Ch 01 — Optical Physics and Link Engineering

Optical Physics and Link Engineering

Every packet traversing a service-provider backbone ultimately rides a beam of infrared light through glass. This first chapter of the optical-transport series covers the photonic underlay that everything else in the series builds on: the physics that govern fibre transmission, the decibel mathematics essential for link engineering, and the formulas that turn span loss, link budget, OSNR, and Shannon capacity into routine design calculations.

Key Concepts at a Glance

Term	One-Line Definition
DWDM	Dense Wavelength Division Multiplexing — packs 96+ independent data channels onto a single fibre by assigning each a distinct infrared wavelength in the C/L-band, spaced as little as 50 GHz apart on the ITU grid [1].
OTN	Optical Transport Network (ITU-T G.709) — the digital framing layer that wraps client signals in ODU containers, adds FEC, and provides per-segment monitoring through up to six TCM levels [3].
EDFA	Erbium-Doped Fibre Amplifier — the long-haul workhorse; amplifies all C-band wavelengths simultaneously in the optical domain via stimulated emission in a short coil of Er³⁺-doped fibre pumped at 980 or 1480 nm [12, 24].
Coherent Optics	Modern transponders that recover amplitude and phase on both polarisations, enabling DP-QPSK through DP-16QAM, DSP-based dispersion compensation, and adaptive capacity-vs-reach optimisation [21, 28].
ROADM	Reconfigurable Optical Add-Drop Multiplexer — uses a Wavelength Selective Switch to remotely add, drop, or pass through individual wavelengths at a node.
OSNR	Optical Signal-to-Noise Ratio — the single most critical metric in optical link design; determines whether a given modulation format can be sustained over a given distance.

Signal Transmission Primer

Every chapter in this series — the dB arithmetic in Ch01, the modulation maths in Ch02, the EDFA noise budget in Ch03, the FEC discussion in Ch04 — assumes a small kit of communication-theory results: signals have a spectrum, modulation produces sidebands, bandwidth and SNR independently set the capacity ceiling, sampling at the Nyquist rate is the pre-condition for digitisation, and FEC closes part of the gap to Shannon. This primer states each result tersely with the equations and numerical anchors needed to read the rest of the series fluently.

Key Insight

The optical layer obeys exactly the same information-theory rules as any other communications channel. OSNR is just SNR measured inside a standardised 0.1 nm reference bandwidth, “baud” is the symbol rate, and the Shannon Capacity Limit later in this chapter is C = B · log₂(1 + SNR) with the noise pulled out of a 0.1 nm slice of spectrum.

Concept	What it says
Spectrum and Carrier	Every signal has two equivalent representations — amplitude vs time (oscilloscope view) and power vs frequency (spectrum-analyser view). The frequency-domain view is where bandwidth, channels, and carriers live. A carrier is the centre frequency around which a modulated signal’s spectrum is parked.
Fourier Decomposition	Any signal — periodic or not — decomposes into a sum (or integral) of sine waves. Periodic signals decompose into discrete harmonics at integer multiples of the fundamental; non-periodic signals (data) decompose into a continuous spectrum. This is why data channels occupy a continuous band rather than discrete spikes.
Bandwidth ≠ Carrier	Capacity depends on B (bandwidth) and SNR, not on the carrier frequency f_c. Two channels of equal width and equal SNR carry equal bit-rates whether their carriers are 5 THz apart or coincident. Higher-frequency bands help only because they tend to have more total spectrum available for allocation.
A Single Tone Carries Zero Bits	An unmodulated sine wave is a constant — fully predicted by amplitude, frequency, and phase. Information requires change. Modulation produces sidebands by the Fourier modulation theorem, and those sidebands are the bandwidth. AM, PSK, QAM, and FM all produce sidebands; only the geometry of the sideband shape differs.
Bandwidth Sets Symbol Rate	Bandwidth caps the symbol rate via Nyquist: R_s ≤ BW / (1 + α) where α is the pulse-shaping roll-off. Spectral occupancy of a digitally-modulated signal is BW ≈ R_s · (1 + α) regardless of constellation order — a 32 GBd QPSK and 32 GBd 16-QAM signal both occupy ~32 GHz.
SNR Sets Bits per Symbol	Constellation density (QPSK = 2 b/sym, 16-QAM = 4 b/sym, 64-QAM = 6 b/sym) is gated by post-FEC SNR. Each extra bit/symbol costs ~3 dB of SNR at high efficiency. Bit rate is R_s × k × n_pol × η_FEC — bandwidth pushes R_s, SNR pushes k, polarisation multiplexing doubles n_pol, FEC sets η_FEC.

Frequency and the Spectrum

A frequency is the rate at which a periodic phenomenon repeats, measured in Hertz (Hz, cycles per second). Light in C-band fibre oscillates at ~193.4 THz. The fundamental relationships are:

f = 1 / T                  (frequency = inverse of period)
λ = c / f                  (wavelength × frequency = speed of light)
λ(nm) = 299 792.458 / ν(THz)   (optical convenience form)

A pure sine wave is y(t) = A · sin(2π · f · t + φ) — three parameters (amplitude, frequency, phase) fully describe it. Once those are known, every future value is known, which is why a pure tone carries no information.

Two equivalent representations of the same signal:

TIME DOMAIN                          FREQUENCY DOMAIN

amplitude                            power
   │                                    │
   │  ╱╲ ╱╲╱╲ ╱╲   ╱╲╱╲                │      ╱╲
   │ ╱  ╲╱    ╲  ╲ ╱    ╲              │     ╱  ╲
   │╱            ╲╱      ╲             │    ╱    ╲
───┼───────────────────────→ time      │   ╱      ╲
   │                                   │  ╱        ╲
                                       └─┴──────────┴─→ freq
oscilloscope view:                       f_c
"what does the signal                  spectrum-analyser view:
 look like over time?"                 "what frequencies are present?"

An unmodulated carrier appears as a single spike on a spectrum analyser. A modulated carrier appears as a hump centred on f_c. The width of that hump is the bandwidth, and the bandwidth is what carries the information.

Fourier and Nyquist — The Physical Intuition

Before the formal mathematics, the rule max baud rate = 2 × bandwidth can be derived from a single physical observation about sine waves. This subsection builds that intuition end-to-end with no equations beyond elementary arithmetic; the formal Fourier and Nyquist treatments follow immediately afterwards.

A sine wave visits exactly two states per cycle

A sine wave is the simplest signal that changes. In one complete cycle it visits exactly two distinct states — a peak and a trough. Not three. Not one. The zero crossings are transitions, not states; the wave is moving through them, not resting at them.

One cycle of a sine wave:

         ● State 1 (peak)
        ╱ ╲
       ╱   ╲
      ╱     ╲
─────╱───────╲─────── zero line   (zero crossings = transitions, not states)
              ╲     ╱
               ╲   ╱
                ╲ ╱
                 ● State 2 (trough)

     ←── one cycle ──→     = 2 distinct states

This is non-negotiable. It is the reason the number 2 appears in every bandwidth formula.

Frequency converts cycles to states-per-second

If a sine wave at frequency f completes f cycles per second and each cycle delivers two states, then it produces 2f states per second:

Frequency	Cycles/sec	States/sec
1 Hz	1	2
10 Hz	10	20
16 GHz	16 × 10⁹	32 × 10⁹

A symbol is a distinct signal level the receiver measures. To carry 32 billion symbols per second (32 GBd), the signal must produce at least 32 billion distinguishable states per second. A 16 GHz sine wave produces exactly that — 2 per cycle × 16 billion cycles. Turn it around: 32 GBd needs at most 16 GHz.

Two samples per cycle is the hard ceiling

Two samples per cycle — signal captured:        One sample per cycle — signal lost:

The wave:    ╱╲    ╱╲    ╱╲                     The wave:    ╱╲    ╱╲    ╱╲
            ╱  ╲  ╱  ╲  ╱  ╲                                ╱  ╲  ╱  ╲  ╱  ╲

Samples:   ●   ●   ●   ●   ●   ●                Samples:   ●         ●         ●
           H   L   H   L   H   L                           0         0         0

Pattern unambiguous — the receiver                Every sample hits the same phase →
reconstructs the full wave. ✓                     looks like a flat line. ✗ aliasing.

Sample slower than 2× per cycle and frequencies above f_s / 2 fold back disguised as lower frequencies — irreversible. This is the physical fact Nyquist proved is universal.

The 32 GBd ↔ 16 GHz baseband identity

Random data contains every possible bit pattern. The hardest pattern is the one that switches every symbol: 1-0-1-0-… At 32 GBd one symbol is 31.25 ps, so one HIGH+LOW pair is 62.5 ps — that pattern is a square wave at 1/(62.5 ps) = 16 GHz. No data pattern can change faster, because the modulator switches at most once per symbol.

