What is the way to increase the information capacity limit?

In addition to implementing the various modulation formats, impairment mitigation techniques, and FEC technologies to approach the limit, it is also desirable to increase the maximum information spectral density by taking optimum values for the parameters in (6-8), including critical fibre characteristics (loss, dispersion and non-linear coefficient), the channel spacing, the effective amplifier noise figure and finally the number of cascaded links.

How long does the Shannon limit apply to transoceanic systems?

For the longest transoceanic systems (>10,000 km and based on direct detection [60,70,71]), whilst the achieved ISDs are usually modest, below 1 b/s/Hz, the results closely approach the Shannon limit.

What is the theoretical limit for the information capacity of a single source?

The capabilities of optically multiplexed OFDM [39], or coherent WDM [27,37] to generate phase coherent high capacity signals from a single source, suggests one way to extend the theoretical ISD limit.

How long has the growth of communication capacity been observed?

Communication capacity has shown a remarkable exponential growth over more than 30 years, with the overall capacity of the core of the network closely tracking the user demand.

What is the effect of XPM on the information capacity of a WDM system?

Note that this equation is applicable to OFDM or coherent WDM techniques, and in a conventional WDM system, the capacity is reduced by a factor of B/∆ν, where ∆ν is the channel spacing in the frequency domain.

What is the maximum performance of a number of measured solid core fibres?

Figure 14 illustrates the predicted maximum performance for a number of measured solid core fibres, all of which demonstrate a maximum information capacity between 6 and 8 b/s/Hz, assuming optimum dispersion management and full compensation of intra-channel non-linearity.

What is the difference between the theoretical limits in figure 11 and the experimental data?

Experimental data is also compared to the theoretical limits in figure 11, but, this time, as a function of transmission distance for direct detection (upper) and coherent detection (lower).

(Open Access) Approaching the Non-Linear Shannon Limit (2010) | Andrew D. Ellis

Q: What are the contributions in this paper?

The authors review the recent progress of information theory in optical communications, and describe the current experimental results and associated advances in various individual technologies which increase the information capacity. The authors confirm the widely held belief that the reported capacities are approaching the fundamental limits imposed by signal-to-noise ratio and the distributed non-linearity of conventional optical fibres, resulting in the reduction in the growth rate of communication capacity. The authors also discuss the techniques which are promising to increase and/or approach the information capacity limit.

Q: What is the optimum case for evaluating the performance limit?

In this paper, the authors consider only coherent detection with signal independent noise, which is the optimum case appropriate to evaluate the performance limit.

JLT-11648-2009.R1

Abstract— We review the recent progress of information

theory in optical communications, and describe the current

experimental results and associated advances in various

individual technologies which increase the information capacity.

We confirm the widely held belief that the reported capacities are

approaching the fundamental limits imposed by signal-to-noise

ratio and the distributed non-linearity of conventional optical

fibres, resulting in the reduction in the growth rate of

communication capacity. We also discuss the techniques which

are promising to increase and/or approach the information

capacity limit.

Index Terms—Information rates, Modulation coding, Non-

linear optics, Wavelength division multiplexing.

I. I

NTRODUCTION

capacity of optical communication links has grown

exponentially since their introduction in the late 1970s,

and each generation has enabled new methods of

communication and services ranging from simple text e-mails

through to the ubiquitous video applications in use today. This

continuing growth has been enabled by many individual

technological advances, including third-window distributed-

feedback lasers, erbium-doped fibre amplifiers, wavelength

division multiplexing (WDM), dispersion management,

forward error correction and Raman amplification. Throughout

this technological evolution, one constant factor has been the

use of single mode optical fibre, although with evolving

designs to control chromatic dispersion and non-linearity.

However, there is now a growing realisation that the

continuing bandwidth demand will shortly push the required

capacity close to the maximum capacity which has been

predicted theoretically for such fibres. The economic and other

consequences of demand exceeding capacity are a matter of

much debate. However, it is generally acknowledged that the

current network architectures and transmission technologies

will not be capable of meeting the customer bandwidth

demand in the medium term. The time at which demand

exceeds supply may be delayed by changes in network

architecture and service pricing, but is likely to occur within

Manuscript received May 18, 2009. This material is based upon work

supported by Science Foundation Ireland under Grant 06/IN/I969, Enterprise

Ireland under grant number CFTD/08/333 and the European Commission

project PHASORS (FP7-ICT-2007-2 22457).

