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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 23 — Nov. 13, 2006
  • pp: 11024–11035
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Frequency domain model to calculate the pump to signal RIN transfer in multi-pump Raman fiber amplifiers

Junhe Zhou, Jianping Chen, Xinwan Li, Guiling Wu, Yiping Wang, and Wenning Jiang  »View Author Affiliations


Optics Express, Vol. 14, Issue 23, pp. 11024-11035 (2006)
http://dx.doi.org/10.1364/OE.14.011024


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Abstract

In this paper, a novel frequency domain model to compute the pump to signal relative intensity noise (RIN) transfer in multi-pump Raman fiber amplifiers (RFAs) is proposed. The analytical expressions for RFAs with single pump and single signal channel are derived as a specific case of the model. The formulas exactly agree with the published results both for the co-pumped and counter-pumped RFAs. Afterwards, the pump to signal RIN transfer in multi-pump RFAs is studied numerically with thorough discussions.

© 2006 Optical Society of America

1. Introduction

RFAs have been intensively investigated due to their promising merits for long haul DWDM systems, such as the low noise figure and the broad gain bandwidth. With multiple pumps, over 100nm bandwidth amplification has been realized [1

1. S. Namiki and Y. Emori, “Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelengthdivision-multiplexed high-power laser diodes,” IEEE J. Sel. Top. Quantum Electron. l 7, 3–16 (2001).

].

Noise figure analysis is one of the key topics in RFAs’ design and analysis. The pump to signal RIN transfer is one of the major causes for the noise in RFAs. If there is intensity modulation to the pump power, the relative intense noise (RIN) will transfer from the pump to the signal wavelength and degrade the system performance. There have been relative papers published on this issue with analytical expressions [2–4

2. C. R. S. Fludger, V. Handerek, and R. J. Mears, “Pump to Signal RIN Transfer in Raman Fiber Amplifiers,” IEEE, J. Lightwave Technol. 19, 1140–1148 (2001). [CrossRef]

], and the results showed that counterpumped RFAs have lower RIN transfer than co-pumped RFAs at high modulation frequency. This is caused by the different walk-off effect between the counter-propagated pump and the co-propagated signal wave, i.e., the group velocity difference between the pump and signal wave is more significant in counter-pumping scheme. This discovery has brought more attention to the counter-pumped RFAs. It was not until the very low RIN (less than −110dB) laser diodes were available that the co-pumped RFAs received comparable attention. The existing analysis gives deep insight into the problem; however, the analysis mentioned above focused on RFAs with single pump and single signal channel, which are not the most commonly used RFAs. To achieve the broadband amplification, usually WDM pumps are required to balance the gain spectrum and the noise figure [5–10

5. X. Zhou, C. Lu, P. Shum, and TH. Cheng, “A Simplified Model and Optimal Design of a Multiwavelength Backward-Pumped Fiber Raman Amplifier,” IEEE Photon. Technol. Lett. 13, 945–947 (2001). [CrossRef]

]. Meanwhile, WDM transmission requires simultaneous transmission of a bunch of signal channels. For the case of multiple pumps and multiple signal channels, there has been no frequency domain model for the pump to signal RIN transfer. Certainly, one may directly calculate RIN transfer in the time domain [11

11. M. Karásek and M. Menif, “Channel Addition/Removal Response in Raman Fiber Amplifiers: Modeling and Experimentation,” IEEE J. Lightwave Technol. 20, 1680–1688 (2002). [CrossRef]

]. However, this approach is very time-consuming and does not give a clear picture of the frequency response. Moreover, it prevents the close formula from being derived.

In this paper, we proposed a novel frequency model to evaluate the pump to signal RIN transfer in WDM pumped RFAs. The model is derived without undepleted pump assumption. And the complicated interactions between the pump-to-pump, pump-to-signal and signal-tosignal are taken into account. It is capable of evaluating the RIN transfer in RFAs with multiple pumps and multiple signals. The calculation of the RIN transfer can be accomplished by evaluating a matrix, which can be computed via Picard method [9

9. Jaehyoung Park, Pilhan Kim, Jonghan Park, H. Lee, and Namkyoo Park, “Closed Integral form expansion of Raman equation for efficient gain optimization Process,” IEEE Photon. Tech. Lett. 16, 1649–1651 (2004). [CrossRef]

] or Runge-Kutta method. With some simplification, the analytical formulas for RFAs with single pump and single signal channel are derived and show exact agreement with the published results [2

2. C. R. S. Fludger, V. Handerek, and R. J. Mears, “Pump to Signal RIN Transfer in Raman Fiber Amplifiers,” IEEE, J. Lightwave Technol. 19, 1140–1148 (2001). [CrossRef]

]. After the demonstration of the validity of our model in the specific case, the detailed simulation results are presented.

