Microring Resonators
Take the arm of a Mach–Zehnder
and, instead of terminating it, bend it around until it bites its own tail: a closed loop of
waveguide, ten microns across, parked next to a straight "bus" waveguide close enough that a little
light couples across the gap. That is a microring resonator, and the closed loop
changes everything. In an MZI, light interferes with a copy of itself once; in a ring, the
circulating light comes back around and interferes with the incoming light again and again,
round trip after round trip. This multi-pass
interference makes the ring
exquisitely selective: at most wavelengths the device is invisible, but at the special wavelengths
where every round trip returns in phase, the ring seizes the light and builds up an intense
circulating field. One device, ten microns of silicon, and you get a wavelength filter, a modulator
platform, a sensor, and — in Module 5 — the storage cell of photonic
weight
banks. The MZI is photonics' transistor; the ring is its tuned circuit.
The resonance condition
A round trip of the ring has physical length L = 2\pi R and optical
length n_{\text{eff}} L. The circulating wave reinforces itself only if
it returns in phase — if the round trip holds a whole number of wavelengths:
- resonant wavelengths satisfy m \lambda_m = n_{\text{eff}}\, L for
integer m (the azimuthal order — how many wave
crests fit around the loop);
- neighbouring resonances are spaced by the free spectral range,
\mathrm{FSR} \;=\; \frac{\lambda^2}{n_g\, L},
where n_g is the group index (dispersion matters — see the
Watch out below);
- each resonance has a linewidth \delta\lambda, summarised by the
quality factor Q = \lambda / \delta\lambda: the
sharper the dip, the higher the Q;
- equivalently Q = \omega_0 \tau: a photon rattles around the ring
for \tau = Q\lambda/(2\pi c) before leaking out — high
Q means long storage.
Numbers, for the standard silicon ring of radius R = 10\ \mu\mathrm{m}
(L \approx 62.8\ \mu\mathrm{m}, n_{\text{eff}} \approx
2.4, n_g \approx 4.2) at
\lambda = 1.55\ \mu\mathrm{m}: the azimuthal order is
m = n_{\text{eff}} L / \lambda \approx 97 — about a hundred crests chase
each other around the loop — and the free spectral range is
\mathrm{FSR} = \frac{(1.55\ \mu\mathrm{m})^2}{4.2 \times 62.8\ \mu\mathrm{m}}
\approx 9.1\ \mathrm{nm}.
A modest Q of 15,500 gives a linewidth
\delta\lambda = \lambda/Q = 0.1\ \mathrm{nm}: the ring distinguishes
colours a thousandth of a percent apart, using a footprint smaller than a red blood cell. That
combination — dense comb of razor-thin lines — is exactly what wavelength-division systems order.
The transmission spectrum — and critical coupling
Watch the bus waveguide's output as you sweep the laser. Off resonance, light strolls past the ring
untouched: transmission ≈ 1. On resonance, the field that couples into the ring, circulates, and
couples back out returns exactly out of phase with the light that never entered — a
destructive interference that carves a sharp dip. Two amplitudes control the fight: the
self-coupling r (how much of the bus field skips the
ring) and the round-trip survival a (how much of the
circulating field outlives one lap of loss). The through-port transmission is
T \;=\; \frac{a^2 - 2ra\cos\varphi + r^2}{1 - 2ra\cos\varphi + (ra)^2},
\qquad \varphi = \frac{2\pi\, n_{\text{eff}} L}{\lambda}.
Play with both sliders below and find the special setting where the dips crash all the way to zero:
At r = a — critical coupling — the escaping circulating
field exactly cancels the bypassing field, and on-resonance transmission is zero: every
photon at that wavelength is dragged into the ring and dissipated by its round-trip loss. Undercouple
(r > a) or overcouple (r < a) and the
cancellation is partial. It is the ring's version of the MZI's extinction problem, and modulator
designers ride this knife edge deliberately: a tiny index change near critical coupling swings the
transmission violently.
All-pass vs add-drop
With one bus waveguide (the all-pass ring above), resonant light has nowhere to go
but round the loop until loss claims it — the through port shows dips, and the ring behaves as a
notch filter. Add a second bus on the far side and you build the add-drop
ring: now the circulating field has a second exit, and resonant wavelengths are handed over to the
new waveguide's drop port as a clean bandpass peak, while everything else sails to
the through port. That is a wavelength-selective railway junction: park one add-drop ring per
wavelength along a bus, tune each to its own channel, and you have a demultiplexer for
wavelength-division traffic — or, driven the other way through the add port, a
multiplexer.
