The MZI in practice

6 min readBuilds on The Mach-Zehnder interferometerOpen in the lab

The Mach-Zehnder interferometer of the previous articles switches light completely, at any wavelength, without loss and at once. Real MZIs are built from couplers that do not divide light exactly in half, waveguides that lose light, and heaters that draw power and take time to respond. This article describes what MZIs are used for on real chips and what limits them there.

What MZIs are used for

Switches. An MZI with a phase shifter routes light to one of its two outputs. Many of them, connected in a network, form a switch fabric that connects any of $N$ inputs to any of $N$ outputs. Silicon switch fabrics of this kind have been built with 32 and 64 ports; the largest thermo-optic one contains 352 MZIs [1].

Modulators. A modulator writes an electrical data signal onto light (Phase shifters). An MZI with fast phase shifters in its arms turns the phase changes into changes of power, which a detector at the other end of a fibre can read [2]. Silicon modulators that deplete a p-n junction of charges are the most common commercial type [3].

Filters and multiplexers. An MZI with unequal arms transmits wavelengths periodically, with the free spectral range $\lambda^2/(n_\text{g}\,\Delta L)$ (The Mach-Zehnder interferometer). It can separate two wavelengths that travel in one waveguide into two waveguides, or combine them. A device that combines light of several wavelengths into one waveguide is a multiplexer, one that separates it a demultiplexer; together they allow many data streams to share one fibre [2].

Sensors. An interferometer measures small changes of refractive index as changes of power [4]. On a chip, the mode of a waveguide reaches into its surroundings (Waveguides), so a substance near one arm changes that arm’s effective index, and the MZI converts the change into a change of power. Biosensing and gas sensing are among the applications studied for silicon photonics [2].

Programmable circuits. Meshes of MZIs set any unitary matrix (Circuits as matrices) and are the core of programmable photonic circuits [5] and of experiments on computing with light [6].

Two figures of merit

Two numbers describe how well an MZI switches. The extinction ratio is the ratio of the largest to the smallest power an output reaches, usually in decibels (Waveguides), $10\log_{10}(P_\text{max}/P_\text{min})$ [3]. An ideal MZI switches an output completely off, so its extinction ratio is infinite. The insertion loss is the power lost on the way through, the ratio of the largest output power to the input power [3]. In a switch, the light that reaches the wrong output is crosstalk; large switch fabrics need crosstalk below −35 dB [1].

Figure 1 shows the simulated spectrum of an MZI with unequal arms, together with three departures from the ideal that can be set: the arm-length difference, the power coupling of both couplers, and a loss in one arm.

Arm-length difference ΔL100 µm

Coupling κ² of both couplers50.0 %

Loss in the upper arm0.00 dB

Power axis

free spectral range: 5.75 nm
extinction, out1 (cross): > 40 dB
extinction, out0 (bar): > 40 dB
insertion loss: 0.00 dB

Solid: out1 (cross); dashed: out0 (bar); light enters at in0. Silicon strip waveguides (group index 4.18), the shorter arm 200 µm. The couplers split the same way at every wavelength (APX-004), which real couplers do not. The extinction ratios are for the heater swept through a full cycle at 1550 nm.

An MZI with arms differing by 100 µm, couplers splitting 50.0 to 50.0 and 0.00 dB loss in one arm. Free spectral range 5.75 nm; extinction ratio > 40 dB at the cross output and > 40 dB at the bar output.

Figure Simulated output spectra of an MZI with light entering in0. Solid: the cross output; dashed: the bar output. The readouts give the free spectral range, the extinction ratio of each output and the insertion loss.APX-004

Couplers that do not split in half

Fabrication never reproduces a design exactly: the width of the waveguides, the gap between them and the thickness of the silicon vary, and with them the coupling of a directional coupler [2]. An MZI whose couplers split 55:45 instead of 50:50 no longer reaches both extremes. With two identical couplers the cross output can still be made dark, but the bar output keeps a remainder of $(1 - 2\kappa^2)^2$ of the input power at its minimum: 1 % for $\kappa^2 = 0.55$ , an extinction ratio of 20 dB. With two different couplers, neither output goes fully dark.

