Waves

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.

15 min read 5 figures What happens at the edge Made into an instrument

Assumes: The mode that will not turn a corner · The channel with no walls

The mode that will not turn a corner ends with a habit: ask where the small part of a field is, because a quantity that is exponentially small is exponentially sensitive, and any mechanism reaching out to where an evanescent tail lives will produce an effect that is unmeasurable and then catastrophic with nothing in between.

Bringing a second guide into the tail is the cleanest such mechanism, and what happens is not a small effect at all.

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. — Fig. 1 The two modes a pair of identical silicon guides supports, against position across them, for guides half a micrometre wide separated by three hundred nanometres. A single guide has one fundamental mode; two guides have two, and neither lives in either guide. The symmetric one spans both; the antisymmetric one has a node exactly between them — checked to be exactly zero rather than nearly so.

There is no leak

The word usually used is “coupling”, which suggests power seeping from one guide to the other. That picture is wrong in a way worth correcting, because the correct picture explains something the wrong one cannot: why the transfer is complete.

A single guide has one bound mode. Two identical guides side by side are one structure, and that structure has two bound modes: a symmetric one, with the field in phase across both guides, and an antisymmetric one, with the field reversed in the second. Neither of them belongs to either guide. The antisymmetric mode has a node exactly halfway between them, by symmetry and to every digit the arithmetic carries; the symmetric one has a maximum there.

Those two modes have different propagation constants, because the symmetric one has more of its field in the higher-index region between the guides and therefore travels a little slower relative to free space — the splitting for the pair drawn is $4.6\times10^{-2}$ per micrometre, computed from the guide’s own dispersion condition.

Now launch light into one guide alone. That field is not a mode of the pair; it is the sum of the two modes in equal parts, which is what makes the field cancel in one guide and add in the other. The two run at different speeds, their relative phase accumulates, and after half a beat they have swapped which guide they cancel in.

So what looks like a transfer is a beat between two solutions, and nothing has leaked anywhere. This is the same reasoning that makes a pair of pendulums exchange their swing: a system with two normal modes, driven into a superposition of them, appears to hand energy back and forth, and the appearance is interference rather than transport.

What that predicts

Power that crosses completely and comes back. The fraction of the power in each of two coupled guides, against distance along them in micrometres, for three gaps between the guides. Light launched into one guide alone does not leak a little into the other: it crosses entirely, at a distance set by the splitting between the two supermodes, and then crosses back, and goes on doing so for as long as the guides run alongside each other. The two powers sum to exactly one at every distance, checked here to twelve figures, because nothing is being lost — the beat is a rotation between two states rather than a leak. A gap of 200 nm transfers in 20.7 µm; A gap of 300 nm transfers in 68.2 µm; A gap of 450 nm transfers in 408.2 µm. So a coupler that divides power in a chosen ratio is made by cutting the interaction region to a chosen length rather than by tuning any strength: half the power at half the coupling length, a tenth at a length whose sine squared is a tenth. That is an unusual design principle and it is why integrated-optic splitters are specified by a dimension rather than by a coefficient. — Fig. 2 The power in each of two coupled guides against distance along them, for three gaps. Light launched into one crosses entirely at a distance set by the splitting, crosses back, and goes on alternating. The two powers sum to exactly one at every distance, to twelve figures, because the beat is a rotation between two states and not a leak. Two hundred nanometres of gap transfers in twenty-one micrometres.

Two predictions follow that a leakage picture does not make, and both are what a coupler is used for.

The transfer is complete. Not a large fraction; all of it. The power in the launched guide goes to exactly zero, which a leak could never produce, and it does so at a definite distance.

And it reverses. Past the coupling length the power comes back, and a coupler cut twice as long as intended transfers nothing rather than transferring more. That is an unusual failure mode and it is why an integrated coupler is specified by a dimension with a tolerance rather than by a coefficient with a margin.

Together they give the design principle. A coupler that splits power in any chosen ratio is made by cutting the interaction region to a length: half the power at half the coupling length, a tenth at the length whose sine squared is a tenth. Nothing is tuned, no material is chosen for a coupling strength, and the whole of the design is a number of micrometres.

