Ray Diagrams for Lenses
A lens is a transparent optical element that refracts light to form an image. Ray diagrams are the geometer's shorthand for predicting exactly where an image will appear, how large it will be, and whether it is real or virtual. By tracing just two or three specially chosen rays from the tip of an object through a lens, you can locate the image with complete precision — no algebra required at the first step. Mastering ray diagrams gives you the intuition to understand cameras, glasses, microscopes, and telescopes before you ever touch the equations. There are two fundamental lens types: converging (convex) lenses, which are thicker in the middle and bring parallel rays to a focus, and diverging (concave) lenses, which are thinner in the middle and spread rays outward. Each has a focal point F on both sides. For a converging lens, parallel rays from infinity all pass through the principal focal point on the far side. For a diverging lens, parallel rays appear to diverge from a virtual focal point on the same side as the incoming light. The distance from the optical centre to the focal point is the focal length f. The nature of the image — real or virtual, upright or inverted, magnified or diminished — depends entirely on how far the object sits from the lens relative to the focal length. Real images are formed by actual ray convergence and can be projected onto a screen. Virtual images exist only when rays appear to diverge from a point and cannot be projected; they are seen by looking back through the lens.
- Identify the Principal Axis and Optical Centre
- Choose Your Three Standard Rays
- Converging Lens: Six Object Positions
- Diverging Lens: Always the Same Outcome
- Sign Convention and Magnification
Identify the Principal Axis and Optical Centre
Draw a horizontal line — the principal axis — passing through the middle of the lens. Mark the optical centre O at the lens's centre. All distances are measured from O. Mark the focal points F on each side at distance f. For a converging lens, mark 2F at distance 2f on each side, since these positions produce special image locations.
Choose Your Three Standard Rays
For any object point (usually the tip of an arrow), you trace three canonical rays: (1) A ray parallel to the principal axis — after passing through the lens it refracts through (or appears to come from) the far focal point F. (2) A ray passing through the optical centre O — it continues in a straight line, undeviated. (3) A ray passing through (or aimed at) the near focal point F — after the lens it emerges parallel to the principal axis. Where any two of these three rays meet (or appear to meet) locates the image tip.
Converging Lens: Six Object Positions
The object's position relative to F and 2F determines the image character. Beyond 2F: image is real, inverted, diminished, between F and 2F on far side. At 2F: real, inverted, same size, at 2F on far side. Between F and 2F: real, inverted, magnified, beyond 2F on far side. At F: rays emerge parallel — image at infinity. Between F and O: rays diverge after the lens; trace them back to find a virtual, upright, magnified image on the same side as the object. This last case is the magnifying glass.
Diverging Lens: Always the Same Outcome
For a diverging lens, the parallel ray refracts outward and appears to come from the near focal point (on the same side as the object). The centre ray passes straight through. These diverging rays never actually meet on the far side, so you extend them backwards. The intersection is always on the same side as the object: the image is always virtual, upright, and diminished, regardless of object distance. This is why diverging lenses correct short-sightedness — they reduce the apparent size and move the effective object closer to the eye's comfortable range.
Sign Convention and Magnification
Adopt the real-is-positive convention (also called the Cartesian convention): distances measured in the direction of incident light are positive; against it are negative. Object distance u is positive when the object is on the incoming side. Image distance v is positive for a real image on the far side, negative for a virtual image on the same side as the object. Focal length f is positive for converging lenses and negative for diverging lenses. The linear magnification m = v/u gives the ratio of image height to object height. A negative m means the image is inverted.
The Key Equation
- f — Focal length (metres (m)): Distance from the optical centre to the focal point. Positive for converging lenses, negative for diverging lenses.
- v — Image distance (metres (m)): Distance from the optical centre to the image. Positive for a real image, negative for a virtual image.
- u — Object distance (metres (m)): Distance from the optical centre to the object. Negative when the object is on the incoming side (real object), using the Cartesian sign convention.
Worked Example
Problem. A converging lens has a focal length of 10 cm. An object is placed 30 cm from the lens. 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 height). This matches the ray diagram prediction for an object placed between 2F and beyond 2F.
Origins · 1 minJohannes Kepler and how this was discovered
Kepler laid the theoretical foundations for geometrical optics in his 1611 work Dioptrice, where he gave the first correct mathematical description of how lenses refract light, explained how converging and diverging lenses form images, and described the ray-based principles that underpin the thin lens model still taught today. Kepler explained how the human eye forms images on the retina using the same lens ray principles — recognising that the brain must 'flip' the inverted retinal image. He did this without ever dissecting an eye himself, reasoning entirely from optics.
Related Concepts
- Refraction of Light — Lens action is entirely based on refraction — Snell's law governs how light bends at each curved glass surface to converge or diverge rays.
- Total Internal Reflection — Understanding the critical angle and refraction limits completes the picture of how light behaves at glass-air boundaries, context for lens design.
- Ray Diagrams for Mirrors — Mirror ray diagrams use the same three-ray technique and sign conventions, making lenses and mirrors natural companions in geometrical optics.
- Optical Instruments — Cameras, microscopes, and telescopes all combine multiple lenses; understanding single-lens ray diagrams is the essential prerequisite for analysing these instruments.