E03 · Electron microscopy / foundations

Electron Optical Column

Drag the condenser, aperture, objective and projector handles; tune their focal lengths. Follow accepted and stopped rays from a finite source through the sample to the screen. Build a sample probe below 100 nm while retaining at least 20 pA, then test how a projector changes the screen without refocusing the specimen.

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

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

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

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

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

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

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

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

Experiment task $\text{—}$

A 100 kV paraxial thin-lens column in the Larmor frame. Uniform discrete source/angle quadrature; accepted ray fraction estimates current. The geometrical diameter is twice the RMS landing coordinate, combined in quadrature with an ideal diffraction scale. This is not a wave-optical PSF or a magnetic-field solver; off-axis rotation, lens aberrations, specimen scattering and dose are omitted.

Physics tutorial

Can you focus the probe without starving its current?

BackgroundDrag the condenser, aperture, objective and projector handles; tune their focal lengths. Follow accepted and stopped rays from a finite source through the sample to the screen. Build a sample probe below 100 nm while retaining at least 20 pA, then test how a projector changes the screen without refocusing the specimen.

Why it mattersExplore how the electron source, column and specimen constrain an instrument before interpreting an image.

Start with the essentials

Focus question: Can you focus the probe without starving its current?
One-sentence intuition: Probe formation and projection are different optical tasks. Throwing away high-angle rays can reduce current while making diffraction worse.

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}_{-}

Coordinates use a consistent paraxial Larmor frame; a real magnetic lens also rotates the beam.

Current acceptance

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

The rays quadrature a uniform source/angle distribution; the current curve has discrete steps.

Probe scale convention

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

This diameter convention is a teaching estimate, not a coherent propagated spot profile.

Depth of field

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

A smaller convergence angle gives more axial tolerance but can increase diffraction.

Common difficulties

Keep the model boundary explicit

Typical misconceptionA larger screen image means a better sample resolution.

Better mental modelProjection changes the screen scale; sample information is limited by the probe and specimen interaction.

Run the experiment

01
Refocus
Select the defocused preset, predict a focal length, then tune the objective until the sample-plane profile contracts.
What to observe: The sample focus depends on the source and both upstream lenses.
02
Spend the current budget
Reduce the aperture, track current and diffraction, and aim for a probe at most 100 nm with at least 20 pA. Check target.
What to observe: A sharp geometrical crossing can still be current-starved or diffraction-limited.
03
Separate projection
Change the projector focal length while holding the upstream optics fixed.
What to observe: The projected width changes while the sample probe stays fixed.