All right, now let's begin to relax one of our approximations, which is to say, let's relax the thin lens approximation. Let's figure out what the power of an actual curved piece of glass would be because that's what we want the design in the end is real-world optics. So to do that, let's just take a single-surface bounding indices n and n prime with a positive radius of curvature by our coordinate system R. Let's first find the height of this surface measured from the radius of curvature and this is not obeying our sign convention, just the distance from the rays of curvature to the surface. Of course, the surface is a sphere so that's an easy thing to do. Once again, we will make our paraxial approximation that is we're still paraxial, just not flat surfaces anymore, and take only the first and second terms or the lowest in quadratic terms of the square root. So that gives us a parabolic approximation of your spherical surface. Now, let's calculate the optical path length through this not-flat surface. And by optical path length theorem we just mean projection and integration along Z axis, so we are just going to calculate nL along the Z axis, and some distance off of the axis, perhaps on this line here for a total thickness of some arbitrary constant d measured from the origin. So we have nL, the distance in air, and then n prime L prime, the distance in the glass. Because we have this expression, we know what L and L prime are, they're that various pieces of that parabola. We can factor out the terms there, we get a term that's a constant and constants don't matter because they don't transform the shape of the wave front. But the one that's left is very interesting. Notice that we've, once again, found an optical path length and it's quadratic, it depends on R squared. Since we already have an expression for the optical path length from a few slides ago and it's R squared over 2f, we can now set this quantity equal to one over 2f, there's one over two there, we find a really important expression. The power of the lens, one of the focal length, is one over the radius of curvature times the difference in the indices. Again, it's just convenient to have the variable and the upside-down variables, let's define one over R as the curvature, that's in units of inverse meters. So the power of a surface is related to the curvature times the difference of the indices or a fraction of the two surfaces that bounce. That's the beginning of what will allow us to design real optics from our thin paraxial first order designs. Now, I mentioned that Fermat's principle was the sophisticated way to design the power of the lens. Perhaps, more fundamental, less-sophisticated way is use Snell's law. So, let's stop for a minute and do that, it's slightly painful and you'll understand from this why you like Optic Studio to do lots of calculations for you just to confirm you get the same answer. And then we begin to get us coordinate systems and useful insight when we do a little more sophisticated ray tracing by hand. So, here's that same surface with positive radius of curvature bounding materials n and n prime. I'm going to shoot it right off the axis with angle u that turns into array of u prime. I'm going to have to draw a normal to the surface up here and define rays i and i prime, our angles, i and i prime, that are my typical Snell's law angles of incidence and let's say I hit the lens at some height y. That's enough for me to solve the problem. So, here in the paraxial version of Snell's law where i and i prime have replaced sine i and sine i prime, I can go figure out what I is relative to u. Alpha is the angle that the normal to the surface makes with the optical axis. So I can replace i with more fundamental variables that relate to my ray. And then I can go use some geometry or similar triangles to figure out that u is y over t and alpha is y over R and so we can make those replacements, many things cancel out. And when I cancel y, I should be able to cancel y because in the paraxial limit, it should not matter what my ray height is, that's what it means to be paraxial. And look at that, I get exactly the same Gaussian lens equation that pops out again and the same power for the surface, one over the radius times the difference in their refractive indices. So, this is not the sophisticated way to go at it, I think Fermat's principle is much simpler but it confirms that Fermat's principle agrees with Snell's law, which is nice. So, that's our first step in transforming thin lenses into curved pieces of glass. The thing that connects those is the concept of the optical path length and from that, we can derive, then, how the optical path length of a curved surface relates to the Gaussian thin lens equation.
