3

Capacity LimitsCapacity Limitsofof

FiberFiber--Optic Communication SystemsOptic Communication Systems

René-Jean Essiambre1, Gerard Foschini1, Peter Winzer1 and Gerhard Kramer2

1 Bell Labs, Alcatel-Lucent, Holmdel, NJ, USA2 Bell Labs, Alcatel-Lucent, Murray Hill, NJ, USAEmail: [email protected]

Presentation at OFC in San Diego, California, USAOptical Fiber Communication (OFC) Conference (OFC), March 2009

2 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009

Acknowledgment

* Early part of this work was supported by DARPA under contract HR0011-06-C-0098

Jim Gordon

Andy Chraplyvy

Bob Tkach

Adel Saleh*Maurizio Magarini

Bert Basch

Torsten Freckmann

Stéphane Colas

Herwig Kogelnik

Seb Savory


Outline

1. Introduction2. Information Theory3. Constellations and Modulation4. “The Fiber Channel”5. Fiber Transmission6. Fiber Nonlinearity Compensation7. Capacity of the Fiber Channel8. Predictions based on Capacity Limit Estimates9. Summary and Outlook10. Acronyms and References


IntroductionIntroduction


Historical Evolution of Fiber-Optic Systems Capacity

What is the ultimate capacity that asingle optical fiber can carry?

Record Capacities

1986 1990 1994 1998 2002 200610

100

1

10

100Sy

stem

cap

acit

y

Year

Single channel

Gb/

sTb

/s

(ETDM)Mult

i-cha

nnel

Optical AmplifierWDM

2010


What are We Trying to Determine?

Transmission of information over fiber-optic networks

Capacity

How to determine a capacity?

Information theory

Impairment

Fiber Kerr nonlinearity

Additive Noise

Is there a fundamental limit to fiber capacityimposed by the Kerr nonlinearity?


What Technologies Are Considered?

Coding

Modulation

Constellations

Electronic signal processing

Optical amplification

• We are including an array of advanced technologies

Fiber loss

Fiber effective area

• We will also estimate the impact on capacity of


Information TheoryInformation Theory


The Birth of Information TheoryOne paper by C. E. Shannon in two separate issues

of the Bell System Technical Journal (1948)

Mathematical theory that calculates the asymptote of the rates that information can be transmitted at an arbitrarily

low error rate through an additive noise channel

Claude E. Shannon (1955)


Definition of the Channel

Example of a “channel”

A “channel” can be defined as that part of a communication system that we are unable or unwilling to change:

• The “waveform channel” is the part of the channel where the signal assumes a continuous (analog) form

• Shannon paid special attention to the additive white Gaussian noise (AWGN) memoryless channel

Informationsource

Sourceencoder

Channelencoder

Digitaldemodulator

Channeldecoder

Sourcedecoder

Outputsignal

Waveform channel

Digitalmodulator

Pulseshaper

Sampler


Additive White Gaussian Noise (AWGN)

A additive white Gaussian noise (AWGN) can be represented by the simple waveform channel

• Adding more noise results in capacity degradation • Shannon theory enabled calculating the asymptote of the

information rate, or capacity, of the AWGN channel• The waveform channel can be replaced by a discrete-time

channel when some conditions are fulfilled

+X Y = X + n

n

Signal field

Noise

Noisy signal field


Shannon’s Formula for Bandlimited Channels

C: Channel capacity (bits/s)

B: Channel bandwidth (Hz)

SNR: Signal-to-noise ratio Signal energy / noise energy (both are per symbol in the same bandwidth and in the same mode of transmission)

C = B log2 (1 + SNR)Shannon capacity:

Shannon formula assumesoptimum constellation and optimum coding!

SNR (dB)

spec

tral

eff

icie

ncy

(bit

s/s/

Hz)

-5 0 5 10 15 20 25 300

12

34

5

67

89

10

Shannon’s l

imit

C / B Capacity per unit bandwidth

“Error

-free”

achievab

le“Er

ror-fr

ee”

not ach

ievable


Definitions of SNR and OSNR

• Classical communication theory uses signal-to-noise ratio (SNR) and “SNR per bit” with signal and noise being of the same polarization and bandwidth

• Optical communication uses optical signal-to-noise ratio (OSNR) where the signal can be in one or two polarizations and the noise is summed over both polarizations. The noise is in a fixed bandwidth of ~12.5 GHz.

