Text from the interactive explanation

Programmable photonic nanojets via phase-only time-reversal: a numerical study

Mikkelsen, Tobias Abilock, Placinta, Cristian, Glückstad, Jesper, Karamehmedović, Mirza

Programmable photonic nanojets · an interactive explainer

Can software alone steer a needle of light that is thinner than its own wavelength, with nothing moving?

Simulated (2D model following the paper’s method) · drag or tap below the block to aim

Green laser light (532 nm) passes a 50-pixel phase strip and a fixed glass block. Only the strip’s pattern changes; the glass never moves.

01

A needle made of light

Light a glass cylinder a few micrometres across with plain, flat laser light, and on its shadow side the light squeezes into a narrow, intense needle: a photonic nanojet . Its waist (its width at half brightness) can be narrower than the light’s own wavelength.

Such needles have been used for super-resolution imaging, nanopatterning, optical trapping and label-free measurement.

Now swap the cylinder for a square block, with the same flat light. No strong jet forms: the shape decides.

The catch: a jet sits wherever the glass puts it. Moving it means reshaping or moving the glass, or reshaping the beam. The authors describe such approaches as typically slow, with modest steering range, or needing complex hardware and calibration.

The paper measures each jet along its own axis: the direction in which the light’s energy flows at the target. The length ℓ and the waist w are the full widths at half maximum (FWHM) of the brightness I ∝ |Hz|² along and across that axis. Every readout on this page uses the same rule.

Simulated (2D model following the paper’s method) · flat light, all strip pixels equal · white bracket = waist, amber dashes = length

Brightness across the jet (1× = the cylinder’s jet). The waist is the width at half of the peak.

02

Your turn: aim it

Same square block, now with a programmable strip above it: a spatial light modulator (SLM, a screen that delays light pixel by pixel). It changes the phase of the light (the timing of the wave’s crests) without blocking any of it.

Use the two classic knobs of beam shaping, tilt and focus , to pour as much light as you can into the amber ring. The score compares the brightness in the ring with the best that any phase pattern on this strip could deliver there.

Each strip pixel j sends a wave that reaches the target with some strength |Gj| and some delay. With every pixel at full brightness and free to choose its timing, the brightest the target can get is when all contributions arrive in step:

Imax = ( Σj |Gj| )²

Phase-only time reversal reaches exactly this bound at the target. Tilt and focus typically reach about half to two-thirds of it here.

Simulated (2D model following the paper’s method) · amber ring at –

The pattern on the strip. Tilt + focus can only make smooth curves; time reversal’s pattern is jagged.

03

Run the light backwards

How does the computer find that pattern? It uses a symmetry of light: in lossless glass, a wave played backwards is still a perfectly valid wave.

So start from the light you want, a bright spot at the target, and play its movie in reverse. Step through the recipe on the right.

Because the backward wave is computed through the glass, the resulting pattern automatically accounts for all the refraction, diffraction and internal bouncing.

Step 1 solves the wave (Helmholtz) equation with a small source s at the target:

∇·( εr−1 ∇H ) + k0² H = s

On the control line the result is Hback = A(x) eiφ(x). The equation has real coefficients, so the complex conjugate A(x) e−iφ(x) is also a solution: the time-reversed wave (with time dependence e−iωt). Phase-only illumination keeps the timing and drops the amplitude:

Hinc(x) = e−iφ(x), |x| ≤ X (paper Eq. 5)

Simulated (2D model following the paper’s method) · –

The 50 numbers on the control line: strength (bars) and crest timing (dots, coloured like the strip).

04

Anywhere below the glass

Time reversal is not limited to a few lucky spots. The map scans a grid of targets every half micrometre below the block. For each one it computes the pattern and the jet, and colours the cell by the jet’s waist.

Inside the dashed line the jet is narrower than one wavelength (532 nm), which covers most of the region. Jets widen with depth and towards the far sides, and they grow dimmer the deeper they go.

Click or tap any cell to send the jet there, or drag the ring in the scene.

Simulated (2D model following the paper’s method)

Dashed white line: waist = 532 nm (one wavelength).

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Write with light

A needle you can steer is a pen. Drag the jet, or let it write on its own: a light-sensitive film below the block records the accumulated brightness, the way nanopatterning writes tiny structures.

The authors suggest the approach could suit microscopy (scanning a focused spot across a sample), nanofabrication and optical trapping of tiny particles.

The film is an illustration: it adds up the simulated brightness over time, but only where the light exceeds about half of the jet’s peak, like a threshold resist.

Simulated light; the amber film exposure is illustrative

05

Does the shape of the glass matter?

Time reversal steers the jet through any glass shape: switch shapes on the right and drag the ring. But are some shapes better at it? Make a prediction first.

