Index Making Pictures

## Image Formation: Lenses

If we can produce image by tracing rays or by using a pinhole camera, why do we need a lens? Let's first consider how a lens refracts light to a sharp focus to make an image.

Figure: Convex lens forming an image
In physical optics, we think of a convex lens forming a real image on a plane. For example we can project an image onto a piece of paper. The lens refracts the light. If the lens is thin, we can approximate it to a plane which refracts light rays towards the horizontal axis through the centre of the lens. Rays parallel to this axis will pass through the focal point (informally, the focus) behind the lens. We often simply call this the focus of the lens. The focus is where parallel rays come together, the distance at which a burning glass produces a tiny image of the sun. There is a focus on each side of the lens. Rays which pass through the focal point in front of the lens will emerge parallel to the axis. A sharp image appears where the rays converge, according to the lens equation 1/o + 1/i = 1/f, where o and i are the object and image distances (from the lens centre) and f is the focal length (also from the lens centre). The image is inverted. For a given lens, the position of the sharp image is determined by the position of the object.

Rays through the centre of the lens are not refracted. These are the only rays which a pinhole camera uses. The lens captures light over its complete diameter. It therefore brings more light to the image than a pinhole camera. The tiny hole in the pinhole camera greatly restricts the amount of light, so a lens is used in conventional cameras to ensure that the film or CCD sensors have enough light to respond.

However, a wide diameter of lens introduces another effect, that of a specific distance to the focal plane. A pinhole camera will produce an image of a given object on any plane (and therefore at different sizes); a lens produces a blurred image, except at the distance where all light rays from the object, whether via the edge or centre of the lens, converge. If our object is truly a point, the ideal lens will produce a point at the image. In a real lens, a disk results which will be tiny only on the focal plane. It becomes progressively larger as we move away from the focal plane. We use the term circle of confusion for this disk. In real lenses the circle of confusion never forms a mathematical point but good lenses have a small circle of confusion when the image is sharply focussed.

An out of focus image is thus one in which each point of the image consists of light from several places on the object, because the circles of confusion overlap the image point.

Real cameras often have a circular iris diaphragm over the lens, to change the effective aperture of it. This masks the outer part of the lens and can be changed to mask more or less of the lens. When the aperture is reduced, less light reaches the image plane but it also makes the camera a better approximation to a pinhole camera. In other words, objects which would otherwise be badly out of focus are now only slightly so. The aperture can thus be adjusted to control the depth of field that looks in focus. With a wide aperture, only objects at a specific depth appear in focus on the image. With a smaller aperture, objects in front of and behind the nominal focal position will also look sharp. With a very narrow aperture, all objects look sharp.

Photographers also talk about wide-angle and telephoto lenses. Camera lenses are contructed from several elements, which allows the angle of view to vary from one lens to another while still giving a good optical performance. A wide-angle lens "sees" a wide angle of view. In other words, the angle subtended at the centre of projection is large. A telephoto lens sees only a small angle. In both cases, the image fills the image plane, so the telephoto lens can be used to photograph something from further away and yet the object will still fill the image plane. More subtly, the distances between objects (going away from the viewer) appear more compressed than with a wide angle lens. The perceived reduction in size with distance is less with a telephto than with a wide-angle lens. Another way of saying this is that the perspective effect is reduced by a telephoto lens but exaggerated by a wide-angle lens.

Strictly an image is only correct when it is viewed from the correct distance, according to the centre of projection. In practice we can tolerate quite a wide variation, unless a wide-angle close-up image is made. Another kind of apparently "distorted" image then results because our eyes cannot focus at the very close distance needed to view it. Of course it is not distorted but, at normal viewing distances, the perspective is too exaggerated to be acceptable.