Classicalcommunication theory

(SNR)

Optical communication(OSNR)

Frequency

Spec

tral

pow

er d

ensi

ty

Signal

NoiseNoise

Signal contains one or twostate(s) of polarization

Noise contains twostates of polarization

Signal contains onemode

(one polarization state)

Noise contains one mode(one polarization state)


Relation Between OSNR, SNR, and SNR per bit ( EB/N0)

The relation between SNR and OSNR depends whether the signal is polarization-division multiplexed (PDM) or not!

SNRpolEs

pol

N0

Pspol

Rs NASE=

SNR:

SNR per bit:

OSNR Ps

2 Bref NASE

OSNR:

Non PDM p = 1

PDM p = 2

SNRpol =2 Bref

p RsOSNR =2 Bref

RB

N0

EBpol

Espol : Energy per symbol in one pol

EBpol : Energy per bit in one pol

N0 : AWGN spectral density (in one pol)

NASE : Spectral density of ASE per state of pol

Pspol : Signal power in one pol

Ps : Signal power (sum of both pols)Rs : Symbol rate RC : Code rate RB : Bit rate (sum of both pols) p : Number of states of pol occupied

by the signalBref : Reference bandwidth (12.5 GHz)M : Number of constellation symbols in

one polSNRpol : SNR in one polSNRB

pol : SNR per bit in one pol

For maximally compact modulation

SNRBpol EB

pol

N0

SNRpol

Rc log2(M)=


Optimum Constellation for the AWGN Channel

Optimum constellation for the AWGN channel (with coding):

• The optimum constellation for the AWGN channel is a bidimensional Gaussian

• This constellation arises in Shannon’s construction, establishing, for the first time, the very existence of optimum coding!

: Probability of having the symbol x

: Signal average power

: Imaginary part of symbol

: Real part of symbol


Optimum versus Discrete Point Constellations

From bidimensional Gaussian to discrete point constellations:

• Discrete point constellations can mimic the optimum bi-dimensional Gaussian probability distribution

• The occupation frequency of each constellation point can vary

Real part of field (n.u.)

Imag

inar

y pa

rt of

fiel

d (n

.u.)

Optimum constellation for AWGN Discrete points constellationapproaching the optimum

•-4 •-3 •-2 •-1 •0 •1 •2 •3 •4-4

-3

-2

-1

0

1

2

3

4


Imag

inar

y pa

rt of

fiel

d (n

.u.)

•-4 •-3 •-2 •-1 •0 •1 •2 •3 •4-4

-3

-2

-1

0

1

2

3

4


Capacity CalculationsChannel capacity (memoryless and constrained input):

For numerical simulations, the capacity formula above is discretized to produce a discrete memoryless channel (DMC) model

The conditional and joint PDFs are related by:

Definitions of the probability density functions (PDFs):

Channel model

PX(x1)

PX(xN)

PY(y1)

PY(yM)

PY|X(ym|xn)

: Probability of choosing x from the input alphabet X , i.e. input distribution: Probability of receiving y from the output alphabet Y , i.e. output distribution: Joint probability of simultaneously having an input x and of receiving y: Conditional probability of receiving y given an input x was sent, i.e. transition probabilities


Constellations and ModulationConstellations and Modulation


Examples of Constellations: 1-D1 bit/symbol 2 bits/symbol 3 bits/symbol 4 bits/symbol

In-p

hase 16 symbols

2-ASK or BPSK 4-ASK 8-ASK 16-ASK


Examples of Constellations: 2-D1 bit/symbol 2 bits/symbol 3 bits/symbol 4 bits/symbol

In-p

hase

Qua

drat

ure

+

QPSK 8-PSK

2-ASK/4-PSK

16-QAM

4-ASK/4-PSK

2-ASK/8-PSK


BER vs SNR per Bit for Various Modulation Formats (No Coding)

0 2 4 6 8 10 12 14 16 1810

10

10

10

10

10

10

10

10

-8

-7

-6

-5

-4

-3

-2

-1

0

SNR per bit (dB)

