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Novel Numerical Method to Model Multichannel Erbium-Doped Fiber Amplifier

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Fiber and Integrated Optics
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Novel Numerical Method to Model Multichannel Erbium Doped FiberAmplifier
Vishnu Priye
a
; Dharmendra K. Singh
b
; Subhash C. Arya
aa
Electronics & Instrumentation Department, Indian School of Mines, Dhanbad, India
b
Electronics &Communication Department, BIT Sindri, Dhanbad, India
To cite this Article
Priye, Vishnu , Singh, Dharmendra K. and Arya, Subhash C.(2006) 'Novel Numerical Method to ModelMultichannel Erbium-Doped Fiber Amplifier', Fiber and Integrated Optics, 25: 5, 375 — 385
To link to this Article: DOI:
10.1080/01468030600817134
URL:
http://dx.doi.org/10.1080/01468030600817134
Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdfThis article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.
Fiber and Integrated Optics
, 25:375–385, 2006Copyright © Taylor & Francis Group, LLCISSN: 0146-8030 print/1096-4681 onlineDOI: 10.1080/01468030600817134
Novel Numerical Method to Model MultichannelErbium-Doped Fiber Ampliﬁer
VISHNU PRIYE
Electronics & Instrumentation DepartmentIndian School of MinesDhanbad, India
DHARMENDRA K. SINGH
Electronics & Communication DepartmentBIT SindriDhanbad, India
SUBHASH C. ARYA
Electronics & Instrumentation DepartmentIndian School of MinesDhanbad, India
We propose a novel numerical method based on genetic algorithm (GA) to solve theevolution equations that are the basis of modeling multichannel erbium-doped ﬁber ampliﬁers (EDFAs). The evolution equations are ﬁrst transformed from an iteration problem to an optimization problem. The unknown parameter related to backward ampliﬁed spontaneous emission (BASE) is coded into a string of bits based on their probable lower and upper limits. By deﬁning proper random variation operators and cost function, the coupled evolution equations are optimized using GA for different values of erbium-doped ﬁber parameters. It is observed that GA is an efﬁcient method for analyzing the EDFA characteristics in the entire C-band.
Keywords
genetic algorithm, evolution algorithm, EDFA modeling, optical ampli-ﬁers
Introduction
The fundamental and intrinsic advantages of the erbium-doped ﬁber ampliﬁers (EDFAs)like low noise, polarization insensitivity, and cross-talk immunity have made their deploy-ment in the actual communication system a reality [1, 2]. Due to the large bandwidth of ampliﬁcation
∼
40 nm each in C-band and L-band [3–5], terabits/s data transmission has
been achieved using dense wavelength division multiplexed (DWDM) systems. In the
Received 13 September 2005; accepted 17 April 2006.Address correspondence to Vishnu Priye, Electronics & Instrumentation Department, IndianSchool of Mines, Dhanbad, 826 004, India. E-mail: vpriye@hotmail.com
375
D o w nl o ad ed B y : [ I N F L I B N E T I ndi a O rd e r] A t : 04 :44 7 J ul y 2010
376 V. Priye et al.
present research scenario, the forward and backward ampliﬁed spontaneous emissions(ASE) have assumed new importance not because they decide the achievable noise ﬁgure[6–9] of the multichannel ampliﬁers, but because they also can be used to pump the EDF
along with 980-nm lasers to achieve L-band ampliﬁers [10–12].
The complete evolution equations that describe the length dependence of pump,signal, FASE (forward ASE), and BASE (backward ASE) represents a two-boundaryvalue problem [7–9] that requires iterative adjustment to the solution as initial conditions
for the BASE is not known. The exact numerical method to solve the evolution equationsis known as the relaxation method [7] and henceforth will be referred to in this article asthe conventional method. In the relaxation method as discussed in Desurvire [7], the ﬁrstintegration over the entire EDF length is done without considering BASE by setting bothFASE and BASE as zero at the input end (
z
=
0) of the EDF. The second integration isdone in the reverse direction starting from the output end (
z
=
l
) to the input end to obtaina set of quasi solutions for pump signal, FASE, and BASE. These quasi solutions are usedto perform back-and-forth integration until a steady state solution is reached. This methodis rigorous but poses a convergence problem and consumes a lot of computation time,when the total C-band spectrum (1525–1565 nm) is subdivided into wavelength strips of 1 nm to incorporate multichannel transmission. To overcome the problem, large numbersof analytical and approximate methods [5, 12–15] have been reported. Recently, black-
box models have been proposed in which gain and noise ﬁgures of EDFA are representedby empirical formulae based on some experimentally determined parameters [15, 16].
In this article, we report a novel numerical method based on genetic algorithm (GA)to solve the evolution equations characterizing the propagation of pump, signal, FASE,and BASE in EDF [17, 18]. GA, also known as evolutionary algorithm, is basically asearch algorithm based on the mechanics of natural selection and natural genetics [19, 20]and recently has been successfully applied in other areas of optical communication tooptimize the synthesis of ﬁber gratings with advance characteristics [21], long-periodgrating ﬁlters [22], multi-layer coatings at oblique light incidence [23], and spectral
equalization of EDFA [24]. We have ﬁrst coded the values of BASE at the input of theﬁber that fall within the expected range. Next we have deﬁned a cost function basedon the behavior of BASE and developed our algorithm based on the evolution equationsto minimize the cost function. The present method yields accurate estimation of pump,FASE, and BASE in the C-band of EDF and can be applied to estimate the BASE andmultichannel ampliﬁcation in EDFA.