That is why the fifty-fifty splitter is the fundamental component of integrated optics. Every interferometer on a photonic chip is two such couplers with a phase shifter between them — a Mach–Zehnder interferometer — and every switch, modulator and filter on such a chip is built from that arrangement — which is two paths whose difference in length decides what emerges, with the couplers doing the splitting and the recombining. The component count of the entire field rests on the completeness of this transfer.

The same structure, in three other subjects

The two-mode picture is not a fact about waveguides, and it is worth collecting the other places it appears because recognising it saves rederiving it.

The lowest note a pipe will carry. The dispersion relation of a guided wave for three cutoffs, in units where the free wave speed is one. Each curve leaves the vertical axis at its own cutoff and bends toward the diagonal, which is the free wave. Above the cutoff the phase velocity is the slope of the line from the origin and always exceeds one, while the group velocity is the slope of the curve and never does: at k = 2 their product is 1.0000, 1.0000, 1.0000, which is one to four decimal places in every case and is an identity rather than a coincidence. Below the cutoff there is no curve, because there is no travelling wave to draw. — Fig. 3 The guide the pair is made of, from the single-guide argument: the dispersion of a single guided mode for three cutoff frequencies. Everything about the coupler comes from one property of this curve — how fast the field outside the core decays, which is fixed by how far the mode’s propagation constant sits above the cladding’s. A mode near its cutoff has a long tail and couples to anything nearby; one far from cutoff has a short tail and does not.

Two pendulums joined by a weak spring. The normal modes are in-phase and out-of-phase, they differ slightly in frequency, and a single pendulum started alone is their sum — so the swing appears to move from one to the other and back. Nobody describes that as a leak.

A molecule’s two atoms. Two identical atomic orbitals brought together give a bonding and an antibonding combination, split by an amount that falls exponentially with the separation because the orbitals’ tails overlap. That splitting is what a chemical bond is, and the exponential is why bond energies depend so sharply on bond lengths.

And two atoms sharing one excitation. The symmetric and antisymmetric combinations have different energies and different radiative rates, one of them enhanced and one suppressed. The excitation appears to hop between the atoms, and what is happening is a beat between two states of the pair.

The pattern is the same in all four: two nearly identical systems, brought near enough to notice each other, have two modes rather than two systems — and the splitting between the modes is what every observable behaviour is made of. What differs between the cases is only what supplies the overlap.

The exponential underneath

A fraction of a micrometre decides a millimetre. The length over which two coupled guides exchange their power completely, against the gap between them — logarithmic in the length, linear in the gap. The coupling coefficient is the overlap of the two guides' evanescent tails, so it falls as the exponential of the gap divided by the tail's own decay length, and the coupling length rises correspondingly. The slope measured off the drawn curve is 11.926 per micrometre, which is the mode's own tail decay rate of 11.926 to every figure carried — the check that the coupling is that overlap and nothing else. A gap of 100 nm gives 6.3 µm; A gap of 200 nm gives 20.7 µm; A gap of 400 nm gives 224.9 µm; A gap of 700 nm gives 8.05 mm. Half a micrometre of gap changes the answer by three decades, which is the practical statement of the single guide's habit: a quantity that is exponentially small is exponentially sensitive. A coupler's gap has to be right to a few tens of nanometres and its length to a few per cent, and the two tolerances are not independent — which is why such devices are made by lithography rather than by assembly. — Fig. 4 The coupling length against the gap between the guides, logarithmic in the length. The slope measured off the curve is 11.93 per micrometre, which is the mode’s own evanescent decay rate to every figure carried — the check that the coupling is that overlap and nothing else. A hundred-nanometre gap couples in 6.3 micrometres and a seven-hundred-nanometre gap in eight millimetres.

The splitting between the two supermodes is the overlap of the two guides’ tails, and a tail is an exponential. So the coupling falls as $e^{-\gamma s}$ with $s$ the gap and $\gamma$ the tail’s decay rate, and the coupling length rises as the same exponential.