BER

16-QAM, 4-ASK64-QAM

BPSK, QPSK, 2-ASK, 4-QAM8PSK

16-QAM

64-QAM

BPSK, 2-ASK

8PSK

1 bit/symbol

2 bits/symbol

3 bits/symbol

6 bits/symbol

QPSK, 4-QAM, 4-ASK

4 bits/symbol

No matter how good the SNR per bit is,there is always a finite probability of error

Some BER curves:


Noisy Channel Coding Theorem (AWGN)

Adding redundant bits to information bitscan improve reliability of detection by detecting

sequences of symbols rather than individual symbols

Given a AWGN channel and a fixed signal power, one can transmit information at a rate R lower than the channel

capacity C with arbitrarily low error probability using coding

1 0 1 1 0 0 0 1 0 0 1 0

Uncoded dataInformation bits Information bits

Detection of bit sequences is no different than detection bit per bit

Uncorrelated sequences

1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1

Coded dataInformation bits Information bits

Redundantbits

Redundantbits

Detection of bit sequences can efficiently retrieve information bits

Correlated sequences


Capacity of Some Modulations (with Optimum Coding)

0 5 10 15 200

1

2

3

4

5

6

SNR (dB)

SE p

er s

ymbo

l (bi

ts/s

ymbo

l)

BPSKQPSK8-PSK16-PSK16-QAM64-QAM

Shan

non

• Each modulation reaches the maximum capacity determined by the logarithm of the number of constellation points in the format

• The optimum constellation for the AWGN channel is a bidimensional Gaussian (Shannon capacity)

Figure courtesy of Maurizio Magarini

BPSK

QPSK

8-PSK

16-PSK

16-QAM

64-QAM



Imag

inar

y pa

rt o

f fi

eld

(n.u

.)

•-4 •-3 •-2 •-1 •0 •1 •2 •3 •4-4

-3

-2

-1

0

1

2

3

4

Multiple Ring Constellations

From bi-dimensional Gaussian to multiple ring constellations:

Ring radii are integer multiples of the inner ring radiusEqual frequency of occupation on each ring

•-2 •-1.5 •-1 •-0.5 •0 •0.5 •1 •1.5 •2

•-2

•-1.5

•-1

•-0.5

•0

•0.5

•1

•1.5

•2

Real part of field (mW1/2)

Imag

par

t of

fie

ld (

mW

1/2 )

Optimum constellation for AWGN(bi-dimensional Gaussian)

Ring constellation used for capacity study(Ex.: 4 rings with 0 dBm average power)

• Ring constellations allow for simpler numerical fiber capacity estimate• Constraints we impose on our multiple ring constellations

Darker area means larger

density of symbols


Capacity of Ring Constellations (with Optimum Coding)

• At low SNR, fewer rings (amplitude levels) are necessary to approach Shannon’s limit

• At high SNR, only multiple rings can approach Shannon’s limit• Capacity continues to increase with SNR as more points can be put on

each ring (larger number of phase values allowed per ring)

1 ring2 rings4 rings8 rings16 rings

0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

7

8

SNR (dB)

SE p

er s

ymbo

l (bi

ts/s

ymbo

l)

Shan

non

PSK2-ASK/PSK4-ASK/PSK8-A

SK/P

SK

bits / symbol1 2 3 4 5

1 ring

Similarly for multiple rings


Modulation and Constellations for High Spectral Efficiency (SE)

• Compact modulation (Nyquist signaling “sin(x)/x” shaped pulses)• Ring constellation structure (amplitude shift keying, ASK)• Number of levels of phase-shift keying (PSK) determined by noise

and nonlinear transmission over optical fibers

-1 -0.5 0 0.5 1Frequency (units of symbol rate)

-40

-30

-20

-10

0

10

Opt

ical

spe

ctru

m(d

Bm/s

ymbo

l rat

e)

RS

Real part of field ( mW1/2 )Im

ag p

art

of f

ield

( m

W1/

2)

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

0

0.20.40.60.8

1

1.21.41.61.8

2

Symbol number

Fiel

d am

plit

ude

( m

W1/

2)

5 10 15 20 25 300

Nyquist pulse

e.g. Sinc sin(x)/x

-5 -4 -3 -2 -1 0 1 2 3 4 5

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (symbol period)

Ampl

itud

e (n

.u.) One pulse Adjacent

pulse

Sampling instant


““The Fiber ChannelThe Fiber Channel””


Difficulties to Define a “Fiber Channel”

• A single parameter can capture the channel: the SNR

Shannon considered the AWGN channel:

Fiber channel capacity depends on many parameters:

• System length• Optical fiber types present• Optical bandwidth allocated from a transmitter to a

receiver • Optical network topologies

• Etc ...

