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Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 4 — Feb. 23, 2004
  • pp: 564–573
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Efficient algorithm and optimization for broadband Raman amplifiers

Xueming Liu and Yanhe Li  »View Author Affiliations


Optics Express, Vol. 12, Issue 4, pp. 564-573 (2004)
http://dx.doi.org/10.1364/OPEX.12.000564


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Abstract

A hybrid genetic algorithm (HGA) assisted by stochastic perturbation and the adaptive technique is proposed. Compared with our previous reports, the proposed HGA can exploit better solutions and greatly shorten the amount of run time. An example shows that the design of multipump Raman amplifiers involves the multimodal function optimization problem with multiple variables. With the new HGA, relationships of the optimal signal bandwidth with the span length and the ON-OFF Raman gain are obtained. A movie demonstrates the detailed interaction in pump-to-signal and signal-to-signal procedures. The corresponding optical signal-to-noise ratio of optimal results is obtained.

© 2004 Optical Society of America

1. Introduction

With their intrinsic merits, fiber Raman amplifiers (FRAs) have rapidly become a critical technology for 40-Gbit/s long-haul transmission and 10-Gbit/s ultra-long-haul transmission [1

1. P. Parolari, L. Marazzi, L. Bernardini, and M. Martinelli, “Double Rayleigh scattering noise in lumped and distributed Raman amplifiers,” J. Lightwave Technol. 21, 2224–2228 (2003). [CrossRef]

3

3. S. Faralli and E. Di Pasquale, “Impact of double Rayleigh scattering noise in distributed higher order Raman pumping schemes,” IEEE Photon. Technol. Lett. 15, 804–806 (2003). [CrossRef]

]. Because the performance of the entire transmission system is strongly influenced by the spectral gain and spectral noise performance [2

2. X. Zhou and M. Birk, “Performance limitation due to statistical Raman crosstalk in a WDM system with multiple- wavelength bidirectionally pumped Raman amplification,” J. Lightwave Technol. 21, 2194–2202 (2003). [CrossRef]

5

5. M. Tang, P. Shum, and Y. D. Gong, “Design of double-pass discrete Raman amplifier and the impairments induced by Rayleigh backscattering,” Opt. Express 11, 1887–1893 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-16-1887. [CrossRef] [PubMed]

], detailed knowledge about the spectral characteristics of the amplifier is key to the advanced design of WDM systems. Amplified spontaneous emission (ASE) and the Rayleigh backscattering effect are two main aspects of FRAs noise performance [4

4. Y. Hadjaret al., “Enhanced double Rayleigh backscattering in second order Raman amplification and system performance implications,” Opt. Commun. 229, 419–423 (2004). [CrossRef]

]. Single Rayleigh backscattering of ASE and double Rayleigh backscattering of optical signals in the transmission fiber grow with increasing distributed gain [6

6. P. B. Hansen, L. Eskilden, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiBiovanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” IEEE Photon. Technol. Lett. 10, 159–161 (1998). [CrossRef]

], and their impairments impose limits on the maximum allowable distributed gain in a system.

Since WDM signals propagating in a fiber experience gain tilts resulting from the energy transfer to longer wavelengths, optimizing the power and wavelength of pumps is used to control gain tilts. So, the design of multipump flat-gain Raman amplifiers presents a grand challenge to numerical simulation. Moreover, optimizing algorithms play a critical role in the design, analysis, and control of WDM transmission systems and are in widespread use in the research of FRAs. Recently, several different approaches to the design of wideband FRAs have been reported, such as the neural network method [7

7. P. C. Xiao, Q. J. Zeng, J. Huang, and J. M. Liu, “A new optimal algorithm for multipump sources of distributed fiber Raman amplifier,” IEEE Photon. Technol. Lett. 15, 206–208 (2003). [CrossRef]

], the simulated annealing algorithm [8

8. M. Yanet al., “Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,” IEEE Photon. Technol. Lett. 13, 948–950 (2001). [CrossRef]

], the traditional genetic algorithm (GA) [9

9. X. Zhou, C. Lu, P. Shum, and T. H. Cheng, “A simplified model and optimal design of a multiwavelength backward-pumped Raman amplifier,” IEEE Photon. Technol. Lett. 13, 945–947 (2001). [CrossRef]

], and the hybrid GA (HGA) [10

10. X. M. Liu and B. Lee, “Optimal design for ultrabroadband amplifier,” J. Lightwave Technol. 21, 3446–3455 (2003).. [CrossRef]

12

12. X. M. Liu and Y. H. Li, “Optimizing the bandwidth and noise performance of distributed multi-pump Raman amplifiers,” Opt. Commun. 230, 425–431 (2004). [CrossRef]

].