The figure checks that identification rather than taking it on trust: the slope measured off the computed curve is the same number as the mode’s own decay rate, to every figure carried. That is the content of the claim “the coupling is the overlap of two tails” — if the coupling had any other origin, the two numbers would differ.

Six hundred nanometres of gap change the answer by three decades. For the silicon guides drawn, the tail’s decay length is eighty-four nanometres, which is a fifth of a wavelength inside the material and is what “tightly confined” means quantitatively — the same number that decides how many modes the guide has room for and how small its étendue is. A coupler’s gap therefore has to be right to a few tens of nanometres, its length to a per cent or two, and the two tolerances are coupled — a gap ten per cent wide needs a length twelve per cent longer to compensate.

That is why such devices are made by lithography and not by assembly, and why the same design in a weakly guiding structure behaves quite differently. A pair of ordinary optical fibres, with an index contrast a hundred times smaller, has a tail several micrometres long and a coupling that barely depends on the gap at all — which is why a fibre coupler is made by fusing and stretching two fibres together rather than by placing them, and why its ratio is set by watching the output while pulling.

What sets the tail, and therefore everything

The decay rate is not a free parameter. It is $\sqrt{\beta^2 - k_0^2n_\text{clad}^2}$ , fixed by the guide’s two indices, its width and the wavelength — the same quantity that decides how tightly the guide holds its light at a bend and whether the guide has a bound state at all.

So one number governs three things that look unrelated. A guide with a long tail bends badly, couples readily to its neighbours, and is close to its cutoff. A guide with a short tail bends tightly, needs its neighbours very close indeed, and is far from cutoff. There is no arrangement that bends well and couples easily, and a photonic circuit’s layout is a negotiation between those two demands: tight bends need strong confinement, and strong confinement needs couplers with gaps at the limit of what lithography resolves.

The dependence on wavelength is the other consequence and it is what limits such a device’s bandwidth. The decay rate depends on the wavelength, so the coupling length does, so a coupler cut to split evenly at 1,550 nanometres splits unevenly at 1,500 — by a few per cent over the band a telecommunications system uses, which is enough to matter and is why broadband splitters are made by other means.

What happens when the two guides are not the same

The completeness of the transfer depends on the two guides being identical, and it is worth seeing what breaks when they are not, because a fabricated pair never is.

If the guides have slightly different propagation constants — because one is a few nanometres wider — then the supermodes are no longer half in each guide, and the transfer is capped. The maximum fraction that crosses is $1/(1 + (\Delta\beta/2\kappa)^2)$ , which is exactly the form that limits any pair of detuned oscillators and is the same algebra with different names on the symbols.

The practical consequence has an unexpected direction. A coupler with a wide gap has a small $\kappa$ , so the same width mismatch is a larger fraction of it, and the transfer is capped lower. A weakly coupled pair is more sensitive to mismatch, not less — which reverses the usual intuition that a gentle interaction is a forgiving one, and is why designers keep gaps small and accept the tighter tolerance on the gap itself.

It also gives the device a use. Make the detuning controllable — by heating one guide, which changes its index — and the coupler becomes a switch: at zero detuning it transfers everything, at a large detuning it transfers nothing, and the transition takes a temperature change of a few kelvin. That is one of the two standard ways of switching light on a chip, and what it is controlling is the ratio of two numbers, one of which is an exponential of a gap.

Where the tail’s power actually is

There is a question the whole account leaves and it is worth answering, because the answer settles what a coupler is doing physically.

An evanescent tail carries no power along the guide — that is what makes it evanescent, and it is why the field beyond a totally reflecting surface transports nothing. So where does the power that ends up in the second guide travel?

Fig. 5 The same tail seen where it decides a bend: the field of a guided mode reaching out past its core, with the radius at which it would have to outrun the cladding marked. That field is what the coupler reaches into. It carries no power along the guide by itself, and the coupler’s transfer is nevertheless made entirely of it.

The answer is that it travels sideways, and slowly. The energy flow in a coupler is not along the guides with a small transverse leak; it is a gentle transverse flow in the gap region, of exactly the magnitude needed to move the power across over the coupling length. Computing it means mapping the Poynting vector of the combined field, and the map shows a flow crossing the gap continuously along the whole length of the coupler.

