So, therefore, it must be the case that we just take all of our paraxial equations,

and instead of using theta or u,

we just put in sine theta or sine u, and we're done.

Unfortunately, that's not how it works.

So this is a little bit less well-known because it's typically not as important.

But it's still important,

and it's called the tangent condition,

and it was actually due to Airy.

So, you might have guessed since we used the marginal ray in

the last derivation that there might be something similar based on the chief ray,

and as a matter of fact, there is.

So, now, let's think about the chief ray.

It, of course, goes through the axis in the aperture stop and

equivalently all pupils and goes to the edge of the field stop.

So, if this particular field stop here,

for example, was the object,

just to make life easy,

we can then ask what happens if we go into the entrance pupil,

go through some arbitrary optical system,

pop out the exit pupil because that's what rays do,

we may come out now with an angle alpha bar,

the angle of the chief ray,

which is different between the exit and the entrance pupil,

and of course, we're going to come up with a different object height.

So, now, let's say that we want to image our object,

and we're not so concerned about reconstructing

a high-fidelity copy of the object as we did

before by looking at the spatial frequencies.

Now, let's think in space.

Let's say we do have that grid of lines like

graph paper here or we have a grid of point sources.

We would like if they're equally spaced in

object space than to be equally spaced over here at the image.

We want no distortion.

Well, that's pretty easy to figure out what to do.

We can relate the angle of the chief ray in the entrance pupil

to the height of the object and

the angle of chief ray in the exit pupil to the height of the object.

That's going to be given by the tangents of

those two angles and the object and image distances, t and t prime.

So we can immediately say, "Well,

we want the ratio of h prime to h to be constant."

That's like saying we want the magnification,

and the last time we were talking sign rule,

we were talking to angular magnification.

Now, we want the magnification to be constant.

So we will simply set h prime over h to be some constant m,

and what we find for a particular geometry of optics,

object distance and image distance,

that now it's the tangent of these angles that must be constant.

Now you see where the problem comes in.

Now, this isn't the same angle we had before.

But the point is sometimes you need to use the sine like we just saw,

and sometimes you need to use the tangent of the angle.

That means you just can't go through

our previous formulas and blindly put in trigonometric Functions.

As a matter of fact, this is the basis of a proof that

when sine and tangent are different than one another,

that is out of the paraxial regime,

you can't have high-fidelity imaging simultaneous with zero distortion imaging,

which is why we can't have a perfect image and condition.

But this gives you some idea of how you generalize from the paraxial to the non-paraxial.