The Mach–Zehnder Interferometer
Electronics has the transistor; photonics has the Mach–Zehnder interferometer
(MZI). It is the one device this entire course keeps reaching for: the modulator that puts bits
onto a laser beam in every datacenter link, the 2×2 switch inside optical routing fabrics, and —
as the next module will show — the elementary "gate" from which programmable photonic matrix
processors are tiled. Its trick is disarmingly simple. Light in a waveguide carries a
phase, but no detector can see phase directly — a photodiode measures only power. The MZI
converts phase into power. It splits light onto two paths, lets the paths accumulate different
phases, and recombines them so that the two copies
interfere: in step,
the light exits one port at full strength; out of step, it exits the other. A phase you cannot see
becomes a brightness you can.
Everything the device needs, the last module already built: a
directional
coupler to split and recombine, two waveguide arms, and enough
coherence that the
reunited copies still remember their common origin.
Anatomy: split, shift, recombine
On a silicon chip the whole structure is a few hundred micrometres of waveguide. Light enters one
port of a 50:50 coupler and emerges as two equal copies. The copies travel down separate arms, one
of which carries a phase shifter — any element that changes the optical path length, by a
controllable phase \varphi (how phase shifters actually work is the
next lesson's business). A second 50:50 coupler then folds the two copies back together onto two
output ports, conventionally called the cross port and the bar
port.
Two copies of one wave, reunited after different journeys — the on-chip descendant of
Young's double slit,
with the far-field fringes replaced by two tidy waveguide ports.
The transfer function
Follow the field with phasors. Write the input amplitude as 1. An ideal
2×2 coupler hands each output 1/\sqrt{2} of the amplitude, tagging the
crossed-over copy with a factor i (a quarter-turn of phase —
the coupler's fingerprint). One arm then multiplies its copy by
e^{i\varphi}. Recombining at the second coupler and collecting terms,
the powers at the two ports come out as:
- with identical ideal 50:50 couplers and arm phase difference
\varphi, the output powers are
P_{\text{cross}} = \cos^2\!\frac{\varphi}{2}, \qquad
P_{\text{bar}} = \sin^2\!\frac{\varphi}{2};
- at \varphi = 0 all power exits the cross port;
at \varphi = \pi all power exits the bar port;
- the two powers always sum to 1 — a lossless interferometer only redistributes
light between its ports, never destroys it;
- a phase difference can come from a path-length difference \Delta L
(\varphi = 2\pi\,n_{\text{eff}}\Delta L/\lambda) or from a phase
shifter — the interferometer cannot tell the two apart.
The curve below is the device's whole personality. Drag the slider to un-balance the couplers —
real fabrication never delivers exactly 50:50 — and watch what happens to the dark states:
With perfect couplers the nulls are total: the interferometer can switch a port fully dark. With
K \neq 0.5 the two interfering copies have unequal amplitudes, their
cancellation is incomplete, and the "dark" port bottoms out at
(1 - 2K)^2. Engineers quote this as the extinction
ratio, P_{\max}/P_{\min} in decibels; a good silicon MZI
reaches 20–30 dB, and chasing the last few dB of extinction is a running theme in modulator and
mesh design.
Worked example: reading the dial
Suppose the phase shifter is set to \varphi = 120^\circ. Then
P_{\text{cross}} = \cos^2 60^\circ = 0.25, \qquad
P_{\text{bar}} = \sin^2 60^\circ = 0.75 .
A quarter of the power exits one port, three quarters the other — the MZI is a
continuously adjustable power divider, not just an on/off switch. Three settings are
worth memorising as reflexes: \varphi = 0 (all cross),
\varphi = \pi (all bar), and
\varphi = \pi/2 (an even 50:50 split). And because
\varphi = 2\pi\,n_{\text{eff}}\Delta L/\lambda depends on wavelength,
a fixed built-in path imbalance turns the same device into a wavelength filter: colours for which
the arms differ by a whole number of wavelengths exit one port, colours half a wavelength out of
step exit the other. One geometry, three careers — switch, dimmer, filter.
