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IIUM Engineering Journal, Vol. 9, No. 2, 2008 Karfaa et al.

THEORETICAL EVALUATION OF NONLINEAR

EFFECTS ON OPTICAL WDM NETWORKS WITH

VARIOUS FIBER TYPES

YASIN M. KARFAA
1*

, M. ISMAIL
1
, F. M. ABBOU

2
, A. S. SHAARI

1
Department of Electrical, Electronics and Systems Engineering, Universiti Kebangsaan

Malaysia, 43600 UKM, Bangi, Selangor, Malaysia.

2
Faculty of Engineering, Multimedia University, 63100 Cyberjaya, Selangor, Malaysia.

*yasin_m_k@yahoo.com

ABSTRACT: A theoretical study is carried out to evaluate the performance of an optical

wavelength division multiplexing (WDM) network transmission system in the presence

of crosstalk due to optical fiber nonlinearities. The most significant nonlinear effects in

the optical fiber which are Cross-Phase Modulation (XPM), Four-Wave Mixing (FWM),

and Stimulated Raman Scattering (SRS) are investigated. Four types of optical fiber are

included in the analysis; these are: single-mode fiber (SMF), dispersion compensation

fiber (DCF), non-zero dispersion fiber (NZDF), and non-zero dispersion shifted fiber

(NZDSF). The results represent the standard deviation of nonlinearity induced crosstalk

noise power due to FWM and SRS, XPM power penalty for SMF, DCF, NZDF, and

NZDSF types of fiber, besides the Bit Error Rate (BER) for the three nonlinear effects

using standard fiber type (SMF). It is concluded that three significant fiber nonlinearities

are making huge limitations against increasing the launched power which is desired,

otherwise, lower values of launched power limit network expansion including length,

distance, covered areas, and number of users accessing the WDM network, unless

suitable precautions are taken to neutralize the nonlinear effects. Besides, various fiber

types are not behaving similarly towards network parameters.

KEYWORD: Nonlinear effects, Nonlinearity standard deviation, BER, Power penalty,

and WDM.

1. INTRODUCTION

High capacity long reach transmission systems employ a large number of wavelength

division multiplexed (WDM) channels with dense channel spacing and high bit rate per

channel. In conventional WDM systems, the main transmission impairments are fiber

losses, group velocity dispersion (GVD) and nonlinear fiber effects [1]. Due to the desired

long distance between amplifiers, high channel powers are required in order to achieve

sufficiently large signal-to-noise ratios (OSNR). The use of erbium doped fiber amplifiers

(EDFA) allows overcoming the first limitation while the others are usually minimized by

optimizing the dispersion compensation with a proper management. All these system

requirements give rise to nonlinear impairments. In long distance transmission links above

100 km including analog signals over optical fibers, high transmission powers are

involved which give rise to nonlinear effects such as self-phase modulation (SPM) in a

single channel system, cross phase modulation (XPM), four wave mixing (FWM),

IIUM Engineering Journal, Vol. 9, No. 2, 2008 Karfaa et al.

stimulated Raman scattering (SRS), and stimulated Brillouin scattering (SBS) between

channels in a WDM system and can be highly detrimental in the presence of dispersion

[2]. These nonlinear effects can limit performance in both digital and analog systems.

2. SYSTEM THEORETICAL ANALYSIS

Avoid using tab or enter buttons whenever possible. (Normal) (what is this??). Figure 1

below shows a typical WC channels optical WDM system. It is assumed that the network

is experiencing all the three nonlinear effects under study; these are XPM, FWM, and

SRS. For this network, the analysis of all the nonlinear effects is possible by taking their

effects on pulses propagating inside an optical fiber and to be included by using the

nonlinear Schrödinger Equation (NLSE), which is used to describe the slowly varying

complex envelope of the optical field when expressing any of the nonlinear effects in the

form that is given by [3] as in Eq. (1) below:

