2:15

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.

8:59

And if we were to add these together to see what kind of net

contrast transfer function there was in an image that was

the sum of many images with different defocus,

we would see that the average CTF starts out fairly strong.

But at this point, it starts to be weakened by the fact

that these two curves are no longer in phase.

And essentially, the signal is damped at high spatial frequency,

because the individual electrons contributing to the image

are contributing images with different defocus value.

And so again, the result is an envelope that describes how

the signal is dampened at higher spatial frequency.

And this is the result of partial temporal coherence.

And so there are at least two envelope functions that

affect the CTF in the electron microscope.

First, there will be an envelope function that we'll call the envelope

due to partial spacial coherence, which reflects the fact that

not all the electrons are coming exactly the same direction down the microscope.

In addition, there will be another envelope that reflects

the fact that not all the electrons have the same energy and

this we will call the envelope due to partial temporal coherence.

In addition to this, there are other miscellaneous problems that cause

an electron microscope image to lose some of the high resolution detail and

each can be characterized as en envelope function.

The net envelope function is the product of all the individual envelope functions.

So the net envelope function is even more severe than any of the components.

So this we'll call the total envelope function.

And it's the total envelope function that defines

how the contrast transfer function is damped more and

more at high spatial frequency.

Now again, what the contract transfer function tells you is how much of

each spatial frequency is transferred, delivered into the final image.

So for instance, this spatial frequency, we might see that because

the contrast transfer function here is approximately negative 0.5,

half of the signal that is present in the sample,

half of the strength of that Fourier component that's present in

the sample is actually delivered into the image.

Here if the CTF reaches a value of say, negative 0.9,

90% of that signal is delivered into the image.

But then as it oscillates, we get to a point where at this spatial frequency,

none of it is delivered to the image.

And so on, as we progress.

And we see that the envelope functions also contribute to that damping.

For instance, if this, at this spatial frequency,

their total contrast transfer function due to oscillations and the damping.

Due to partial, spatial, and temporal coherence and

other factors causes the CTF to be only 0.2.

It means that only 20% of the signal present in the sample

at that spatial frequency is being transferred into the image.

And this is why, when you look at the power spectrum of an actual EM image,

you see that these oscillations of the CTF or the ton rings decrease in

intensity as you go to higher and higher spatial frequency.

Their intensity is lost, because if we were to plot the contrast transfer

function not only is it oscillating, but it's also being damped at higher and

higher spatial frequencies until the variations are almost undetectable.

Now these envelope functions also depend on defocus.

So going back to this picture,

a picture of viruses at high defocus delta z and

this picture at low defocus delta z.

If I were to plot the contrast transfer function for

this image as a function of spatial frequency,

the envelope functions that limit the contrast

transfer function would attenuate more in the high

defocus image than they would in the low defocus image.

14:33

And as a result, the CTF here in addition

to oscillating more rapidly would also

attenuate more quickly at high spacial

frequency than in this case, closer to focus.

So the advantages of taking pictures far from focus is that

you have stronger, low resolution frequencies here.

So, it's easier to see your particles.

But the high resolution details are more difficult to recover for two reasons.

First of all, the contrast transfer function is oscillating rapidly.

And second of all,

the envelope functions dampen the signal at those high spatial frequencies.

In comparison, images at close to focus, the high resolution details are going

to be more easily recovered, because the oscillations are slower and

because the envelope functions are more generous.

But this comes at the cost of a loss of low spatial frequency signal,

which makes it difficult to see the particles.