Two-ring coupled resonator filter

Drop- and through-port spectra for a series-coupled double microring add-drop filter, from the exact transfer-matrix solution. Two identical rings, identical bus couplers. Drag anything; the traces recompute on every frame.

H_d = j·κ₁²κ₂·x / D(x)
H_t = t₁ − t₂(1+t₁²)·x + t₁·x² / D(x)
D(x) = 1 − 2t₁t₂·x + t₁²·x² x = a·e^−jφ φ = 2πn_effL/λ t²+κ²=1

drop |H_d|² thru |H_t|²

hover the plot to read values

κ₁ = κ₃0.35

κ₂0.065

loss3.0 dB/cm

span300 GHz

radiusµm

n_g

λ₀nm

FSR: —
3-dB BW: —
peak drop: —
thru null: —
split: —

Lossless point couplers and a round-trip phase linearized about λ₀ with the group index. Round-trip field transmission is a = 10^(−αL/20) with L = 2πR. The maximally flat condition is κ₂ = κ₁²/(1+t₁²); the resonance splitting is FSR·asin(κ₂)/π, set by the inter-ring gap alone. Coupler dispersion is not modelled, so a filter that is maximally flat at λ₀ will not be a few tens of nm away.