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Weighted Undepleted Pump Model for
Broadband Counter-Pumped Raman Fiber
AmplifiersJoo M. Ferreira, Rogrio Nogueira, Paulo Monteiro, and Armando N. Pinto
AbstractWe develop a low complexity computational
model for the gain profile and amplified spontaneous emission
noise for broadband counter-pumped Raman fiber amplifiers.
The proposed model is based on two adjustment parameters
used to account for the interactions between the pumps. The
obtained results show a good agreement between experimental
measurements and detailed numerical simulations, for differ-
ent combinations of pump wavelengths and pump powers, with
a processing time several times lower than the time taken by adetailed numerical model.
Index TermsCounter-pump; Gain; Optical amplifiers; Ra-
man amplifiers; Raman scattering.
I. INTRODUCTION
As optical networks become more complex, new tools and
protocols such as network planning tools or routing
wavelength assignment (RWA) protocols have to be developed
and implemented in order to help operators to plan, optimize
and operate the networks [1]. However, in order to developand implement such tools and protocols, it is necessary to
estimate correctly and in a fast way the figures of merit of
network components, like the gain and noise figures of Raman
fiber amplifiers (RFAs) [2,3]. Nowadays, transmission systems
can have more than 80 signals and tens of other resources
like amplifiers and transponders. Therefore, it is necessary to
develop and implement models with low computational effort
in order provide reliable results in a useful time interval.
RFAs are suitable for long and ultra-long haul amplification
due to their low noise figures [4]. When modeling a
transmission link with Raman amplification, it is known that
Manuscript received November 22, 2011; revised June 14, 2012; accepted
June 20, 2012; published July 17, 2012 (Doc. ID 157908).Joo M. Ferreira (e-mail: [email protected]) and Paulo Monteiro
are with Nokia Siemens Networks Portugal, S.A. Rua Irmos Siemens,
2720-093 Amadora, Portugal, and are also with the Department of Electronics,
Telecommunications and Informatics, University of Aveiro, Campus de Santiago,
3810-193 Aveiro, Portugal, and are further affiliated with the Instituto de
Telecomunicaes, Campus de Santiago, 3810-193 Aveiro, Portugal.
Rogrio Nogueira is with Nokia Siemens Networks Portugal, S.A. Rua
Irmos Siemens, 2720-093 Amadora, Portugal, and is also with the Instituto de
Telecomunicaes, Campus de Santiago, 3810-193 Aveiro, Portugal.
Armando N. Pinto is with the Department of Electronics, Telecommunications
and Informatics, University of Aveiro, Campus de Santiago, 3810-193 Aveiro,
Portugal, and is also with the Instituto de Telecomunicaes, Campus de
Santiago, 3810-193 Aveiro, Portugal.Digital Object Identifier 10.1364/JOCN.4.000595
there are three major interactions [5], i.e., pump-to-pump,
pump-to-signal and signal-to-signal interactions. When consid-
ering network planning tools or RWA protocols, the signal-
to-signal interaction has to be taken into account even if the
transmission link does not have RFAs [6]. For this reason this
interaction is usually accounted for in the transfer function
of the transmission fiber. Several models have been developed
to predict and describe the RFA behavior. In [7], the authors
stated that the composite Raman gain can be expressed as the
logarithmic sum of each Raman gain created by each pump
wavelength with a weighting factor. These weighting factors
are found after solving the propagation equations. In [8], a
model suitable for the small signal regime of a counter-pumped
RFA with a single pump was proposed. This approach is known
as the undepleted pump model. An approximated analytical
expression to model the noise figure in dispersion managed
fibers was presented in [9]. A model for the gain and noise
figure for a multi-pump RFA without considering the pump-to-
pump interactions was developed in [10]. However, as shown
in [11], the pump-to-pump interactions tend to be relevant as
the number of pumps and the pump power increase. In [11], the
authors also presented an analytical approximated solution
for the gain considering the pump-to-pump interactions by
iteratively solving the propagation equations for the pumps.
