Optics 068 · Diffraction, Fourier optics, and computational imaging

Phase Retrieval Challenge

Recover an invisible phase plate from real simulated intensity records: scan one CCD, confront twin ambiguity, replay blind iterations and break an incorrect support prior.

Recorded-amplitude residual

\varepsilon_A

$0.50$

Simulation-only phase score / missing area

\sigma_\phi,m

$50\%$

Iteration / acquired record count

k,N_z

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Phase Retrieval Challenge

Measurement mode index (named buttons below)

h

1

Iteration count (scrubbable)

k

60

Assumed support radius (not the physical aperture)

R_S

0.50\mathrm{mm}

Camera defocus scan half-range

\Delta z

20\mathrm{mm}

Unknown specimen phase strength

\alpha

2\mathrm{rad}

Read-noise SD / fixed clear-aperture peak

\eta

0\%

Independent initial-guess seed

s

1

Camera position index (named buttons below)

j

1

An intensity camera cannot see phase. First fit one focal image: the conjugate-inverted answer remains equally valid. Then move one camera to three known defocus positions and compare the recovered phase with the separately labelled synthetic answer key. Replay iterations on the error plot, drag the assumed support circle, or change a blind seed. Wrong support and read noise can prevent a good fit. The solver receives only detector magnitudes, propagation distances and a calibrated unit-amplitude aperture—not the answer key.

Synthetic pure-phase specimen with known unit amplitude, ideal paraxial lens and finite-grid point sampling. Additive Gaussian read noise, not photon statistics. Truth is scoring-only; no uniqueness or global-convergence guarantee.

Start with the essentials

Focus question: Can intensity-only measurements recover a complex field despite ambiguity and noise?
One-sentence intuition: At focus, the conjugate-inverted object gives identical data: $|\mathcal F_u U|^2=|\mathcal F_u\{U^*(-\mathbf r)\}|^2$ . Distinguish an exact alternative answer from a stagnated mixture. Multiple known planes can reject the twin, but never determine global piston.

Core mathematical model

The camera really moves

\begin{aligned}M_j&=\begin{pmatrix}-\delta_j/f&f\\-1/f&0\end{pmatrix}\\d_j(\mathbf r)&=e^{-i\pi\delta_j r^2/(\lambda f^2)}\\\Delta x_D&=\frac{\lambda f}{N\Delta x_O}\end{aligned}

Object is one focal length before the lens; camera is one focal length plus signed defocus behind it. No artificial phase plate is inserted. Coordinates are millimetres; positive defocus is farther from the lens.

Blind amplitude projection

\begin{aligned}g_j&=\mathcal F_u(d_j U_k)\\h_j&=\sqrt{I_j}e^{i\arg g_j}\\U_{k+1}&=\mathcal P_S\!\left[\frac1{N_z}\sum_j d_j^*\mathcal F_u^{-1}h_j\right]\end{aligned}

Measured magnitudes replace predicted ones. Back-propagate each plane and average, then impose unit amplitude within the assumed circular support and zero outside. This is a GS-like parallel projection, not HIO or guaranteed global optimization.

Fit and truth are different tests

\begin{aligned}\varepsilon_A^2&=\frac{\sum_j\|\,|g_j|-\sqrt{I_j}\,\|^2}{\sum_j\|\sqrt{I_j}\|^2}\\\sigma_\phi^2&=\langle\operatorname{wrap}(\phi_k-\phi_{\rm true}-c)^2\rangle\end{aligned}

Residual uses only acquired data. Phase score is simulation-only, removes a best global piston and uses overlapping supported pixels. Missing true area is reported separately; a twin is not silently relabelled correct.

Noise is not phase

\begin{aligned}I_j^{\rm rec}&=\max(0,I_j^{\rm ideal}+n_j)\\n_j&\sim\mathcal N(0,\eta^2 I_{\rm clear}^2)\end{aligned}

Fixed-seed additive read noise with one clear-aperture peak reference. Clipping biases dark pixels upward; no shot-noise or photon-budget claim. Fixed logarithmic display does not renormalize each picture.

Common difficulties

Small data error means correct phase?

Typical misconceptionMore iterations always reveal the unique physical phase.

Better mental modelNo. Nonconvex constraints can stagnate; a Fourier twin fits exactly. A wrong support discards actual object pixels. Synthetic truth is withheld and used only for evaluation.

Run the experiment

01
Scene 1: One image, two valid phases
Compare one focus record with three defocus records. Keep the same specimen, seed, aperture and iteration count.
What to observe: Defocus diversity rejects the focus twin for this example; global piston remains unobservable.
02
Scene 2: Scan three planes, recover phase
Replay the error curve. Change the blind seed and compare fit error with the separately labelled phase score.
What to observe: A software support is not a physical optical stop. Only the camera moves; detector records do not change when you change the reconstruction seed.
03
Scene 3: Wrong prior and noisy data
Drag the assumed support circle inward, add noise, then inspect the actual moving CCD in overview, side and detail views.
What to observe: A small residual does not prove correct phase. Wrong support and clipped read noise can bias a reconstruction.