The authors are with the Tyndall National Institute and Department of

Physics, University College Cork, Ireland (phone: +353-21-490-4858; fax:

+353-21-490-4880; e-mail: andrew.ellis@tyndalll.ie).

the next decade.

In this paper, we will review the limits to the information

capacity in optical fibre communications, and investigate the

modern technologies currently being researched to approach

this limit. We will also outline the fundamental issues required

to increase the limits of current optical networks imposed by

signal-to-noise ratio and fibre non-linearities. The paper is

organised as follows. In section II, we briefly review historical

trends in communication bandwidth provision, before

considering in section III the classical theoretical predictions

of the limits to communication capacity of linear transmission

channels. In section IV, the more recent development of the

use of information theory to predict the capacity of non-linear

channels in optical fibre communications is reviewed. Then in

section V we describe recently proposed technologies

including the multi-carrier transmission techniques of

orthogonal frequency division multiplexing (OFDM) and

coherent WDM, and discuss the potential of these technologies

to allow significant increases in the maximum capacity of an

installed optical fibre network.

II. B

ACKGROUND

1980 1990 2000 2010

100

10k

100k

10M

100M

Core Bit Rate/Net Access Rate

Year

Net Access Rate (bit/s)

Fig. 1. Evolution of telecommunication network capacities in response to

changing access technologies and consumer applications. Circles: Bandwidth

of available access network connection. Candlesticks: Ratio of maximum

capacity of managed link in core network to access rate. Solid line: Growth

trend for net access rate, Dashed Lines: Approximate bounds of ratio of core

to access capacity

The evolution of demand in telecommunications networks may

be traced by plotting the bandwidth available to the user (net

access rate, circles) against the date of introduction of various

access technologies, as shown in figure 1, starting from the

introduction of the 1.2kb/s modem for use in Bulletin Board

Systems in 1978 [1] through to Passive Optical Networks at

contended bit rates up to 10 Gb/s [2] for video and gaming

Approaching the Non-Linear Shannon Limit

Andrew D. Ellis, Jian Zhao, Member, IEEE, David Cotter

JLT-11648-2009.R1

applications. The solid line shows a steady growth rate of 15%

per annum in net access rate. To date overall network

capacities, such as transatlantic link capacities, have increased

at a faster rate [3] due to increasing numbers of users with

access to advanced communications services.

Accompanying the increase in available access rates has been

a steady increase in the bandwidth within the core of the

network. The candlestick symbols in figure 1 illustrate as a

function of time the ratio of two parameters: the numerator is

the maximum capacity in the core network which may be

independently configured (for example, a T1 or T2 carrier in

1978, and a 40 Gb/s SONET wavelength or 100 GbE

wavelength in 2012); and the denominator is the available

access rate. Despite the exponential growth in bandwidth

demand, this ratio has remained remarkably constant,

representing the continual design trade off that is made

between, on one hand, complexity (favouring coarse

bandwidth granularity in the core network), and on the other

hand, reliability (favouring fine granularity). Over a period of

three decades since the late 1970s to the present day, these

ratios have consistently fallen within a band of values 500-

5,000 (horizontal dashed lines, figure 1). Despite profound

developments in the underlying technologies over this period,

it would appear that the basic cost-driven design trade-offs

have remained unchanged. Therefore, extrapolating these

bands of values into the future suggests that the network

should be able to support 100 Gb/s transport in the core

network today [4] and 1 Tb/s transport as early as 2017.

However, to maintain the current core network architecture,

this would require a total number of wavelengths deployed

similar to today, typically 160, but carrying information at an

information spectral density exceeding 30 b/s/Hz (which

represents an immense technical challenge).

1983 1988 1993 1998 2003 2008 2013

Bit Rate Distance Product

(Tbit/s.km)

Year Reported

Fig. 2. Evolution of maximum reported transmission capacity for single

wavelength (diamonds, open symbols for optical time division multiplexing),

wavelength division multiplexing (triangles), single and multi-banded OFDM

(filled circles) and coherent detection (open circles).

