How Do Electromagnetic Lenses Magnify the Image?

[Dario56]

In optical microscope both objective and eyepiece are used to magnify the sample image. Magnification is determined by laws of geometrical optics (intersection of optical beams from the same point of the sample)

In electron microscope, electromagnetic lenses are used to magnify the sample image. How do such lenses enable magnification of sample image? Since the way image is formed isn't determined by laws of geometrical optics.

 

[vanhees71]

The idea behind an electron microscope is that electrons, which we usually first learn to be "particles", have also wavelike properties due to quantum mechanics. As light, which is nothing else than electromagnetic waves with wavelengths of between about 400 and 800 nm (nano-meters) our eyes are sensitive too, is described by a wave equations, which can be derived from Maxwell's equations, within quantum theory electrons are described by a wave equation, the Schrödinger equation.

The same math that explains, how the rules of geometrical optics follow from wave optics based on Maxwell's equations, also explains the particle aspects of the electrons. Since electrons carry electromagnetic charge (one negative elementary charge) you can deflect them with electric and magnetic fields, and you can tailor your fields such that they act like lenses do for light.

The advantage of electron microscopes compared to optical ones is that it is easy to manipulate electrons which have a much smaller wave length. This is important, because the wave length is also a measure for the ability to resolve small structures, i.e., the smaller structures you want to observer the higher resolution you need and thus you need waves with low wave length.

Electromagnetic waves of shorter wave lengths are UV-, X-, and finally $\gamma$ rays, but the electromagnetic radiation of such shorter wave lengths is not easily manipulated, because they just go through the material pretty unaffected and thus it is hard to construct lenses for them.

For electrons it's easier, because to get short wavelengths you just need electrons with higher momenta (larger velocities, because $p=m v$). The Einstein-de Broglie formula then tells you that the wave length of electrons with a momentum $p$ is $\lambda=h/p$, where $h$ is Planck's quantum of the action, ruling quantum physics. Further, as already mentioned above, you can use electric and magnetic fields as lenses for the electron waves, and this enables to construct electron microscopes using indeed ideas very similar to optics, enabling in this way pictures which can resolve much smaller structures than with an optical microscope.

 

[BvU]

Did you google 'electron microscope' and 'electron optics' ?

$\$

 

[Dario56]

Did you google 'electron microscope' and 'electron optics' ?

$\$

Yes, wasn't able to find an answer.

 

[Dario56]

The idea behind an electron microscope is that electrons, which we usually first learn to be "particles", have also wavelike properties due to quantum mechanics. As light, which is nothing else than electromagnetic waves with wavelengths of between about 400 and 800 nm (nano-meters) our eyes are sensitive too, is described by a wave equations, which can be derived from Maxwell's equations, within quantum theory electrons are described by a wave equation, the Schrödinger equation.

The same math that explains, how the rules of geometrical optics follow from wave optics based on Maxwell's equations, also explains the particle aspects of the electrons. Since electrons carry electromagnetic charge (one negative elementary charge) you can deflect them with electric and magnetic fields, and you can tailor your fields such that they act like lenses do for light.

The advantage of electron microscopes compared to optical ones is that it is easy to manipulate electrons which have a much smaller wave length. This is important, because the wave length is also a measure for the ability to resolve small structures, i.e., the smaller structures you want to observer the higher resolution you need and thus you need waves with low wave length.

Electromagnetic waves of shorter wave lengths are UV-, X-, and finally $\gamma$ rays, but the electromagnetic radiation of such shorter wave lengths is not easily manipulated, because they just go through the material pretty unaffected and thus it is hard to construct lenses for them.

For electrons it's easier, because to get short wavelengths you just need electrons with higher momenta (larger velocities, because $p=m v$). The Einstein-de Broglie formula then tells you that the wave length of electrons with a momentum $p$ is $\lambda=h/p$, where $h$ is Planck's quantum of the action, ruling quantum physics. Further, as already mentioned above, you can use electric and magnetic fields as lenses for the electron waves, and this enables to construct electron microscopes using indeed ideas very similar to optics, enabling in this way pictures which can resolve much smaller structures than with an optical microscope.

