The packet that bends with nothing pushing it

The packet that moves at another speed found that a group of waves travels at its group velocity, not at the speed of its crests. The packet that will not keep its shape found the second thing a dispersion relation says: a packet spreads, at a rate fixed by its own bandwidth, because its components travel at different speeds and drift apart. Later essays found the exceptions that nonlinearity allows — the pulse two failures keep alive, the solitons hidden in a hump, the even train that breaks into a rogue wave — each a balance between spreading and a medium that steepens.

This essay finds an exception with no medium and no nonlinearity at all. In a perfectly linear equation, with nothing but free propagation, there is a packet that does not spread. That is surprising enough. What is stranger is what it does instead: its peak accelerates, sideways, along a parabola, as if it were falling — while nothing pulls on it. Berry and Balazs found it in 1979, as a curiosity of quantum mechanics, and called it a nonspreading wave packet. It is now made routinely with light, and the reason it can exist turns out to be the reason rainbows have bright edges.

The equation every free packet obeys #

A quantum particle with no force on it obeys the free Schrödinger equation; in units where Planck’s constant and the mass are one,

$$i\frac{\partial\psi}{\partial t} = -\frac{1}{2}\frac{\partial^2\psi}{\partial x^2}.$$

The same equation, with $t$ replaced by the distance along the beam, governs the transverse profile of a light beam in the paraxial approximation — a beam whose rays make small angles with its axis — and the profile of a short pulse in a medium with ordinary dispersion. Every result about free quantum packets is therefore also a result about beams of light, and the other way round.

Its best-known solutions spread. A Gaussian packet of width $w_0$ widens to $w_0\sqrt{1 + t^2/w_0^4}$: the narrower it starts, the faster it spreads, because sharpness has to be paid for in a spread of momenta, and a spread of momenta is a spread of velocities. That seems to be the end of the matter for a free particle: whatever its shape, it is made of plane waves moving at different speeds, and they must drift apart.

A packet that keeps its shape #

Berry and Balazs looked for a solution whose intensity keeps its shape, and found one built from the Airy function,

$$\psi(x,t) = \mathrm{Ai}\!\left(x - \frac{t^2}{4}\right)\exp\!\left[\frac{it}{2}\left(x - \frac{t^2}{6}\right)\right].$$

The intensity is $|\mathrm{Ai}(x - t^2/4)|^2$: the same function at every moment, merely shifted, by an amount that grows as the square of the time.

The Airy function decays rapidly on one side and oscillates on the other, with lobes that get narrower and dimmer going left. Its brightest lobe sits at $x = -1.02$. At later times the whole pattern has moved right by $t^2/4$: to $-0.77$ at $t = 1$, $-0.02$ at $t = 2$, $2.98$ at $t = 4$. That is uniform acceleration — the motion of a body falling under a constant force — in a problem with no force. Nothing about the shape has changed: the lobes have the same widths and the same heights. The packet neither spreads nor slows; it bends.

The rays that make the bend #

Two puzzles need answering, and the second answers the first. How can a free wave keep its shape, when its components move at different speeds? And how can anything accelerate with nothing pushing it?

The answer is in the rays. In the limit of short wavelengths, a wave behaves as a family of rays, and for a free particle every ray is a straight line in the plane of position and time: each component of the packet has its own velocity and keeps it. The Airy packet’s phase, read off from the formula, assigns each point of the tail a velocity that grows the further left it starts. Rays starting further left travel faster to the right. A family of straight lines, each starting further back and travelling faster, does not cross itself at random; its members touch a curve that bends, and the curve they all touch — their envelope — is the parabola $x = t^2/4$.

An envelope of rays is a caustic, the place where neighbouring rays crowd together and the light is brightest. The angle the rainbow has to be found the rainbow at a caustic, the angle at which the rays leaving a raindrop pile up and turn back, and Airy himself worked out the pattern of light near that caustic in 1838: a bright fringe, with dimmer fringes on the lit side and darkness on the other. The wave pattern near any fold caustic is the Airy function. So the Airy packet’s bright lobe is a caustic travelling along a curved envelope of straight rays, and no ray is accelerating. The lobe accelerates because it is not made of the same rays from one moment to the next: it is wherever the rays happen to be crowding at that moment, and the crowding point moves along the parabola while each ray goes straight.

