Lenses and Image Formation

A lens is a piece of glass with curved faces, and everything it does follows from the refraction you met in Lesson 3 happening twice — once at each surface. This is the longest lesson in the module because it is the one whose skill genuinely transfers: once you can construct an image with two straight lines, every question about what kind of image a lens produces answers itself, and you stop having to remember a table of cases. By the end you should be able to draw that construction, read the image's position, size and orientation off it, put numbers to it when a question gives you numbers, and say what changes when the lens is a diverging one instead.

A lens is two refracting surfaces

Take a ray heading towards a lens parallel to its axis and follow it through. At the first surface it crosses from air into glass, and because the surface is curved, the normal there is tilted rather than square to the ray — so the ray bends towards that tilted normal. At the second surface it crosses back out into air and bends away from the second normal. Both bends are in the same rotational sense, because the two faces curve oppositely, so the deviations add rather than cancel. That is the difference between a lens and the parallel-sided block from Lesson 3, where the two bends cancelled and the ray came out parallel to how it went in.

What makes a lens more than a beam-deflector is the shape of the curve. On a convex surface, the further from the axis a ray strikes, the more steeply the surface is tilted, so the more strongly that ray is bent. Rays near the axis are barely deviated and rays near the rim are turned sharply, and a spherical surface gets the graduation close enough to right that all of them are steered to very nearly the same point. So the lens is not doing anything a flat boundary could not do at any individual point; the cleverness is entirely in arranging the tilt so that every ray's bend is exactly the one that brings it to the common meeting place.

Two shapes matter. A converging lens — also called convex, and thicker in the middle than at the edges — brings parallel rays together. A diverging lens — concave, thinner in the middle than at the edges — spreads them apart. The thick-middle and thin-middle test is the reliable way to tell them apart, and it works for the lens in a magnifying glass as well as for the ones in spectacles.

The vocabulary you need before you can draw anything

Four terms, and each is defined by where it sits on the diagram rather than by anything abstract. The principal axis is the straight line through the middle of the lens, perpendicular to it — the line of symmetry, and the reference line every measurement is taken along. The optical centre is the point where that axis passes through the middle of the lens. The principal focus, written F, is the point on the axis where rays arriving parallel to the axis are brought together after passing through the lens. And the focal length, written f, is the distance from the optical centre to F, measured in metres or centimetres.

Three facts about F save trouble later. There is one on each side of the lens, because light can arrive from either direction, and both are the same distance out — so a diagram shows two F marks, symmetrically placed. The point at twice that distance, marked 2F, is not a focus of anything; it is drawn purely because it turns out to be the boundary between two kinds of image, so it earns its place on the diagram as a landmark. And a fatter, more strongly curved lens bends rays harder, so it has a shorter focal length: short focal length means powerful lens, which is the opposite way round from how it sounds.

The two rays that do all the work

To find the image of an object you do not need to trace the thousands of rays that actually leave it. You need two, chosen because their paths are known without any calculation, and you take them from the same point on the object — conventionally the tip of an arrow standing on the axis. The first is the ray that leaves the tip travelling parallel to the axis: by the definition of the principal focus, it must pass through F on the far side. The second is the ray that leaves the tip heading for the optical centre: at the middle of the lens the two faces are parallel to each other, so it behaves like the parallel-sided block and carries straight on undeviated.

Draw both and see where they cross. Every other ray from that same point on the object arrives at the same place — that is what a lens does — so the crossing point is the image of the tip. Since the object sits on the axis and the axis is a line of symmetry, the foot of the image sits on the axis too, and you can draw the whole image as an arrow from the axis up or down to the crossing point. A third ray is available if you want a check: one that passes through F on the near side leaves parallel to the axis, which is the first rule run backwards. Two are enough to fix a point, so two is what the figures use.

The word real in that caption is doing precise work, and it is the contrast with the virtual images of Lesson 2. Here the rays genuinely converge and cross at the image position, so light is actually arriving there: put a screen at that point and a focused picture appears on it, which is exactly what happens in a camera and at the back of your eye. A virtual image, by contrast, is where diverging rays only appear to have come from, and no screen will catch it. Real images from a single lens always come out inverted; virtual ones always come out upright. That pairing is worth holding, because it lets you check any answer for internal consistency.

What happens as you move the object

Redraw the same two rays with the object at different distances and the character of the image changes in a fixed sequence. With the object a long way off — far beyond 2F — the image is real, inverted, very small, and sits just outside F. Bring it in to somewhere between 2F and infinity and the image is still real, inverted and smaller, now between F and 2F: that is the case in the figure above, and the one a camera works in. At exactly 2F the two distances match and the image is real, inverted and the same size as the object, at 2F on the other side. Between F and 2F the image is real, inverted and larger than the object, beyond 2F on the other side — the arrangement a projector uses.

