97

CHAPTER 3

IMPACT OF EDFA GAIN SATURATION ON

DYNAMIC RWA

3.1 INTRODUCTION

In an optical communication system, the optical signals from the

transmitter are attenuated by the optical fiber as they propagate through it.

Other optical components, such as multiplexers and couplers, also add loss.

After some distance, the cumulative loss of signal strength causes the signal

to become too weak to be detected. Before this happens, the signal strength

has to be restored. Prior to the advent of optical amplifiers, the only option

was to regenerate the signal. A regenerator converts the optical signal to an

electrical signal, cleans it up, and converts it back into an optical signal for

onward transmission.

Optical amplifiers offer several advantages over regenerators.

Regenerators are sensitive to the bit rate and modulation format used by the

communication systems, whereas, optical amplifiers are transparent to the bit

rate or signal formats. Thus a system using optical amplifiers can be more

easily upgraded, for example, to a higher bit rate, without replacing the

amplifiers. In contrast, in a system using regenerators, such an upgrade would

require all the regenerators to be replaced. Furthermore, optical amplifiers

have fairly large gain bandwidths, and as a consequence, a single amplifier

can simultaneously amplify several WDM signals. In contrast, separate

regenerators are required for each wavelength.

98

Optical amplifiers have become an essential component in optical

transmission systems and networks to compensate for system losses.

Amplifiers are used in three different configurations as shown in Figure 3.1.

An optical preamplifier is used just in front of a receiver to improve its

sensitivity (Olsson 1989). A power amplifier is used after a transmitter to

increase the output power. A line amplifier is used typically in the middle of

the link to compensate for link losses. The design of the amplifier depends on

its configuration. A power amplifier is designed to provide the maximum

possible output power. A preamplifier is designed to provide high gain and

the highest possible sensitivity, that is, the least amount of additional noise. A

line amplifier is designed to provide a combination of all of these.

Figure 3.1 Different Configurations of Optical Amplifier

There are four types of optical amplifiers: Erbium Doped Fiber

Amplifiers, fiber Raman amplifiers, fiber Brillouin amplifiers and

semiconductor optical amplifiers. The most common optical amplifier

available today EDFA. EDFAs are used in almost all amplified WDM

systems, whereas Fiber Raman amplifiers are used in addition to EDFAs in

many ultra-long haul systems. The EDFA has a gain bandwidth of about 35

nm in the 1.55 µm wavelength region. The great advantage of EDFA is its

capability of simultaneously amplifying many WDM channels. EDFAs

spawned a new generation of transmission systems, and almost all-optical

fiber transmission systems installed over the last few years use EDFAs

instead of regenerators.

Transmitter

Preamplifier Line amplifier Power amplifier

Receiver

99

The key physical phenomenon behind signal amplification in an

EDFA is stimulated emission of radiation by atoms in the presence of an

electromagnetic field. Spontaneous emission process does not contribute to

the gain of the amplifier since the emitted photons may have the same energy

as the incident optical signal but they are emitted in random directions,

polarizations, and phase. This is unlike the stimulated emission process,

where the emitted photons not only have the same energy as the incident

photons but also the same direction of propagation, phase and polarization.

This phenomenon is usually described by saying that the stimulated emission

process is coherent, whereas the spontaneous emission process is incoherent.

Spontaneous emission has a deleterious effect on the system. The

amplifier treats spontaneous emission radiation as another electromagnetic

field, and the spontaneous emission also gets amplified, in addition to the

incident optical signal. The amplified spontaneous emission appears as noise

at the output of the amplifier. In some amplifier designs, the ASE noise can be

large enough so as to saturate the amplifier. In this chapter, the impact of gain

saturation and the ASE noise generated by the EDFAs are considered during

the dynamic lightpath establishment process. Ramamurthy et al (1999)

considered the impact of EDFA gain saturation and the ASE noise during the

establishment of a lightpath. However, they have used an approximate EDFA

gain model. Deng et al (2004) also use a simple gain model. In this work,

EDFA gain is numerically obtained by solving a transcendental equation

called the Saleh equation. In order to obtain the gain quickly, a fast and

accurate approach is also suggested.

3.2 ERBIUM DOPED FIBER AMPLIFIERS

An EDFA is shown in Figure 3.2. It consists of a length of silica

fiber, whose core is doped with ionized atoms (ions), Er3+

, of the rare earth

100

element Erbium. This fiber is pumped using a signal from a laser, typically, at

a wavelength of 980 nm or 1480 nm. In order to combine the output of the

pump laser with the input signal, the doped fiber is preceded by a wavelength-

selective coupler.

Figure 3.2 An Erbium doped Fiber Amplifier

At the output, another wavelength-selective coupler may be used if

needed to separate the amplified signal from any remaining pump signal.

Usually, an isolator is used at the input and/or output of any amplifier to

prevent reflections into the amplifier. A combination of several factors has

made the EDFA the amplifier of choice in today’s optical communication

systems: the availability of compact and reliable high power semiconductor

pump lasers, polarization independence, easy coupling of light into

transmission fiber, device simplicity and negligible crosstalk when amplifying

WDM signals.

