Depth of Field - in depth.

Don't know about you but this confuses the hell out of me - Bruce.

Out in the Fields

By David M.

If you care to examine them, older lenses for 35mm cameras have an extra row of tiny numbers around the focus ring. Older Hasselblad lenses have a pair of elegant red pointers that move in and out as the aperture ring is moved. Large format lenses have nothing. (This isn’t unusual. If you buy a simplified digital camera that will do everything for you, you’ll get a novel telling you how to do it. When you buy a complicated LF camera, you get a camera.)

What do these numbers and pointers do? And why are they on different rings? And what does it all mean? Depth of field is a widely misunderstood thing, barnacled with myths and rules of thumb. Let us try to project some light into the depths.

To begin, let us think of a lens as a hole in the front of the camera with a lump of glass in it. For the moment, we shall ignore the glass. Our lens’s job is to focus the light from one point in the scene onto another point onto the ground glass, the film or the sensor. (We shall henceforth use the word film.)

What could be simpler? One point out here – and another point in there.  For each point in the scene, we can focus a corresponding point onto the film. All we have to do is move the lens back and forth and any point in the scene can be brought to sharp focus on the film. But sadly, we can’t focus all the scene’s points, all at once. As this point here becomes sharp, that point there becomes blurred. We don’t want blurred photographs, do we? What shall we do?

An Imperfect World

Two things will help us, both of them exploiting the imperfections of the photographic process. In photography, imperfection can be better than perfection, although for the moment we are imagining perfect lenses.

Firstly, our sensor or film cannot resolve infinitely fine detail. Each sensor element has a finite size and any detail smaller than that cannot be resolved. Each grain of silver in film has a finite size too, although they are not arranged quite so neatly. (Many people believe this is an advantage.)

As a result, we may be able to focus very, very sharply, but infinite sharpness is always beyond us.We can see that even the sharpest image must be made up of tiny spots, not infinitely small points. Sharper images will have tinier spots or more spots, or both. All kinds of factors influence how sharp we want our photographs to be. A palm-sized selfie on the screen of an iPad-clone is less demanding than an atrium-sized mural of a forest.

Now, let’s imagine taking a real photograph with a camera and a lens. Light is reflected from the subject in all directions and some of it comes towards the camera. We can imagine a narrow cone of light coming from one point in the scene and through the hole in the camera (with its lump of glass). We can imagine another cone inside the camera, where the lens focuses the light onto the film. We’ll return to the inside of the camera later.

Chubby Cones and Clever Clogs

Almost all lenses have a way of adjusting the size of the hole, although because we are such clever-clogses, we say aperture rather than hole. If the hole is large, we shall have a chubby cone and if it’s small, a very thin one. Let’s consider what happens.

We shall need some numbers to do this. We’ll just make them up. So, here we are, contemplating a landscape, with (for instance) an interesting tree in the middle. Let’s imagine that the tree is twenty metres away. Now let’s imagine that the hole in our camera is four millimetres wide. (We are purposely ignoring the focal length of the lens.)

An excerpt from the British Journal of Photography almost 100 years ago and
proving just how dumbed-down readers have become in the interim - present
company excepted, of course. :)

We focus on the interesting tree. The cone of light is twenty metres long and four millimetres wide at its fattest point. Halfway along, at ten metres, it will be only two millimetres wide, and at fifteen metres it will be one millimetre wide. This is smaller than a blade of grass, so we shall be able to distinguish blades of grass fifteen metres away from the camera and five metres in front of the interesting tree.

We say that we can resolve objects a millimetre wide. If we reduce the size of the hole to two millimetres (half of four) this one-millimetre spot will reduce to half a millimetre, about the size of a grain of sand and the one-millimetre point will be ten metres away, so that blades of grass there will be resolved, too. (Don’t forget that we are using a single point as an example, and a complete image is made up of all the other points too.)

Still More

We might think that this is enough resolution for anyone, but can we push things further? Half a millimetre is smaller than the twigs on our tree, so they would still be resolved if we moved our point of focus beyond the tree to a more distant point. Let’s say forty metres, twice as far. Our cone is now a forty metres long but only two millimetres at its widest point. At twenty metres, where our tree is placed, it is one millimetre wide, still smaller than a blade of grass or an interesting twig. We can still be happy with that. 

