Optics 072 · Diffraction, Fourier optics, and computational imaging

Ptychography & Coherent Diffraction Imaging

Lensless imaging from overlapping diffraction records: a moving specimen, Poisson counts and known-probe PIE amplitude/phase recovery.

Measured-amplitude fit

\varepsilon_A

$0.50$

Simulation-only complex truth error

\varepsilon_{\rm truth}

$50\%$

Pass / scans / selected electrons

k,N_{\rm scan},C_j

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Ptychography & Coherent Diffraction Imaging

Scan step

\Delta x

8\,\mu\mathrm m

Probe diameter

D

28\,\mu\mathrm m

Acquired scan count

N_{\rm scan}

25

Starting passes (play advances)

k_0

30

PIE update step

\beta

0.70

Incident expected electrons per illumination

N_{\rm in}

200000

Zero: expectation; one: Poisson counts

n

0

Synthetic specimen phase strength

a_\phi

1.70\,\mathrm{rad}

Synthetic specimen amplitude absorption

a_A

0.55

Independent reconstruction seed

s_0

1

Inspected recorded-frame index

j_0

0

Unobservable recovered-phase display offset

\phi_0

0\,\mathrm{rad}

A camera sees diffraction—not the specimen. Inspect recorded intensities from overlapping scans, then replay recovered amplitude and invisible phase. The scanner really moves; every iterate uses only recorded data. Remove overlap, acquire one frame, or starve the camera of photons. A convincing fit is not proof of a unique answer.

Synthetic thin scalar specimen, known probe/positions, finite Fourier modes. Not blind reconstruction; constant phase and dark regions remain unobservable.

Physics tutorial

Audit the physics of Ptychography & Coherent Diffraction Imaging

BackgroundA localized known probe multiplies the unknown thin specimen. Far away, a camera records only $I_j=|\mathcal F\{P\,O_j\}|^2$ . The specimen moves between exposures; shared illuminated regions must agree across many intensity constraints. There is no imaging lens between specimen and detector.

Why it mattersHow does overlapping illumination make a lensless amplitude-and-phase reconstruction possible?

Start with the essentials

Focus question: How does overlapping illumination make a lensless amplitude-and-phase reconstruction possible?
One-sentence intuition: Intensity loses the constant phase: $|\mathcal F\{e^{i\phi_0}P O_j\}|^2=I_j$ . Turn the phase-offset knob: phase colors rotate while predicted counts stay unchanged. Disconnected illuminated islands can have separate offsets.

Core mathematical model

Record exit-wave intensity

\begin{aligned}\psi_j&=P O_j\\\mu_{jq}&=\frac{N_{\rm in}|\mathcal F\psi_j|_q^2}{n^2\sum |P|^2}\\D_{jq}&\sim\operatorname{Poisson}(\mu_{jq})\end{aligned}

Finite discrete Fourier modes, unit efficiency. Normalization conserves transmitted power without undoing absorption. Noise off records expectations.

Measured modulus, estimated phase

\psi'_j=\mathcal F^{-1}\!\left[\sqrt{D_j/a}\,e^{i\arg(\mathcal F\psi_j)}\right]

Only measured electron amplitudes constrain the detector plane. Scaling is calibrated. No true detector phase is supplied.

Original soft-probe PIE

\begin{aligned}W&=\frac{|P|}{|P|_{\max}}\frac{P^*}{|P|^2+0.01}\\O_j&\leftarrow O_j+\beta W(\psi'_j-\psi_j)\end{aligned}

Chapter Eq. 6 with user step. Shared object pixels accumulate overlapping constraints. Probe is known, not jointly estimated: not blind ePIE.

Conjugate sampling

\Delta x_D=\frac{\lambda z}{n\Delta x_O}=2.078125\,\mathrm{mm}

532 nm, 250 mm propagation and 64 micrometre patch width. Scalar paraxial far field, not high-NA continuous pixel integration or metrology.

Common difficulties

A low residual proves the object?

Typical misconceptionFit a diffraction image and the entire specimen is known.

Better mental modelOne frame, missing regions and phase ambiguities admit alternatives. Overlap couples estimates but does not guarantee uniqueness or convergence. Truth is separate scoring-only data.

Run the experiment

01
Scene 1: Overlapping scan · phase recovered
Pause and inspect scans. The stage moves opposite the probe coordinates; the illuminated lab-axis point stays fixed.
What to observe: Neither diffraction phase nor truth enters PIE. Known probe and exact positions are calibration assumptions, not recovered here.
02
Scene 2: Disconnected illumination · ambiguity
Replay zero to thirty passes. Compare acquired/predicted diffraction, then recovered amplitude/phase with separately labelled answers.
What to observe: Absorption reduces expected total electrons. Every frame retains its physical scale; displays use a separate fixed logarithmic scale.
03
Scene 3: Photon-starved diffraction
Try disconnected illumination, one frame, low Poisson counts and different seeds. Rotate the unobservable phase offset.
What to observe: Unilluminated recovery pixels are dark. Truth scoring removes one constant phase only, never feeding the solver.