Different choices of system and fiber parameterslead to different “fiber capacities”


Overview of Literature on Capacity Limit of Optical Fibers

Fiber capacity estimates (that include fiber Kerr nonlinearities):

• Empirical approaches: [49]-[52]

• Approximate solutions assuming:

Fiber nonlinearity is low [48]-[54], [56], [57], [59] Fiber nonlinearity is considered as multiplicative noise [48], [49], [54] Average zero dispersion [55]

• Analysis limited to:

Specific nonlinear propagation effects [50] Confined to specific binary formats [58] Not maximally compact modulation [60]

These approaches do not generally explicitly take into account the impact of spectral confinement due to optical filtering in

optically routed networks


Optically-Routed Networks

In optically-routed networks (rings/mesh), neighboring WDM channels are not known but are transported over the same fiber!

RxRx

Tx

Tx

RxRx

(a) Point-to-Point

(b) Ring (c) MeshRx

Tx

RxRxTx

RxRx

Tx

ROADMl1l2

Tx

Rx


The “Fiber Channel”

Optical path ElectricalElectrical

DSP E/OData Data’DSPO/E

Tx Rx

fiber type 1 fiber type 2 ROADM

• The optical path incorporates:

Ideal distributed Raman amplification Square optical filtering from ROADMsStandard single-mode fiber (SSMF)

• Arbitrary complex electronic processing is allowed at either ends (transmitter and receiver) of the optical path

We define the “fiber channel” within a point-to-point connection in an optically-routed network

We do not consider the presence of optical regenerators


Optical Elements and Fields

In-band noise RxTx

In-band signal(WDM channel of interest)

Out-of-band signal

Out-of-band noise

The WDM channel of interest co-propagates with other fieldsIn-band noise from distributed amplificationOut-of-band signals (other WDM channels)

Out-of-band noise (in other WDM channels bands)


Optical Spectrum Layout in Wavelength-Division Multiplexing

FrequencyRS

WDM channel of interest

NoisePow

er

WDM frequency bandNeighboring

WDM channels

Neighboring WDM channels

Guardband

In-band Out-of-bandOut-of-band

B

• Channel spacing is limited by signal bandwidth

• The ‘in-band’ fields (signal and noise) travel from the transmitter to the receiver

• The ‘out-of-band’ fields (signal and noise) are generally not available to the transmitter or the receiver


Fiber TransmissionFiber Transmission


FiberNonlinearities

NoiseFilteringEffects

Three phenomena are at play simultaneously during propagation• Each physical effect influences the other• Some phenomena are deterministic while others are stochastic• Nonlinear transmission over fibers is not simply a transfer function!

Physical Phenomena at Play

•Chromatic Dispersion•Optical Filtering

•Amplified Spontaneous Emission•Double Rayleigh Scattering•Shot noise

• Intra-channel nonlinearities• Inter-channel nonlinearities


Source of NoiseSources of Noise

• Shot noise at receiver

• Double Rayleigh Backscattering (DRB)

• Amplified spontaneous emission (ASE)

Dependent on signal powerIs smaller than other sources of noise for sufficiently long fiber links

Proportional to signal power Can be minimized by inserting a sufficient number of isolators

Independent of signal power Dominates shot noise for sufficiently long (i.e. lossy) transmission linesFundamental limitation for noise and possibly fiber nonlinearity

Consider ASE here as the most fundamental source of noise!

Noise


Amplified Spontaneous Emission

• Amplified spontaneous emission (ASE) is quantum noise

• AWGN of Shannon is gaussian noise

• Can ASE be represented by AWGN?

• An answer by Jim Gordon in 1963 who says [14,15]:

AWGN approximates well ASE even at low signal levels!

“From this and from the Gaussian distribution of the output noise, it is clear that the amplification of the Gaussian noise input may be considered to have proceeded in a perfectly classical manner provided that we include the extra effective input photon to account for the response of the amplifier to the input zero-point fields. This result is valid for arbitrarily small input noise.”

