The Geometric Optics Approximation
Now that we have the concepts of rays and wavefronts we move on to the subject of geometric optics. One approximation that geometric optics makes is that the waves (rays) travel in straight lines until they hit a surface. When the ray encounters a surface it can either bounce back (reflect) or bend (refract) but then continues to travel in a straight line.
When is geometric optics applicable? If a laser shined light onto a screen with two slits, as we understand it the light would diffract, and we would see bright and dark fringes. But how can we imagine light as traveling in straight lines when it can interfere like this? It is important to realize that the idea of the rays traveling in straight lines is only valid if the wavelength of the wave (light) is much smaller than any of the objects or slits the wave will encounter. If the slit in our diffraction problem is much greater than the wavelength of incident light, then we can ignore the effects of diffraction. Mathematically, the geometric optics approximation can be written as \[\lambda \ll d\] where \(d\) is the size of any slit or object the light encounters and \(\lambda\) is the wavelength of the light.
How We See Things
Before going too much further it is worth considering how we see things. First, we must establish that eyes only see light rays that fall into them. Next, consider a tree. The tree does not give off its own visible light – we know this because we would not be able to see the tree at night time, without any lights on. On a bright day we see the tree because the sun gives off light which hits the tree and reflects off in many directions; some of it reaches our eyes.
The light that is reflected is not the same as the incident light – otherwise everything outside would be the same color as the sun! Instead objects reflect certain colors preferentially, and absorb others. The tree leaves, for example, reflect green strongly and absorb most of the other colors. When sunlight (which is a combination of all the colors) falls on the tree, the green is strongly reflected while colors like red are absorbed. Most of the light that reaches our eyes is green light, this is why the tree appears green.
If we can “see” a particular point on the tree, it means some of the rays that reflected from that point enter our eyes. The act of seeing only the top point of the tree is shown below on the left:
Here multiple rays have been drawn that enter the eye. Our diagram is not meant to suggest that a disproportionate amount of light enters the eye from the top of the tree. We draw a higher density of light rays in this region because we are more interested in light that reaches the eyes than what happens to light going in other directions. If we look at the whole tree, then the picture on the left can be drawn for every point on the tree that reflects light to our eyes. As shown on the right, all points on the tree reflect light in many angles, so our eyes receive light from all points on the tree. The phenomenon of light scattering in all directions when it hits an object is called diffuse reflection.
Because multiple rays enter our eyes from different locations at slightly different angles, our brain can judge how far away the top of the tree is from us. In doing this, our brain assumes that the light rays traveled to us in a straight line. This approximation is the same as the one we made at the beginning of this section.
Now consider an object that produces its own light rays, like a light bulb. It has rays going off in all different directions, and we can see the light bulb if some of those rays enter our eyes. Our brain tells us where the light bulb is by assuming that the light rays travel in straight lines.
For the purposes of figuring out where something is or how the light rays travel, it is not important whether the object creates its own light or if it reflects light; in both cases the object has light bouncing off it at all angles.
These considerations about how we see things raise an interesting possibility. The only information we have access to for sight is the light that reaches our eyes. If the light takes a bent and twisted path to our eyes, then we will judge objects to be at different places, or to be different sizes than they are. This is exactly what happens when we look in a mirror and see an image of ourselves! Our study of optics is essentially the study of how light given off by objects (whether this light is created by the object or simply reflected) can be manipulated into appearing like it comes from somewhere else. We call this somewhere else an image. This is illustrated below.
Almost every image used in this section adopts the convention of using solid rays to represent real light that actually exists, while using dotted rays to show light rays we imagine to exist. The dotted rays are traced back to the image point by our brains, but the solid rays are the actual light rays that we see. To locate where the image of a point is, trace the light rays back to where they appear to originate from. To figure out the entire image of an object then we must find the image of each point on the object individually.