Formulation
The general evolution equations based on the three-level atomic rate equations that de-scribe the evolution of co-directional pump power at pump wavelength, forward andbackward ASE, and signals at discrete wavelengths within the bandwidth of the EDFAare described in detail in Desurvire [7]. For the sake of continuity they are reproducedbelow:
dqdz
=−
qρ
0
σ
pa
b
0
ρ(r)ρ
0
ψ
p
(r)AB
2
πrdr
(1)
dp
±
k
dz
=±
ρ
0
σ
a
(λ
sk
)
b
0
ρ(r)ρ
0
ψ
sk
(r)
C
−
DB
2
πrdr
(2)
D o w nl o ad ed B y : [ I N F L I B N E T I ndi a O rd e r] A t : 04 :44 7 J ul y 2010
Multichannel Erbium-Doped Fiber Ampliﬁer 377
where
A
=
1
+
j
η
sj
1
+
η
sj
(p
+
j
+
p
−
j
)ψ
sj
(r)
(3)
B
=
1
+
qψ
p
(r)
+
j
(p
+
j
+
p
−
j
)ψ
sj
(r)
(4)
C
=
η
sk
qψ
p
(r)
1
+
η
p
+
j
(p
+
j
+
p
−
j
1
+
η
sj
ψ
sj
(r)
[
p
±
k
+
2
p
o
k
]
(5)
D
=
1
+
j
η
sj
1
+
η
sj
(p
+
j
+
p
−
j
)ψ
sj
(r)
+
η
p
1
+
η
p
qψ
p
(r)
p
±
k
(6)The different notations are the following:
q
=
P
p
(z)/P
sat
(λ
p
)
=
normalized pump power;
p
±
k
=
P
s
(z)/P
sat
(λ
s
)
=
normalized forward and backward propagating signal and ASE power;
P
p
(z)
=
pump power;
P
s
(z)
=
signal or ASE power;
P
sat
(
λ
s
or
λ
s
)
=
saturation power at pump or signal wavelength;
p
ok
=
equivalent input noise power;
ρ(r)
=
radial distribution of Erbium ions;
σ
a
=
absorption cross-section at pump and signal wavelength;
ψ
p
or
ψ
s
=
modal ﬁeld at pump or signal wavelength;
η
p
or
η
j
=
excitation efﬁciency at pump wavelength or signal at a wavelengthcorresponding to
j
.The terms
A
,
B
,
C
, and
D
come into the evolution equation by considering the rateequations of pump power and signal powers in the total emission and absorption band of EDFA. The expressions of
A
and
B
given in Eqs. (3) and (4) contain terms that representthe evolution of pump and forward and backward signal powers. In the summation,the value of
j
corresponds to the number of individual wavelengths considered. Forexample,
j
=
25 implies that the C-band is divided into 25 equally spaced wavelengths.The expression
C
depicted in Eq. (5) incorporates equivalent input noise power andnormalized forward and backward propagating signal and ASE power besides pumppower and other constants. Similarly, Eq. (6) for
D
shows the multiplication of normalizedforward and backward propagating signal and ASE power and summation of pump andforward as well as backward signal power. When the transition is degenerate, there exist
D o w nl o ad ed B y : [ I N F L I B N E T I ndi a O rd e r] A t : 04 :44 7 J ul y 2010
378 V. Priye et al.
Table 1
EDF parameters used in this articleErbiumconcentrationNumerical aperture(NA)CoreradiusErbium-dopedradius1
.
15
×
10
25
ions/m
3
0.61 1.64
µ
m 1.64
µ
m
Table 2
Data required in solving the rate and propagation equationsScanningwavelengthrangesScanningwavelength step(
λ
)Central signalwavelength(
λ
c
)Er
3
+
metastablelifetime(
τ
)Pumpwavelength1525–1565 nm 1 nm 1550 nm 10 ms 980 nm
g
1
and
g
2
sublevel in the lower and upper states; then we have relation
g
1
σ
a
(λ)
=
g
2
σ
e
(λ)
or
σ
e
(λ)σ
a
(λ)
=
g
1
g
2
=
η(λ)η
s
=
σ
es
σ
as
>σ
ep
σ
ap
=
η
p
This expression is used in gain coefﬁcient for multilevel laser systems. The parameter
η(λ)
is of central importance in the modeling of EDFAs. Evaluation equations based onthe three-level atomic rate equation that requires 980-nm pump wavelength has
σ
ap
=
2
.
17
×
10
−
25
and
σ
ep
=
0
.
0. This relationship shows that if the ratio of the stimulatedemission to absorption at the signal wavelength is not greater than that at the pumpingwavelength, a positive signal gain coefﬁcient can never be achieved. For Er
3
+
glasspumped as a three-level pumping scheme we have
η
ep
=
0; hence, the condition isalways veriﬁed.The multichannel wavelengths are chosen by the value of
j
. For example, if
j
=
40,the entire C-band (1525–1565 nm) is divided into 40 channels separated by 1 nm. Inthe conventional method, termed “relaxation method” in Desurvire [7], the differentialequations are solved using the fourth-order Runge Kutta method and making iterativeforward and backward integration with the initial condition that BASE and FASE arezero at ﬁber input. The parameters and data required for solving rates and propagationequations used in the present article are given in Tables 1 and 2, respectively.
Genetic Algorithm Modeling
Genetic algorithm (GA) is adaptive heuristic search algorithm based on the evolutionaryideas of natural selection and genetics. As such, they represent an intelligent exploitation
D o w nl o ad ed B y : [ I N F L I B N E T I ndi a O rd e r] A t : 04 :44 7 J ul y 2010

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