A loss in one arm has a similar effect. The light arriving from the two arms then has unequal amplitudes, and, as in the article Interference, unequal waves cannot cancel completely.

Wavelength

A directional coupler divides light evenly only near the wavelength it was designed for, because its cross-over length depends on the wavelength. The widget, like the lab, ignores this (APX-004). In a real switch fabric of $32 \times 32$ ports built from MZIs with directional couplers, the crosstalk stayed below −20 dB only over a range of 3.5 nm [1]. For a switch, a narrow working range is a limit; for a filter, the wavelength dependence of an unbalanced MZI is the purpose.

Heat, speed and power

Thermo-optic phase shifters draw power continuously while they hold a state. In the $32 \times 32$ switch, a phase change of $\pi$ took about 18 mW per heater and the chip consumed 1.9 W [1]. Heaters respond in 10–100 µs, and heat reaching neighbouring waveguides shifts their phases [5]: thermo-optic switch fabrics switch in tens of microseconds [1]. Electro-optic switches change state in about a nanosecond, but their free charges absorb light, and the $32 \times 32$ electro-optic fabric lost 13–19 dB on the chip [1].

Size and accumulated loss

Every MZI in a mesh or a switch fabric adds its length and its loss to the light’s path. The $64 \times 64$ thermo-optic switch occupies 21.7 mm × 9.6 mm, and its insertion loss is 12–18 dB [1], while switch fabrics are generally wanted with less than 10 dB [1]. Meshes with a shorter optical path, such as the rectangular mesh of Clements et al., lose correspondingly less [7].

Calibration

Because each MZI’s phases and couplers deviate from the design in a different way, a large circuit does not work as designed when it is switched on. Each element’s phase offset has to be measured and set, and kept set while the temperature of the chip changes. The $64 \times 64$ switch relied on a calibration method for its switching elements, and the $32 \times 32$ electro-optic switch on an array of power monitors to adjust each element [1]. In a mesh, all path lengths must also match to within the coherence length of the light [7].

What engineers do about it

Each limit has known remedies, each with its own cost. Couplers that depend less on wavelength and fabrication, such as MMIs, replace directional couplers; the $64 \times 64$ switch uses them [1]. Heaters on both arms allow the phase to be set in either direction [1]. Monitors and control electronics keep each MZI at its working point. Semiconductor optical amplifiers integrated with the MZIs compensate loss, at the price of noise [1]. The Toolbox pages describe these components, and the lab’s real-world mode is planned to show the same effects on the reader’s own circuits.

These limits also bound what a mesh of MZIs can compute. Claims that optical computing is generally faster or more efficient than electronics leave them out; a fair comparison includes the power of the lasers, the phase shifters, the detectors and the electronics that convert between the two domains.

Key idea

Every limit of a real MZI is a departure from one of the ideal assumptions: exact splitting, no loss, independence of wavelength, and a phase that stays where it is set.

How do we model itOptional · the extinction ratio of an imperfect MZI

With couplers of power coupling $\kappa_1^2$ and $\kappa_2^2$ ( $t_{1,2} = \sqrt{1 - \kappa_{1,2}^2}$ ) and the phase difference $\Delta\varphi$ , the matrix product of the article Circuits as matrices gives, for light of amplitude 1 at in0,

\begin{aligned} b_0 &= t_1 t_2\, e^{i\Delta\varphi} - \kappa_1 \kappa_2, \\ b_1 &= i\left(\kappa_2 t_1\, e^{i\Delta\varphi} + t_2 \kappa_1\right). \end{aligned}

