Engineering & Science
Mach-Zehnder Interferometer (MZI): Phase Transmittance & Extinction Ratio
Calculate integrated silicon-photonics Mach-Zehnder bar-port transmittance T = COS(π * Δn_eff * L / λ)^2 and finite coupler Extinction Ratio ER = -20 * LOG10(ABS(1 - 2 * κ)) (dB).
Why a 3% directional-coupler split error caps MZI extinction at 24.44 dB!
Integrated Mach-Zehnder Interferometer (MZI) modulators and biosensors split light across two waveguide arms, imprint an effective-index perturbation Δn_eff over active length L (phase shift ΔΦ = 2π * Δn_eff * L / λ), and interfere the two arms at an output coupler: T = COS(ΔΦ / 2)^2 = COS(π * Δn_eff * L / λ)^2! At telecom λ = 1.55 μm with thermo-optic or carrier-depletion shift Δn_eff = 0.0005: L = 500 μm gives T = 0.7645, L = 775 μm hits the exact 50% quadrature bias point (ΔΦ = π/2, T = 0.5000), and L = 1,550 μm hits complete destructive null (ΔΦ = π, T = 0)!
In real wafer fabrication, etched 50:50 directional couplers drift slightly to power split κ ≠ 0.50 (e.g., 47:53 split κ = 0.47). Because unequal arm amplitudes cannot cancel to zero, maximum achievable Extinction Ratio across two identical couplers is bounded by ER (dB) = -20 * LOG10(ABS(1 - 2 * κ))—capping a 47:53 coupler at 24.44 dB!
| Silicon Photonics MZI Arm Length L (λ=1.55 μm, Δn=0.0005) | Phase Shift ΔΦ (rad) | Ideal Transmittance T = cos^2(ΔΦ/2) | Modulator Bias Point |
|---|---|---|---|
| Partial Phase Tuning (L = 500 μm) | 1.0134 rad | 0.7645 (76.45%) | High-Transmission Slope |
| Quadrature Linear Bias (L = 775 μm) | 1.5708 rad (π / 2) | 0.5000 (50.00%) | Maximum Linear Small-Signal Gain |
| Half-Wave Null Length L_π (L = 1,550 μm) | 3.1416 rad (π) | 0.0000 (0.00%) | Destructive OFF State |
Ideal MZI Optical Power Transmittance T = COS(π * Δn_eff * L / λ)^2
Evaluates COS(PI() * B3 * B4 / B2)^2 with both wavelength B2 and active arm length B4 expressed in micrometers.
=IF(OR(B2 <= 0, B4 < 0), "Invalid MZI Inputs", ROUND(COS(PI() * B3 * B4 / B2) ^ 2, 4))Coupler-Limited Extinction Ratio ER = -20 * LOG10(ABS(1 - 2 * κ)) (dB)
Returns "Ideal Infinite ER (50:50 Split)" at exact κ = 0.50 and computes decibel extinction limit for realistic 0 < κ < 1 fabrication tolerances.
=IF(OR(B5 <= 0, B5 >= 1), "Invalid Split Ratio", IF(B5 = 0.5, "Ideal Infinite ER (50:50 Split)", ROUND(-20 * LOG10(ABS(1 - 2 * B5)), 2)))Try the example
Use the original values or change them to test your understanding.