Contribution to the achievement of SDGs
by Furukawa Electric Group

Efficient Numerical Scheme
for Simulating Complex Raman Amplifiersin Optical Telecommunication SystemsZoltán Papp*1, *2, Zoltán Várallyay*1, Ryuichi Sugizaki *3, Shigehiro Takasaka*3, Norihiro Ohishi*3,Junji Yoshida*3, Daichi Ogata*3, Shunya Hayashi *3, Toshio Kimura*3
This paper introduces a numerical scheme that efficiently predict Raman amplifier behaviors with diverse pump configurations. The scheme handles bothRaman pumps in both directions and multiple incoherent pumps with broad spectra.
Incorporating frequency-dependent loss, Raman gain, and inter-signal Raman gain, it enables
accurate evaluation of signal evolution and optimization for flat gain in wavelength-division multiplexing (WDM) systems. For backward pumps, which interact with all signals and pumps dueto the nonlinear Raman gain effect, we employ an iterative optimization method that converges
to a solution with acceptable small error after a few optimization cycles. This approach effectively addresses the inherent boundary value problem associated with backward propagating
pumps. Validation with measurements demonstrates the scheme's accuracy, empoweringresearchers and engineers to design and optimize Raman amplifiers for superior performance in
optical telecommunication systems.

ABSTRACT1.

INTRODUCTION

Furukawa Electric Co., Ltd. has been a pioneer in Raman
amplifier technology since 2000 actively contributing to
its research, development, and implementation1), 2). Raman
amplifiers can play a significant role in boosting signal
power within long-haul optical telecommunication links.
Unlike their rear-earth doped counterparts, Raman amplifiers offer several advantages: firstly, they can amplify signals across any telecommunication band with the selection of the appropriate pump wavelength. Secondly, theygenerate significantly less amplified spontaneous emission (ASE), minimizing signal noise. Moreover, various
pump configurations, including the “WDM pump”
scheme, can be employed to achieve a broad and flat
gain profile for amplified signal channels, ensuring optimal performance in WDM systems 1).
Designing and optimizing these amplifiers require accurate numerical simulations that can handle various pumpconfigurations and the nonlinear Raman gain effect in thatevery propagating signal or pump channels affects every
other signal or pump channels. This paper presents anovel approach and efficient numerical scheme for simulating Raman amplifiers with diverse pump scenarios,
including Raman pumps in both forward and backward
directions, and incoherent pumps with broad spectral
Simulation Group 1: General Simulation, FETI Ltd.
Physical Doctoral School, Szeged University, Hungary*3
Photonics Laboratory, R&D Division, Furukawa Electric Co., Ltd.
*1
*2

regions in the forward direction.
Our proposed scheme addresses the challenges associated with these complex pump configurations. For
backward-propagating Raman pumps, an iterative method tackles the inherent boundary value problem, ensuring
accurate propagation calculations. Additionally, the software incorporates frequency-dependent loss and Ramangain coefficients, capturing the realistic behavior of various optical fibers. Furthermore, it accounts for inter-signal Raman gain, a critical factor for accurate gain prediction in densely spaced WDM systems.
This simulation tool excels in evaluating signal evolution
and achieving flat gain for WDM signals, employingdiverse combinations of Raman pumps. Its effectivenessis demonstrated through comparisons with actual measurements reported in our previous work 3), showing
excellent agreement between simulated and experimental
results.
This paper presents a numerical scheme implemented
in MATLAB that efficiently handles diverse pump configurations, including Raman pumps in both forward andbackward directions and incoherent pumps with broad
spectral regions in the forward direction. The scheme
incorporates key nonlinearities such as frequency-dependent loss/gain and inter-signal Raman gain, enabling
accurate prediction of WDM signal evolution and optimization for flat gain (i.e., minimizing gain variations acrossdifferent signal channels) using various pump combinations. The scheme’s accuracy is validated by comparing
our simulations to published experimental results, demFurukawa Electric Review, No. 55 2024

53

Efficient Numerical Scheme for Simulating Complex Raman Amplifiers in Optical Telecommunication Systems

onstrating excellent agreement. This work provides avaluable tool for researchers and engineers involved indesigning and optimizing Raman amplifiers for enhanced
performance in optical telecommunication systems.

2.

THEORY

Several prior studies 4)-8) have simulated the stimulated
Raman effect using various approaches, primarily differing in their handling of backward propagating signals andequation formulations. This work introduces frequency
dependency of Raman gain by offering two approaches:
Cordina’s estimation 9) and utilizing measured effective
core area data to scale the Raman gain coefficient.
In our model, the power evolution is calculated by integrating Eq. (1), which contains the frequency dependent
effective core area data in this description equivalently toRef. 4) except that we do not include backward Reyleighscattering here.