That also answers why the shape holds. The caustic is a property of the family of rays, not of any one of them, and the family is arranged so that its crowding looks the same at every moment, only displaced. Spreading is what happens to rays that diverge from a common region; these rays converge onto a moving fold, and keep converging.

What Newton would say #

A packet whose peak falls with no force on it looks like a violation of the first law, and of Ehrenfest’s theorem, which the average that obeys Newton found exact: the average position of any quantum packet moves as Newton says a particle would under the average force. With no force, the average position must move at constant velocity.

It does. For the ideal packet the question cannot even be asked, because its tail stretches to minus infinity with lobes that fade too slowly for the average to exist; that is also why the ideal packet carries infinite energy and could never be made. The physical version is the Airy function with its tail cut down by a factor $e^{ax}$, and for that one the centroid exists, and the figure computes it, propagating the packet exactly. The brightest lobe accelerates along the parabola, out to 24 by $t = 10$. The centroid does not move, to within four thousandths. Ehrenfest’s theorem holds; the bending is a property of where the packet is brightest, not of where it is on average. The long, dim tail on the left carries the opposite motion, drifting back as the head runs forward, and keeps the average still.

The cost of a finite tail #

The factor $e^{ax}$ that makes the energy finite also makes the packet mortal.

The head is continually rebuilt from the tail, and a truncated tail eventually runs out. The main lobe’s peak then fades, as $e^{-at^2/2}$: slowly at first, then faster. Compared with a Gaussian packet of the same width, the Airy lobe wins handsomely at first — at $t = 2$ the Gaussian has fallen to 0.43 of its peak while the Airy lobes have kept 0.95 and 0.87 — and still wins at $t = 6$. With a larger truncation factor it eventually loses, because the Gaussian fades only as one over the time and the Airy lobe as a Gaussian in time. A smaller $a$ keeps the lobe going longer, and demands more energy in a longer tail. A real Airy beam is a trade between how far it bends without spreading and how much light it must waste in its tail to do so.

The snapshots show the structure of that trade. At the start most of the light is in the bright head and the first few side lobes; by $t = 7$ the head has travelled twelve units and lost more than half its height, while the tail behind it has spread into a long, low ripple. The centroid, marked by the dotted line, sits far behind the head the whole time. What the eye follows is not where the light is but where it is concentrated.

A head that grows back #

The rays picture predicts something else, and it is the property that has made Airy beams useful.

The head at any moment is made of rays that were in the tail a moment before. Block the head, and the rays that would have formed it later are still on their way, coming from further back. The figure blots out the main lobe entirely at the start, leaving only the side lobes, and propagates what remains: by $t = 4$ a new main lobe has formed where the unobstructed packet’s would be, with almost half the intensity. The beam heals. In optics this lets an Airy beam carry a bright spot round an obstacle in its path, or through a scattering medium that knocks out parts of the beam, and recover on the far side.

How big the bend is in a real beam #

The dimensionless units hide how large the effect is. For a light beam, $x$ is measured in units of a chosen transverse length $x_0$, the width of the main lobe roughly, and $t$ in units of $kx_0^2$, the distance over which a beam of width $x_0$ would diffract appreciably; $k$ is the light’s wavenumber. Take near-infrared light of one micrometre wavelength and a main lobe fifty micrometres wide. Then $kx_0^2$ is sixteen millimetres, and after ten centimetres of travel, about six units, the lobe has moved sideways by ten units: about half a millimetre, ten times its own width, with its width unchanged. A Gaussian beam fifty micrometres wide would have spread to six times that width over the same distance.

The same scaling says why the effect needs a narrow lobe to be seen. The sideways shift grows as the square of the distance divided by the cube of the lobe width; a lobe a millimetre wide bends by less than a hundredth of its width over a metre. Accelerating beams are a phenomenon of fine structure — tens of micrometres for light, nanometres for electrons — exactly where ordinary beams spread fastest.

Pulses that arrive late on a curve #

Because the same equation governs a pulse in a dispersive medium, with time and position exchanging roles, the Airy shape works in time as well as in space. An optical pulse shaped as an Airy function of time, sent into a fibre with ordinary dispersion, keeps its shape over distances where a Gaussian pulse of the same duration would smear out, and its peak arrives progressively later — or earlier — along a parabola in delay against distance: a pulse whose arrival time accelerates. Combining the two, an Airy profile in both space and time makes a packet of light that neither spreads nor diffracts for a while, and curves in both, which has been demonstrated under the name of an Airy light bullet. The trick in every case is the same cubic phase, and the same caustic of straight rays in a plane that now includes time.