At F itself the two construction rays come out parallel to each other and never cross, so there is no image at all: the lens sends out a parallel beam, which is how a spotlight or a lighthouse is built. And inside F, closer to the lens than the focus, the emerging rays diverge, so again they never cross in front — but traced backwards they meet on the object's own side, giving a virtual, upright, magnified image. That is a magnifying glass, and it explains the one thing people always get wrong about them: a magnifying glass only magnifies while it is held closer to the object than its focal length. Move it further away and the image flips over and shrinks.

Putting numbers to it

Items in these batteries ask what kind of image you get rather than exactly where it lands, so the constructions above are the core skill. But one relationship makes every one of those qualitative answers checkable instead of memorised, and it is worth having for that reason alone. It ties the focal length of the lens to the object distance and the image distance, and it is usually written with all three as reciprocals — one over each — because that is the form in which it comes out simplest.

The signs in that legend are a convention, not a discovery, and it is the convention almost every textbook uses: distances to things the light really goes to count as positive, and a negative answer for v is the equation's way of telling you the image came out virtual and on the same side as the object. To use the equation for v you rearrange it by subtracting one over u from both sides, giving one over v equal to one over f minus one over u — and then, having found one over v, you must remember to turn it upside down to get v itself. That last step is the one people drop.

The other quantity worth a number is magnification, which is simply how many times bigger the image is than the object. It equals the ratio of the two heights, and it also equals the ratio of the two distances, which is the useful form because distances are what the equation above gives you.

Work with the sizes of the numbers and ignore the signs when you are asked only whether the image is bigger or smaller: if the image distance exceeds the object distance the image is enlarged, and if it is shorter the image is reduced. That single comparison reproduces the whole table above without your having to recall it.

Compare those two examples side by side, because between them they contain the misconception this lesson exists to break. The same lens, unchanged, gave a same-size inverted real image in one case and a doubled upright virtual image in the other. A converging lens does not magnify; a converging lens with the object inside its focal length magnifies. Everything depends on where the object is, which is why the construction is worth more than any remembered list.

The diverging lens

A diverging lens is thinner in the middle, and its two surfaces bend rays outwards rather than inwards. Parallel rays arriving along the axis therefore leave spreading apart, and if you trace those spreading rays backwards they all appear to come from a single point on the incoming side of the lens. That point is this lens's principal focus, and because the light never actually goes there, its focal length is counted as negative — which is exactly what the sign convention in the legend above was set up to handle.

The construction rules are unchanged, which is the point worth taking from the next figure. The ray arriving parallel to the axis leaves along the line that traces back through F, and the ray heading for the optical centre goes straight on. The only difference is that the emerging rays diverge, so they never cross in front of the lens and you have to trace them back to find where they appear to meet.

That last clause is the whole of what you need to remember about diverging lenses, and it is a relief after the five cases of the converging one: the image is always virtual, always upright, always smaller than the object, and always between the lens and F, no matter where the object sits. Move the object closer and the image grows a little towards full size but never reaches it; move it further away and the image shrinks towards F. There is no arrangement that makes a diverging lens produce a real image or an enlarged one on its own.

Lens power, in dioptres

Opticians do not talk about focal lengths; they talk about power, which is just one divided by the focal length in metres. The reason for preferring it is that powers add: put two lenses together and the combination's power is the sum of the two, which is a great deal easier than combining focal lengths. The unit of power is the dioptre, symbol D, and a dioptre is one per metre.

Two things fall out of the definition. Because the focal length is on the bottom, a short focal length gives a big power — a strong, fat lens — and a long focal length gives a small one. And because f carries a sign, so does P: a converging lens has positive power and a diverging lens negative power, which is why a prescription written as −2.5 D tells you immediately that the lens is a diverging one without any further information.

Why it matters in aviation

A head-up display is a lens problem. Its symbols are generated on a small source inside the unit and projected through optics that leave them focused at infinity, so the light reaching the eye arrives as a nearly parallel bundle, in the same state as light from the runway beyond the glass. The eye can therefore hold one focus for both, and the symbols can be read without the pause and refocusing that would be needed if they sat at arm's length like a panel instrument. The construction in this lesson is what arranges that: an object at the focal point of a converging lens forms no image at any finite distance and the light leaves as a parallel beam.

Lenses do more ordinary work elsewhere on the aircraft. Enhanced vision cameras form a real image on a sensor exactly as the figures here form one on a screen. Landing and taxi lights use a reflector and a lens together for the opposite purpose: rather than forming an image, they take the output of a small bright source and concentrate it into the narrow set of directions where it is wanted, which is why such a light is intense along the aircraft's path and dim to the side.

What the test asks

The headline distractor is that a converging lens always magnifies, usually dressed up as a question about a magnifying glass or a projector. It magnifies only with the object inside the focal length, and beyond 2F it does the opposite. Expect a second item pairing image type with orientation incorrectly — a real upright image, or a virtual inverted one — neither of which a single lens produces. A third relies on forgetting that a lens works by refraction at both surfaces and offers reflection as the mechanism. And in the numeric items, the two recurring slips are finding one over v and forgetting to invert it, and mixing centimetres with metres in the power formula, which throws the answer out by a factor of a hundred.