Applying a Numerical Simulation Technique to the Design of WDM-Pumped
Raman Amplifiers, and Methods for the Automatic Determination of Pump Powers
by Koji Fujimura * , Atsushi Oguri * , Takeshi Nakajima * ,Yoshihiro Emori * , Shu Namiki * and Misao Sakano *In the design of Raman amplifiers, it is necessary, with such characteristics as
pump wavelength, pump power, signal power given, to arrive at a numericalprediction of the corresponding output signal power, Raman gain, noise figure, multipath
interference, etc. Further, if the values for pump wavelength and pump power that are required
to obtain the various target characteristics can be obtained, the efficiency with which the Raman
amplifier can be designed is significantly increased. This paper first describes the numericalsimulator used to estimate amplifier characteristics. It then proposes a Raman amplifier structurethat gradually increases the number of pump wavelengths, and applies the numerical simulator
to its design. Finally it discusses a technique for predicting the pump power to be applied inorder to obtain the desired values of output signal power and Raman gain.

ABSTRACT1.

INTRODUCTION

The use, in an optical transmission system using
wavelength division multiplexing (WDM), of a distributed
WDM-pumped Raman amplifier for optical amplification
is effective in reducing noise and broadening gain
bandwidth 1), 2). We have developed a numerical simulator
for the prediction of Raman amplifier characteristics, in amodel incorporating the input and output characteristicsof the signal, the noise figure, multipath interference (MPI)
due to Rayleigh scattering, and nonlinear phase shift. We
have also made proposals relating to Raman amplifier
structure and pump techniques using this numericalsimulator 3), 4). Recently we have proposed and designed
an amplifier in which the pump wavelength is gradually
increased (an upgraded Raman amplifier) using thatsimulator, and have evaluated its characteristics 5).
In designing a Raman amplifier, the crucial point is
to obtain the pump wavelength and pump power forachieving the various characteristics required. In thesimulator described above, the values in the design are
selected by human agency, and human decisions andmodifications of the design values are reiterated until
the characteristics obtained by the simulator satisfy therequired characteristics. It is to be hoped that, by further
improving the efficiency of the design system, it will bepossible to determine automatically the pump wavelengthand pump power that satisfy the characteristics. Asof the present time a number of techniques havebeen proposed to determine pump wavelength and* Fitel-Photonics Lab., R&D Div.

pump power by assigning the characteristics to besatisfied in terms of signal output and gain 6)~8). Further
a superposition rule was developed to determine
the pump wavelength for obtaining the target gain
characteristics 2). We have developed a technique using
linear approximation for the automatic determination ofpump power 9). This is a technique that exhibits a strong
convergence to true values even when the initial value
departs from the solution.
This paper first describes the numerical simulator used
to predict Raman amplifier characteristics, and then, as
an example of a design using it, reports on a bandwidth
upgradable Raman amplifier. Finally it discusses anautomatic design technique for predicting pump power.

2.

BASIC EQUATIONS AND DESIGNSUPPORT SYSTEM

2.1 Structure of the Raman Amplifier
Light incident on an optical fiber excites the opticalphonons of the glass molecules, loses energy and isscattered as low-frequency light. The phenomenon is
known as Raman scattering, and in silicate fibers thefrequency shift is normally distributed in the vicinity of
13 THz. If light is present in the scattered light band,stimulated emission occurs, allowing use as a Raman
amplifier.
Figure 1 shows the basic structure of a Raman
amplifier. It involves an optical signal input of a pluralityof wavelengths, pumping lasers having different
wavelengths, optical fiber as the gain medium, anda coupler for the signal light and pumping light. In

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Applying a Numerical Simulation Technique to the Design of WDM-Pumped Raman Amplifiers, and Methods for the Automatic Determination of Pump Powers����� ������

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Figure 1

Basic structure of Raman amplifier.