2. Theoretical model

Wave and noise propagation in FRAs are characterized by a variety of physical effects [15

15. A. A. B. Tio, P. Shum, and Y. D. Gong, “Wide bandwidth flat gain Raman amplifier by using polarization-independent interferometric filter,” Opt. Express 11, 2991–2996 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-2991. [CrossRef] [PubMed]

, 16

16. H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photon. Technol. Lett. 11, 530–532 (1999). [CrossRef]

], the major influences of which, for the gain profile design of backward-propagating multipump Raman amplifiers, are pump-to-pump, signal-to-signal, and pump-to-signal stimulated Raman scattering (SRS), as well as the fiber loss experienced by both pump and signal waves [7

7. P. C. Xiao, Q. J. Zeng, J. Huang, and J. M. Liu, “A new optimal algorithm for multipump sources of distributed fiber Raman amplifier,” IEEE Photon. Technol. Lett. 15, 206–208 (2003). [CrossRef]

12

12. X. M. Liu and Y. H. Li, “Optimizing the bandwidth and noise performance of distributed multi-pump Raman amplifiers,” Opt. Commun. 230, 425–431 (2004). [CrossRef]

, 17

17. X. M. Liu, “Powerful solution for simulating nonlinear coupled equations describing bidirectionally pumped broadband Raman amplifiers,” Opt. Express. (this issue).

]. In the steady state, the coupled equation can be described as

±dPkdz=[αk+j=1k1gR(vjvk)ΓAeffPjj=k+1n+mvkvjgR(vkvj)ΓAeffPj]Pk,(k=1,2,,n+m).
(1)

Because backscattering powers of pumps and signals are usually approximately 30 and 20 dB lower than their original powers, respectively [18

18. X. M. Liu and B. Lee, “A fast and stable method for Raman amplifier propagation equations,” Opt. Express 11, 2163–2176 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163. [CrossRef] [PubMed]

], the backscattering pumps and backscattering signals are ignored in simulating ASE waves. Furthermore, forward and backward noise powers are also less than input signal powers by ~30 dB [18

18. X. M. Liu and B. Lee, “A fast and stable method for Raman amplifier propagation equations,” Opt. Express 11, 2163–2176 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163. [CrossRef] [PubMed]

]. Then the model equations for ASE waves include such physical effects as attenuation, SRS, spontaneous Raman scattering, Rayleigh scattering, thermal noise, and so on; namely [12

12. X. M. Liu and Y. H. Li, “Optimizing the bandwidth and noise performance of distributed multi-pump Raman amplifiers,” Opt. Commun. 230, 425–431 (2004). [CrossRef]

, 16

16. H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photon. Technol. Lett. 11, 530–532 (1999). [CrossRef]

, 17

17. X. M. Liu, “Powerful solution for simulating nonlinear coupled equations describing bidirectionally pumped broadband Raman amplifiers,” Opt. Express. (this issue).