Simulated (2D model following the paper’s method) · target –

This page’s re-simulation · paper’s six targets

Reveal the paper’s answer to run this comparison.

Bars: score relative to the square, then mean waist and mean length. Same input power for every shape. As in the paper, the square beats the best superformula shape, but by a different margin than the paper’s ≈40%: this page scores the peak brightness, while the paper averages brightness over the jet area, and the shapes are scaled differently. The paper compared the square only with superformula shapes; the cylinder and flower bars are this page’s additions.

Superformula explorer (paper Eq. 7, drawing only, not simulated)

r(θ) = ( |cos(nθ/4)|m₂ + |sin(nθ/4)|m₃ )−1/m₁

06

Two needles from one pattern

Time reversal adds up. Pretend two spots glow at once, record their combined crest timing, and a single pattern makes two jets. A weight sets how strongly each pretend source shines.

Challenge: balance the two jets while they sit at the same depth. Then choose “One deeper” (or drag a ring down) and find the weight that balances them again. What happens as the depth difference grows?

Paper: targets at the same depth need no extra weight; the larger the depth difference, the larger the weight needed. In its examples (Fig. 6), right-hand weights of ×2, ×6 and ×12 take a pair from left-dominated to balanced to right-dominated. A formula that predicts the right weight is left for future work.

This page’s model shows the same trend; its exact numbers depend on the setup. Here the brightness ratio grows roughly with the square of the weight.

Sback(r) = Σp wp exp( −|r − rp|² / 2σ² ) (paper Eq. 9)

Simulated (2D model following the paper’s method) · blue and amber rings: the two targets (the amber one has the adjustable weight)

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07

Imperfect glass

Real glass is never perfect. Here the pattern stays the one designed for the ideal block, and you “manufacture” a flawed block: wrong refractive index, slightly wrong size, or slightly misplaced.

The flaws are drawn to scale: tiny, but so is the wavelength. In this model a sharp jet survives every flaw. Index errors barely matter; size and position errors mostly nudge the jet.

The authors tested 15 random flaws of each type at 6 targets. Jets stay robust ; misplacement causes the largest spread (peak-intensity CV 0.32, versus 0.12 for index and 0.09 for size errors).

Simulated (2D model following the paper’s method) · target – · below the block: recomputed for the flawed block; inside it: the ideal block’s field, dimmed · amber dashes: ideal outline

Paper’s Table 1 · square block · mean ± standard deviation

CV (coefficient of variation) = standard deviation ÷ mean: the relative spread. Position errors spread the jet’s brightness and length the most.

For a cylinder the light’s phase delay scales with index × radius (φ ∼ k₀nR), so index and size errors should act alike, and the paper finds matching statistics, a check that the numerics behave.

08

What this brings closer

One fixed glass block, one programmable phase pattern: a needle of light thinner than its wavelength, steered by software alone, its pattern computed by running light backwards.

No moving parts, and no light thrown away. The authors see the setup as potentially suitable for microscopy, nanofabrication and trapping systems .

This is a numerical study : everything here, in the paper and on this page, is a 2D simulation. The next steps the authors name are a 3D solver (the framework generalizes to 3D in principle) and a predictive rule for weighting several jets.

Simulated (2D model following the paper’s method) · one pattern, two jets circling below the fixed block

09

Sources & method

License and figures

The paper is published under CC BY 4.0. Figures 3, 4, 5 and 6 shown above are taken from it; white margins were trimmed, panels were re-arranged side by side, and the images were resized and compressed. No other changes were made.

What comes from where

Items tagged From the paper, the ≈40% square advantage, the shape study and Tables 1 and 2 are the paper’s results.

All live scenes, readouts, the steering map, the shape bar chart, the two-jet weights and the flawed-block readouts are this page’s own re-simulation. Its numbers can differ from the paper’s.

The light-sensitive film in “Write with light” is an illustration.

How this page simulates

Its own re-implementation of the paper’s method: a 2D finite-difference frequency-domain (FDFD) solution of the wave equation at λ₀ = 532 nm, glass index n = 1.49, 30 grid points per micrometre, with absorbing boundaries.

Geometry: an 8 µm square block (also an r = 4 µm cylinder, the paper’s best superformula shape scaled into the same 8 µm box, and a six-lobed superformula “flower”); control line 0.3 µm above the block, 8.3 µm wide, split into 50 segments.

The field of each segment was precomputed. By reciprocity the same data give the backward phase for any target, so patterns and jets are computed live in the browser. Below the block the field is carried down exactly on the simulation grid.

Jet waist and length use the paper’s rule: full widths at half maximum along and across the local energy-flow direction.

The incoming flat light drawn before the strip and the wave animation (the field’s real part over one optical cycle) are visual aids; the colour scales are normalised separately inside and outside the glass, as in the paper’s figures.