◼ How it works
The electrode is a coplanar waveguide (CPW): a center signal line of width w with
two ground planes separated by gaps s, on a SiO₂ buffer over an X- or Z-cut LiNbO₃ substrate. The
microwave drive travels along this line as a traveling wave while the optical mode
co-propagates in a Ti-diffused waveguide beneath it. The model below is computed exactly as the
script runs.
1 · Quasi-static CPW. Conformal mapping gives the modulus
k = w/(w+2s), k′ = √(1−k²), and the line impedance
Z₀ = (30π/√ε_eff)·K(k′)/K(k), where K is the complete elliptic integral. A partial-fill factor
η = 1 − e−2t_ox/s blends buffer and substrate: ε_eff = ½[1 + η·ε_SiO₂ + (1−η)·ε_LN].
Then C = 1/(Z₀·v_p), L = Z₀/v_p, with v_p = c/√ε_eff.
2 · Frequency-dependent RLGC. Conductor loss follows the skin
effect, R(f) ∝ √(π·f·μ₀/σ)/(w+2t); dielectric loss G(f) = ω·C·[η·tanδ_SiO₂ + (1−η)·tanδ_LN].
The complex propagation constant and characteristic impedance are
γ = √[(R+jωL)(G+jωC)] and Z_c = √[(R+jωL)/(G+jωC)].
3 · S-parameters. The electrode is cast as an ABCD matrix
(A = D = cosh γℓ, B = Z_c·sinh γℓ, C = sinh γℓ/Z_c) and converted to S₁₁/S₂₁ referenced to the
50 Ω port. The RF attenuation is α = 8.686·Re(γ) dB/cm and the microwave index n_RF = c·Im(γ)/ω.
4 · Optical index. Bulk n_o and n_e of LiNbO₃ (and n of SiO₂)
come from Sellmeier equations; the Ti diffusion raises the core by Δn. A slab eigenvalue solver
returns the guided n_eff and group index n_g. TM uses n_e with r₃₃ (strong);
TE uses n_o with r₁₃ (weak).
5 · Velocity matching & EO bandwidth. A velocity mismatch
Δβ = (ω/c)(n_RF − n_opt) causes walk-off between the microwave and optical waves. The traveling-wave
EO response is m(f) = |(1 − e−uℓ)/(uℓ)| with u = α + jΔβ. The −3 dB roll-off of m(f) sets
the modulation bandwidth; shrinking |n_RF − n_opt| and the loss α widens it.
6 · Half-wave voltage. Vπ = λ·g /(N·n³·r·Γ·L), with N = 2 for a
push-pull MZM (else 1), g the electrode gap, r the active EO coefficient, Γ the optical/RF overlap
and L the electrode length. The frequency-dependent value is Vπ(f) = Vπ_DC / m(f), so the EO
roll-off directly raises the drive voltage at high frequency.
Design goal: low RF loss, Z_c near 50 Ω, and velocity matching (n_RF ≈ n_opt) together
maximize the EO bandwidth for a given Vπ·L product.