The beam that does not spread and the price it pays

The fan of plane waves inside every beam found that any beam of light is a sum of plane waves travelling at slightly different angles, and that a beam spreads because those waves fall out of step as they travel. A narrow beam needs a wide fan of angles and so spreads fast; that is diffraction, stated as a property of a sum. The essay noted one exception in passing: a beam made of plane waves all travelling at the same angle to the axis, a cone, does not change its shape as it goes, and in a sentence it named the price. This essay makes the beam, computes its price, and follows what it is used for.

The device is an axicon, a lens with a conical surface instead of a spherical one, and its effect on a laser beam looks like a defeat of the rule that every beam must spread. Measured in the computation behind the figures, it is a rearrangement of where the light goes, and every number in it follows from the one fact that each ring of the beam carries the same power.

A cone of plane waves #

A conical surface bends every ray that passes through it by the same angle towards the axis, whatever its distance from the axis. Light from a parallel laser beam leaves an axicon as a cone of plane waves, each converging on the axis at the same small angle α, half a degree in the figures. Where the waves from opposite sides of the cone cross they interfere, and because the cone is symmetric about the axis the pattern depends only on the distance r from the axis. Summing plane waves arriving from every direction round a cone gives the Bessel function $J_0(k\alpha r)$, with $k = 2\pi/\lambda$, and the intensity is its square.

The first figure is computed, not drawn from the formula: the Fresnel diffraction integral of the light leaving a ten-millimetre axicon is evaluated across the beam at 29 centimetres. It gives the Bessel pattern: a bright central core whose first dark ring is at $2.405/k\alpha$, 28 micrometres from the axis for red light and half a degree, and around it ring after ring, each fainter at its peak than the last but wider in circumference.

A Gaussian beam focused by a lens to a spot of the same size, drawn dashed, has no rings. Its light is all in the central spot, as the light of a focused beam usually is. The rings are the first visible sign that the axicon’s beam is a different kind of object.

A core that keeps its width #

The second figure follows the width of the core with distance. Over the 57 centimetres within which the cone of light overlaps itself — the distance $R/\alpha$ at which rays from the axicon’s edge reach the axis — the core’s half-width at half its peak stays between 13 and 14 micrometres. A Gaussian beam with that half-width at its waist has a Rayleigh range of 2.7 millimetres: its width grows by a factor of $\sqrt2$ within that distance of the focus and doubles within five. The axicon’s core keeps its width over about two hundred of those ranges.

This is the feature that gave such beams their name, non-diffracting, when James Durnin demonstrated one in 1987 with an annular slit and a lens. The name is a description of the core’s width, not of the light. The reason the core keeps its width is that it is never made from itself. At a distance z from the axicon the light arriving on the axis comes from a thin ring of the axicon at radius αz, the only part of the aperture whose rays reach the axis there. A little further on, a slightly larger ring supplies it. The core at each distance is formed afresh by light that has, until then, been travelling in the outer parts of the beam, and when the cone runs out of aperture at $R/\alpha$, the core runs out of light and the beam ends.

How little of the light is in the core #

That supply has a cost, and the third figure measures it. Integrating the computed intensity outwards from the axis, the power enclosed within a radius grows almost in a straight line. Each ring of the Bessel pattern, though dimmer than the one inside it, is wider by just enough to carry the same power; the brightness of the n-th ring falls as $1/n$ and its circumference grows as $n$. At 29 centimetres the beam’s light extends out to 2.5 millimetres from the axis, through about sixty-nine rings, and the core, being one ring’s worth, holds a little over one per cent of the power.

So the core is narrow and long-lived because it is poor. A Gaussian beam could hold all its power in a 28-micrometre spot, and would spread in millimetres; the Bessel beam keeps 28 micrometres for half a metre by holding ninety-nine per cent of its power outside the spot. The brightness no lens can increase found that no passive optic can increase the brightness of light — the power per area per solid angle — and the axicon does not. It trades concentration for length.

