E39 · Electron wavefront

Electron Ptychography: Recover Phase from Overlap

What do overlapping probe positions add to a set of diffraction patterns? Redundancy. A diffraction pattern records intensity and loses the phase. When neighboring probe positions overlap, each part of the object shows up in several patterns. An iterative algorithm adjusts a phase guess until it agrees with all of them, which can recover the lost phase, though not always uniquely.

Record a diffraction pattern at every position of an overlapping scan, then refine a phase estimate over repeated iterations against the stored counts.

Total recorded electrons $\text{—}$

Adjacent intensity overlap $\text{—}$

Maximum cumulative incident fluence $\text{—}$

Measured amplitude relative RMS $\text{—}$

More measurements and target

Recorded frame count $\text{—}$

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

Completed iterations $\text{—}$

Committed probe calibration error $\text{—}$

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

Scan step in 0.1 nm cells

s

4

Calibrated in-focus intensity sigma

\sigma

0.35\,\mathrm{nm}

Recorded probe defocus

\Delta f

60\,\mathrm{nm}

Incident electrons per recorded frame

N

10000

Advanced settings

Assigned specimen phase scale

\phi_0

0.8\,\mathrm{rad}

Counts: 0 expectation, 1 seeded statistics

1

Iterations for next reconstruction

n

30

Joint object and probe update

1

Assumed probe defocus calibration error

\delta f

0\,\mathrm{nm}

Selected recorded scan position

j

50\,\%

Recorded-count display gain only

G

1

Learning goal and model limits

Predict · record · explain

Reconstruct from stored counts

Target: after at least 20 iterations, bring the measured-amplitude residual below six percent. Adjacent probe positions must overlap by at least 50 percent, with at least 1000 incident electrons per frame. These checks are diagnostics; passing them does not prove that the solution is unique.

Acquisition sliders plan the next scan. Reconstruction reads stored counts. Drag the probe to browse scan positions; gain changes the display.

Model assumptions

The model uses a coherent 200 kV beam and a single-slice teaching object that changes only the phase. ePIE jointly updates a finite probe and the unit-modulus object from stored counts and known integer positions. Independent width and defocus calibration initialize the probe; a frozen-probe option supplies a mismatch comparison. Unit incident norm and unit object modulus fix the amplitude scale. Constant phase and opposite affine phase ramps in object and probe remain ambiguous; regular scans can have further ambiguities. The displayed object phase is relative to its illuminated circular mean. ADF and integrated CoM use these same electrons. The latter is a weak-phase, finite-detector comparison with a free integration constant, not a general quantitative phase inversion. Multislice scattering, position refinement, partial coherence, damage, background and real-material resolution are omitted. A small residual alone does not prove that the object is unique or correct. Angles and specimen size are enlarged schematically.

Start with the essentials

Focus question: What do overlapping probe positions add to a set of diffraction patterns?
One-sentence intuition: Compare the measured diffraction plane with the reconstructed object plane, and do not confuse what was computed with what was measured.

Core mathematical model

How the stored diffraction counts arise

\begin{aligned}\Psi_j&=\mathcal F\!\left[P\,O_j\right]\\C_j&\sim\operatorname{Poisson}\!\left(N|\Psi_j|^2\right)\end{aligned}

The known probe and positions define each patch of the exit wave. Because the detector samples the complete discrete pattern, a unit-modulus object conserves the expected number of electrons.

Joint object and probe updates

\begin{aligned}\Psi_j^c&=\sqrt{C_j/N}\,e^{i\arg\Psi_j}\\O_j&\leftarrow O_j+\beta\frac{P^*}{\max|P|^2}(\psi_j^c-\psi_j)\end{aligned}

The object step is 0.7 with a unit-modulus constraint. Joint mode adds a probe step of 0.15 after two initialization passes, using the previous object value. Each pass normalizes the probe to the independently known incident intensity.

Residual against the measured amplitudes

R=\sqrt{\frac{\sum_{j,k}(|\Psi_{j,k}|-\sqrt{C_{j,k}/N})^2}{\sum_{j,k}C_{j,k}/N}}

The residual measures how well the estimate matches the recorded amplitudes. A sparse scan can fit well and still leave the phase unconstrained.

Common difficulties

What a low residual does not prove

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

Better mental modelThe residual describes only this model. The model uses a coherent 200 kV beam and a single-slice teaching object that changes only the phase. ePIE jointly updates a finite probe and the unit-modulus object from stored counts and known integer positions. Independent width and defocus calibration initialize the probe; a frozen-probe option supplies a mismatch comparison. Unit incident norm and unit object modulus fix the amplitude scale. Constant phase and opposite affine phase ramps in object and probe remain ambiguous; regular scans can have further ambiguities. The displayed object phase is relative to its illuminated circular mean. ADF and integrated CoM use these same electrons. The latter is a weak-phase, finite-detector comparison with a free integration constant, not a general quantitative phase inversion. Multislice scattering, position refinement, partial coherence, damage, background and real-material resolution are omitted. A small residual alone does not prove that the object is unique or correct. Angles and specimen size are enlarged schematically.

Run the experiment

01
Predict the overlap
Choose the scan step, the probe width and the number of electrons per frame.
What to observe: With the counts per frame fixed, a smaller step gives more overlap and also more total dose.
02
Record the scan
Acquire the dataset and browse the stored diffraction frames.
What to observe: Moving the acquisition sliders afterwards does not rewrite the stored frames.
03
Refine the phase estimate
Choose the number of iterations and the assumed probe calibration, then reconstruct.
What to observe: Only the recorded counts, the positions and the calibration constrain the object update.
04
Make it fail
Compare the sparse, low-count and incorrect-probe examples before you check the target.
What to observe: A small residual can come with a phase that is wrong or only weakly constrained.

Core mathematical model

How the stored diffraction counts arise

Joint object and probe updates

Residual against the measured amplitudes

Common difficulties

What a low residual does not prove

Run the experiment

Predict the overlap

Record the scan

Refine the phase estimate

Make it fail