Text from the interactive explanation

3D-printed fiber-based zeroth- and high-order Bessel beam generator

Innem V. A. K. Reddy, Andrea Bertoncini, Carlo Liberale

Optica · 2022 · Reddy, Bertoncini & Liberale

Light leaving an optical fiber spreads out almost at once. These researchers 3D‑printed a structure about as wide as a hair onto the fiber’s end — and the light came out as a thin needle that holds its width far longer than an ordinary beam, or as a hollow, twisting vortex.

Could a fiber’s tip become a tool that images deep inside the body, grips microscopic particles — or, as the authors suggest, one day pulls them in like a tractor beam?

No twist: ring 11.2 µm, lens focus 100 µm, 1060 nm light — the needle measured at about 230 µm long.

A 3D model of the printed tip (cut open along its length, like the paper’s Fig. 1) with a simulation of its light, in true proportions. The light spreads from the ring across the air gap, starts to overlap before the lens, builds into the needle — brightest about one focal length past the lens — and then parts into a widening cone. The real light is infrared and invisible; it is shown in false colour, like the paper’s camera images. Add twist and the needle becomes a hollow vortex tube. Drag to turn the model.

1

Why light won’t stay thin

Light is a wave. Squeeze a laser beam into a tiny spot and the waves in it must travel in a spread of slightly different directions — so after the spot, the beam fans out again. Physicists call this diffraction .

Predict: squeeze a beam to half its width. How far does it stay narrow?

Stays narrow for≈ 49 µm

“Narrow” here means the beam is less than √2 times its smallest width (the Rayleigh range, counted on both sides of the spot).

The paper’s benchmark: an ordinary beam 3.4 µm wide stays narrow for only about 50 µm — less than the width of a hair.

That matters for fibers. A fiber can carry light into places nothing else fits — inside the body, into microscopic fluid channels — but at its tip the light (here about 3.6 µm wide) starts to spread within a few tens of micrometres.

So how could a beam be thin and long at the same time?

2

Build a beam out of a ring

In 1987, Durnin and colleagues found a way around diffraction. Shine light through a thin ring , then through a lens placed one focal length away. Each point of the ring becomes a straight, flat wave, and every one of them is tilted toward the centre by the same angle. Together they form a cone.

Where the tilted waves overlap they interfere : crests meet crests at the centre and make a bright core, and around it they alternately cancel and reinforce in rings. Add the waves one ring‑point at a time and watch.

Side view (a slice through the middle)

Camera image across the beam

A bright core with rings: a Bessel beam

Inside the needle

With the full ring you have made a Bessel beam (named after the mathematical function that describes its rings). Because every wave has the same tilt, the overlap pattern looks the same at every depth — the beam is called “diffraction‑free” — until the waves run past each other and part into a ring again. Its core is a needle of light.

The tilt angle is roughly the ring radius divided by the lens’s focal length, R/f. Steeper tilts make the interference stripes closer together, so the core gets narrower. Working it through gives the paper’s Eq. (1): core width (full width at half maximum) ≈ 0.36 λ f / R, where λ is the wavelength.

The ring decides everything. What happens when you change it?

3

Tune the needle

Three numbers shape the needle: the ring’s radius R, its width Δ, and the lens’s focal length f. The paper’s formulas, with λ = 1.06 µm infrared light:

core width ≈ 0.36 λ f / Rneedle length ≈ f² λ / (R Δ)

Try making the core sharper. What happens to its length? Then find a way to win length back.

Wave simulation from the ring to beyond the needle · widths stretched

Core width3.4 µm

Needle length237 µm

vs. ordinary beam4.8× longer

Length ÷ width69

Recreate what they printed

Pick a challenge, then move the sliders.

Notice the trade‑off. A bigger ring tilts the waves more steeply: a sharper core, but they cross for a shorter distance. A longer focus does the opposite.

Divide length by width and R and λ cancel: length ÷ width = f / (0.36 Δ) . Only a longer focus or a thinner ring makes the needle more needle‑like. Check it with the “Length ÷ width” readout — move R and it stays put.

Each side of the ring, after the lens, is a band of light about λ f / Δ wide (a thinner ring spreads more in the gap and fills more of the lens). The bands cross the axis at an angle of about R/f, so they overlap over a length of roughly (λ f / Δ) ÷ (R/f) = f² λ / (R Δ) — the paper’s Eq. (2). The needle sits about one focal length beyond the lens.

On a lab bench these knobs are bulky lenses and slits. The question was how to fit all of it on the end of a fiber.

4

Printing a ring‑maker on a fiber

Earlier fiber versions needed special hollow fibers, tiny cones that are hard to make precisely, or complex setups — and none had produced the twisted beams you’ll meet next. The team printed the whole optical train onto a standard fiber’s flat end in a single step using two‑photon lithography : a laser focused into liquid resin hardens it only at the tiny focal point, where two photons can be absorbed at once. Moving that point traces out 3D shapes with layers a few hundred nanometres thin.

The model below is cut open along its length, like the paper’s Fig. 1. Slide the cutting disc through the printed tip — or drag along the model — to see its cross‑section and what the light does inside.

Cross-section at the cut

Ring radius at its end11.2 µm

The three printed parts each do one job. The aligner (a tapered, holey waveguide inspired by photonic‑crystal fibers) catches the fiber’s light and squeezes it into a single clean spot, forgiving a slightly off‑centre or tilted print. The ring‑maker keeps the same outside width, but inside it is an inverse taper: its solid core opens around a hollow centre into a ring‑shaped core that grows along its length, hemmed in by three rings of air holes whose size roughly doubles from ring to ring. The lens , held by a slotted cage exactly one focal length away, turns the ring into the cone of tilted waves. Because it is all printed, each of R, Δ and f is just a number in the design file.

