So we've learned that each of the four terms in N, the conjugate matrix, carries information about the system where the focal planes is the system in focus. It turns out we can write the conjugate matrix in an extremely compact form using the logic that we've just gone through, and that form is important enough that we should know it. So I've written out the conjugate matrix here between an object at plane 0 and and an image at plane plus one. And let's assume that we really do have a conjugate condition and so we know that N12 is equal to zero. If we now look at the top equation, we find that we have an object height, y, multiplied by a number and that gives us the image height, yk + 1. We already have a name for that number. It's the magnification. Now notice the script here is the magnification, not bold M, the system matrix, so it's the older version of the symbol we're used to. So that means we can simply write down N11 is the system magnification. That's pretty important obviously. And there's two ways we can figure out what N22 is, one by observation. You can find that the determinant of all these matrices is equal to one if the refractive index that you're exiting into is the same refractive index that you started in. However, another way to do that is to just think about launchig rays here and hopefully you can do this in your head now. Let's imagine we launched a ray off of the axis of the object as we set y0 equal to 0. Now we have a relationship between the object angle, a single number, and the image angle. We have a name for the relationship between the image angle and the object angle for an axial ray. It's the angular magnification which we know is one over the transverse or regular magnification. And of course, if you multiply these two number together, you get one and proving indeed the determinate of the conjugate matrix is one. Finally, if we think about a case where we launched an axial ray as I've shown here, so that means that the initial angle is zero. The ray moves through the whole system and finally comes out at an angle. The ratio between the angle that you exit the system and the right height that you shot into the system for an axial ray is the focal length. Pause for a minute and draw out for a single lens with your graphical retracing and prove to yourself that that's true. Remember, we have a thick lens system that's equivalent now. We're using similar ideas that we used in analyzing thin lenses, but now we've got a thick system. So if you did that, you find that the distance from the thin lens to the point you intersect the axis is the focal length and that means that the exit angle is indeed just given by the ratio of the focal length and the incident ray height. This is the definition of focal length. Focal length is not given by where the light hits the axis because remember for thick lenses, you have to measure back to the principal planes. How do you know where the principal plane is? The first thing you do, and we now have the tools for that, is you shoot array through the system that starts out with zero incident angle. You find the exit angle. The ratio between those two is the focal length, and now you measure backwards one focal length from the back focal plane, and that's your back principle plane. So overall now, we have learned that the N11 and N22 terms are related to the magnification and the N21 term is related to the effective focal length or effective system power, and that means we can write this conjugate matrix in a beautifully compact form. Assuming we are actually at the conjugate condition, this term is zero. The two diagonal terms are m and one over m, and the final off-diagonal term is minus the system power. So if you then wanted to use this you could, Symbolically or numerically trace your way through a complex system with the abcd matrices and then set the result equal to this right here and that gives you the magnification and the system power, which is two pretty useful things about the system. You already know where the image plane is because you got there by setting this term equal to zero. That's a fair number of the questions we wanted to ask about a system or number, but now we're getting all of those from one matrix analysis of the overall [INAUDIBLE] system. So again, I always like to go back to basics to make sure that I believe things. So if I go ahead and calculate N for a single lens, that's a single transfer, a single refraction, and one more transfer, I get this matrix here. If I set this term equal to zero, I notice the Gaussian thin lens equation pops out, and when I do that, these terms indeed turn into the magnification and the overall lens power. I want you to go through that algebra and make sure you believe it. So the point is, we now have a tool that allows us to symbolically, for example, look all the way through a lens system, capturing all the degrees of freedom, the focal length of the lenses, the distances between them. Then through setting terms like this, we can constrain the magnification, we can constrain where the image plane is. We can constrain the overall system focal length or power. And that takes the degrees of freedom we had, focal lengths and distances,, and reduces them to a smaller set of degrees of freedom that give us the design properties we want. That is the process of first order lens design.
