A lens is a piece of clear material with at least one curved surface. Light refracts when it enters and again when it leaves, and the curve makes rays that hit different parts of the lens bend by different amounts. Think of a lens as a stack of tiny prisms: the ones near the edge bend light a lot, and the ones near the middle hardly bend it at all.
A converging (convex) lens is thicker in the middle. Its prisms point so that every ray bends toward the axis, and rays that arrive parallel to the axis all meet at the focal point, a distance f(the focal length) from the lens. A diverging (concave) lens is thinner in the middle. It bends every ray away from the axis, so parallel rays spread out as if they came from a focal point on the incoming side.
A lens has a focal point on each side, at the same distance f. We give f a sign: positive for converging and negative for diverging.
Two rays are enough to find an image. A ray parallel to the axis bends through the far focal point, and a ray through the center of the lens goes straight on. Where the rays meet is the tip of the image. For a converging lens the answer depends on where the object is, exactly as with a concave mirror:
| Object position | Image | Example |
|---|---|---|
| d_o > 2f | Real, inverted, smaller | A camera, an eye |
| d_o = 2f | Real, inverted, the same size | |
| f < d_o < 2f | Real, inverted, larger | A projector, a slide viewer |
| d_o = f | No image: the rays emerge parallel | A lighthouse or spotlight lens |
| d_o < f | Virtual, upright, larger | A magnifying glass |
A real image is where light actually meets, on the far side of the lens. A screen placed there shows it. A virtual image is where light only seems to come from, so no screen can catch it. A diverging lens works the same way at every distance: its image is always virtual, upright, smaller, and between the lens and the object.
It is the same pair of equations you met for curved mirrors:
The signs again carry the meaning. f is positive for converging and negative for diverging. d_i is positive for a real image (on the far side of the lens) and negative for a virtual one (on the object’s side). The magnification mis negative for an inverted image, and |m| > 1 means the image is larger.
Start with a converging lens and f = 10 cm. Slide the object in from far away and watch the real image grow and move away, then jump to the object’s side as a virtual, magnified image when you cross f. Then switch to a diverging lens.
Pick a lens, set its focal length, and slide the object toward and away from it. Two rays leave the tip of the object: one parallel to the axis, and one through the center of the lens, which goes straight on.
Opticians don’t describe a lens by its focal length. They use its power, in diopters (D):
A shorter focal length is a stronger lens, so it has a bigger power. A converging lens has a positive power and a diverging lens a negative one. And powers of thin lenses held together simply add.
Your eye is a converging lens system. The cornea and the lens inside the eye bend light so that the world forms a real, inverted image on the retina, the light-sensing layer at the back. Your brain flips it back over. For a sharp picture, the image has to land on the retina.
In a nearsighted eye the focus lands in front of the retina, so distant things are blurry. The eye is effectively too strong for its length. A diverging glasses lens spreads the light a little before it enters the eye and pushes the focus back onto the retina. In a farsighted eye the focus would land behind the retina, so close things are blurry. A converging glasses lens pulls the focus forward.
Pick an eye type, look at where the focus lands, and then choose a glasses lens that makes the picture sharp. Try the nearsighted eye with a converging lens, and see what goes wrong.
Light from a distant object enters the eye as parallel rays. The eye’s lens (with the glasses, if any) focuses them, and the picture is sharp only if the focus lands on the retina. Start with a nearsighted eye, then add a glasses lens until the picture is sharp. (The vertical scale is exaggerated.)
Project 6-1-4 can use lenses in pairs. Two simple results cover most of what you’ll build:
A magnifier. Hold the object at the focal point, and the rays leave the lens parallel, so your relaxed eye sees a large virtual image. How much bigger it looks (its angular magnification) is about
where 25 cm is the usual closest comfortable viewing distance. A lens with f = 5 cm magnifies about 5×.
A simple telescope. The objective, a long-focal-length converging lens, makes a real image of the distant object at its focal point. The eyepiece, a short-focal-length lens, acts as a magnifier on that image. When the lenses are one total focal length apart,
Four questions. They are not graded. Answer them to find out what still needs work.
You now have all three of light’s tricks: bounce, bend, and focus. Next you’ll put them to work.