],

±dPASE,k±dz=αkPASE,k±+γkPASE,k
+PASE,k±j=1k1gR(vjvk)ΓAeffPj±[1+2hvkPASE,k±(1+(eh(vjvk)kBT1)1)Δv],
PASE,k±j=k+1n+mvkvjgR(vkvj)ΓAeff[Pj±+4hvk(1+(eh(vkvj)kBT1)1)Δv]
(2)

In Eqs. (1) and (2), the indexes k=1, 2,…,n and k=n+1,…,n+m represent pump and signal waves, respectively. Pk, vk, γk, and αk are the power, frequency, Rayleigh scattering, and attenuation coefficient for the kth wave, respectively. P ASE,K is the ASE noise power in one mode in the frequency resolution Δv, and its superscripts ‘+’ and ‘-’ denote forward- and backward-propagating ASE waves, respectively. h, kB, and T are Planck’s constant, Boltzmann’s constant, and temperature, respectively. Aeff is the effective area of the optical fiber. The factor of Γ accounts for polarization randomization effects, whose value lies between 1 and 2. gR(vj-vk) is the Raman gain coefficient from wave j to k. The frequency ratio vk/vj describes vibrational losses. The minus and plus signs on the left-hand side of Eq. (1) describe the backward-propagating pump waves and forward-propagating signal waves, respectively. The frequencies vk are enumerated in decreasing order of frequency (k=1, 2,…, n+m).

3. Multimodal function for multi-pump Raman amplifiers

Model equations of multipump Raman amplifiers are multivaritate problems; e.g., the design of the four-pump Raman amplifier has eight variables (including four wavelengths and four powers). Then their design problems comprise the multimodal function optimizations, and Fig. 1 demonstrates an example with four pumps. Figure 1 shows the contour of optimal signal bandwidth Δλ with pump wavelengths λ 2 and λ 3, where the wavelengths of two other pumps are specified as λ 1=1434.72 and λ 4=1497.75 nm. Figure 1(b) shows the magnification of the white dashed frame in Fig. 1(a). The color scale in the inset of Fig. 1 illustrates the distribution of Δλ. In optimizing Fig. 1, we assume that Γ=2, L=40 km; there are 55 signal channels spaced at 200 GHz, and the signal power of each channel is 1 mW; the gain spectrum gRv) and attenuation spectrum α(v) of the fiber are from Ref. [10

10. X. M. Liu and B. Lee, “Optimal design for ultrabroadband amplifier,” J. Lightwave Technol. 21, 3446–3455 (2003).. [CrossRef]

]; the gross Raman gain can compensate the loss of signals (i.e., G ON-OFF > αL); and the gain peak-to-peak ripple ΔG is less than 1.1 (i.e., ΔG<1.1).

From Fig. 1, we can see that there are multiple global maxima and many local maxima in the four-pump Raman amplifier. To reach the global maxima, the parameters of the pump spectra have to carefully accounted for. For example, in the cases of A and B global maxima in Fig. 1(b), although their wavelengths λ 1, λ 2, λ 4 are equal and the difference of λ 3 is small (λ 3=1467.25 nm for A and λ 3=1467.81 nm for B), their powers are greatly different (P 1, 2, 3, 4=216.89, 160.11, 88.72, 151.38 mW for A and P 1, 2, 3, 4=240.87, 167.33, 91.22, 151.27 mW for B). If λ 4=1497 nm instead of λ 4=1497.75 nm, the signal bandwidth is Δλ=61.3 instead of 82.5 nm, although other parameters are the same as the case of A. Therefore, the signal bandwidth is sensitive to the parameter set of pumps, whose values should be optimized in order to obtain the global optima.

4. Algorithm

Fig. 1. Contour of optimal signal bandwidth Δλ with pump wavelengths λ 2 and λ 3 in the four-pump Raman amplifier, where wavelengths of two other pumps are specified as λ 1=1434.72 and λ 4=1497.75 nm. (b) Magnification of the white dash frame in (a). The color scale in the inset of Fig. 1 illustrates the distribution of Δλ. n j (j=1, 2, …, 7) denote the center of seven clusters for simulating Fig. 3. D is the normalized niche radius. The global maximum Δλ=82.5 nm.

