A lens bends light as it passes through glass; a mirror turns it back. Different physics, same geometry: both bring parallel rays to a focus, a distance f away, and one equation covers both.
To find an image, follow just three rays from the top of the object: one parallel to the axis, one through the centre, one through the focal point. Where they meet is the top of the image.
A real image is where light really arrives, so a screen catches it, and it is upside down. A virtual image is only where the light seems to come from: no screen, but your eye sees it, upright.
Put an object nearer a converging lens or a concave mirror than its focal length and the image is virtual, upright and bigger. That is a magnifying glass, and a shaving mirror.
The object distance u, the image distance v and the focal length f are tied together for every thin lens and every mirror. Put the object at 2f and the image is at 2f; at infinity and the image is at f; at f and the image has gone to infinity. A negative v means a virtual image on the object’s side of a lens, behind a mirror; a negative f means the element diverges.
A camera lens focuses a distant scene a focal length behind it. For something nearer, the image forms further back, v = uf/(u − f), so the lens has to move away from the sensor. For a 50 mm lens, an object a metre away needs the lens 2.6 mm further out than at infinity, and one at half a metre needs 5.6 mm. That small travel is the whole of focusing; on a phone it is done by a coil moving the lens a fraction of a millimetre.
A lens with a focal length of 10 cm is a magnifying glass when you hold it less than 10 cm from the page: the image is virtual, upright and bigger. It magnifies the angle your eye sees by about 25/f, where 25 cm is how close an eye can focus comfortably: 2.5× for 10 cm, or 3.5× if you bring the image right in to your near point. A shorter focal length, a more strongly curved lens, magnifies more, and is held closer.
A convex mirror always makes a virtual, upright, smaller image, so it shows a wider scene in the same small mirror. With a focal length of −60 cm, a car 100 cm behind appears 37.5 cm behind the mirror, 0.38 of its size. It also makes things look further away than they are, which is why they carry a warning. A UN regulation sets a minimum radius of curvature of 1.2 m for these mirrors, a focal length of 60 cm, so that the view does not shrink too far.
A spherical mirror brings rays near the axis to a focus at R/2, but rays far from the axis cross early. For a mirror of radius 12 cm and a ray 6 cm from the axis, that is 0.93 cm early, a blur many times bigger than a good image. A parabola has no such error for rays parallel to its axis, which is why telescopes and car headlights use parabolic mirrors. Switch on “Real curved surface” in the wide spherical mirror example to see it.
A flat mirror makes a virtual image as far behind the mirror as you are in front, the same size, upright. Look at it: your right hand is still on the image’s right as you see it. What has been reversed is front to back, so that the image faces you. It looks left-right reversed because we imagine turning around to face the image by swinging about a vertical axis; turn by flipping over instead and you would call it up-down.
The power of a lens is 1/f with f in metres, in dioptres (D). Spectacles of −2 D have f = −50 cm; a reading lens of +2.5 D has f = 40 cm. Two thin lenses together have the sum of their powers, which is why an optician can add a lens to a trial frame and read off the total. A short-sighted eye cannot focus on things further than its far point; a lens whose focal length equals that distance, diverging, makes distant objects look as though they were there.
The Sun is half a degree across, so a lens or mirror of focal length f makes an image of it 0.9 mm wide for each 10 cm of f: about f/107. All the sunlight falling on the lens is gathered into that spot, which is how a magnifying glass lights paper: the bigger the lens relative to the spot, the hotter it gets.
These are thin lenses and mirrors, with rays close to the axis. A real lens has thickness, so its focal length is measured from principal planes rather than the surface; it bends different colours by different amounts (chromatic aberration); and it blurs rays far from the axis. A good camera lens has a dozen elements to correct these. The picture is stretched differently across and along the axis so that it fits, which keeps straight lines straight and image positions right but makes angles look wrong. Objects are real and on the axis side of the element; virtual objects are not shown.
One where the light really arrives. The rays from each point of the object actually cross at a point of the image, so a screen placed there shows it, and so does a camera sensor. A real image from a single lens or mirror is always upside down.
One where the light only seems to come from. The rays leaving the lens or mirror spread out, and your eye traces them back to a point they never reach. You can see it, but no screen can catch it. A flat mirror, a magnifying glass and a shaving mirror all give virtual images, which are always upright.
Because the rays cross. A ray from the top of the object, after being bent towards the axis by a converging lens, ends up below the axis on the far side of the focus, so the top of the object makes the bottom of the image. Inside the focal length the rays have not yet crossed, and the image is the right way up.
The distance at which a lens or mirror brings parallel rays, light from a very distant object, to a point. For a mirror it is half the radius of curvature. For a lens it depends on the curvature of its surfaces and on the glass: the lens maker’s equation. A stronger curve gives a shorter focal length and a more powerful lens.
It depends on whether it is a lens or a mirror, which is why people mix them up. A concave mirror curves inwards and converges light, like the bowl of a spoon. A concave lens is thinner in the middle and diverges it. A convex lens converges and a convex mirror diverges.
To make a virtual, upright image the object must be inside the focal length. Between one and two focal lengths you get a real, upside-down, enlarged image instead, which is how a projector works. Beyond the focal length a magnifying glass flips the world upside down, as you will have noticed holding one at arm’s length.
A convex mirror shows a much wider field of view than a flat one the same size, because it always makes a smaller, upright, virtual image. The price is that objects look smaller and so further away than they are, which is the reason for the warning printed on them.
The unit of lens power: one over the focal length in metres. A lens of 1 dioptre brings parallel light to a focus a metre away; +4 dioptres, 25 cm. A diverging lens has a negative power. Powers of thin lenses placed together add, which is why spectacle prescriptions are given this way.
A mirror reflects every colour the same way, so it has no colour fringes; it can be supported from behind, while a lens can only be held by its edge and sags; and light passes through only the surface, not the thickness of glass. Its surface can be a parabola, which focuses all the parallel rays to a single point.
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