Illustrated Physics
physics, drawn
Waves

The coupler that does not care about the colour

Two identical guides side by side swap their light back and forth, and a coupler cut to the length of one swap works perfectly at one wavelength and badly at every other. Make the guides unequal, and sweep the inequality from one sign to the other along their length, and the light no longer swaps — it follows a single mode of the pair as that mode moves from one guide to the other. The transfer is then nearly complete across hundreds of nanometres of wavelength and immune to the widths being made wrong, and it costs length.
13 min read 5 figures The same equation againMade into an instrument

Assumes: Two tails that swap everything · The crossing that never happens

Two tails that swap everything found that two identical waveguides brought close together do not leak light into one another a little at a time. They exchange all of it and exchange it back, periodically, because the pair has two modes of its own — a symmetric one and an antisymmetric one, with slightly different speeds — and light launched into one guide is an equal mixture of both, beating between them. A directional coupler is cut to the length of half a beat.

That is also the device’s weakness, and it was left there as the open question. The beat length is set by the overlap of the two guides’ evanescent tails, a tail is an exponential whose reach depends on the wavelength, and so a coupler cut to transfer everything at one colour transfers less at every other. It is equally sensitive to anything else that changes the beat: the gap between the guides, and above all any difference between them, which caps how much light can cross at all.

The fix that is now standard in silicon photonics does not refine the beat. It removes it, by building the coupler so that the light never beats in the first place. The idea is borrowed directly from the crossing that never happens: make two modes approach, couple them, and sweep them past one another slowly enough that a system prepared in one of them stays in it while its character changes completely.

A cut coupler works at one colour

The comparison is made for silicon guides half a micrometre wide in silica, 250 nanometres apart, at the telecommunications wavelength of 1550 nanometres. Each guide’s mode, and the coupling between two of them, is solved from the transcendental condition for a slab waveguide at every wavelength, and the light’s passage along each coupler is integrated from the coupled-mode equations rather than taken from a formula.

A cut coupler works at one colour, a tapered one at all of them. The fraction of light that crosses from one silicon guide to its neighbour, 0.5 µm wide and 250 nm apart, against wavelength from 1300 to 1800 nm. The directional coupler is two identical guides cut to 37.6 µm, the length that transfers everything at 1550 nm; at other wavelengths the evanescent tails reach further or less far and the transfer falls to 3 per cent at the edge of the range. The tapered couplers have their widths swept 200 nm in opposite directions along their length, so the two guides' propagation constants cross in the middle. directional, 37.6 µm: above 95 per cent over 94 nm; tapered, 300 µm: above 95 per cent over 313 nm; tapered, 800 µm: above 95 per cent over 438 nm. Each curve comes from integrating the coupled-mode equations along the coupler, with the guide modes and their coupling solved afresh at every wavelength.
Fig. 1 The fraction of light crossing from one guide to the other against wavelength, for a directional coupler of identical guides cut to transfer everything at 1550 nm, and for two tapered couplers 300 and 800 µm long.

Two identical guides at this spacing transfer everything in 37.6 micrometres at 1550 nanometres. At 1400 nanometres the tails are shorter, the coupling weaker, and the same length transfers less; at 1800 nanometres the tails reach further, the light has beaten past full transfer and is on its way back, and only 3 per cent crosses. The band over which the cut coupler passes more than 95 per cent is 94 nanometres wide.

The sensitivity is steep because the coupling is an exponential of the gap measured in units of the tail’s decay length, and the decay length grows with wavelength. A field outside a guide is a wave squeezed below the frequency at which it could propagate in the cladding, and the further below that it is squeezed the faster it dies; longer wavelengths are squeezed less, decay more slowly, and reach the neighbouring guide with more of their strength. Across the 500 nanometres drawn, the coupling of these guides changes by more than a factor of four, and a device whose whole function is to be exactly half a beat long cannot survive a factor of four in its beat.

The tapered couplers are the same guides with one change. Along the coupler, one guide is made progressively narrower and the other progressively wider, by 200 nanometres from end to end, so that at the start the first guide is the wider and at the end the second is. A 300-micrometre version passes more than 95 per cent over 313 nanometres; an 800-micrometre version over 438. Neither has any sense of a design wavelength.

Following a mode through a crossing

What happens inside a tapered coupler is best seen by watching the two modes of the pair along its length, not the two guides.