On the other hand, figure 2 illustrates the evolution of the

available fibre transmission capacity reported from

experiments carried out in research laboratories worldwide.

Whilst a long term growth trend of around 60% per annum

was observed from the early 1990s, this has saturated recently,

prompting a move towards the adoption of coherent detection

techniques, where the additional degree of freedom (optical

phase) is expected to allow for greater capacity increases [5].

However, even with such innovations, capacity increases of

about 25 times the current capacity, as required around 2017,

appear to be very challenging.

III. S

HANNON LIMIT

Real Field Component

æ æ

a) b)

Imaginary

Filed Component

Real Field Component

Fig. 3. a) Ideal transmitted constellation (continuous) and b) discrete point

approximation (after [8]).

The Shannon limit to information capacity on a

communications link [6,7] is well known. Shannon proved that

reliable communication over a discrete memory-less channel is

possible if the communication rate R satisfies R < C, where C

is the channel capacity, and is given by













ave

1log

(1)

where P

ave

is the average signal power and equals C⋅E

, where

is the average energy per bit, N

the noise spectral density

and B the channel bandwidth. The proof presented in [6]

assumes arbitrary line and error correction coding. For a linear

channel degraded by additive white Gaussian noise, the

optimum constellation in phase and quadrature components of

the optical field may be calculated. The optimum field takes

arbitrary continuous values, with the probability of each value

following a Gaussian distribution [5]. Such a continuous bi-

Gaussian distribution may be emulated in practice by a

discrete-point constellation [8, 9], as shown in figure 3. In a

well designed discrete point constellation the density of points

reduces with distance from the centre of the constellation, in a

manner approaching the optimum distribution. This

approximation may be improved further by varying the

probability of occupancy of each point in the constellation.

Many different constellations may be considered for optical

transmission (as shown for example in figure 4), ranging from

single quadrature formats typically generated with a single

amplitude modulator, including (a) binary phase shift keying

(BPSK), (b) amplitude shift keying (ASK) and (c) quaternary

ASK (4-ASK), to formats consisting of in-phase and

quadrature components, including (d and e) M-ary phase shift

keying (QPSK and 8PSK respectively), (g and h) quadrature

amplitude shift keying (QAM) constellations (typically

JLT-11648-2009.R1

generated using a dual parallel Mach Zehnder modulator), and

(f) hybrid amplitude phase shift keying (APSK) (typically

generated using an amplitude modulator and a phase

modulator in series).

Real Field Component

Imaginary

Filed Component

Fig. 4. Some examples of signal constellations with one (a,b), two (c,d) and

three (e-h) bits per symbol.

To calculate the performance of each constellation, we first

determine the impact of noise on each constellation point. For

a system using coherent detection, the noise and signal are

combined as a vector addition and the noise is independent of

the signal amplitude [10,11]. On the other hand, for direct- and

differentially- detected signals, the noise level after detection

is dependent on the signal intensity [12]. In this paper, we

consider only coherent detection with signal independent

noise, which is the optimum case appropriate to evaluate the

performance limit. In the following, we calculate the bit error

rate (BER) performance of a given constellation assuming

hard decision detection, by calculating the probability that a

given transmitted bit crosses an imaginary boundary (the

decision threshold) between it and its nearest neighbour [13].

We use the constellation of figure 4c as an example.

Real Field Component

Detection

Probability

Decision Threshold

Imaginary

Field Component

Fig. 5. An example of BER calculation. a) Constellation diagram for 4-ASK,

b) Probability of detecting a given field amplitude given that symbol 2 was

transmitted.