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the case of WDM pumping, superposition of the gain
characteristics of each pump wavelength makes it
possible to achieve the target gain characteristic-e.g. an extremely flat gain characteristic--over a broad
bandwidth. Thus in designing a Raman amplifier it isnecessary to determine the pump wavelengths and pump
powers for obtaining the target characteristic.
2.2 Basic Equations
In WDM transmission, the propagation along the fiber of
the time average value of optical power is given by thefollowing differential equation 2), 10), 11)dPν±= -αν Pν±+εν Pν +Pν± Σ ERµ, ν(Pµ++Pµ-)µ>ν
dz-Pν± Σ ν ERν, µ(Pµ++Pµ-)
µ<ν
µ


1
+2hν Σ ERµ,ν(Pµ++Pµ-) 1+ ∆ν
µ>ν
 h(µ -ν )  exp  kT  -1 


±

±

(1)

where:
Pν+ is the average power of the light (the subscriptν is the frequency of that light and the plus andminus signs indicate propagation in the forward
and backward directions respectively),εν is the backward Rayleigh scattering coefficient oflight of frequency ν ,ERµ,ν is the nonpolarized Raman gain efficiency fromlight of frequency µ to light of frequency ν ,∆ν is the infinitesimal bandwidth around frequency
ν,h is Planck’s constant,
k is Boltzmann’s constant, andT is absolute temperature.

αν is the fiber loss constant at frequency ν and includes
the spontaneous Raman scattering effect, and may be
represented byαν = αν*+4hν Pν± Σ ERν, µ 1+
µ<ν



1




 h(ν -µ ) exp  -1 
 kT 

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Figure 2

Flowchart of design support system of
WDM-pumped Raman amplifier.

scattering, the 3rd and 4th that of stimulated Raman
scattering, and the 5th that of spontaneous Raman
scattering.
2.3 Design Support System
In order to determine the pump wavelengths and pump
powers that would give the target gain characteristics, wehave configured a design support system that provides
numerical solutions to Equations (1) and (2). Since insolving these differential equations we must deal withlight propagating in both the forward and backward
directions, we have to solve problems having boundary
values at two points--the input and output sides. To dothis we have used a 4th-order Runge-Kutta method thatsolves iteratively for the forward and backward directions.
Figure 2 shows a flowchart of the design support
system. When the fiber condition, signal condition andpump condition are input to the solver, characteristics
of output signal power, noise figure (NF), multipath
interference (MPI), optical signal-to-noise ratio (OSNR)
and nonlinear phase shift are output. It is then determined
whether or not these match the design specifications, andif they are found to match the design is complete. If they
do not match, the pump conditions are modified and theprocess is repeated until they do.
In Figure 3(a) through (f) are shown typical output
windows for Raman gain, NF, net gain, MPI, OSNR andnonlinear phase shift.

(2)

where:
αν* is the fiber loss constant not including theRaman scattering effect.
So summarizing, from Equation (1), the 1st term
represents the effect of fiber loss, the 2nd that of Rayleigh

3.

BANDWIDTH-UPGRADABLE RAMANAMPLIFIERS

3.1 Background and Amplifier StructureWith increases in the traffic volume on opticalcommunication systems, it is anticipated the signal
bandwidth used in WDM system will become broader.

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Applying a Numerical Simulation Technique to the Design of WDM-Pumped Raman Amplifiers, and Methods for the Automatic Determination of Pump Powers

Rather than providing bandwidth for future demand
from the outset, however, it will be more desirable, fromthe standpoint of decreasing initial invsetment, to effectbandwidth upgrading as the need arises. Since with
Raman amplifiers the amplification bandwidth can beeasily increased by WDM-pumping, making it possible toaddress these needs by adding pump wavelengths.
An example of a Raman amplifier with upgradable
amplification bandwidth has already been reported,
in which, by adding 3 pump wavelengths, an upgrade
was effected in one step from C band or L band to C+Lband 2). To achieve a more flexible response to market
demand, however, it would be preferable to upgradeusing a narrower amplification width.
Accordingly in this paper we report on a Raman
amplifier unit with an amplification bandwidth that is
upgradable from approximately 20 nm to 75 nm at 8-nm

(1-THz) intervals, as shown in Figure 4, using backward
pumping of an 80-km TrueWave® RS nonzero dispersionshifted fiber. There are two methods of upgrading theamplification bandwidth: a) by adding pump wavelengths
toward the shorter wavelengths and b) by adding pumpwavelengths toward the longer wavelengths, and we
conducted investigations into which of them is mostadvantageous in terms of pumping light cost.

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Output windows of design support system for WDM-pumped Raman amplifier.

† TrueWave® is a registered trademark of OFS Fitel.