The light follows one supermode through the crossing. Inside a tapered coupler 800 µm long at 1550 nm. Above: the fraction of the light in the first guide along the coupler, from integrating the coupled-mode equations, and dashed, the share of the first guide in the local supermode the light was launched into. They agree to within 8.3 percentage points along the whole length: the light does not beat between the guides but follows the supermode as that supermode changes from being in the first guide to being in the second. Below: the two supermodes' propagation constants relative to their average, which approach each other and repel across a gap of twice the coupling, 0.084 per micrometre, at the point where the guides are equally wide; dashed, the two single guides' constants, which cross.
Fig. 2 Inside a tapered coupler 800 µm long at 1550 nm. Above: the light in the first guide along the coupler, from the integration, and dashed, the first guide’s share of the pair mode the light was launched into. Below: the propagation constants of the pair’s two modes, which repel across a gap, and dashed, those of each guide alone, which cross.

At the start the first guide is much wider than the second, so its propagation constant is much larger, and the two guides are badly mismatched. A mismatched pair barely couples: its modes are almost exactly the modes of the separate guides, and light launched into the first guide is almost entirely in one mode of the pair. That is the difference from the identical pair, where light in one guide is always an even mixture of two modes that must beat.

Moving along, the widths approach each other and the two guides’ constants — the dashed lines in the lower panel — approach and cross in the middle, where the guides are equal. The pair’s modes do not cross. Coupled by the overlap of the tails, they repel, and the gap between them at the point of closest approach is twice the coupling. On each side the upper pair mode is mostly in whichever guide is wider, so as the crossing is passed the upper mode’s weight slides from the first guide to the second.

The light goes with it. The upper panel shows the fraction in the first guide falling smoothly from one to nearly nothing and staying there, tracking the share the local pair mode would have to within about eight percentage points — the ripple being the small amount of the other mode picked up near the crossing, where the gap between the modes is smallest and following is hardest. There is no beat because there is only one mode carrying the light. The coupler has stopped being an interferometer and become an adiabatic transfer: the same thing a ball bouncing between walls that close slowly does, keeping an invariant of its motion while the motion itself changes.

Slow enough, and how long that is

Following a mode through a crossing works only if the crossing is passed slowly compared with the rate set by the gap. Too fast, and the light cannot keep up; it jumps the gap and stays in the first guide, which is the outcome the two-level crossing calls a diabatic passage.

Robustness is bought with length. At 1550 nm, the fraction of light transferred against the length of the coupler. The directional coupler of two identical guides transfers all of its light at 37.6 µm and then gives it back, oscillating for ever with length. The tapered coupler, with the same guides, gap and 200 nm width swing, transfers little when short, because the crossing is swept too fast for the light to follow, and approaches complete transfer as it lengthens: above 95 per cent from 195 µm, 5 times the directional coupler's length. The solid curve is the coupled-mode integration; the dots are the Landau–Zener formula for the chance of jumping the gap, from the coupling and the rate at which the taper sweeps the detuning, and the two agree.
Fig. 3 The fraction transferred at 1550 nm against coupler length: for the directional coupler, which oscillates, and for the tapered coupler, which approaches complete transfer as it lengthens. The dots are the Landau–Zener formula for the chance of jumping the gap.

The directional coupler’s transfer oscillates with length for as long as it is drawn: complete at 37.6 micrometres, nothing at twice that, complete again at three times. The tapered coupler with the same guides and the same 200-nanometre swing transfers little when short, because the detuning is swept past the gap too quickly, and rises towards completion as it lengthens, passing 95 per cent at 195 micrometres — five times the directional coupler.

The dots are not a fit. The chance of a two-level system jumping an avoided crossing is given by the Landau–Zener formula, e−2πκ2/∣Δ˙∣e^{-2\pi\kappa^2/|\dot\Delta|}, from the coupling κ\kappa and the rate Δ˙\dot\Delta at which the detuning is swept, and here the rate is the width swing’s effect on the propagation constants divided by the coupler’s length. The formula was derived in 1932 for atoms in a changing field; applied to a pair of silicon guides, with the length playing the part of time, it tracks the integration of the coupled-mode equations. What remains of the tapered coupler’s oscillation is the partial beat left by the finite mismatch at its ends.

That length is the price. Adiabatic transfer is robust because it is slow, and in a waveguide slowness is distance — the same bargain a taper that matches every note strikes when it spreads a change of impedance over a distance instead of making it at a step.

