Engineering & Science
Pockels Electro-Optic Modulator Half-Wave Voltage (Vpi) & Transmission
Calculate transverse lithium-niobate (LiNbO3) Pockels modulator half-wave voltage V_pi = (lambda_0 * d) / (2 * n_e^3 * r_33 * L) and crossed-polarizer optical transmission I / I_0 = SIN((PI()/2) * (V / V_pi))^2.
See how waveguide electrode Aspect Ratio (d / L) drops Pockels V_pi from kilovolts to ~1 Volt!
In non-centrosymmetric crystals like Lithium Niobate (LiNbO3, extraordinary index n_e = 2.14, Pockels coefficient r_33 = 30.8 pm/V at telecom wavelength lambda_0 = 1,550 nm), an applied electric field E = V / d shifts refractive index linearly: delta_n = -(1/2) * n_e^3 * r_33 * (V / d). Bulk longitudinal Pockels cells require 2–4 kilovolts to reach a pi-radian half-wave retardation, making 40-GHz telecom modulation impossible!
Transverse integrated waveguide Mach-Zehnder / amplitude modulators solve this via geometry: applying voltage across a tiny electrode gap d = 10 um (0.00001 m) while propagating light along interaction length L = 20 mm (0.02 m) scales V_pi = (lambda_0 * d) / (2 * n_e^3 * r_33 * L) down by d / L = 1 / 2000 to just V_pi = 1.284 V! Biasing at the quadrature point V = 0.642 V (~V_pi / 2) yields linear 50.02% optical transmission!
| Modulator Drive Bias (LiNbO3 Waveguide d = 10 um, L = 20 mm) | Half-Wave Voltage V_pi (V) | Applied Voltage V (V) | Crossed-Polarizer Transmission (%) |
|---|---|---|---|
| Quadrature Linear Bias (~V_pi / 2) | 1.284 V | 0.642 V | 50.02% (Max Linear Slope) |
| Full Half-Wave ON State (V = V_pi) | 1.284 V | 1.284 V | 100.00% Peak Transmission |
| Invalid Crystal Geometry (L <= 0 or n <= 0) | Guarded Optical Error | N/A | "Invalid Modulator Inputs" |
Transverse Pockels Modulator Half-Wave Voltage V_pi (Volts)
Evaluates (lambda_m * d_m) / (2 * n_e^3 * (r_33_pmV * 10^-12) * L_m) for telecom/laser waveguide geometry.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0), "Invalid Modulator Inputs", ROUND((B2 * B3) / (2 * (B4 ^ 3) * (B5 * 0.000000000001) * B6), 3))Electro-Optic Amplitude Modulator Transmission (%) = 100 * SIN((PI()/2) * (V / V_pi))^2
Computes normalized intensity transmission at drive voltage B7 relative to exact unrounded V_pi.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0), "Invalid Modulator Inputs", ROUND(100 * (SIN((PI() / 2) * (B7 / ((B2 * B3) / (2 * (B4 ^ 3) * (B5 * 0.000000000001) * B6)))) ^ 2), 2))Try the example
Use the original values or change them to test your understanding.