3.2.1 Principle of Operation

The energy levels of Erbium ions in silica glass are shown in

Figure 3.3 and are labeled E1, E2 and E3 in order of increasing energy. Several

other levels in Er3+

are not shown. Each energy level that appears as a discrete

line in an isolated ion of Erbium is split into multiple energy levels when

these ions are introduced into silica glass. This process is termed as Stark

splitting. Moreover, glass is not a crystal and thus does not have a regular

Residual pump Wavelength-selective

coupler

Signal out Isolator

Erbium fiber

1550 nm

Signal in

Pump

101

structure. Thus the Stark splitting levels introduced are slightly different for

individual Erbium ions, depending on the local surroundings seen by those

ions. Macroscopically, when viewed as a collection of ions, this has the effect

of spreading each discrete energy level of an Erbium ion into a continuous

energy band. This spreading of energy levels is a useful characteristic for

optical amplifiers since they increase the frequency or wavelength range of

signals that can be amplified. Within each energy band, the Erbium ions are

distributed in the various levels in a nonuniform manner by a process known

as thermalization. It is due to this thermalization process that an amplifier is

capable of amplifying several wavelengths simultaneously.

Figure 3.3 Energy levels in the Er3+

ions of Silica Glass

The fourth energy level E4 is present in fluoride glass but not in

silica glass. The difference between the energy levels is labeled in terms of

the wavelength in nm of the photon corresponding to it. The upward arrows

indicate wavelengths at which the amplifier can be pumped to excite the ion

into the higher energy level. The 980 nm transition corresponds to the band

gap between the E1 and E3 levels. The 1480 nm transition corresponds to the

gap between the bottom of the E1 band to the top of the E2 band. The

E1

E4

1530 nm 980 nm 1480 nm

980 nm

(Fluoride glass

only)

E3

E2

102

downward transitions represent the wavelength of photons emitted due to

spontaneous and stimulated emission. In the case of Erbium ions in silica

glass, the set of frequencies that can be amplified by stimulated emission

from the E2 band to the E1 band corresponds to the wavelength range

1525-1570 nm, a bandwidth of 50 nm, with a peak around 1543 nm. This is

one of the low attenuation windows of standard optical fiber used by optical

systems.

The ionic population in the three levels is denoted by N1, N2 and

N3. In thermal equilibrium, N1 > N2 > N3. The population inversion condition

for stimulated emission from E2 to E1 is N2 >N1 and can be achieved by a

combination of absorption and stimulated emission as follows. The energy

difference between the E1 and E3 levels corresponds to a wavelength of

980 nm. So if optical power at 980 nm-called the pump power-is injected into

the amplifier, it will cause transition from E1 to E3. Since N1 > N3, there will

be a net absorption of the 980 nm power. This is called pumping.

The ions that have been raised to level E3 by this process will

quickly transit to level E2 by the spontaneous emission process. The lifetime

for this process, τ32, is about 1 µs. Atoms from level E2 will also transit to

level E1 by the spontaneous emission process, but the lifetime for this process,

τ21, is about 10 ms, which is much larger than τ32. Moreover, if the pump

power is sufficiently large, ions that transit to the E1 level are rapidly raised

again to the E3 level only to transit to the E2 level again. The net effect is that

most of the ions are found in level E2, and thus population inversion is

achieved between the E2 and E1 levels. Therefore, if a signal in the

1525-1570 nm bands is injected into the fiber, it will be amplified by

stimulated emission from the E2 to E1 level.

103

Several levels other than E3 are higher than E2 and, in principle, can

be used for pumping the amplifier. But the pumping process is more efficient

because it uses less pump power for a given gain, at 980 nm than these other

wavelengths. Another possible choice for the pump wavelength is 1480 nm.

Pumping at 1480 nm is not efficient as 980 nm pumping. The degree of

population inversion that can be achieved by 1480 nm pumping is also lower.

The higher the population inversion, the lower the noise figure of the

amplifier. Thus 980 nm is preferred to realize low noise amplifiers. However,

higher power pump lasers are available at 1480 nm, compared to 980 nm, and

thus 1480 nm pump finds applications in amplifiers designed to yield high

output powers. Another advantage of the 1480 nm pump is that the pump

power can also propagate with low loss in silica fiber that is used to carry

signals. Therefore, the pump laser can be located remotely from the amplifier

itself. This feature is used in some systems to avoid placing any active

components in the middle of the link.

An EDFA can be operated in three configurations. In co-directional

pumping, the pump signal is injected from the same direction as the signal

flow. In counter-directional pumping, the signal and pump are injected in

opposite directions. In dual pumping, two pumps are used. In practice, most

amplifiers deployed in real systems are more complicated. Two stage designs

are more commonly used. The two stages are optimized differently. The first

stage is designed to provide low noise, and the second stage is designed to

produce high output power. Apart from EDFAs operating in the C band

(1530-1565 nm), EDFAs have also been developed to operate in the L band

(1565 to 1625 nm). There are significant differences in the design of C- and

L-band EDFAs. These amplifiers are usually realized as separate devices,

rather than as a single device.

