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E31 · Electron wavefront

Aberration Corrector: Tune the Electron Probe

Tune signed residual aberrations, record a Ronchigram, and compare the calculated specimen-plane probe.

Interactive modelAberration Corrector: Tune the Electron Probe
Recorded frame count—\text{—}
Total recorded electrons—\text{—}
Recorded incident electrons per frame—\text{—}
Calculated coherent Strehl—\text{—}
Calculated half-energy radius—\text{—}
Recorded convergence semi-angle—\text{—}
Experiment target—\text{—}

Physics tutorial

Aberration Corrector: Tune the Electron Probe

BackgroundTune signed residual aberrations, record a Ronchigram, and compare the calculated specimen-plane probe.

Why it mattersCan you focus a wider cone without losing its coherent peak?

Start with the essentials

Focus question
Can you focus a wider cone without losing its coherent peak?
One-sentence intuition
Compare conjugate planes without confusing prediction and measurement.

Core mathematical model

Signed wave aberration

χ=2πλ[C102θ2+C122(θx2−θy2)+C213θxθ2+C304θ4]\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 radians, all coefficients are lengths; spherical aberration is entered in micrometres. Positive defocus has the sign defined here.

Probe and recorded Ronchigram

P=F−1 ⁣[Ae−iχ]IR=∣F ⁣[Peiϕ]∣2\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 propagates the normalized probe through one fixed synthetic amorphous phase sheet.

Calculated coherent peak

S=∣∑j∈Ae−iχj∣2NA2S=\frac{\left|\sum_{j\in A}e^{-i\chi_j}\right|^2}{N_A^2}

This on-axis Strehl compares the same aperture with zero aberration. It is not inferred from the camera.

Common difficulties

Scientific boundary

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

Better mental modelCoherent 200 kV thin phase-screen model. Fixed-axis astigmatism and coma, signed defocus and third-order spherical aberration; coefficients are residuals, not corrector currents. Ronchigrams are simulated recorded diffraction counts. Probe intensity, aperture phase, Strehl and encircled radius are separately calculated references, never reconstructed from counts. No chromatic spread, source-size blur, higher aberrations, dose damage or material calibration. Angles and planes are magnified schematically.

Run the experiment

  1. 01

    Predict the limitation

    Adjust residual aberrations and convergence. Predict how the probe changes.

    What to observe: A small aperture can hide aberrations while broadening the probe.
  2. 02

    Commit a measurement

    Acquire a Ronchigram, then change a slider.

    What to observe: Recorded counts and calculated references remain tied to their stored lens settings.
  3. 03

    Compare planes

    Inspect the measured diffraction plane and the separately calculated specimen-plane probe.

    What to observe: An aperture phase plot is a model diagnostic, not another detector.
  4. 04

    Check and challenge

    Check the target, then compare astigmatism, spherical aberration and low-count cases.

    What to observe: Lens correction and sufficient counts are separate requirements.