Why every ring carries the same power #

The equal shares of power come straight from the shape of the Bessel function. Far from the axis, $J_0(x)$ behaves like $\sqrt{2/\pi x}\,\cos(x - \pi/4)$: an oscillation whose amplitude falls as the inverse square root of the distance. The intensity, its square, falls as $1/x$. A ring of the pattern at distance x has a circumference proportional to x and a width fixed by the period of the oscillation, π in these units. Its power is intensity times area, $(1/x)\times x \times \pi$, which does not depend on x at all. Every ring carries the same power, the core no more than any of them.

That is the arithmetic of a cone of plane waves, and no design of axicon can change it, because it follows from the angle alone. A beam with a core that never spreads and finite power cannot exist: the rings would have to go on for ever, each with its equal share, and their sum would be infinite. A real axicon’s beam has as many rings as fit within the light that has not yet passed the axis, and as the beam travels the outer rings are used up feeding the core and the beam’s edge moves inward. That is why the line ends where it does, and why the number of rings at the middle of the line, sixty-nine here, sets the core’s share.

A line of light against a point of it #

The fourth figure compares the brightness along the axis with that of a lens of the same aperture. The axicon’s axis brightens steadily with distance: each point is fed by a ring of the aperture of radius αz, and a larger ring has a larger circumference, so it delivers more light to the axis. The intensity grows in proportion to the distance until the ring reaches the aperture’s edge, where the light runs out and the axis goes dark within a few centimetres. The ripples near the end are the edge of the aperture making its own diffraction rings.

A lens puts the same light to a point. At its focus, 29 centimetres away, the intensity is a hundred and eighty-eight thousand times the incident, fifty times the axicon’s best, and three centimetres beyond the focus it has fallen five hundred times. The two optics have the same aperture and pass the same power. One concentrates it into a short, intense focus; the other spreads it along a long, modest line. The integral of the on-axis brightness along the axis, a measure of how much light has been put on the axis altogether, is far closer between them than their peaks.

A core that grows back #

The same supply explains the property most often advertised for such beams, that they heal themselves. Block the middle of the axicon with an opaque disc and the near part of the line disappears: the light that would have formed the core there came from the disc. But the light from outside the disc reaches the axis further on, at distances beyond b/α for a disc of radius b, and there the core forms exactly as before, as the last figure computes: dark to seventeen centimetres, then back, within a few centimetres, to nearly its unobstructed brightness.

An obstacle placed in the beam downstream does the same thing, for the same reason. It blocks the core and some of the rings at one distance; behind it, the light that will form the core further on is still travelling in rings outside the obstacle’s shadow, and the core reappears after a distance set by the obstacle’s size divided by the cone angle. That is why such beams are useful for looking into or cutting through scattering material, where a Gaussian focus would be scattered away and the Bessel core is rebuilt from the light around it.

The same line behind a coin #

An axicon is not the only way to make a cone of waves. Any ring of light that sends its waves inward at one angle does it, and the oldest example was found by accident. The wave that comes from the rim followed the light diffracted round the edge of an obstacle, and the edge of an opaque disc is a ring: every point on it sends light inward towards the axis behind it at a slightly different angle for each distance. On the axis those contributions arrive in step, which is why there is a bright spot at the centre of a disc’s shadow — the spot Siméon Poisson predicted in 1818 as an absurd consequence of the wave theory and François Arago then saw. That spot is not a point. It persists along the axis behind the disc, for as far as the shadow extends, with a narrow width set by the disc’s size and the distance: a short, faint Bessel line made by a coin. The difference from the axicon is only that a disc’s edge sends its light inward at an angle that changes with distance, so the spot widens as it goes, where the axicon’s fixed angle keeps its core the same.