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Applying a Numerical Simulation Technique to the Design of WDM-Pumped Raman Amplifiers, and Methods for the Automatic Determination of Pump Powers�����������

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Schematic of bandwidth-upgradeable
Raman amplifier.

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Figure 4

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Figure 6 Pump power required at each stage of upgrading.

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Figure 5

Simulated Raman gain profiles.

3.2 Simulation Results
Figure 5 shows the results of Raman gain profiles for thetwo methods of upgrading--toward shorter and toward
longer wavelengths. As can be seen both methods havebeen confirmed to provide adequate flatness of theamplification bandwidth.
Figure 6 shows the results of simulations of pump
power when upgraded toward shorter and longer
wavelengths. A comparison of the specified pump power
required by the two methods of upgrading shown in
Figure 6 demonstrates that from the standpoint of thecost of pumping light, the shorter wavelength method of

upgrading is superior for two reasons:
The first is that the total of the specified power requiredat all pump wavelengths was only 662.7 mW whenupgraded in the shorter wavelength direction, or about
half the 1277.2 mW that was required when upgradedin the longer wavelength direction. This is because
when upgraded in the shorter wavelength direction, thepump having the longest wavelength, which required thegreatest pump power, was fixed at 1495.2 nm, whereas
when upgraded in the longer wavelength direction, it
shifted at each stage of the upgrade.
The second reason is that when upgraded in the longer
wavelength direction, pump power is required for two
extra pumps, at 1480.5 and 1487.8 nm. That is because
the longest pump wavelength used is 1480.5 nm for 7
pumps and 1487.8 nm for 8. It is for these reasons that
upgrading in the shorter wavelength direction is morecost effective. It can further be seen that when upgradedin the shorter wavelength direction the specified power
required for pump wavelengths other than the longest(1495.2 nm) is comparatively small--no more than 80 mW.
3.3 Experimental Results
To verify the results of simulation, experiments werecarried out for upgrading in the shorter wavelength
direction under conditions similar to those of thesimulation. As Figure 7 shows it was possible to obtain
Raman gain similar to that in the simulation in Figure5(a). Figure 6(a) compares the pump power obtained

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Applying a Numerical Simulation Technique to the Design of WDM-Pumped Raman Amplifiers, and Methods for the Automatic Determination of Pump Powers�����������

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Raman gain profiles from experiments.

experimentally (solid lines) and by simulation (dashedlines), confirming a good agreement between theexperimental and simulated values.

4.

AUTOMATIC DESIGN AND CONTROL

4.1 Guideline for Automatic Design
As was shown in Figure 2, to obtain the target output
signal power and gain profile for a WDM-pumped
Raman amplifier it is necessary to determine thewavelength and input power for each pump. It is therefore
extremely effective to develop a procedure for automatic
determination of the distribution of the pump powers. This
automatic determination may be obtained by following
the flowchart shown in Figure 8, whereby signal condition
and fiber condition are input together with the target
value for output signal power, and the optimum pumpconditions are determined by solving the inverse problem
by numerical calculation.
Methods previously used to solve the inverse problem
include a genetic algorithm 6) , simulated annealing
algorithm 7), and neural networks 8). In all these methods
the optimum settings for pump wavelengths and pump
powers are determined simultaneously. And since it is anonlinear problem there are disadvantages, in that the

optimization takes time and a unique solution cannot be
obtained. Again, a method that uses a superposition ruleto determine the optimal pump wavelengths has also
been developed in the past 2), 12).
As a guideline for design, we propose a method
in which pump powers are optimized after the pumpwavelengths have been determined. For determining
the optimal pump wavelengths a superposition rule isused. And since the wavelength of the 14xx nm pumping
laser is in practice fixed, it is effective to optimize thepump powers setting with respect to the existing pumpwavelength settings.
4.2 Algorithm
Following is an explanation of the algorithm for automatic
determination of optimal pump powers after determining
optimum pump wavelengths. First we ignore the Rayleighscattering term and spontaneous Raman scatteringterm in Equation (1), which have virtually no effect on
the optimization of pump power. Also, for simplicity
we assume a single signal light and a single forward
pump. As the initial condition we take the longitudinal
distribution of signal power to be PS0(z) and that of pump
power to be PP0(z). Further, let the amounts of change
in power from the initial to target values be definedas ε (z) and ν (z), and let the longitudinal distribution oftarget signal power and pump power be PS(z) and PP(z),respectively. Thus we have:PS(z) = PS0(z) + ε (z)PP(z) = PP0(z) + η (z)(3)

