Directional Couplers and Splitters
Every circuit technology has its wiring primitives. Electronics has the junction and the solder
blob: touch two wires together and current happily divides. Photonics cannot copy that — crossing
two waveguides at a shallow angle mostly just scatters light — and yet it owns a trick electronics
would envy. Run two waveguides parallel, a few hundred nanometres apart, never
touching, and light flows from one to the other through the gap: completely, then back
again, in a smooth periodic exchange you can dial to any split you like by choosing how long the
guides run together. This device — the directional coupler — plus its stolid
cousins the Y-junction and the MMI, are the taps, tees and combiners of every photonic chip. They
are also the last thing this module needs to build: with a coupler in hand, the next module snaps
two of them around a pair of phase-shifted arms and gets the Mach–Zehnder interferometer, the
transistor of this course.
Leaking across the gap
The mechanism has been waiting since the modes lesson. A guided mode carries an
evanescent tail — real, exponentially decaying field reaching a few hundred
nanometres beyond the core. Bring a second, identical waveguide inside tail's reach and that field
is no longer decaying into empty cladding: it lands in a guide, and drives it the way a
periodic push drives a swing, on resonance, because identical guides share the same
n_{\text{eff}}. Cycle by cycle, amplitude accumulates in the neighbour.
It is the exact optical analogue of
quantum
tunnelling between two wells through a classically forbidden barrier — the cladding gap
is the barrier, and the transfer rate falls exponentially with its width. Coupled-mode
theory condenses all of it into one number, the coupling coefficient \kappa
(per unit length), and two equations for the field amplitudes a and
b in the guides:
\frac{\mathrm{d}a}{\mathrm{d}z} = i\kappa\, b, \qquad
\frac{\mathrm{d}b}{\mathrm{d}z} = i\kappa\, a .
Launch everything in guide A (a = 1, b = 0) and the solution is pure
trigonometry:
- after the guides run together for a length L, the powers are
P_{\text{cross}} = \sin^2(\kappa L), \qquad
P_{\text{through}} = \cos^2(\kappa L);
- the split is set by the product \kappa L:
\kappa L = \pi/4 gives a 50:50 splitter,
\kappa L = \pi/2 transfers 100% across, and beyond
that the power swings back;
- the two powers always sum to 1 — an ideal coupler redistributes light, it does not lose
it;
- \kappa falls roughly exponentially with the gap
(it is set by evanescent-tail overlap) and grows with wavelength, since longer waves have
fatter tails.
Reading the sin² dial
The plot below is the coupler's whole design space. Length runs along the axis; the slider sets
\kappa (in practice: the gap — a wider gap means a smaller
\kappa and a longer, gentler exchange). Every coupler ever laid out is a
point on one of these curves:
Work the standard example. A silicon coupler with a 200 nm gap has, say,
\kappa = 0.30\ \mu\text{m}^{-1}. For a 50:50 split:
\kappa L = \frac{\pi}{4} \;\Longrightarrow\;
L = \frac{\pi}{4 \times 0.30} \approx 2.6\ \mu\text{m};
full transfer takes twice that, L \approx 5.2\ \mu\text{m}. Micrometres!
High index contrast strikes again: the same exchange in weakly-guiding silica takes millimetres. The
other standard order is the tap: skim off 1% to monitor a laser. Then
\sin^2(\kappa L) = 0.01, so
\kappa L = \arcsin(0.1) \approx 0.10 rad — a 0.33 µm brush at this
\kappa, more practically a longer run at a wider gap. You can integrate
the coupled-mode equations yourself and watch the analytic answer emerge from raw numerics:
// March da/dz = iκ·b, db/dz = iκ·a with complex amplitudes a, b (re, im).
const kappa = 0.30; // µm⁻¹
const dz = 0.001; // step, µm
let aRe = 1, aIm = 0, bRe = 0, bIm = 0; // all light starts in guide A
const marks = [0.5, 1.3, 2.6, 3.9, 5.2]; // lengths to report, µm
let next = 0;
for (let z = 0; z <= 5.2 + dz / 2; z += dz) {
if (next < marks.length && z >= marks[next]) {
const pA = aRe * aRe + aIm * aIm, pB = bRe * bRe + bIm * bIm;
const analytic = Math.sin(kappa * z) ** 2;
console.log("L = " + z.toFixed(1) + " µm through: " + pA.toFixed(3) +
" cross: " + pB.toFixed(3) + " analytic sin²(κL): " + analytic.toFixed(3));
next++;
}
// i·κ·b means (re,im) → (−κ·b_im, κ·b_re): rotate by 90° and scale.