To obtain a global optima of as much as possible and shorten the amount of run time in optimizing the design of multipump flat-gain Raman amplifiers, from the old version of our HGA, an new HGA assisted by the adaptive technique and stochastic perturbation is proposed. In the code of pumps, we divide the wavelength range λfa of each pump into fixed range λf and variable range λa, which are shown in Fig. 2. Figure 2 illustrates the distribution of wavelength λ and power P of four pumps, where the y coordinate represents the pump power and the abscissa denotes the pump wavelength. Then it is a nine-dimensional optimization problem in the case of the four-pump Raman amplifier, including four wavelengths, four powers, and variable range λa in the code of each chromosome. From Fig. 2, we can see that λ 1∈[λ 0, λ 0+λf+λa], λ 2∈[λ 0+λf+λa, λ 0+2·(λf+λa)], λ 3·[λ 0+2·(λf+λa), λ 0+3∈(λf+λa)] and λ 4∈[λ 0+3·(λf+λa), λ 0+4·(λf+λa)]. To see the detailed procedure of encoding the chromosome, we use Fig. 1 as an example. In simulating Fig. 1, we assume that λf=14 nm, λ 0=1430 nm, λa∈[0, 2 nm] (λa has a different value for each chromosome per generation), and P 1, 2, 3, 4∈[0, 600 mW]. Here, λf is from the experiential value and λ 0 is calculated from the signal spectra and λf. In fact, the simulated results show that we can obtain the global maximum when λf∈[14 nm, 19 nm]. Because λfa of the pumps is an adaptive variable in each chromosome and λf is less sensitive to its initial value, the adaptive technique makes our proposed HGA greatly robust and flexible. Then, this technique increases the exploring capability of HGA.

Fig. 2. Distribution of wavelength λ and power P of four pumps, where the y coordinate represents pump power and abscissa denotes the pump wavelength.

From the new HGA, a simulated example is tested. In the simulation, we assume that the population size N=800, the number of clusters Nc=7, the normalized niche radius D=0.2, and the other parameters are the same as in Fig. 1. The parameter set here is also the same as in Figs. 46 except for the specified values in those figures. The normalized Euclidean distance dij between two individuals i and j is calculated as follows:

dij=k=14(λk(i)λk(j)λfa)2.
(3)

During the HGA procedure, the distance dij among the centers of all clusters must be more than the critical distance D. After the optimization, all centers of the seven clusters reach the global maxima, and the results are shown in Fig. 1(b). Seven white points represent the centers of the clusters (i.e., maxima), the region in each circle is considered a cluster, and the centers of the other clusters must be out of this region. The optimized parameter set is tabularized in Table 1. The signal spectra of the center of each cluster are demonstrated in Fig. 3. In Fig. 1(b), Fig. 3, and Table 1, n j (j=1, 2,…,7) denote the centers of seven clusters. The dots in each curve of Fig. 3 represent the channels. From Fig. 3 and Table 1, we can see that ① this case of four-pump Raman amplifiers is a six-variate problem; ② the pump spectra of the seven solutions differ greatly, although the optimal results of Δλ are the same; ③ to reach the specified bandwidth, there are many combinations of parameter sets. The numerical results also show that ① although the old version of our HGA can obtain the global optima [10

10. X. M. Liu and B. Lee, “Optimal design for ultrabroadband amplifier,” J. Lightwave Technol. 21, 3446–3455 (2003).. [CrossRef]

12

12. X. M. Liu and Y. H. Li, “Optimizing the bandwidth and noise performance of distributed multi-pump Raman amplifiers,” Opt. Commun. 230, 425–431 (2004). [CrossRef]

], the optimized solutions include some local optima instead of all global optima; ② to obtain the same number of global optima, the old HGA costs twice the run-time as compared with the new HGA.

Table 1. Power, wavelength, and bandwidth of the optimal results in Fig. 3.

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Fig. 3. Signal spectra of centers of seven clusters based on the proposed HGA.

5. Optimal results and noise performance

In the following simulations, we employ the predictor-corrector method and the “pure” shooting algorithm to solve Eqs. (1) [18

18. X. M. Liu and B. Lee, “A fast and stable method for Raman amplifier propagation equations,” Opt. Express 11, 2163–2176 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163. [CrossRef] [PubMed]

, 20

20. X. M. Liu and B. Lee, “Effective shooting algorithm and its application to fiber amplifiers,” Opt. Express 11, 1452–1461 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1452. [CrossRef] [PubMed]

] and assume that: γ=10-7 m-1; Nc, D, N, Γ, gRv), α(v) and signal spectra are the same as Fig.3.

