In our previous lectures, we introduced first the simple thin lens equations.

Which relate object and image distances to a single lens.

We then graduated to the y-u per axial ray tracing, in which,

in a tabular format, we pushed a ray consisting of a height and

angle descriptor through a system.

And most recently, we introduced an a, b, c, and d matrices.

Which takes that tabular format, the system of linear equations, and

captures them as two-by-two matrices.

The real advantage of that last approach is we can now cascade

an entire system description up.

Multiply out those matrices either numerically or symbolically.

And end up with descriptions of entire optical systems.

That's a very powerful approach for first order design.

And so here we're going to explore how we can use system descriptions

given by these matrices to put constraints on a system.

And it turns out the conjugate matrix N, that we defined earlier,

is the way to do that.

And each of the four terms in that matrix have very important properties.

So we're going to go through those one at a time.

So I've written out here the matrix equation for N.

And remember, what it does is it relates the object given by its object height and

the angle coming off of the object to the image given by the image height and

the angle coming in to the image.

So what we're going to do is look at each of the four terms of N.

And set them = 0 one at a time.

And we'll discover that enforces a constraint or a condition on the overall

conjugate condition on the system that we're looking at.

So let's look at the first one.

If the term N12 = 0, then just looking at the top equation here,

we see that the image height y depends only on the object height, y0.

And it has no dependence on the object angle.

So pictorially, that's the situation that's shown here.

We launch a bunch of rays from the object with arbitrary angles.

And they all come back to the same image height, independent of the angle.

And that, of course, is our imaging condition.

So if we calculate or someone gives us a conjugate matrix N,

all we have to do to enforce the system to be in focus is set this term = 0.

And that's an example of how we'll use this for design.