104

3.2.2 Gain Saturation in EDFA

An important consideration in designing amplified systems is the

saturation of the EDFA (Desurvire et al 1989). Gain saturation is the signal

power dependent gain compression that an amplifier suffers when the input

signal strength becomes large. Depending on the pump power and the

amplifier design itself, the output power of the amplifier is limited. As a

result, when the input signal power is increased, the amplifier gain drops. This

is called gain saturation. This behavior can be captured approximated by the

following equation:

G = 1 +G

Gln

P

P max

in

sat

(3.1)

Here, Gmax is the unsaturated gain, and G the saturated gain of the

amplifier, Psat

is the internal saturation power of the amplifier, and Pin is the

input signal power. Deng et al (2004) have used equation (3.1) in their work

to model the EDFA. The saturated gain is less than the unsaturated gain.

EDFA gain saturation can degrade the received signal quality and increase the

receiver BER.

3.2.3 Gain Dispersion in EDFA

Gain dispersion refers to the wavelength dependence of the gain of

an EDFA. Since the population density at the various levels within a band is

different, the gain of an EDFA becomes a function of the wavelength. When

such an EDFA is used in a WDM communication system, different WDM

channels undergo different degrees of amplification. The wavelength

dependence of gain is referred to as gain dispersion. This is a critical issue in

WDM systems with cascaded amplifiers.

105

One way to improve the flatness of the amplifier gain is to use

fluoride glass fiber instead of silica fiber, doped with Erbium (Rajiv

Ramaswami and Sivarajan 1998). Such amplifiers are called Erbium Doped

Fluoride Fiber Amplifiers (EDFFAs). However, there are certain drawbacks

to using fluoride glass. The noise performance of EDFFA is poorer than

EDFA. Fluoride fiber is difficult to handle. It is brittle, difficult to splice with

conventional fiber, and susceptible to moisture. Another approach to flatten

the EDFA gain is to use a filter inside the amplifier. Long period fiber

gratings and dielectric thin film filters are currently the leading candidates for

this application (Giles 1997 and Dung et al 1998). Most commercially

available amplifiers are able to provide gain flatness of less than 1 dB ripple

across the nominal band (Deng et al 2004).

3.3 FIBER RAMAN AMPLIFIERS

Stimulated Raman scattering (SRS) is one of the nonlinear

impairment that affects signals propagating through the optical fibers. The

same nonlinearity can be exploited to provide amplification as well. SRS is

an interaction between light waves and the vibrational modes of silica

molecules. If a photon of energy hν1 is incident on a molecule having a

vibrational frequency νm, the molecule can absorb some energy from the

photon. In this interaction, the photon is scattered, thereby attaining a lower

frequency ν2 and a corresponding lower energy hν2. The modified photon is

called a Stokes photon. Because the optical signal wave that is injected into a

fiber is the source of the interacting photons, it is often called the pump wave,

since it supplies power for the generated wave. This process generates

scattered light at a wavelength longer than that of the incident light. If another

signal is present at this longer wavelength, the SRS light will amplify it and

the pump wavelength signal will decrease in power. Consequently, SRS can

106

severely limit the performance of a multichannel optical communication

system by transferring energy from short-wavelength channels to neighboring

higher-wavelength channels. This is a broadband effect that can occur in both

directions.

The Raman gain spectrum is fairly broad and the peak of the gain is

centered about 13 THz below the frequency of the pump signal used. Using

pumps around 1460-1480 nm provides Raman gain in the 1550-1600 nm

window. A few key attributes distinguish Fiber Raman amplifiers from

EDFAs. Unlike EDFAs, Raman effect can be used to provide gain at any

wavelength. Raman amplification relies on simply pumping the same silica

fiber used for transmitting the data signals, so it can be used to produce

lumped or discrete amplifiers, as well as distributed amplifiers. In the lumped

case, the fiber Raman amplifier consists of a sufficiently long spool of fiber

along with the appropriate pump lasers in a package. In the distributed case,

the fiber can simply be the fiber span of interest with the pump attached to

one end of the span. Fiber Raman amplifiers are used in addition to EDFAs in

many ultra long haul systems.

3.4 FIBER BRILLOUIN AMPLIFIERS

The operating principle behind fiber Brillouin amplifiers is

essentially the same as for fiber amplifiers except that the optical gain is

provided by Stimulated Brillouin Scattering (SBS). SBS arises when light

waves scatter from acoustic waves. The resultant scattered wave propagates

principally in the backward direction in single mode fibers. This backward

scattered light experiences gain from the forward propagating signals, which

leads to depletion of the signal power. Fiber Brillouin amplifiers are also

pumped optically and a part of the pump power is transferred to the signal.

SBS differs from SRS in three important aspects which affect the operation of

107

the fiber Brillouin amplifiers considerably: amplification occurs only when

the signal beam propagates in a direction opposite to that of the pump beam,

the stokes shift for SBS is smaller (~10 GHz) by three orders of magnitude

compared with that of SRS and depends on the pump frequency and Brillouin

gain spectrum is extremely narrow.

Fiber Brillouin amplifiers can be used as a preamplifier to improve

the receiver sensitivity. The narrow bandwidth of fiber Brillouin amplifiers

can be used to advantage in coherent and mutichannel light wave systems.

3.5 SEMICONDUCTOR OPTICAL AMPLIFIERS

Semiconductor Optical Amplifiers (SOAs) actually preceded

EDFAs, although they are not as good as EDFAs for use as amplifiers.