One more printed piece changes the light’s character completely.

5

Twist the light

Put a spiral staircase — a spiral phase plate — on the ring‑maker’s end face. Its thickness climbs as you go around, so light passing one side is delayed by whole wavelengths relative to the other. The beam’s wavefront becomes a corkscrew. The number of twists, l, is called its topological charge ; the twisting light carries orbital angular momentum , and at the very centre the waves cancel, leaving a dark core inside a bright ring.

These high‑order Bessel beams (vortex beams) had never been made from an optical fiber before. The team printed staircases with up to 20 twists, using 0.94 µm light, the wavelength their printed steps were best matched to.

Idealized Bessel profiles from the paper’s design (ring radius 16.25 µm, lens focus 60 µm). The right‑hand side shows the light’s electric field at one instant, colour‑coded orange for one direction and blue for the other.

The printed staircase (illustration)

Puzzle: same size, different twist

More twist makes a bigger ring. But a fiber that carries twisted light (a “ring‑core” fiber) has one fixed ring size. Can you make a 20‑twist beam exactly as wide as the 4‑twist one? Change the right‑hand design.

Left, fixed: l = 4, R = 5.6 µm, f = 70 µm → 19.9 µm

How do you know the twist is pure?

The team tested each beam by sending it through a programmable “untwister” that applies the opposite twist. Only when the counter‑twist exactly cancels the beam’s own does a bright spot reappear at the centre. Try it on a beam with l = +16.

6

Holding things with a needle of light

Light pushes and pulls gently on whatever it passes through. Tiny beads are drawn toward the brightest part of a beam, so a focused beam can hold them — optical trapping . The team dipped a printed fiber tip into a drop of water full of 5 µm plastic beads. With the laser on, beads were pulled into the bright core and settled into rings; moving the fiber dragged them along; with the laser off, they drifted away. Because the needle is long, this worked with the probe 60 µm away from the glass surface.

Laser on · beads held: 0

Drag in the picture to move the fiber. Illustrative model (beads drift randomly and are drawn up the brightness gradient of the needle’s cross‑section); time is sped up.

The authors look further: vortex beams like the ones in chapter 5 are an ideal shape for so‑called tractor beams , which can pull objects toward the source. Their approach, they write, would let optical fibers exert such pulling forces — a proposal for future work, not something tested here.

7

What they actually showed

Everything below was measured on printed structures, printed on fiber tips, or in arrays on glass slides for easier testing.

1.53–6.15 µm

Needle cores of chosen widths, each matching the design formula, from ring radii of 16.25 down to 6.5 µm.

~230 µm vs ~50

A 3.46 µm needle held its width about 4–5× farther than an ordinary beam of the same width.

~1 mm

The longest needle, from a small ring (5.6 µm) and a long‑focus lens (150 µm).

l = 1 → 20

The first high‑order (vortex) Bessel beams made from an optical fiber; ring sizes followed the formula, and 20 is not a limit.

same ring, 4 vs 20

Two designs gave equally wide rings with very different twist — useful for feeding ring‑core fibers.

~0.45 dB

Low loss through the printed structure: roughly 90% of the light gets through.

pure twist

Untwisting tests showed little leakage into other twist values.

beads held

Microbeads trapped and moved in water with the fiber‑made needle.

↺

Back to the tip

The opening question was whether a fiber’s end could shape light into something useful far beyond it. The answer here is a structure printed in one step, almost lossless, whose needle length, core width and twist are all chosen in the design file.

Your needle: ring radius 11.2 µm, ring width 4 µm, focus 100 µm → a 3.4 µm core about 237 µm long.

What the authors expect it could enable (proposals, not results of this paper):

Probes inside the body

Thin endoscopic probes for imaging such as optical coherence tomography, where a long needle keeps a sharp focus over depth.

Fiber tweezers & tractor beams

Trapping and moving particles from a fiber — and, with vortex beams, fibers that might pull objects toward them.

Twisted‑light communication

Plugging standard fibers straight into ring‑core fibers that carry many twist values at once to raise data capacity.

Perfect optical vortices

Rings whose size doesn’t change with twist, by leaving off the lens or adding a second one.

Plug‑in tips & tougher materials

Printed terminals that any fiber plugs into, and glass‑like printable materials for harsh environments where polymer may struggle.

Source

Innem V. A. K. Reddy, Andrea Bertoncini and Carlo Liberale, “3D‑printed fiber‑based zeroth‑ and high‑order Bessel beam generator,” Optica 9(6), 645–651 (2022). doi.org/10.1364/OPTICA.453839 · read it at Optica (open access)

Supplementary figures S2, S4, S5 and S6: Reddy, Bertoncini & Liberale, Supplement 1 to Optica 9, 645 (2022), doi.org/10.6084/m9.figshare.19755295, licensed CC BY 4.0. Images converted to WebP (S5 and S6 also resized); content unmodified.

Charts re‑plot values read from the paper’s Figs. 4 and 7. Beam images on this page are simulations unless labelled measured. The 3D model follows the shapes in the paper’s Fig. 1 and Supplement Fig. S2 with representative sizes; the light inside the printed waveguides is drawn as their guided ring or spot. From the ring‑maker’s end face on, the light is a wave simulation: a ring of light (radius R, width set by Δ, optionally twisted) spreads across one focal length of air, passes a thin lens, and propagates on (exact angular‑spectrum propagation). Its core widths and ring sizes agree with the paper’s Eqs. (1) and (3) to within about 6%, and its needle lengths with Eq. (2) to within about 20%. Chapter 2’s waves are a simplified model of tilted beams. The staircase is an illustration, and the bead trapping an illustrative model.