(1)
where P is the propagating power channels in Watt
units, both in forward and backward direction that is indicated by the ± sign in the superscript. z is the length
coordinate along the fiber, v is the frequency of the specific channel that is evaluated. α the background loss ofthe medium. CR0 is the measured Raman gain coefficientefficiency corrected by a scalar multiplication with a value
very close to 1 which scalar multiplication can take into
consideration the uncertainty of the measurement. We
investigate the effect of this multiplication factor later
below when we compare the difference between measurements and simulations. A0 is the effective core area ofthe fiber at the frequency where the measurement wasdone. One can see that the Raman gain coefficient efficiency (CR0 ) scales with the ratio of effective core areas at
new and original frequencies, enabling calculation ofRaman gain at different frequencies. ξ represents the frequency of that channel that influences channel v.
Other possible method to implement the frequency
dependence of the Raman gain along with the measured
frequency dependent effective core area is to use the following estimation by Cordina 9) :(2)Where f 0 is the frequency where the Raman gain coefficient efficiency was measured, f is the frequency that
influences other channels depending on the Δ f frequency
difference. ε is a material dependent coefficient that is for
AllWave fibers, for example, has a fitting value of 2.8.
Boundary values of all signals and pumps are essential
for the solution of Eq. (1), forward and backward propagating signals need to have the initial values at the begin-ning and at the end of the fiber, respectively. Solutions arecarried out by a second order Runge-Kutta (RK2) method
combined with a matrix formalism that contains all thepossible interactions between the propagating channels.
Key part in every Raman amplifier simulation is the handling of backward propagation. We studied several methods4) -8) to treat this problem using different estimation
techniques to determine the starting values for backward
signal channels which was used to integrate the equationin the forward direction. Iteration techniques used to minimize the difference between the boundary conditions andthe actual powers at the end of the fiber. Our backward
propagation method can be described in five steps which
are the followings:
1.	Propagation only in the forward direction (all channels,except backward pumps propagate in the forward
direction).
2.	Use the obtained power values at the end of the fiberfor doing now backward propagation including thebackward pumps.
3.	The result of the backward propagation provides initialvalues for the backward pumps at z=0 m and use
them as starting values for the 1st iteration step.
4.	Apply the Nelder-Mead simplex method, to find theexact starting values for the backward pumps after afew iteration steps which provide acceptable error
compared to the specified boundary values at z =L.
5.	Use the starting values from the previous (4th ) step, tomake the final propagation with and without amplification to calculate on/off gain.
This method uses the experiences from the above referenced papers combined with our matrix formalism of
signal interactions to achieve high accuracy within shortcomputational time.

3.

RESULTSWe differentiate two kinds of pumps in these simulations:narrow-linewidth laser diodes, effectively modeled as single-frequency “Dirac delta” functions (c-pumps, or coherent pumps), and broad-spectral pumps modeled by sampling their spectra in several frequency points and theiroverall spectral distribution is Gaussian (i-pumps, or incoherent pumps). During solver testing, simulated and measured On/Off gains were compared for various pumpconfigurations and signal channels. On/Off gain, the ratioof amplified to unamplified signal powers in decibels, is acommon performance metric for Raman amplifiers.
Figure 1 compares simulations and measurements for
2 dBm input signals amplified by a 1423 nm c-pump in an
80 km AllWave fiber with varying pump powers. Excellent
agreement is observed. The experiment was repeatedusing a 1425 nm i-pump (right-hand side of Figure 1).
Although the measured spectrum was sampled for thesimulation, the correlation between measured and simulated results is good, with discrepancies attributable toinherent uncertainties in Raman gain measurements, as
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Efficient Numerical Scheme for Simulating Complex Raman Amplifiers in Optical Telecommunication Systems

1550 mW measurement
50 mW simulation
100 mW measurement
100 mW simulation
150 mW measurement
150 mW simulation
200 mW measurement
200 mW simulation

6200 mW measurement

10On/Oﬀ Gain / dB

On/Oﬀ Gain / dB

200 mW simulation
300 mW measurement
300 mW simulation
400 mW measurement
400 mW simulation

504

201530

1540

1550

1560

1570

1530

1540

Wavelength / nm

1550

1560

1570

Wavelength / nm

Figure 1 On/Off gain measurement-simulation comparison using c-pump(left) and i-pump(right) with different pumping powers.