These interactions can also be estimated by numerical or semi-
analytical models as shown in [12], where a semi-analytical
algorithm for calculating the gain of counter-pumped RFAs
with multiple pumps and signals was proposed. In this case,
the interactions between the pumps and signals were solved
using the RungeKutta method. In [13], a semi-analytical
model considering the depleted regime of an RFA with a single
pump and with frequency dependent losses was presented.
These analytical or semi-analytical model approaches tend to
imply a significant number of approximations, which, in most
cases, diminish the generality of the models. On the other
hand, some detailed numerical models have been published
in the literature. In [14], a detailed numerical model based
on the application of the average power analysis technique
to solve the propagation equations of RFAs was presented.
This model shows a similar accuracy and is computationally
more efficient than the fourth-order RungeKutta routine [15].
However, it is still not suitable for treating large networks
due to the computation power required, especially when large
numbers of pumps and signals are considered. In [16,17], the
authors developed a closed integral form of coupled Raman
equations; these equations allowed a reduction of processing
time while maintaining good accuracy. Nevertheless, an
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iterative numerical method is still necessary to obtain the
results, and the time needed for this process may preclude their
utilization in some more complex scenarios.
We develop a low complexity computational model for the
gain of the RFA, suitable for use with multiple pumps and
with a large number of signals. To achieve this, we customize
the undepleted pump model with the introduction of two
new parameters, in order to account for the pump-to-pumpinteractions. These parameters can be adjusted based on
experimental measurements, simulation results, or analytical
expressions. Using these two parameters, an accurate black
box model for the gain profile is obtained, without increasing
the computational complexity of the undepleted pump model.
Based on this approach, we also present and discuss a method
to estimate the amplifier spontaneous emission (ASE) noise
and optical signal-to-noise ratio (OSNR). The obtained results
for the gain profile and ASE noise show a good agreement
with experimental measurements and detailed numerical
simulations, with a computational effort much lower than the
detailed numerical model. With this model it is possible to
estimate the gain and ASE noise of the amplifier based on asimple input/output relation, and to the best of the authors
knowledge it is the first time that the problem has been
addressed in this way.
This paper is organized as follows. In Section II, we
introduce and explain the proposed model for the gain of
RFAs and the approaches used to adjust the parameters. In
Section III, an extensive comparison with experimental mea-
surements and detailed numerical simulations are presented.
In Section IV, we present a method to estimate the ASE and
the OSNR based on the proposed model. The conclusions are
presented in Section V.
II. WEIGHTED MODEL
To account for the pump-to-signal interaction, let us
consider the undepleted pump model, where a small signal
regime is assumed [8]. The output signal power is given by
Ps(L)=Ps(0)exp
Cs,pPp(L)LeffsL
, (1)
where Ps(0) is the input signal power, the effective fiber length
is given by Leff=1exp(pL)
p, Pp(L) is the input pump power,
p is the fiber attenuation at the pump wavelength, s is
the fiber attenuation at the signal wavelength and L is the
transmission length. The Cs,p is the Raman gain efficiency
of the fiber, considering the wavelength of the signal and the
pump. Generically, for waves i and j centered at frequencies iand j , respectively, the Raman gain efficiency is given by [ 18]
Ci,j =
gR (j i)
2 Aeffif j i 0,
i
j
gR(i j)
2 Aeffifj i < 0,
(2)
where Aeff is the fibers effective area, gR () is the Raman
gain coefficient of the fiber and is the frequency spacing
between the two waves. The Raman gain coefficient can be
obtained using the technique presented in [18,19].