Slower patterns fill the band below 16 GHz:

Pattern	Cycle period	Frequency
1-0-1-0… (perfect alternation)	2 symbols	16 GHz — fastest possible
1-1-0-0-1-1-0-0…	4 symbols	8 GHz
1-1-1-0-0-0…	6 symbols	5.3 GHz
1-1-1-1-0-0-0-0…	8 symbols	4 GHz
Long runs of identical symbols	many	down to 0 Hz (DC)

Random data contains all of these simultaneously, so its baseband spectrum extends from DC to 16 GHz — and slower patterns are statistically far more common than faster ones, which is why the spectrum slopes downward (50 % of adjacent bit pairs are identical, only 0.4 % of any 8-bit window is perfect alternation).

What happens when you cut the bandwidth

16 GHz available → fastest pattern survives → 32 GBd works:

   ┌──┐  ┌──┐  ┌──┐    Sharp transitions, every symbol distinct.
   │  │  │  │  │  │    ✓ Receiver decodes cleanly.
───┘  └──┘  └──┘  └──


8 GHz available → 16 GHz components removed → max baud drops:

   ╱╲   ╱╲   ╱╲        Transitions rounded — symbols blur.
  ╱  ╲ ╱  ╲ ╱  ╲       Inter-symbol interference (ISI): one symbol's
─╱────╳────╲╱────╲─    tail leaks into the next.  ✗ 32 GBd fails.
      ↑ symbols overlap at the receiver

You cannot push 32 GBd through an 8 GHz channel. The fastest patterns require 16 GHz; if those frequencies are absent the patterns cease to exist on the wire.

The argument in plain English

1. To transmit 32 billion symbols per second, the signal must distinguish 32 billion states per second.
2. The hardest data pattern is alternation — switch every symbol.
3. That alternating pattern is a wave: 1 cycle = 2 symbols = 62.5 ps → 16 GHz.
4. No data pattern can change faster: 16 GHz is the fastest the modulator can ever produce.
5. Slower patterns fill below 16 GHz; random data covers DC → 16 GHz.
6. A channel that does not pass DC → 16 GHz cannot carry 32 GBd. The physics is one-way: max baud = 2 × bandwidth.

What Fourier and Nyquist actually proved

The intuition above is something an engineer could observe by experiment. Fourier (1807) [39] proved rigorously that any signal — not just square waves — decomposes into sine waves, so the “fastest pattern = highest frequency” logic holds for every modulation scheme, not just NRZ. Nyquist (1928) [32] proved the converse: a band-limited channel passing 0 → B Hz can carry at most 2B independent states per second, and no clever waveform can cheat this ceiling.

They arrived at the same relationship from opposite directions — Fourier by analysing signals, Nyquist by analysing channels — and together they elevated the rule of thumb to a fundamental law of information transmission. The remainder of this section formalises both results.

Two Nyquist Theorems — Signalling vs Sampling

The number 2 appears in two different DWDM relationships that look identical but solve different problems:

"32 GBd needs 16 GHz of bandwidth"        → baud   ≤ 2 × BW
"16 GHz signal needs ≥ 32 Gsamples/sec"   → f_s    ≥ 2 × f_max

Both are called Nyquist. Both descend from the same physical fact (a sine wave has 2 distinguishable values per cycle — a peak and a trough). But they are two distinct theorems sizing two different parts of the transponder + line system. Confusing them leads to wrong specs at design time.

Theorem 1 — Nyquist Signalling (1928) [32]

What it says. Maximum symbol rate ≤ 2 × channel bandwidth. A signal at R_s symbols/sec needs a channel of at least R_s / 2 Hz of baseband, or R_s Hz of passband on a carrier.

What it sizes. The optical channel — the ITU grid slot, ROADM passband, MUX/DEMUX filter width. This is the physics of the transmission medium.

What it answers. “How wide must the wavelength’s spectral slot be?”

Theorem 2 — Nyquist-Shannon Sampling (1928 / Shannon 1949) [32]

What it says. Minimum sampling rate ≥ 2 × highest signal frequency. Below that rate, frequencies above f_s / 2 fold back as aliases — the wrong waveform is reconstructed and the corruption is irreversible.

What it sizes. The transponder silicon — specifically the DAC at the transmitter (which generates the analog drive for the modulator) and the ADC at the receiver (which digitises the received electrical waveform for the DSP). This is the speed of the electronics.

What it answers. “How fast must the ADC and DAC clocks run?”

Where each theorem acts

           Sampling theorem        Signalling theorem        Sampling theorem
           sizes THIS              sizes THIS                sizes THIS
                ↓                       ↓                         ↓
          ┌──────────┐           ┌────────────┐             ┌──────────┐
Digital   │          │  Analog   │            │   Analog    │          │  Digital
symbols ─→│   DAC    │─→ voltage→│  OPTICAL   │─→ voltage  ─→│   ADC    │─→ symbols
from DSP  │ ≥ 2·fmax │  to mod   │  CHANNEL   │  from rcvr  │ ≥ 2·fmax │   to DSP
          └──────────┘           │ ≥ R_s/2 BW │             └──────────┘
                                 │ (fibre +   │
                                 │  ROADMs)   │
                                 └────────────┘

The two theorems operate independently. The DAC does not know the channel width — it only needs to generate the analog waveform fast enough to represent every cycle. The channel does not know the DAC speed — it only needs to be wide enough to pass the signal’s spectrum.

Numerical example — a 32 GBd transponder

Question	Theorem	Calculation	Answer
What ITU slot?	Signalling	Baseband 32/2 = 16 GHz, passband 32 GHz, +α≈0.15 → ~34 GHz occupied	50 GHz fixed-grid slot [1] (16 GHz guard)
What ADC/DAC rate?	Sampling	f_max = 16 GHz, so f_s ≥ 32 Gsa/s	64+ Gsa/s typical (2× oversampling)

Why oversampling is standard

The sampling theorem gives the bare minimum (2 × f_max). Production transponders run ADCs and DACs at 2–4× the baud rate because:

• Real anti-aliasing filters have finite roll-off; oversampling relaxes their cutoff requirements.
• Pulse shaping (root-raised-cosine, etc.) needs many samples per symbol to define the pulse shape.
• Coherent DSP blocks (CD compensation, MIMO equalisation, timing recovery, carrier-phase recovery — see Coherent DSP Internals) need oversampled data to converge.

Baud rate	Nyquist minimum	Typical production rate
32 GBd (100G class)	32 Gsa/s	64 Gsa/s
64 GBd (400G class)	64 Gsa/s	96–128 Gsa/s
96 GBd (800G class)	96 Gsa/s	128–160 Gsa/s

The takeaway

Both theorems are needed simultaneously and neither replaces the other. The signalling theorem governs network planning — it sets ITU grid choices and ROADM passband targets. The sampling theorem governs hardware procurement — it sets DAC/ADC silicon specs. They share the number 2 because they share a root in sine-wave geometry, but they answer different design questions for different stakeholders.

Fourier Decomposition — Why Bandwidth Exists

Fourier (1807) [39] proved that any signal — periodic or not — can be expressed as a sum (or integral) of sine waves at different frequencies, amplitudes, and phases:

f(t) = A₀ + Σₙ Aₙ · sin(2π · fₙ · t + φₙ)

This is exact, not an approximation. The decomposition is what bandwidth physically is — the set of sine-wave frequencies that the signal contains.

Periodic vs non-periodic — harmonics or continuous spectrum

The structure of the decomposition splits cleanly along periodicity:

Signal type	Decomposition	Spectrum
Periodic (repeats every T)	Discrete harmonics at integer multiples of f₀ = 1/T	Spikes at f₀, 2f₀, 3f₀, ...
Non-periodic (e.g. random data)	Continuous Fourier integral	Smooth spectrum across a band

This distinction is precise and operationally important. A periodic square wave has a Fourier series — a discrete set of harmonics. Random data has a Fourier transform — a continuous spectrum. Both produce bandwidth, but only the periodic case has “harmonics” in the strict sense.

Square wave — the canonical periodic-signal example

A square wave at fundamental f decomposes into odd-harmonic sines with 1/n amplitude:

square(t) = (4/π) · [ sin(2πft) + ⅓·sin(6πft) + ⅕·sin(10πft) + ⅐·sin(14πft) + ... ]

Adding harmonics progressively reconstructs the square:

1 harmonic only:                3 harmonics:                ∞ harmonics:

       ╱╲                          ╱──╲                     ┌──────┐
      ╱  ╲                        ╱    ╲                    │      │
   ──╱────╲──                  ──╱──────╲──              ───┘      └───
            ╲  ╱                          ╲    ╱
             ╲╱                            ╲──╱

A perfect square wave with infinitely sharp transitions requires infinite bandwidth. In practice, transmission systems retain enough harmonics to keep the eye open at the receiver and rely on pulse shaping or matched filtering to manage the truncation.

