focuses the same distance from the lens but above or below the axis.
4. All of those focus points lie on one flat surface through the principal focus, at right angles to the principal axis, called the focal plane.
   1. A camera aimed at a distant view has its sensor sitting on the focal plane.
5. This is how a focal length is usually found in practice: hold the lens so that it throws a sharp patch of light from a far-off window onto a wall, then measure from the centre of the lens to the wall with a metre rule.
6. The object has to be genuinely far away, because light from something close by is still spreading out noticeably when it reaches the lens and comes to a focus beyond the focal plane.