2 3
2 2 21 1 1 1

1 21 31 1 1 2 3 12 3

1
9

2 2 6

A A Ai
A i A A A A

z T T
ω

α
β β γ ψ

∂ ∂ ∂  + + − = +
 ∂ ∂ ∂

( 1 )

Since three of the nonlinear effects are considered here, so, Eq. (1) contains a last term

on the R.H.S ( ωψ ) to account for SPM and SBS that are not included in this study; where

n
α

represents the n
th

channel’s attenuation parameter, n2
β

represents the n
th

channel’s

second order dispersion parameter, n3
β

represents the n
th

channel’s third order dispersion

parameter, n
γ

represents the n
th

channel’s nonlinearity coefficient where

( eff
cAn /

02
ωγ =

), nR
T

represents the n
th

channel’s Raman shift, mn
d

represents the

walk-off parameter between channels m and n .

Fig. 1: Optical Communication System for Multiple Transmitted Channels and Various

Types of Fiber (OF: optical fiber, can be SMF, DCF, NZDF, or NZDSF).

2.1 Theoretical Analysis of XPM

XPM is a fatal nonlinear impairment in WDM systems as a nonlinear phase noise

results after the interaction of optical amplified spontaneous emission (ASE) and fiber

Kerr nonlinearities [4]; it is one of the dominant degradation effects for bit rates as much

as 10 Gbps in WDM systems although it can be of use as in case of the optical signal

regeneration for future high-speed optical communication systems; when the interaction

IIUM Engineering Journal, Vol. 9, No. 2, 2008 Karfaa et al.

takes place between continuous optical waves, constant phase changes and constant

polarization rotations are induced, but, if the intense wave is pulsed or intensity

modulated, the induced nonlinear phase shift becomes time dependent, leading to phase

modulation and polarization modulation of the co-propagating waves. Thereby, new

spectral components are generated on their optical spectra resulting in an XPM-induced

spectral broadening. In some researches where the channel spacing is reduced, the

degradation due to XPM and FWM increased and the optimum dispersion compensation

scheme is significantly affected by these effects. The dispersion is the middle parameter

that increasing it minimizes FWM but increases XPM and vice versa. Therefore, since

FWM can be minimized using a high local dispersive transmission fiber such as a SMF

and thus the main inter-channel nonlinear effect is XPM. The mathematical expression to

calculate XPM induced crosstalk power is derived following literature models [5-7]. The

XPM power fluctuation due to one channel interference is [5]:

( ) ( )
( )

( )

, , 0

sin / 2
, 4

j X k j k

D
P L P P e

j k

α
ω

ω γ ω
α β ω ω

−
=

+ ∆
(2)

where
2

/ cλ ωλ∆ = ∆ is the wavelength spacing, and 2
2

2 /
C

D cπ β λ= − is the

chromatic dispersion,
2

β is the group velocity dispersion. By taking the magnitude and

ignoring
2α being too small:

( ) ( )
( )2

, , 0

sin / 2
, 8

j X k j k

D
P L P P e

D k

α
ω

ω πγ ω
λω

−
=

∆
(3)

Considering the contributions from
C

W number of channels, Eq. (2) can be rewritten

as:

( ) ( )
( )2

, , 0

sin / 21
, 8

j X k j k

C C

D
P L P P e

W D k

α
ω

ω πγ ω
λω

−
=

∆
∑ (4)

The total noise variance due to XPM-induced power fluctuation for an IM/DD system

XPM
P is obtained by applying the integration of Eq. (4) for the interval

1 2
( , )ω ω as follows:

( ) ( )
2

2 2

, ,
,

XPM d j X k
P R P L H d

ω ω ω= ∆∫ (5)

2.2 Theoretical Analysis of FWM

FWM occurs when the channels in a WDM system are equally spaced, so that the new

waves generated by FWM will fall at channel frequencies and, thus, will give rise to

crosstalk; its severity is maximized when dispersion is minimized by some proper

techniques, so they are inversely proportional because the FWM products add coherently

in each span. FWM may result when three light signals at different wavelengths interact in

the fiber to create a fourth light signal at either a different or same wavelength causing a

distortion to the desired signal. The new optical waves frequencies are judged by fijk=fi+fj-

fk, which is θ
+
; or it results when two propagating waves at frequencies f1 and f2 mix and

generate sidebands at 2f1-f2 and 2f2-f1, which is θ
-
. These sidebands co-propagate with

initial waves and grow at their expense, besides creating inband crosstalk that cannot be