Rectangular pulse → sinc spectrum (the non-periodic case)

A single rectangular pulse of width T is non-periodic. Its Fourier transform is the sinc function:

Pulse, time domain:            Spectrum (Fourier transform):
                                                 power
        ┌────┐                                     │   ╱╲
        │    │                                     │  ╱  ╲       sin(πfT)
   ─────┘    └─────                                │ ╱    ╲     ────────
                                                   │╱      ╲╱╲    πfT
                                                   └──┬───┬──→ freq
                                                      0  1/T  2/T

   width T                                         first null at f = 1/T
                                                   main-lobe width = 2/T

The pulse-width × spectral-width relationship is inverse:

Pulse width T	Main-lobe width	Implication
Narrow (fast data)	Wide	More bandwidth needed
Wide (slow data)	Narrow	Less bandwidth needed

This is the formal mathematical reason “faster data needs more bandwidth”. A 32 GBd signal has symbol period T = 31.25 ps and a sinc main-lobe extending to 1/T = 32 GHz — directly setting the channel-width requirement that ITU flex-grid slot-widths track.

Bandwidth-limited pulses → ISI

If a channel passes only a fraction of the spectrum required by the pulse, the high-frequency content (sharp transitions) is removed and the time-domain pulse spreads:

Full bandwidth — sharp pulses:        Bandwidth-limited — energy spreads:

  ┌──┐  ┌──┐     ┌──┐                  ╱╲   ╱╲        ╱╲
  │  │  │  │     │  │                 ╱  ╲ ╱  ╲      ╱  ╲
──┘  └──┘  └─────┘  └───              ─╱────╳────╲────╱────╲──
                                            ↑
                                       inter-symbol interference (ISI):
                                       symbols overlap, decision boundaries blur

Modern coherent transponders sidestep this by transmitting root-raised-cosine (RRC) pulse-shaped waveforms designed to fit inside R_s · (1 + α) of bandwidth and to satisfy the Nyquist zero-ISI criterion at the ideal sample instants. The pulse is engineered to fit the band; the band is not engineered to pass arbitrary harmonics.

Note

The “harmonics” framing of bandwidth is most accurate for square-wave NRZ signalling on legacy direct-detect systems. For coherent QAM with pulse shaping, the relevant model is the continuous Fourier transform of the pulse-shaped baseband signal, which yields a finite, well-defined occupied bandwidth without invoking infinite harmonics. Both viewpoints land on the same engineering conclusion: occupied bandwidth scales with symbol rate.

Frequencies, the Spectrum, and Carriers

A carrier is the centre frequency of a modulated signal. When a vendor specifies “this transponder operates on ITU channel 42, 193.200 THz”, the signal’s spectrum is centred at 193.200 THz. The carrier itself does not carry information; the deviations of the modulated waveform around the carrier do.

Optical and RF systems use carriers because frequency-translation lets multiple independent signals share the same physical medium simultaneously, each parked at a different centre frequency. This is frequency-division multiplexing; DWDM applies the idea to fibre, with each lambda being one carrier:

DWDM C-band, 50 GHz spacing (sketch — 8 of 96 channels shown):

   power
     │
     │   ▄▄    ▄▄    ▄▄    ▄▄    ▄▄    ▄▄    ▄▄    ▄▄
     │  ████  ████  ████  ████  ████  ████  ████  ████
     │ ██████████████████████████████████████████████████
     │_│50│__│50│__│50│__│50│__│50│__│50│__│50│__________  frequency →
       GHz   GHz   GHz   GHz   GHz   GHz   GHz
        f₁    f₂    f₃    f₄    f₅    f₆    f₇    f₈

Each channel is a self-contained modulated signal centred on its own carrier; the guard bands between carriers keep them from spilling into each other and the mux/demux filter them apart at the receiver.

Three numbers fully describe what a channel is doing on a fibre or wire:

Quantity	Symbol	Optical example
Carrier frequency (where the channel sits)	f_c	193.1 THz (ITU C-band anchor)
Signal bandwidth (how wide the hump is)	B	~32 GHz for 32 GBd at α=0.0
Signal-to-noise ratio inside B	SNR	e.g. 18 dB at the receiver

Capacity (Shannon) depends on B and SNR. Multiplexing (which slot the channel sits in, what filter passes it, what frequency the optic tunes to) depends on f_c. The two are independent.

Optical bands as ranges of carrier frequencies

The “bands” later in this chapter — O, E, S, C, L, U — are named ranges of carrier frequency available on standard single-mode fibre. Each band is a spectrum allocation inside which many DWDM channels can sit. Numbers from ITU-T G.694.1 [1] / G.694.2 [2] are tabulated below in §Fibre Physics and Transmission Bands.

A statement like “this 100G signal is on C-band channel 42 at 193.200 THz” specifies the band (C), the carrier (193.200 THz), and — implicit in the modulation choice — the bandwidth (32 GBd ≈ 32 GHz).

Why a Carrier Is Needed

A baseband data stream cannot propagate as light unless it is translated to optical frequencies. Three reasons drive the use of a carrier:

1. The medium only carries light. Single-mode fibre transmits in the 1260–1675 nm window (~179–238 THz). Baseband I/Q from the DSP sits at 0 to ~R_s/2 (e.g. DC to ~16 GHz for 32 GBd). The carrier laser provides the optical frequency that the fibre actually propagates; modulation lifts the baseband signal onto that carrier.
2. Frequency-division multiplexing. Without distinct carriers, every channel would occupy the same baseband slot and overlap. With carriers spaced on the ITU grid, channels coexist on one fibre and are separated at the receive end by optical filtering (mux/demux, ROADM WSS).
3. Coherent detection. A coherent receiver mixes the inbound signal with a local oscillator (LO) laser at approximately the same frequency. The beat product at the photodiode strips out the optical carrier and recovers the baseband I/Q for downstream DSP. This requires a stable, well-defined carrier frequency at the transmitter to mix against.

The carrier is therefore an enabler — of physical propagation, of multiplexing, and of coherent recovery — but it carries no information itself. All information lives in the modulation, i.e. in the sidebands.

Signal vs Noise vs Bandwidth

SNR is the dimensionless ratio of signal power to noise power in the band of interest:

SNR_linear = P_signal / P_noise
SNR_dB     = 10 · log₁₀(P_signal / P_noise)

Bandwidth B is the spectral width over which the signal energy is distributed. Noise is the unwanted power present inside that same band — for the optical layer, dominated by amplified spontaneous emission (ASE) from EDFAs (see [[03-optical-amplifiers-edfa-raman-cl-band]]).

The three quantities couple through Shannon’s law C = B · log₂(1 + SNR), derived in §Shannon Capacity Limit below: capacity scales linearly with bandwidth and only logarithmically with SNR. They are independent levers, not a single “signal quality” axis — bandwidth and SNR can rise or fall independently, and a link’s capacity tracks both.

Note

SNR and OSNR are dimensionless. This is why the literature freely writes “OSNR margin = 4.5 dB” without milliwatt context — only the ratio matters, not the absolute power.

Why Bandwidth — Not Carrier Frequency — Sets the Data Rate

This is the most-confused point in modulation engineering and the one most worth stating precisely. The Shannon equation is:

C = B · log₂(1 + SNR_linear)

f_c does not appear. Capacity depends on bandwidth B and SNR; it does not depend on where in the spectrum the band is parked. A 32 GHz-wide signal at 193.1 THz and the same signal at 195.9 THz carry identical capacity if their SNRs are equal.

A worked comparison:

Scenario	Carrier f_c	Bandwidth B	SNR	Capacity C (Shannon)
(a) “10 GHz signal”	10 GHz	5 GHz	18 dB	≈ 30 Gbit/s
(b) “30 GHz signal” — same width	30 GHz	5 GHz	18 dB	≈ 30 Gbit/s
(c) “30 GHz signal” — wider	30 GHz	15 GHz	18 dB	≈ 90 Gbit/s
(d) C-band lambda	193.1 THz	32 GHz	18 dB	≈ 192 Gbit/s
(e) L-band lambda	187.5 THz	32 GHz	18 dB	≈ 192 Gbit/s

(a) vs (b): same bandwidth, different carrier — same capacity. (b) vs (c): same carrier, wider bandwidth — three-times capacity. (d) vs (e): very different carriers (>5 THz apart), same bandwidth — same capacity.

Warning

“Higher frequency = higher data rate” is true only when the higher-frequency band has more allocated bandwidth — not because the carrier f_c itself contributes capacity. A 30 GHz carrier is not faster than a 10 GHz carrier on its own merits; the comparison depends entirely on which slot widths are available at each.

Why a band rather than a single tone

1. A single tone is a constant; only changes carry information. Hartley (1928) [35]. Modulating amplitude, phase, or frequency creates sidebands; the sidebands carry the data.
2. Capacity scales linearly with B, only logarithmically with SNR. Doubling B doubles C. Doubling SNR (a +3 dB step) buys at most one extra bit/s/Hz at high SNR. Bandwidth is the cheapest lever.
3. FDM only works because each channel is finite-width. Lambdas are slots, not lines.
4. Filtering against noise requires finite signal bandwidth. A matched filter integrates energy across the signal band and rejects out-of-band noise. A single tone gives the receiver no spectrum to filter against.