Which makes “no power flows in an evanescent field” a statement about a single mode rather than about a region. One mode’s tail carries nothing; a superposition of two modes whose tails overlap carries something, because the flow is a cross term between them and a cross term needs two. That is the same distinction the energy accounting of two interfering waves turns on, and it is the reason a statement true of each of two things separately can be false of them together.

A slab, two identical guides, and one wavelength

The guide is a slab and a real one is a rectangle. A slab confines in one direction and a waveguide on a chip confines in two, which changes the mode profile, the effective index and the numerical prefactor of the coupling. The exponential is unchanged, because it comes from the tail in the direction of the gap, and every number here should be read as the right exponent with a prefactor of order one.

The two guides are identical. They are not, in practice: lithography leaves them differing in width by a few nanometres, which detunes their propagation constants and caps the transfer at a fraction below one — by the same $1/(1 + (\Delta\beta/2\kappa)^2)$ that limits any pair of detuned oscillators. A coupler with a wide gap has a small $\kappa$ and is therefore more sensitive to that mismatch, which is a second reason gaps are kept small.

The coupled-mode treatment is first order in the overlap. At very small gaps the two guides’ modes are not weakly perturbed versions of the single guide’s, the supermodes have to be computed for the pair as a whole, and the coupling coefficient computed from the isolated modes is wrong by tens of per cent. Below about a hundred nanometres the figures are qualitative.

The wavelength is a single value. A pulse has a bandwidth, the coupling length depends on the wavelength, and a pulse split by a coupler therefore splits unevenly across its own spectrum — which is a mild effect over a telecommunications band and a severe one for a pulse short enough to have a wide one.

And the guides are taken as lossless and parallel. A real coupler has bends at each end to bring the guides together and separate them again, those bends couple too, and the effective length is the parallel section plus a contribution from the approaches that has to be computed or measured.

Two profiles where there is only ever one field

The supermode figure draws two field profiles at one instant and the thing they do is beat. What is actually present at any point along the coupler is a single field, the sum of the two with a relative phase that depends on where along the guides it is looked at, and that field has no symmetry at all except at the two ends of each beat. The two profiles are a basis rather than a description of anything present.

The transfer figure draws power against distance and cannot show the phase. The light emerging from the second guide is not merely present; it carries a definite phase relative to what would have emerged from the first, and that phase is what makes the interferometer built from two such couplers work. A plot of power is blind to the quantity the device exists to manipulate.

And the length figure draws a smooth exponential over gaps down to eighty nanometres, where the treatment that produced it has stopped being accurate. The curve is right in its slope over the whole range and right in its value only over the upper part, and there is nothing in the drawing to say where the change happens.

Still open: how to make a coupler that does not care about the wavelength

The coupling length depends on the wavelength through the tail, so an ordinary directional coupler is a narrowband device, and that is a real limitation for anything that has to work across a telecommunications band or, worse, across the visible.

Several approaches exist and none is fully satisfactory. Adiabatic couplers bring two guides of slightly different widths gradually together and apart, so that the power transfers by following a supermode rather than by beating; they are much less sensitive to wavelength and to fabrication, and they are five to ten times longer. Multimode interference couplers use a wide section supporting several modes whose interference reproduces the input at particular lengths; they are robust and lossier. And subwavelength-grating structures engineer the effective index of the gap region so that its dispersion cancels the coupling’s; they work over remarkable bandwidths and require lithography at the limit of what is available.

What the field does not have is a splitter that is simultaneously short, broadband, tolerant of fabrication and low-loss, and the reason is the one this essay is about: the quantity that has to be controlled is an exponential of a dimension, and an exponential is not controlled by being careful.

The habit worth carrying away is the one the supermode figure is. When two systems appear to exchange something, look for the modes of the pair rather than of either. A leak is a transport picture and it predicts a partial transfer; a splitting is an interference picture and it predicts a complete and reversible one. The second is right here, and it is right for coupled pendulums, for two atoms sharing an excitation, for a molecule’s bonding and antibonding levels, and for every case where two nearly identical things are brought near enough to notice each other.