Now, in addition to oscillating, the contrast transfer function of an electron microscope is damped at high spatial frequencies. And so, if we plot the CTF as a function of spatial frequency. We see it not only oscillating, but also being damped at higher spatial frequencies. That's a contrast of 1 and minus 1. We see that it's damped and the shape of the damping. We call an envelope function. [SOUND] And now, I'd like to talk about the causes of this damping. So let's imagine that our sample has a profile something like this. Okay. A high density, a peak and a subpeak next to it. And then let's imagine that we image that with an electron microscope and one electron is heading straight down the column in this direction. And because of that, it will produce an image on the image plane. That closely resembles the sample. But what happens at the next electron that comes down the microscope, comes down at a different angle than the former? And so instead, it passes at an angle say, like that. Then it's going to produce an image on the image plane that looks like that and these two images together are going to look something like. 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. 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. In this case, the envelope functions would be more generous. 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.
