In a uniform material, light travels in straight lines. We draw a path of light as a ray, a line with an arrow. Optics is mostly the study of what happens to rays when they reach a surface, and the simplest thing a surface can do is send them back. That is reflection.
Not every surface reflects the same way. A very smooth surface, like a mirror, sends parallel rays all off in the same direction. That is specular reflection, and it is what makes a clear image. A rough surface, like paper or a wall, sends the rays off in every direction. That is diffuse reflection, and it is why you can see a wall from anywhere in the room but can’t see your face in it. Both kinds follow the same rule at the tiny scale of each bump. The bumps just point every which way.
To describe a bounce, you need a reference line. The normal is a line drawn at 90° to the surface, at the point where the ray strikes. The angle between the incoming ray and the normal is the angle of incidence. The angle between the outgoing ray and the normal is the angle of reflection. The law says:
The incident ray, the reflected ray, and the normal all lie in the same plane. And always measure from the normal, never from the surface.
Tilt the mirror and compare how far the mirror turned with how far the beam turned. Then change the incoming angle and check that the angles on both sides of the normal are always equal.
A fixed beam strikes a mirror. Tilt the mirror and watch the reflected beam. The dashed gray beam shows where it went when the mirror was upright.
When light from an object reflects off a mirror and reaches your eye, your brain assumes the light came straight from its source. It traces the reflected rays backward in straight lines. They appear to meet at a point behind the mirror. That point is the image. In a flat mirror, the image has three features:
Surprisingly, only half your height. To see your head, the ray from the top of your head must reflect into your eye. By the law of reflection, it hits the mirror exactly halfway between your eye level and the top of your head. To see your feet, the ray reflects halfway between your eyes and your feet. The mirror only has to span those two points.
A mirror at 45° to a horizontal ray has a normal at 45° to the ray, so the angle of incidence is 45° and the ray turns by 90°. A periscope uses two parallel mirrors at 45°. The first turns the ray up by 90° and the second turns it back to horizontal. The ray leaves in the same direction it entered, but shifted over. That is how you see over a wall or around a corner.
More generally, two mirrors with an angle α between them turn a ray by a total of
Parallel mirrors (α = 0) turn the ray by nothing, which is the periscope. Mirrors at right angles (α = 90°) turn the ray by 180°, sending it straight back toward where it came from. That’s how a bicycle reflector and the reflectors left on the Moon work.
Take a flat mirror and curve it, and the normal now points in a different direction at every point. Rays still obey the law of reflection at each point, but they no longer stay parallel. A concave mirror curves inward, like the inside of a bowl. A convex mirror bulges outward, like the back of a spoon.
A few terms describe a curved mirror. The center of curvature C is the center of the sphere the mirror is part of, and R is its radius. The principal axis runs through the middle of the mirror and C. A concave mirror brings rays that arrive parallel to the axis together at the focal point F, halfway between the mirror and C. So the focal length is
A convex mirror spreads parallel rays apart, and they look as if they came from a focal point behind the mirror. We make that distinction with a sign: f is positive for a concave mirror and negative for a convex mirror.
To find an image, draw rays from the tip of the object. Two are easy: a ray parallel to the axis reflects through F (or appears to come from F behind a convex mirror), and a ray through F reflects parallel to the axis. Where the reflected rays meet is the image tip.
For a concave mirror, the image depends on where the object is:
| Object position | Image | Example |
|---|---|---|
| d_o > 2f (beyond C) | Real, inverted, smaller | You, far from a spoon’s bowl |
| d_o = 2f (at C) | Real, inverted, the same size | |
| f < d_o < 2f (between F and C) | Real, inverted, larger | A projector-style image |
| d_o = f (at F) | No image: the rays emerge parallel | A flashlight or headlight reflector |
| d_o < f (inside F) | Virtual, upright, larger | A makeup or shaving mirror |
Drawing rays gives you the idea. An equation gives you numbers. For a curved mirror, with object distance d_o and image distance d_i measured from the mirror:
The signs carry the meaning. f is positive for concave and negative for convex. d_i is positive for a real image, in front of the mirror, and negative for a virtual image behind it. The magnification m is negative for an inverted image and positive for an upright one. If |m| is greater than 1, the image is larger. You’ll meet this same equation again in 6-1-3 for lenses.
Start with a concave mirror and f = 10 cm. Move the object from far away toward the mirror, and watch the image grow, then blow up near the focal point, then jump behind the mirror. Then switch to a convex mirror and see that nothing like that ever happens.
Pick a mirror, set its focal length, and slide the object toward and away from it. Watch the image flip, grow, shrink, and jump behind the mirror. Two of the three principal rays are drawn from the tip of the object.
Four questions. They are not graded. Answer them to find out what still needs work.
You now know how light bounces. Next you’ll see what it does when it passes through something instead.