The dark lines a wave cannot avoid

Two waves of light that overlap make fringes: bands where they arrive in step and add, and bands half a cycle out of step where they cancel. What a thousand slits buy followed what happens as more waves join — the bright bands sharpen and the dark regions widen — but always with the waves from a regular grating, arriving in a fixed pattern of steps. What has not been asked is what the dark places look like in general, when several waves of the same colour overlap from arbitrary directions with arbitrary phases.

The answer is that the dark places stop being fringes and become points. Two waves cancel along whole lines, because two arrows of equal length point in opposite directions along a whole family of positions. Three arrows of comparable length can add to zero only at isolated positions, where their directions happen to close into a triangle. At each such point the light is exactly zero, and round each one something remarkable happens to the phase. John Nye and Michael Berry worked out the general theory in 1974, prompted by radio echoes from the bottom of the Antarctic ice sheet that showed features nobody could explain, and called the points dislocations, by analogy with the defects in a crystal. Nye had been modelling the radio echoes with pulses of ultrasound in a tank, bouncing them off a rough surface, and the recorded wave trains kept showing places where a crest seemed to split into two, like the extra half-plane of atoms in a crystal dislocation. The split crests were the visible signature of a phase winding round a point where the echo was exactly zero. They are now usually called optical vortices.

Points where three waves cancel #

The simplest case is three plane waves of equal strength, travelling in directions 120° apart. Their brightness forms a hexagonal pattern of bright spots, and between the bright spots are points of exact darkness. The lines of equal phase, drawn every 60°, show what is special about those points: lines of every phase meet there. Going once round a dark point, a path crosses every phase from zero to a full cycle, so the phase winds round the point by one turn. Round half the points it winds one way and round the other half the other way, and the two kinds alternate across the lattice.

The phase of a wave at a point is the angle of an arrow — the arrow that turns once per cycle and whose length is the amplitude. Where the arrow has zero length, its angle is undefined, and nothing forbids the angles nearby from taking every value. What makes the winding unavoidable is that it is an integer. Going round a closed loop, the phase must come back to its starting value up to whole cycles, so the total turning round any loop is a whole number of cycles. If the number is not zero, the loop must enclose a point where the phase is undefined — a zero of the amplitude.

Why points, and why three #

The reason the dark places are points can be put as a count of conditions. The amplitude of a wave at a point is a complex number — an arrow with two components — and for the light to be exactly zero both components must vanish. Two conditions on a position in a plane, which has two coordinates, are satisfied at isolated points; in space, with three coordinates, along lines. That is the same counting that makes two energy levels avoid crossing unless two separate conditions are met at once, and it is why a generic zero of a wave is neither a whole region nor nothing at all.

Two waves are the exception that makes fringes look normal. The sum of two arrows can vanish only if the arrows are exactly equal in length, and for two plane waves of unequal strength they never are, anywhere: the “dark” fringes of unequal beams are only dim. For two beams of exactly equal strength the length condition holds everywhere at once, and the zeros spread out into whole lines in a plane and whole sheets in space — the fringes of textbook interference. That is a coincidence of the special case. Add a third wave, or make the two beams’ strengths vary across the field, as every real beam’s does, and the sheets break up into lines, which is what light generically does.

Why they cannot be removed #

An integer cannot change by a small amount, so a small change to the light cannot change the winding round any loop. That is the whole of why vortices are robust. Nudge one wave’s direction, change its phase a little, weaken it slightly: the dark points move, but the winding round any loop that stays away from them does not change, so they cannot simply vanish.

They can disappear in pairs. A point of winding +1 and a point of winding −1 have a total winding of zero, and a loop round both of them sees no net turning; if they move together and meet, they can annihilate without any loop far away noticing. The figure shows exactly that.

Make one of the three waves stronger. Its amplitude is $a$ while the others stay at one. Three arrows of lengths 1, 1 and $a$ can close into a triangle — add to zero — only if each is shorter than the sum of the other two, the triangle inequality. As $a$ approaches 2 the triangles that close become flatter and rarer, the positions where they close come together in pairs of opposite winding, and at $a = 2$ each pair meets and annihilates. Beyond it the strong wave can no longer be cancelled anywhere, and the pattern has no dark points at all, only dim ones.