Since PS0(z), PP0(z), PS(z) and PP(z) all satisfy Equation(1), they may be substituted, yielding a differential
equation relating to ε and ν . Then by ignoring the
2nd-order terms for ε and ν we obtain the linear
differential equation
 dε (z)  dz   -α S+gSPPP0(z) dη (z)  =  -gPSPP0(z) dz 

-gSPPS0(z)   ε (z) -α P-gPSPS0(z)   η (z) ε (z)  η (z)

=F
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Flowchart for automatic design of
WDM-pumped Raman amplifier.

(4)

where: α S and α P are the fiber loss coefficients for thesignal light and pumping light respectively, g SP is theRaman gain efficiency between the pumping light andthe signal light, and may be expressed as gPS = (ν P/ν S)gSP(ν S and ν P being the frequencies of the signal light andpumping light respectively).
Since the members of matrix F in Equation (4) are
functions of z, it cannot be solved by direct analysis. Thus
the fiber is divided into N sectors, with a sector length sominute that PS0(z) and PP0(z) can be considered constantwithin that sector.
In this way, if differential equation (2) is solved for thesector from z=zi to z=zi+1, we obtain the simultaneous
equation

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Applying a Numerical Simulation Technique to the Design of WDM-Pumped Raman Amplifiers, and Methods for the Automatic Determination of Pump Powers(5)

and if Equation (5) is applied iteratively from z=0 to z=L,
we obtain the simultaneous equation ε (L)  ε (0)  ε (0)  = ANAN-1 ... A2A1 
=A

η
(L)
η
(0)




 η (0)

(6)

 ε 1(L)  ε 1(0)  .. 
 .. 
 . 
 . 
 ε m(L)
 ε m(0)
 η 1(L) =A  η 1(0) 
 . 
 . 
 .. 
 .. 
 η n(L)
 η n(0)
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If this is expanded to m signal light of wavelengths andn pumping wavelengths, we obtain the simultaneous
equation(7)

Equation (7) represents a relational expression for the
input and output sides of the amount of change in signal
power and pump power respectively. If the total numberof the equations is greater than that of the unknown
quantities, those unknown quantities can be found by thelinear least squares method.
Up to this point the discussion has concerned the linear
domain within which the amount of change is assumed
to be minute, but can be extended into a method capableof addressing larger amounts of change. The amountof change is determined not in one step only as thedifference between the initial and target values, but withina range in which linear approximation is useful. It is thencontinued, by iterative steps, until it matches the target
value. The usefulness of this method depends on thewidth of the steps in the change in power. Since it is alinear problem, the calculation can be carried out in avery short time.
If a little more time is expended, a design of extremely
high accuracy can be achieved. It is sufficient to apply thedesign support system once or twice in the final step toreach a direct numerical solution to Equation (1), to revise
the longitudinal distribution of signal power and pump
power to a precise value. We have called the method inwhich Equation (1) is solved in the final step the “highaccuracy procedure” and that in which it is not solved in
the final step the “high-speed procedure.”
4.3 Verifying Accuracy
To verify the accuracy of the design method describedabove, we derived Raman gain spectra from thepump power allocation optimized using the method
and compared them with those obtained by thesuperposition rule. For the Raman amplifier we used
100 km of dispersion-shifted fiber (DSF), 10 backward
pumps, and 97 signal channels from 1530 to 1610 nm
at 100-GHz spacing. We assumed an input signal powerof 0 dBm/ch and pump wavelengths already given.
Pump powers were optimized so that Raman gain of 20������������� ������������� ���� ���� ε (zi+1)  ε (zi)  =Ai 

η
(z
)
i+1


 η (zi)

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Raman gain spectra.