IIUM Engineering Journal, Vol. 9, No. 2, 2008 Karfaa et al.

filtered optically or electronically and channels beat against each other to form

intermodulation products. The mathematical expression to calculate FWM induced

crosstalk power is derived with following the models done by [8-11]:

4 2 2 1/ 2

1/3 2

2 2 2

4 10 ( )
[ ]

9 ( )

ijk ijk A ijk

FWM

ijk eff C ijk

f G L N U
P P

cA D

λ
γ

ω α β

×
=

∆ + ∆
∑ (6)

where
ijk

f is the frequency for channels i , j , k respectively, λ is the operating

wavelength of the optical signal, 1
jkl

G = for two-tone products, Gjkl = 2 for three-tone

products, L is the length of fiber line,
A

N is the number of EDFAs,
ijk

U is the degeneracy

factor and takes values of 3 or 6, c is the speed of light in free space,
eff

A is the fiber core

effective cross-section area,
f∆

is the channel spacing,
C

D is the chromatic dispersion,

β∆
is the phase matching coefficient for FWM to occur. Figure 2 below shows the power

spectral representation of the FWM, XPM, and SRS nonlinear effects as it appears in the

output for a regular input to the system when
C

W frequency channels are transmitted in

optical WDM system.

Fig. 2: Nonlinear Impairments in an Optical WDM Network as a Limiting Power

Degradation Factor.

2.3 Theoretical Analysis of SRS

SRS causes the optical power from one mode (with a higher frequency) to be

transferred in the forward direction to the same or other modes, at a lower frequency. It

depends critically upon the optical power density within the fiber and hence only becomes

significant above threshold power levels; therefore, the launched power into the system is

limited because of these Raman effects [12]. Beside that, the large-capacity and long haul

network transmission essentially use the erbium-doped fiber amplifiers (EDFAs) to

compensate the fiber attenuation without using electrical regenerators. The expansion of

the optical network dimensions to thousands of kilometers with the feasibility of the

concept of transparent optical networks introduces some problems like accumulated ASE

noise and aggravation of fiber nonlinearity effects, where SRS is one of them. Even the

mitigation techniques of SRS will give some undesired results, for example, the mitigation

by having a large core and short fiber which reduces both the optical power density in the

IIUM Engineering Journal, Vol. 9, No. 2, 2008 Karfaa et al.

core and the interaction length of the optical field in the fiber, but this leads to a

multimode core, and can consequently lead to a significant degradation of the output beam

quality. Other techniques have their drawbacks as well. Anyhow, there are many attractive

features as a frequency converter, beam clean-up, pulse compression, etc. For the above

network where the occurrence of SRS problem is assumed, the mathematical expression to

calculate SRS induced crosstalk power is derived with following the models done by [13-

16].

When channel 0 with the shortest wavelength
Shortest

λ is affected by a longer
longest

denoted by channel i, the probabilities for the channels in ON state are:

0
P O (channel 0 is ON) =

ON
P (7)

i
PO (channel i is ON) =

ON
P (8)

Assuming both channels are random and independent from each other, the probability

that both channels are in ON state is:

0rbON i ON ON ON
P P O PO P P P= × = × = (9)

SRS happens when both channels involved are in ON state, of which the probability is

shown as above. The average SRS power depletion in channel 0 due to interference from

channel i,
0D i

µ is defined as:

0 0D i P rbON
D Pµ = × (10)

where
0P

D is the fractional power lost by channel 0 due to all other channels, and is

defined as:

1
6

1 0

1.4 10
4

CW
C i eff i i

i eff

W POL
D

ωλ γ

πλ

−
−

∆
= × ×∑ (11)

where
C

W is the wavelength channels number,
i

P is the optical power carried in the i
th

channel,
i

PO is the
i

P at ON state,
eff

A is the effective core area of the fiber,
eff

L is the

effective fiber length given by (1 exp( )) /
eff

L Lα α= − − , where α is the fiber loss

coefficient and L is the fiber length. The Raman gain coefficient
i

γ that is coupling the

first and i
th

channel can be expressed using the triangular Raman gain as follows:

13
1.5 10

i P

i v
γ γ

∆
=

×
for

13
1.5 10i v∆ < × and 0

i
γ = otherwise (12)

Here,
P

γ is the peak Raman gain coefficient with a value of 126 10−× cm/W2 and v∆ is

the channel frequency spacing. The mean and standard deviation of
0P

D are related

according to the following relation [16]:

Case I: Walk-off length
W

L is very large.

0 0

2(2 1)

3 ( 1)

C
D i D i

C C

W W
σ µ

−
=

−
(13)

IIUM Engineering Journal, Vol. 9, No. 2, 2008 Karfaa et al.

Case II: Walk-off length
W

L is very small.

0 0
( 1)

W
D i D i

C C

W W

α
σ µ=

−
(14)

The total SRS induced crosstalk noise power due to interactions of all channels
SRS

P is

given by:

1
3

1/ 4
1

1 0

2.2 10
2 2

CW
C i i effi A

SRS

i eff b C

W LPO N
P K

A R D

λ γ ω

λ π

−
−

∆
= × × ×∑ (15)

where 1/ 4
1

2(2 1)

3 ( 1)

C C

W
K

W W

−
=

−
, 1/ 4

( 1)

C C

W W

−
for large and small walk of respectively.

Then, the bit error rate for the system due to nonlinearities is given by [3]:

_1 _ 0

4 ( ) 2 2

DD
NL

NL ASE ASE

I II I
BER erfc erfc

P σ σ

    −−
 = +   

   +     

(16)

where
NL

P is to account for the calculated nonlinear crosstalk noise power; so it can be

XPM
P ,

FWM
P , or

SRS
P ;

D
I is given by [3]:

0 1 1 0

0 1

I I
I

σ σ

+
=

+
(17)

where
1

I is the transmitted current for bit ‘1’,
0

I is the transmitted current for bit ‘0’,
D

I is

the threshold current level for the decision circuit, 1_ASE
σ

is the Amplified Spontaneous

Emission (ASE) noise for transmission of bit ‘1’, 0_ASE
σ

is the ASE noise for

transmission of bit ‘0’. The accumulated ASE_ASE beat noise for bits ‘1’ and ‘0’ is given

by [17]:

2 2 2

_ 0 2

4 ( ) (2 )
ch e

ASE ASE d ASE accn e

B
R P B B

B
σ

−
= − (18)

where
d

R is the photo detector responsivity,
ch

ASE accn
P

−
is the accumulated ASE noise for

any channel,
0

B is the optical bandwidth, and
e

B is the electrical bandwidth.

3. MATERIALS AND METHOD

The first step of the method of evaluating the nonlinear effects is by conducting the

theoretical analysis with modeling the optical WDM network under three considered

nonlinear effects; these are: XPM, FWM, and SRS. Then, the second step is to use the

findings of other researchers for specifying the best model among those. The third step is

to derive the relevant expressions for finding the XPM, FWM, and SRS induced crosstalk.

The fourth step is to simulate the theoretical expressions of the three nonlinear effects with

IIUM Engineering Journal, Vol. 9, No. 2, 2008 Karfaa et al.

the mathematical calculations using the Matlab software. The fifth step is to analyze the

results of the simulations. And the last step is to conclude the outcome of the evaluation of

nonlinear effects. For all of the three models of nonlinear crosstalk derivations, the same

basic equation is used, which represents the pulses propagating inside, that is the

Nonlinear Schrödinger Equation (NSE), which is commonly used to describe the slowly

varying complex envelope of the optical field when expressing the XPM, FWM, and SRS

in the form [3] (hanging sentence??). With taking NSE equation, then some other models

are considered or followed to ease the derivations with some approximations. Those

models are indicated to when used. (poorly written; consider rephrasing) The model of this

optical WDM system is shown in Fig. 1. This system is suffering from nonlinear effects.