TABLE I
PUMP WAVEL ENG TH S AN D INPUT PUMP POWERS OF RFA
CONFIGURATIONS
Pumps
1426 nm 1444 nm 1462 nm 1487 nm
Conf. 1 182.7 mW 237.5 mW 88 mW 233 mWConf. 2 355.4 mW 256.8 mW
Conf. 3 397.7 mW 279 mW 83.8 mW Conf. 4 25500 mW 25500 mW
We can extend the undepleted pump model to support
multiple pumps, where the contribution of the pump for the
signal output power in Eq. (1) is replaced by a sum over all the
pumps. In this case, the output signal power is given by
Ps(L)=Ps(0)exp
N
i=1
Cs,iPi(L)Leff,i
sL
, (3)
where Leff,i =1exp(iL)
iis the effective length for each pump,
Pi(L) is the input pump power of pump i,
i is the fiberattenuation at the pump wavelength i and N is the number
of pumps.
In order to assess the accuracy of the undepleted pump
model given by Eq. (3) we perform measurements and detailed
numerical simulations in a transmission system amplified
with a counter-pumped RFA. The detailed numerical model
used in this work is the one developed in [14]. To model
the Raman gain coefficient, gR(), we use a set of 14
Gaussian functions with the parameters shown in [18]. The
RFA implemented, configuration 1 in Table I, has four pumps
centered at 1426 nm, 1444 nm, 1462 nm and 1487 nm, with
attenuation coefficients of 0.26 dB/km, 0.25 dB/km, 0.23 dB/km
and 0.23 dB/km, respectively. The pumps P1, P2, P3, P4 arecoupled to the fiber using an optical multiplexer, followed by
an optical coupler. The transmission fiber is a single mode
fiber (SMF) with 80 km, with an effective area of 80 m2
and an attenuation coefficient of 0.2 dB/km around 1550 nm.
Since we are interested in measuring the pump-to-signal and
pump-to-pump interactions, a tunable laser operating at 1 mW
is used as the probe to be amplified. We measure the On/Off
gain from 1510 nm to 1610 nm using an optical spectrum
analyzer (OSA). The On/Off gain of the RFA for a wavelength
s, GdB,s, in decibels is defined as
GdB,s = 10log10Ps,On(L)
Ps,Off(L)
, (4)
where Ps,On(L) is the signal output power with amplification
(when the pumps are On) and Ps,Off(L) is the signal output
power without amplification (when the pumps are Off).
Figure 1 shows experimental measurements (crosses) for
the RFA On/Off gain compared with the undepleted pump
model (dotted line), Eq. (3), and with detailed numerical results
(full line) [14]. We can observe that the detailed numerical
model accurately describes the experimental data. When
comparing the undepleted pump model with the experimental
or numerical model results, we can observe a large deviation.
This is due to the non-linear effect of stimulated Raman
scattering (SRS), which transfers energy between the pumps,
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Fig. 1. On/Off gain measurements and comparison with numerical
models for a counter-pumped RFA with four pumps and a total pump
power of 745 mW: experimental measurements (crosses), detailed
numerical results (full line), undepleted pump model (dotted line),
linear regression of the detailed numerical model (dashed line), linear
regression of the undepleted pump model (dashdot line). The insetrefers to the pump wavelengths and pump powers of the RFA.
known as pump-to-pump interactions. This fact is highlighted
by observing the linear regression of the results obtained with
each model; see Fig. 1. It can be seen that the undepleted pump
model presents a gain with a negative tilt (dashdot line in
Fig. 1), whereas the real gain profile presents a positive tilt
(dashed line in Fig. 1).