As $\Delta\varphi$ runs through a cycle, the powers range over

\begin{aligned} (t_1 t_2 - \kappa_1\kappa_2)^2 &\le |b_0|^2 \le (t_1 t_2 + \kappa_1\kappa_2)^2, \\ (\kappa_2 t_1 - t_2\kappa_1)^2 &\le |b_1|^2 \le (\kappa_2 t_1 + t_2\kappa_1)^2. \end{aligned}

For identical couplers, $\kappa_1 = \kappa_2 = \kappa$ , the cross output’s minimum is zero, while the bar output’s minimum is $(t^2 - \kappa^2)^2 = (1 - 2\kappa^2)^2$ and its maximum 1. Its extinction ratio is therefore

\text{ER}_\text{bar} = -20 \log_{10} |1 - 2\kappa^2|\ \text{dB},

20 dB for $\kappa^2 = 0.55$ and 40 dB for $\kappa^2 = 0.505$ . A loss in one arm multiplies that arm’s amplitude by $a < 1$ ; the minimum of the cross output becomes $\kappa^2 t^2 (1 - a)^2$ , no longer zero. The simulated curves of Figure 1 agree with these expressions.

In short

MZIs are used as switches, modulators, wavelength filters, sensors and as the elements of programmable meshes.
Couplers that do not split exactly in half, loss in one arm and the wavelength dependence of couplers limit how completely an MZI switches, measured by its extinction ratio.
Thermal phase shifters cost milliwatts each and respond in tens of microseconds; large circuits of MZIs are millimetres long, lose several decibels and must be calibrated.

Build it in the labTwo-colour demultiplexerOpen the unbalanced MZI, switch one laser off, and use the spectrum view to see its free spectral range.Open the lab

References

1Cheng, Q., Yao, C., Calabretta, N., Stabile, R., Suzuki, K., Kawashima, H., Tang, W., Glick, M., Chu, T., Ikeda, K., Matsumoto, R., Namiki, S., Bergman, K., Penty, R. (2023). Photonic switch fabrics in data center/high-performance computing networks. In: Glick, Liao & Schmidtke (eds.), Integrated Photonics for Data Communication Applications, Elsevier, ch. 8, pp. 265–301. doi:10.1016/B978-0-323-91224-2.00003-5
2Chrostowski, L., Hochberg, M. (2015). Silicon Photonics Design. Cambridge University Press. doi:10.1017/CBO9781316084168
3Reed, G., Thomson, D., Zhang, W., Gardes, F., Mastronardi, L., Li, K., Matsuo, S., Kanazawa, S., Vivien, L., Lafforgue, C., Bowers, J. E., Koos, C., Romagnoli, M., Lončar, M., Zhang, M., Abel, S., Liao, L. (2023). Optical modulators. In: Glick, Liao & Schmidtke (eds.), Integrated Photonics for Data Communication Applications, Elsevier, ch. 3, pp. 69–121. doi:10.1016/B978-0-323-91224-2.00011-4
4Saleh, B. E. A., Teich, M. C. (2019). Fundamentals of Photonics (3rd edition). Wiley. link
5Bogaerts, W., Pérez, D., Capmany, J., Miller, D. A. B., Poon, J., Englund, D., Morichetti, F., Melloni, A. (2020). Programmable photonic circuits. Nature 586, 207. doi:10.1038/s41586-020-2764-0
6Shen, Y., Harris, N. C., Skirlo, S., Prabhu, M., Baehr-Jones, T., Hochberg, M., Sun, X., Zhao, S., Larochelle, H., Englund, D., Soljačić, M. (2017). Deep learning with coherent nanophotonic circuits. Nature Photonics 11, 441. doi:10.1038/nphoton.2017.93
7Clements, W. R., Humphreys, P. C., Metcalf, B. J., Kolthammer, W. S., Walmsley, I. A. (2016). Optimal design for universal multiport interferometers. Optica 3, 1460. doi:10.1364/OPTICA.3.001460 (open access)

Approximations used on this page: APX-004 wavelength-independent coupling ratio.