Thanks. However, my question wasn't answered.

 

[vanhees71]

Then I misunderstood the question :-(.

 

[Dario56]

Then I misunderstood the question :-(.

Let me explain what I mean :)

I do know that EM lenses deflect electron beams, but what does that have to do with magnification? Optical lenses magnify images by deflecting optical beams coming from some point in the object making them intersect at different position in comparison with real object. Geometrical optics laws determines magnification or as you said wave optics actually determines it as geometrical optics laws are derived fundamentally from wave optics.

However, image in electron microscope isn't formed as intersection of electron beams like in optical microscopes and so I don't understand how do they magnify images.

I know how images are formed. In TEM for example, electrons leave bright spots on special material. Higher kinetic energy of electrons, brighter the spot, but that doesn't explain how is image of sample analyzed magnified from its original size.

 

[Drakkith]

Per wikipedia's article on SEM:

Magnification in an SEM can be controlled over a range of about 6 orders of magnitude from about 10 to 3,000,000 times.[27] Unlike optical and transmission electron microscopes, image magnification in an SEM is not a function of the power of the objective lens. SEMs may have condenser and objective lenses, but their function is to focus the beam to a spot, and not to image the specimen. Provided the electron gun can generate a beam with sufficiently small diameter, a SEM could in principle work entirely without condenser or objective lenses, although it might not be very versatile or achieve very high resolution. In an SEM, as in scanning probe microscopy, magnification results from the ratio of the dimensions of the raster on the specimen and the raster on the display device. Assuming that the display screen has a fixed size, higher magnification results from reducing the size of the raster on the specimen, and vice versa. Magnification is therefore controlled by the current supplied to the x, y scanning coils, or the voltage supplied to the x, y deflector plates, and not by objective lens power.

Hope this helps.

 

[Dario56]

Per wikipedia's article on SEM:

Magnification in an SEM can be controlled over a range of about 6 orders of magnitude from about 10 to 3,000,000 times.[27] Unlike optical and transmission electron microscopes, image magnification in an SEM is not a function of the power of the objective lens. SEMs may have condenser and objective lenses, but their function is to focus the beam to a spot, and not to image the specimen. Provided the electron gun can generate a beam with sufficiently small diameter, a SEM could in principle work entirely without condenser or objective lenses, although it might not be very versatile or achieve very high resolution. In an SEM, as in scanning probe microscopy, magnification results from the ratio of the dimensions of the raster on the specimen and the raster on the display device. Assuming that the display screen has a fixed size, higher magnification results from reducing the size of the raster on the specimen, and vice versa. Magnification is therefore controlled by the current supplied to the x, y scanning coils, or the voltage supplied to the x, y deflector plates, and not by objective lens power.

Hope this helps.

Yes, that is for SEM. For TEM it is different though.

 

[Lord Jestocost]

Regarding TEM: Have a look at Fig. 1 in https://onlinelibrary.wiley.com/doi/full/10.1002/smll.201906198

 

[Drakkith]

Yes, that is for SEM. For TEM it is different though.

To my entirely untrained eyes it looks to me like the magnification for TEM is determined by how tightly the parallel beam of electrons is focused after passing through the sample and how far the sensor is placed from the final focal point. The parallel beam passes through the sample, is focused by the magnetic lens, converges down to it's smallest diameter at the focal point, then diverges before finally being picked up by the detector. If the final magnetic lens focuses the beam more tightly, then the focal point moves away from the detector and the cone of electrons is larger when it strikes the detector, making the image larger. If the beam is focuses less strongly then the opposite appears to happen.

But that's mostly from me looking at simplified diagrams from wikipedia and the like. I could be mistaken.