6.1 Electro-optic modulators

A modulator is an interferometer whose arms are built in a Pockels crystal. A voltage across one or both arms changes the refractive index, shifts the phase of the light in that arm, and moves the output between bright and dark. Thin-film LN confines light to a sub-micron ridge, so the gold electrodes can sit a few microns apart (median gap 5.0 µm over 144 TFLN device rows derived); the same index change then needs a few volts, and the electrodes can carry a microwave signal along the chip at nearly the speed of the light.

Transfer function and half-wave voltage

Mach-Zehnder intensity transfer (lossless, balanced splitters):

• V_π is the voltage that moves T from maximum to minimum; φ_b is the static bias phase.
• Push-pull drive puts opposite fields in the two arms; one signal electrode between two grounds (G-S-G) does this on x-cut film. For the same electrode, the push-pull MZM V_π is half the single-arm phase-shifter V_π.
• IQ modulator: two MZMs nested in an outer interferometer with a 90-degree phase between them; each child MZM is biased at null.

Half-wave voltage-length product for an x-cut film, TE mode, field along the crystal z axis:

• g: electrode gap; Γ: overlap between the normalized optical intensity and the applied field E_z (Γ = (g/V) integral of E_z |E_opt|² dA / integral of |E_opt|² dA). The full tensor overlap is evaluated by the eo-modulator-atlas FEM engine (chapter 2.2, L3).
• Worked value: λ = 1.55 µm, g = 5 µm template, Γ = 0.5 template, n_e = 2.1376 at 1550 nm, 21 C derived, r₃₃^S = 30.8 pm/V at 633 nm P-comp (Turner 1966 via Weis and Gaylord Table 4): V_πL = 2.58 V cm derived.
• r₃₃ choice moves the result: the Kaminow 1980 value 28.8 pm/V P-comp gives 2.75 V cm derived (+7%). No primary r₃₃ measurement at 1550 nm is available; the 633 nm clamped value is used.
• Atlas cross-check: median telecom push-pull V_πL for TFLN MZM/IQ device rows is 2.44 V cm (n = 66 rows) derived: eo-modulator-atlas device rows at commit eb97f96, telecom band, push-pull convention, MZM and IQ.

Implied overlap from the atlas (derived): over 71 x-cut TFLN push-pull MZM/IQ device rows with gap, wavelength, and V_πL, Γ_implied spans 0.15 to 6.12, with 45 of 71 rows between 0.45 and 0.67 derived. Values above 1 are not physical overlaps; 8 rows exceed 1 and mark devices outside the lateral-gap model or its inputs: four operating points of the Shen 2024 slow-light MZM (6.12, 3.21, 1.84, 1.31), two Yu 2024 devices that claim a material enhancement (1.33, 1.47), and two Xue 2026 devices (1.03, capacitively loaded, thick silica buffer). The formula assumes the 633 nm clamped r₃₃ at all wavelengths and Zelmon n_e at the row wavelength.

Modulator families

Counts: 209 TFLN device rows at atlas commit eb97f96 derived; 31 rows have no electrode type.

Ring and photonic-crystal modulators

Resonance shift per volt: Δ λ = λ Δ n_eff / n_g. A resonator stores light for the photon lifetime τ = Q_L / ω₀, so its modulation response rolls off on the scale of the linewidth Δ ν = ν₀ / Q_L; the exact factor depends on detuning.

The Lorentzian linewidth ν₀/Q_L at 1550 nm is 34 GHz for Q_L = 5,700 and 10.7 GHz for Q_L = 18,000 derived; measured bandwidths can exceed it when the device is operated detuned (atlas note on Zhang 2023 (b): headline 104 GHz at the -8 dB detuning point, authors' photon-lifetime limit 35 GHz).

How V_π and bandwidth are measured, and why papers disagree

• Convention. V_π can be quoted per phase-shifter arm, for a push-pull MZM, or as a differential (V+ minus V-) value; these differ by factors of 2. Values are kept in the convention each paper states; none are converted. Example: the authors of Aimone 2026 (a) quote 1.1 V cm (differential reference); 3.26 V × 7.5 mm gives 2.45 V cm derived.
• Frequency. V_π measured below 1 GHz (DC, kHz or MHz triangle) and at RF differ: below about 20 kHz, charge relaxation in TFLN enhances the low-frequency response (relaxation-enhanced band 1-20 kHz, response measured 1 mHz to 1 MHz) reported. V_π values are split at 1 GHz into DC and RF values.
• Bandwidth reference. "3 dB EO bandwidth" is 20 log₁₀ of the detected modulation amplitude relative to a stated reference (DC, 1 GHz, or an unstated low-frequency plateau). Methods differ: VNA/LCA EO S21, optical sideband ratio, link EOE response.
• Instrument limits. 67 of 145 TFLN/TFLT bandwidth values in the atlas are lower bounds ("over 67 GHz", "over 110 GHz"); the most frequent bound values are 110 GHz (15 rows), 67 GHz (13), 50 GHz (12) derived. Medians over bounded data understate true bandwidth.
• Extrapolation. Several headline bandwidths are extrapolated beyond the measured range (e.g. 220 / 218 GHz for C / O band at 7 mm, extrapolated reported). Extrapolated bandwidths are a separate class from measured ones.
• Disclosure. Of 209 TFLN device rows, V_πL is stated or derivable in 64%, 3 dB bandwidth in 60%, on-chip insertion loss in 25%, RF loss in 11%, Z₀ in 10%, n_RF in 10% derived.