1. Lenses, apertures and resolution

2. Lenses in TEM vs. optical microscope
An electron beam is used to view the sample
Electromagnetic lenses made from metal
To magnify and focus our image we change the current running through a coil around a soft metallic core, which changes the strength of the resulting magnetic field surrounding the electron beam
Light is used to see the sample
Glass lenses
We physically move the lenses up and down to change the focus and intensity of the image
To increase the magnification we have to change lenses

3. Ray diagram
A ray diagram is a diagram that shows how light rays pass through a lens
Electrons passing through the middle of the lens are unaffected while all other electron paths are bent when passing through a lens
The strength of the lens determines where electrons are focused (stronger lenses have shorter focal lengths)
The focal plane is where parallel rays intersect after passing through the lens.
The image formed by the lens is rotated by 180°

4. Principal optical elements
Lens plane (the plane where the lens is located)
Object plane (the plane containing the viewed object)
Image plane (the plane containing the image, which always lies below the lens)
Focal plane (the plane in which the parallel rays are brought to a focus)
1 𝑓 = 1 𝑑 𝑑 𝑖
For convex lenses the focal plane lies behind the lens, while for convex lenses the focal plane lies in front of the lens
f – distance of focal plane from the lens plane
d0 – distance of object plane from lens plane
di – distance of image plane from lens plane

5. Magnification and Focus
Magnification in convex lenses can be described as: 𝑀= 𝑑 𝑖 𝑑 0
In a TEM we change the magnification by changing the strength of the lens without actually changing the lens
If we make the lens stronger than we shorten the focal length, which means that the magnification becomes smaller
Compared to other microscopes, TEM can also gain information from images that are out of focus (overfocused, underfocused)
Typically we tend to operate the objective lens at a fixed strength, while moving the object plane closer to the lens thus making d0 smaller and M larger
In theory we could achieve unlimited magnification, but unfortunately there are other factors limiting the resolution of the microscope

6. Electromagnetic LENSES
Electromagnetic lenses are constructed from two parts:
A cylindrically symmetrical core made from a soft magnetic material (polepiece) with a hole (bore) drilled through it
A coil of copper wire surrounding each polepiece
When a current is passed through the copper coil a magnetic field is created. The strength of the field controls the electron trajectories
Soft refers to the materials magnetic properties and not mechanic.
Soft iron is usually used as a material for the core.
In most lenses there are two polepieces which can be part of the same piece of iron or two separate pieces.
Because of resistive heating of the coil the lenses need to be cooled by a water recirculating system.

7. Objective lenses
A TEM has several lenses (multi-lens system) from which most of them are weak. The strongest lens is the objective lens which forms the images magnified by other lenses. There are several different types of objective lenses:
Split polepiece objective lens (A): Can produce a broad electron beam for TEM and a fine beam for AEM and STEM. The lens has also enough space for other instruments (x-ray spectrometer) to reach the sample. The sample can also be tilted and rotated
Immersion objective lens (B): Due to its short focal length, this lens is capable of producing very high resolutions. There is not enough space to move the sample or do additional analysis

8. Objective lenses
Snorkel objective lens (C)

9. Superconducting lenses
To overcome the limitation of ferromagnetic polepieces (saturized magnetization) superconducting lenses can be used.
Superconductor lenses can generate intense fields which are promising for forming fine probes with high-energy electrons, their aberrations are smaller and they do not require any cooling making the lenses smaller
Unfortunately superconductors generate fixed fields and cannot be varied, which means that they are not very flexible.
These lenses became popular with the discovery of high-Tc superconductors

10. Electron path through magnetic fields
In a magnetic field, electrons do not travel in a straight line but rather spiral in a helical trajectory
The electrons rotate under the influence of the rotationally symmetrical magnetic field
This effect causes the sample image to rotate on the display screen as we change the focus or magnification of the electromagnetic lense
For electrons with higher energies, we must use stronger lenses to get similar ray paths
The optic axis is sometimes referred as the rotation axis.
If we wish to deflect the beam or tilt it we need to use an additional electromagnetic field.
If we wish to blank the beam we need to apply an electrostatic field.

11. Apertures and diaphragms
Apertures are inserted into the lens in order to limit the collection angle of the lens. This allows us to control the resolution of the image formed by the lens, the depth of field, the depth of focus, the image contrast and many other things
The apertures are circular holes in metal disks made from either Pt or Mo
Diaphragms come in several forms. They can be either individual disks, or they can be a series of different apertures in a single metal strip
Often the diaphragm collects contamination caused by the electron beam. These contaminations are known to cause astigmatism

12. Lens PROBLEMS
Electromagnetic lenses have many imperfections which limit the resolution the microscope but paradoxically help us get a better depth of field and focus.
The main defects that electromagnetic lenses experience are:
Spherical aberration
Chromatic aberration
Astigmatism

13. Spherical aberration
This defect occurs when the lens field behaves differently for off-axis rays
A point object is imaged as a disk of finite size, which limits our ability to magnify details because they are degraded by the imaging process
To correct this aberration we can create a diverging (concave) lens which spreads out the off-axis beams in such a way that they re-converge to a point rather than a disk.
The further the electron beam is off axis the stronger the bend

14. Chromatic aberration
This term refers to the frequency, wavelength or energy of the electrons
The problem occurs due to the variation of the electron energy that pass through the sample. The objective lens bends electrons with lower energy stronger and thus electrons from a point once again form a blurred disc
This aberration gets worse for thicker specimens
To correct this aberration by using thinner samples or through energy-filtering
In reality electrons aren’t monochromatic

15. Astigmatism
Astigmatism occurs when the electrons sense a non-uniform magnetic field as they spiral around the optic axis. This defect arises because we can’t create perfectly cylindrically symmetrical polepieces
This is easily corrected using stigmators, which are small octupoles that introduce a compensating field to balance the inhomogeneities causing astigmatism

16. Depth of focus and field
Small apertures need to be used to minimize their aberrations, which in turn give us a better depth of focus and better depth of field
The depth of field refers to the distance along the axis on both sides of the object plane within which we can move the object without detectable loss of focus in the image
The depth of focus refers to the distance along the axis on both sides of the image plane within which the image appears focused