The light that spreads in two beams

The gap a repeat opens found that a wave in a periodic medium has a band of allowed frequencies and a gap where none travels, and the mode that lives in the mistake, the frequency a lattice cannot carry, the end that knows how the middle was cut, the mirror that works from every direction and the sandbars that reflect the sea followed what the gap does: traps light at defects, reflects it perfectly, protects states at edges. Two tails that swap everything, in another part of the collection, found two waveguides side by side exchanging their light completely and back again, as their evanescent tails overlap.

Put many such guides side by side, and the periodic medium is no longer a stack that a wave crosses but a row along which light travels, hopping sideways as it goes. The band now describes how light spreads across the row, not whether it gets through, and the answer differs sharply from ordinary diffraction. This essay follows light launched into a single guide of such a row, finds that it spreads in two beams rather than one hump, and finds in the band’s curvature a direction where beams do not spread and a direction where they un-spread.

Hopping between neighbours #

A single-mode waveguide carries light along its length at a fixed propagation constant; its field extends a little way outside its core as an evanescent tail. Two guides close enough for their tails to overlap exchange light at a rate set by the overlap, the coupling $C$, measured per unit length: two tails that swap everything found the light passing completely from one to the other in a distance $\pi/2C$ and back again.

In a row of many identical guides each couples to its two neighbours, and the amplitude $a_n$ in guide $n$ obeys a simple rule: as the light travels a short distance along the row, each guide gains amplitude from its neighbours in proportion to theirs, with a quarter-cycle phase shift. The rule is the discrete version of the wave equation, and it is exactly the equation for a quantum particle hopping between the sites of a one-dimensional lattice, with distance along the guides playing the part of time. Light launched into one guide spreads across the row the way an electron placed on one atom spreads through a crystal.

Two beams, not one hump #

The rule can be solved exactly. The amplitude in guide $n$ after a distance $z$ is a Bessel function, $i^n J_n(2Cz)$, and the power is its square. The figure plots it at four distances. At first the light leaks into the neighbouring guides; then it forms two beams moving outward, one to each side, at a steady two guides per coupling length. Most of the power sits at the leading edges of the beams, and the guides between are dim, with the centre almost dark: after seven coupling lengths the starting guide holds under three per cent of the light, and the brightest guides, twelve away on either side, hold eight per cent each.

This is called discrete diffraction. It was predicted for coupled waveguides in 1965 and seen in 1973 in arrays of guides in gallium arsenide, and from the late 1990s Hagai Eisenberg, Yaron Silberberg and their colleagues used arrays written into semiconductor chips to explore everything else the band makes possible. The same pattern appears for electrons launched at one site of a chain, for cold atoms in optical lattices, and for photons in the arrays used to build quantum circuits.

Why two beams #

The difference from ordinary diffraction is visible when the two are set side by side at the same overall width. Light from a narrow source in a slab of glass spreads into one smooth hump, brightest straight ahead; in the row of guides it spreads into two bright fronts with a dim, oscillating middle. The explanation lies in the band.

Any pattern of light across the row can be written as a sum of Bloch waves, each advancing in phase by a fixed step $k$ from one guide to the next. Each Bloch wave travels along the row with a propagation constant $2C\cos k$ added to the single guide’s, and moves sideways at a speed equal to the slope of that relation, $-2C\sin k$ guides per unit length. Light launched into one guide contains every phase step equally. In a uniform slab the corresponding relation between sideways wavenumber and propagation constant is a parabola, whose slope grows without limit, so the sideways speeds are spread evenly and the light fills a smooth hump. In the row the relation is a cosine, whose slope has a maximum, $2C$, at $k = \pm\pi/2$. Many phase steps near there share nearly the same sideways speed, so their light arrives together at the maximum distance, and the pattern piles up at its fronts — the same pile-up at an extreme speed that produces the bright edge of a caustic, or the rainbow’s bright edge at its limiting angle.

There is a second way to say the same thing that connects it to the walk that comes home. A particle taking random steps between neighbouring sites spreads diffusively, its width growing as the square root of the number of steps, into a Gaussian hump. The light in the array is not taking random steps; it is taking every possible sequence of steps at once, with amplitudes that interfere. That is a quantum walk, and interference makes it spread ballistically — its width grows in proportion to the distance, as $\sqrt{2}\,Cz$ — with its weight pushed out to the fronts. The two-beam pattern is the signature of a quantum walk, and arrays of coupled waveguides are one of the ways quantum walks are built.

