We're now going to start understanding

the aberrations of lenses that don't depend on color.

Up to now, when we do our first order design, the paraxial approximation.

When we approximated sine theta as theta,

we made all of our lenses perfect.

At least if they were infinitely thin.

That is, if a thin line deflex a ray

like the power times height as we saw in our thin lens equations,

then you can have perfect imaging.

Of course, lenses are not infinitely thin and sine theta does not equal theta.

So, in almost all real-world cases you have to

deal with the fact that you cannot actually do that perfect imaging,

but that's just a formalism you use for design.

So, let's start out with some basics.

It turned out a lot of people thought about

this and Maxwell of Maxwell's equations fame was one of them.

He wrote down some useful way of thinking about,

what would you mean by aberration-free or perfect imaging?

He wrote down three laws.

One of them is that all the rays that come from a point at

an object can converge back to a point of the image,

something we've taken for granted up to now.

We're going to call the deviations from that ray aberration,

the rays don't go where they're supposed to.

Another condition would be if you have an object's surface that's

a plane like a piece of paper that you're imaging let say,

that the image would also be on a plane.

That's also something we've taken for granted,

but it turns out lenses don't like to do this and it's

often the case that the image surface wants to be

curved and that would be a problem if you have to put like

a digital camera there that's flat as it can't be in focus over the whole field.

We'll call that field curvature.

The third thing is that if you imaged maybe on your piece of paper,

a perfect grid, a piece of graph paper,

you would like the image to also come out as a rectilinear grid to have

the lines that were straight in the object be straight

in the image and separated by equal space distances.

It turns up that's something that we've taken it for granted up to now.

Real lenses also don't do and that's called distortion.

So, one of the reasons he, Maxwell,

wrote these three rules down is then so he could do a nice proof that it's impossible.

That with real optics you can't actually satisfy all these constraints.

Our goal and as designers is simply to get close enough.