2. Mathematical modeling

The power evolution in RFAs is governed by the following equations [12

12. G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, New York, 2001)

]:

s(i)Piz+1vg,iPit=j=1,jin+mg(νi,νj)PiPjαiPi
i=1......n+m
(1)

where s(i) is the sign function that indicates the direction of the wave transmission, 1 indicates the forward propagation and -1 indicates the backward propagation, respectively, m is the pump number, n is the signal channel number, vg,i is the group velocity at frequency νi, αi is the attenuation coefficient at frequency νi, g(νi, νj) is the Raman gain coefficient between the frequencies νi and νj:

g(νi,νj)={gr(νiνj)2Aeff(νi>νj)(νjνi)gr(νjνi)2Aeff(νi<νj)
(2)

where gr(νi, νj) is the Raman gain spectrum, Aeff is the effective area of the fiber, factor 2 stands for the polarization effect.

To derive the equations governing the RIN transfer in frequency domain in WDM pumped RFAs, we assume that the amplitude of the pump modulation is relatively small compared with the steady state pump power Pi(z), whose derivative of t is zero. The RIN power on each pump as well as the consequent transferred RIN power on each signal channel is denoted as ΔPi(z,t), which has the nonzero derivative of t, ΔPi(z,t)+Pi(z) satisfies equation (2), and if the second order term is omitted, we have:

s(i)ΔPi(z,t)z+1vg,iΔPi(z,t)t=j=1,jin+mg(νi,νj)Pi(z)ΔPj(z,t)
+j=1,jin+mg(νi,νj)Pj(z)ΔPi(z,t)αiΔPi(z,t)
i=1......n+m
(3)

Taking Fourier transform on Eq. (3) and rewriting it in matrix form, one has:

ΔP(z,ω)z=AΔP(z,ω)
(4)

where:

ΔP(z,ω)=(ΔP1(z,ω)ΔPm(z,ω))
A=(s(1)(j=1,j1m+ng(ν1,νj)Pjα1jωvg1)s(1)g(ν1,νm)P1s(n+m)g(νn+m,ν1)Pn+ms(n+m)(j=1,jn+mn+mg(νn+m,νj)Pjαn+mjωvg,m+n))

The solution of the equation has the following form:

ΔP(L,ω)=MRINΔP(0,ω)
MRIN=limΔz0k=1LΔz(I+A(kΔz)Δz)
(5)

M RIN can be evaluated numerically via forward Euler method, Runge-Kutta method or Picard method. For the co-pumped RFAs, ΔP(0,ω) is known and the RIN transfer ΔP s(L, ω) can be simply calculated by multiplying the matrix. For the counter-pumping scheme, the RIN transfer can also be evaluated. We achieve this by separating the vector ΔP(0,ω) into [ΔPp(0,ω)ΔPs(0,ω)] , ΔP(L,ω) into [ΔPp(L,ω)ΔPs(L,ω)] and the matrix M RIN into [M11M12M21M22] respectively. Since the RIN on the signal at the input end of the fiber is zero, ΔP s(0, ω)=0, therefore the Eq. (5) becomes:

[ΔppoutΔpsout]=[M11M12M21M22][Δppin0]
(6)

and the signal RIN at the output end is

Δpsout=M21M111Δppout
(7)

3. Analytical formula for RIN transfer in RFAs with single pump and single signal channel

Before investigating the RIN transfer in WDM pumped RFAs, a specific case of RFA, i.e, one pump and one signal wavelength RFA, will be studied using the model proposed above. The case has already been studied by C. Fludger et al. [2

2. C. R. S. Fludger, V. Handerek, and R. J. Mears, “Pump to Signal RIN Transfer in Raman Fiber Amplifiers,” IEEE, J. Lightwave Technol. 19, 1140–1148 (2001). [CrossRef]

] with analytical formulas. With the undepleted pump assumption, the same expressions can be derived using Eq. (4).

