The Mach-Zehnder interferometer

5 min readBuilds on Phase shifters and Splitters and combinersOpen in the lab

A Mach-Zehnder interferometer (MZI) divides light between two paths, called arms, and recombines it at a second coupler. The power at each of its two outputs depends on the difference in phase that the light gains in the two arms. A change of this phase difference by half a cycle moves the light from one output to the other. On a chip, an MZI consists of two directional couplers, two waveguides and a heater; it is used as a switch, as a tunable splitter and as a modulator, and it is the basic element of programmable photonic circuits [1]. This article assembles the MZI from the parts of the previous articles and shows how its outputs depend on the phase difference.

Two couplers and two arms

Each part of the MZI was introduced in an earlier article. The first coupler, a 50:50 directional coupler, divides the light entering in0 equally between the two arms, with the light in the lower arm a quarter cycle ahead (Splitters and combiners). The arms are waveguides; a heater on the upper arm adds the phase $\Delta\varphi$ to the light in it (Phase shifters). The second coupler combines the light of the two arms, and where the light goes depends on the phase difference between them (Interference). Two detectors measure the power at out0 and out1.

The same arrangement is a classic instrument of free-space optics, in which beamsplitters and mirrors play the roles of the couplers and arms [2].

Figure 1 shows an MZI as the lab draws it, with light entering at in0.

Phase difference Δφ (heater)0

out0 (bar)0 %

out1 (cross)100 %

fields at the end of the upper (solid) and lower (dashed) arm

Solid curve: out1 (cross); dashed: out0 (bar). Ideal 50:50 couplers, no loss; powers as a percentage of the input at in0.

With a phase difference of 0 between the arms, 0 % of the light leaves at out0 and 100 % at out1.

Figure A Mach-Zehnder interferometer with a heater on the upper arm; light enters at in0. The brightness shows the power in each part, the clock the fields at the end of the two arms, and the curves the output powers for every phase difference.APX-001, APX-004

Balanced arms

An MZI whose two arms have equal optical path lengths is balanced. With the heater off, the light arrives at the second coupler with the same phase relation it had after the first: the lower arm a quarter cycle ahead. Each output of the second coupler receives two contributions, one from each arm.

At out1, the cross output, the light from the upper arm crosses in the second coupler, and the light from the lower arm crossed in the first. Each contribution has crossed once and gained one quarter cycle, so they arrive in step and interfere constructively. At out0, the bar output, one contribution has crossed twice and gained half a cycle, the other has not crossed at all. They arrive half a cycle apart and cancel. All the light leaves at out1.

The heater changes this. At $\Delta\varphi = \pi$ , the light of the upper arm is delayed by half a cycle: the contributions at out1 now cancel and those at out0 add. All the light leaves at out0. The heater does not block light in either arm, and the light does not choose an arm: it travels in both arms at all times, and the phase difference between them decides at which output its two parts interfere constructively.

The transfer function

For every phase difference, the output powers of a balanced MZI with ideal 50:50 couplers are

P_\text{cross} = P_\text{in}\cos^2\frac{\Delta\varphi}{2}, \qquad P_\text{bar} = P_\text{in}\sin^2\frac{\Delta\varphi}{2}, \tag{1}

the cross output at out1 and the bar output at out0 for light entering in0 [3]. A relation of this kind, between what enters a device and what leaves it, is its transfer function. Since $\cos^2 + \sin^2 = 1$ , the two powers always add up to the input power: the MZI loses no light; it distributes it. Table 1 lists four points of Eq. (1).

Phase difference $\Delta\varphi$	Cross output (out1)	Bar output (out0)
$0$	100 %	0 %
$\pi/2$	50 %	50 %
$\pi$	0 %	100 %
$2\pi$	100 %	0 %

Table Output powers of an ideal, balanced MZI for light entering in0, from Eq. (1).

A phase difference of $\pi$ moves all the light from one output to the other, which makes the MZI a switch. Between $0$ and $\pi$ it divides the light in any ratio, which makes it a tunable splitter. With a heater of 100 µm, a phase difference of $\pi$ requires a temperature rise of roughly 40 K (Phase shifters).