Slow only where it matters

A linear taper is wasteful. It changes the widths at the same rate everywhere, including near its ends, where the guides are so mismatched that their pair modes are far apart and the light could follow a much faster change. The only place slowness is needed is near the crossing, where the gap between the pair modes is smallest.

Creeping through the crossing and hurrying elsewhere shortens the coupler. At 1550 nm, the fraction transferred against length for two tapers with the same guides, gap and 200 nm width swing. The linear taper changes the widths at a steady rate along its length and stays above 95 per cent from 280 µm. The shaped taper changes them quickly near its ends, where the guides are far from matched and the pair modes far apart, and slowly near the middle, where they cross and the gap between the modes is smallest — keeping the rate at which the light's mode turns constant along the whole length. It stays above 95 per cent from 40 µm, 86 per cent shorter. Both come from integrating the coupled-mode equations along the taper.
Fig. 4 The fraction transferred at 1550 nm against length for a linear taper and for a shaped taper with the same guides, gap and width swing. The shaped taper hurries near its ends and creeps through the crossing, turning the light’s mode at a constant rate.

The shaped taper redistributes the same total change so that the angle describing how the pair mode is split between the guides advances at a constant rate along the coupler: fast where the modes are far apart, slow where they are close. With the same guides, gap and 200-nanometre swing, it passes 95 per cent from 40 micrometres, against 280 for the linear taper — a seventh of the length, and barely more than the directional coupler’s 37.6. It is broadband as well: a 60-micrometre shaped taper passes more than 95 per cent over 375 nanometres of wavelength.

So far that looks like an unqualified improvement, and the next figure shows that it is not. Its lesson is the one the whole subject turns on.

Immunity to being made wrong

The robustness that matters most in manufacture is not to wavelength but to the guides themselves. A lithographic process makes nominally identical guides that differ by a few nanometres in width, and those few nanometres change their propagation constants.

A few nanometres of mismatch ruin a cut coupler and not a slow one. At 1550 nm, the fraction transferred when the two guides are not made as drawn: one wider and the other narrower by the amount across, a fabrication error of a few nanometres being ordinary. The directional coupler cut to 37.6 µm transfers everything only when the guides match, and with a 10 nm mismatch it transfers 67 per cent, with 20 nm 15 per cent: a detuning between the guides caps how much can cross. The linear tapered coupler 800 µm long transfers 99.7 and 99.8 per cent at the same errors, because a mismatch only moves the point along the taper where the two guides' constants cross, and the slow taper is slow everywhere. The shaped taper 60 µm long, which reaches full transfer in a fraction of the length by being slow only in its middle, transfers 82 and 64 per cent: a mismatch moves the crossing out of the slow middle into a part of the taper that hurries, where the light cannot follow.
Fig. 5 The fraction transferred at 1550 nm when one guide is wider and the other narrower than designed, by the amount across: for the directional coupler cut to 37.6 µm, the linear taper 800 µm long, and the shaped taper 60 µm long.

For the directional coupler a mismatch is fatal. Guides that differ in width have different propagation constants, their pair modes are no longer split evenly between them, and the maximum fraction that can cross falls below one however the length is chosen. At a 10-nanometre mismatch the coupler cut to 37.6 micrometres passes 67 per cent; at 20 nanometres, 15 per cent.

For the tapered coupler a mismatch is almost invisible. It adds a constant offset to the difference between the guides, which moves the point along the taper where their constants cross — a little earlier or a little later — and the light crosses wherever the crossing is. At 10 and 20 nanometres the 800-micrometre coupler passes 99.7 and 99.8 per cent. As long as the crossing stays inside the taper, the transfer does not care where.

The shaped taper is the third curve, and it gives back most of what the short length bought. At a 10-nanometre mismatch it passes 82 per cent and at 20 nanometres 64 — better than the directional coupler, far worse than the linear taper. The reason is exactly the design that made it short. It is slow only in its middle, where the crossing was meant to be, and a mismatch moves the crossing towards one end, into a part of the taper built to hurry. The light meets the gap where it cannot follow. The robustness of adiabatic transfer to fabrication error comes from being slow everywhere the crossing might be, and a design that removes slowness from places the crossing was not expected to be removes the tolerance to its being there.

That is a general property of optimised designs, not a peculiarity of couplers. An optimisation for the nominal case spends every margin it can find, and the margins it spends were the tolerance. The mode that will not turn a corner meets the same trade in bends, where the tightest bend a design can survive at the nominal width is the one that fails first when the width is wrong.