With the undepleted pump assumption, Eq. (4) for the co-pumped RFA becomes:

ΔPp(z,ω)z+jωvg,pΔPp(z,ω)=αpΔPp(z,ω)
ΔPs(z,ω)z+jωvg,sΔPs(z,ω)=gPp(z)ΔPs(z,ω)+gPs(z)ΔPp(z,ω)αsΔPs(z,ω)
Pp(z)=Pp0exp(αpz)
Ps(z)=Ps0exp(0zgPp0exp(αpl)αsdl)
(8)

the solution of the first equation in (8) is:

ΔPp(z,ω)=exp[[jωvg,pαp]z]ΔPp(0,ω)
(9)

Substituting (9) into the second equation of (8), one has:

ΔPs(z,ω)z=[jωvg,sαs+gPp0exp(αpz)]ΔPs(z,ω)
+gPs0exp[0zgPp0exp(αpl)αsdl]exp[[jωvg,p+αp]z]ΔPp(0,ω)
(10)

The solution of (10) is:

ΔPs(L,ω)=exp(0L[jωvg,sαs+gPp0exp(αpz)]dz)
*gPs0ΔPp(0,ω)0Lexp(((jωvg,p+αp)+jωvg,sz)z)dz
=gPs0ΔPp(0,ω)exp(jωvg,sL)Gvg,s(1exp((j2πfbvg,sαp)L))αpvg,s+j2πfb
(11)

where:

G=exp[0LgPp0exp(αpl)αsdl]=gPp0LeffαsL
Leff=1exp(αpL)αp
b=j2π(1vg,svg,p)
jω[1vg,p+1vg,s]=j2πfbvg,s
(12)

Hence, one has:

ΔPs(L,ω)Ps(L)=ΔPs(L,ω)Ps0G
=ΔPp(0,ω)Pp0lnGLeffexp(jωvg,sL)vg,s[1exp[(j2πfbvg,sαp)L]]αpvg,s+j2πfb
(13)

By defining

rs=ΔPs(L,ω)Ps(L)2
rp=ΔPp(0,ω)Pp(0)2
(14)

one has:

rs=rp(lnG)2[vg,sLeff]2(αpvg,s)2+(2πfb)2
*[12cos[2πfbvg,sL]exp(αpL)+exp(2αpL)]
(15)

Surprisingly, Eq. (15) is exactly the same as the equation derived in Ref. [2

2. C. R. S. Fludger, V. Handerek, and R. J. Mears, “Pump to Signal RIN Transfer in Raman Fiber Amplifiers,” IEEE, J. Lightwave Technol. 19, 1140–1148 (2001). [CrossRef]

].

Similarly, the equation for counter-pumping scheme can also be derived using Eq. (4), which also exactly agrees with the corresponding equation in Ref. [2

2. C. R. S. Fludger, V. Handerek, and R. J. Mears, “Pump to Signal RIN Transfer in Raman Fiber Amplifiers,” IEEE, J. Lightwave Technol. 19, 1140–1148 (2001). [CrossRef]

].

4. Numerical results and discussion

In our simulation, a standard single mode fiber with the length of 50km is used as the gain media. The second order dispersion coefficient β 2 and the third order dispersion coefficient β 3 are -20.4071711919 ps 2/km and 0.17348694743 ps 3/km at the wavelength of 1550nm. 80 channels of signals are launched into the fiber with 100GHz channel spacing. The input signal power for each channel is –10dbm. Different pumping schemes are investigated upon this piece of fiber sharing the same pumping wavelength, i.e. 1425nm, 1440nm, 1450nm, 1465nm and 1490nm. The data of Raman gain spectrum comes from the published literature [13

13. C. Fludger, A. Maroney, N. Jolley, and R. Mears, “An analysis of the improvements in OSNR from distributed Raman amplifiers using modern transmission fibers,” in Optical Fiber Communication Conference Technical Digest (Optical Society of America, 2000) 100–102.