Noise


Distributed Raman Amplification

Distributed amplification maximizes delivered OSNR!

Signal power evolution:Noise

Distributed amplification

…

0 LA 2 LA Namp LA3 LA (Namp-1) LA

Pin

Distance

Pow

er (

dB)

Discreteamplification

OSNR for distributed amplification:

( = L )

~ 58: Photon energy at signal wavelength: Reference bandwidth (~12.5 GHz): Fiber launch power (in dB): Phonon occupancy factor (~ 1.1 @ 1550 nm): Fiber loss coefficient: Number of amplifiers: Amplifier spacing: Path length


Discrete versus Distributed Amplification

• The OSNR at fixed nonlinear phase stays nearly constant for a wide range of amplifier spacings

• There is a fundamental advantage of distributed over discrete amplification at fixed nonlinearity for moderate and large amplifier spacings

Signal power is 0 dBm / channel

(c)

0 110 10 10

220

25

30

35

Amplifier spacing (km)

OSN

R at

con

stan

tno

nlin

ear

phas

e (d

B)

0

1

2

3

(b)

Non

linea

r ph

ase

(rad

ians

)O

SNR

(dB)

20

25

30

35

(a)

Discrete amplification

Distributed amplification

~ 9

dB

Fundamental advantage of

distributed over discrete

amplification for 100-km amplifier

spacing

Optical Signal-to-Noise Ratio(OSNR)

Noise

OSNR at fixed

Nonlinear phase: (see [44] for instance)

: Fiber nonlinear coefficient: Signal power evolution: System length


Fiber Dispersion

• Fiber dispersion can be considered as an all-pass filter• All-pass filters introduce memory in the channel• All-pass filters can be perfectly compensated without loss of

information (linear medium)

Effect of fiber dispersion:

In the absence of fiber nonlinearity,there is no loss of information associated to fiber dispersion

FilteringEffects

: Fiber dispersion: Distance of propagation


Optical Filtering at ROADMs

• Non-flat filters are not cascadable as the passband narrows• Square filters do not narrow the band when cascaded cascadable

Optical filter cascadability:

Square passband filters provides maximum cascadability

Non-flat passband filters Flat passband filters

No bandpassnarrowing

Bandpassnarrowing

FilteringEffects

H(f)

Concatenationbandwidth

OriginalBandwidth X 10

H(f)

X 10Concatenation

bandwidth

OriginalBandwidth


Propagation for Distributed Amplification

Generalized Nonlinear Schrodinger Equation (GNSE):

Additive white Gaussian noise

FiberNonlinearities

: Electrical field

: Fiber dispersion

: Nonlinear coefficient

: Spontaneous emission factor

: Phonon occupancy factor

: Photon energy at signal wavelength

: Fiber loss coefficient


Fiber Propagation Model

Waveform channel model for fiber propagation:

• Unlike for the AWGN channel, the fiber channel requires a large number of different operations to be performed in succession

• Unclear how to develop a general theory to evaluate the capacity of such a channel

Inputfield

Outputfield

+n

Noise

+e i NL

All-passfilter

FiberNonlinearity

Fiberdispersion

E(0,t)

Succession of (infinitely) small step sizes

E(z,t)

FiberNonlinearities

44 | OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009

Classification of Fiber Nonlinearities (Single Polarization)

INTER-CHANNEL

Signal-SignalSignal-Noise

XPM-inducedNPN

NPN WDM nonlinearities

XPM FWM

INTRA-CHANNEL

Signal-SignalSignal-Noise

SPMParametric

amplification

MI

NPN

SPM-induced NPN

Isolated-pulseSPM

IXPM IFWM

List of Acronyms

• NPN: nonlinear phase noise• WDM: Wavelength-division

multiplexing• XPM: Cross-phase modulation• SPM: Self-phase modulation• MI: Modulation instability• FWM: Four-wave mixing• IXPM: Intra-channel XPM• IFWM: Intra-channel FWM

Nonlinear interactions with strong memory

FiberNonlinearities


Fiber Nonlinearity Fiber Nonlinearity CompensationCompensation

46 | OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009

Nonlinear Transmission Compensation at Tx and Rx

Propagating fields can be classified by:• In-band and out-of-band fields• Available and non-available fields