A. Adjusting the On/Off Gain Tilt
It is known that the SRS between pumps changes thepump power distribution within the fiber, resulting in an
amplification of the pumps with higher wavelengths and in
the depletion of pumps with lower wavelengths. We propose
to include this effect in the undepleted pump model, Eq. (3), by
adjusting the input pump power, in order to account for the
pump-to-pump interactions and therefore obtain the correct
output signal power
Ps(L)=Ps(0)exp
N
i=1
Cs,iPSRSi Leff,i
sL
, (5)
where PSRSi
is the corrected input pump power of pump i
to account for the pump-to-pump interactions. In order toestimate the PSRS
i, we use
PSRSi
=Pi(L)exp
N
j=1
Ci,jPj (L)1exp(jDint)
j
, (6)
where the Dint parameter can be understood as an interaction
length between the pumps, in order to account for the
amplification or depletion of the pumps due to SRS between
the pumps. Note that when j = i, Ci,j = 0. In order to estimate
Dint, we start by writing the amplifier On/Off gain using
Gs =
Ps,On(L)
Ps,Off(L)= exp
Ni=1
Cs,iPSRSi Leff,i
, (7)
where Gs is the On/Off gain for a wavelength s. To simplify this
equation let us consider the same attenuation coefficient for all
the pump wavelengths; therefore, Leff,i = Leff. Using Eq. (6),
we obtain
ln(Gs)Leff
Ni=1
Cs,iPi(L)exp
Leff
Nj=1
Ci,jPj (L)
, (8)
where Leff =1exp(Dint)
. Considering the first term of the
Taylor expansion of the exponential in Eq. (8), we obtain
ln(Gs)
Leff
Ni=1
Cs,iPi LeffNi=1
Nj=1
Cs,i Ci,jPiPj. (9)
Expanding the exponentials of Leff and Leff , we can write
Dint ln(Gs)L
Ni=1
Cs,iPi
LN
i=1
Nj=1
Cs,i Ci,jPiPj. (10)
This expression relates Dint with the On/Off gain for the signals. By substituting the On/Off gain, Gs, with an experimental
or numerical result, it is possible to obtain Dint. However,
this derivation is made assuming some approximations, the
most relevant one being the assumption that the first-order
Taylor series is a good approximation for the exponentials. To
overcome this, the chosen wavelength, s, should be the one
which adjusts best the On/Off gain tilt given by Eq. ( 5), when
compared with the detailed numerical model or experimental
measurements. Therefore, a set of wavelengths has to be
tested, and the one which provides the best description for the
tilt is chosen.
Let us consider the amplifier of configuration 1. By
measuring the On/Off gain for different wavelengths in theamplifier region of interest and assessing the gain tilt based
on Eqs. (5), (6) and (10), and comparing with the detailed
numerical model, we verify that the wavelength which best
adjusts the tilt is 1571 nm. Using this wavelength to calculate
the On/Off gain (see Fig. 2), we can see that the gain tilt is well
adjusted. This fact is highlighted by the linear regressions of
both models. However, an offset in the gain is still present.
B. Adjusting the On/Off Gain Offset
We can adjust the offset in the On/Off gain of the RFA by
including another parameter in the model, i.e.,
Ps(L)=Ps(0)exp
N
i=1
Cs,iPSRSi L
eff,i
sL
, (11)
where Leff,i
is given by
Leff,i =1exp((i +SRS)L)
i +SRS, (12)
where SRS accounts for a gain reduction due to the depletion
of the pumps.
We can write the On/Off gain difference between the model
with the adjusted tilt, Gt, obtained with Eq. (5), and the
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598 J. OPT. COMMUN. NETW./VOL. 4, NO. 8/AUGUST 2012 Ferreira et al.
(a)
(b)
Fig. 2. On/Off gain measurements and comparison with numerical
models for a counter-pumped RFA with four pumps and a total pump
power of 745 mW: experimental measurements (crosses), detailed
numerical results (full line), weighted undepleted pump model (dotted
line), linear regression of the detailed numerical model (dashed line),
linear regression of the weighted undepleted pump model (dashdot
line), when adjusting (a) the tilt and (b) the tilt and offset.
experimental or numerical result, Gs, for the wavelength s, as
Gs = exp
N
i=1
Cs,iPSRSi
Leff,i
exp
N
i=1
Cs,iPSRSi
Leff,i
. (13)
Solving Eq. (13) in order to Leff,i
, we obtain
Leff,i =GsN
i=1(Cs,iP
SRSi
)+Leff,i. (14)
MakingLeff,i 1/i, and Leff,i
1/(i +SRS), we obtain
SRS =1
GsN
i=1 Cs,iPSRSi
+1i
i. (15)
The results obtained based on the measurement of the
reference wavelength considered previously are presented in
Fig. 2(b). As we can see, the results show that both the tilt and
the offset are well adjusted.
