Like astigmatism, it has this character of rh squared,

still a third-order aberration,

the sum of the powers here is three.

But it's got some different trigonometric functions

and that's just generated from that ray aberration polynomial,

where we took all any possible function,

took the derivative of the x and y ray coordinates with respect to

rh and theta and kept the terms that obey the symmetry conditions.

The difference between Petzval and astigmatism,

and they are actually quite related,

but the difference here is that we see the same behavior,

it looks like a defocus,

that defocus depends on h squared,

and we'll look at that in more detail now,

but the two terms are equal in x and y.

I put a three here before in the case of astigmatism.

That meant the defocus wasn't the same in x and y.

Here it is.

So, what does it mean? What does it do?

Well, let's imagine we started with a point with h equal to zero,

like all aberrations say of spherical,

that means we get perfect imaging if we only had Petzval because if h is equal to zero,

x and y, the ray errors are equal to zero.

Hey, life is good. Now, let's move our object point off some small distance each.

Our image point is going to move off

the axis by whatever the magnification of a system is,

but there's going to be defocus, how much defocus?

Before times h squared.

That's going to be equal to,

I think it was a2, our defocus term here.

So, that would give us for all our rays that we will launch into the pupil,

a linear ray aberration term and that's a defocus,

and it's the same in x and y.

So, we would find once again, a perfect focus,

it's just shifted, it's defocused,

it's not in the perfect image plane.

If we shifted to even larger h,

we'd find quadratically more defocus.

So, we're going to find a focal surface here which

where the rays actually do come to a perfect point,

if this is the only aberration we have,

issues they don't do it at the right place,

they do it on a curved,

quadratically curved in this cubic aberration view of the world,

this third-order aberration, we find a quadratic focal surface here, the Petzval surface.

This is a pain because if I have a piece of film or I have a silicon detector,

they are not universally,

but almost universally flat and I can't bring all of

my light into focus even though it's in focus

somewhere where the focus is depends on the field arc.

So, that's called field curvature in a general sense and you can

see it's related quite closely to astigmatism and we'll come back to that.