The angle that sets the phase step #

A phase step between neighbouring guides sounds abstract, but it is set by something simple: the angle at which light enters the row. A plane wave arriving at a small angle to the guides’ axis reaches each guide slightly later than its neighbour, and the delay, measured in phase, is the wave’s wavenumber times the guide spacing times the sine of the angle. Launching straight on gives a phase step of zero; tilting the input beam increases it; and at a particular angle, set by the wavelength and the spacing, it reaches $\pi/2$ or $\pi$. The band is therefore a map from the angle of the input to the behaviour of the beam inside: how fast it walks across the row and whether it spreads, stays the same width or contracts.

That map has a feature a uniform medium lacks. In a slab of glass a beam tilted further always moves sideways faster. In the array the sideways speed peaks at $k = \pi/2$ and falls again beyond it, reaching zero at $k = \pi$, where the beam goes straight down the row again despite its steep entry. Past the peak, tilting the input more makes the beam move sideways more slowly — anomalous refraction, the kind the ray on the wrong side of the normal found in materials engineered to bend light the wrong way.

A band with three kinds of curvature #

The slope of the band sets how fast a beam moves sideways. Its curvature sets how fast the beam spreads, for the same reason that the curvature of a relation between frequency and wavenumber sets how fast a pulse spreads in time.

For a broad beam, containing a narrow range of phase steps around some $k$, the band can be treated as locally a parabola with curvature $-2C\cos k$. At $k = 0$ — a beam launched straight down the row, all guides in phase — the curvature has the same sign as in a uniform medium, and the beam spreads like an ordinary beam. At $k = \pm\pi/2$ — a beam launched at the angle that makes it cross the row fastest — the curvature vanishes, and to this order the beam does not spread at all. At $k = \pm\pi$ — neighbouring guides in antiphase — the curvature is reversed, and the beam behaves as if diffraction ran backwards: a beam that would spread, contracts.

The mass a curve decides found exactly this in the bands of a crystal, where the curvature of an electron’s band is its inverse effective mass, and near the top of a band the mass is negative. The light in the array is the optical version of an electron in a crystal, and its reversed diffraction at $k = \pi$ is the negative mass of an electron at the top of its band: pushed one way, it responds the other.

A beam that does not spread #

The figure launches the same broad beam — a Gaussian across a few guides — with three phase steps, and tracks its width. Launched straight down the row or in antiphase, it widens at the same rate, to six and a half guides after ten coupling lengths, because the band’s curvature at $k = 0$ and $k = \pi$ has the same size and opposite sign, and width does not care about the sign. Launched at a phase step of $\pi/2$, it moves across the row at the maximum sideways speed and barely widens at all, from one and a half to just over two guides, spreading only through the band’s much weaker third-order curvature.

A beam that crosses a medium without diffracting is something no uniform medium can offer. In free space the only way to prevent spreading is to make the beam wide — the fan of plane waves inside every beam found that spreading is set by the range of directions in the beam, and a narrow beam needs a wide range — or to use special beam shapes that carry infinite power in principle. The array offers a narrow beam that does not spread, as long as it travels at the right angle, because at that angle the band has no curvature for diffraction to act through.

Diffraction run backwards #

The reversed curvature at $k = \pi$ can do something more striking than slowing the spreading: it can undo it.

The figure launches a narrow beam straight down the row and lets it spread for four coupling lengths. Then it reverses the sign of the field in every second guide. That changes nothing about the power in any guide, but it shifts every Bloch wave’s phase step by $\pi$, moving the whole beam from the region of ordinary diffraction to the region of reversed diffraction. The phase differences the beam accumulated while spreading now unwind: the beam contracts, and four coupling lengths later it is back to its starting width, three-quarters of a guide.

In experiments the shift of phase step is made not by reaching into each guide but by the geometry: launching the light at the right angle, or building a section of array whose guides follow a zigzag so that light acquires the extra phase as it crosses. Arrays built this way have focused light and imaged it with no lens at all, combining sections of ordinary and reversed diffraction as a lens designer combines glasses of opposite dispersion. The general name is diffraction management, and it is the spatial counterpart of the dispersion management that keeps pulses short in optical fibre, where lengths of fibre with opposite dispersion are joined so that each undoes the spreading of the last.

