How Lenses Bend and Focus Light

Light changes speed when it passes from one transparent material into another. If it enters at an angle, the change in speed also changes its direction. This bending is called refraction.

A lens has curved surfaces, so rays strike different parts at different angles. The two surfaces work together to redirect the rays in an organised pattern. The exact path depends on the lens shape, the materials involved and the colour of the light.

A ray travelling through the optical centre of a thin lens changes direction very little in a simple diagram. Rays parallel to the main axis behave more dramatically. Their paths reveal whether the lens is converging or diverging. The distance from the centre of the lens to the focal point is called the focal length. A strongly curved lens usually bends light more and has a shorter focal length than a gently curved lens.

A convex lens is thicker in the middle than at the edges. It bends parallel rays inward so they meet at a focal point. If a distant object sends nearly parallel rays into the lens, a real image can form on the other side. A screen or camera sensor can capture that image.

A concave lens is thinner in the middle. It spreads parallel rays outward. The rays appear to come from a focal point on the same side as the object, producing a virtual image that cannot be projected onto a screen by that lens alone.

The object distance matters. With a convex lens, an object beyond the focal length can produce an inverted real image. An object placed closer than the focal length produces an upright, magnified virtual image. This is how a simple magnifying glass works. The same piece of glass can therefore behave differently because you moved the object. Cameras and eyes adjust image formation in different ways. A camera can move lens elements, while the human eye changes the shape of its flexible lens to focus objects at different distances.

You can predict an image by tracing a few principal rays. Real lenses can have imperfections, but the simple model explains the main behaviour. Where the rays meet tells you the image position. Its orientation and size can then be read from the diagram.

• •Draw the main axis and mark the focal points.
• •Send one ray parallel to the axis.
• •Refract that ray through or away from the focal point.
• •Draw a ray through the optical centre.
• •Find where the rays or their backward extensions meet.

Ray diagrams can be checked with a simple formula for thin lenses. In one common convention, 1/f = 1/u + 1/v, where f is the focal length, u is the object distance and v is the image distance, with real distances counted as positive. Suppose a convex lens has a focal length of 10 cm and an object sits 30 cm away. Then 1/v = 1/10 minus 1/30, which is 2/30, so v is 15 cm. A real image forms 15 cm behind the lens.

Magnification is the image distance divided by the object distance, so here it is 15 divided by 30, giving 0.5. The image is inverted and half the height of the object. Now move the object to 5 cm, inside the focal length. This time 1/v = 1/10 minus 1/5, which gives minus 1/10, so v is minus 10 cm. The negative sign marks a virtual image on the same side as the object, magnified two times and upright. Be careful: some textbooks use a different sign convention.

In a healthy eye, the cornea and lens bring light from distant objects to a focus on the retina. In short sight, the eyeball is usually slightly too long or the focusing too strong, so distant images form in front of the retina and look blurred. A concave lens spreads the rays a little before they enter the eye, pushing the focus back onto the retina. In long sight, the focus falls behind the retina, and a convex lens adds the extra converging power that is missing.

Opticians measure lens power in dioptres, which is one divided by the focal length in metres. A lens with a focal length of half a metre has a power of 2 dioptres. Converging lenses are written as positive and diverging lenses as negative, so a prescription of -2.00 means a diverging lens with a focal length of 0.5 metres. With age, the eye's lens becomes stiffer and less able to change shape for close work, a condition called presbyopia, which is why many people need reading glasses in middle age.

Glass does not bend every colour by the same amount. Violet and blue light slow down slightly more in glass than red light does, so they refract more strongly. A simple convex lens therefore brings blue light to a focus a little closer to the lens than red light. When you look through a cheap magnifier or an old telescope, bright edges can show coloured fringes because the colours do not all meet in the same place. This flaw is called chromatic aberration, and it troubled astronomers from the earliest days of the telescope.

Seventeenth century astronomers reduced the problem by using very gently curved lenses with long focal lengths, which led to absurdly long telescopes held up on masts and ropes. Isaac Newton concluded that the colour problem could not be cured with lenses and built a telescope that used a curved mirror instead, since a mirror reflects all colours in the same way. In the eighteenth century opticians found a better answer: pair a convex lens of crown glass with a weaker concave lens of flint glass. The two glasses spread colours by different amounts, so their errors largely cancel. Modern camera lenses build on the same idea.

Most of your eye's focusing is not done by the lens at all. The curved front surface, the cornea, provides roughly two thirds of the eye's total power of about 60 dioptres, because light bends most strongly where it passes from air into the eye. The lens inside adds the rest and fine-tunes it for distance. Refraction depends on the difference in speed between two materials, and the jump from air to the watery tissue of the cornea is large. That arrangement works beautifully on land, but it makes the eye depend on having air in front of it.

Open your eyes underwater and the cornea loses most of its power, since light travels at almost the same speed in water as in the cornea itself. The image falls far behind the retina, leaving you severely long-sighted, which is why everything looks blurred. A swimming mask or goggles fixes this simply by trapping a layer of air in front of your eyes, restoring the air-to-cornea boundary. Fish face the problem permanently. Their corneas do little focusing, so their lenses are nearly spherical and packed with protein so that they bend light strongly, doing almost all the work themselves.

Lenses form images through refraction at curved surfaces. Convex lenses bring rays together, while concave lenses spread them apart. Trace a few important rays and check the object distance, and you can predict where an image forms and what it will look like.