Post on 03-Oct-2021
1
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: rjessiam@alcatel-lucent.com
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 KogelnikSeb Savory
OThL1.pdf
© 2009 OSA/OFC/NFOEC 2009
978-1-55752-865-0/09/$25.00 ©2009 IEEE
2
3 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
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
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11 IntroductionIntroduction
3
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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
100
Syst
em c
apac
ity
Year
Single channel
Gb/
sTb
/s
(ETDM)Mult
i-cha
nnel
Optical Amplifier�WDM
2010
6 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
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 capacity limit imposed by the Kerr fiber nonlinearity?
4
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22 Information TheoryInformation Theory
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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)
5
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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
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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
6
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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 lim
it
C / B � Capacity per unit bandwidth
“Error
-free”
achievab
le“Er
ror-fr
ee”
not ach
ievable
12 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
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)
7
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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!
SNRpol �Es
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 �
EBpol
N0
SNRpol
Rc log2(M)=
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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
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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
Real part of field (n.u.)
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
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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
9
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33 Constellations and ModulationConstellations and Modulation
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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
10
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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
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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:
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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
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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
12
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Real part of field (n.u.)
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 radius� Equal 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
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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
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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 )
Imag
par
t of
fie
ld (
mW
1/2
)
-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
1.21.4
1.61.8
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
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44 ““The Fiber ChannelThe Fiber Channel””
14
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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”
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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
15
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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
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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:
� Distributed optical amplification � Optical filtering from ROADMs� Various fiber types
• 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
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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 fields� In-band noise from distributed amplification� Out-of-band signals (other WDM channels)
� Out-of-band noise (in other WDM channels bands)
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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
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55 Fiber TransmissionFiber Transmission
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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
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Source of NoiseSources of Noise
• Shot noise at receiver
• Double Rayleigh Backscattering (DRB)
• Amplified spontaneous emission (ASE)
� Dependent on signal power� Is 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 lines� Fundamental limitation for noise and possibly fiber nonlinearity
Consider ASE here as the most fundamental source of noise!
Noise
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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
19
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Distributed 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
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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
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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
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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
21
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Propagation for Distributed Amplification
Generalized Nonlinear Schrodinger Equation (GNSE):
Additive white Gaussiann noise
FiberNonlinearities
: Electrical field
: Fiber dispersion
: Nonlinear coefficient
: Spontaneous emission factor
: Phonon occupancy factor
: Photon energy at signal wavelength
: Fiber loss coefficient
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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 if NL
All-passfilter
FiberNonlinearity
Fiberdispersion
E(0,t)
Succession of (infinitely) small step sizes
E(z,t)
FiberNonlinearities
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43 | 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 amplificatio
n
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
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66 Fiber Nonlinearity Fiber Nonlinearity CompensationCompensation
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45 | 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 noise
Accessible atTx or Rx RO
ADM
sN
ot available
WDM channel of interest
Out
-of-
band
In-b
and
Neighboring WDM channel
In-band noise
Neighboring WDM channel
Out-of-band noise
FiberNonlinearities
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Reverse Fiber Propagation of Signal
Equation of (forward) propagation:
Reverse propagation equation is obtained by:
z � - z
�2� - �2
or equivalently:
�� � �
• Perfect backward propagation can be achieved if the evolution ofall fields involved is known
• In optically-routed networks, the neighboring WDM fields data are not known!
0
FiberNonlinearities
24
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77 Capacity of the Fiber ChannelCapacity of the Fiber Channel
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Fiber Capacity Estimate
• Use ideal distributed amplification in transmission fiber(local gain = local loss)
• Low-loss optical fibers(0.2 dB/km)
• Multiple rings constellations(with equal radii and equal frequency of occupation)
• Ideal coding(as always when using 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
• 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:
25
49 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
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
Real part of field (mW1/2 )
No impairments
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Real part of field (mW1/2 )
Noise only
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Real part of field (mW1/2 )
Noise and fiber nonlinearity
�XPM
Average phase rotation due to
XPM
Spreading of points
* A low number of points is used here for clarity
50 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
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
Real part of field [ mW1/2]
Pave = -3 dBm
-2 -1 0 1 2
-2
-1
0
1
2
Real part of field [mW1/2]
Pave = 0 dBm
-0.5 0 0.5
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Real part of field [mW1/2]
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
26
51 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
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
52 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
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
27
53 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
88 Predictions based on Capacity Predictions based on Capacity Limit EstimatesLimit Estimates
54 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
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:
28
55 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
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
56 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
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
29
57 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
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.
58 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
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
30
59 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
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
60 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
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
31
61 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
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
62 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
99 Summary and OutlookSummary and Outlook
32
63 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
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!
64 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
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
33
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1010 Acronyms and ReferencesAcronyms and References
66 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
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
34
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Selected References (1/7)
Digital communication
Information Theory
68 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
Selected References (2/7)Noise
Coding
Coding for optical communications
35
69 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
Selected References (3/7)Modulation and constellations
Raman amplification
70 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
Selected References (4/7)Fiber-optic communication
36
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Selected References (5/7)Fiber-optic communication (con’t)
72 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
Selected References (6/7)Capacity calculations applied to optical communication
37
73 OFC Tutorial | March 2009 All Rights Reserved © Alcatel-Lucent 2009
Selected References (7/7)
Recent record in high spectral efficiency fiber transmission
Capacity calculations applied to optical communication (con’t)
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