Now to understand what's going on here, let's think about.

If, if we were to a Fourier transform of this original sample,

there would be at least two important waves.

One, the, the major wave would have a maximum around the top

of this sample and it would have minimum at the extremities.

The next high resolution Fourier component would have three maximum here,

representing these three peaks and so it would be a higher frequency component.

Now, if we look at the images formed by these two electrons,

the first electron would carry its low frequency

component in just the right position.

The higher frequency component would also be in just the right position.

Now the second electron,

the low frequency component would be shifted

somewhat as would the higher frequency component.

And when we compare these two in detail,

we see that the low frequency components match thoroughly well.

There displacement is small compared to their wave length.

And so they match fairly well to give a major bump in the sum, but the high

frequency components are completely shifted with respect to each other.

This one and this one are completely shifted.

And so, it together,

those two high frequency components hardly contribute anything.

And as a result, the sum image, the net image that we record has the low frequency

components, but it's missing the high frequency components.

In other words, because the electrons were coming down

at subtly different angles, the image became blurred.

And when an image is blurred, it affects the higher frequency

components much more severely than the lower frequency components.

And so one of the reasons that the contrast transfer function

of an electron microscope is damped at higher spatial frequencies

is due to only partial spatial coherence of the electron gun.

Remember, spatial coherence was a measure of how uniformly each

electron that the gun produced, whether or

not it comes down the microscope in exactly the same direction.

If some of the electrons are coming down at a slightly different angle,

then the pictures that each one produces will be slightly shifted with

respect to each other and that's partial spatial coherence.

And because of that, we can actually plot the,

the impact of that shift as a function of spatial frequency.

And if we do,

we find that those small shifts don't matter much at low spatial frequency.

But as we get to higher and higher spatial frequency, they matter more and more and

more until the high frequency components are essentially eliminated from the image,

because each electron is not coming down in exactly the same direction.

Remember that each electron individually comes down the microscope and

contributes to the ultimate image and

so the next electron that comes, contributes its image.

And if these images aren't perfectly superimposed,

the sum becomes blurred and that dampens the high frequency components.

Now, in an analogous fashion, if the different electrons

coming down the microscope come with different energies,

so let's suppose the first electron comes this way.

And if we draw the objective lens,

if this first electron has energy e,

then the scattering that occurs at

the sample will be focused by the objective

lens to form an image at a particular plane.

Let's call it here.

[SOUND] The image plane.

If however, the next electron comes down the microscope with a different energy.

Let's say, e plus delta.

It is scattered from the sample again, just like the first electron.

But in this case, because the energy electron is different, if the energy

is higher, it will be, it will be less focused by the objective lens.

And so, it won't be focused until a plane that's lower.

Okay.

So this is the image plane for a higher energy electron.

Now of course, there is in a microscope, a detector further on down.

And the detector as we have discussed before is conjugate

to some plain up here where the image is being formed.

And what we see is that the, the first electron with energy e,

it's going to contribute in over-focused image to the detector.

But the second electron with higher energy is going to

contribute an under-focused image to the detector.

And so the net image is going to be the sum of different

images with different focus values.

And if you were to plot, let's plot

the CTF as a function of spatial frequency

again for the lower electron e.

If its contrast transfer function, looked like this.

The contrast transfer function of the higher energy electron,

which is now under focused in comparison is going to look like this.