6.2 Traveling-wave design: velocity, impedance, RF loss

At 100 GHz the drive signal on the chip has a wavelength of about 1.4 mm (c / (f n_RF) with n_RF = 2.2 derived), shorter than the electrode. The electrode is therefore a transmission line: the microwave and the light race down the chip together. If they travel at the same speed and the microwave does not fade, every slice of the electrode adds phase in step, and the modulator stays fast however long it is.

Model

Voltage on a line with propagation constant γ = α + j ω n_RF / c, load reflection Γ_L = (Z_L - Z₀)/(Z_L + Z₀), source reflection Γ_S = (Z_S - Z₀)/(Z_S + Z₀):

The light sees the voltage at the time it reaches z, t = t0 + n_g z / c. The modulation it accumulates, normalized to the electrode length:

EO response (dB) = 20 log₁₀ |m(f) / m(0)|, with m(0) = V_S Z_L / (Z_S + Z_L). derived from transmission-line theory; the model class is the one in Chung, Chang, Adler 1991 [Chung et al. 1991] and Alferness 1982 [Alferness 1982].

Single-effect limits (derived here from the matched-load form, Z_S = Z_L = Z₀):

Worked numbers derived, using the Chen 2022 (c) device (capacitively loaded, n_RF 2.35, n_g 2.2, Z₀ 50.5 ohm, RF loss 2.947 dB/cm at 67 GHz, L = 10 mm) reported:

• Velocity term alone: f_3dB = 0.443 c / (0.15 × 10 mm) = 88.5 GHz.
• Loss term alone, loss scaling as √(f): 6.4 dB is reached at 67 GHz × (6.4 / 2.947)² = 317 GHz.
• Reported 3 dB bandwidth: > 67 GHz (bound) reported.

Design levers:

• Velocity. In a quasi-TEM line n_RF = c √(L' C') and Z₀ = √(L' / C'); adding capacitance per length (T-rail loading) raises n_RF and lowers Z₀ together. Of 28 TFLN/TFLT device rows that report both indices, 23 have |n_RF - n_g| ≤ 0.10; the others are Chen 2022 (c) (0.15), Renaud 2022 (3um-738) (0.16), Mao 2022 (a) (0.28), Lin 2026 (a) (0.50, lumped), Chen 2023 (a) (n_g 4.0, approx.) derived.
• RF loss. Narrow gaps raise conductor loss; T-rail segments keep the rails wide while bringing the field close to the waveguide reported. Low-k underfill was used to reduce RF loss and reach velocity matching with a narrow gap reported.
• Impedance. A source at 50 ohm into a lower-Z₀ line divides the voltage; load mismatch adds a counter-propagating wave whose contribution averages out once ω (n_RF + n_g) L / c >> 1.

Slow light and loaded electrodes

In a slow-light waveguide the phase from a given index change grows with the optical group index, which lowers V_πL; the electrode must then slow the microwave to the same group velocity. A 1 mm SiN-loaded TFLN MZM with slow-light waveguides and capacitively loaded slow-wave electrodes reached 0.21 V cm with bandwidth > 110 GHz reported.

• Atlas RF fields: RF index, optical group index, Z₀, and RF loss in power dB/cm at a stated frequency (amplitude coefficients converted). A cross-section's RF index and impedance are not the effective values of a periodically loaded line, so cross-section values do not give the loaded-line n_RF or Z₀.
• Measured RF loss is usually quoted at one frequency; the √(f) scaling (conductor skin effect) is a model assumption, and dielectric loss scales as f.
• Instrument roll-off, probe and cable de-embedding, and the photodiode response enter measured EO S21 and are removed by the authors with stated or unstated calibrations.