Since the figure shows a single-line transmission system with multiple channels, so, it is

assumed that at each one certain time, there is a certain type of fiber to be used, and

replace by other types alternatively. Tables 1 and 2 for the values of system parameters

that are used in the simulation are shown below. (last two sentences are poorly written;

consider rephrasing)

Table 1: Input parameters used in the simulation of all nonlinear crosstalk effects.

Parameter Values

Number of channels,
C

W 128

Number of nodes, N 20

Number of amplifiers,
A

N 20

Optical fiber length, L 200

Input Power,
in

P 0

Bit rate,
b

R 10

Optical bandwidth,
o

B 20

Electrical bandwidth,
e

B 10

Operating frequency, v 1.9355×10
14

Frequency channel spacing, ω∆ 50 to 100

Wavelength spacing, λ∆ 0.4 to 0.8

Nonlinear refractive index,
2

n 2.6×10
-20

Group index,
g

n 1.46

Basic operating light wavelength, λ 1550

Receiver’s responsivity,
d

R 1

Load resistence,
L

R 50

Inversion factor,
e

F 3

Amplifier’s gain, G 20

IIUM Engineering Journal, Vol. 9, No. 2, 2008 Karfaa et al.

Raman gain coefficient,
R

g 6×10
-13

Temperature, T 300

Table 2: Parametric characteristics of many types of optical fibers.

Fiber type 2( )
eff

A mµ ( / / )CD ps km nm

( / )dB kmα 1( )W kmγ −

SMF 80 17 0.2 2.5

DCF 20 –80 0.29 3

NZDF 72 –3 0.23 3.9

NZDSF 50 4.5 0.25 3.84

4. RESULTS AND DISCUSSION

A Matlab software is used to validate the three analytical expressions for the nonlinear

crosstalk noises XPM, FWM, and SRS. Some of the system parameters used are number

of channels, C
W

= 128 channels, input power = 0-20 dBm, fiber loss coefficient = 0.2-0.25

dB/km, channel spacing range 10-100 GHz (0.08-0.8 nm). The XPM power penalty and

crosstalk noises due to FWM, and SRS are shown in Fig. 3, 4 and 5 respectively versus

number of channels and input powers respectively. Figure 3 is for the network when it is

impaired by XPM, and shows the XPM power penalty versus input power, whereas Fig. 4

shows the standard deviation of FWM induced crosstalk versus number of channels when

the optical WDM network is experiencing FWM nonlinear effect. Figure 5 is for the

network which is influenced by SRS. Four types of optical fiber are used in the

calculations for the graphic relations: single-mode fiber (SMF), dispersion compensation

fiber (DCF), non-zero dispersion fiber (NZDF), and non-zero dispersion shifted fiber

(NZDSF). These fibers differ from each other in the specifications that are expressed by

all the necessary parameters for the computations to satisfy the algorithm of finding the

different fiber nonlinearity originated noise phenomenon’s (like XPM, FWM, and SRS in

this case). The used parameters in the calculations are fiber attenuation, chromatic

dispersion, nonlinearity refractive index coefficient, fiber nonlinearity factor, and fiber

core cross-sectional area. The differences in the parameters’ values among the various

fiber types continue to appear finally in the graphic relations of power penalties versus

input powers, and so, they vary in the graphs. Using the mathematical meaning for this, it

seems that different types of fiber respond differently to the effect of changing the input

powers and the relevant changes in the power penalties due to fiber nonlinearities. Lastly,

Figure 6 shows BER versus received power in the presence of all the three nonlinear

effects: XPM, FWM, and SRS. Standard fiber SMF is used, while the parameters are fixed

at 8 nodes, 100 GHz channels spacing, while the rest of the parameters are as in Tables 1

and 2.