Tx Rx

Neighboring WDM channel

Out-of-band noiseAccessible at

Tx or Rx ROAD

Ms

Not available

WDM channel of interest

Out

-of-

band

In-b

and


In-band noise


Out-of-band noise

FiberNonlinearities


Reverse Fiber Propagation of Signal

Equation of (forward) propagation:

Reverse (or backward) propagation equation is obtained by:

z - zor equivalently:

2 - 2

• Perfect backward propagation can be achieved if the evolution of all fields involved is known

• In optically-routed networks, the neighboring WDM fields data are not known!

0

FiberNonlinearities

• C. Paré et al. Opt. Lett., Vol. 21, pp. 459-461 (1996)• R-J Essiambre et al., ECOC Tu3.2.2 (2005), OFC OWB1 (2006), Phton. Technol. Lett., Vol. 18, pp.

1804-1806 (2006)• K. Roberts et al. Photon. Technol. Lett., Vol. 18, pp. 403-405 (2006)

For paper using backward propagation, see for instance:


Capacity of the Fiber ChannelCapacity of the Fiber Channel


Fiber Capacity Estimate

• Use ideal distributed Raman amplification(local gain = local loss)

• Moderate-loss optical fibers(0.2 dB/km)

• Multiple rings constellations(with equal radii and equal frequency of occupation)

• Ideal coding(use mutual information, i.e. Shannon)

• Full access to the entire in-band field at transmitter and receiver(coherent detection)

• Back-propagation of nonlinear fiber transmission at Tx and Rx (digital signal processing)

• Ideal virtually square response optical filters for optical routing(in reconfigurable optical add-drop multiplexers, ROADMS)

• Single-polarization of signal and noise

• Assume negligible polarization-mode dispersion (PMD)

Elements considered to obtain a fiber capacity estimate:


Procedure to Calculate a Capacity EstimatePictorial representation of a 4-ring constellation*:

• The various PDFs are obtained by fitting each cloud of the output distribution• The capacity estimate is calculated from the constrained memoryless channel

formulas given previously (information theory part)

(Average power is 0 dBm for each constellation)

Calculation of capacity estimate:

Constellation after back-rotation of each individual pointOriginal constellation

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Real part of field (mW1/2 )

Imag

par

t of

fie

ld ( m

W1/

2 )

Input

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2


No impairments

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2


Noise only

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2


Noise and fiber nonlinearity

XPM

Average phase rotation due to

XPM

Spreading of points

* A low number of points is used here for clarity


Evolution of Constellation with Signal PowerSymbol rate = 25 Gb/s, 50.0 GHz spacingSymbol rate = 25 Gbaud, 50 GHz channel spacing

Average phase

rotation due to XPM

-0.2 -0.1 0 0.1 0.2-0.2

-0.1

0

0.1

0.2

Imag

par

t of

fie

ld [

mW

1/2 ]

Pave = -21 dBm

-0.2 -0.1 0 0.1 0.2

-0.2

-0.1

0

0.1

0.2

Pave = -18 dBm

-0.2 0 0.2

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Pave = -15 dBm

XPM

-0.5 0 0.5

-0.5

0

0.5

Pave = -12 dBm

Pave = -9 dBm

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

Real part of field [mW1/2]

Pave = -6 dBm

-1 0 1-1.5

-1

-0.5

0

0.5

1

1.5


Pave = -3 dBm

-2 -1 0 1 2

-2

-1

0

1

2


Pave = 0 dBm

-0.5 0 0.5

-0.6

-0.4

-0.2

0

0.2

0.4

0.6


Imag

par

t of

fie

ld [

mW

1/2 ]

Evolution of the constellation with increasing power (at fixed noise level)• At low signal launch powers, the clouds are large because of the low SNR• At high signal launch powers, the clouds are large because of fiber

nonlinearity


Capacity Estimates Results including Fiber Nonlinearity

• With a sufficient number of rings, one can approach Shannon limit very closely• Because of fiber nonlinearity, the capacity reaches a maximum at some SNR• Maximum capacity can be increased by increasing the number of rings• Increasing the number of rings beyond 16 brings marginal capacity increase

