Let's say, you've built an optical system or designing an optical system, where you've used one of these prisms to get an image upright, let say. You will have a problem then, it just sort of making a pencil and paper diagram of this. In that the light is coming out of the plane, that it's all folded, it quickly becomes a three-dimensional problem. You might be trying to figure out, given the numerical aperture, and the field of your object, how big do I need to make these prisms, so I don't been yet, I don't clip any of the race. That's so hard in 3D. So, there's a simple technique to deal with that and it's called the tunnel diagram. So, after the the bouncing pencil technique to figure out what's upright and what's left, and how's left right changed, the tunnel diagram allows you to now do a graphical, or other simple geometric optical ray-trace without having to live in three- dimensions. So it's a technique that makes designing with prisms much simpler. So let's take this particular arrangement, this is another cover arrangement called the Porro prism. If you've ever held a set of binoculars, and they have kind of some bumps along the tube, the bumps were to hold these prisms. Because what they do again, is they flip up and down. So, they are used to get an upright image for the eyeball. So, the idea is we simply take two right angle prisms, they're glued together on their hypotenuse here, and make one, two, three, four bounces. But notice unlike the previous ways, where we could upright an image with an even number of reflections, here, we continue on the same same path. That's the useful thing instead of binoculars, it kind of has to be a linear. I have to have the 0.2 binoculars out at the world. I have to have the image continue on towards my eyeball. So, I can't make a 90-degree bend. That's a pretty good example of that sets awkward to do paper and pencil design to figure out, where are all these apertures and do I hit them with rays from a real object. So the first thing to do is you unfold the system. We take every time we have a reflection. What we do, is we simply take the ray, and we unfold it. We imagine bending it, so it was straight. We duplicate the whole prism system that the ray is actually going through. So in the case of this first prism here, we come through a triangular piece of glass that's from the hypotenuse where you enter to this leg where you first hit. Then, we add the next piece of glass, which is from a leg to leg of this triangle, and that's right here. But we place it up on the same optical path as the ray sees it, but with the ray going straight. Then we get to this second band and we're going to go out to this last little triangular bit of class from the leg to the hypotenuse. That's right here. What we did is we unfolded this right angle prism and turned it into a cube. You can see of course, here's the optical path which is now straightened out. The optical path, every time it hits one of these surfaces, instead of bending because it reflects, simply goes from the glass to the same glass, so it has no optical effect. But now, we have nothing but a set of square apertures we have to get through. Now, that's much more easy to design. Here is the second prism, notice that it's in a second different plane, and so, we see these sort of imaginary lines, where we formally reflected in a different plane. Now, we end up with this rectangular block of glass that we're going through. That's a much easier thing to design with. Perfect. Now, we can reduce it to our traditional two-dimensional diagram, we don't do 3D diagram here, and we see that oh, we're just going to refract through the glass with some tilted right here. That would have been pretty hard to draw for this fully three-dimensional system. Actually, you can do one more thing if you want to make this particularly simple. If you remember the concept of reduced distance, that we can at least paraxially, when we're doing design, take any block of a glass, so let say, thickness 2A here, and we can make it shorter by the ratio of the index to air, and then replace the index with air, maybe index one. Now, we get a system that the ray comes out at the same angle, but we don't even have to deal with the refraction. So, these tunnel diagrams because this is now like a tunnel that the light has to get through, are the ways to simplify ray-tracing, they're particularly useful when you're doing your first order design.