Creation is the same thing in reverse: lower the amplitude back below 2 and the pairs reappear, born together at the places where they died. Phase singularities obey a conservation law of the same kind as electric charge — net winding is conserved — and for the same reason that a superfluid’s vortices come in one size, they are quantised. In helium the quantum is a whole cycle of the condensate’s phase round a vortex core; in light it is a whole cycle of the optical phase round a dark line.

Random light is full of them #

Regularity is not needed. Any field made of three or more waves going different ways has vortices, and a random one has a great many.

The figure adds two dozen plane waves arriving from random directions with random phases — a fair model of the speckle that coherent light makes when it scatters off any rough surface — and finds the dark points by counting the winding round every small square of a fine grid. There are 46 of them in a window four and a half wavelengths across, with the two senses nearly balanced; the imbalance of two is a pair cut in half by the window’s edge. Berry and Mark Dennis showed in 2000 that a random wave made of components travelling equally in every direction has, on average, $k^2/4\pi$ vortices per unit area, which is π per square wavelength. The count here is a little under that and within the scatter 46 points allow.

So every speckle pattern — the grainy sparkle of a laser pointer on a wall — is threaded with lines of perfect darkness running through the space in front of the wall, and every dark grain in the pattern a camera records is, at its centre, one of these points. Nye’s radio echoes from the ice sheet were full of them for the same reason: the echo was a sum of reflections from a rough bed.

Light has a further property the scalar picture leaves out — the direction in which it shakes — and that has singularities of its own. In a field of mixed polarisation there are points where the light is exactly circularly polarised, so the orientation of the polarisation ellipse is undefined, and round which the ellipse’s axis turns by half a cycle; and lines where it is exactly linear, so its handedness is undefined. Nye found these in 1983 by the same counting, and they are as generic as the dark points. The blue sky’s polarisation pattern has its own version: a handful of points — named for Arago, Babinet and Brewster — where the skylight is unpolarised and the direction of its polarisation is undefined, round which that direction turns, and which every map of the sky’s polarisation must contain.

The same points, elsewhere #

Phase singularities are not an optical curiosity. Any wave with an amplitude and a phase has them. The tides in an ocean basin are waves of sea level with a daily phase, and there are points where the tidal range is zero and the time of high water turns round through the whole day as one goes round them — the amphidromic points of tide tables, charted in the nineteenth century before anybody called them vortices. Sound fields in rooms have them. The electron waves in a magnetic field have them, and so does the wave function of an electron passing round a solenoid, whose phase winds by an amount set by the flux inside.

In a ray picture they are invisible. The surface every ray is normal to found that rays from a point are the normals of a family of wavefronts, and that near a vortex the wavefront is a helical ramp climbing one wavelength per turn — normal to every ray, but not a single surface. The axis of the ramp is the dark line, and it is where geometrical optics, which knows nothing of phase cycles, has nothing to say.

Writing a twist into a beam #

A beam can be given a single vortex on its axis deliberately. The cleanest way is to pass it through a plate of glass whose thickness rises steadily round the centre by exactly one wavelength’s worth of optical path, a spiral staircase that delays each part of the beam by an amount proportional to its angle round the axis. The more common way uses a grating with a defect in it.

Interfere a twisted beam with a plain, tilted one and record the pattern. Far from the axis it is an ordinary set of straight fringes, a grating. At the axis the phase winding adds one extra fringe on one side, so a fringe splits into two like the tines of a fork. Now shine a plain beam onto the recorded fork. It is a hologram, and like every hologram it reconstructs the wave that made it: the first diffracted order comes out with its phase winding once round its own axis, and the order on the other side winds the opposite way. Forks with more tines make higher windings. Vladimir Bazhenov, Marat Vasnetsov and Marat Soskin made the first such gratings in 1990, and a computer-controlled liquid-crystal screen displaying a fork is now the standard way to make twisted light in a laboratory.