However, they are finding other applications in switches and wavelength

converter devices. The two major types of SOAs are the resonant, Fabry-

Perot amplifier (FPA) and the nonresonant, traveling-wave amplifier (TWA).

TWAs are more widely used than FPAs because they have a larger optical

bandwidth, high saturation power, and low polarization sensitivity.

Population inversion in an SOA is achieved by forward biasing a pn

junction. In practice, a simple pn junction is not used, but a thin layer of a

different semiconductor material is sandwiched between the p-type and n-type

regions. Such a device is called a heterostructure. The SOAs have a more

rapid gain response, which is on the order of 1 ps to 0.1 ns. This results in

both advantages and limitations. The advantage is that SOAs can be used for

implementation of both switching and signal processing functions in optical

networks. The limitation is that the rapid carrier response causes the gain at a

particular wavelength to fluctuate with the signal rates up to several Gb/s.

108

Since this affects the overall gain, the signal gain at other wavelengths also

fluctuates. This gives rise to crosstalk effects when a broad spectrum of

wavelengths must be amplified.

In this work, only EDFAs are considered. Amplifiers are not

perfect devices. They introduce noise, and this noise accumulates as the signal

passes through multiple amplifiers along its path due to the analog nature of

the amplifier. The gain of the amplifier depends on the total input power. For

high input powers, the EDFA tend to saturate and the gain drops. Further, the

gain of the EDFA is not flat over the entire passband, thus some channels see

more gain than others. Ideally, a sufficiently high output power is required to

meet the needs of the network applications. The gain should be flat over the

operating wavelength range, and the gain should also be insensitive to

variations in the input power of the signal.

In chapter 2, it is assumed that the EDFAs always deliver the

desired small signal gain irrespective of the input signal power or wavelength.

However, the ASE noise generated in the EDFAs has been taken into account.

In this chapter, ASE noise, EDFA gain saturation and gain dispersion have

been considered during lightpath establishment for a connection request. This

requires the online computation of the gain of the EDFAs used in the network.

This chapter also presents a fast and accurate approach to obtain the signal

gain of EDFAs. EDFA gain dynamics originating from adding and dropping

optical signals at EDFAs (Sun et al 1996 and Sun et al 1997) is not

considered. Ramamurthy et al (1999), Deng et al (2004) and Yang and

Ramamurthy et al (2005) have also not considered EDFA gain dynamics in

their work. However, Tian and Kinoshita (2003) have demonstrated that with

the combination of electrical feedforward and feedback automatic gain control

(AGC), the power excursion caused by fast channel adding/dropping

processes can be minimized.

109

3.6 GAIN MODEL OF EDFA

The gain models proposed by Desurvire (1989) and Morkel and

Laming (1989) are based on numerical integration and also involve

measurement of amplifier parameters which is time consuming and also

inaccurate. Saleh et al (1990) proposed a gain model for EDFAs which is

based on a single transcendental equation. In this work, the model proposed

by Saleh et al (1990) is used to obtain the gain of the Erbium doped fiber

amplifiers. Saleh et al (1990) describe an analytic method for fully

characterizing the gain of an EDFA that is based on easily measured

monochromatic absorption data. The analytic expressions, which involve the

solution of one transcendental equation, can predict the signal gains and pump

absorption in an amplifier containing an arbitrary number of pumps and

signals from arbitrary directions.

The derivation of the Saleh equation is presented below. Saleh et al

(1990) considered a two level system with an arbitrary number of input

beams. A two level EDFA uses the 2/15

4 I and the 2/13

4 I levels of the Erbium

atom. When embedded in glass fiber, these fine structure levels are Stark

broadened. This allows pumping at a wavelength near 1480 nm to provide

gain in the 1525-1560 nm range. They have considered an amplifier of length

L with a density of active atoms ρ within an active volume of cross-sectional

area A. An arbitrary number N of light beams of wavelength λk and power

Pk(z, t) travel through the amplifier in a direction indicated by uk. Signal and

pump beams are treated identically, i.e., a given beam can be either a pump or

a signal. For beams entering at z = 0, have uk= 1, while beams entering at

z = L have uk = -1. In this derivation, power is expressed in units of photons

per unit time i.e., the actual power normalized by the photon energy. The rate

equation for the fractional population of ions in the upper state N2(z, t) is

110

N

j2 2j

j 1

P (z, t)N (z, t) N (z, t) 1u

z A z=

∂∂= − −

∂ τ ρ ∂∑ (3.2)

where τ is the spontaneous lifetime of ions at the upper energy level. The

fractional population of the lower state N1(z, t) is obtained from N1(z, t) +

N2(z, t) = 1. The light in the fiber is subject to absorption or stimulated

emission at rates that depend on wavelength. The cross section for stimulated

emission and absorption at wavelength λk are ekσ and a

kσ , respectively. The

change of power in the kth

beam is described by

( )[ ] )t,z(P)t,z(Nuz

)t,z(Pk

a

k2

a

k

e

kkkk σ−σ+σΓρ=∂

∂ (3.3)

where Γk is the confinement factor at wavelength λk.

