Lenses, Mirrors, and Imaging

Lenses and mirrors form images by bending or reflecting rays, with image position and size predicted by the paraxial imaging equations.

Definition

The process by which converging or diverging optical elements map points of an object to corresponding image points, described in the paraxial approximation by linear relations among object distance, image distance, focal length, and magnification.

Scope

This topic covers image formation by thin and thick lenses and by plane and curved mirrors in the paraxial regime. It includes the lens-maker's equation, the thin-lens and mirror equations relating object and image distances to focal length, lateral and angular magnification, the sign conventions for real and virtual objects and images, cardinal points of a system, and the combination of multiple elements. It treats imaging as the ideal mapping of object points to image points before aberrations are considered.

Core questions

• Given an object and a lens or mirror, where is the image and how large is it?
• How does the shape and refractive index of a lens determine its focal length?
• How do the focal lengths and spacings of several elements combine into a single system?
• When is an image real and inverted versus virtual and upright?

Key concepts

• focal length
• thin-lens equation
• lens-maker's equation
• lateral magnification
• principal planes
• real and virtual images
• converging and diverging lenses

Key theories

Thin-lens and mirror equations

For paraxial rays the reciprocal of the image distance plus the reciprocal of the object distance equals the reciprocal of the focal length, with the focal length set by surface curvatures and refractive index through the lens-maker's equation.

Cardinal points and system combination

A general optical system is characterized by its focal, principal, and nodal points, allowing any sequence of lenses and mirrors to be treated as a single equivalent element for paraxial imaging.

Clinical relevance

The imaging equations govern the prescription of spectacle and contact lenses to correct myopia, hyperopia, and presbyopia, and they underlie the design of the objective and eyepiece systems in microscopes, telescopes, and surgical loupes.

History

Practical lens combinations for telescopes were assembled by Galileo and Kepler in the early seventeenth century, but a complete paraxial theory of imaging, including the systematic use of principal and focal points, was given by Gauss in 1841, providing the framework still used in optical design.

Key figures

• Carl Friedrich Gauss
• Johannes Kepler
• Galileo Galilei

Seminal works

• hecht2017
• bornwolf1999

Frequently asked questions

Why is the image from a simple magnifier upright while a camera image is inverted?

When the object is closer to a converging lens than its focal point the image is virtual and upright, as in a magnifier; when the object is beyond the focal point the rays converge to form a real, inverted image, as on a camera sensor.

What sets a lens's focal length?

The focal length depends on the curvatures of the two surfaces and on how much the lens material refracts light relative to its surroundings, as quantified by the lens-maker's equation.