So let's do an example of that. These diagrams, by the way, are taken from the Newport lens catalog, it's a classic optical lens supplier to the research community. And these are the quantities they will give you for their lenses. They tell you what they call here the front focal length, that's the distance from the apex to the focal point. And the effective focal length, the actual one over the power of the lens. And of course, EFL and EFL are the same front and back, because it doesn't matter which way you go. But front focal length and back focal length are not necessarily the same, and are not necessarily the same as the effective focal length. So let's look at a Plano convex lens with positive and negative power. In this case, of course, the rays only bend, will have power really at the front surface paraxially. So what that means, and you should work through this is that if we have a ray coming in, we only get power on that front surface and then it just goes right on to the axis. So that turns out to mean that the front focal length in this case, and the effective focal length, the one over the power of the lens, are the same. But the back focal length and the effective focal length are not the same. And the same thing happens over here on the negative lens. So we'll run some homework problems and quizzes on having you do this yourself. But these are the kinds of quantities that you'd get from manufacturer. And they show you how really that, when you're thinking about the equivalent paraxial properties of a lens, a thick lens now. You have a focal length, which is before the only quantity we have. Now we add these two principle planes as the next set of quantities which describe how a thick lens looks like a thin lens. So how would we trace through a real thick lens using our graphical techniques, or later on our more sophisticated numerical techniques in the paraxial limit using our thin paraxial techniques, through this thick lens? So here's how to think about that and we get a very, very simple rule, but let's not just take it as a rule, let's figure it out. I'm using the same two rays I had before, 1 and 2, but I'm going to image turning number 2 around backwards. So, it's exactly the same ray, but of course, light goes both ways, reciprocity. So I'm going to imagine now rays 1 and 2 from before are both coming forward at this lens. Ray 1 bends at the first surface, bends at the second surface, hits the axis at the back focal point. Ray 2 comes from the front focal point, bends at the front surface, bends at the second surface and comes out as an axial ray. Now, let's think about this in terms of virtual objects and images. I see rays 1 and 2 here apparently converging to a point right here. So this is my virtual object. I have rays 1 and 2 converging to a virtual object which appears to be right at the front principal point. Conversely, if I look at whether 1 and 2 is coming out of my lens, they appear to come from a virtual image that's at the rear principal plane. So this suggests something kind of interesting. If I wanted to, for example, let's trace ray 2 through this system. But I'm trying to use not the real curved surfaces, but just principal planes and focal lengths, this equivalent, that thick paraxial thin lens system. I would apparently take ray 2 in, remember, I don't have any glass in my paraxial ray trace so it just continues to the front principal plane. It then teleports over to the rear principal plane where I then apply all of the lens power, capital, or whatever the focal length. So I bend this ray 2 at this plane, of course it's coming from the front focal plane so it comes out axially. Similarly, ray 1 comes in, it goes through this point because I don't know this surface exists anymore, I've replaced it with it's equivalent description. It goes right to the point P, teleports to P prime, and then bends at P prime, of course, because it comes in axially, goes down through the rear focal point. The rule is that then you take your curved surfaces of glass, you do this construction to find the principal planes or for a lens from a catalog, you just take the principal planes that the catalog manufacturer gives to you. And if you would like to retrace a system with that real thick lens in it, you replace no matter how complex it is, we're sowing singlets that it could be a multi-elemental system. You replace that whole system with two principal planes and a focal length. Throw away the real glass now, this is how you go back to paraxial thin line tracing from a thick optic. And you shoot rays in following your thin lens conventions. When you get to the front principal plane, you teleport those rays with their positions and ray angles intact. And then at the back principle plane, you apply the length vocal length there, apply the lens power, as if the thin lens lived there. Of course if you happen to be going backwards, the same convention applies, except you teleport from P prime, right to P. So formally, the principle planes are conjugates and they know they're conjugates because an object here is apparently in focus since I've got an image here. So that's our definition of conjugates and their unity plus one magnification. So the formal phrase for the principal planes, another more sophisticated way to think of them is they're the conjugates with unit magnification. Ray tracing, you simply replace the lens with P and P prime, and you simply teleport between this gap. Now warning, sometimes P prime can be over to the left of P. These planes can be anywhere for unusual negative lens systems, etc. But it doesn't matter, you just follow the rule. You shoot your rays in until you hit P, you teleport to P prime, you'll apply the lens as if it lived on P prime. There's occasionally a useful concept called the Nodal Points. And so I'll mention these so that you have the jargon. The nodal points are the case that your incident and exiting refractive index are the same. The nodal points are simply the intersection of the principal planes with the optic axis. Sometimes you use that, but the other reason I thought it was maybe worth mentioning is our very first ray in graphical ray tracing was the one where the ray came in and it hit the lens on axis. How you replace that concept for thick Gaussian optics is the ray comes and it hits the front principal plane at the axis or its hits the nodal point. Your tailboards over and exits the course at the same angle. So, this is how you'd start a retrace of a lens, thick lens replaced by its principal planes if you are doing graphical retrace, because you always draw this ray first, because it's the easy one. Let's do a little more complex example to emphasize how we can expand and collapse optical systems with this concept. We can take a single lens. We can take multiple lenses and collapse them too nothing more than P and P prime and that equivalent focal length. We can take that equivalent focal length of a thin lens and expand it out into multiple elements. This is kind of like an accordion for our optical system, going between the thin world and the thick. So we've so far only dealt with a singlet. Now let's think about two singlets, this is an air spaced doublet in the jargon of optics. Let's see how we replace that two line system with a single equivalent thin lines described by principle clients and focal lengths. So if I was doing a retrace of this, I would come into this lens with that Axial ray, it's my ray number one from the previous slides. The ray would bend to the front circles, it would bend again in the back circles. And if I projected that rays back, I would find the equivalent principal plane of P1 and P1 Prime So I can replace this thick lens with P1, P1 Prime and effective focal length F1. Tracing on through, that ray would continue. I'm going to continue with this formerly exit right here, and I'm going to now bend it through the second lens. Well, if I have, let's say, a principal plane and equivalent focal length distance, equivalent focal length for the second lens. I could trace this ray through by teleporting it over to the second or back principal plane, and then applying the power of the second ones. And finally, the ray would come out and it'd hit the axis. So this point right here is the back focal point of this two thick lens system. So I can put a little circle around that and label that on my diagram, there's my back focal point. And it's pretty obvious at that point, but this back focal distance has little to do with the actual effective focal length of this two lens system. Because really this is a thick line system, but I don't know where to measure back to. But I can find where to measure back to by projecting this ray back through the system until it intersects the projection of the incident axial ray, and that is now P Prime. My back focal plane for this two lens system. I could do the same thing on the backwards as I've drawn this system it's essentially symmetric, so I would find that P lived somewhere around here. I could then take both of these pieces of glass away and replace them with P and P Prime and this distance, which is the effective focal length, have this two lens combination. If I begin to move these lens around in some sort of zoom system like the camera lens we looked at in the beginning of this class. Then you can see how as I move this lens around it will change where this ray intersected the axis. That would change the effective focal length of this whole system. And now you can start to see how you could make a lens with a variable focal length, and actually design and understand it. You could derive equations out of this. The graphical retracing is a somewhat tedious way to do that, but it would be a great way to start getting a feel for how the system would work, what its limits might be, etc.