The thermal Achilles heel
Everything that makes the ring sharp makes it twitchy. The same thermo-optic coefficient that the
last
lesson used as a tuning knob acts here as an uninvited hand on the dial: a silicon
ring's resonance drifts by roughly
\frac{d\lambda}{dT} \approx 70\ \mathrm{pm/K}.
Set against a high-Q linewidth this is brutal. A
Q = 50{,}000 ring has
\delta\lambda \approx 31\ \mathrm{pm} — so a temperature change of
half a kelvin shoves the resonance a full linewidth off the laser. A neighbouring processor
core waking up will do that a hundred times over. Every serious ring deployment therefore carries a
micro-heater and a feedback servo that locks the resonance to its channel, burning milliwatts to
stand still — and "how much tuning power per ring?" becomes a first-order term in the energy budget
of any ring-based system. Run the design numbers yourself:
const lambda = 1.55e-6; // m
const ng = 4.2; // group index of the Si wire
const c = 3e8; // m/s
function ringReport(radius_um: number, Q: number): void {
const L = 2 * Math.PI * radius_um * 1e-6; // circumference, m
const fsr_nm = (lambda * lambda) / (ng * L) * 1e9;
const dl_pm = (lambda / Q) * 1e12; // linewidth, pm
const tau_ps = (Q * lambda) / (2 * Math.PI * c) * 1e12;
const dT_off = dl_pm / 70; // kelvins to drift one linewidth
console.log("R = " + radius_um + " um, Q = " + Q);
console.log(" FSR = " + fsr_nm.toFixed(2) + " nm, linewidth = " + dl_pm.toFixed(0) + " pm");
console.log(" photon lifetime = " + tau_ps.toFixed(1) + " ps");
console.log(" drifts one linewidth per " + dT_off.toFixed(2) + " K");
}
ringReport(10, 15500);
ringReport(10, 50000);
ringReport(5, 50000); // smaller ring: wider FSR, same twitchiness
Climb to the gallery ringing the dome of St Paul's Cathedral in London, whisper at the wall, and a
friend with an ear to the stone a hundred feet away hears you clearly: the sound skims the curved
wall in repeated grazing reflections, staying trapped near the perimeter instead of spreading into
the dome. Lord Rayleigh worked out the physics in 1910 and the name stuck — closed-loop resonances
that hug a curved boundary are whispering-gallery modes, and a microring's hundred
circulating wave crests are precisely Rayleigh's whisper, shrunk a millionfold and made of light.
The family tree is broad: droplets of water, spheres and toroids of silica on chips hold
whispering-gallery modes with Q beyond 10^{9}
(a photon circulating for microseconds — tens of thousands of kilometres of travel in a
droplet-sized orbit), and such resonators can detect the landing of a single virus particle by the
picometre twitch of their resonance. The microring is the industrialised, lithography-friendly
member of the dynasty: modest Q, but stamped out ten thousand at a time
next to transistors.
The resonance condition uses n_{\text{eff}}, so it is tempting to
difference two neighbouring resonances and derive
\mathrm{FSR} = \lambda^2 / (n_{\text{eff}} L). Tempting, and wrong — in a
silicon wire it overestimates the FSR by nearly a factor of two. The slip is assuming
n_{\text{eff}} stays put while \lambda changes.
It doesn't: waveguide dispersion drags the effective index with wavelength, and the resonance
spacing depends on how fast the round-trip phase changes with
\lambda, which brings in the derivative. Carrying it through gives the
group index, n_g = n_{\text{eff}} - \lambda\, dn_{\text{eff}}/d\lambda
— for a standard silicon wire n_{\text{eff}} \approx 2.4 but
n_g \approx 4.2, because dn_{\text{eff}}/d\lambda
is large and negative. The division of labour is clean: n_{\text{eff}}
fixes where the resonances sit; n_g fixes how far apart
they are. The same distinction — phase velocity versus group velocity — governs pulse propagation
everywhere in this course, and mixing the two indices is the single most common numerical error in
ring design homework.
Where this goes next
You now hold both of photonics' fundamental interference machines: the MZI (single-pass, broadband,
big) and the ring (multi-pass, resonant, tiny). The next lesson puts
them both to work at the job that pays silicon photonics' bills — writing bits onto light — where
the MZI's robustness and the ring's compactness fight a genuinely close match. Later, rings return
as WDM
channel filters and weight banks, where their comb-like spectrum becomes the
addressing scheme of an entire photonic accelerator.