E03 · Electron microscopy / foundations

Electron Optical Column

Can you focus an electron probe to a small spot and still keep enough current? Yes, if you treat the optics above and below the specimen as separate jobs. The condenser lens, aperture and objective lens set how small the probe is and how much current it carries. The projector lens sits after the specimen, so adjusting it and the screen only changes how much of that current the screen catches. The specimen stays in focus.

Drag the condenser, aperture, objective and projector along the column and tune their focal lengths. Follow the rays from a finite source: the aperture stops some, and the rest pass through the sample to the screen. Focus a probe smaller than 100 nm at the sample while keeping at least 20 pA. Then adjust the projector and screen width so the screen catches at least 90 percent of that current, without refocusing the specimen.

Combined probe scale $0\,\mathrm{nm}$

Accepted probe current $0\,\mathrm{pA}$

Ideal diffraction scale $0\,\mathrm{nm}$

Fraction of probe intercepted $0\,\mathrm{\%}$

More measurements and target

Geometrical RMS diameter $0\,\mathrm{nm}$

Accepted ray fraction $0\,\mathrm{\%}$

RMS convergence angle $0\,\mathrm{mrad}$

Depth-of-field estimate $0\,\mathrm{\mu m}$

Projected RMS diameter $0\,\mathrm{\mu m}$

Screen-intercepted current $0\,\mathrm{pA}$

Current outside screen $0\,\mathrm{pA}$

Experiment task $\text{—}$

Simulated examples; each plot identifies its data source.

Condenser focal length

f_c

80\,\mathrm{mm}

Objective focal length

f_o

70\,\mathrm{mm}

Aperture diameter

D_a

400\,\mathrm{\mu m}

Projector focal length

f_p

70\,\mathrm{mm}

Advanced settings

Condenser position

z_c

80\,\mathrm{mm}

Objective position

z_o

250\,\mathrm{mm}

Aperture position

z_a

160\,\mathrm{mm}

Source diameter

d_s

20\,\mathrm{nm}

Incoming current

I_0

200\,\mathrm{pA}

Projector position

z_p

420\,\mathrm{mm}

Active screen width

W_s

1000\,\mathrm{\mu m}

Learning goal and model limits

Instrument settings

Electron Optical Column

Close down the aperture while watching current and diffraction. Aim for a probe of at most 100 nm carrying at least 20 pA, then catch at least 90 percent of that current on the screen. Press Check target.

Model assumptions

The column runs at 100 kV and uses paraxial thin lenses in the Larmor frame. Rays are traced in one transverse plane from 205 equally weighted combinations of source position and angle. The histograms add up the current these rays carry to the specimen and to the finite screen. They are ideal geometric distributions, not acquired counts or a two-dimensional image. The Gaussian probe-size reference is shown separately. The geometric diameter is twice the RMS landing position, added in quadrature to an ideal diffraction scale. There is no wave-optical point spread function and no magnetic-field solver. Off-axis rotation, lens aberrations, specimen scattering and dose are left out.

Physics tutorial

Can you focus an electron probe to a small spot and still keep enough current?

BackgroundDrag the condenser, aperture, objective and projector along the column and tune their focal lengths. Follow the rays from a finite source: the aperture stops some, and the rest pass through the sample to the screen. Focus a probe smaller than 100 nm at the sample while keeping at least 20 pA, then see how the projector changes the screen image without refocusing the specimen.

Why it mattersBefore you trust an electron image, you need to know what the source, the lens column and the specimen itself allow.

Start with the essentials

Focus question: Can you focus an electron probe to a small spot and still keep enough current?
One-sentence intuition: Forming the probe and projecting it onto the screen are different optical tasks. Blocking high-angle rays can cost current and make diffraction worse at the same time.

Core mathematical model

Free flight and a thin lens

\binom{r}{\theta}_{z+L}=\begin{pmatrix}1&L\\0&1\end{pmatrix}\binom{r}{\theta}_z,\quad \binom{r}{\theta}_{+}=\begin{pmatrix}1&0\\-1/f&1\end{pmatrix}\binom{r}{\theta}_{-}

The coordinates are paraxial and measured in the Larmor frame. A real magnetic lens also rotates the beam.

Current through the aperture

I_p=I_0\frac{N_{\mathrm{accepted}}}{N_{\mathrm{incident}}}

The rays sample a uniform spread of source positions and angles at fixed points, so the current curve changes in discrete steps.

Probe diameter estimate

d_g=2\sqrt{\langle r^2\rangle},\quad d=\sqrt{d_g^2+(0.61\lambda/\alpha_{\mathrm{RMS}})^2}

This diameter is a teaching estimate, not the profile of a coherently propagated spot.

Current on a finite screen

I_b=\frac{I_0}{205}\sum_{j\in b}1,\qquad I_p=I_{\mathrm{screen}}+I_{\mathrm{missed}}

Every ray carries the same current, and each bin adds up the rays that land in it. The probe current splits into the part the screen catches and the part that misses it. The histogram has no count noise, diffraction propagation or detector response.

Depth of field

\mathrm{DOF}\simeq d/(2\alpha_{\mathrm{RMS}})

A smaller convergence angle tolerates more focus error along the axis, but it can increase diffraction.

Common difficulties

Screen size and sample resolution

Typical misconceptionA larger screen image means a better sample resolution.

Better mental modelThe projector only changes the scale on the screen. How much detail you get from the sample is set by the probe and by how it interacts with the specimen.

Run the experiment

01
Refocus the probe
Select the defocused preset and predict a focal length. Then tune the objective until the profile at the sample plane narrows.
What to observe: The sample focus depends on the source and both upstream lenses.
02
Trade current for probe size
Close down the aperture while watching current and diffraction. Aim for a probe of at most 100 nm carrying at least 20 pA, then press Check target.
What to observe: Rays can cross at a sharp geometric focus and still carry too little current or be limited by diffraction.
03
Adjust only the projection
Select the clipped-projection preset. Leave the upstream optics alone, and adjust the projector or widen the screen until it catches at least 90 percent of the probe current.
What to observe: The full width of the projected rays changes, but the probe at the sample stays the same. Narrow the active screen and the caught current falls, while the full projected width does not change. The screen histogram is not renormalized, so current that misses the screen simply does not appear.