Encoding Information in Light
A voltage on a wire is one number: high or low, and that's your bit. A light wave is a far richer
object. Write down the field of a monochromatic beam,
\mathbf{E}(t) \;=\; A\,\cos\!\bigl(2\pi\nu t + \varphi\bigr)\,\hat{\mathbf{p}},
and count the knobs: an amplitude A, a phase
\varphi, a carrier frequency \nu (the
wavelength), and a polarisation direction \hat{\mathbf{p}} — the
Jones vector from the
optics prerequisite. Add a fifth knob the equation hides: which spatial mode of the fibre
or waveguide the light occupies. Five independent degrees of freedom, where copper has one.
The last
lesson celebrated light as a carrier; this one asks the engineer's follow-up: how do
you actually write bits onto it, and how many bits will each knob hold? The accounting
tool is
information theory: a symbol drawn from M distinguishable
states carries \log_2 M bits.
The five knobs
| Degree of freedom | What you vary | Typical use |
|---|
| Amplitude | A (power levels) |
On–off keying (OOK), PAM-4 in datacenter links |
| Phase | \varphi relative to a reference |
BPSK, QPSK, QAM in coherent long-haul links |
| Polarisation | which of two orthogonal Jones vectors |
Polarisation multiplexing — doubles every coherent link |
| Wavelength | which of ~80+ carrier colours |
WDM — parallel channels in one fibre |
| Spatial mode | which core / transverse mode |
Space-division multiplexing (research, multi-core fibre) |
The first two knobs are varied fast, symbol by symbol, and define the modulation
format. The last three are usually held fixed per channel and used to run many channels
in parallel: two polarisations × 80 wavelengths × (someday) several spatial modes. The
distinction matters for computing too: Module 5 will use wavelength as a data-parallelism
axis in photonic matrix engines, and phase — the hardest knob to see, since
no
detector reads phase directly — as the very substance of interference-based
computation.
Constellations: the amplitude–phase plane
Amplitude and phase together are one complex number,
A e^{i\varphi}, so every modulation format can be drawn as a set of
points in the complex plane — a constellation diagram. The horizontal axis
(I, in-phase) and vertical axis (Q,
quadrature) are the real and imaginary parts of the field. A format with
M constellation points carries \log_2 M bits
every symbol:
OOK — light on, light off — uses only amplitude, and a bare photodiode can read
it. QPSK keeps the amplitude constant and puts the information entirely in phase:
four phases, 45^\circ, 135^\circ, 225^\circ, 315^\circ, two bits per
symbol. 16-QAM uses both knobs at once for four bits per symbol. Long-haul
systems routinely run 16-QAM on both polarisations; short datacenter links stick to amplitude-only
PAM-4 because the receiver is vastly cheaper. Why the split? Phase is invisible to a photodiode —
reading a phase format requires coherent detection: interfering the signal with a local
laser so that phase becomes power, exactly the trick the Mach–Zehnder interferometer will make
mechanical in Module 3.
Counting bits: the link budget of knobs
Multiply the axes together and the numbers get impressive fast. The rate of one wavelength channel
is
R \;=\; \underbrace{R_s}_{\text{symbols/s}} \;\times\;
\underbrace{\log_2 M}_{\text{bits/symbol}} \;\times\;
\underbrace{2}_{\text{polarisations}},
and a fibre carries one such channel per wavelength. Worked example — a modern coherent
transponder: R_s = 64 Gbaud, 16-QAM
(\log_2 16 = 4 bits/symbol), two polarisations:
R = 64 \times 4 \times 2 = 512\ \mathrm{Gb/s}
\quad\text{per wavelength},
and 80 wavelengths make 41 Tb/s per fibre. Try the format zoo yourself:
interface Format { name: string; points: number; }
const formats: Format[] = [
{ name: "OOK", points: 2 },
{ name: "QPSK", points: 4 },
{ name: "16-QAM", points: 16 },
{ name: "64-QAM", points: 64 },
];
const baud = 64e9; // symbols per second
const pols = 2; // polarisation multiplexing
const wavelengths = 80; // WDM channels
for (const f of formats) {
const bitsPerSymbol = Math.log2(f.points);
const perLambda = baud * bitsPerSymbol * pols; // b/s on one wavelength
const perFibre = perLambda * wavelengths;
console.log(
f.name.padEnd(7) + bitsPerSymbol + " bits/symbol " +
(perLambda / 1e9) + " Gb/s per λ " +
(perFibre / 1e12).toFixed(1) + " Tb/s per fibre");
}
If 16-QAM gives 4 bits per symbol and 64-QAM gives 6, why not 4096-QAM and be done with it? Because
the constellation points must remain distinguishable at the receiver, and noise blurs each
received point into a fuzzy cloud. Packing more points into the same transmit power squeezes them
closer together — go from M to 4M points (two
extra bits) at fixed mean power and the spacing roughly halves, so you need about
6\ \mathrm{dB} more signal-to-noise ratio, roughly
3 dB per additional bit. That is Shannon's bargain in miniature: each extra bit
per symbol costs a doubling of signal power (or a halving of noise, or a shorter link).
High-order QAM is therefore a short-reach, high-SNR luxury; transoceanic cables run modest formats,
and — a warning shot for Module 5 — analog optical computing faces exactly the same
arithmetic: every extra bit of precision you demand from an analog optical signal doubles the
required SNR. Remember this vignette when we price photonic neural networks.
Phase is meaningless without a reference — so where does the receiver get one? It runs its own
laser, the local oscillator, and interferes it with the incoming signal; the beat between
them turns phase differences into measurable photocurrents. But two free-running lasers drift
apart, the fibre adds its own slowly wandering phase, and the two polarisations get scrambled into
each other en route (in Jones-calculus language: the fibre applies an unknown, drifting
2×2 unitary). The modern solution, which arrived around 2008 and rebuilt the entire long-haul
industry, is to sample the interfered fields with fast ADCs and let digital signal
processing undo everything in software — estimate the phase drift, invert the polarisation
unitary, even cancel the fibre's dispersion. Coherent optics is thus a genuinely hybrid technology:
exquisitely analog physics up front, a small supercomputer of DSP behind the photodiodes. Keep that
pattern in mind — an analog optical heart wrapped in electronic correction — because it is the
architecture of nearly every photonic computer in this course too.
Where this goes next
Five knobs, each a channel; constellations to pack bits densely; parallel axes to multiply
throughput. All of it exploits light's gift for carrying structure without disturbance.
The next
lesson turns to the particles themselves — photon versus electron — and finds the
deep physical reason the very properties that make light a five-lane highway make it nearly
impossible to build a light-controlled light switch. Later, in Module 8, today's formats return in
earnest inside real
transceivers.