Why higher-frequency bands often do allow more data — without breaking the rule

Higher-frequency regions of the electromagnetic spectrum tend to have more total bandwidth available for allocation. The C-band offers ~4.8 THz; combined C+L ~11 THz; proposed S+C+L expands further. A 30 GHz radio carrier in millimetre-wave can sit inside a 5 GHz allocation; a 1 GHz radio carrier in VHF might sit inside a 0.2 MHz allocation. The centre frequency is a label; the available bandwidth around it is the prize.

Baseband vs Passband and the Double Sideband

Modulation acts mathematically as a multiplication of the baseband signal by the carrier. A useful trigonometric identity drives the sideband geometry:

cos(A) · cos(B) = ½ · cos(A − B) + ½ · cos(A + B)

When a baseband component at frequency f_d multiplies a carrier at f_c, the result splits into two components — one at f_c − f_d (lower sideband) and one at f_c + f_d (upper sideband). The baseband spectrum is therefore mirrored around f_c:

Baseband (one-sided, DC to R_s/2):   Passband (two-sided, mirrored ±f_c):

power                                  power
  │█                                     │       ╱╲
  │██                                    │  ╱╲  ╱  ╲  ╱╲
  │███                                   │ ╱  ╲╱    ╲╱  ╲
  │████                                  │╱              ╲
  │█████                                 │
  └────────→ freq                        └─┴────┴────┴─→ freq
   0    R_s/2                              f_c-R_s/2  f_c  f_c+R_s/2

Width = R_s/2 (or R_s·(1+α)/2 with         Width = R_s   (or R_s·(1+α) with
       pulse shaping)                              pulse shaping)

A 32 GBd signal occupies ~16 GHz one-sided at baseband and ~32 GHz two-sided around the optical carrier. This factor-of-two is the “double sideband” — it is real spectrum, real photons at both sideband frequencies, not a math artifact.

A note on negative frequencies

The Fourier transform of a real signal produces a result symmetric about DC: F(−f) = F*(f). The negative-frequency content is mathematically present but carries no information not already in the positive side. In baseband one-sided plots, the negative side is suppressed because it contains no new physics. After modulation onto a carrier, the upper- and lower-sideband split is physical — both sidebands are present in the optical signal — and a passband spectrum analyser sees both.

Sampling and the Nyquist Rate

A continuous waveform is reconstructed losslessly from discrete samples if and only if it is sampled at f_s ≥ 2 · B, where B is the bandwidth of the signal (Nyquist 1928 [32]; Shannon 1949). Sample slower and frequencies above f_s / 2 fold back into the captured spectrum disguised as lower frequencies — aliasing — irreversibly.

graph LR
    A["Analogue<br/>optical signal<br/>Bandwidth B"] -->|"Sample at ≥ 2B<br/>(Nyquist)"| B["Discrete<br/>samples<br/>(numbers)"]
    B -->|"DSP<br/>decision"| C["Symbol<br/>(constellation point)"]
    C -->|"Demap<br/>k bits/symbol"| D["Coded<br/>bit stream"]
    D -->|"FEC<br/>decode"| E["User<br/>bits"]
    style A fill:#1D9E75,stroke:#0F6E56,color:#fff
    style B fill:#378ADD,stroke:#185FA5,color:#fff
    style C fill:#7F77DD,stroke:#534AB7,color:#fff
    style D fill:#BA7517,stroke:#854F0B,color:#fff
    style E fill:#D85A30,stroke:#993C1D,color:#fff

Warning

“Bandwidth” here is the bandwidth of the signal, not the channel. A 32 GBd signal occupies ~32 GHz of optical spectrum (more with roll-off), and the receive ADC must run at ≥ 64 GSa/s to capture it without aliasing. This is why every 100G+ coherent transponder integrates a multi-tens-of-GSa/s ADC.

The receiver coherently mixes the inbound 1550 nm light with a local-oscillator laser to bring the modulation down to electrical baseband; a high-speed ADC samples that baseband; from there the entire signal lives as numbers in a DSP. See [[02-modulation-coherent-and-dwdm-architecture]] for the front-end architecture and [[06-coherent-dsp-internals]] for the post-sampling DSP chain (chromatic-dispersion compensation, polarisation tracking, carrier-phase recovery, symbol detection).

Symbols, Bits, and Baud

A symbol is one waveform decision per signalling interval. The symbol rate (also baud, abbreviated GBd or Gbaud) is symbols per second. The bit rate is:

bit_rate = R_s · k · n_pol · η_FEC

  R_s     = symbol rate (baud)
  k       = bits per symbol = log₂(M) for an M-ary constellation
  n_pol   = 1 (single-pol) or 2 (PDM, dual polarisation)
  η_FEC   = net code rate (≈ 0.93 for 7 % FEC, ≈ 0.80 for 25 % SD-FEC)

The dominant rule for spectral occupancy is that bandwidth depends on R_s and pulse shaping, not on k:

BW_occupied ≈ R_s · (1 + α)        (digital QAM with RRC roll-off α)

A direct consequence: at a fixed symbol rate, swapping QPSK for 16-QAM does not change the channel width. It changes the bit rate (more bits per symbol) at the cost of higher required SNR (denser constellation, smaller decision regions).

Modulation	Baud R_s	k (bits/sym)	n_pol	Bit rate	Bandwidth	Required OSNR (post-FEC)
OOK	32 GBd	1	1	32 Gbit/s	~32 GHz	~12 dB
DP-QPSK	32 GBd	2	2	128 Gbit/s	~32 GHz	~12.5 dB
DP-8QAM	32 GBd	3	2	192 Gbit/s	~32 GHz	~16 dB
DP-16QAM	32 GBd	4	2	256 Gbit/s	~32 GHz	~19 dB (SD-FEC); ~16 dB with PCS [29]
DP-16QAM	64 GBd	4	2	512 Gbit/s	~64 GHz	~22 dB (3 dB symbol-rate penalty over 32 GBd row)
DP-64QAM	96 GBd	6	2	1152 Gbit/s	~100 GHz	~25 dB

Read across the first four rows: the bandwidth column is constant. Read down: bit rate climbs with constellation density, bandwidth climbs only when the symbol rate climbs. These are the two independent levers.

Worked numerical example — 100G to ~34 GHz

A complete chain from client rate to occupied bandwidth, in numbers:

Step 1 — Client signal:                 100 GbE = 100 Gbit/s user data
Step 2 — OTU4 wrapping (~12% overhead): 100 → 111.81 Gbit/s
Step 3 — FEC encoding (~15% overhead):  111.81 → ~128 Gbit/s line rate
Step 4 — DP-QPSK (4 b/sym, dual-pol):   128 / 4 = 32 GBd
                                        symbol period T = 1/32×10⁹ = 31.25 ps
Step 5 — Pulse shaping (RRC, α ≈ 0.15):
            BW_baseband = 16 × (1 + 0.15) = 18.4 GHz one-sided
            BW_passband = 2 × 18.4       = 36.8 GHz two-sided
            At −10 dB:                     ~34 GHz occupied
Step 6 — Fits in 50 GHz ITU slot:        50 − 34 = 16 GHz total guard band

Tip

A claim like “100G QPSK at 100 GBd” is wrong by inspection. With dual polarisation, QPSK carries 4 bits/symbol; 100 Gbit/s user data lands at ~32 GBd after FEC overhead. The full modulation-format coverage is in [[02-modulation-coherent-and-dwdm-architecture]].

Shannon’s Channel Capacity Law (primer)

Shannon (1948) [31] proved that every channel has a hard ceiling on the rate at which bits can be sent reliably:

C = B · log₂(1 + SNR_linear)

C in bit/s; B channel bandwidth in Hz; SNR_linear as a plain ratio (not dB). Doubling bandwidth roughly doubles capacity; doubling SNR (+3 dB) buys only one extra bit/s/Hz at high SNR — diminishing returns set in fast. The §Shannon Capacity Limit sub-section later in this chapter is the optical-domain instantiation of this same equation.

Worked example — required SNR for a target bit rate

Solve Shannon for the SNR needed to push a target rate through a given bandwidth:

SNR_required_dB = 10 · log₁₀ ( 2^(C/B) − 1 )

Target bit rate C	Bandwidth B	Spectral eff. C/B	2^(C/B) − 1	Required SNR (dB)
100 Gbit/s	32 GHz	3.125 bit/s/Hz	7.71	8.9 dB
200 Gbit/s	32 GHz	6.25 bit/s/Hz	75.1	18.8 dB
400 Gbit/s	64 GHz	6.25 bit/s/Hz	75.1	18.8 dB
400 Gbit/s	75 GHz	5.33 bit/s/Hz	39.3	15.9 dB
800 Gbit/s	96 GHz	8.33 bit/s/Hz	321	25.1 dB

These are theoretical minima — the ideal-coding floor. Production transponders carry a 2–6 dB implementation penalty above Shannon, which is why the OSNR thresholds quoted later in the chapter (DP-16QAM ~22 dB, DP-QPSK ~12 dB) sit above the Shannon numbers above. The gap is the implementation tax.

