Mach–Zehnder interferometer (MZI)

A Mach–Zehnder interferometer (MZI) splits light into two arms, delays one arm by a phase $\mathrm{\Delta }\varphi$ relative to the other, and recombines them, so that the power in one output follows ${\cos }^2(\mathrm{\Delta }\varphi \mathrm{/}2)$ and the power in the other ${\sin }^2(\mathrm{\Delta }\varphi \mathrm{/}2)$: the light moves completely from one output to the other as $\mathrm{\Delta }\varphi$ goes from 0 to $\pi$. Driving the phase electrically makes the MZI a switch or a Mach–Zehnder modulator; making one arm longer by $\mathrm{\Delta }L$ makes it a wavelength filter whose transmission repeats every $\text{FSR}={\lambda }^2\mathrm{/}(n_g\mathrm{\Delta }L)$, which is 5.6 nm for a 100 µm path difference in a silicon wire at 1550 nm. Ludwig Zehnder described the arrangement in 1891 and Ludwig Mach refined it in 1892; in waveguide form it is the most common interferometer in photonic integrated circuits.

Transfer function and port convention

The couplers decide which output is which. A 2×2 coupler, whether a directional coupler or a 2×2 MMI coupler, gives the field that crosses to the other waveguide a 90° phase shift relative to the field that stays. With two 3 dB couplers of this kind the outputs are

$$\begin{array}{rl}{\displaystyle P_{\text{cross}}}& {\displaystyle =P_{\text{in}}{\cos }^2(\mathrm{\Delta }\varphi \mathrm{/}2)}\\ {\displaystyle P_{\text{bar}}}& {\displaystyle =P_{\text{in}}{\sin }^2(\mathrm{\Delta }\varphi \mathrm{/}2)}\end{array}$$

so a balanced MZI ($\mathrm{\Delta }\varphi =0$) sends all the light to the cross port, and a phase difference of $\pi$ moves it to the bar port. With a Y-branch splitter and combiner there is a single output, $P_{\text{out}}=P_{\text{in}}{\cos }^2(\mathrm{\Delta }\varphi \mathrm{/}2)$; at $\mathrm{\Delta }\varphi =\pi$ the recombined field has the antisymmetric shape that the single-mode output waveguide cannot guide, and the light radiates into the cladding and substrate instead of reaching a second port. Some texts label the ports the other way round. The physics is the same in every convention: the two outputs are complementary, they sum to the input less the losses, and at $\mathrm{\Delta }\varphi =\pi \mathrm{/}2$, the quadrature point, they are equal and the response to a small phase change is steepest.

The phase difference comes from a difference in optical path length, through the physical length, the effective index, or both:

$$\mathrm{\Delta }\varphi =\frac{2\pi }{\lambda }(n_{\text{eff},1}{L}_1-n_{\text{eff},2}{L}_2)$$

In a modulator or switch the arms are equal and a phase shifter of length $L$ changes one arm's effective index by $\mathrm{\Delta }{n}_{\text{eff}}$, giving $\mathrm{\Delta }\varphi =2\pi \mathrm{\Delta }{n}_{\text{eff}}L\mathrm{/}\lambda$. Moving the light fully from one port to the other takes $\mathrm{\Delta }{n}_{\text{eff}}L=\lambda \mathrm{/}2$, which is 0.775 µm at 1550 nm: an index change of $7.75\times {10}^{-3}$ over 100 µm, or $7.75\times {10}^{-4}$ over 1 mm. In a modulator the voltage that produces it is the half-wave voltage $V_{\pi }$.

Spectral response: the asymmetric MZI

When one arm is longer by $\mathrm{\Delta }L$ in the same waveguide, $\mathrm{\Delta }\varphi =2\pi n_{\text{eff}}\mathrm{\Delta }L\mathrm{/}\lambda$ depends on wavelength, and each output is a raised-cosine filter, periodic in frequency. Its period, the free spectral range, depends on how fast $\mathrm{\Delta }\varphi$ changes with wavelength, which brings in the group index $n_g$ in place of $n_{\text{eff}}$:

$${\text{FSR}}_{\lambda }=\frac{{\lambda }^2}{n_g\mathrm{\Delta }L},{\text{FSR}}_{\nu }=\frac{c}{n_g\mathrm{\Delta }L}$$

The distinction matters most in high-index waveguides: a silicon wire has $n_{\text{eff}}\approx 2.4$ but $n_g\approx 4.3$, so using the effective index would overstate the FSR by about 80%. Worked values at 1550 nm:

The Directional Coupler and MZI Calculator plots both outputs of an MZI against wavelength for ideal, fixed or coupled-mode couplers and an arm loss difference, and reports the free spectral range and the extinction ratio of each port.