The same equation on a chain of atoms #

The rule the guides obey — each site gains amplitude from its two neighbours at a fixed rate — is the tight-binding model of solid-state physics, written for light. An electron in a chain of atoms, each weakly overlapping its neighbours, obeys the same equation with time in place of distance along the guides, and an electron placed on one atom spreads through the chain in the same two beams, at a maximum speed set by the width of its band. Spin waves in a chain of magnetic atoms, excitations hopping between the molecules of a crystal, and atoms tunnelling between the wells of an optical lattice all do the same. The waveguide array is the easiest of these to watch, because the “time” is a distance along a chip and the whole history of the spreading can be photographed from above, by the light that scatters out of the guides as it goes.

That is why arrays of guides have become a test bench for band physics. Effects that are fleeting or hidden in solids — Bloch oscillations, localisation, edge states of the kind the end that knows how the middle was cut found at the boundary of a dimerised chain — can be built into an array by choosing the guides’ spacings and widths, and then seen directly in the light’s path.

When the guides are not identical #

Everything so far assumes a perfect row. Make the guides slightly different from one another, at random, and the picture changes completely. The walk that interference can stop found that waves in a disordered medium can be brought to a halt by interference among their scattered paths — Anderson localisation — and a disordered array of waveguides is one of the cleanest places to see it. Light launched into one guide spreads ballistically for a short distance, then stops spreading: its profile freezes into an exponentially decaying peak around the starting guide, and it stays there however far it travels. This was observed in 2008, in arrays written with random variations of the guides’ widths, by Yoav Lahini and colleagues. The two-beam pattern of the perfect row and the frozen peak of the disordered one are the two extremes of the same equation, and arrays of guides with controlled disorder have become a laboratory for studying the transition between them.

A gradient across the array — guides whose propagation constants increase steadily from one side to the other — does something different again: the light oscillates back and forth across the row instead of spreading, the optical version of the Bloch oscillation that the mass a curve decides found electrons in crystals are too quickly scattered to show. Waveguide arrays show it cleanly, because nothing scatters the light.

Where the model stops #

The equation used here keeps only nearest-neighbour coupling, assumes every guide identical and single-moded, and ignores loss and nonlinearity. Real arrays have weak coupling to next-nearest neighbours, which slightly distorts the band and makes the non-spreading angle only approximately non-spreading; guides are made with small unavoidable differences, which at long enough distances begin to localise the light; and at high intensity the guides’ refractive index depends on the light’s power, which in an array produces discrete solitons — beams that hold together by balancing the nonlinearity against discrete diffraction, and that exist in both the ordinary and the reversed-diffraction regions with opposite signs of nonlinearity. The coupled-mode picture also fails if the guides are so close that their modes are strongly distorted, when the full wave equation across the whole array must be solved.

What the pictures cannot show #

The figures show power, guide by guide, and the widths computed from it; they do not show the phase, which is where the band lives. Two beams with identical power profiles but different phase steps behave completely differently, as the width figure shows, and nothing in a photograph of the light leaving the array distinguishes them. The figures also cannot show how small these structures are: a typical array has guides a few micrometres wide and a few micrometres apart, written by lasers or etched into chips, with coupling lengths of a millimetre or so, and the entire two-beam pattern of the first figure fits within a millimetre across and a centimetre along.

Still open: how far a quantum walk can be scaled #

Arrays of coupled waveguides carrying single photons, or pairs of photons, are used to build quantum walks whose interference patterns are hard for classical computers to predict when many photons are involved. The two-beam pattern of one photon is easy to compute; the joint pattern of many indistinguishable photons launched into different guides is not, and measuring it is one of the proposed demonstrations that a quantum device can do something a classical computer cannot. How large such arrays can be made while keeping every guide identical enough, every coupling precise enough and loss low enough for the interference to survive — and whether the tasks they perform are useful beyond the demonstration itself — are questions of active research in integrated quantum optics.

The habit worth carrying away is to read a medium’s behaviour off the shape of its band rather than its presence. Light hopping between neighbouring guides has a sideways speed with a maximum, so it spreads in two beams that pile up at that speed, and the band’s curvature, −2C cos k, gives it ordinary diffraction at k = 0, none at π/2 and reversed diffraction at π — so a periodic row can carry a beam without spreading it, and gather up a beam it has spread. A uniform medium offers only the first; the repeat supplies the other two.