Ray diagrams for lenses

A lens bends light in a completely predictable way for three special rays, and knowing just these three lets you locate the image of any point without measuring a single angle.

• Ray parallel to the principal axis. A convex lens refracts it through the far focus $F_2$; a concave lens refracts it so it appears to spread out from the near focus $F_1$, on the same side as the incoming ray.
• Ray through (or aimed at) a principal focus. For a convex lens, a ray aimed at the near focus $F_1$ emerges parallel to the axis; for a concave lens, a ray aimed towards the far focus $F_2$ emerges parallel to the axis.
• Ray through the optical centre $O$. It passes straight through with no bending at all, at any angle — the fastest ray to draw and the one worth using every single time.

Any two of these three rays, drawn from the same point on the object (usually the arrow tip), are enough: refraction bends them according to the rule above, and the point where the two outgoing rays genuinely intersect is the real image of that point. When the outgoing rays instead spread apart and never meet, the eye still perceives an image because it extrapolates the rays backward in straight lines — that backward crossing point, marked with dashed construction lines since no real light actually reaches there, is the virtual image. For a convex lens, where those two rays cross depends entirely on where the object sits relative to $F_1$ and $2F_1$: outside $2F_1$ they cross close in, between $F_1$ and $2F_1$, giving a diminished real image; between $F_1$ and $2F_1$ they cross far out, beyond $2F_2$, giving an enlarged real image; and once the object moves inside $F_1$, the refracted rays diverge on the far side and only their backward extensions meet, on the same side as the object, giving an enlarged virtual image (the magnifying-glass case). For a concave lens, the outgoing rays always diverge, whatever the object distance, so the backward extensions always cross between $O$ and $F_1$ on the object's own side — a single unvarying virtual, erect, diminished image, exactly parallel to how a convex mirror always gives one type of image.

The construction is not a lucky trick — it works because refraction at a thin lens obeys the same two bending rules everywhere on the lens, so tracing any two representative rays from a point is enough to pin down where every other ray from that same point must also end up, because all real rays from a single object point genuinely reconverge at one image point (or appear to, for a virtual image).

Step 1. A thin lens is built from many tiny refracting surfaces, but its overall effect on light from a single point reduces to just three checkable behaviours: rays through $O$ pass straight through, rays parallel to the axis bend to (or appear to come from) a focus, and rays aimed at a focus bend to travel parallel to the axis afterwards. These are not separate physical rules — they are three convenient special cases of the same refraction law that governs every ray, chosen because their bending is trivial to draw.

Step 2. Because a real point source sends out light in every direction, not just along these two special rays, the two we draw are only a sample. The reason the sample is enough is that all the rays leaving one object point, after passing through the same lens, are guaranteed by the lens's geometry to reconverge (or appear to diverge from) a single common point — the image. Two rays already fix that point uniquely, since two non-parallel straight lines meet at exactly one location; a third ray, if drawn, would simply pass through the same point as a check.

Step 3. Whether that common point is reached by actual light (a real image, drawn with solid lines, always found on the side of the lens the light exits) or only by extending the emerging rays backward (a virtual image, drawn with dashed lines, found on the same side as the object) depends purely on whether the two chosen rays are converging or diverging after they leave the lens — convex lenses can produce either, depending on object distance, while a concave lens's diverging rays never converge, so it produces only virtual images.

An object is placed between the focus $F_1$ and twice the focal length $2F_1$ on one side of a convex lens. Locate the image with a ray diagram and state its nature.

Step 1. Draw the principal axis, mark the optical centre $O$, the two foci $F_1$ and $F_2$ equidistant from $O$, and $2F_1$, $2F_2$ at twice that distance on each side. Draw the object as a vertical arrow with its foot on the axis and tip $A$, standing between $F_1$ and $2F_1$ on the left.

Step 2. From $A$, draw Ray 1 parallel to the axis; after the lens it bends to pass through $F_2$. From $A$, draw Ray 2 straight through $O$, undeviated — these are the two easiest rays to draw accurately.

Step 3. Because the object lies between $F_1$ and $2F_1$, the two refracted rays do not meet until they are beyond $2F_2$ on the far side. Mark that meeting point $A'$, drop a perpendicular to the axis, and join it to $O$'s side of the axis to complete the image $B'A'$. Since the two rays actually cross on the far side, real light reaches $A'$: the image is real, inverted, and enlarged, formed beyond $2F_2$ — the standard result CBSE expects for this exact object position, matching the image-formation table without needing a formula.

Competitive stretch

Now place the same object inside $F_1$ (closer to the lens than the near focus) and repeat the construction. Ray 1 (parallel to axis, bending through $F_2$) and Ray 2 (through $O$, undeviated) now spread apart on the far side of the lens and never cross there. Extending both rays backward (dashed) shows they meet on the same side as the object, farther from the lens than the object itself. Because no real light travels along those dashed extensions, this is a virtual, erect, and enlarged image — the magnifying-glass configuration, the one case where a convex lens behaves like a concave lens's virtual-image family, only enlarged instead of diminished.

• Attempting only the two focus-related rays and skipping the ray through $O$, which is undeviated and by far the easiest to draw accurately — always include it when possible.
• Mixing up which focus, near $F_1$ or far $F_2$, each rule applies to for a convex lens versus a concave lens; the direction of the rule flips between the two lens types.
• Drawing a virtual image's backward extensions as solid lines instead of dashed — no real light travels along a backward extension, so it must be dashed.
• Assuming the ray through the optical centre has to be drawn horizontally; it only needs to be undeviated, and can pass through $O$ at any angle the object's position requires.

Ask the tutor: If picking the right pair of rays for a given lens problem still slows you down, ask the AI tutor to time-drill a few convex- and concave-lens ray diagrams with you.