You can run the phasor arithmetic yourself. The program below pushes a field through
coupler–phase–coupler as three 2×2 complex matrix multiplications — the exact calculation, not the
summary formula — and checks it against \cos^2(\varphi/2):
type C = { re: number; im: number }; // a complex number
const c = (re: number, im = 0): C => ({ re, im });
const add = (a: C, b: C): C => c(a.re + b.re, a.im + b.im);
const mul = (a: C, b: C): C => c(a.re * b.re - a.im * b.im, a.re * b.im + a.im * b.re);
type M = C[][]; // 2x2 complex matrix
const apply = (m: M, v: C[]): C[] => [
add(mul(m[0][0], v[0]), mul(m[0][1], v[1])),
add(mul(m[1][0], v[0]), mul(m[1][1], v[1])),
];
const s = 1 / Math.sqrt(2); // ideal 50:50 coupler:
const coupler: M = [ [c(s), c(0, s)], // straight-through amplitude s,
[c(0, s), c(s)] ]; // crossed amplitude i*s
function mziCrossPower(phiDeg: number): number {
const phi = (phiDeg * Math.PI) / 180;
const phase: M = [ [c(Math.cos(phi), Math.sin(phi)), c(0)], // e^{i phi} on the top arm
[c(0), c(1)] ];
let v: C[] = [c(1), c(0)]; // all light enters port 1
v = apply(coupler, v);
v = apply(phase, v);
v = apply(coupler, v);
return v[1].re * v[1].re + v[1].im * v[1].im; // |amplitude|^2 at the cross port
}
for (const deg of [0, 60, 90, 120, 180]) {
const exact = mziCrossPower(deg);
const formula = Math.cos((deg * Math.PI) / 360) ** 2;
console.log(deg + "° matrix: " + exact.toFixed(4) + " cos²(φ/2): " + formula.toFixed(4));
}
That the device is three matrix multiplications is not a curiosity — it is the pivot on which the
whole course turns. The next module will read the MZI as its matrix and start composing.
The layout was proposed independently in the early 1890s by Ludwig Zehnder and by Ludwig Mach —
son of Ernst Mach, of shock-wave and speed-of-sound fame. Their motivation had nothing to do with
computing: with the two arms widely separated, you could park a candle flame or a gas cell in one
arm and read tiny refractive-index changes as fringe shifts, which made the instrument a workhorse
of wind tunnels and flow physics for a century. Its more famous cousin, built by Albert
Michelson, folds the geometry in half with
mirrors so the light retraces its own path — the design that ruled out the luminiferous aether in
1887 and, scaled to four-kilometre arms, detected gravitational waves at LIGO in 2015. Integrated
photonics settled on the Mach–Zehnder rather than the Michelson form for a prosaic engineering
reason: it is a feed-forward device. Light goes in one end and out the other, no light is
reflected back toward the source, and the two clean output ports are exactly what you want when
the next stage is another chip component rather than a human squinting at fringes.
A destructive-interference null looks like disappearing light, and "light + light = darkness"
is regularly written up as if energy were being annihilated. It never is. Look again at the
transfer function: P_{\text{cross}} + P_{\text{bar}} = 1 at
every phase. When the cross port goes dark, the interference is simultaneously
constructive at the bar port — the coupler's i fingerprint
guarantees the two ports are always exactly out of step. Interference redistributes energy; it
cannot destroy it. The confusion usually comes from diagrams that draw only one output. Whenever a
single-output interferometer seems to "lose" light at its null, the missing power has gone
somewhere real — out the unlabelled second port, or back toward the source, as in a Michelson.
Keep this reflex handy: in Module 4 it becomes the reason a lossless photonic mesh is described by
a unitary
matrix — unitarity is energy conservation, written in linear algebra.
Where this goes next
The MZI turns phase into power; everything now hinges on how well you can set that phase.
The next
lesson opens up the phase shifter itself — heaters, junctions and Pockels materials,
and the speed–power–loss trade-offs that decide whether an MZI is a millisecond-scale trimmer or a
100-gigabit modulator. From there, a resonant cousin (the microring) and then Module 4, where
MZIs assemble into meshes that multiply matrices at the speed of light.