dB was realized throughout the signal bandwidth. The
Raman gain spectrum used as the initial value averaged
approximately 2 dB, a difference of 18 dB from the target
value.
Figure 9 shows the Raman gain spectra resulting fromthe design. The results from the high-accuracy procedure
are in virtually full agreement with the results from thesuperposition rule, with a difference of 0.03 dB at themost. From these results it can be seen that our method
provides an effective means of determining the optimum
pump power distribution with respect to given pumpwavelengths.
We then attempted verification using the results
obtained from the experiments in Section 3 above.
Optimum pump powers were determined taking 42 mW
as the initial value of pump power for each pump andthe Raman gain spectra obtained in the experiments as
the target value. This was then compared with the pump
powers used in the experiments.
Figure 10(a) shows the Raman gain obtained
experimentally, which becomes the target value, andthe initial value of Raman gain. The pump powers
optimized by the high-accuracy procedure differedfrom the experimental values by 2 mW in total, and by
a maximum of 6 mW with respect to any one of thepumps. This discrepancy, based on a total pump powerof 390 mW, was 0.5% and 1.5% respectively. The Ramangain obtained by the high-accuracy procedure agreed
with the experimental results to within 0.2 dB. Figure10(b) shows the Raman gain spectra obtained by thehigh-speed procedure using step widths of 0.5, 1 and
2 dB, the Raman gain spectra obtained by the highaccuracy procedure, and the Raman gain spectra fromexperimental results. It can be seen that the smaller
the step width is made the closer the approach to theexperimental results and the greater the accuracy. Whatis more the Raman gain spectra obtained at a 0.5-dB step
width is in substantial agreement with the results obtainedby the high-accuracy procedure.
4.4 Feedback Control Simulation
Finally let us present an example of feedback control
simulation for the high-speed procedure. Since the highspeed procedure uses linear approximation for nonlinear
equations, it is useful with respect to control from thestandpoint of the calculation time. Let us consider a

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Applying a Numerical Simulation Technique to the Design of WDM-Pumped Raman Amplifiers, and Methods for the Automatic Determination of Pump Powers������������� ����

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Figure 11 Output signal power before failure, after failure
and after control restoration by one step only.

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���������� ����Figure 12 Difference in output signal power from before
the failure, obtained by control using a onestep procedure and for step widths of 1 and 0.5
dB.

Figure 10 Raman gain spectra.

system in which are connected two Raman amplifiers set
to a net gain of 0 dB, each using 100 km of dispersionshifted fiber (DSF) with 10 backward pumps. The signal
condition was 97 signal channels from 1530 to 1610
nm at 100-GHz spacing, and an input signal power of 0
dBm/ch. That is to say the target output signal power was
0 dBm/ch.
Now let us assume a condition in which one of thelaser diodes in the first of these Raman amplifiers fails,
resulting in the output signal power of the second of theRaman amplifiers deviating from the initial value. To copewith this a process is undertaken whereby the pump
powers of the second Raman amplifier are controlled so
as to return the output signal power to the initial value.
First the output signal power of the second Raman
amplifier is monitored, and the amount of change in thesignal power input to it is predicted using Equation (7).
The distribution of the pump power required to return
output signal power to the initial value in then similarly
determined using Equation (7).
Figure 11 shows the output signal power before thefailure, after the failure and after control restoration basedon a procedure of one step only. Control simulations
were also carried out for step widths of 0.5 and 1 dB.
Figure 12 shows these results in terms of the difference in
output signal power from before the failure. As in Figure10(b), it can be seen that the smaller the step width ismade the better the output signal is restored. In the case����of a 0.5-dB step width, the flatness (difference between
maximum and minimum values) was restored from 1.99
dB to 0.54 dB. This is extremely close to that value of 0.50
dB before the failure.

5.

SUMMARY

A numerical simulator has been developed for predicting
the characteristics of Raman amplifiers. This simulator
has been applied to configuring Raman amplifiers
proposed by the authors, having gradually increased
numbers of pump wavelengths. Experimental andsimulated values were found to be in good agreement,
and through comparisons of methods for upgradingin the shorter wavelength direction and the longer
wavelength direction, it was found that the shorter
wavelength upgrade was desirable in terms of pump lightcost.
An automatic design method was also developed
for predicting pump powers on the basis of linear
approximation. It was confirmed that even when the initialvalue of pump power diverged from the optimum value(18 dB for Raman gain), this design method showed
powerful converging performance.

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Applying a Numerical Simulation Technique to the Design of WDM-Pumped Raman Amplifiers, and Methods for the Automatic Determination of Pump Powers

ACKNOWLEDGMENT
In conducting the research reported here the authors
would like to express their thanks to Mr. Akira Aikawa
and to Mr. Hiroyuki Katayama of Furukawa Information
Technology Corp. for their cooperation in building thesimulator.

REFERENCES
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