Glossary entry

Mach–Zehnder interferometer (MZI)

A two-arm interferometer that splits light, applies a phase difference between the arms, and recombines them. The most common interferometric building block in integrated photonics.

A Mach–Zehnder interferometer splits an input beam into two arms, applies a controllable phase difference $\Delta\phi$ between them, and recombines them at a second beam combiner. The output intensity depends on $\Delta\phi$ :

P_\text{bar} \;=\; P_\text{in} \cos^2(\Delta\phi / 2), \qquad P_\text{cross} \;=\; P_\text{in} \sin^2(\Delta\phi / 2).

For balanced 50/50 splitters (MMI or directional couplers), the two output ports show complementary intensity patterns as $\Delta\phi$ varies — full transmission at one port and zero at the other when $\Delta\phi = 0$ , and vice versa at $\Delta\phi = \pi$ .

Spectral response (asymmetric MZI). When the two arms differ in optical path length by $\Delta L$ :

\Delta\phi(\lambda) \;=\; \frac{2\pi \, n_g \, \Delta L}{\lambda},

where $n_g$ is the group index. The transmission is a cosine-squared (sinusoidal-in-frequency) filter with free spectral range:

\text{FSR} \;=\; \frac{\lambda_0^2}{n_g \, \Delta L}.

Typical applications:

Application	Variant	Phase shifter
Mach–Zehnder modulator	Balanced arms	Voltage-driven (Pockels, plasma dispersion)
Tunable filter	Asymmetric, $\Delta L > 0$	Thermo-optic for fine tuning
Interleaver (50 / 100 GHz separator)	Cascaded asymmetric stages	Static
WDM (de)multiplexer	Cascaded MZI tree	Static
Optical switch	Balanced	Thermo-optic or electro-optic
Pulse shaper / spectrometer	Many parallel asymmetric stages	Static
Optical phased array beamformer	Many balanced MZIs with relative phase control	Thermo-optic
Coherent receiver 90° optical hybrid	Balanced with 90° hybrid coupler	Static

Imbalanced amplitude. For real beam splitters with intensity coupling ratios $\kappa_1$ , $\kappa_2$ (deviations from 50/50), the extinction ratio at the cancellation port is limited to

\text{ER} \;=\; \frac{(1 + \sqrt{\kappa_1 \kappa_2 / [(1-\kappa_1)(1-\kappa_2)]})^2}{(1 - \sqrt{\kappa_1 \kappa_2 / [(1-\kappa_1)(1-\kappa_2)]})^2}.

Achieving high ER ( $> 30$ dB) requires balanced splitting to within $\pm 0.1$ dB, which is challenging for narrow-band directional couplers but achievable with broadband MMI couplers (see multimode interference coupler).

The MZI is conceptually identical to a Michelson interferometer with the round-trip folded out — both rely on the same principle of two-beam interference, but the MZI is in transmission rather than reflection, naturally suited to integrated photonics geometries.