A 100 GHz FSR is what an interleaver needs to separate the odd and even channels of a 50 GHz grid, since neighboring channels then fall on opposite outputs. A 10 GHz FSR is a delay of one symbol at 10 Gbaud, the delay-line interferometer that demodulates differential phase-shift keying by interfering each symbol with the one before it. A single stage has a cosine passband with no flat top; cascading stages, choosing unequal coupler ratios in a lattice of them, or adding a ring resonator to one arm flattens the passband and steepens its edges, and trees of stages with $\mathrm{\Delta }L$, $2\mathrm{\Delta }L$ and so on make WDM demultiplexers. A heater on one arm tunes the filter, which otherwise drifts with temperature as the effective index changes.

Extinction ratio and imbalance

An output goes fully dark only if the two paths into it carry equal amplitudes. For couplers with power coupling ratios ${\kappa }_1$ and ${\kappa }_2$, the extinction ratio of the bar port, its maximum over its minimum as $\mathrm{\Delta }\varphi$ is swept, is limited to

$${\text{ER}}_{\text{bar}}={\left(\frac{1+a}{1-a}\right)}^2,a=\sqrt{\frac{{\kappa }_1{\kappa }_2}{(1-{\kappa }_1)(1-{\kappa }_2)}}$$

With two identical couplers this reduces to $1\mathrm{/}(1-2\kappa {)}^2$, and the cross port still goes fully dark, because the two paths into it, through one coupler's bar and the other's cross, carry equal amplitudes $\sqrt{\kappa (1-\kappa )}$ whatever $\kappa$ is; with unequal couplers neither port is perfect. The numbers set the fabrication tolerance for high-contrast switches and modulators:

A loss difference between the two arms limits the extinction in the same way. For arm powers in the ratio $r$ (in linear units) reaching an ideal second coupler, the fields at the canceling port are $1$ and $\sqrt{r}$, and $\text{ER}=[(1+\sqrt{r})\mathrm{/}(1-\sqrt{r}){]}^2$: 0.1 dB of imbalance caps the extinction at 45 dB, 0.5 dB at 31 dB, 1 dB at 25 dB, and 3 dB at 15 dB. A 30 dB switch therefore needs its couplers within about 0.3 dB of 50:50 and its arms within 0.5 dB of each other. That is demanding for directional couplers, whose split drifts with wavelength and with the gap, and easier with broadband MMI couplers; designs that need more use a tunable coupler (itself a small MZI), an attenuator in one arm for trimming, or two stages in cascade. Typical imbalance figures for the couplers themselves are in the MMI coupler and directional coupler entries.

Uses

The 90° optical hybrid of a coherent receiver is built from the same couplers and phase shifts. A mesh of Mach–Zehnder interferometers can perform matrix-vector multiplication by interference, the basis of optical computing; why that looks free but is not is covered in Why light moves data and electrons still compute.

Common questions

What is a Mach–Zehnder interferometer used for?

In integrated and fiber optics: modulators for coherent and many direct-detect transmitters, thermo-optic switches, wavelength filters and interleavers, and sensors. In free space it was built to measure refractive-index changes in gases, such as the airflow in a wind tunnel, and it remains a standard layout for interference experiments with single photons.

What is the difference between a Mach–Zehnder and a Michelson interferometer?

Both split light into two paths and recombine it. A Michelson sends each beam to a mirror and back through the same beamsplitter, so the path difference is twice the mirror displacement and one output returns toward the source. An MZI uses a separate splitter and combiner and passes through each arm once, so both outputs are accessible without a circulator and each arm can be modified on its own, which suits waveguides.

How do you calculate the FSR of an MZI?

Divide the square of the wavelength by the group index times the path difference: $\text{FSR}={\lambda }^2\mathrm{/}(n_g\mathrm{\Delta }L)$. A 100 µm path difference in a silicon wire ($n_g\approx 4.3$) gives 5.6 nm at 1550 nm, or 697 GHz from $c\mathrm{/}(n_g\mathrm{\Delta }L)$.

What limits the extinction ratio of an MZI?

Unequal splitting and unequal loss in the arms. With identical couplers the bar port reaches 30 dB only if the split is within about 0.3 dB of 50:50, and 0.5 dB of loss difference between the arms caps it at 31 dB on its own.

References: M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999). B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019). K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic Press, 2006). L. Chrostowski and M. Hochberg, Silicon Photonics Design (Cambridge University Press, 2015).