In this example, also shown in figure 5, additive white

Gaussian noise gives a Gaussian probability density function

for the received signal values (figure 5b). For the second bit

), erroneous detection occurs by crossing either of two

decision thresholds towards its nearest neighbours (e

and e

The probability of an error for this bit is thus:













−













−

22 N

(2)

where e

is the field amplitude of the i

constellation point and

Q is related to the complimentary error function by

( )













erfc

1 x

(3)

Note that here, Q represents a mathematical function, and

should not be confused with the “Q-factor” used in optical

communications. Note that the constellation points located at

the two ends (furthest from the centre in the general case) have

fewer nearest neighbours and therefore smaller error

probabilities. For example, a transmitted e

would only be

erroneously detected if it crossed the threshold between itself

and e

The total BER is then given by the sum of the BER for each

bit

multiplied by the probability P

that this bit is

transmitted, that is

∑

⋅=

PBER

(4)

The transmitted signal power is related to the geometric

distribution of the constellation points, such that the mean

energy per bit of the transmitted bit E

is:

∑

⋅=

iib

ePE

(5)

which in turn gives an electrical signal-to-noise ratio of E

Here, the signal-to-noise ratio is the parameter commonly used

in communication theory, and in an optical system limited by

amplified spontaneous emission noise (ASE), represents the

photon number per bit entering the optical pre-amplifier. For a

coherently detected signal, the electrical signal-to-noise ratio

(snr) is related to optical signal-to-noise ratio (OSNR) by

OSNR=R⋅snr/(2B

ref

), where R and B

ref

are the transmission

capacity and reference noise bandwidth (e.g. 12.5 GHz,

corresponding to 0.1nm at 1550nm wavelength) respectively,

and the ASE noise is assumed to be randomly polarised. BER

performance results for a few common modulation formats are

shown in table I [11,14], where m represents the number of

constellation points (log

(m) bits per symbol) assuming

=1/m.

The performance of each format may then be compared to the

Shannon capacity limit by calculating the required snr for a

given BER and error correction code, and calculating the net

information spectral density (number of transmitted bits per

hertz), taking the symbol rate and the number of bits per

symbol into account.

JLT-11648-2009.R1

Table I. Error probabilities for a few common modulation formats as a

function of electrical signal-to-noise ratio.

Format Bit Error Probability

ASK

(figure 4b)

(

)

snrQ

Bi-polar MASK

(figure 4a&c)

( )













−

⋅

−

snr

)(log.6

log

BPSK

(figure 4a)

(

)

snrQ 2

MPSK (m>4)

(figure 4e)

( )

























msnrQ

sinlog.2

log

Rectangular QAM

(log

(m)=even)

(figure 4d)

(

)













−













− snr

m 1

log3

)(log

Due to the current limitation in the electronic bandwidth, it is

impractical to modulate the full optical bandwidth available.

Current technologies to achieve the maximum possible

information throughput involve WDM where the available

optical bandwidth is split into frequency bands, each of which

is modulated separately. In this case, the information spectral

density C/B also depends on the combined width of the guard

bands between WDM channels. Figure 6 shows the

information spectral density (ISD) as a function of snr in a

WDM system with 20% guard bands for uni-polar ASK

(circles), PSK (squares), and QAM (diamonds) formats.

NRZ

BPSK

QPSK

16-QAM

16-ASK

16-PSK

256-QAM

Signal-to-Noise Ratio, E

(dB)

ISD per Polarisation (b/s/Hz/pol)

Fig. 6. Information spectral density of uni-polar ASK (circles), PSK (squares)

and QAM (diamonds) showing the maximum system capacity as a function of

electrical signal-to-noise ratio for a BER of 10

-12

in a WDM system with 20%

guard bands between channels. The solid line represents the Shannon

theoretical limit [3, 6].

Whilst the benefits associated with modulation in both

quadratures, using either M-PSK or QAM, are apparent from

figure 6 and the expressions in table I and recent record

spectral density results [15] strong forward error correction

(FEC) is essential to enable operation close to the fundamental

Shannon limit [16]. Furthermore, the use of higher order

modulation formats suggests that the capacity increase is only

obtained at the expense of requiring higher snr and

implementation complexity. These requirements could be

reduced somewhat by combining functions traditionally

performed separately, for example, it has been demonstrated

that demodulation and error correction may be performed

simultaneously, with some performance benefit [17].