Note

Spectral efficiency C/B is the right axis for comparing modulation formats — same units on both sides. Halving B while holding C constant doubles C/B, and the required SNR climbs roughly 3 dB per extra bit/s/Hz at high efficiency.

SNR vs OSNR

Optical engineers do not quote raw SNR. They quote Optical Signal-to-Noise Ratio (OSNR), the same idea with one practical adjustment: noise is measured inside a standardised reference bandwidth — 0.1 nm, which at 1550 nm is approximately 12.48 GHz — rather than across the full electrical bandwidth of the receiver [33, 34].

The reference window is required because an OSA sees noise spread across the entire C-band; OSNR figures only compare across instruments and operators if everyone integrates noise across the same slice. IEC 61280-2-9 [33] defines the procedure; ITU-T G.697 [34] specifies how OSNR is reported on amplified DWDM links.

Conversion between RBWs is a one-line scaling:

OSNR(RBW₂) = OSNR(RBW₁) − 10 · log₁₀(RBW₂ / RBW₁)

Warning

Always check the reference bandwidth before comparing OSNR numbers. A vendor data sheet quoting “OSNR sensitivity 12 dB” at 0.5 nm is roughly 5 dB worse than the same figure at 0.1 nm — the larger window collected five-times more noise.

FEC in One Paragraph

A coded link operates below the SNR an uncoded receiver would require, because Forward Error Correction adds redundant bits at the transmitter that the receiver uses to fix errors. The dB difference between the uncoded SNR threshold and the coded threshold for the same target BER is the Net Coding Gain (NCG) — typically ~6 dB for ITU-T G.709 hard-decision FEC, 8–9 dB for super-FEC, and 11–12 dB for modern soft-decision (LDPC) FEC. NCG buys reach: every dB of NCG is roughly one more EDFA span at the same modulation. The cost is bandwidth — a 7 % FEC overhead means every 100 user-bits ride 107 line-bits, captured by η_FEC in the symbol-rate equation. The OTN/FEC discussion is in [[04-otn-sdh-and-network-design]]. Take-away: every OSNR threshold quoted later in this chapter is a post-FEC threshold; strip the FEC and the same modulation needs roughly 6–11 dB more OSNR to hit the same BER.

Key Insight

FEC does not reduce noise. It changes the rules of the bit-decision game so that the same noisy waveform produces fewer surviving bit errors after decoding. Shannon’s law still rules — FEC simply moves the operating point closer to the theoretical limit instead of leaving the link 10+ dB short of it.

Fibre Physics and Transmission Bands

Key Insight

Optical networking operates in the infrared spectrum (1260-1625 nm) where silica glass exhibits its lowest attenuation [7, 21]. The C-band (1530-1565 nm) dominates DWDM because it sits at the absolute minimum of the fibre loss curve and aligns with the EDFA gain window.

Silica glass fibre is not uniformly transparent across all wavelengths. The useful transmission window is divided into five bands, each with distinct loss characteristics and roles in modern networks.

Band	Wavelength Range	Attenuation	Role in DWDM Systems
O-band	1260-1360 nm	~0.35 dB/km	Short-reach and legacy systems
E-band	1360-1460 nm	Variable	Historically blocked by the water peak; G.652D low-water-peak fibre opens it for future use
S-band	1460-1530 nm	~0.25 dB/km	Raman pump wavelengths; expansion band for next-generation systems
C-band	1530-1565 nm	~0.20 dB/km	Primary DWDM band — lowest loss, EDFA gain window
L-band	1565-1625 nm	~0.22 dB/km	DWDM extension band adding ~96 channels beyond C-band

C-band and L-band together provide approximately 192 channels — the backbone of modern long-haul transport [1]. The “water peak” near 1383 nm in the E-band was an artefact of hydroxyl (OH-) ions trapped during older fibre manufacturing processes [7]. Modern G.652D low-water-peak fibre virtually eliminates this absorption, opening the E-band and S-band for future DWDM expansion as traffic demands grow beyond C+L capacity.

graph LR
    subgraph Fiber Optical Bands
        O["O-band<br/>1260-1360 nm<br/>0.35 dB/km"]
        E["E-band<br/>1360-1460 nm<br/>Water peak"]
        S["S-band<br/>1460-1530 nm<br/>~0.25 dB/km"]
        C["C-band<br/>1530-1565 nm<br/>0.20 dB/km"]
        L["L-band<br/>1565-1625 nm<br/>0.22 dB/km"]
    end
    O --> E --> S --> C --> L
    style C fill:#7F77DD,stroke:#534AB7,color:#fff
    style L fill:#D85A30,stroke:#993C1D,color:#fff
    style O fill:#378ADD,stroke:#185FA5,color:#fff
    style E fill:#888780,stroke:#5F5E5A,color:#fff
    style S fill:#BA7517,stroke:#854F0B,color:#fff

C-band and L-band (highlighted) are where DWDM operates — the lowest-attenuation window of silica fibre.

The continuous attenuation curve below makes the band table concrete and shows why the water peak shaped legacy band assignments:

xychart-beta
    title "SMF Fibre Attenuation vs Wavelength (G.652D)"
    x-axis "Wavelength (nm)" [1260, 1300, 1340, 1380, 1420, 1460, 1500, 1530, 1550, 1565, 1590, 1625]
    y-axis "Attenuation (dB/km)" 0.15 --> 0.45
    line "G.652D SMF" [0.35, 0.33, 0.31, 0.27, 0.26, 0.23, 0.21, 0.20, 0.19, 0.20, 0.21, 0.23]
    line "Legacy (water peak)" [0.35, 0.33, 0.34, 0.45, 0.30, 0.23, 0.21, 0.20, 0.19, 0.20, 0.21, 0.23]

The absolute minimum sits near 1550 nm at ~0.19 dB/km — the reason C-band dominates DWDM. At 1550 nm the 0.20 dB/km of modern G.652.D fibre decomposes as approximately 96 % Rayleigh scattering (intrinsic to the glass matrix, scaling as 1/λ⁴ and therefore dominant at short wavelengths) and approximately 4 % material absorption (residual OH⁻ ions plus the silica IR absorption tail, dominant beyond 1600 nm) [21, 23]. Modern manufacturing has reduced OH⁻ contamination to parts-per-billion, virtually eliminating the historical 1383 nm water peak. The L-band’s ~0.02 dB/km penalty over C-band is negligible for backbone designs — when evaluating fibre plant for C+L deployment, the real constraint is amplifier availability rather than fibre loss.

Tip

Every additional 0.05 dB/km of attenuation costs 5 dB of budget on a 100 km span — the difference between needing 12 amplifier sites versus 10 on a 1000 km route. Translate fibre loss directly into amplifier site count when comparing fibre vendors or routes.

Core Optical Terminology

Term	Definition
Lambda	A single wavelength of light. One lambda equals one data channel in DWDM.
Frequency (v)	Inversely related to wavelength: v = c / lambda, where c is approximately 3 x 10^8 m/s.
ITU Grid	The international channel plan (ITU-T G.694.1) [1]. Channels are spaced by frequency — fixed-grid spacings are 12.5 / 25 / 50 / 100 GHz; 75 GHz is supported only on the flex-grid (G.694.1 §7) [1]. ITU anchor: 193.1 THz (1552.524 nm); 1550.12 nm = 193.4 THz is one valid grid channel near band centre [1].
OSNR	Optical Signal-to-Noise Ratio. The single most critical metric in optical link design — it determines whether a given modulation format can be sustained over a given distance.
BER	Bit Error Ratio. The fraction of bits received incorrectly. Pre-FEC BER thresholds gate whether a link is viable.

Decibel Mathematics — dB vs dBm

Key Insight

dB is a dimensionless ratio between two power levels. dBm is an absolute power measurement referenced to 1 milliwatt. The distinction matters because every optical link budget mixes both: component losses and gains (dB) applied to actual signal power (dBm).

The Distinction

dB (decibel) expresses a ratio. It answers “how much larger or smaller is one power relative to another?” Fibre loss, splice loss, amplifier gain, noise figure, and OSNR are all ratios, so they are all expressed in dB.

dB = 10 x log10(P2 / P1)

dBm (decibel-milliwatt) expresses absolute power referenced to 1 mW. It answers “how much optical power is actually present?” Launch power, received power, and receiver sensitivity are real measurements, so they use dBm.

dBm = 10 x log10(P / 1 mW)

dBm-to-Milliwatt Conversion Reference

dBm	Milliwatts	Practical Context
+20	100 mW	High-power EDFA output
+10	10 mW	Strong signal
+3	2 mW	Double the 1 mW reference
0	1 mW	Reference point
-3	0.5 mW	Half the reference
-10	0.1 mW	1/10 of reference
-20	10 uW	Weak signal
-30	1 uW	Very weak — approaching receiver sensitivity

Conversion formulae: P(mW) = 10^(dBm/10) and dBm = 10 x log10(P in mW).

