Ring & platform
nm
µm
µm
–
–
dB/cm
dB
%
%
K⁻¹
Thermal tuning & WDM plan
mW/FSR
–
GHz
GBd
Summary
Channel plan
Inverse fit — measured all-pass spectrum
nm
pm
dB
Both branches reproduce the measured spectrum exactly — a single power
spectrum cannot distinguish under- from overcoupling (r and a are interchangeable). Uses λ₀ and ng
from the platform panel to convert FSR → round-trip length.
What this means
Transmission spectrum (ring tuned to λ₀)
Coupling design map — extinction & Q vs κ₁²
N-ring mux — channel spectra on the grid
Method: closed-form ring transfer functions after Yariv (Electron. Lett. 36, 321, 2000)
and Bogaerts et al. (Laser & Photon. Rev. 6, 47, 2012): all-pass
T = (a²−2ra·cosφ+r²)/(1−2ra·cosφ+(ra)²) and the add-drop Eqs. (5)–(6), with self-coupling r = √(1−κ²),
single-pass amplitude a from α·L plus excess coupler/bend loss, and round-trip phase φ from first-order
dispersion via ng. FWHM, finesse and loaded Q use Eqs. (7)–(8), (20)–(23);
Qi = 2πng/(λα) and 1/QL = 1/Qi + 1/Qc. Heater power scales
as PFSR·Δλ/FSR, with N·PFSR/2 for an N-ring mux with random fab offsets. Spectra assume the
ring is thermally trimmed so a resonance sits on λ₀; κ and a are taken wavelength-flat (good over 1–2 FSR,
optimistic across a full O-band grid); crosstalk uses the Lorentzian-skirt approximation of the drop line.
Get neff and ng for your cross-section from the
slab waveguide mode solver; mapping coupler gap → κ² needs a coupled-mode or
FDTD solve.
Why can't a measured notch tell you if the ring is under- or overcoupled?
The all-pass power spectrum depends on r and a only through their product r·a (which sets
the linewidth) and their difference |r−a| (which sets the depth) — swap r and a and T(λ) is bit-identical. So one
spectrum always fits two physically different rings: low loss with weak coupling, or high loss with strong
coupling, with very different intrinsic Q. Labs break the tie with a coupler-gap sweep across several rings
(κ trends with gap, loss doesn't), by tracking linewidth vs wavelength, or with a phase-sensitive measurement.