C. Analytical Expressions for Dint and SRS
We have shown that it is possible to obtain accurate
results for the amplifier gain based on a customization of
the undepleted pump model with two extra parameters.
These parameters are estimated based on a comparison
with experimental or numerical results; we call this method
weighted by comparison (WBC). Nevertheless, it can beuseful to have analytical expressions for Dint and SRS.
Considering that the weighted undepleted pump model
should converge to the undepleted pump model when the
number of pumps is one, we expect that Dint and SRSshould approach zero for a single pump configuration. Besides
that, Dint should be inversely proportional to the total
pump power [20], and should account for the pump-to-pump
interactions. The parameterSRS accounts for the pump power
losses due to the SRS effect between pumps. Therefore, we
can derive an expression in dB/km based on the depletion of
the first pump, which is the most attenuated pump. Based on
the principles stated, and only on known parameters of the
system, we derive analytical expressions which better adjustthe results obtained from simulations considering different
scenarios:
Dint N1
PtotalN
j=2|C1,j |
, (16)
SRS 10
Llog10
PSRS
1
P1(L)
, (17)
where C1,j is the Raman gain efficiency of the fiber between
the first pump P1 and the pump j; PSRS1 is the remaining
power after accounting for the depletion of the first pump,
which is the most attenuated pump; and P1(L) is the inputpump power of the first pump. With this approach, we are
able to propose a model for the output signal power of RFAs
with different numbers of pumps and powers, without the need
to compare with experimental measurements or numerical
simulations to adjust the parameters. We call this approach
weighted by analytical expressions (WAE); it is given by
Eqs. (6) and (11), and the parameters Dint and SRS are
obtained with the analytical expressions (16) and (17).
Figure 3 shows the results obtained with the WAE model
for the RFA of configuration 1. As we can see, the results
obtained present a good accuracy; however, it is slightly worse
than the accuracy obtained when Dint and SRS are obtained
through comparison with experimental or detailed numericalsimulation, WBC. Nevertheless, based on WAE the results
are obtained without any comparison with experimental or
numerical data, which makes this method simpler to use.
III. PERFORMANCE ASSESSMENT
In order to assess the proposed model in scenarios with
different numbers of pumps, pump wavelengths and powers,
we experimentally implement counter-pumped RFAs with two,
three and four pumps; see Table I. We measure the On/Off
gain from 1510 nm to 1610 nm when considering the amplifier
with configuration 1 and from 1510 nm to 1570 nm for
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Fig. 3. On/Off gain measurements and comparison with numerical
models for a counter-pumped RFA with four pumps and a total pump
power of 745 mW: experimental measurements (crosses); detailed
numerical results (full line); WAE (dotted line); linear regression of the
detailed numerical model (dashed line); linear regression of the WAE
(dashdot line).
configurations 2 and 3, using an OSA. The obtained results for
the considered configurations are shown in Figs. 3 and 4.
To assess the error of the gain profile we measure the mean
absolute error (MAE) between the estimated and measured
gain profiles. Figure 5 shows the MAE calculated between
1510 nm and 1610 nm for configurations 1, 2 and 3. As we
can observe, the detailed numerical model describes correctly
the experimental measurements for all the setups considered,
with an MAE below 0.5 dB. When considering the undepleted
pump model, we verify that it presents a higher error than
the detailed numerical model, and this error increases withthe number of pumps and pump powers; it reaches an MAE
of 2.8 dB in configuration 1. When comparing the weighted
undepleted pump model, WBC and WAE with the detailed
numerical model, we verify that for some configurations
the MAE is almost the same as the detailed numerical
model, which highlights the accuracy of the proposed models.