]. The gain profile of the RFA is optimized using the matrix-based algorithms [9

9. Jaehyoung Park, Pilhan Kim, Jonghan Park, H. Lee, and Namkyoo Park, “Closed Integral form expansion of Raman equation for efficient gain optimization Process,” IEEE Photon. Tech. Lett. 16, 1649–1651 (2004). [CrossRef]

], so that the net gains at the different signal channels are almost the same.

Since the RIN transfer in multi-pump RFAs is determined by the matrix, as derived in Eq. (5) and (7), the total RIN transfer for one signal channel is the linear combination of the RIN transfer induced by different pumps. Hence, we present our simulation results of the RIN transfer assuming that only one pump is regarded as the noise source and other pumps have no RIN. The RIN transfer is defined as 10logrsrp as Ref [2

2. C. R. S. Fludger, V. Handerek, and R. J. Mears, “Pump to Signal RIN Transfer in Raman Fiber Amplifiers,” IEEE, J. Lightwave Technol. 19, 1140–1148 (2001). [CrossRef]

] suggested, where rs and rp are defined in Eq. (14). Since rs is proportional to rp, the RIN transfer value 10logrsrp will always be the same despite different pump noise level. However, as mentioned in part 2, the pump noise power should be relatively weaker than the pump power. Usually this assumption is valid.

4.1 co-pumping

The Raman gain profile of the co-pumping scheme is illustrated in Fig. 1. The gain profile has been equalized and the corresponding pump powers are 440.8mW, 312.9mW, 116.7mW, 180mW, and 39.1mW. The maximum gain ripple is 0.9dB. For comparison, we also calculated the Raman gain profile in a wider range, i.e. from 1515nm-1605nm with 110 channels. It can be seen that the gain difference between the two is quite slight. In fact, during simulation, we discover that RIN transfer difference between the different signal configurations is also slight. Without loss of generality, 80 channels of signal will be used in the later simulations.

Fig. 1. Net gain spectrum of the co-pumped 50km RFA

Although the gain spectrum has been flattened, it is not the case for RIN transfer. Different signal wavelength suffers different RIN transfer. As illustrated in Fig. 2, the RIN transfer induced by pump 1 shows that longer wavelength suffers less RIN transfer, this can be explained by the fact that signal channel at longer wavelength has larger gain from this pump, and also by the fact that signal channel at longer wavelength has larger velocity difference to the pump than the shorter wavelength signal channel.

Fig. 2. RIN transfer of the co-pumped 50km RFA from pump 1 to different signal channels

In Fig. 3, Fig. 4 and Fig. 5, the RIN transfer from the pumps to the signal channels at the wavelengths of 1530nm, 1561nm and 1594nm is demonstrated. Different pump induce different RIN transfer on the signal channels. For signal channel at 1530nm, pump 1 causes the most significant RIN transfer while for signal channel at 1561nm, pump 4 plays the major role. From the figures, it can be inferred that the pump providing more gain on the signal will cause more RIN transfer. One more interesting phenomenon is that the RIN transfer from pump 4 to the signal channel at 1530nm reaches maximum at the frequency of about 200MHz. This has not been observed in single pump case and it might be caused by the complex coupling between the pumps and the signals, which transfer the RIN from pump 4 to pump 3.

Fig. 3. RIN transfer of the co-pumped 50km RFA at 1530nm
Fig. 4. RIN transfer of the co-pumped 50km RFA at 1561nm
Fig. 5. RIN transfer of the co-pumped 50km RFA at 1594nm

4.2 Counter-pumping

For the counter-pumping scheme, the Raman gain spectrum is illustrated in Fig. 4. The gain profile has also been equalized and the corresponding pump powers are 397.1mW, 288.2mW, 111.4mW, 180.1mW, and 47.4mW. The maximum gain ripple is about 1dB.

Fig. 6. net gain spectrum of the co-pumped 50km RFA

In Fig. 7, RIN transfer from pump 1 to different signal wavelength is illustrated. Since the difference of the group velocity between the pump and the signal wavelength is larger than that of the co-pumping scheme, the decrease of the RIN transfer versus frequency is much faster. This is consistent with the published results. Like co-pumping scheme, flat net gain profile does not guarantee flat RIN transfer, i.e. different signal wavelength experiences different RIN transfer.

