E31 · Electron wavefront

Aberration Corrector: Tune the Electron Probe

Can a wider cone of electrons focus to a sharper probe? Only if the leftover aberrations are tuned small. A wider convergence angle can give a sharper probe, but the outer rays carry uncorrected wave aberrations that spread intensity out of the central peak. A Ronchigram that looks well tuned still does not prove the resolution you will get in an experiment.

Adjust the residual aberrations, each with its sign, record a Ronchigram, and compare it with the calculated probe at the specimen.

Ronchigram recorded electrons $\text{—}$

Highest detected test frequency $\text{—}$

Usable diameter at the fixed test frequency $\text{—}$

Calculated half-energy radius $\text{—}$

More measurements and target

Recorded frame count $\text{—}$

Recorded incident electrons per frame $\text{—}$

Calculated coherent Strehl $\text{—}$

Recorded convergence semi-angle $\text{—}$

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

Convergence semi-angle

\alpha

20\,\mathrm{mrad}

Signed residual defocus

C_{10}

12\,\mathrm{nm}

Residual astigmatism, fixed x axis

C_{12}

8\,\mathrm{nm}

Residual third-order spherical aberration

C_{30}

80\,\mathrm{\mu m}

Tested field radius

r

20\,\mathrm{nm}

Ronchigram electrons / calibration total

N

10000

Advanced settings

Residual coma, fixed positive x axis

C_{21}

400\,\mathrm{nm}

Counts: 0 expectation, 1 seeded statistics

1

Recorded-count display gain only

G

1

Learning goal and model limits

Predict · record · explain

Correct the coherent probe

Target: reach a calculated Strehl ratio of at least 0.8 and a half-energy radius of at most 0.075 nm, with a convergence semi-angle of at least 15 mrad, at least 5000 recorded electrons, and a detected field diameter of at least 30 nm at the fixed test frequency. These values come from the known lens model; the aberrations are not fitted from the counts.

The lens, field and electron controls record a new simulated calibration immediately. Gain changes the display.

Model assumptions

This is a coherent 200 kV model with a thin phase screen as the specimen. It includes astigmatism and coma along fixed axes, signed defocus and third-order spherical aberration. The coefficients are residual aberrations, not corrector currents. Ronchigrams are simulated recorded diffraction counts. The probe intensity, aperture phase, Strehl ratio and encircled radius are calculated separately as references and are never reconstructed from the counts. A separate known-grating scan shares a fixed total electron count across five field positions and six frequencies. Its measured modulation and three-standard-error threshold define a sampled information limit and a usable field at 2.5 inverse nanometres. An assumed quadratic field-curvature law changes local focus; these are teaching calibration results, not microscope specifications. The model leaves out chromatic spread, source-size blur, higher-order aberrations, dose damage and material calibration. Angles and planes are magnified schematically.

Start with the essentials

Focus question: Can a wider cone of electrons focus to a sharper probe?
One-sentence intuition: The Ronchigram is measured in the diffraction plane and the probe is calculated in the specimen plane; compare the two, but do not mistake the calculation for a measurement.

Core mathematical model

Wave aberration with signed coefficients

\begin{aligned}\chi&=\frac{2\pi}{\lambda}\left[\frac{C_{10}}2\theta^2+\frac{C_{12}}2(\theta_x^2-\theta_y^2)\right.\\&\quad\left.+\frac{C_{21}}3\theta_x\theta^2+\frac{C_{30}}4\theta^4\right]\end{aligned}

Angles are in radians and every coefficient is a length; spherical aberration is entered in micrometres. The sign of positive defocus follows this formula.

The probe and the recorded Ronchigram

\begin{aligned}P&=\mathcal F^{-1}\!\left[Ae^{-i\chi}\right]\\I_R&=\left|\mathcal F\!\left[Pe^{i\phi}\right]\right|^2\end{aligned}

A unitary discrete Fourier transform carries the normalized probe through one fixed, synthetic amorphous phase sheet.

Height of the coherent peak

S=\frac{\left|\sum_{j\in A}e^{-i\chi_j}\right|^2}{N_A^2}

This on-axis Strehl ratio compares the peak with that of the same aperture without aberrations. It is calculated, not inferred from the camera.

Common difficulties

What a sharp calculated probe does not prove

Typical misconceptionA low residual or sharp calculated probe certifies an experimental image.

Better mental modelThe residuals and the calculated probe describe only this model. This is a coherent 200 kV model with a thin phase screen as the specimen. It includes astigmatism and coma along fixed axes, signed defocus and third-order spherical aberration. The coefficients are residual aberrations, not corrector currents. Ronchigrams are simulated recorded diffraction counts. The probe intensity, aperture phase, Strehl ratio and encircled radius are calculated separately as references and are never reconstructed from the counts. A separate known-grating scan shares a fixed total electron count across five field positions and six frequencies. Its measured modulation and three-standard-error threshold define a sampled information limit and a usable field at 2.5 inverse nanometres. An assumed quadratic field-curvature law changes local focus; these are teaching calibration results, not microscope specifications. The model leaves out chromatic spread, source-size blur, higher-order aberrations, dose damage and material calibration. Angles and planes are magnified schematically.

Run the experiment

01
Predict what limits the probe
Adjust residual aberrations and convergence. Predict how the probe changes.
What to observe: A small aperture can hide aberrations, but it makes the probe wider.
02
Record a measurement
Acquire a Ronchigram, then change a slider.
What to observe: Each lens change records fresh counts and references; changing display gain preserves them.
03
Compare the two planes
Inspect the measured diffraction plane and the separately calculated specimen-plane probe.
What to observe: The aperture phase plot is a view into the model, not a second detector.
04
Measure the field at a fixed dose
Compare the corrected, wide-field and low-electron presets. Open the recorded field and dose test.
What to observe: The same three-standard-error rule tests every grating. A sharp reference probe can coexist with an undetectable modulation or a poor field edge.
05
Check the target, then test the limits
Check the target, then compare astigmatism, spherical aberration and low-count cases.
What to observe: Correcting the lens and collecting enough counts are two separate requirements.

Core mathematical model

Wave aberration with signed coefficients

The probe and the recorded Ronchigram

Height of the coherent peak

Common difficulties

What a sharp calculated probe does not prove

Run the experiment

Predict what limits the probe

Record a measurement

Compare the two planes

Measure the field at a fixed dose

Check the target, then test the limits