In the previous lectures, we started with graphical ray tracing. And by shooting the three principal rays, one, two, and three, we learned things like the system magnification and thin lens equations, which told us something about image conditions, and magnifications, and useful properties of a system. When we went to Optic Studio, we saw rays moving through potentially complex optical systems, refracting at every surface. And in that case, those rays were really carrying information about the entire optical field, its phase, and solving Maxwell's equations. What we want to start doing now is, filling in the gap between those, moving from the simple graphical techniques towards more sophisticated views of what a lens is and what it means to do ray tracing? The first thing we need to do, is actually calculate the optical path length of a lens. That's a function that describes how the lens transforms the phase fronts that are incident on it. And remember, optical path length, which we derived or defined before, is simply, N times L, or more formally, the integral of N along the ray, and that tells us something about the phase that's accumulated along the ray. Now, we could do that through a lot of techniques but the technique which is really convenient for doing this calculation is Fermat's principle. And this is an example of where knowing these deeper laws, like Fermat's principle, allows you to do things very, very simply and quickly. So, what we are going to do is, we are going to imagine launching some light off of an object in multiple directions. It goes through a lens and converges back to an image. Fermat's principle states that no matter what direction we shoot the ray, by the time they all get back to the image, the total optical path length, or transit time, should be invariant to the ray direction, it should be stationary using the language of mathematics. And this kind of makes physical sense. If you think about a set of waves converging toward the image here, if they weren't all in phase, we wouldn't have constructive interference at the image point, and actually generate a converging spherical wave there, so that's the physical interpretation of it. The calculation is quite simple. On the left hand side, we're going to simply calculate the optical path length of a ray that goes right down the axis, there's NT, with the minus sign, of course, because this is a minus distance, negative distance, and then continues on to the image N prime T prime. Now, I could have some thickness of the lens right there and that's just a constant, constants don't matter because they don't change the shape of the wave front, so I'm going to set that equal to zero. I could have put a constant here, wouldn't have made any difference. I'm beginning to set that optical path length equal to a ray, to that of a ray that goes up to some arbitrary distance R and back down. So I calculate new distances based on that hypotenuse of that triangle, the same thing over on the right hand side. And now, I add in a correction factor. This must be what the lens does, what extra phase, what extra optical path length it adds in order to make this condition true, in order to make Fermat's principal happen. Because we're going to be paraxial here, or first order design as in the name of the course, we are going to take these angles to be sufficiently small that we can take sine theta and tan theta equal to theta. Snell's law simplifies N theta equals N theta, and in the case of these square root functions here, we'll take the binomial approximation and just keep the first term. So, that's all the same approximation at the small angle paraxial approximation. So, when we do that, we get terms will cancel out, you see the NT here and the N prime T prime, and so we can solve for the optical path length of the lens. Of course, it's quadratic because that's all we kept here. Wasn't really, doesn't really matter what the shape of the lens was, we're only retaining the quadratic terms so, unsurprisingly, we find a quadratic form for the optical path length. The interesting bit is what's in parentheses here, because this is an old friend. That looks like the Gaussian thin lens equation, which we already know, so we can replace the bit in parentheses with one over the focal length. So, the optical path length of a paraxial thin lens is minus R squared over 2f. That's an important expression, we're going to use that a lot. Let's pause for a minute and make sure this makes sense. Remember, I said I dropped an arbitrary constant right in the axis here, the term that we left out here, so I can put any constant I want in here. Constant optical path lengths aren't interesting, it's only their shape that matters. So, let me imagine, I put just S 0, some constant in here, that would represent the optical path length N times L at the center of the lens. For positive focal length lens, F, then we see that quadratically going away from the axis, the lens gets thinner. The optical path length decreases. That makes sense, that's what a lens should do. If, of course, the focal length was negative, the opposite would occur, and it would get thicker going away from the origin, and it does that proportional to 1 over F. So, the quantity 1 over F is powerful enough, is important enough that we need a name for it, we're going to use this a lot. We call it the power. It's just a convenient variable. It's got units of one over meters, and that's useful enough, we give it its own name, in optics, called diopters. And if you have eyeglasses, you've heard of diopters. One diopter of correction is a relatively weak correction, for example. So, a powerful lens, a large number here, that meaning a short focal length, would mean that the change of the optical path length with radius was vast. Powerful lenses are very curved, weak lenses, small power, few diopters, a one-diopter eyeglass lens, for example, are relatively shallow in their curvature. So, powerful lenses are lots of curvature, weak lenses are little curvature. That power concept is useful in lots of ways, but here's the first one. We can restate our Gaussian thin lens equation by replacing 1 over F with this power, Phi, and we can now interpret that as a transformation of the radius of curvature of the lens because of the wavefront. Here, we have a radiating spherical wave and at the lens, it would have a radius of curvature equal to T. It gets transformed into a converging spherical wave with radius of curvature T prime. What this equation tells us, a different way of thinking about the Gaussian thin lens equation, is that the lens simply adds power, in the case of a positive lens, that transforms this radius of curvature, which would be negative in our sign convention, to this radius of curvature, which would be positive. If, for example, the object is just at the focal length, such that the radius of curvature of the emerging spherical wave is equal to the power of the lens, then we get an infinite radius of curvature, which is consistent with having the object at the front focal plane. This is a slightly more sophisticated way to think about what lenses do, they transform radii of curvature.