Fig. 7. RIN transfer of the counter-pumped 50km RFA from pump 1 to different signal channels

The RIN transfer from pump 1 to pump 5 to the signal channel at 1530nm, 1561nm and 1594nm is plotted in Fig. 8, Fig. 9 and Fig. 10. Similarly, it can also be inferred that in counter-pumping scheme the pump providing more gain on the signal will cause more RIN transfer.

Fig. 8. RIN transfer of the counter-pumped 50km RFA at 1530nm
Fig. 9. RIN transfer of the counter-pumped 50km RFA at 1561nm
Fig. 10. RIN transfer of the counter-pumped 50km RFA at 1594nm

5. Conclusion

We have proposed a novel frequency domain model to compute the pump to signal RIN transfer in WDM pumped RFAs without undepleted pump assumption. When the pump and signal numbers are reduced to one and the undepleted pump assumption is used, the analytical expressions derived from the proposed model exactly agree with the published results. The verified model is used to compute the RIN transfer in multi-pump RFAs. The results show that the forward pumping scheme suffers more severe pump to signal RIN transfer than counter-pumping scheme. This is consistent with the published results. The cutoff frequency of RIN transfer is around 10M Hz for the forward pumping scheme and around 1kHz for the backward pumping scheme respectively. RIN on different pump wavelengths result in different RIN transfer on signals. In the results of Fig. 2–5 and Fig. 6–10, it can be roughly seen that the more gain the pump produces, the more severe RIN transfer is. However, as shown in Fig. 3, complex pump interactions may affect the RIN transfer and result in the shift of the peak RIN transfer frequency. Different signal wavelengths also suffers different RIN transfer, it also roughly depends on the gain of the pumps. By adjusting the pump RIN, one may balance the noise figure in RFAs, and this is a rather interesting topic worth studying.

Acknowledgments

This work is partially supported by NSFC (ID: 60377013, 90204006, 60507013), Ministry of Education, China (ID: 20030248035)

References and links

1.

S. Namiki and Y. Emori, “Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelengthdivision-multiplexed high-power laser diodes,” IEEE J. Sel. Top. Quantum Electron. l 7, 3–16 (2001).

2.

C. R. S. Fludger, V. Handerek, and R. J. Mears, “Pump to Signal RIN Transfer in Raman Fiber Amplifiers,” IEEE, J. Lightwave Technol. 19, 1140–1148 (2001). [CrossRef]

3.

M. D. Mermelstein, C. Headley, and J.-C. Bouteillier, “RIN transfer analysis in pump depletion regime for Raman fiber amplifiers,” Electron. Lett. 38, 403–405, (2002). [CrossRef]

4.

Bruno Bristiel, Shifeng Jiang, Philippe Gallion, and Erwan Pincemin, “New Model of Noise Figure and RIN Transfer in Fiber Raman Amplifiers,” IEEE Photon. Technol. Lett. 18, 980–982 (2006). [CrossRef]

5.

X. Zhou, C. Lu, P. Shum, and TH. Cheng, “A Simplified Model and Optimal Design of a Multiwavelength Backward-Pumped Fiber Raman Amplifier,” IEEE Photon. Technol. Lett. 13, 945–947 (2001). [CrossRef]

6.

M. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and X. Li, “Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,” IEEE Photon. Technol. Lett. 13, 948–950 (2001).. [CrossRef]

7.

V.E Perlin and H.G. Winful, “Optimal design of flat-gain wide-band fiber Raman amplifiers,” IEEE, J. lightwave technol. 20, 250–254 (2002). [CrossRef]

8.

V.E. Perlin and H.G. Winful, “On distributed raman amplification for ultrabroad-band long-haul wdm systems,” IEEE, J. lightwave technol. 20, 409–417 (2002). [CrossRef]

9.

Jaehyoung Park, Pilhan Kim, Jonghan Park, H. Lee, and Namkyoo Park, “Closed Integral form expansion of Raman equation for efficient gain optimization Process,” IEEE Photon. Tech. Lett. 16, 1649–1651 (2004). [CrossRef]

10.