The same principle in other hands

The pattern — sweep a mismatch through zero slowly, and let the system follow its eigenstate — recurs wherever there are two coupled modes and a parameter that can be swept.

In atomic physics it is how population is moved completely between two quantum states with pulses whose timing and strength need not be exact: a frequency sweep through resonance, slow enough to be adiabatic, inverts the atom whatever the precise pulse area, where a pulse timed to half a Rabi oscillation must be exact. In nuclear magnetic resonance the same sweep inverts spins across a sample in which the field varies from place to place. The coupler is the spatial version, and in all of them the trade is identical: a process that works for a range of conditions because it is slow, against one that is fast and works only when everything is right.

In the channel with no walls and its fibres the principle appears as the adiabatic taper, which changes a fibre’s diameter gradually enough that light stays in its fundamental mode as that mode spreads out of the core and into the cladding. A photonic lantern uses it to split a multimode fibre into single-mode ones without losing light, which is impossible for a sudden junction and possible for a slow one because the light is never asked to jump between modes — provided there are at least as many single-mode fibres as the multimode fibre has modes, since the number of modes is an invariant no passive device can reduce.

What the model assumes

The coupled-mode picture is approximate at small gaps. Each guide’s mode is solved on its own and the coupling taken from the overlap of tails, which is accurate when the tails are weak where the other guide is. At a 250-nanometre gap in silicon the error is of order ten per cent in the coupling, which shifts the lengths drawn without changing any comparison.

The guides are slabs. A real silicon waveguide confines light in both directions across it, its modes have two polarisations with very different couplings, and a coupler designed for one polarisation behaves differently for the other. Adiabatic couplers are often designed to work for both, which is harder than it sounds.

Nothing is lost. Every curve conserves power to within the integration’s accuracy, a thousandth, and a real coupler loses light to scattering from rough sidewalls — a loss proportional to length, which is the second price of an adiabatic design beyond the space it takes.

The field that slides from one guide to the other

The propagation constants in the second figure are drawn as two smooth curves, and what they represent is a field distribution across the pair that changes continuously along its length: a single hump sitting in the first guide, widening across the gap near the crossing, and settling into the second guide. That is the physical picture of following a mode, and no plot of two numbers along the coupler shows it.

Nor do the pictures show the design space. The swing of the widths, the shape of the taper along its length and the gap can all be varied, and the lengths quoted belong to the simplest choice — a linear taper at a fixed gap. Optimised tapers reach the same transfer in a fraction of the length, and they are found by searching that space rather than by any single formula.

Still open: the shortest coupler that does not care

Adiabatic couplers are robust and long; directional couplers are short and fragile. Much current work asks whether the trade is fundamental. Shortcuts to adiabaticity — tapers shaped so that the light ends in the right guide without having followed a mode the whole way, borrowed from techniques developed for driving quantum systems quickly — reach broadband transfer in a few times the length of a directional coupler in simulation and in some fabricated devices. Inverse design, in which an optimiser shapes the whole coupler region with no presumption of guides or tapers, produces devices a few micrometres across that transfer light over wide bands.

What is not known is how short a device can be while keeping the tolerance that made adiabatic couplers worth their length. The shaped taper in the figures shows the trade in its simplest form: it recovered the length and the bandwidth and gave back most of the tolerance, because the tolerance was exactly what slowness bought. Designs optimised over many fabrication variants at once — robust inverse design — try to keep both, and pay in length again. Whether a coupler can be simultaneously short, broadband and insensitive to how it was made — or whether some quantity connecting the three is bounded — is an open question in integrated photonics.

The habit worth carrying away is to ask whether a device works by interference or by following. Interference is fast and depends on everything being exactly as designed; following is slow and depends only on the direction of a sweep. A beat between two modes rewards precision; a slow passage through their crossing rewards patience, and the choice between them is a choice of what is cheap to control.

Part 6 of 6

This essay is one argument about Guided waves. The others:

Two guides, and the two solutions they have. The transverse field of the two modes a pair of identical slab guides supports, against position across them, for guides half a micrometre wide separated by a gap of 300 nanometres at a wavelength of 1550 nanometres. A single guide has one fundamental mode; two guides side by side have two, and neither of them lives in one guide. The symmetric one is a single hump spanning both, the antisymmetric one has a node exactly between them — checked here to be exactly zero rather than nearly so — and they have slightly different propagation constants because the symmetric one has more of its field in the high-index gap region. That difference, computed from the slab's own dispersion condition, is 4.60e-2 per micrometre. Everything else about a coupler follows from it: a wave launched into one guide alone is the sum of the two supermodes in equal parts, they run at different speeds, and the interference between them moves the power from one guide to the other and back. There is no leakage in the account anywhere — only two solutions beating. Two tails that swap everything Part 5 — Bring two guides close enough for their evanescent tails to overlap and they do not leak a little power into each other. They exchange all of it, and then exchange it back, over a length fixed by the splitting between two modes that belong to neither guide — so a coupler is cut to a length rather than tuned to a ratio, and the length depends exponentially on a gap of a few hundred nanometres.