Under steady state conditions, the population densities and optical

power are independent of time. Then remembering that 2

ku = 1 and obtaining

N2(z, t) by setting ( )0

t

t,zN2 =∂

∂ in equation (3.2), equation (3.3) can be

rewritten as

N

jkk k jIS

j 1k K

dP (z)u 1dP u

P (z) P dz=

= − α +

∑ (3.4)

where αk and ISkP are the absorption constant and intrinsic saturation power of

the kth

beam respectively and are given by

akkk σΓρ=α

( )τσ+σΓ

=ak

ekk

ISk

AP (3.5)

111

Finally equation (3.4) is integrated to obtain

−= α−

IS

k

outinLin

k

out

kP

PPexpePP k (3.6)

where inkP and out

kP are the input and output powers of the kth

beam and where

Pin = ∑=

N

1j

injP

Pout = ∑=

N

1j

outjP (3.7)

are the total input and output powers, respectively. For uk = 1, inkP and out

kP

are the powers at z = 0 and z = L, respectively, whereas for uk = -1, the

locations of the input and output beams are reversed. Equation (3.6) can be

reduced to a single transcendental equation by summing it over k:

Pout = ∑=

−N

1k

PB

koutkeA

Ak = ISkink PPLin

k eePα− ,

Bk = IS

kP

1 (3.8)

Equation (3.8) is referred to as the Saleh equation. If αk, ISkP and the input

powers are known, this equation can be solved for Pout. Once solved, the result

can be inserted into equation (3.6) to compute directly the output power outkP

at each wavelength. The gain at wavelength λk can then be calculated since

inkP and out

kP are known.

Pin =

Pout =

112

The above results are applicable even when the amplifier is pumped

in the 980 nm region. In this case, ekσ = 0 in equation (3.5).

3.7 FAST AND ACCURATE SIGNAL GAIN ESTIMATION

Deng et al (2004) use the approximate EDFA gain model given by

equation (3.1). Ramamurthy et al (1999) also use an approximate EDFA gain

model. In this work, EDFA gain is obtained using the Saleh model. The gain

of an EDFA can be obtained by solving the transcendental equation (3.8)

mentioned above. It can be solved using standard numerical techniques like

the bisection method and the Newton-Raphson method. The above mentioned

numerical techniques require an initial guess for the root. Newton-Raphson

method is the method of choice as it converges quadratically near the root

when provided with a proper guess for the root. Equation (3.8) involves

exponential quantities, and hence arithmetic overflow can occur at high input

power if the initial guess for the root is far away from the actual root (Roudas

et al 1999). Therefore a proper initial guess for the root is essential.

It is observed, while solving the transcendental equation (3.8), that

an initial guess of Pin/xini (where xini ≥ 1) for the root enables the numerical

method to converge to the solution without any arithmetic overflow error. For

the EDFA reported by Bononi and Rusch (1998), Newton-Raphson method

converges to the desired root quickly when xini takes a value of 1.01. It is

further observed that for the above EDFA, the actual root can be expressed as

Pin/xfin, where xfin varies from 1.01 to 1.08. The above methods hold good for

multisignal propagation also. This approach is used in this work to obtain the

EDFA gain when pumped at 1480 nm or 980 nm for single and multi-

wavelength propagation. This guess converges to the actual root without any

overflow errors.

113

The above approach has been used to obtain the gain of the EDFA

reported by Bononi and Rusch (1998). The amplifier is pumped at 980 nm.

The pump power is 18.4 dBm. The signal wavelengths are 1552.4 nm and

1557.9 nm. The absorption constants are [0.257, 0.145, 0.125] m-1

and the

intrinsic saturation power are [0.44, 0.197, 0.214] mW at [980, 1552.4,

1557.9] nm, respectively. The fluorescence life time is 10.5 ms. The length of

the EDFA can be adjusted to obtain the desired gain to suit the specific

requirements.

Figure 3.4 shows the EDFA gain for the two different signals

mentioned above assuming that only one signal propagates through the

amplifier at a time. The length of the EDFA is 18 m. It can be seen that

different wavelengths are amplified differently. This is referred to as gain

dispersion. To obtain the signal gain for each input signal power, the initial

guess for the root is selected as Pin/xini, where Pin is the sum of the signal and

pump power. The value of xini is chosen as 1.01. Table 3.1 presents various

power of the input signal at 1552.4 nm and the corresponding gain obtained.

It also indicates the values of xfin that resulted when the numerical method

converges to the actual root.

114

0

2

4

6

8

10

12

14

16

18

-40 -30 -20 -10 0 10 20

Input signal power (dBm)

Gai

n (

dB

)

1557.9nm

1552.4nm

Figure 3.4 Signal Gain versus input Signal Power at Wavelengths

of 1552.4 nm and 1557.9 nm

Table 3.1 Number of iterations required by Newton-Raphson method

to converge to the desired root for different input Signal

power. Pump power is 18.4 dBm, Signal wavelength is

1552.4 nm and guess for the root (i.e., Pout) is Pin/xini with xini

= 1.01

Input signal

power (dBm)

Signal gain

(dB) xfin

Number of

iterations required

-40 16.809 1.030112 20

-30 16.803 1.030105 20

-20 16.743 1.030032 21

-10 16.197 1.029368 23

0 13.101 1.025444 18

10 6.527 1.015328 6

115

Table 3.2 presents the various input power and the corresponding

gain obtained for the input signal at 1557.9 nm. It also indicates the value of

xfin that resulted when the numerical method has converges to the actual root.

