weighted undepleted pump model for broadband counter-pumped raman fiber amplifiers

7/28/2019 Weighted Undepleted Pump Model for Broadband Counter-Pumped Raman Fiber Amplifiers

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Ferreira et al. VOL. 4, NO. 8/AUGUST 2012/J. OPT. COMMUN. NETW. 595

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

1943-0620/12/080595-08/$15.00 2012 Optical Society of America


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596 J. OPT. COMMUN. NETW./VOL. 4, NO. 8/AUGUST 2012 Ferreira et al.

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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(a)

(b)



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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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)


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.

REFERENCES

[1] A. Morea, F. Leplingard, T. Zami, and N. Brogard, New trans-

mission systems enabling transparent network perspectives,

Compt. Rend., vol. 9, no. 910, pp. 9851001, 2008.

[2] F. Leplingard, T. Zami, A. Morea, N. Brogard, and D. Bayart, De-

termination of the impact of a quality of transmission estimator

margin on the dimensioning of an optical network, in Optical

Fiber Communication Conf., 2008, OWA6.[3] K. Manousakis, K. Christodoulopoulos, E. Kamitsas, I. Tomkos,

and E. Varvarigos, Offline impairment-aware routing and wave-

length assignment algorithms in translucent WDM optical net-

works, J. Lightwave Technol., vol. 27, no. 12, pp. 18661877,

2009.

[4] J. Bromage, Raman amplification for fiber communications sys-

tems, J. Lightwave Technol., vol. 22, no. 1, pp. 7993, 2004.

[5] C. Headley and G. Agrawal, Raman Amplification in Fiber

Optical Communication Systems. Academic Press, 2005.

[6] M. Zirngibl, Analytical model of Raman gain effects in massive

wavelength division multiplexed transmission systems, Elec-

tron. Lett., vol. 34, no. 8, pp. 789790, 1998.

[7] S. Namiki and Y. Emori, Ultrabroad-band Raman amplifiers

pumped and gain-equalized by wavelength-division-multiplexed

high-power laser diodes, IEEE J. Sel. Top. Quantum Electron.,

vol. 7, no. 1, pp. 316, 2001.

[8] S. Chinn, Analysis of counter-pumped small-signal fibre Raman

amplifiers, Electron. Lett., vol. 33, no. 7, pp. 607608, 1997.

[9] A. Kobyakov, M. Vasilyev, S. Tsuda, G. Giudice, and S. Ten,

Analytical model for Raman noise figure in dispersion-managed

fibers, IEEE Photon. Technol. Lett., vol. 15, no. 1, pp. 3032,

2003.

[10] X. Zhou, C. Lu, P. Shum, and T. Cheng, A simplified model

and optimal design of a multiwavelength backward-pumped fiber

Raman amplifier, IEEE Photon. Technol. Lett., vol. 13, no. 9, pp.

945947, 2001.

[11] S. Cani, L. de Calazans Calmon, M. Pontes, M. Ribeiro,M. Segatto, and A. Cartaxo, An analytical approximated so-

lution for the gain of broadband Raman amplifiers with mul-

tiple counter-pumps, J. Lightwave Technol., vol. 27, no. 7, pp.

944951, 2009.

[12] J. Zhou, J. Chen, X. Li, and W. Jiang, A novel algorithm for

backward-pumped Raman amplifier, Fiber Integr. Opt., vol. 24,

no. 6, pp. 529535, 2005.

[13] M. Santagiustina, Exact, implicit, integral solution of depletion

and saturation in Raman and Brillouin fiber amplifiers, in Eu-

ropean Conf. on Lasers and Electro-Optics, and the Int. Quantum

Electronics Conf., CLEOE-IQEC 2007, 2007.

[14] B. Min, W. Lee, and N. Park, Efficient formulation of Raman

amplifier propagation equations with average power analysis,

IEEE Photon. Technol. Lett., vol. 12, no. 11, pp. 14861488, 2000.


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