Finding a dark point #

A dark point is, by definition, hard to see: it is a place where there is nothing to see. What gives it away is its phase, which can be made visible by adding a reference. Interfere the light with a plain tilted wave and the result is a set of fringes, which bend wherever the light’s phase changes; round a vortex the phase changes by a whole cycle, so a whole extra fringe appears, and the fringe pattern has a fork in it at exactly the dark point. The fork’s direction gives the sign of the winding and the number of extra tines gives its size. This is how vortices in speckle, in the focus of a lens and in the beams of every twisted-light experiment are counted, and it is the same pattern as the grating in the figure above, read in the opposite direction.

The energy of the light shows the vortex in a second way. Where the phase turns round a point, the direction in which the wave’s energy flows — perpendicular to the surfaces of equal phase — turns round it too, so the energy circulates round each dark line like water round a drain. The circulation is what carries the angular momentum, and it is why a twisted beam can turn things.

The dark core and the twist it carries #

A beam with a vortex on its axis must be dark on the axis, and the darkness is not a shadow of anything. The phase changes by $2\pi\ell$ round a circle of radius $r$, so the faster the winding the faster the phase must change round small circles, and a wave cannot sustain a phase gradient steeper than its wavelength allows without its amplitude going to zero. The brightness rises from the axis as $r^{2\ell}$ and peaks at a radius that grows with $\ell$, so beams of higher winding have larger dark cores — a doughnut whose hole grows with the twist.

In 1992 Les Allen and colleagues at Leiden pointed out that such a beam carries angular momentum about its axis that has nothing to do with polarisation: $\ell\hbar$ per photon, from the twist of its wavefronts. A small particle trapped in the dark ring of a focused twisted beam is set spinning round the axis, and the effect was demonstrated in 1995. It is the light-borne cousin of angular momentum stored in fields that seem to be doing nothing, and it has made twisted light a tool for rotating microscopic objects, for carrying information in the winding number, and for testing how angle and angular momentum trade against each other.

What the pictures cannot show #

Every figure is a slice: the field in a single plane, where vortices are points. In three-dimensional space they are lines — threads of darkness running through the light, curving, looping, and occasionally reconnecting, much as vortex lines do in a fluid. A speckle field in front of a wall is a tangle of such lines, and a twisted beam carries one straight down its axis. Nothing drawn here shows the lines themselves or how they knot, and knotted vortex lines in light were made deliberately for the first time in 2010.

The fields are also perfectly monochromatic. A vortex is a zero of a single-frequency wave; in light of several colours the zeros of the different colours sit in different places, and white light has no exact darkness, though its colours fan out in a characteristic spiral round each near-zero. And the drawing of a fork shows the band edges of an idealised recording, where a real hologram has finite contrast and so also diffracts some light into unwinding orders.

The domain of the argument is any scalar wave of one frequency in which three or more components overlap. Inside it, zeros of the amplitude are generic, quantised and conserved in pairs. For two waves alone, or for fields so smooth that only one wave dominates everywhere, they need not exist at all.

Still open: what the tangle of dark lines looks like #

In three dimensions the vortex lines of a random wave form a tangle, and its statistics are not fully known. Simulations by Kevin O’Holleran, Dennis and colleagues found that most of the length of vortex line in a large random field belongs to lines that wander through the whole volume rather than to small closed loops, and that the lines behave in some respects like random walks and in others not; the fraction that is knotted, and how the answer depends on the field’s spectrum, are measured mostly by simulation. Whether such tangles can be engineered for use — optical knots carrying information, or fields designed to trap atoms along a chosen curve of darkness — is being explored.

The points themselves need only counting to understand. Where three or more waves of one frequency overlap, the amplitude vanishes at isolated points in a plane — lines in space — round which the phase turns through a whole number of cycles; the number cannot change continuously, so the points survive any small disturbance and are created and destroyed only in opposite pairs, and random light holds about π of them per square wavelength. A fork in a grating writes one into a beam, and the beam’s dark core is not a shadow but the price of its twist.