-20 dBm measurement-4 dBm measurement-20 dBm simulation-4 dBm simuation-8 dBm measurement

0 dBm measurement-8 dBm simulation

0 dBm simulation

On/Oﬀ Gain / dB

98deviations in the calculated, stimulated Raman gain values. Interestingly, we observed a tendency for the solver
to overestimate gain in certain cases. To address thisissue and to obtain better agreement with the measurements, we reduced the Raman gain and recalculated theaverage difference across various gain efficiencies asshown in Figure 3. As evident in the figure, lowering thegain by 2-3% significantly improves the solver’s accuracy,
particularly in the low input power regime. At 0.96-0.97
multiplication factor we get average deviations for all
input signals that are not exceeding the 0.3 dB average
error. This is a promising correction to get even better
accuracy, therefore we implemented this correction factor
to our code.
Average diﬀerence between
measurement and simulation / dB

discussed later. Still the difference between the measurements and simulations after 80 km of amplification
remains well below the 0.4 dB error.
We further explore multi-pump configurations, employing five c-pumps as secondary pumps and four i-pumps
as primary pumps to amplify a 47-channel signal acrossthe C and L bands in a 40 km AllWave fiber. Theoretically,
secondary Raman pumps amplify the broad-spectrum
primary pumps, contributing to a flat gain profile desirable
in telecommunication systems. Figure 2 compares measured and simulated On/Off gains for various input signalpowers. The results exhibit good agreement, with noticeable discrepancies only at longer wavelengths. The average difference remains always below 0.5 dB. This close
match validates the simulation’s accuracy, particularly
considering inherent uncertainties in Raman gain measurements.

-20 dBm

1.2-10 dBm-8 dBm

1.0-6 dBm
-4 dBm0.8-2 dBm
0 dBm

0.6
0.4
0.2
0.0
0.90

0.92

0.94

0.96

0.98

1.00

Multiplication factorFigure 3 Average difference between the measurements
and simulations using a multiplication factor for themeasured Raman gain coefficient efficiency data.
Calculations are repeated for several input signalvalues. Minimum deviation from the measurements are
at 3% smaller Raman gain values than the original one.

76
1530

1550

1570

1590

1610

Wavelength / nm
Figure 2 Comparison of measurement and simulation for
On/Off gain in multi pump system with different signal
input powers.

Accounting for measurement uncertainties in Ramangain coefficients can further enhance accuracy. These
uncertainties, typically ranging from 1-3%, can induce

We extensively tested our code’s backward pump solving capabilities with diverse configurations and the statistic of the computational time is illustrated in Figure 4.
Note, that most of the calculations require 200-1000
steps with the RK2 method for the 40-80 km propagation
lengths and the number of the propagating signals
including the sampling points of the i-pumps gives morethan 200 frequency channels that are propagating simulFurukawa Electric Review, No. 55 2024

55

Efficient Numerical Scheme for Simulating Complex Raman Amplifiers in Optical Telecommunication Systemstaneously. Figure 4 encompasses various setups, including 1-6 second-order forward c-pumps, 1-4 first-orderforward i-pumps, and fiber lengths ranging from 40 to
80 km. Figure 4 presents the average simulation time for30-40 different Raman amplifier configurations, alongwith the 95% confidence interval, indicating the typicalrange within which most simulations are completed using
an average laptop computer Today. This information aidsin understanding the code’s computational efficiency
under different load scenarios.
500

Calculation time / s43-427
400

ability of our tool, establishing it as a valuable resource
for researchers and engineers designing and optimizing
Raman amplifiers for enhanced performance in practical
optical telecommunication systems.
This work opens doors for further exploration of complex pump configurations and optimization algorithms
within Raman amplifier design. Our plan to integrate thisRaman Amplifier code with FETI’s propagation simulator(PropSim) 10) will give further opportunities in complex andnonlinear optical design. The ability to achieve high fidelity simulations while maintaining close correlation withexperimental results paves the way for significant
advancements in optical fiber-based system designs.
* Allwave is a registered trademark or trademark of OFS Fitel, LLC in
Japan and other countries.

300
200

REFERENCES

29-157

100
6-20

18-48

01

23

4

Number of backward pumps
Figure 4 Average calculation time and 95% confidence interval
as functions of the number of backward pumps.
Blue numbers are representing the low and upper
end values of the confidence interval. Gray dashedline is just for guiding the eyes and emphasizing theexponential growth of the average values.

4.

CONCLUSION

In conclusion, we have presented a modified and efficient
numerical scheme for simulating complex Raman amplifiers in optical telecommunication systems. This scheme
effectively handles diverse pump configurations, including Raman pumps in both directions and incoherent
pumps with broad spectra.
Furthermore, the software incorporates frequencydependent loss and Raman gain coefficients, along withinter-signal Raman gain, offering a comprehensive simulation environment. By leveraging this scheme, we demonstrated the ability to accurately evaluate signal evolution and achieve flat gain performance for WDM signals
using various pump combinations.
The effectiveness of our proposed scheme is underscored by the excellent agreement between our simulations and actual measurements reported in our previous
work 3). This agreement validates the accuracy and reli-

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