Why Two or Three Rays Are Enough
The question
A lit object sends light out in every direction, and countless rays strike the lens. So why does a ray diagram draw only two or three of them, and how can such a small sample tell us where the image is?
Short answer: every ray that leaves one point of the object and passes through the lens arrives at the same image point. Where those rays meet is fixed by the lens and the object, not by which rays you pick. Two lines cross at exactly one point, so any two rays are enough to find it. We pick the special rays because their paths after the lens can be drawn without any calculation, and a third ray is a check on the drawing.
The rest of this notebook shows that this is true, using pictures, a short proof, and an interactive simulator.
1 dependency successfully precompiled in 5 seconds
Main.NB.principal_y
draw_ray! (generic function with 1 method)
1. There really is a lot of light
Start with the tip of the arrow (the object). It scatters light in every direction. The figure below draws 120 rays leaving the tip, evenly spaced in angle.
- Most of them miss the lens (faint lines). They play no part in forming the image.
- Every ray that does pass through the lens is bent, and the bent rays all meet at one point.
- Convex lens: the rays actually cross on the far side. That is a real image; put a screen there and you see a sharp spot.
- Concave lens: the rays spread apart after the lens. Traced backwards (dashed), they all seem to come from one point on the object's side. That is a virtual image; your eye sees the light as if it came from there.
2. Any two of them give the same point, so choose easy ones
All of the rays through the lens meet at one point, so you can find that point with any two of them. Most rays are awkward to draw, though: you would need to work out the angle each one bends through. Three rays can be drawn with a ruler alone, because we know exactly what the lens does to them:
| Ray | Convex (converging) lens | Concave (diverging) lens |
|---|---|---|
| ① Parallel ray | arrives parallel to the axis, leaves through the far focal point F′ | arrives parallel to the axis, leaves as if it came from the near focal point F′ |
| ② Centre ray | goes straight through the centre of the lens, undeviated | goes straight through the centre of the lens, undeviated |
| ③ Focal ray | passes through the near focal point F, leaves parallel to the axis | heads toward the far focal point F, leaves parallel to the axis |
In the figure below, the coloured special rays are drawn on top of the faint "ordinary" rays. They meet at exactly the point where all the others meet. Nothing about them is privileged except that they are easy to draw.
3. Why every ray meets at the same point
Put the lens at $x = 0$ and the object's tip at $(-u,\ h)$. Pick any ray from the tip and let it hit the lens at height $y$. Its slope before the lens is
$$s = \frac{y - h}{u}.$$
A thin lens with focal length $f$ (positive for convex, negative for concave) changes a ray's slope by an amount proportional to how far from the centre the ray strikes:
$$s' = s - \frac{y}{f}.$$
That one rule covers all three special rays: a ray through the centre ($y = 0$) is not bent, and a ray arriving parallel ($s = 0$) leaves with slope $-y/f$, which aims it at the focal point.
After the lens, at a distance $v$ beyond it, the ray's height is
$$y + s'v \;=\; y\left(1 + \frac{v}{u} - \frac{v}{f}\right) \;-\; \frac{h\,v}{u}.$$
Now choose $v$ so that the bracket is zero:
$$\boxed{\;\frac{1}{v} = \frac{1}{f} - \frac{1}{u}\;}$$
(Here $u$ is a positive distance. In the Cartesian sign convention, where a real object has $u < 0$, the same equation reads $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$; the lens lab below uses that form.)
At that distance $y$ disappears: every ray, wherever it hit the lens, arrives at the same height $h' = -hv/u$. This is the thin-lens equation, and the image point it gives does not depend on which ray you used. If $v$ comes out negative (always the case for a concave lens), the rays only meet when extended backwards, so the image is virtual.
