Up to now, our first-order ray tracing has involved the distance of the object and the image from the lens. So if you're tracing a multiple lens system, you would find that as rays move through the system, you'd continually have to find where the image of lens one would be. Then relate to that to where lens two was, and you'd find it's actually quite awkward. So we now want to move towards a more sophisticated view of ray tracing, where we simply deal with the distances between elements. We're really only concerned with where is the object and where is the image, at the very first element, and the very last element in the system. In the intermediate spaces, we simply want to direct the ray, what's its height, what's its angle, as it moves through the system. Notice the height and angle at least if we're in a single plane which we'll deal with here, we'll assume that we have rotationally symmetric system so we can stay in a single plane. Only two variables that specify array, which are its height and angle. So we now want to move to a system where we track those two variables for array as it moves all the way through a system. So let's derive those quantities. Let's start with our single lens imaging system, which we're familiar with. We'll define for the ray a paraxial angle before and after the lens. We can relate those angles to the height of the ray at the lens y, and the object and image distances through simple triangles. We're quite used to that. If we then write the thin lens equation but replace the object and image distances, now with these triangular relationship up here, such that we end up with angles and the ray height but the lens, we get a new equation. This equation actually is in a sense gone backwards, because here we have, n' u' after the lens, equals n u before the lens. That's just our paraxial Snell's law. We find that the modification when you refract through a lens is that the right angle has changed, proportional to the height, that the ray hits the lens and proportional to the power of the lens. So we'll call this our refraction equation. It gets rays across lenses. The height y stays the same, but the angle changes from u to u prime. Then, it's very very simple geometry. Now, to simply move the ray to some new surface at a distance d away. Again, we're going to move to d, the distance between surfaces, instead of t, the distance to objects or images. So just a little bit of first-order trigonometry. It says that, "the ray height changes like the ray angle times the distance the ray travels." Notice, I've reminded you here of the sign convention, that if we're going from a surface k forward, that's d-prime moving forward a positive number. But if we were to look at surface k plus one and ask how far backwards do we have to go, that would be d no prime, because we'll always have primes after the surfaces, and no primes before, and that would be a minus quantity. So these two values are equal, though of opposite sign. So with these two things, these two equations, we can refract changing the angle at a lens, keeping the height of the ray the same, then we can keep the angle of the ray the same as we move between upticks and change the height. We'll call this the transfer equation, because it transfers us between surfaces. Now, most of the time we have rays traveling through vacuum or air. So we'll take the refractive index as one. But if we're within a lens, imagine we're treating two surfaces that are the front and the back of the lens, then we need to keep track of the refractive index. That actually can sometimes be awkward to keep track of all of the refractive indices, and you can. Those equations we just showed you have a refractive index in them. But as an alternative approach, one can actually scale the refractive index out of the equations. I actually like to do this in my own work, but it's up to you. So let's start with Snell's law, and sine theta, equals m prime, sine theta prime, go to our paraxial approximation. Notice the variable u here, we'll use to remind ourselves that we're in a paraxial regime, so you use the paraxial version of theta. So let's define sets we see in u, here seems to be an interesting quantity. Let's define u hat, a scaled angle would to be the refractive index, times the paraxial ray angle. Second, let's look at our Gaussian thin lens equation. Here we notice that wherever we see a distance, it's divided by the local refractive index. That suggests that maybe we should scale distance by the local refractive index. So we multiply the ray angle by n, we divide distance by m. If you make those two substitutions and rewrite the last two equations, the refractive index drops out. We now get a ray angle changes only by refractive power of the lens. The u hat going to be u hat prime here doesn't change because we've embedded Snell's law, the dependence on refractive index, into the definition of the scale of u. Here, the refractive index that we multiplied into the end and divide it out of the d, actually cancel out, and so this equation is written unchanged. So when I'm working, I actually like to work in the scaled coordinate systems. I don't pay attention to refractive index, and I work with these variables. Then when I'm done, I come back and simply scale all distances and angles by the refractive index. However, if you don't like that, just work with the previous equations that had refractive index included in them.