Focal length: definition, properties, and practical effects

Definition and conventions

For a thin converging lens or a concave mirror, the focal length is the distance from the optical centre (or vertex) to the focal point where parallel rays from a distant source meet. For diverging lenses or convex mirrors the focal length is treated as negative under the usual sign convention; it refers to a point from which light appears to diverge. Optical texts and instrument makers commonly use the following relations:

• Thin lens equation: 1/f = 1/do + 1/di, relating focal length (f) to object distance (do) and image distance (di).
• Lens maker's formula (for a lens in air): 1/f = (n−1)(1/R1 − 1/R2), where n is refractive index and R1, R2 are surface radii of curvature.

Optical power is the reciprocal of focal length (measured in metres) and is expressed in diopters; higher power means shorter focal length and stronger convergence.

Characteristics and related quantities

Focal length controls several important optical properties:

• Magnification: the ratio of image size to object size depends on f and the distances involved.
• Field of view: shorter focal lengths produce wider angular views, longer focal lengths narrow the view.
• Depth of field: with other factors equal, shorter focal lengths yield greater depth of field for a given framing and aperture.
• Effective focal length (EFL): in multi-element systems or when using sensors smaller than a full-frame reference, the working focal length can be quoted as an effective value for comparison.

Uses, examples and distinctions

In photography the focal length is typically given in millimetres and distinguishes wide-angle, standard and telephoto lenses. Typical examples: 24 mm (wide), 50 mm (standard), 200 mm (telephoto). In eyeglasses and contact lenses, focal length underpins prescriptions expressed as diopters. In telescopes and microscopes the focal length of objective and eyepiece elements determines overall magnification and image formation.

History and development

The notion of focal length grew out of early studies of refraction and image formation by 17th-century scientists and instrument makers as optical theory and glassmaking advanced. Over time, the simple focal-length concept was extended to complex, multi-element objectives and to systems that correct aberrations; modern optical design uses ray tracing and optimization to achieve desired focal properties across a field of view.

Practical notes and further reading

When comparing optics, be aware of differences between nominal focal length and effective focal length, and between geometrical focal point and perceived image scale on a detector. For introductory material on lenses see lens basics, for mirror optics see curved mirror resources, for system-level discussions consult general optical device references, and for experimental behaviour of light and focal points see light and focal point. These resources provide practical examples and diagrams that complement the relationships outlined here.