The check below computes eight randomly chosen rays numerically and records where each one crosses the image plane. They all land on the same point.
let u = 30.0, h = 6.0, strikes = (-9.3, -4.1, 0.7, 6.2)
rows = [begin
img = image_of(u, h, f)
(lens = f > 0 ? "convex, f = $f" : "concave, f = $f",
ray_hits_lens_at = y,
image_distance_v = img.v,
ray_height_at_v = y + outgoing_slope(y, u, h, f) * img.v,
predicted_h′ = img.h)
end for f in (12.0, -15.0) for y in strikes]
slate_table(rows; default_format = (kind = :fixed, digits = 6),
format = (ray_hits_lens_at = :fixed, image_distance_v = (kind = :fixed, digits = 3)))
end| lens | ray_hits_lens_at | image_distance_v | ray_height_at_v | predicted_h′ |
|---|---|---|---|---|
| convex, f = 12.0 | -9.30 | 20.000 | -4.000000 | -4.000000 |
| convex, f = 12.0 | -4.10 | 20.000 | -4.000000 | -4.000000 |
| convex, f = 12.0 | 0.70 | 20.000 | -4.000000 | -4.000000 |
| convex, f = 12.0 | 6.20 | 20.000 | -4.000000 | -4.000000 |
| concave, f = -15.0 | -9.30 | -10.000 | 2.000000 | 2.000000 |
| concave, f = -15.0 | -4.10 | -10.000 | 2.000000 | 2.000000 |
| concave, f = -15.0 | 0.70 | -10.000 | 2.000000 | 2.000000 |
| concave, f = -15.0 | 6.20 | -10.000 | 2.000000 | 2.000000 |
4. Every point of the object makes its own image point
The object is not only its tip: every point on it sends out its own spray of light, and each spray is focused to its own image point. The figure shows rays from three points on the arrow, each in its own colour. Each colour converges to its own point, and together those points make up the image arrow.
So why do diagrams trace rays from the tip only? The base of the arrow sits on the axis, and a point on the axis is imaged onto the axis. Once you have the image of the tip, you know where the image arrow starts and ends, and the points in between land on a straight line between those two points.
5. Try it yourself: the lens lab
The simulator below traces the light from the tip of the object, sent out in all directions, through a thin lens. Things to try:
- Turn up "Light rays from the tip". More light makes the image point brighter, but it does not move it.
- Click anywhere on the lens to trace a ray of your own, or press Pick 2 random rays. Whichever two rays you pick, they cross at the image.
- Switch the special rays ①②③ on and off. Any two of them give the image; the third one checks your drawing.
- Make the lens small. A special ray can then pass outside the actual glass, but it still lands on the image point. A construction ray only has to obey the lens rule; it does not have to be one of the rays that really gets through.
- Drag the orange object, or press ▶ Move the object, and watch the convex lens's image switch from real and inverted to virtual and upright as the object passes the focal point F.
- Tick light from the whole object to see every point of the arrow imaged at once.
Summary
- An object sends out light in all directions. Of the rays that pass through the lens, every one from a given object point goes to the same image point: a real crossing for a convex lens with the object beyond F, or a virtual one, found by tracing backwards, for a concave lens or a convex lens with the object inside F.
- That point is fixed by $u$ and $f$ alone, through $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$. It does not depend on which rays you look at.
- Two lines meet at one point, so two rays fix the image. We draw the parallel, centre and focal rays because their paths after the lens are known without calculation, and we add the third to check the drawing.
- The other rays are not ignored. They land on the same spot and make the image bright, but they cannot change where it is.
- A special ray can miss the real glass and still be a valid construction line.
Check yourself
- A ray from the tip of an object hits a convex lens near its top edge. Without drawing it, which point does it pass through after the lens?
- Why can a ray diagram for a concave lens never give a real image of a real object?
- If you cover the bottom half of a convex lens, does half of the image disappear?
Answers
- It passes through the image of the tip, the same point the special rays meet at, because every ray from the tip does.
- A concave lens always makes rays from a real object spread apart, so they never actually meet. Only their backward extensions meet, which gives a virtual image.
- No. Every part of the lens still sends light from every object point to the right image point. The whole image is still there, only dimmer. Try it in the lab by shrinking the lens.