Where this goes next

Two guides, and the two solutions they have. The transverse field of the two modes a pair of identical slab guides supports, against position across them, for guides half a micrometre wide separated by a gap of 300 nanometres at a wavelength of 1550 nanometres. A single guide has one fundamental mode; two guides side by side have two, and neither of them lives in one guide. The symmetric one is a single hump spanning both, the antisymmetric one has a node exactly between them — checked here to be exactly zero rather than nearly so — and they have slightly different propagation constants because the symmetric one has more of its field in the high-index gap region. That difference, computed from the slab's own dispersion condition, is 4.60e-2 per micrometre. Everything else about a coupler follows from it: a wave launched into one guide alone is the sum of the two supermodes in equal parts, they run at different speeds, and the interference between them moves the power from one guide to the other and back. There is no leakage in the account anywhere — only two solutions beating. Two tails that swap everything Bring two guides close enough for their evanescent tails to overlap and they do not leak a little power into each other. They exchange all of it, and then exchange it back, over a length fixed by the splitting between two modes that belong to neither guide — so a coupler is cut to a length rather than tuned to a ratio, and the length depends exponentially on a gap of a few hundred nanometres. Two levels that refuse to cross. On the left, the energies of a two-state system as one state is swept past the other, for couplings of 0, 0.05, 0.15, 0.35 electronvolts. With no coupling at all the two levels cross, which is the dotted pair. With any coupling whatever they do not: the eigenvalues are plus and minus half the square root of the squared detuning plus four times the squared coupling, so the closest approach is exactly twice the coupling — measured off the drawn curves as 0.000, 0.100, 0.300, 0.700 eV against 2V, agreeing to 0.0e+0. Far from the crossing the curves rejoin the uncoupled lines, so the repulsion is local: it is largest where the two states are degenerate and dies away as the square of the coupling divided by the detuning. On the right is what makes this more than a picture of two hyperbolas — the character of the upper state, meaning how much of the first basis state is in it. It swaps completely across a region whose width is set by the coupling, so the level that arrives as one thing leaves as the other. That exchange of identity is why the phrase avoided crossing is misleading: nothing is avoided, the labels are. The crossing that never happens Two energy levels swept past one another do not cross. Any coupling between them, however small, opens a gap of exactly twice the coupling — and the two levels exchange their identities across it, so the state that arrives as one thing leaves as the other while the labels are what avoided anything. A graded junction against a quarter-wave layer. Reflectance against wavelength for four ways of joining a medium of index 1 to one of 1.52. The bare interface reflects 4.26 per cent at every wavelength. A single quarter-wave layer of index √(n₀n_s) takes that to zero at 550 nm exactly and rises symmetrically either side, which is the shape of every single-layer coating and every single-section transformer. The remaining curves are graded junctions of 150 nm and 400 nm, built as 120 thin layers whose index climbs geometrically and put through the same matrix product. They do not have a design wavelength at all. Below a cutoff they are flat and negligible; above it they climb steeply toward the bare value, and the cutoff is set by the length: 398 nm for the 150 nm taper, 1062 nm for the 400 nm taper. Roughly, a taper works for every wavelength shorter than about twice its own optical length, which is the statement that a reflection needs a partner a quarter of a wavelength further in to cancel against. The engineering versions of this are everywhere: the horn on a loudspeaker, the moth's eye, the flared transition between two waveguides, and the graded layer that lets an ultrasound probe reach tissue across a hundredfold impedance step. The taper that matches every note A quarter-wave layer cancels a reflection at one wavelength and only near it. Spread the same change of impedance over a distance instead, and the reflection vanishes for every wavelength shorter than about twice that distance — not by cancelling one echo against another, but by leaving no step anywhere for an echo to come from.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Adiabatic followingAvoided crossingCoupled mode theoryDirectional couplerEvanescent waveIntegrated photonicsLandau zenerSupermode