How it works · 02
Refraction and lenses
How glass bends light to a point, why focal length sets how much of the world a picture takes in, and why lens design took a century of maths.
Try it: focal length and angle of view
- Horizontal angle of view
- Diagonal angle of view
- Full-frame equivalent
- Lens moves out to focus
Bending light
Light slows down in glass, and when it crosses a surface at a slant it changes direction. Snell’s law of 1621 gives the rule: the sine of the angle times the refractive index stays the same on both sides. A curved surface meets each ray at a different angle, so a convex lens can bend a whole fan of rays from one point back together at another point. That is how a lens forms a bright image through a wide opening, where a pinhole must stay tiny. Lenses were ground for spectacles in the 1280s and fitted to the camera obscura by the 1550s.
Focal length and focusing
A lens has a focal length f: the distance behind it at which rays from a very distant point meet. For a subject at distance u, the image forms at v, where 1/f = 1/u + 1/v. Closer subjects focus further back, so a lens moves away from the film to focus close. A 50 mm lens focused at 3 m moves out less than a millimetre; focused at 25 cm it moves out 12.5 mm, which is why close-up work needs extension tubes or macro lenses.
Angle of view
How much of the scene fits on the frame depends on the focal length and the size of the film or sensor. The angle of view is 2 arctan(d / 2f), where d is the frame’s width (or diagonal). On 35mm full frame, a 50 mm lens takes in about 40° across, close to how much a person attends to without moving their eyes; 24 mm is wide (74°), 200 mm is narrow (10°). A smaller sensor sees less through the same lens, so it needs a shorter focal length for the same view: that ratio is the crop factor (see formats). The zoom lens changes focal length without changing focus.
Aberrations and lens design
A single lens is imperfect. Blue light bends more than red, so colours focus at different distances (chromatic aberration); the achromatic lens fixed this by pairing two glasses. Rays through the edge of a lens focus nearer than rays through the centre (spherical aberration), and off-axis points smear into commas and lines. Petzval in 1840 was the first to design a lens by calculation; Abbe, Zeiss and Schott made it a science with new glasses; the cooke triplet and tessar showed that a few well-chosen elements could correct most faults at once. Anti-reflection coatings later made lenses with many elements practical.