Lenses

Real and virtual image formation through thin converging and diverging lenses.

Conceptual context

A lens is a transparent medium bounded by two curved surfaces that redirect light by refraction. Unlike mirrors, which work by reflection, lenses transmit light and bend it towards a focal point or diverge it away from one. The study of lenses forms the backbone of applied geometrical optics: from corrective eyeglasses to projection systems, image formation depends on understanding how the geometry and material of the lens determine the ray paths.

This leaf focuses on the thin lens model, where the glass thickness is negligible compared to working distances. This model predicts the position, size and orientation of the image formed from experimentally measurable quantities.

🟢 Essential level

When a beam of parallel rays passes through a converging lens, the rays cross at a point called the focus. The distance between the lens center and the focus is the focal length $f$. If the lens is diverging, the rays spread out and their back-extensions meet at a virtual focus.

The image position $d_i$ depends on where the object $d_o$ is placed relative to the focus. If the object is beyond the focus of a converging lens, the image is real (projectable onto a screen) and inverted. If the object lies between the lens and the focus, the image is virtual, upright and magnified. The lateral magnification $m$ quantifies how much larger or smaller the image is compared to the object and whether it is inverted.

A pocket magnifying glass illustrates the simplest case: bringing the lens close to a small piece of text so the paper stays within the focal distance makes the letters appear enlarged, upright, and on the same side of the lens as the object — a virtual image. If instead the object is moved beyond the focus, as in a projector, the lens forms a real, inverted image on a screen, the same way a camera projects the scene upside down onto its sensor.

🔵 Formal level

Basic formulas

The fundamental relation governing image formation in a thin lens is:

Thin lens equation

This equation connects the focal length $f$ with the object distance $d_o$ and the image distance $d_i$. Signs follow the convention: positive distances in the direction of light propagation. For a converging lens $f$ is positive; for a diverging one, negative. If $d_i$ is positive the image is real; if negative, virtual.

Image size and orientation are obtained through the lateral magnification:

Lateral magnification

The negative sign means that when $d_i$ and $d_o$ are both positive, $m$ is negative: the real image is inverted. A magnitude of $m$ greater than one means the image is enlarged; less than one, reduced.

Real student questions

• Why does the image invert when the object moves beyond the focus? Because the rays cross the optical axis as they converge. Geometrically, the upper ray from the object ends below at the image and vice versa.

• Can a diverging lens form a real image? Not with a real object. Only with a virtual object (created by another lens) can a diverging lens produce a real image under certain conditions.

• What happens if the object is placed at the center of curvature? In a converging lens with $d_o$ equal to twice $f$, the image forms at the same distance on the other side, with $m$ equal to minus one: a real, inverted image of the same size.

• Why are myopia-correcting lenses diverging? The myopic eye converges too strongly, forming the image in front of the retina. A diverging lens reduces total convergence, shifting the image back onto the retina.

Final synthesis

The thin lens transforms the position and size of an object through refraction controlled by its geometry and material. The lens equation, lateral magnification and lensmaker's equation form a closed system that fully predicts the image for any paraxial configuration. Mastering the signs of these quantities is equivalent to mastering the classification of real and virtual images.