Table 3.2 Number of iterations required by Newton-Raphson Method to

converge to the desired root for different input signal power.

Pump power is 18.4 dBm, Signal wavelength is 1557.9 nm and

guess for the root (i.e., Pout) is Pin/xini with xini = 1.01

Input signal

power (dBm)

Signal

gain (dB) xfin

Number of

iterations required

-40 16.04 1.030112 20

-30 16.04 1.030106 20

-20 15.99 1.030044 20

-10 15.56 1.029464 20

0 12.874 1.02571 16

10 6.527 1.015236 6

It can be observed that xfin varies from 1.01 to 1.04. As the

amplifier approaches saturation, xfin also approaches the lower bound of 1.01.

Figure 3.5 shows the signal gain for the two signals when they propagate

through the amplifier together.

116

0

2

4

6

8

10

12

14

16

18

-40 -30 -20 -10 0 10 20

Input signal power (dBm)

Sig

nal

gai

n (

dB

)

1552.4nm

1557.9nm

Figure 3.5 Signal Gain versus Signal Power when Signals at 1552.4 nm

and 1557.9 nm co-propagate through the Amplifier

Table 3.3 shows the input powers of the two signals and the

corresponding signal gains obtained when the two signals are traveling

through the amplifier together. The pump power is 18.4 dBm. The initial

guess for the root is Pin/xini, where xini =1.01.

It can be observed that even in the case of two signals traveling

through the amplifier, the value of xfin ranges from 1.01 to 1.04. This is valid

even if the length of the EDFA is altered. However, if the pump power is

changed, the above range for xfin does not hold good.

Table 3.4 shows the number of iteration required for different initial

guesses for the root. Two values of input power are considered. The length of

the EDFA is 18 m. Signal wavelength is 1552.4 nm and the pump power is

18.4 dBm.

117

Table 3.3 Number of iterations required by Newton-Raphson method

when the signals at 1552.4 nm and 1557.9 nm propagate

together in the amplifier. Pump power is 18.4 dBm and

guess for the root (i.e., Pout) is Pin/xini, with xini = 1.01

Input

power of

the signal

at 1552.4

nm (dBm)

Input

power of

the signal

at 1557.9

nm (dBm)

Gain for

the signal

at 1552.4

nm (dB)

Gain for

the signal

at

1557.9nm

(dB)

xfin

Number

of

iterations

required

-40 -40 16.808 16.045 1.030111 21

-30 -30 16.797 16.034 1.030098 21

-20 -20 16.688 15.934 1.029964 21

-10 -10 15.756 15.074 1.028810 21

0 0 11.449 11.129 1.023151 14

10 10 4.437 4.396 1.011357 4

Table 3.4 Number of iterations required for different guesses of the root

Input signal power (dBm) Number of iterations required

Guess for the root

Pin/1.01 Pin/10.01 10Pin Pin/2

-40 dBm 20 188 211 99

0 dBm 18 202 225 111

118

It can be seen that an initial guess of Pin/xini with xini =1.01 requires

the minimum number of iterations compared with the other initial guesses.

Hence this approach is called fast and accurate.

The guess of Pin/xini is found to hold true even for 1480 nm

pumping. Saleh et al (1990) consider an EDFA pumped at 1480 nm to

amplify a 1550 nm signal. The length of the EDFA is 387 cm. The absorption

constants are [0.792, 0.876] m-1

and the intrinsic saturation power being

[0.549, 0.272] mW at [1480, 1550] nm, respectively. The general guess of

Pin /xini, with xini =1.01 has been used in the Newton-Raphson method and the

results obtained for different signal/pump power matches well with the results

of Saleh et al (1990) and are shown in Figure 3.6. When the signal power is

0 dBm, xfin varies from 1.140 to 2.7943 for the pump powers shown in

Figure 3.6. For the signal power of –26.7 dBm, xfin varies from 1.188 to

26.288.

-20

-15

-10

-5

0

5

10

15

-40 -30 -20 -10 0 10 20

Pump power (dBm)

Gai

n (

dB

)

Signal power = 0dBm

Signal power = -26.7dBm

Figure 3.6 Signal Gain versus input Signal Power using 1480 nm

pumping

119

3.8 IMPACT OF EDFA GAIN SATURATION ON THE

BLOCKING PERFORMANCE OF 15-NODE NETWORK OF

INTERCONNECTED RINGS

In this section, the impact of EDFA gain saturation on the blocking

probability of 15-node network of interconnected rings is studied. The

network topology is shown in Figure 3.7. Table 3.5 presents the system

parameters used in this study. The absorption and emission cross sections at

the signal wavelength (λ) are determined using the curve fitting technique

(Desurvire 1994). First the normalized experimental absorption and emission

line shapes are obtained by a linear superposition of eight Gaussian line

shapes. Ramamurthy et al (1999) have used only a single line shape in their

gain model. The normalized absorption cross section line shapes at various

signal wavelengths are obtained using (Desurvire 1994)

λ∆

λ−λ=λ ∑

=2

ia

2

ia8

1iia

)(2log4expa)(Ia (3.9)

where aai, λai and ∆λai refer to peak intensity, center wavelength and line width

of the ith

Gaussian line shape respectively and are presented in Table 3.6.

