---
title: "Electron Optical Column Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/electron-column/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/electron-column/"
last_modified: "2026-10-25"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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?

**Background** 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 see how the projector changes the screen image without refocusing the specimen.

**Why it matters** Before 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 misconception** A larger screen image means a better sample resolution.

**Better mental model** The 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

1. 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.

2. 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.

3. 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.

Physical conclusion

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.

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)