The same trick works for any wave. Ultrasound engineers build annular transducer arrays that launch cones of sound, so that a narrow beam keeps its width through a greater depth of tissue; electron microscopists have made electron Bessel beams with annular apertures; and radio engineers have made them with circular arrays of antennas. In each case the long, narrow core is bought, as in the figures, with the power in the rings.

What the line is used for #

Light-sheet microscopes, which image living tissue by illuminating one thin plane of it at a time from the side, use Bessel beams swept sideways to make a sheet that is thin over a long distance; a Gaussian sheet as thin would be as short as its Rayleigh range. Laser machining uses them to drill long, narrow holes in glass and to cut transparent materials along a line, because the core keeps its width through the whole thickness. The depth a broad spectrum can see described optical coherence tomography, which can be given an extended depth of focus with an axicon. And optical tweezers made from Bessel beams hold particles at several points along the axis at once.

In every one of these uses the rings are a nuisance that has to be managed. A light sheet made from a Bessel beam illuminates the tissue on either side of the plane being imaged with the light in its rings, which blurs the image and bleaches fluorescent markers outside the plane; much of the engineering of Bessel light-sheet microscopes goes into suppressing or computationally removing the rings’ contribution. The price named in the third figure is paid by every instrument that uses the beam.

Where the description holds #

The figures treat a perfect axicon lit by a perfectly uniform plane wave. A real laser beam is Gaussian, brighter at the centre, so the inner rings of the axicon are fed more than the outer ones and the core’s brightness along the axis rises and then falls rather than climbing to the edge. Real axicons have a slightly rounded tip, which sends the central light off in a broad cone and spoils the core’s first centimetres. The Fresnel integral used is a paraxial approximation, accurate for a half-degree cone and not for steep ones, and the light is treated as a scalar wave, which is accurate for small angles: for steep cones the polarisation of the light changes the core’s shape. None of these changes the arithmetic of the rings, which follows from the cone alone.

What the figures cannot show #

The figures show intensity in a plane and along a line. They cannot show the beam’s structure in three dimensions: that the rings are cylinders, nested round the core along the whole length of the line, and that the light in them is moving inward, towards the axis, while the light in the core is moving outward, towards the rings beyond. The beam is a steady flow of light through the core rather than a static pattern, which is the self-healing argument in a different form. Nor can they show the beam’s phase, which in the core advances along the axis slightly faster than a plane wave’s would — the core travels with a phase velocity greater than that of light and a group velocity, for a short pulse, slightly less — a property that has led to claims of light pulses travelling faster than light along the axis, which on inspection are claims about where the light is being fed in from, not about anything travelling faster.

Still open: how dark the rings can be made #

The ideal Bessel beam carries infinite power in its infinitely many rings, and every real one is a compromise between the length of its core, its width and how much light its rings waste. Shaped beams designed since — Airy beams that curve, beams whose core can be made to vary in a chosen way along the axis, combinations of Bessel beams of slightly different cone angles that cancel the rings’ light in some regions — try to keep the long, narrow core while reducing what goes to waste. How far the rings can be suppressed while keeping the core’s length, and whether a beam can be found that is long, narrow and efficient all at once, runs into theorems about how much light can be confined in a region of given length and width. Those theorems set bounds; the best achievable trade for practical apertures, and how close the shaped beams in use come to it, is being worked out for each application.

The habit worth carrying away is to ask where a beam’s power is, not just where its brightest part is. An axicon turns a laser beam into a cone of plane waves that interfere into $J_0(k\alpha r)^2$, a core 56 μm across that keeps its 13–14 μm half-width for 57 cm, 200 Rayleigh ranges of a matched Gaussian, because every point of it is fed afresh by a ring of the aperture — and since every ring of the pattern carries the same power, the core holds only about one per cent of it, 1.16% among 69 rings, and the axis reaches 3,476 times the incident intensity where a lens of the same aperture reaches 188,000. The core does not escape diffraction. It is made of diffraction, continuously, from light that was somewhere else.