The reduction in ISD due to the FEC overhead is shown in

figure 7 for various modulation formats employing coherent

detection. In this figure it is assumed that the baseline system

was designed to give a BER of 10

-12

at a 15.5dB snr (sufficient

for direct detection of an on-off keyed signal). The required

FEC overhead for error-free operation is calculated and then

subtracted from the net information capacity. A simplified

approximation was used to calculate the FEC overhead, where

each FEC was assumed to require 7% overhead [18] for every

-3

of BER to be corrected. For example, we assumed an

overhead of 21% for the correction of a BER of 3·10

-3

. From

figure 7, it is shown that the calculated overhead results in a

negligible decrease in capacity for 2-bit per symbol uni-polar

signal with coherent detection. However, as the number of bits

per symbol is increased, the BER degradation increases,

requiring larger overheads, and, eventually, the required

additional FEC overhead outstrips the additional capacity

offered by an extra bit per symbol, at a fixed snr. By changing

to bi-polar, the required snr is greatly reduced, allowing, in

this example, 3 bits per symbol. Exploiting both quadratures

with M-PSK further reduces the required snr (see figure 6) and

4 bit per symbol M-PSK is possible at 15.5dB snr with

appropriate FEC. QAM exhibits still further performance

enhancement, resulting in 5 bits per symbol without significant

reduction in throughput due to FEC overhead.

Whilst it is likely that FEC circuits will become available

which will require less overhead than assumed here, including

current proprietary FEC circuits, it will still be the case that an

optimum ISD will exist for a given snr and modulation format.

Transmitted Bits per Symbol (b/s/Hz/pol)

Information Spectral Density

after FEC Overhead (b/s/Hz/pol)

Fig. 7. Illustration of the limitation in the net information capacity as a

function of the number of transmitted bit per symbols for uni-polar M-ASK

(circles), bi-polar M-ASK (down triangles), M-PSK (squares) and QAM

(diamonds) assuming a snr of 15.5dB.

It is clear that any required guard band between WDM

JLT-11648-2009.R1

channels reduces the ISD. The guard bands may be avoided by

employing orthogonal frequency division multiplexing

(OFDM) techniques [19,20], such as no-guard-interval OFDM

[21-23], coherent WDM [24-27] direct detection OFDM

[28,29] and coherent optical OFDM [30-35].

Frequency

Amplitude

Fig. 8. Illustration of overlapping modulation sidebands of OFDM signal.

In all of these multi-carrier systems, the frequency spacing

between the orthogonal sub-carriers is equal to the symbol rate

per subcarrier. A typical example of the orthogonal carriers are

shown in figure 8, where the peak of the spectrum of a given

channel corresponds to nulls in the spectra of all of the other

sub-channels, and in particular, the first null in the spectrum of

the adjacent sub-channel. Ideally, matched filters are used to

separate each sub channel [19], and this may be implemented

efficiently using Fast Fourier Transform algorithms for low

sub-channel data rates (e.g. 100Mb/s), with the digital signal

processing (DSP) complexity scaling approximately linearly

with the total capacity (∝ N⋅logN, where N is the channel

number) [28-33]. However, for a system with a high symbol

rate per channel (e.g. 40 Gb/s), the practical implementation of

precise matched filters proves difficult, and may be

approximated in the optical domain using asymmetric Mach

Zehnder interferometers [23,24] or with simple digital filters

[22]. The impact of any residual crosstalk may then be

minimised using appropriate optimisation of the relative

phases of each sub-channel [25] or cancelled using post-

detection signal processing [26, 36]. In all cases, the net result

is the straightforward generation of a signal with a capacity per

polarisation equal to the number of bits per symbol (or

log

(m)), with the potential suitability for ultra-high total

capacities (between 300 and 1,080 Gb/s and beyond [37-39])

which are difficult to achieve using single carrier modulation.