Similarly, the normalized emission cross section line shape at various signal

wavelengths have been obtained using

λ∆

λ−λ=λ ∑

=2

ie

2

ie8

1iie

)(2log4expa)(Ie (3.10)

The values of aei, λei and ∆λei are presented in Table 3.7. The parameters listed

in Table 3.6 and 3.7 have been obtained from Desurvire (1994). The results

obtained using equation (3.9) is normalized to unity and hence it is multiplied

120

by 5.3×10-25

( peak absorption cross section value of the Erbium doped fiber)

to obtain the actual absorption cross section at the signal wavelengths. The

results obtained using equation (3.10) is multiplied by 5.8×10-25

(peak

emission cross section value of the Erbium doped fiber) to obtain the actual

emission cross section at the signal wavelengths (Desurvire 1994).

Absorption constants and the intrinsic saturation power at various

wavelengths are obtained with the help of the EDFA parameters listed in

Table 3.5 and using equation (3.5). Table 3.8 lists the absorption constants

and the intrinsic saturation power for the various wavelengths used in this

work.

Figure 3.7 15-node Network of interconnected Rings

9

10

11

7

8

2

5

15 6

14

3

4

1

13

12

121

Table 3.5 System Parameters and their Values used to Study the

Impact of EDFA Gain Saturation

Parameters Values

Multiplexer loss (Lmx) 4 dB

Demultiplexer loss (Ldm) 4 dB

Switch characteristic loss (Ls) 1 dB

Waveguide/fiber coupling loss (Lw) 1 dB

Switch loss (Lsw)

(for a P × P switch)

(2log2P)Ls+ 4 Lw

Tap loss (Ltap) 1 dB

Fiber loss (Lf) 0.2 dB/km

Desired input EDFA gain (Gin) 22 dB

Desired output EDFA gain (Gout) 18 dB at nodes 1, 6, 7 & 13

16 dB, elsewhere

Pump wavelength (λp ) 980 nm

Pump input power ( Ppin

) 18.4 dBm

Core radius of EDFA ( r) 1.277 µm

Core area of EDFA (A) 5.1295 × 10-12

m2

Overlap factor of EDFA (Γ) 0.5

Florescence time of EDFA (τ) 10.5 ms

Ion density of EDFA (ρ) 1.1413 × 1024

ions/m3

Number of wavelengths 8

Wavelength Spacing 100 GHz

Bit rate per channel 2.5 Gb/s

Optical Bandwidth ( B0) 36 GHz

Electrical bandwidth ( Be) 2 GHz

ASE factor (nsp) 1.5

Bandwidth

currentthermalRMS ,

thη 5.3 × 10-24

Hz

A

122

Table 3.6 Parameters used to find Normalized Absorption Line Shape

i aai λai (nm) ∆λai

(nm)

1 0.03 1440 40

2 0.31 1482 50

3 0.17 1492 29

4 0.37 1515 29

5 0.74 1530 16.5

6 0.28 1544 17

7 0.30 1555 25

8 0.07 1570 35

Table 3.7 Parameters used to find Normalized Emission Line Shape

i aei λei (nm) ∆λei

(nm)

1 0.06 1470 50

2 0.16 1500 40

3 0.30 1520 25

4 0.37 1530 12.5

5 0.38 1542.5 13

6 0.49 1556 22

7 0.20 1575 45

8 0.06 1600 60

123

Table 3.8 Absorption Constant and Intrinsic Saturation Power at

various Wavelengths

Wavelength

( nm)

Intrinsic saturation

power ( mW)

Absorption

constant (m-1

)

1546.99 0.182 0.168

1547.80 0.185 0.165

1548.60 0.187 0.162

1549.40 0.190 0.158

1550.20 0.193 0.154

1551.00 0.195 0.151

1551.80 0.197 0.147

1552.60 0.199 0.143

980 0.44 0.257

Figure 3.8 shows the gain characteristics of the input EDFA used in

this work at two different wavelengths. It can be seen that gain dispersion is

negligible. Gain dispersion is crucial in WDM systems with cascaded

amplifiers.

124

0

5

10

15

20

25

-50 -40 -30 -20 -10 0 10 20

Signal power (dBm)

Gai

n (

dB

)

Signal wavelength = 1551 nm

Signal wavelength = 1547.80 nm

Figure 3.8 Gain versus input Signal Power for the input EDFA at

Signal Wavelengths of 1547.8 nm and 1551 nm

Figure 3.9 shows the gain characteristics of the output EDFA at

1546.99 nm. EDFA of 20 m length is used at the nodes 1, 6, 7 and 13.

EDFA of 18 m length is used at the remaining nodes.

0

5

10

15

20

-40 -30 -20 -10 0 10 20

Signal power( dBm)

Gai

n (

dB

)

L= 20 m L= 18 m

Figure 3.9 Gain versus input Power for the output EDFA at a Signal

Wavelength of 1546.99 nm

125

Figure 3.10 shows the gain characteristics of the output EDFA at

1552.6 nm for length L =20 m and L =18 m.