Nevertheless, we verify that when using the WBC, the error
is typically smaller than when using the WAE. Regarding
configuration 2, the WBC presents a slightly higher error
than the undepleted pump model. This happens because the
depletion of the first pump is overestimated by the model, due
to the fact that, in this configuration, the first pump has a
higher input power than the second pump and the wavelengthseparation between them is small, about 18 nm, which means
that the depletion is negligible.
In Fig. 6, we can observe the MAE as a function of the total
input power, between the gain estimated from the undepleted
pump model and the weighted model using the two methods to
adjust the parameters, and the detailed numerical model for
configurations 1 and 4. The power is increased equally in all
the pumps. Considering the WBC approach, we verify that the
MAE obtained is below 0.4 dB for all cases tested. Regarding
the WAE approach, we conclude that it tends to be suitable for
a total pump power up to 900 mW and a wavelength separation
below 60 nm between the first and last pumps, considering
an MAE up to 1 dB. When comparing the WBC and WAE
(a)
(b)
Fig. 4. On/Off gain measurements and comparison with numerical
models for a counter-pumped RFA in (a) configuration 2 with a
total pump power of 612 mW and (b) configuration 3 with a
pump power of 760 mW: experimental measurements (crosses);
detailed numerical results (full line); undepleted pump model (dotted
line); weighted undepleted pump model based on comparison with
experimental or detailed numerical simulation, WBC (dashed line);
weighted undepleted pump model based on analytical expressions,
WAE (dashdot line). The inset refers to the pump wavelengths and
pump powers of the RFA.
Fig. 5. MAE of the On/Off gain for the different models for
configurations 1, 2 and 3, when compared with experimental results.
approaches with the undepleted pump model, we verify that
the results are significantly improved.
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Fig. 6. MAE between the On/Off gain obtained with the undepleted
pump model, WBC and WAE, and the detailed numerical model for
different total input pump powers for configurations 1 and 4; see
Table I.
TABLE II
PROCESSING TIMES (ms) OF THE MODELS IN AN INTEL
T5500 PROCESSOR AT 1.66 GHz WIT H 2 GB OF RAM
Model Conf. 1 Conf. 2 Conf. 3
Detailed numerical model 7300 2500 4700Undepleted pump model 90 35 50WAE; WBC 90 40 50
We also calculate the MAE as a function of the fiber length,
between the gain estimated from the undepleted pump model
and from the WBC and WAE, and the detailed numerical model
for configuration 1. When considering the WBC, the MAE isbelow 0.3 dB for all the distances considered, up to 200 km.
For the WAE, the results present an MAE below 0.5 dB up to
180 km for configuration 1. The results are similar for all the
other configurations.
The processing time is also an important metric for
assessing the applicability of a simplified model. Table II
shows the processing time of each model. When considering
configuration 1, the processing time to calculate the On/Off
gain of 100 signals, one by one, by the detailed numerical
model, with a numerical resolution of 200 m and a convergence
criterion of 1106, is more than 7.3 s while the WAE takes
less than 100 ms. This means that the WAE is more than
70 times faster than the detailed numerical model for this
configuration. When considering configurations 2 and 3, to
calculate the On/Off gain of 60 signals, one by one, the detailed
numerical model takes more than 2.3 s and 4.7 s, respectively.
When considering the WAE it takes less than 40 ms and
50 ms, respectively, to obtain these results. As we can observe,
the WAE is faster than the detailed numerical model even
in simpler configurations. Also, when considering the WAE,
the processing time is almost the same despite the complexity
of the RFA configurations. As expected, when comparing the
WAE with the undepleted pump model, the computational time
is similar. Regarding the WBC and WAE approaches, their
computational times are the same after finding the parameters
Dint and SRS.
