Curved mirrors

Converging and diverging mirrors

Flat mirrors produce a lot of interesting and strange results. (See the problem Alice and Aslan for example.) But things can get even more interesting with curved mirrors. Consider a mirror that is a piece of a sphere — something like the bowl of a spoon.

Let's consider an object standing in front of the a mirror curved towards the object as shown at the right. From every point on our object (the red box), light is hitting it (not shown) and then spraying out in all directions (only shown are the rays going towards the mirror). Some of the light rays from the object run into the mirror and are reflected by it. This mirror takes light rays that are spreading apart and reflects them so that they come together (or spread apart less) where an eye can observe them. This is called a converging mirror. 

If the mirror is curved the other way, it will spread out the rays instead of bringing them together. This is called a diverging mirror. (Don't worry about the fact that the eye sees only "one ray" in this case. There really are an infinite number of rays between every pair of rays! It's just not convenient to show them all.)

A word about language and notation: In many texts and websites curved mirrors and lenses are referred to as "convex" or "concave", This means "the top and bottom bends away from you when you look at it" or "the top and bottom bends toward you when you look at it".  Since, for a mirror, if you look a convex mirror from the other side it's concave and vice versa. Therefore, "concave" and "convex" don't just refer to the object, but to how you are looking at hit. This can be very confusing. We'll try to stick to the language "converging" and "diverging". That tells what the object is doing to the light so it seems more direct.

How rays behave at the mirrors

When each ray hits the curved mirror, it will reflect off. How will it behave? Here's the principle:

When a light ray hits a curved mirror, since the light particles are much smaller than the mirror, we can consider only a very small piece of the mirror where the particle hits. A small piece of a large sphere can be treated as flat so we can use our rule for a flat mirror — the angle of incidence (measured from the normal) is equal to the angle of reflection (also measured from the normal).

Forming images: the converging mirror

Something particularly interesting happens with a converging mirror. To figure out what happens to the light we look at a few convenient rays.

Its easy to figure out what happens to a few of the rays that intercept the mirror, and from these we can get a good idea about what happens to the rest of the rays. These convenient are sometimes called principal rays, but there is nothing special about them; it's just that it is easy to figure out where they go, so we will call them the easy-to-figure-out rays.

Here's the diagram.  We've put in an observer (eyes and a brain) so we can talk about what the observer sees if they look at the mirror. (And we've only show the rays from the object that are headed towards the mirror since they're the only ones that form an image.)

We've drawn a line through the center of the sphere of which the mirror is a part. Since it is a radius of the sphere it is perpendicular to the mirror.  This gives us one of the "easy lines":

1. The ray to the center line at the mirror — This ray is drawn in purple. The center line shows the perpendicular to the surface so it's easy to get the reflected ray; it goes off at the same angle below the center line as it came in above the center line.

A second "easy ray" is the one that goes through the center.

2. The ray through the center of the sphere — This ray is drawn in green. It follows a radius from the center of the sphere to the mirror. It therefore hits the mirror normal to the surface so the angles of incidence and reflection are 0. It goes right back along the line it came in on.

The third "easy ray" is not so easy to prove. This is the ray that travels to the mirror parallel to the center line. If it hits the mirror not too far away from the center line (compared to the radius) it reflects back to a point halfway between the mirror and the center of the sphere. The point along the center line a distance R/2 from the mirror (where R is the radius of the sphere) is called the focal point.  Our third result is

3. The ray parallel to the center line — This ray is drawn in red. It reflects from the mirror and passes through the focal point.

Note that this has the interesting implication that if we bring in a parallel beam of rays they will all reflect so as to go through the focal point; something like this:

In this image, a beam of parallel rays comes onto the mirror from the left. All reflect off and pass through the focal point. Note that the rays that are farther off the axis are shown as going "through" the mirror. The mirror shown is spherical. The rays shown indicate where the mirror would really have to be to send all the rays exactly through the focal point. The rays that are too far from the axis would actually miss the focal point by a little, blurring it. This blurring is called spherical aberration.

The interesting thing about the image is that all of the rays leaving a particular point on the object (the object point) cross through a single point after reflecting off the mirror.  This is called the image point. So:

For every point on the object, the rays spraying out from that point are reflected from the mirror to pass through a single point in space. For every point on the object there is an image point. Go to the Physlet Physics Optics page and move the source point (white box) up and down on the object to see how an image of the object is built up in empty space.

So the mirror creates a real image (it would perhaps better be called a virtual object). An observer looking at the light rays reflected from the mirror will infer that there is an object floating in the air at the point of the image!

Wait. What? Do you believe that? This is a good example of "the implications game". Our basic principles are few and solid. Light travels in straight lines. Rays reflect off at a mirror so the angle of incidence equals the angle of reflection. Our brains infer where an object is by tracing the rays the two eyes receive back and seeing where they intersect. That's all we have used.  If we don't believe the result we have to figure out what went wrong with our reasoning.

In this case, nothing is wrong. There DOES appear to be an object that is apparently floating in mid-air. Sometimes our brains are confused by conflicting messages and the fact that part of the image may disappear when it moves across the edge of the mirror (or other factors) may lead us to see the image as not floating (though there are lots of good examples that work just fine). An example that works well is shown in the figure at the right. The ring in the center of the object (which has hidden curved mirrors) appears to be floating in air. There is no ring actually there! (The real ring is on the mirror a few inches below.)

If we put a screen at the image — a piece of paper, say — each point on the screen reflects the spot of rays it receives and sprays it back in all directions. Anyone looking at the screen will then see the image on it. Or, you can use a camera to actually take a picture of the real image ("virtual object") as we have done of the ring above.

In the follow-ons, we look at the geometry of the real image and derive the equations that specify the properties of the image.

Joe Redish 4/11/12