0

2

4

6

8

10

12

14

16

18

20

-40 -30 -20 -10 0 10 20

Signal power (dBm)

Gai

n (

dB

)

L = 18m

L = 20m

Figure 3.10 Gain versus input Power for the output EDFA at a Signal

Wavelength of 1552.6 nm

Calls arrive at the network following a Poisson process. The traffic

over the entire network is uniformly distributed. The call durations are

exponentially distributed with a mean of 1. The number of wavelengths on

each link is 8 and they are: [1546.99, 1547.80, 1548.60, 1549.40, 1550.20,

1551.00, 1551.80 and 1552.60] nm. The signal power per channel is assumed

to be 1 mW at the transmitter. The overall gain of the wavelength routing

node compensates for the transmission losses incurred between and within

each wavelength routing node, so no in-line EDFAs are used. External

intensity modulation is assumed at the transmitters. The discrete event

simulation used in this chapter is similar to that discussed in chapter 2. For

every dynamically arriving connection request, a route and a free wavelength

is determined. The BER at the receiver for this lightpath is estimated. The

estimation of BER considers the various transmission impairments. The gain

126

of the EDFAs present in the signal route is determined using the fast and

accurate approach explained in section 3.7. In addition to the impairments

considered in chapter 2, the impact of EDFA gain saturation is also taken into

account to estimate the BER. Fixed routing of calls and first-fit wavelength

assignment is adopted. A worst case situation of mark state is assumed in all

the propagating channels. This ensures that the EDFA gain saturation effect

is severe. Further, worst case assumption of mark in all the channels will

increase the in band crosstalk in the network.

Figure 3.11 shows the impact of a realistic EDFA on the blocking

probability performance of the network. An EDFA which suffers gain

saturation is referred to as a realistic EDFA. It can be seen from the figure that

a realistic EDFA blocks more calls when compared with an ideal EDFA due

to gain saturation even in the absence of in-band crosstalk. Gain saturation

leads to a reduction of the small signal gain which an EDFA is supposed to

provide. Inadequate gain leads to reduced signal power and results in

increased BER. The impact of gain saturation and demux/mux in-band

crosstalk on blocking probability performance is also shown in Figure 3.11.

Switch crosstalk ratio is set to 0 and filter crosstalk ratio is set to -25 dB. It is

observed that the presence of gain saturation and in-band crosstalk results in

more number of calls being blocked compared to the case of an ideal EDFA

with no in-band crosstalk.

127

0

0.1

0.2

0.3

0.4

0.5

0 10 20 30 40 50 60

Network load (Erlangs)

Blo

ckin

g p

robab

ilit

y

Realisitic EDFA & No crosstalk

Ideal EDFA & no crosstalk

Realisitic EDFA and demux/mux inband crosstalk

Figure 3.11 Impact of Realistic EDFA on 15-node Network of

interconnected Rings

It can be concluded from the above discussions that the gain

saturation in EDFAs can have severe impact on the received signal quality. It

can lead to unsatisfactory BER at the receiver of the destination node and

hence leads to increased blocking probability.

3.9 CONCLUSIONS

Erbium doped fiber amplifiers are not ideal. Gain saturation, gain

dispersion and ASE noise are some of the imperfections of the EDFA which

can affect the signal quality over a lightpath. Evaluation of the on-line BER of

a lightpath requires the determination of the gain of an EDFA. The gain of the

EDFA is governed by a transcendental equation which can be solved using

numerical techniques like Newton-Raphson method. This technique requires

an initial guess for the root. If the guess is far from the actual root, overflow

errors can occur at high input powers. In this chapter, a guideline to choose

128

the initial guess for the root is suggested. This is validated for the EDFA

appearing in the literatures. This guess helps to obtain the EDFA gain with

minimum iterations. It is also observed that EDFA gain saturation can have a

severe impact on the BER of a lightpath.

The use of a realistic EDFA accounting for gain saturation effects

can lead to an increase in the blocking probability of the network by 0.025

even in the absence of crosstalk. In other words, the number of calls blocked

increases by 2.5% at a traffic load of 50 Erlangs when a realistic EDFA is

used. When the crosstalk effect is included it is seen that there is a drastic

change in the performance. The number of calls blocked increases by almost

30% at a traffic load of 50 Erlangs. It can be concluded that the gain

saturation effects has a large impact on the signal quality and BER and a fast

and accurate model for online calculation is essential in the process of

dynamic RWA.

In chapters 2 and 3, crosstalk introduced by the switches and

demux/mux are considered. Fiber nonlinearities are not considered. Among

the nonlinearities, stimulated Raman scattering is regarded as the ultimate

power limiting phenomenon in multiplexed optical communications systems.

If two or more signals at different wavelengths are injected into a fiber, SRS

causes power to be transferred from the lower wavelength channels to higher

wavelength channels as long as the wavelength difference is within the

bandwidth of the Raman gain. This effect is called as SRS-induced crosstalk.

In chapter 4, the impact of SRS-induced crosstalk is considered while

establishing lightpaths. The combined effect of all the above mentioned

crosstalk effects is also studied.