IV. N

LINEAR LIMITS

The above discussion applies equally to optical fibre, wireless

and copper based transmission systems, and in the absence of

any further signal degradation, performance approaching the

Shannon limit would be possible using forward error

correction. Wireless systems, particularly those employing

OFDM, experience non-linearity due to the saturation

characteristics of power amplifiers [40]. On the other hand,

periodically-amplified optical fibre based systems are

characterised by distributed non-linear effects in the fibre

itself. The most predominant non-linear effect arises from the

intensity dependent refractive index (Kerr effect) and results in

a number of phenomena such as self-phase modulation [41],

cross-phase modulation [42] and inter- [43] and intra-channel

[44] four wave mixing. Whilst many techniques to mitigate

the impact of non-linearity have been developed, including

most significantly dispersion management [45-49], the impact

of these non-linearities in terms of the information theoretical

limits have only recently been addressed [50-52]. The key

simplification introduced by Mitra and Stark [50] was to

equate a non-linear communication channel to a linear channel

with multiplicative noise, for which analytical results can be

obtained. It was found that, in contrast to linear channels with

additive noise, the capacity of a non-linear channel does not

grow indefinitely with increasing signal power, but has a

maximal value. This is a fundamental feature which

distinguishes non-linear communication channels from linear

ones. In making use of this new analytical approach, it is

assumed that any deterministic effects, such as chromatic

dispersion and self-phase modulation, which depend only on

the channel of interest, may be fully compensated. This

compensation may take the form of efficient modifications of

the transmitted or received signals based on prior knowledge

of the signal format itself. For example, reduction in dispersion

penalties are observed using pre-chirp [53,54] or electronic

dispersion compensation [55-57], whilst non-linear penalties

may be lowered by reducing phase noise (or timing jitter) by

modulating the received signal with a phase proportional to the

received intensity [58-60]. For multi-level formats, these

techniques may also be applied predicatively at the transmitter

[61]. Full non-linearity compensation may be applied at the

expense of complexity, either by optical phase conjugation

[62,63], or via emulation of back propagation using look-up

tables [64].

Assuming ideal compensation of all intra-channel effects other

than noise, cross-phase modulation (XPM), which causes

multiplicative noise, appears to be the principal source of

impairments that fundamentally limits the information capacity

of an optical communication system. XPM induces random

fluctuations in the target channel which are exponentially

related to the intensity of the neighbouring WDM channels. It

has been shown that the intensity scale for these fluctuations is

given by [50,65]

eff

XPM













∆

ln.2

(6)

where B, D, ∆λ and

are the channel bandwidth, local

dispersion, WDM channel spacing and fibre non-linear

coefficient respectively. N

is the number of WDM channels

and L

eff

is the non-linear effective length of the system given

by N

[1-exp(-

L)]/

, for a system with lumped amplifiers

where

is the loss coefficient and N

is the number of

amplifiers. Here we have assumed equally spaced channels of

equal intensities. A related amount of information is lost from

the channel of interest due to the random crosstalk induced by

XPM. The net effect is to reduce the information capacity of a

Approaching the Non-Linear Shannon Limit

Figures

Citations

The GN-Model of Fiber Non-Linear Propagation and its Applications

Capacity Trends and Limits of Optical Communication Networks Discussed in this paper are: optical communication network traffic evolution trends, required capacity and challenges to reaching it, and basic limits to capacity.

Capacity Trends and Limits of Optical Communication Networks

Roadmap of optical communications

Ultra-high-density spatial division multiplexing with a few-mode multicore fibre

References

A mathematical theory of communication

Digital Communications

The Mathematical Theory of Communication

On the design of low-density parity-check codes within 0.0045 dB of the Shannon limit

Synthesis of band-limited orthogonal signals for multichannel data transmission

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Frequently Asked Questions (20)

Q1. What are the contributions in this paper?

Q2. What is the key simplification introduced by Mitra and Stark?

Q3. What is the main source of impairments that limits the information capacity of an optical communication system?

Q4. What is the effect of the increased BER on the snr?

Q5. What is the way to increase the information capacity limit?

Q6. What is the optimum case for evaluating the performance limit?

Q7. What is the reason why wireless systems experience non-linearity?

Q8. How can the impact of residual crosstalk be minimised?

Q9. What is the frequency spacing between the orthogonal carriers?

Q10. How long does the Shannon limit apply to transoceanic systems?

Q11. What is the theoretical limit for the information capacity of a single source?

Q12. How long has the growth of communication capacity been observed?

Q13. What is the effect of XPM on the information capacity of a WDM system?

Q14. What is the maximum performance of a number of measured solid core fibres?

Q15. What is the effect of the calculated overhead on the BER?

Q16. What is the difference between linear and additive noise?

Q17. What is the difference between the theoretical limits in figure 11 and the experimental data?

Q18. How does the evolution of bandwidth in telecommunications networks be traced?

Q19. How many telecommunications networks can support 100 Gb/s?

Q20. What is the way to achieve a high symbol rate?