Why dB and dBm Arithmetic Works

In the linear domain, output power is the product of input power and gain: P_out = P_in x G. Taking 10 x log10 of both sides converts multiplication into addition:

dBm_out = dBm_in + dB_gain    (amplification)
dBm_out = dBm_in - dB_loss    (attenuation)

The result stays in dBm because the starting point was an absolute reference. This is the entire reason the optical industry uses logarithmic units — a multi-span link budget with dozens of loss and gain elements reduces to simple addition and subtraction.

The arithmetic also has a strict “invalid combination” — two absolute powers in dBm cannot be added directly:

dBm + dBm = INVALID

To combine two optical signals (e.g. wavelengths summing in a mux), convert to mW first, add, then convert back:

0 dBm + 0 dBm  =  1 mW + 1 mW  =  2 mW  =  10·log₁₀(2)  =  +3.01 dBm

Two equal-power signals therefore sum to +3 dB above either alone — not back to 0 dBm. This is why combining 96 channels at 0 dBm each in a C-band mux yields a total fibre power near +20 dBm, and why EDFA output saturation is specified as a total dBm figure rather than per-channel.

Warning

The most-common dB-arithmetic error in link-budget spreadsheets is summing per-channel dBm values directly. A 96-channel system at +0 dBm/channel carries +19.8 dBm of total optical power on the fibre, not 0 dBm. Link-budget sheets should track per-channel and total separately.

flowchart LR
    A["Launch power<br/>+1 dBm<br/>(1.26 mW)"] -->|"Fiber loss<br/>-24 dB<br/>(div 251)"| B["After fiber<br/>-23 dBm<br/>(0.005 mW)"]
    B -->|"EDFA gain<br/>+25 dB<br/>(x316)"| C["After amp<br/>+2 dBm<br/>(1.58 mW)"]
    C -->|"More fiber<br/>-12 dB<br/>(div 15.8)"| D["At receiver<br/>-10 dBm<br/>(0.1 mW)"]
    style A fill:#1D9E75,stroke:#0F6E56,color:#fff
    style B fill:#E24B4A,stroke:#A32D2D,color:#fff
    style C fill:#1D9E75,stroke:#0F6E56,color:#fff
    style D fill:#BA7517,stroke:#854F0B,color:#fff

In dB math: +1 - 24 + 25 - 12 = -10 dBm. In linear math: 1.26 mW / 251 x 316 / 15.8 = 0.1 mW. Same answer — dB is simpler.

Per-Component Loss and Gain Values

Component	dB Value	Linear Ratio	Physical Meaning
SMF loss/km (C-band)	0.20 dB	x0.955/km	95.5% of power passes each kilometre [7]
SMF loss/km (L-band)	0.22 dB	x0.951/km	Slightly higher loss than C-band
SMF loss/km (O-band)	0.35 dB	x0.923/km	Highest loss among the useful bands
Fusion splice	0.05 dB	x0.989	98.9% of power passes through
Connector pair	0.30 dB	x0.933	93.3% of power passes through
EDFA gain (typical)	20-30 dB	x100-1000	Amplifies signal 100- to 1000-fold [12, 13, 24]
EDFA noise figure	4.5-6 dB	x2.8-4.0	Degrades SNR by this factor
MUX/DEMUX loss	4-7 dB	x0.20-0.40	20-40% of power passes through
ROADM node loss	8-12 dB	x0.06-0.16	6-16% of power passes through
System margin	3-5 dB	x0.32-0.50	Reserve for ageing and degradation

Essential Optical Engineering Formulas

Every optical link design revolves around a small set of formulas. Each is presented below with its derivation context and a worked example.

Wavelength-Frequency Conversion

v (THz) = 299,792.458 / lambda (nm)
lambda (nm) = 299,792.458 / v (THz)

Parameter	Value	Notes
lambda	1548.52 nm	Transponder card reading
v	299,792.458 / 1548.52 = 193.600 THz	ITU channel C21 on the 100 GHz grid

Span Loss Calculation

Span_loss (dB) = (alpha x L) + (N_splices x splice_loss) + (N_connectors x conn_loss)

Variable	Example Value	Description
alpha	0.20 dB/km	G.652D fibre attenuation in C-band [7]
L	65 km	Span length
N_splices x splice_loss	3 x 0.05 = 0.15 dB	Fusion splice losses
N_connectors x conn_loss	2 x 0.30 = 0.60 dB	Connector pair losses
Total span loss	13.75 dB	Only 4.2% of original power arrives

Why Fibre Attenuation Is dB/km, Not dBm/km

Fibre attenuation is consistently quoted in dB/km — never dBm/km. The unit choice is not arbitrary: it reflects the underlying physics of how silica absorbs and scatters light.

The unit logic

Each kilometre of fibre absorbs a fixed percentage of whatever power enters it, not a fixed number of milliwatts. A percentage is a dimensionless ratio. Ratios are expressed in dB. Therefore the unit is dB (ratio) per km, not dBm (absolute) per km.

The same fibre, two different launch powers:

Launch power	Power into 1 km @ 0.2 dB/km	Power lost	Fraction lost
+10 dBm (10 mW)	9.55 mW	0.45 mW	4.50 %
−20 dBm (0.01 mW)	0.00955 mW	0.000 45 mW	4.50 %

Same fibre, same 0.2 dB/km, same 4.5 % loss per km — but the absolute power lost differs by a factor of 1000. The fibre does not absorb a fixed mW per km; it absorbs a fixed fraction of the photons present per km. This is why the dB ratio is the natural unit.

Why dBm/km would be physically nonsensical

A unit of “0.2 dBm/km” would imply every km removed a fixed 0.01 mW of optical power regardless of how much was launched:

Launch 10 mW   → after 1 km: 10 − 0.01 = 9.99 mW       (negligible loss)
Launch 0.01 mW → after 1 km: 0.01 − 0.01 = 0 mW        (signal destroyed)

A fibre that selectively destroys weak signals while barely affecting strong ones is not a passive medium — it would imply a non-linear absorption process inverse to incident power. Real silica behaves the opposite way: absorption is linear, photon-count-proportional, and scales with launched power.

How α is measured in practice

α (dB/km) = ( P_in(dBm) − P_out(dBm) ) / L (km)

dBm − dBm = dB. Absolute minus absolute gives a ratio. Divide by length and the unit is dB/km — the consistent ratio per unit length.

Note

The 1/λ⁴ Rayleigh scaling is what makes shorter wavelengths (O-band, 1310 nm) intrinsically lossier than C-band (1550 nm) — at half the wavelength, scattering is 16 times stronger. The C-band’s status as the DWDM workhorse is set by this physics, not by chance.

Link Budget (Power Budget)

P_received (dBm) = P_launch (dBm) - total_losses (dB) - margin (dB)

If the received power falls below the receiver sensitivity threshold, the link fails and amplification is required.

Unamplified example: 65 km span, launch power +2 dBm, receiver sensitivity -22 dBm, margin 4 dB.

Step	Value
Span loss	13.75 dB
P_received	+2 - 13.75 - 4 = -15.75 dBm (26.6 uW)
Sensitivity check	-15.75 > -22 — link works without amplification
Remaining headroom	6.25 dB beyond minimum sensitivity

Amplified example: 120 km span, launch +1 dBm, sensitivity -18 dBm, margin 3 dB. Span loss = 24.75 dB. Without amplification: received = -26.75 dBm — below threshold. With a mid-span EDFA at km 60 providing 25 dB gain: received = +1 - 12.3 + 25 - 12.45 - 3 = -1.75 dBm — passes comfortably.

The diagram below traces optical power node-by-node through a 3-span amplified link. Each EDFA restores power with a deliberate 0.2 dB over-compensation per span to account for connector ageing and splice degradation across the 15–20 year life of the plant:

flowchart LR
    A["+1 dBm<br/>1.26 mW<br/>Launch"] -->|"80 km fibre<br/>-16.8 dB"| B["-15.8 dBm<br/>26 µW"]
    B --> C["EDFA 1<br/>+17 dB"]
    C -->|"Output"| D["+1.2 dBm<br/>1.32 mW"]
    D -->|"80 km fibre<br/>-16.8 dB"| E["-15.6 dBm<br/>28 µW"]
    E --> F["EDFA 2<br/>+17 dB"]
    F -->|"Output"| G["+1.4 dBm<br/>1.38 mW"]
    G -->|"80 km fibre<br/>-16.8 dB"| H["-15.4 dBm<br/>29 µW"]
    H --> I["Pre-amp<br/>+17 dB"]
    I --> J["+1.6 dBm<br/>At receiver"]
    style A fill:#1D9E75,stroke:#0F6E56,color:#fff
    style B fill:#E24B4A,stroke:#A32D2D,color:#fff
    style C fill:#BA7517,stroke:#854F0B,color:#fff
    style D fill:#1D9E75,stroke:#0F6E56,color:#fff
    style E fill:#E24B4A,stroke:#A32D2D,color:#fff
    style F fill:#BA7517,stroke:#854F0B,color:#fff
    style G fill:#1D9E75,stroke:#0F6E56,color:#fff
    style H fill:#E24B4A,stroke:#A32D2D,color:#fff
    style I fill:#BA7517,stroke:#854F0B,color:#fff
    style J fill:#7F77DD,stroke:#534AB7,color:#fff

Point	dBm	mW	Event
Launch	+1.0	1.26	Transponder output
After span 1	-15.8	0.026	16.8 dB loss in 80 km fibre
After EDFA 1	+1.2	1.32	17.0 dB gain
After span 2	-15.6	0.028	16.8 dB loss
After EDFA 2	+1.4	1.38	17.0 dB gain
After span 3	-15.4	0.029	16.8 dB loss
After pre-amp	+1.6	1.45	17.0 dB gain, receiver-ready

Note

The 0.2 dB/span over-compensation is standard practice. It builds in margin for the inevitable increase in splice and connector losses as the fibre plant ages — typically 0.1-0.3 dB over a 15-20 year life cycle.