Somehow, we seem to have conjured extra resolution out of the scene by exploiting the imperfections of the process. If we change the size of the hole, the cone will get fatter or thinner and as a result, those little millimetre-sized spots will move nearer of further away as we do it. As we adjust the hole, we’re simultaneously adjusting the amount of resolution in the subject, as well as controlling the amount of light. But this isn’t all.

Imagine the scene on the other side of the point of focus (behind the tree). We can visualise another very narrow cone getting wider as it gets further away. There will be a point where it, too is half a millimetre wide, and consequently, the grass and twigs there will be resolved just as well as the half-millimetre point in front of the subject. As if by magic, we find that we have twice as much resolution as we’d imagined.

The distance between the point of acceptable resolution at the back and the one at the front is called the Depth of Field. Those little mystery numbers around the lens tell you where the people who made your camera think that the image will be acceptably sharp. They are guidelines rather than rules. Some photographers think that they are a bit optimistic.

Rounding Off Infinity

If we focus our lens on infinity we can adjust the size of the hole so that it will resolve details (like the grass and twigs) at whatever distance we like in front of infinity. Fortunately, in practical photography, infinity lies at the horizon, rather than beyond the edge of the known Universe and this has its advantages. Instead of focusing on infinity itself, we adjust our focus so that the second half-millimetre point lies on the horizon. Now everything from the distant half-millimetre point at “infinity” to the nearest half-millimetre point in front of the interesting tree will be resolved sufficiently well to be called sharp.

The distance we focus on will vary with the size of the hole we have chosen and the resulting narrowness of the cone. The distance we focus on to achieve this is called the Hyperfocal Distance. It’s very popular with street photographers. (Hence, “f8 and…”)

We promised to come back, to deal matters inside the camera. There’s a second cone of light inside the camera. Let’s make up some more numbers to explore what happens behind the lens.

We’ll keep the same size of hole, but we’ll choose a distance of 50mm from lens to film, as this is the size of the commonest 35mm camera lens. (Please note that to keep things simple, we are still pretending that we have a Platonically ideal lens with no thickness at all, in exactly the same plane as the imaginary hole.)

We can see that this cone is very much shorter and chubbier. Twenty-five millimetres from the film plane, it’s still two millimetres wide and it’s half a millimetre wide at six and a quarter millimetres from the film. While we are focusing, the lens can move more than six millimetres so it’s a dimension to be taken seriously. Half a millimetre of resolution on the film, enlarged onto 10x8 paper will produce an image made up of four-millimetre-wide spots – entirely unacceptable. We would get sharper results from a pinhole camera. (If you haven’t already, why not try a pinhole camera?).

The Evil Twin

We can see that inside the camera, very small movements make a very big difference. This is the evil twin of depth of field – Depth of Focus. If we care to imagine two different lenses, perhaps a 100mm and 25mm, we can see that the cone inside the camera is even shorter and fatter for the 25mm lens and longer and thinner for the 100mm one. As a result, very tiny movements of the 25mm lens will have a much greater effect on the resolution of the image. A lens with a short focal length has less depth of focus than a long one.

So far, we’ve discussed all this in terms of absolute sizes. We’ve discovered that a four-millimetre hole will resolve the same size spot at the same distance away from the lens. Why then do photographers often repeat the cliché that “wider lenses have more depth of field”? The answer lies in the way photographers like to complicate matters. We have been using the entirely accurate, but not-very-technical term “hole”.

To baffle non-photographers, we call the hole an “aperture” and we give it a special number. The same actual size of hole has a different special number on lenses with different focal lengths, because the f-number is calculated by dividing the actual size of the hole by the focal length of the lens. So we if we divide 25mm by 50mm we get ½, or, more helpfully 1:2, which we generally write as f2. A 50mm f2 lens has nominally a hole of 25mm.

The fact that they are really fractions, with the top half removed, explains why smaller holes seem to have bigger numbers; f16 means that the hole is one-sixteenth of the focal length of the lens. The reason we do this is so that we can directly compare how much light different lenses allow onto the film. It saves us the trouble of carrying round a notebook, a pencil and a slide rule.

And it explains why lenses with a short focal length seem to have more depth of field. The same actual size of hole has a different and bigger (that is bafflingly, smaller) f-number on lenses with shorter focal lengths, so when we compare our different lenses, we are deceived by the cunning numerical tricks we have played on ourselves. 

There is another explanation, but that story will have to wait for another bedtime.