IV. AS E N OISE MODEL AND OSNR
Some planning tools and RWA algorithms are based not only
on the amplifier gain profile, but also on the signal OSNR
after the transmission links. Therefore, it is also important
to develop and improve the ASE estimation and the OSNR
prediction for the transmission systems.
The ASE noise of the amplifier can be related to the
amplifier gain [21,22]. Therefore, we can apply the proposed
approach to better estimate the ASE profile of the RFA. Let us
consider a multi-pumped RFA. The net gain, G, is given by
G = exp(si gna lsL)
i
gi, (18)
where i refers to the pump of the configuration imple-
mented. When considering the undepleted pump model, g i =
exp
Cs,iPi(L)Leff,i, and when considering the proposed model
we can estimate g i using g i = expCs,iPSRSi
Leff,i, where
PSRSi is the corrected input pump power given by Eq. (6)and L
eff,iis given by Eq. (12). Based on [21] and considering
the proposed model, the ASE noise power at the end of the
amplifier, L, can be written as
PASE(L)= hB0
iE i ln(g i)
i ln(g i)
1+
Leffexp(L)i ln(g i)
G
1+
Leffi ln(gi )
, (19)
where E i is the spontaneous factor, given by
E i = 1+ 1
exp
hikT
1
, (20)
and h is Plancks constant, k is Boltzmanns constant, T is
the temperature in Kelvin, i is the frequency difference
between the i-th pump and the signal and B0 is the reference
bandwidth.
Figure 7(a) shows experimental measurements for the ASE
noise of configuration 1, from 1520 nm to 1590 nm. The ASE
is measured without signals and the reference bandwidth is
2.5 GHz. These experimental results are compared with results
obtained with the detailed numerical model, and with results
obtained by the ASE model when using the undepleted pumpmodel and the WAE model to estimate the gain of the amplifier.
We verify a good agreement between the experimental data,
the detailed numerical model and the proposed model.
This improved estimation of the amplifier gain and ASE
noise allows a better estimation of the OSNR of the
transmission system. To assess this, we measure the output
signal power and the ASE power experimentally with an OSA,
from 1520 nm to 1590 nm. Figure 7(b) shows the experimental
results measured and the comparison with the undepleted
pump model, with the WAE and with the detailed numerical
model. As we can observe, the OSNR is better described by the
proposed approach than the results obtained by the undepleted
pump model.
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(a)
(b)
Fig. 7. (a)ASE noise measured andestimated based on theundepleted
pump model and on the WAE for the amplifier of configuration 1; (b)
OSNR measurements and estimation based on the undepleted pump
model and based on the WAE for configuration 1.
V. C ONCLUSION
A novel model to estimate the On/Off gain and ASE
noise of a transmission link with RFA amplification with
multiple pumps is proposed. This model is based on the
customization of the undepleted pump power with two extra
parameters, Dint and SRS, to account for the pump-to-pump
interactions. Two methods are proposed to adjust these
parameters, one based on a comparison with experimental
data or detailed numerical simulations, WBC, and the otherbased on analytical expressions, WAE. The second method
is simpler to implement, because it can be used without
comparison with experimental or numerical simulations. When
comparing the WAE with experimental measurements and
detailed numerical simulations, we found that this model
provides accurate results for a total input pump power up to
900 mW, for a wavelength separation between pumps of up to
60 nm and for a transmission distance of up to 180 km.
The proposed model presents an improved computational
performance without losing significant accuracy for the most
used pump configurations. In fact, it was verified that the
model can be more than 70 times faster than the detailed
numerical model. Therefore, it can be used in network planning
tools, in applications which have to estimate the gain and the
ASE noise or to design RFAs with a limited computational
power.
ACKNOWLEDGMENTS
This work has been partially supported by the Fundao
para a Cincia e a Tecnologia (FCT), under the re-
search grant SFRH/BDE/51095/2010, and the OSP-HNLF-PTDC/EEA-TEL/105254/2008 and CONTACT-PTDC/EEA-
TEL/114144/2009 projects.
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