Where OSNR Degradation Actually Comes From

The OSNR formula derived below carries an underlying causality that is worth stating outright: passive fibre adds zero noise. All OSNR degradation in an amplified line system originates in the EDFAs, never in the fibre itself. The role of fibre attenuation is indirect — it forces amplification, and amplification adds ASE. This re-framing changes how a designer reasons about span-length tradeoffs and why every chapter in this series spends so much time on amplifier noise figure.

Numerical proof — passive fibre preserves OSNR exactly

Send a signal-and-noise pair through 80 km of fibre with no amplifier:

At launch:                      After 80 km @ 0.2 dB/km (16 dB loss):
  P_signal = 0 dBm               P_signal = 0 − 16 = −16 dBm
  P_noise  = −40 dBm             P_noise  = −40 − 16 = −56 dBm
  OSNR     = 0 − (−40) = 40 dB   OSNR     = −16 − (−56) = 40 dB   ← unchanged

Signal and noise both attenuate by 16 dB. The ratio is preserved. Fibre adds zero noise.

Where degradation enters — the amplifier

The signal at −16 dBm must be restored to a usable level. An EDFA with G = 16 dB and NF = 5.5 dB amplifies the signal back to 0 dBm — and amplifies the existing noise by the same 16 dB — but also injects fresh ASE noise that was not there before:

flowchart LR
    A["TX<br/>P_sig = 0 dBm<br/>P_noise ≈ 0<br/>OSNR ≈ ∞"] -->|"80 km fibre<br/>−16 dB on signal AND noise<br/>0 noise added"| B["After fibre<br/>P_sig = −16 dBm<br/>OSNR ≈ ∞"]
    B -->|"EDFA<br/>G = 16 dB, NF = 5.5 dB<br/>+16 dB on existing<br/>+ ASE injection"| C["After EDFA<br/>P_sig = 0 dBm<br/>OSNR = 36.5 dB"]
    style A fill:#1D9E75,stroke:#0F6E56,color:#fff
    style B fill:#BA7517,stroke:#854F0B,color:#fff
    style C fill:#E24B4A,stroke:#A32D2D,color:#fff

The OSNR went from effectively infinite to 36.5 dB. The fibre did not cause this drop. The amplifier did.

How span length couples into ASE

EDFA ASE power scales with the linear gain factor as P_ASE ∝ (G − 1) [42]. Doubling the span loss doubles the dB gain required, which raises (G − 1) by ~16× — far more than linear in span length:

Parameter	60 km span	120 km span
Span loss (@ 0.2 dB/km)	12 dB	24 dB
EDFA gain required	12 dB	24 dB
G_linear	15.85	251.2
(G − 1) (ASE driver)	14.85	250.2
ASE multiplier vs 60 km	1.00×	≈ 17×
OSNR after 1 span (P_ch = 0 dBm, NF = 5.5 dB)	40.5 dB	28.5 dB
Δ OSNR per span	—	−12 dB

A 12 dB increase in span loss costs 12 dB of OSNR per span — not a marginal penalty.

Same total distance, different span lengths

The ASE-vs-gain relationship inverts the usual intuition that “fewer amplifier sites is cheaper”. Cover the same 80 km with one long span vs two shorter ones:

Topology	Per-span loss	Per-span gain	(G − 1) per span	Total ASE	OSNR (P_ch = 0 dBm, NF = 5.5 dB)
1 × 80 km	16 dB	16 dB	38.8	38.8 units	36.5 dB
2 × 40 km	8 dB	8 dB	5.31	10.62 units	41.5 dB
Ratio	—	—	—	3.65× less	+5 dB

Same total fibre, same total loss, different number of amplifier sites — and the two-shorter-span topology produces 3.65× less ASE noise and 5 dB better OSNR. This is why some SP backbone designs deliberately deploy more amplifier sites at shorter span lengths: the amplifier-shelter CAPEX is recovered through better OSNR margin, which sustains higher modulation orders, which monetises into more capacity per fibre.

Design implications

The “fibre is noiseless, EDFAs add the noise” framing reorganises three classes of optimisation:

1. Raman amplification helps because it provides distributed gain inside the noiseless fibre itself, reducing the gain required from each lumped EDFA and therefore reducing the ASE that EDFA contributes. See [[03-optical-amplifiers-edfa-raman-cl-band]].
2. Lower-loss fibre helps disproportionately. G.654E ULL fibre at 0.17 dB/km vs G.652.D at 0.20 dB/km saves 2.4 dB of EDFA gain over an 80 km span — translating to roughly 40 % less ASE per span at the same launch power. The reach gain compounds across a long chain.
3. Reach is fundamentally amplifier-limited, not fibre-limited. A hypothetical noiseless amplifier would permit arbitrary reach on real fibre. The entire reach ceiling of optical networks is set by amplifier noise figure and the cumulative ASE that follows from forcing repeated amplification. See [[05-fibre-nonlinearities-and-power-optimisation]] for the second-order constraint — at high launch powers the ASE-vs-NLI tradeoff sets a launch-power optimum, beyond which the system worsens.

Key Insight

Span length couples into OSNR exponentially through (G − 1), not linearly. Two 40 km spans beat one 80 km span by ~5 dB of OSNR with the same total fibre — and that 5 dB is the difference between sustaining DP-16QAM and being forced down to DP-8QAM on long-haul routes.

OSNR — Definition and Engineering Approximation

The fundamental definition is a pure ratio of signal power to noise power, both measured within a standard 0.1 nm (12.5 GHz) reference bandwidth:

OSNR (dB) = 10 x log10(P_signal / P_noise)

For planning cascaded EDFA chains with identical spans, a shortcut approximation applies:

OSNR ~ 58 + P_ch - NF - 10 x log10(N_spans) - span_loss

Note

The constant “58” is not arbitrary — it derives from the physics of spontaneous emission noise. Each EDFA adds ASE noise proportional to P_ASE = 2 × n_sp × h × v × Δv × (G − 1), where h is Planck’s constant (6.626 × 10⁻³⁴ J·s) and v ≈ 193.4 THz [21, 24]. Evaluated in dBm, the term 10 × log₁₀(2 × h × v × Δv) equals approximately −58 dBm.

Worked example: 8-span amplified link, 80 km spans, channel power 0 dBm, noise figure 5.5 dB.

Parameter	Value
Span loss	(0.2 x 80) + 0.2 + 0.6 = 16.8 dB
OSNR	58 + 0 - 5.5 - 10 x log10(8) - 16.8 = 26.67 dB
Linear SNR	10^(26.67/10) = 464:1
DP-16QAM threshold	~22 dB
Margin	26.67 - 22 = 4.67 dB — Pass

Note

OSNR thresholds quoted in this chapter (e.g., DP-16QAM ~22 dB) are conservative SD-FEC planning numbers [21, 28]. Modern oFEC / OpenZR+ implementations require approximately 17–20 dB OSNR for DP-16QAM at 200 G — vendor data sheets remain the authoritative source for any specific transponder generation [18, 19].