Without fiber nonlinearity

0 5 10 15 20 25 30 350

1

2

3

4

5

6

7

8

SNR (dB)

Spec

tral

eff

icie

ncy

(bit

s/s/

Hz) With fiber

nonlinearity

Distance = 1000 km

Shan

non

limit1 ring

2 rings4 rings8 rings16 ringsShannon


Capacity Estimate Results versus Record Experiments

Post-deadline paper Th.3.E.2, NEC, AT&T and Corning at ECOC 2008 (PDM-RZ-8PSK, distance = 662 km)

Post-deadline paper Th.3.E.4, KDDI at ECOC 2008 (PDM-OFDM-16QAM, distance = 640 km)

Post-deadline paper Th.3.E.5, Alcatel-Lucent at ECOC 2008 (PDM-16QAM, distance = 315 km)

Distance = 500 km

SNR (dB)

Spec

tral

eff

icie

ncy

(bit

s/s/

Hz)

1 rings2 rings4 rings8 rings16 ringsShannon

0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

7

8

9

10

1 ring2 rings4 rings8 rings16 ringsShannon Sh

anno

n lim

it


Predictions based on Capacity Predictions based on Capacity Limit EstimatesLimit Estimates


A Prediction of Fiber Capacity Limits

SSMF fiber capacity limit?~560 Tbits/s-1300-1620 nm

(320 nm)

~140 Tbits/s~8 Tbits/sC+L bands(80 nm)

Capacity limit ?Capacity(2008)

Capacity of commercial systems for two amplification bands

80 m2Effective area

100 GbaudSymbol rate

2.6x10-20 m2/WNonlinear index

0.2 dB/kmFiber loss

SSMFFiber type

1000 kmOptical path length

Optical path parameters

~1 dB /yearHistorical rate of increase in SE

~2021Year to reach a SE of 14 bits/s/Hz

~0.8 bits/s/HzCommercial systems (2009)

~14 bits/s/HzSpectral efficiency (2 pol)

~7 bits/s/HzSpectral efficiency (1 pol)

Estimate of spectral efficiencies (SE)

Based on current fiber capacity estimates and historical rateof growth of spectral efficiency, one can extrapolate the

total fiber capacity as:


Estimate of Year to Reach Fiber Capacity Limits

One estimates the fiber capacityto reach its limits near 2025!

1

10

100

1000

10000

100000

1000000

1990 2000 2010 2020 2030

Year

Gb/

s

W DMResearch

W DMCommercial

Increase in number of

WDM channels

Increase in SE = 1 “dB” / year

WDM Research

WDM

Commercial

2021Capacity limit of C+L bands(140 Tbits/s)

2025Capacity limit in 1300-1620 nm

band(560 Tbits/s)

80 m2Effective area

100 GbaudSymbol rate

2.6x10-20 m2/WNonlinear index

0.2 dB/kmFiber loss

SSMFFiber type

1000 kmOptical path length

Optical PathParameters

Based on current fiber capacity estimates and historical data

Data from Bob Tkach


Physical Phenomena Impacting Capacity NOT Discussed Here

• Minimum loss coefficient that monomode fibers can have due to fundamental material and waveguide properties

(ultimate low-loss optical fibers)

Fundamental limit in fiber loss:

Fundamental limit in nonlinear coefficients:

• Monomode fibers with minimum nonlinear refractive index• Monomode fibers with maximum effective area

Other physical effects:

• Raman scattering• Brillouin scattering


Let’s Assume a Simple Scaling* for Nonlinear Transmission

Defining a nonlinear phase spectral density:

Using this simplified model, on can calculatecapacity scaling with fiber parameters

: Nonlinear phase spectral density

: Nonlinear phase

: Channel spacing

: System length

: Fiber nonlinear coefficient

: Signal power

• Let’s assume that is a good indicator of nonlinear transmission• From capacity limit estimate results: ~ 3.2 rad/THz/pol

* This scaling is intended to be used as a crude model for extrapolating capacity and should not be considered as an exact model.