const daRe = -kappa * bIm * dz, daIm = kappa * bRe * dz;
const dbRe = -kappa * aIm * dz, dbIm = kappa * aRe * dz;
aRe += daRe; aIm += daIm; bRe += dbRe; bIm += dbIm;
}
One more detail hides in the numerics: the crossed light emerges with its phase a quarter turn
behind (that factor i in the equations). A directional coupler is
not just a power divider but a phase-respecting two-port — the crossed path always picks up
90° relative to the through path. File that fact carefully; it is exactly what makes the
interferometer of the next module route light where it is told.
The stolid cousins: Y-junctions and MMIs
The directional coupler's precision is also its weakness: the split depends on
\kappa L, and \kappa moves with fabricated
gap, wavelength and temperature. When a design just needs a rugged, set-and-forget 50:50, two
alternatives trade adjustability for tolerance. The Y-junction splits by brute
symmetry — one guide forks into two mirror-image branches, and equal halves follow from geometry
alone (though it only divides; run it backwards with mismatched inputs and the unmatched part
radiates away as loss). The multimode interference coupler (MMI) is subtler and
ubiquitous: a short, deliberately wide section supporting many modes, in which the input
field's superposition of
modes drifts in and out of phase and periodically reassembles into sharp copies of the input — one
copy, then two, then more, at predictable depths (the "self-imaging" effect, a guided-wave cousin
of the Talbot carpet). Chop the slab where two images form and you have a 50:50 splitter set by one
robust dimension — the slab width — rather than a fussy nanometre gap. MMIs tolerate fabrication
error and temperature far better than directional couplers, at the cost of a little excess loss and
no continuous tunability: workhorse 50:50s are MMIs; precision or variable ratios go to directional
couplers.
Hang two pendulums from a common slightly-flexible beam — two children on a swing set will do — and
start only one swinging. Over the next half minute the first swing dies while the second builds
from nothing, then the exchange runs in reverse, indefinitely. The beam's tiny flex is the coupling;
the energy sloshes at a rate set by its stiffness, exactly as \kappa sets
the sloshing between waveguides. The clean way to see it — in the swings and in the
coupler — is to change basis: the symmetric motion (both together) and the antisymmetric motion
(opposed) are the system's true normal modes, each with its own frequency, and "all the energy in
swing A" is just those two modes in phase. They drift apart and the energy appears in swing B. In
coupler language: the pair of guides supports an even and an odd supermode with slightly different
effective indices, and the full transfer length is where their accumulated phases differ by π —
L_\pi = \lambda / \,[2(n_{\text{even}} - n_{\text{odd}})]. Same device,
two languages; the supermode view returns when meshes get large.
Three couplers' worth of classic errors. One: seeing power vanish from guide A and
booking it as attenuation. It is in guide B — and if the coupler runs longer, it comes back:
the exchange is periodic, not a leak. A "3 dB coupler" left twice as long is not a 6 dB coupler; it
is a straight wire (100% crossed at \kappa L = \pi/2… and returned at
\pi). Design means landing on the right point of a sine, not draining a
tank. Two: treating the labels "50:50" as guaranteed. The ratio rides on
\kappa, which moves exponentially with gap and noticeably with wavelength
and temperature — a coupler cut for 50:50 at 1.55 µm is 55:45 not far away, which is precisely the
coupler-imbalance problem the Mach–Zehnder lesson inherits (and why its "dark" ports are never
perfectly dark). Three: forgetting the crossed path's built-in 90° phase lag. Add
powers and the books balance; add amplitudes while ignoring the i
and every interference calculation that follows comes out wrong.
Where this goes next — the module banked
Take stock of what this module built: light pinned by total internal reflection, quantised into
modes priced by n_{\text{eff}}, carried across oceans in fibre and
across chips in silicon, hauled on and off through gratings and tapers, and now split, tapped and
recombined at will — a complete passive wiring kit, with dB budgets to audit it. What the kit still
lacks is anything active: every device so far does the same thing to every photon forever.
The next
module begins with the device that changes that — two of today's couplers bracketing
two waveguide arms, where a controllable phase turns wiring into computing.