Refraction Through Spherical Lenses
A lens is a transparent optical device, typically made of glass or plastic, that refracts (bends) light to either converge it to a point or spread it outward. Lenses work because light changes speed when it crosses the boundary between two media — for example, from air into glass — causing it to change direction. This bending follows Snell's Law at each curved surface. Lenses are everywhere: in your eyes, spectacles, cameras, microscopes, and telescopes. Understanding how they form images is fundamental to optics and explains much of how we see and record the world. There are two principal types of spherical lens. A convex lens (also called a converging lens) is thicker at the centre than at the edges. It bends parallel rays inward so they meet at a real focal point on the far side. A concave lens (also called a diverging lens) is thinner at the centre than at the edges. It bends parallel rays outward so they appear to come from a virtual focal point on the same side as the incoming light. The focal length, f, is the distance from the lens centre to this focal point and is positive for convex lenses and negative for concave lenses. The nature of the image — whether it is real or virtual, upright or inverted, magnified or diminished — depends on both the type of lens and the object's distance from it. Real images can be projected onto a screen; virtual images cannot, but they appear to the eye as if light is coming from a certain location. Mastering the qualitative rules for image formation is the first step before applying the thin-lens equation quantitatively.
- Why Light Bends at a Curved Surface
- The Convex (Converging) Lens — Three Key Rays
- The Concave (Diverging) Lens — Always Diverging
- Applying the Thin-Lens Equation
Why Light Bends at a Curved Surface
When a ray of light travels from one medium to another of different optical density, it bends at the interface according to Snell's Law: n₁ sin θ₁ = n₂ sin θ₂. A spherical lens has two such curved surfaces. Because the surfaces are curved, different parts of the wavefront strike at different angles, so each part bends by a different amount. A convex surface curves toward the incoming light, steering rays toward the principal axis; a concave surface curves away, steering rays away from it. The combined effect of both surfaces gives the lens its overall converging or diverging power.
The Convex (Converging) Lens — Three Key Rays
To locate an image formed by a convex lens, trace any two of three standard rays from the tip of the object: (1) A ray parallel to the principal axis refracts and passes through the far focal point F₂. (2) A ray passing through the optical centre of the lens continues in a straight line without bending. (3) A ray passing through the near focal point F₁ refracts and emerges parallel to the principal axis. Where these rays (or their extensions) meet, the image forms. If the object is beyond 2F, the image is real, inverted, and diminished. At 2F the image is real, inverted, and same size. Between F and 2F the image is real, inverted, and magnified. Inside F the image is virtual, upright, and magnified — this is how a magnifying glass works.
The Concave (Diverging) Lens — Always Diverging
For a concave lens, parallel rays diverge after refraction as if they originate from the virtual focal point on the same side as the object. Using the same three-ray technique: (1) A ray parallel to the axis refracts and diverges as if coming from F on the near side. (2) A ray aimed at the optical centre passes straight through. (3) A ray aimed at the far focal point emerges parallel to the axis. The refracted rays always diverge and never actually cross on the far side, so they must be extended backwards. The result is always a virtual, upright, and diminished image, regardless of object distance. Concave lenses are used in spectacles for short-sightedness (myopia).
Applying the Thin-Lens Equation
For a thin lens — one whose thickness is negligible compared to its focal length — all distances are measured from the optical centre. The thin-lens equation (also called the lens formula) links object distance u, image distance v, and focal length f. By convention, distances are positive in the direction light travels (real is positive). If you substitute measured values and find v is positive, the image is real and on the far side; if v is negative, the image is virtual and on the same side as the object. The linear magnification m = v/u tells you the ratio of image height to object height; a negative magnification means the image is inverted.
The Key Equation
- f — Focal length (metres (m)): Distance from the optical centre to the principal focus; positive for a convex lens, negative for a concave lens.
- v — Image distance (metres (m)): Distance from the optical centre to the image; positive if the image is real (on the far side), negative if virtual (on the same side as the object).
- u — Object distance (metres (m)): Distance from the optical centre to the object; negative for a real object in the real-is-positive convention used here (some textbooks use the New Cartesian sign convention where u is negative).
Worked Example
Problem. An object is placed 30 cm in front of a convex lens of focal length 10 cm. Find the image distance and state the nature of the image.
Answer. The image forms 15 cm on the far side of the lens. It is real, inverted, and diminished (half the object's size). This is consistent with the object being between 2F and F: wait — 30 cm = 3F, which is beyond 2F (20 cm), so the image is indeed real, inverted, and diminished, as expected for an object beyond 2F.
Origins · 1 minIbn al-Haytham (Alhazen) and how this was discovered
Ibn al-Haytham, writing in his monumental work Kitāb al-Manāẓir (Book of Optics, c. 1011–1021), laid the scientific foundations of optics. He correctly explained that vision occurs because light rays travel from objects into the eye — overturning the ancient Greek emission theory. He investigated refraction experimentally, studied lenses and their magnifying properties, and used ray diagrams to analyse image formation. His work directly influenced European scientists such as Roger Bacon and later Johannes Kepler, making him the intellectual ancestor of the modern understanding of lenses. Ibn al-Haytham spent years under house arrest in Cairo, during which time he wrote the Book of Optics — one of history's most productive periods of scientific confinement.
Related Concepts
- Snell's Law and Refraction — Refraction through lenses is governed at each surface by Snell's Law; understanding n₁sinθ₁ = n₂sinθ₂ explains why light bends when entering glass.
- Reflection and Mirrors — Curved mirrors form images using similar ray-diagram techniques and the mirror formula parallels the thin-lens equation, making comparison instructive.
- Optical Instruments — Microscopes, telescopes, and cameras all rely on combinations of converging and diverging lenses; understanding single-lens refraction is the prerequisite.
- Total Internal Reflection — Total internal reflection is another consequence of Snell's Law at glass–air interfaces, deepening understanding of how light behaves inside transparent media like lenses.