IIUM Engineering Journal, Vol. 9, No. 2, 2008 Karfaa et al.

Fig. 3: Penalty due to XPM versus Input Power for various fiber types where ASE

noises are not counted.

Fig. 4: Standard deviation of FWM induced crosstalk versus Input Power for various

fiber types.

IIUM Engineering Journal, Vol. 9, No. 2, 2008 Karfaa et al.

Fig. 5: Standard deviation of SRS induced crosstalk versus Input Power for various

fiber types.

Fig. 6: BER due to nonlinear effects (XPM-FWM-SRS) versus Received Power using

standard SMF fiber.

5. CONCLUSION

In this paper, a theoretical study is carried out to investigate the transmission of signal

for N nodes optical WDM network in the presence of nonlinear crosstalk due to XPM,

FWM, and SRS. The performance analysis is done by calculating the XPM power penalty,

and the standard deviation of the noises due to FWM and SRS that the optical signal is

experiencing when transmitting through 128-144 channels. The power penalty and

standard deviation to evaluate the nonlinear crosstalk noise power in the analysis are

chosen because those give an indication of signal transmission in a problem of

compensation requirement or an extra budget to pay back for the lost power, since the

IIUM Engineering Journal, Vol. 9, No. 2, 2008 Karfaa et al.

BER of the system must be kept at a level of
9

10
−

or below, which is preferable as it means

that more packets are transmitted with no errors. When the BER is high, the performance

of the network deteriorates as it indicates corrupted and lost data. The results using Matlab

show that a higher BER and nonlinear crosstalk occurs in a WDM network when the

number of nodes is large. This is true for large number of channels; that means small

channel spacing (except SRS that reduces with increasing number of channels or reducing

channel spacing). The dispersion is working proportionally with XPM and oppositely with

FWM and SRS. The noises due to nonlinear effects are not same for different types of

fiber, and that indicate the nature of the limitations due to nonlinearity and due to fiber

type. Narrowing channel spacing is also not preferable as it increases the BER. In

conclusion, to transmit signal in optical WDM networks, physical impairments such as

XPM, FWM, and SRS should be taken into account as those directly affect the BER.

The standard single-mode fiber (SMF) is showing better results in the performance

evaluation than the other tested three types of fiber (DCF, NZDF, and NZDSF). Then,

since the DCF fiber is used in the dispersion management and dispersion map as a solution

to go close to the trade-offs in the way to keep the negative effects of the fiber

nonlinearities to the minimum at the acceptable level that helps to improve the system’s

performance, so, this DCF fiber is the best to use in alternative mixing techniques to play

this turn.

6. ACKNOWLEDGEMENT

Dr. Hairul Azhar Abdul Rashid, the senior lecturer in Faculty of Engineering,

Multimedia University kindly provided technical support, printing facilities, downloading

reference papers, and many useful discussions.

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dispersionmap for 10 Gbit/s WDM links over standard-fiber employing duobinary format,” In

Transparent optical networks, 9
th
ICTON Intern. Conference. Proc. no.1, 79-82. 2007.

[2] B. S. Marks, C. R. Menyuk, A. L. Campillo, and F. Bucholtz, “Analysis of interchannel
crosstalk in a dispersion-managed analog transmission link,” Journ. of Lightwave Technol.

Vol. 24, no. 6, 2305-2310, 2006.

[3] G. P. Agrawal, “Fiber Optic Communication Systems,” John Wiley & Sons: New York, 2002.

[4] X. Li, F. Zhang, Z. Chen, and A. Xu, “Optimum dispersion mapping of 40 Gb/s RZ-DPSK
WDM transmission systems with XPM-induced nonlinear phase noise,” In Intern. Nano-

Optoelectronics Workshop i-NOW '07 Proc. 156-157, 2007.

[5] B. Xu, and M. B. Pearce, “Comparison of FWM- and XPM-induced crosstalk using the
Volterra series transfer function method,” Journ. of Lightwave Technol. Vol. 21, no. 1, 40-53,

2003.