Unequal arms

If one arm is longer than the other by $\Delta L$ , the arms add a phase difference of their own, $2\pi n_\text{eff}\,\Delta L/\lambda$ , which depends on the wavelength. As the wavelength changes, the output powers of Eq. (1) run through their cycle: the MZI transmits some wavelengths at one output and others at the other. Such an unbalanced MZI is a wavelength filter. The spacing between two neighbouring wavelengths of full transmission is the free spectral range,

\text{FSR} = \frac{\lambda^2}{n_\text{g}\, \Delta L}, \tag{2}

where $n_\text{g}$ is the waveguide’s group index, which accounts for the change of the effective index with wavelength [3]. For silicon strip waveguides, $n_\text{g} \approx 4.18$ [3], and an arm-length difference of 100 µm gives a free spectral range of 5.7 nm around 1550 nm. The lab’s two-colour demultiplexer uses this effect to send two wavelengths 1.6 nm apart to different outputs.

Why the MZI matters

An interferometer turns a phase difference into a difference in power, which a detector can measure [2]. The MZI does this on a chip, in a few hundred micrometres, with a phase that a heater or an electrical signal sets. A thermo-optic phase shifter makes it a switch; a phase shifter driven by the plasma dispersion effect makes it a fast modulator [3].

An MZI with a phase shifter at one input can set both the splitting ratio and the phase between its outputs, and meshes of such elements can apply any unitary matrix to the light in several waveguides [4] [5]. The article Circuits as matrices develops the description that makes this precise.

Key idea

A detector measures the power of light, not its phase. An interferometer converts a phase difference into a difference in power, and so makes it measurable and usable.

How do we model itOptional · the matrices behind Eq. (1)

With the coupler matrix of the article Splitters and combiners, an ideal 50:50 coupler and the arms with the heater are

C = \frac{1}{\sqrt 2}\begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix}, \qquad D = \begin{pmatrix} e^{i\Delta\varphi} & 0 \\ 0 & 1 \end{pmatrix}.

Light passes the first coupler, then the arms, then the second coupler, so the MZI’s matrix is the product with the first part on the right:

S = C\,D\,C = i\,e^{i\Delta\varphi/2} \begin{pmatrix} \sin\frac{\Delta\varphi}{2} & \cos\frac{\Delta\varphi}{2} \\[2pt] \cos\frac{\Delta\varphi}{2} & -\sin\frac{\Delta\varphi}{2} \end{pmatrix}. \tag{3}

For light entering in0, the output amplitudes are the first column: $|S_{10}|^2 = \cos^2(\Delta\varphi/2)$ at out1 and $|S_{00}|^2 = \sin^2(\Delta\varphi/2)$ at out0, which is Eq. (1). The common factor $i\,e^{i\Delta\varphi/2}$ does not change the powers, but it matters when the MZI is part of a larger interferometer. The lab computes the same product for its MZI switch.

For unequal arms, $D = \operatorname{diag}(e^{i\beta L_0}, e^{i\beta L_1})$ with $\beta = 2\pi n_\text{eff}/\lambda$ , and the phase difference $\beta \Delta L$ changes with $\lambda$ through both $\lambda$ itself and $n_\text{eff}(\lambda)$ . To first order, a change $\delta\lambda$ changes it by $2\pi n_\text{g} \Delta L\, \delta\lambda / \lambda^2$ ; setting this equal to $2\pi$ gives Eq. (2).

In short

An MZI splits light into two arms and recombines it; the phase difference $\Delta\varphi$ between the arms decides how the power divides between the two outputs.
For a balanced MZI with 50:50 couplers, the outputs receive $\cos^2(\Delta\varphi/2)$ and $\sin^2(\Delta\varphi/2)$ of the input power; a phase difference of $\pi$ switches the light completely.
With unequal arms, the phase difference depends on the wavelength, and the MZI becomes a filter with the free spectral range $\lambda^2/(n_\text{g}\,\Delta L)$ .

Next · FoundationsCircuits as matricesHow any circuit of waveguides, couplers and phase shifters is described by a matrix of complex numbers, and how connecting circuits multiplies their matrices.Read next Build it in the labMZI switchOpen the example, turn the heater and route the light from one output to the other.Open the lab

References

1Bogaerts, 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
2Saleh, B. E. A., Teich, M. C. (2019). Fundamentals of Photonics (3rd edition). Wiley. link
3Chrostowski, L., Hochberg, M. (2015). Silicon Photonics Design. Cambridge University Press. doi:10.1017/CBO9781316084168
4Reck, M., Zeilinger, A., Bernstein, H. J., Bertani, P. (1994). Experimental realization of any discrete unitary operator. Physical Review Letters 73, 58. doi:10.1103/PhysRevLett.73.58
5Clements, 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-001 lossless components by default, APX-004 wavelength-independent coupling ratio, APX-007 wavelength-independent phase shift.