Every doubling of span count costs exactly 3 dB of OSNR — a logarithmic penalty that explains why subsea systems with 60+ amplifier hops must use DP-QPSK while metro links of 2–4 spans can sustain DP-16QAM. The cascade and design lookup below operationalise the formula for quick feasibility checks:

flowchart LR
    TX["TX<br/>P = 0 dBm<br/>OSNR = ∞"] -->|"Span 1<br/>-17 dB"| A1["EDFA 1<br/>G = 17 dB<br/>adds ASE"]
    A1 -->|"Span 2<br/>-17 dB"| A2["EDFA 2<br/>G = 17 dB<br/>adds ASE"]
    A2 -->|"Span 3<br/>-17 dB"| A3["EDFA 3<br/>G = 17 dB<br/>adds ASE"]
    A3 -->|"..."| AN["EDFA N"]
    AN --> RX["RX<br/>P = 0 dBm<br/>OSNR = ?"]
    style TX fill:#1D9E75,stroke:#0F6E56,color:#fff
    style A1 fill:#BA7517,stroke:#854F0B,color:#fff
    style A2 fill:#BA7517,stroke:#854F0B,color:#fff
    style A3 fill:#BA7517,stroke:#854F0B,color:#fff
    style AN fill:#BA7517,stroke:#854F0B,color:#fff
    style RX fill:#D85A30,stroke:#993C1D,color:#fff

OSNR by span count, assuming NF = 5.5 dB, span loss = 17 dB, channel power = 0 dBm:

Spans	10·log₁₀(N)	OSNR (dB)	Linear Ratio	Supports
1	0.0	35.5	3548:1	Any format
4	6.0	29.5	891:1	DP-16QAM (needs ~22 dB)
8	9.0	26.5	447:1	DP-16QAM (with 4.5 dB margin)
12	10.8	24.7	295:1	DP-8QAM (needs ~18 dB)
16	12.0	23.5	224:1	DP-8QAM (with 5.5 dB margin)
24	13.8	21.7	148:1	DP-QPSK (needs ~12 dB)
40	16.0	19.5	89:1	DP-QPSK (with 7.5 dB margin)

Key Insight

Every doubling of span count costs exactly 3 dB of OSNR. This logarithmic scaling is why subsea systems with 60+ amplifier hops must use DP-QPSK while metro links of 2-4 spans can sustain DP-16QAM — the modulation ceiling falls predictably with hop count.

Tip

Use this lookup as the working design aid: count amplifier hops, read the OSNR, and verify the target modulation has at least 3 dB of margin before committing to a line-system design.

Shannon Capacity Limit

C = B x log2(1 + SNR_linear)
SNR_linear = 10^(OSNR_dB / 10)

A 400G channel occupying 75 GHz bandwidth at 25 dB OSNR reaches a theoretical Shannon limit of 623 Gbps [28, 30]. The actual 400G payload uses 64% of this limit — typical for production coherent systems, reflecting the gap between ideal and achievable coding.

System Capacity and Fibre Latency

Formula	Example	Result
Total capacity = N_channels x per_channel_rate	88 ch x 200G DP-QPSK	17.6 Tbps per fibre pair
Same fibre upgraded	88 ch x 400G DP-16QAM	35.2 Tbps — doubled, same infrastructure
One-way latency = distance x 5 us/km	Riyadh-Jeddah, 950 km	4.75 ms one-way, ~9.52 ms RTT

Note

Light in glass travels at roughly 200,000 km/s — two-thirds of vacuum speed due to the refractive index (~1.47). The 5 µs/km rule accounts for this and is accurate enough for all planning purposes.

See Also

• Modulation, Coherent Detection, and DWDM Architecture
• Optical Amplifiers — EDFA, Raman, and Wideband C+L
• OTN, SDH, and Network Design
• MPLS Traffic Engineering — RSVP-TE, SR-TE, and Fast Reroute
• Tier-1 SP Architecture & L3VPN
• Internet Infrastructure — IGW, IXP, CDN, DPI

References

Standards (ITU-T)

1. ITU-T G.694.1 — Spectral grids for WDM applications: DWDM frequency grid (10/2020). https://www.itu.int/rec/T-REC-G.694.1
2. ITU-T G.694.2 — Spectral grids for WDM applications: CWDM wavelength grid (12/2003). https://www.itu.int/rec/T-REC-G.694.2
3. ITU-T G.709/Y.1331 — Interfaces for the Optical Transport Network (06/2020). https://www.itu.int/rec/T-REC-G.709
4. ITU-T G.798 — Characteristics of optical transport network hierarchy equipment functional blocks (12/2017). https://www.itu.int/rec/T-REC-G.798
5. ITU-T G.872 — Architecture of optical transport networks (10/2017). https://www.itu.int/rec/T-REC-G.872
6. ITU-T G.873.1 — Optical Transport Network — Linear protection (03/2022). https://www.itu.int/rec/T-REC-G.873.1
7. ITU-T G.652 — Characteristics of a single-mode optical fibre and cable (11/2016). https://www.itu.int/rec/T-REC-G.652
8. ITU-T G.654 — Characteristics of cut-off shifted single-mode optical fibre and cable (03/2020). https://www.itu.int/rec/T-REC-G.654
9. ITU-T G.655 — Characteristics of non-zero dispersion-shifted single-mode optical fibre and cable (11/2009). https://www.itu.int/rec/T-REC-G.655
10. ITU-T G.657 — Characteristics of a bending-loss insensitive single-mode optical fibre and cable (11/2016). https://www.itu.int/rec/T-REC-G.657
11. ITU-T G.707/Y.1322 — Network node interface for the synchronous digital hierarchy (SDH) (01/2007). https://www.itu.int/rec/T-REC-G.707
12. ITU-T G.661 — Definitions and test methods for the relevant generic parameters of optical amplifier devices and subsystems (07/2018). https://www.itu.int/rec/T-REC-G.661
13. ITU-T G.662 — Generic characteristics of optical amplifier devices and subsystems. https://www.itu.int/rec/T-REC-G.662
14. ITU-T G.975 — Forward error correction for submarine systems (10/2000). https://www.itu.int/rec/T-REC-G.975
15. ITU-T G.975.1 — Forward error correction for high bit-rate DWDM submarine systems (02/2004). https://www.itu.int/rec/T-REC-G.975.1
16. ITU-T G.698.2 — Amplified multichannel dense wavelength division multiplexing applications with single channel optical interfaces. https://www.itu.int/rec/T-REC-G.698.2

Standards (IEEE / OIF / MSA)

17. IEEE 802.3 — Ethernet; specifically 802.3bs (400 GbE), 802.3ck, 802.3df (800 GbE). https://standards.ieee.org/standard/802_3-2022.html
18. OIF 400ZR Implementation Agreement (OIF-400ZR-01.0). https://www.oiforum.com/documents/
19. OpenZR+ MSA specification. https://www.openzrplus.org/
20. OIF Common Management Interface Specification (CMIS). https://www.oiforum.com/documents/

Books

21. G. P. Agrawal, Fiber-Optic Communication Systems, 5th ed., Wiley, 2021.
22. G. P. Agrawal, Nonlinear Fiber Optics, 6th ed., Academic Press, 2019.
23. I. P. Kaminow, T. Li, A. E. Willner (Eds.), Optical Fiber Telecommunications VI-A & VI-B, Academic Press, 2013.
24. P. C. Becker, N. A. Olsson, J. R. Simpson, Erbium-Doped Fiber Amplifiers — Fundamentals and Technology, Academic Press, 1999.
25. C. Headley & G. P. Agrawal (Eds.), Raman Amplification in Fiber Optical Communication Systems, Academic Press, 2005.
26. V. Alwayn, Optical Network Design and Implementation, Cisco Press, 2004.
27. R. Ramaswami, K. N. Sivarajan, G. H. Sasaki, Optical Networks: A Practical Perspective, 3rd ed., Morgan Kaufmann, 2009.

Papers

28. K. Roberts et al., “Beyond 100 Gb/s: Capacity, Flexibility, and Network Optimization,” J. Lightwave Technol. 35 (2017).
29. J. Cho et al., “Probabilistic Constellation Shaping for Optical Fiber Communications,” J. Lightwave Technol. 37, 1590 (2019).
30. A. D. Ellis et al., “Performance limits in optical communications due to fiber nonlinearity,” Adv. Opt. Photon. 9, 429 (2017).

Foundational Papers (signal theory)

31. C. E. Shannon — A Mathematical Theory of Communication, Bell System Technical Journal, vol. 27, pp. 379–423, 623–656, July & October 1948. (The capacity law C = B · log₂(1 + SNR).)
32. H. Nyquist — Certain Topics in Telegraph Transmission Theory, Transactions of the AIEE, vol. 47, pp. 617–644, April 1928. (The sampling-rate ≥ 2B principle.)
33. IEC 61280-2-9 — Fibre optic communication subsystem test procedures — Part 2-9: Digital systems — OSNR measurement for DWDM systems. https://webstore.iec.ch/publication/5854
34. ITU-T G.697 — Optical monitoring for dense wavelength division multiplexing systems (02/2024). https://www.itu.int/rec/T-REC-G.697
35. R. V. L. Hartley — Transmission of Information, Bell System Technical Journal, vol. 7, no. 3, pp. 535–563, July 1928. (Information rate is proportional to bandwidth; a single-frequency tone carries no information.)
36. J. Fourier — Théorie analytique de la chaleur, Firmin Didot, Paris, 1822 (work of 1807). (Decomposition of arbitrary functions into sine and cosine series — the mathematical foundation of spectral analysis.)
37. E. Desurvire — Erbium-Doped Fiber Amplifiers: Principles and Applications, Wiley-Interscience, 2nd edition, 2002. (Canonical reference for ASE noise and the (G − 1) scaling.)