Impact of Fiber Loss Coefficient on Spectral Efficiency

Lowering fiber loss increases spectral efficiency and is most valuable for long-haul transmission

PDM over SSMF with effective area of 80 m2

0 0.05 0.1 0.15 0.2 0.25 0.3

Fiber loss coefficient (dB/km)

Spec

tral

eff

icie

ncy

(bit

s/s/

Hz)

500 km1000 km2000 km4000 km8000 km

02468

101214161820222426


Impact of Fiber Effective Area on Spectral Efficiency

Increasing the fiber effective area improves spectral efficiency and is most valuable for long-haul transmission

PDM over SSMF with loss coefficient of 0.2 dB/km

40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

14

16

18

20

22

24

Fiber effective area ( m2)

Spec

tral

eff

icie

ncy

(bit

s/s/

Hz)

•

•

500 km1000 km2000 km4000 km8000 km


Relative Increase in SE with SNR Improvement for Shannon Limit

At high SE, large improvement in SNR produces only small relative gain in SE

Relative improvement in spectral efficiencyby improving the SNR

Spectral efficiency (bit/s/Hz)

Rela

tive

incr

ease

in S

E (%

)

Diminishing return on SNR improvement

SNR (dB)

SE (

bits

/s/H

z)

-5 0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

10

Shan

non’

s lim

it

0 5 10 150

20

40

60

80

100

120

140

160 •

3 dB6 dB9 dB12 dB15 dB

Improvementin SNR


Relative Increase in SE as a function of SNR for Shannon Limit

At high SE, large improvement in SNR produces only small relative gain in SE when operating at high SNR

Relative improvement in spectral efficiency (SE) as a function of SNR obtained by improving the SNR

Diminishing return on SNR improvement

SNR (dB)

SE (

bits

/s/H

z)

-5 0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

10

Shan

non’

s lim

it

0 5 10 15 20 25 30 35 400

20

40

60

80

100

120

140

160

SNR (dB)

Rela

tive

incr

ease

in S

E (%

) 3 dB6 dB9 dB12 dB15 dB

Improvementin SNR


Summary and OutlookSummary and Outlook


Summary

• Shannon’s information theory allows to determine an asymptote of the channel information rate for a signal impaired by additive white Gaussian noise

• Determining the limiting information rate in point-to-point fiber transmission in optically-routed network can allow to set limits on optical network capacity• Achieving capacity requires an array of advanced

technologies• Many important open issues remain to be addressed to

solve the problem of maximizing fiber capacity in optical networks!


Some Directions in Fiber Capacity Evaluation

• Constellation optimization for nonlinear transmission

• Including receiver with full memory

• Alternative modulations

• Full dispersion mapping exploration

• Comparison of different fiber configurations

• Advanced nonlinearity compensation schemes

• Dual polarizations


Acronyms and ReferencesAcronyms and References


List of Acronyms

ASE Amplified spontaneous emissionASK Amplitude shift keyingAWGN Additive white Gaussian noiseBER Bit error ratioDMC Discrete memoryless channelDRB Double Rayleigh BackscatteringDSP Digital signal processingE/O Electronic to optical conversionETDM Electronic time division multiplexingFEC Forward error controlFFE Feed-forward equalizerFWM Four-wave mixingGNSE Generalized Nonlinear Schrodinger Equation IFWM Intra-channel four-wave mixingIXPM Intra-channel cross-phase modulationMI Modulation instabilityNPN Nonlinear phase noiseO/E Optical to electronic conversion

OSNR Optical signal-to-noise ratioPCM Pulse-coded modulation ?PDF Probability density functionPDM Polarization-division multiplexingPPM Pulse-position modulation ?PSK Phase shift keyingQAM Quadrature amplitude modulationQPSK Quadrature phase shift keyingROADM Reconfigurable optical add/drop

multiplexerRx ReceiverSE Spectral EfficiencySPM Self-phase modulationSNR Signal-to-noise ratioSSMF Standard single-mode fiberTx TransmitterWDM Wavelength division multiplexingXPM Cross-phase modulation


Selected References (1/7)

Digital communication

Information Theory


Selected References (2/7)Noise

Coding

Coding for optical communications


Selected References (3/7)Modulation and constellations

Raman amplification


Selected References (4/7)Fiber-optic communication


Selected References (5/7)Fiber-optic communication (con’t)


Selected References (6/7)Capacity calculations applied to optical communication


Selected References (7/7)

Recent record in high spectral efficiency fiber transmission

Capacity calculations applied to optical communication (con’t)


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