In making the translation from the paraxial world, in where we lived up to now, we didn't have to pay attention when we wrote down the trigonometric function of an angle, whether it was sine or tan because we approximated both of them as simply the angle itself, and now we do. In a detailed discussion of this, very quickly gets you into lens design topics which are outside the scope of this class. But I want to give you some idea of when you choose one or the other sine or tan and why. This is a matter of fact is the basis of Maxwell's proof that you can't have everything you want at once. So let's start with the first of these and it's called the Abbe sine condition, and it's pretty important and it's one of the ways if you're designing a lens, it's one of the constraints that you can put on the lens to get good imaging. So let's imagine that we have our marginal ray that of course is launched from a field point somewhere, and it goes through the edge of the aperture stop or the pupil somewhere, that's what it means to be a marginal ray. There's some angle that of course that marginal ray makes with the axis in the field stop as it goes through the edge of the pupil. What would our condition on the angle B when we get, let say out to the image plane that of course is ideally conjugate to this field stop? What will be the condition on the angle there, alpha prime let's say. In the praxia world, it would be that the two are given by a ratio, two angles, which is the angular magnification. But we know that we can't use that anymore because we have a trig function here. We can't be using raw angles anymore. It's going to be sine or tangent. So to understand this, we have to use a little Fourier Optics, very little but if you have taken or plan to take that class, here's one of the many important links. Let's imagine we have some object here or the image of an object in the field stop, but now rather than think about it as a point radiating, let's take its Fourier transform to ask what are the spatial frequencies. What are the frequency components that make up that object, and maybe here's one of them right here. In the plane, transverse to the optical axis that sinusoid, which is one of the Fourier components of my object, has a particular period and I'll call that lambda sub x, imagining that this is an x axis, and one over that number, of course, would be the spatial frequency. Frequency is one over period. Well, that sinusoid on the boundary, this is a sinusoid in the electric field at frozen time, and I'm looking at the electric field, would correspond to a plane wave launched off into the space. I can figure out what plane wave by just drawing the crests of that plane wave such that the wavelength, the distance between those crests is whatever the local wavelength this, vacuum wavelength of the refractive index. Once I fix the local wavelength and once I fix this period on the boundary, I can only fit one plane wave. So there's a one-to-one relationship here, and that plane wave has to go off at a particular angle. You can do just, you can draw a triangle in here and you can find out that the period on the boundary is the local wavelength vacuum over n, over the sine of the angle, and that's just a little quick trigonometry. So now, if I think about that I have a whole set of spatial frequencies, one over these periods, that all are made up from the Fourier transform of my object, and I want to transfer that object out to image space and get a possibly scaled but faithful High Fidelity representation of the image. Then it must be the case that it's not the angles that are now related by the angular magnification, that's what we hadn't praxia land. It's at the spatial frequencies must all scale from the object to the image. Because if the spatial frequency scale, then I can do an inverse transform in the image space and get the same, the same object backward or the image of that object. But if those spatial frequencies are different, some of them scale differently than others, then when I do my inverse Fourier transform, I've screwed up the inverse transform. I don't reconstruct the object. So this is the analogy now to the angular magnification we had before, and you can see if we set sine alpha equal to alpha, we recover the angular magnification. But we learned that it's actually spatial frequencies that have to be transferred to the system in order to faithfully reconstruct images and objects. Once we set those equal and divide out the constants, you see that instead of alpha the angle being related to the magnification, it's sine alpha. So this is the Abbe sine, condition for high-quality, high-fidelity imaging from object to image plane. The concept of angular magnification gets replaced by this rule here, that it's the sine of the angles that have to be conserved between those two planes, and that's one of the two ways that you generalize from paraxial to non-paraxial optics.
