2.6 EO combs and frequency shifters

Drive a phase modulator inside a ring at exactly the ring's free spectral range (FSR) and every sideband lands on the next resonance; the sidebands cascade into a comb whose lines are spaced by the drive frequency. In a TFLN microring this produced more than 900 lines at about 10 GHz spacing reported. The same physics in a lithium niobate disk resonator inside a microwave cavity appeared in the same Nature issue (Rueda et al. 2019; companion conference paper doi:10.1364/nlo.2019.nm2a.5).

Two coupled rings form a photonic molecule with two supermodes split by the coupling. A microwave at that splitting moves light from one supermode to the other, shifting its frequency by up to 28 GHz at about 90% on-chip efficiency reported.

Resonant EO comb, steady state. Dispersion-free ring, drive Ω = 2π FSR, round-trip amplitude transmission r, peak round-trip phase modulation β; the intracavity-modulator comb generator of Kourogi et al. 1993:

Far from the pump the line power decays at a constant rate per line, set by the complex-time singularity of eq. 2.18, r exp(iβ cos t) = 1; for high finesse F = π√r/(1 − r) the amplitude envelope is exp(−π|k|/(βF)) derived:

Numerical Fourier analysis of eq. 2.18 at r = 0.99, β = 0.1: fit over lines 5 to 40 0.8715 dB/line, asymptotic 0.8715, high-finesse form 0.8730 derived. Doubling the drive voltage halves the slope; doubling the finesse does the same. Published definitions of β differ (per pass or per round trip, peak or rms voltage); here β is the round-trip peak phase.

Dispersion limits the span: line k is detuned from its resonance by the accumulated dispersion, and the cascade stops where that detuning exceeds the modulation coupling (design treatment: Buscaino et al. 2020). Set D₂ > 0 in the comb figure to see the span shrink.

Reported comb generators

Frequency shifter, coupled-mode. Two rings with coupling μ have supermodes split by 2μ. A push-pull microwave at Ω = 2μ couples the supermodes at a rate G proportional to the drive voltage. Steady state, zero detuning, input into supermode a, output from supermode b (EO frequency shifter):

It reaches 1 at G = κ/2 when κ_ex = κ (no intrinsic loss) derived. Reported: shifts up to 28 GHz at about 90% on-chip, cascaded to 119.2 GHz with a 29.8 GHz drive reported (arXiv v1 stated about 99%).

                                              
 b ---------------  ω₀+μ    shifted ω+Ω       
         ^                [########........]  
         |  Ω = 2μ + δ                        
         |  δ =  0.00 κ                       
         |  G = 0.250 κ                       
 a ------+--------  ω₀−μ    in ω              
                          [###.............]  
  efficiency  51.8 %, carrier   −7.1 dB       

Modulation index. β = πV_p/V_π,rt with V_π,rt the voltage for a π round-trip phase derived. Whether a paper quotes a single-pass or round-trip index, peak or rms voltage, must be read from its methods.

Line counts. They depend on the optical spectrum analyzer's dynamic range and on the threshold used; the comb figure prints the threshold with the count. At r = 0.99, β = 0.314 rad (slope 0.2780 dB/line), 423 lines lie within 60 dB of the pump line (N = 4096 to 32768 FFT points give the same count) derived. The estimate 2⌊60/slope⌋ + 1 = 431 refers the threshold to the envelope extrapolated to k = 0, which lies 1.27 dB below the pump line.

Analytic envelope. The residue of eq. 2.18 at its pole t = π/2 − ia gives the dashed envelope in the comb figure derived; lines with |k| ≥ 2 lie on it, the pump line above it:

Lumped model. Microwave loss and velocity mismatch on the ring electrode reduce β at high drive frequency; eq. 2.18 assumes a lumped, uniform phase per round trip. The coupled-mode lines (comb figure, D₂ > 0) uses κ = −2 ln r / T_rt; at D₂ = 0 its lines lie within 0.08 dB of eq. 2.18 for |k| ≤ 200 derived.

Contrast with Kerr combs. EO combs need no phase matching and no anomalous dispersion; their span is set by βF and dispersion, their spacing by an electronic oscillator, which makes them locked by construction (Kerr solitons: 2.5).

Numerical checks: resonant comb

FFT on N = 8192 points over one drive period; Parseval error = |Σ_k|c_k|² / ⟨|E|²⟩ − 1|. β → 0 leaves a single line. derived

Numerical checks: frequency shifter

Eq. 2.19 at G = κ/2, κ_ex = κ: 1.000000000000. At G = 0: 0.0. Detuning ±0.7κ at G = 0.4κ, κ₀ = 0.1κ: difference 0. Time-domain steady state (κt = 60) minus eq. 2.19: 2e−13. derived