A falling frame turns it into a standing wave #

There is one more way to see why the packet must exist, and it is the most economical. Describe the free particle from a frame that accelerates at the packet’s own rate, a half in these units, the way the fall that does not depend on what is falling replaced gravity with an accelerating lift. In that frame a free particle feels a uniform force, as everything in an accelerating lift feels a weight, and the Schrödinger equation acquires a linear potential. A particle in a linear potential has a stationary solution: the Airy function, a fixed pattern decaying on the uphill side and oscillating on the downhill side. Seen from the accelerating frame, the Berry–Balazs packet is that stationary pattern, standing still. Seen from the laboratory, the stationary pattern of the falling frame is a pattern that falls.

Made with light, then with electrons #

For thirty years the Airy packet was a mathematical curiosity, because its tail made it impossible to prepare exactly. In 2007 Siviloglou, Christodoulides and colleagues made it with light, using the fact that the Fourier transform of an Airy function is a pure cubic phase: shine a laser through a spatial light modulator that imposes a phase proportional to the cube of position, focus it with a lens, and the far field is an Airy beam, truncated by the laser’s own Gaussian profile. The beam’s main lobe curved sideways as it travelled by a measurable amount over tens of centimetres, along a parabola, and healed after obstructions.

The applications followed the properties. A curved bright lobe can be used to sweep small particles along a curved path in a liquid, a technique nicknamed optical snowblowing. Intense Airy pulses ionise air along a curved channel, making a plasma filament that bends. Light-sheet microscopes use Airy beams to illuminate a thin slice of tissue over a longer distance than a Gaussian beam of the same thickness could, because the Airy lobe does not spread. And in 2013 the same mathematics was realised with electrons: a beam of electrons passed through a nanofabricated mask with a cubic phase pattern came out as an Airy electron beam, bending in free space inside an electron microscope, a matter wave obeying the free Schrödinger equation exactly as Berry and Balazs had described.

Why this is not as strange as it looks #

The deeper lesson is that “free particles move in straight lines” is a statement about rays, and a wave is not a ray. A wave packet’s bright spot can follow any curve that a family of straight rays can have as an envelope, and the parabola is the simplest. More elaborate phases give packets that follow other curves — circles, even — for as long as the rays forming the caustic last, and they have been made. What Newton’s first law constrains is the average, and the average obeys it exactly.

The connection with the quarter cycle a turning point costs and the bounce that comes in fixed heights is closer than it looks. A neutron above a mirror under gravity has stationary states that are Airy functions, because near a smooth turning point every wave is one. The free Airy packet is the same function made to move, as the falling frame above showed: the equivalence principle, applied to a wave, turns the standing pattern above a floor into a pattern bending through empty space.

What the pictures cannot show #

The figures solve the free equation in one dimension with a single truncation factor; real Airy beams are two-dimensional — the product of two Airy functions, curving diagonally — and are truncated by the laser’s profile rather than by an exponential. The propagation is exact for the paraxial equation, which fails for beams so tightly focused that their rays make large angles with the axis; nonparaxial accelerating beams follow circular rather than parabolic paths and have been studied separately. And the centroid figure tracks an average over the whole packet, including a tail extending far to the left of the plotted range; on any finite screen, a measurement of the average would miss part of the tail and see a small drift.

Still open: how far an accelerating beam can carry a useful spot #

Applications want an accelerating lobe that is both narrow and long-lived, and those two demands fight: a narrower lobe needs a wider spread of transverse momenta and so a faster-fading head for the same power, and every extra metre of propagation must be paid for in tail. How close practical beams can come to the limits, whether shaping the truncation more cleverly than an exponential can extend them, and whether accelerating packets of light, electrons or even neutrons can be used to steer or probe in places a straight beam cannot reach, are being explored in optics laboratories and electron microscopes now.

The habit worth carrying away is to ask whether a bright spot is a thing or a place where things are crowding. The Airy packet’s lobe is a caustic — the moving envelope of a family of straight rays — so it can follow a parabola, $x = t^2/4$, without spreading and with no force, while the packet’s centroid stays put as Ehrenfest’s theorem requires. Its tail pays for the bend, and blocking the head only removes what the tail will send again.