Can a microscope restore images without knowing how far the object is out of focus? The authors jointly optimize a circular phase mask and a Richardson-Lucy deconvolution algorithm with a Wiener-Butterworth back projector using simulated blur and shot noise. They fabricate the mask with two-photon grayscale lithography and experimentally restore etched smile patterns displaced by up to ±10 mm, as well as three fibers at different depths. A nearly invariant Bessel-like point spread function makes one restoration model useful across depth. The demonstrated low-numerical-aperture system retains a

A tiny printed surface reshapes light. A restoration algorithm turns its predictable blur into a clear image—even as the object moves in depth.

First, make blur consistent across depth. Then, teach the restoration how to undo it.

Keep the smile readable as it moves.

A normal microscope has a narrow sweet spot. Move the sample, insert the phase mask, then restore the image. The pictures below are measured snapshots from the paper.

At the sweet spot.

At zero defocus, the ordinary microscope resolves the smile.

Move to +4 mm . Can you keep the smile clear?

Discrete measured snapshots, every 2 mm. Conventional images are available across ±4 mm.

The mask alone does not deliver a sharp picture. It produces a ringed, nearly constant blur. Knowing that blur lets the same restoration work when the sample’s depth is unknown.

A point tells the whole story.

Imagine photographing one luminous point. Its image—the point spread function—shows how the optics spread light. Compare how that pattern changes with depth.

At 0 mm, ordinary optics concentrate light into a compact central spot.

Computed from the paper’s published phase profile. Intensity uses one fixed normalization across all depths for each optical system. The amber color scale reveals faint rings; it does not renormalize individual slices.

A Bessel-like central core persists across depth. The surrounding rings spread light into a pattern that the restoration can recognize.

The surface and the algorithm learn together.

Rather than choosing a mask first and repairing its images later, the researchers optimize both parts of the imaging system around the quality of the final restored image.

3,200 sketches

Images are blurred at different depths, with simulated shot noise.

Optics + parameters

The phase profile and restoration parameters are optimized end to end.

A familiar algorithm

Richardson–Lucy deconvolution uses a Wiener–Butterworth back projector.

Learning designs the optical surface and a small set of algorithm parameters. At image recovery, an iterative deconvolution algorithm performs the restoration.

The optimization favors a useful spatial-frequency band while controlling noise amplification. The mask’s optical transfer function cuts off around 0.7 of the full-aperture diffraction cutoff. The restored images therefore recover the details carried by this designed imaging band. The accelerated restoration stabilized in about 15 iterations, compared with about 2,000 for conventional Richardson–Lucy using the same forward and backward projector.

A learned surface, written in polymer.

Light acquires a different phase as it crosses different thicknesses. A smooth, circularly symmetric height profile turns the optimized phase into a physical optical element.

The shape is the function.

The raised center and rim surround an annular valley. Rotate the profile, then reveal the writing process.

The complete radial surface is visible. The 15 µm height modulation is exaggerated 25× to reveal its shape.

Interactive rendering of the design profile. The writing spot illustrates fabrication; its path is conceptual.

A Nanoscribe Quantum X writes IP-S photoresist with two-photon grayscale lithography. Dynamic laser power controls the local printed height.

Two precompensation cycles correct repeatable surface errors. The resulting mean absolute height difference from the design was 118.9 nm .

This fabrication method makes the learned optical surface real—connecting computational design to a working microscope.

The print used a 25×, NA 0.8 objective, 1 µm slicing, 0.2 µm hatching and a 20 cm/s scan speed. Laser power varied from 8 to 50 mW. After precompensation, the reported height-error standard deviation was 190.1 nm. The surface rendering shows the designed 0–15 µm modulation; the measured mask height map is in Figure 3 below.

Three depths. One restored image.

Three tapered fibers are placed about 9 mm apart along the optical axis. Conventional focus selects one fiber. The phase mask and restoration make all three readable at once.

The near fiber is in focus; the other depths appear blurred. Try choosing another fiber.

An extended-depth 2D image brings objects at different depths into view. It does not assign them reconstructed 3D positions.

For samples that extend or move along the optical axis, a readable image can persist without continuously refocusing.

The paper demonstrates this at NA ≈0.022. Higher numerical aperture improves resolution while shortening the absolute depth range.

Defocus-Insensitive Microscopy via Spatial Phase Modulation Using End-to-End Learning

Panu Hildén, Sebastian Kalt, Markus Nyman, Sami Hamriti, Martin Wegener, Carsten Rockstuhl and Andriy Shevchenko.

Figures and measured image crops: Hildén et al., ACS Photonics (2026), CC BY 4.0. Images are cropped and resized; their data are unchanged.

Graphical abstract: AI-generated conceptual illustration. Optical bench and fabrication views are conceptual schematics. Point spread function visualization is computed from Equation 4 and the published phase coefficients.