Jian Chen, Xueming Liu, Chao Lu, Yixin Wang, and Zhaohui Li, “Design of Multistage Gain-Flattened Fiber Raman Amplifiers,” IEEE, J. lightwave technol. 24, 935–944 (2006). [CrossRef]

11.

M. Karásek and M. Menif, “Channel Addition/Removal Response in Raman Fiber Amplifiers: Modeling and Experimentation,” IEEE J. Lightwave Technol. 20, 1680–1688 (2002). [CrossRef]

12.

G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, New York, 2001)

13.

C. Fludger, A. Maroney, N. Jolley, and R. Mears, “An analysis of the improvements in OSNR from distributed Raman amplifiers using modern transmission fibers,” in Optical Fiber Communication Conference Technical Digest (Optical Society of America, 2000) 100–102.

OCIS Codes
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators
(290.5910) Scattering : Scattering, stimulated Raman

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: July 14, 2006
Revised Manuscript: September 6, 2006
Manuscript Accepted: September 21, 2006
Published: November 13, 2006

Citation
Junhe Zhou, Jianping Chen, Xinwan Li, Guiling Wu, Yiping Wang, and Wenning Jiang, "Frequency domain model to calculate the pump to signal RIN transfer in multi-pump Raman fiber amplifiers," Opt. Express 14, 11024-11035 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11024


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References

  1. 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. l 7, 3-16 (2001).
  2. C. R. S. Fludger, V. Handerek, and R. J. Mears, "Pump to Signal RIN Transfer in Raman Fiber Amplifiers," IEEE, J. Lightwave Technol. 19, 1140-1148 (2001). [CrossRef]
  3. M. D. Mermelstein, C. Headley, and J.-C. Bouteillier, "RIN transfer analysis in pump depletion regime for Raman fiber amplifiers," Electron. Lett. 38, 403-405, (2002). [CrossRef]
  4. Bruno Bristiel, Shifeng Jiang, Philippe Gallion, and Erwan Pincemin, "New Model of Noise Figure and RIN Transfer in Fiber Raman Amplifiers," IEEE Photon. Technol. Lett. 18, 980-982 (2006). [CrossRef]
  5. X. Zhou, C. Lu, P. Shum, and TH. Cheng, "A Simplified Model and Optimal Design of a Multiwavelength Backward-Pumped Fiber Raman Amplifier," IEEE Photon. Technol. Lett. 13, 945-947 (2001). [CrossRef]
  6. M. Yan, J. Chen, W. Jiang, J. Li, J. Chen, and X. Li, "Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps," IEEE Photon. Technol. Lett. 13, 948-950 (2001). [CrossRef]
  7. V.E Perlin, H.G. Winful, "Optimal design of flat-gain wide-band fiber Raman amplifiers," IEEE, J. lightwave technol. 20, 250 - 254 (2002). [CrossRef]
  8. V.E. Perlin, H.G. Winful, "On distributed raman amplification for ultrabroad-band long-haul wdm systems," IEEE, J. lightwave technol. 20, 409-417 (2002). [CrossRef]
  9. Jaehyoung Park, Pilhan Kim, Jonghan Park, H. Lee, and Namkyoo Park, "Closed Integral form expansion of Raman equation for efficient gain optimization Process," IEEE Photon. Tech. Lett. 16, 1649 - 1651 (2004). [CrossRef]
  10. Jian Chen, Xueming Liu, Chao Lu, Yixin Wang, and Zhaohui Li, "Design of Multistage Gain-Flattened Fiber Raman Amplifiers," IEEE, J. lightwave technol. 24, 935-944 (2006). [CrossRef]
  11. M. Karásek and M. Menif, "Channel Addition/Removal Response in Raman Fiber Amplifiers: Modeling and Experimentation," IEEE J. Lightwave Technol. 20, 1680-1688 (2002). [CrossRef]
  12. G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, New York, 2001)
  13. C. Fludger, A. Maroney, N. Jolley, and R. Mears, "An analysis of the improvements in OSNR from distributed Raman amplifiers using modern transmission fibers," in Optical Fiber Communication Conference Technical Digest (Optical Society of America, 2000) 100-102.

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