[6] A. V. T. Cartaxo, “Cross-phase modulation in intensity modulation direct detection WDM
systems with multiple optical amplifiers and dispersion compensators,” Journ. of Lightwave

Technol. 17(2): 178-190, 1999.

[7] I. B. Djordjevic, A. Stavdas, C. Skoufis, S. Sygletos, and C. Matrakidis, “Analytical modeling
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Modeling and Simulation. Vol. 23, no. 4, 226-233, 2003.

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[8] R. Ramaswami, and N. S. Kumar, “Optical networks: a practical perspective,” 2
nd

edition:

Academic Press, 2002.

[9] M. Eiselt, “Limits on WDM Systems due to four-wave mixing: a statistical approach,” Journ.
of Lightwave Technol. Vol. 17, no. 11, 2261-2267, 1999.

[10] S. Song, T. C. Allen, K. R. Demarest, and R. Hui, “Intensity-dependent phase-matching
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[11] A. R. Chraplyvy, “Limitations on lightwave communications imposed by optical-fiber
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[12] M. K. Abdullah, Z. A. T. Al-Qazwini, M. D. A. Samad, and M. Mokhtar, “Optimum transmit
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[14] F. Forghieri, R. W. Tkach, and A. R. Chraplyvy,” Effect of modulation statistics on Raman
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Lightwave Technol. Vol. 18, no. 7, 915-921, 2000.

[17] G. Keiser, “Optical fiber communications, “ 3
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edition, McGraw-Hill, 2000.

NOMENCLATURE

in
P Input power (launched power) W

rec
P Received power W

NL
P Overall nonlinearity lost Power W

ON
P Power when channel is ON state W

0D i
µ Average power depletion W

0P
D Fractional power loss W

0D i
σ Standard deviation of power loss dB

C
W Number of wavelength channels -

N Number of nodes -

A
N Number of amplifiers -

G Gain of the amplifier dB

R
G Raman gain coefficient dB

β∆ Phase-matching coefficient -

IIUM Engineering Journal, Vol. 9, No. 2, 2008 Karfaa et al.

R
T Raman frequency shift THz

P
γ Peak Raman gain 2cmW −

i
γ Coupling Raman gain coefficient for channels 1, i -

L Optical fiber length m

in
P Input power dBm

b
R Bit rate Gbps

0
B Optical bandwidth GHz

e
B Electrical bandwidth GHz

v Operating frequency Hz

f Any frequency of any channel Hz

ω∆ Frequency channel spacing GHz

λ∆ Wavelength spacing nm

λ Basic operating light wavelength nm

2
n Nonlinear refractive index

2 1
m W

−

g
n Group index

2 1
m W

−

d
R Responsivity of the receiver

1
AW

−

L
R Load resistance Ohm

e
F Inversion factor -

U Degeneracy factor -

T Temperature K
ο

d Walk-off parameter -

W
L Walk-off length m

c Speed of light in free space
1

ms
−

eff
A Cross-sectional area of fiber core (effective)

2
m

eff
L Effective Optical fiber length m

BER Bit error rate dB

XPM Cross-phase modulation -

FWM Four-wave mixing -

SRS Stimulated Raman scattering -

IIUM Engineering Journal, Vol. 9, No. 2, 2008 Karfaa et al.

D
I Decision level of current for logic circuitry A

1
σ Standard deviation of lost power for bit
σ Standard deviation of lost power

Greek letters

α Fiber loss coefficient (attenuation)
γ Nonlinearity coefficient of the fiber
Ψ To account for unstudied nonlinear effects
ω Any angular frequency
β Dispersion of any order
σ Standard deviation of lost power

Subscript

i Channel number

j Channel number

k Channel number

n Index

m Index

accn Accumulated

ASE Amplified spontaneous emission

o Optical

e Electrical

C
Chromatic

eff
Effective

rbON
Probability of both channels is ON

NL Nonlinear

D
Decision level

W
Walk-off

Superscript

ch Any channel