Alex Sludds

Vernier two-ring filter

Two series-coupled rings of unequal radius. Only the resonances that happen to line up survive; everything else is suppressed by the ring that does not resonate there. The composite free spectral range is far larger than either ring's, at the cost of side modes that never fully disappear.

|Hd|² =  (κ₁²κ₂)²·a₁a₂  /  |D|²     D = 1 − t₁t₂(x₁+x₂) + t₁²x₁x₂
xi = ai·e−jφi    φi = 2πni(λ)Li/λ    FSRV = FSR₁·FSR₂ / |FSR₁−FSR₂|
drop |Hd|² thru |Ht|²
A tall aligned resonance at the design wavelength with suppressed side modes on either side.
hover the plot to read values
µm
µm
0.35
0.065
3.0 dB/cm
30 nm
nm
FSR₁
—
FSR₂
—
Vernier FSR
—
side mode
—
3-dB BW
—

Both rings are assumed tuned into alignment at λ₀: the mode order of each is rounded to the nearest integer and its index adjusted to put a resonance exactly there, which is what a thermal tuner does for you in practice. Dispersion is first order, from dn/dλ = (neff − ng)/λ₀. Side mode suppression is measured inside the plotted span, so widen the span until you can see the next Vernier order before trusting it. Making the radii closer together buys a larger composite FSR and immediately costs you side mode suppression — that trade is the whole design problem.