5.1. Optimized Results for Bandwidth Δλ with L and GON-OFF

To reveal the influence of some major parameters on the FRA bandwidth, we plot the figures that show the relationships between the optimal signal bandwidth Δλ and the span length L or the ON-OFF (or gross) Raman gain G ON-OFF in Figs. 4 and 5, respectively. In calculations, we assumed that G ON-OFF >αL and ΔG<1.1 dB in Fig. 4, and L=50 km and ΔG<1.1 dB in Fig. 5.

The red solid curves in Figs. 4 and 5 are the fitted lines based on all circles (each circle corresponds to an optimized global maximum), respectively. We can see that, from Figs. 4 and 5, the optimal signal bandwidth Δλ approximately linearly decreases with increasing L and G ON-OFF, and the relationship is that Δλ=-0.88322×L+114.59 nm and Δλ=-7.09127×G ON-OFF +71.35 nm in our optimizations, respectively. Therefore, extending the fiber span length L and increasing the gross Raman gain G ON-OFF are done at the cost of decreasing signal bandwidth Δλ.

Table 2. Power, wavelength and bandwidth of the optimal results at L=80 km in Fig. 4.

table-icon
View This Table
| View All Tables
Fig. 4. Relationship of optimal signal bandwidth Δλ versus span length L. Circles are the optimized global maxima based on the new HGA, and the red solid curve is their fitted line.
Fig. 5. Relationship of optimal signal bandwidth Δλ versus ON-OFF Raman gain G ON-OFF. Circles are the optimized global maxima based on the new HGA, and the red solid curve is their fitted line.

5.2. A movie for the detailed procedure of FRA

To give a clearer understanding of the relationship of pump-to-signal and signal-to-signal, we present a movie that demonstrates the evolution of all channels transmitting along the fiber. In simulations, we assumed that L=40 km, Δλ≥82.5 nm on the conditions of ΔG<1.1 dB and G ON-OFF >αL, and ΔG of all signal is less than 2.5 dB. Other parameters are the same as in Fig. 4. The red dots in the movie represent the signal channels. The optimal parameters of four pumps are that P 1, 2, 3, 4=207.66, 186.44, 111.28, 139.68 mW, and λ 1, 2, 3, 4=1432.68, 1446.77, 1466.14, 1498.65 nm. The movie shows that ① there are strong interactions of signal-to-signal and pump-to-signal; ② the pump-to-signal interaction can compensate the attenuation of signals and increase signal power when z>~30 km; ③ SRS effects of signal-to-signal make the power of higher-frequency (i.e., shorter wavelength) waves flow into that of lower-frequency (i.e., longer-wavelength) waves.

Fig. 6. (1745 KB) Movie showing the procedure of all channels transmitting along the fiber.

5.3. Optical signal-to-noise ratio

Figure 7 exhibits the corresponding optical signal-to-noise ratio (OSNR) of L=40 and 50 km in Fig. 4. In calculating Eq. (2), a midpoint shooting algorithm is used, which is from Ref. [17

17. X. M. Liu, “Powerful solution for simulating nonlinear coupled equations describing bidirectionally pumped broadband Raman amplifiers,” Opt. Express. (this issue).

] [In calculating Eqs. (1) and (2), different shooting algorithms are adopted. Equation (1) is simulated by a “pure” shooting algorithm that has faster computational speed, and Eq. (2) is done with a mid-point shooting algorithm that has better stability. Their detailed comparisons are shown in Ref. [17

17. X. M. Liu, “Powerful solution for simulating nonlinear coupled equations describing bidirectionally pumped broadband Raman amplifiers,” Opt. Express. (this issue).

]). From Fig. 7, we find that the OSNR in FRAs decreases with extending span length L, and OSNR of shorter-wavelength signals is less than that of higher-wavelength signals. To equalize OSNR tilt, one can use the bidirectionally pumping scheme [17

17. X. M. Liu, “Powerful solution for simulating nonlinear coupled equations describing bidirectionally pumped broadband Raman amplifiers,” Opt. Express. (this issue).

].

6. Discussions

Fig. 7. Corresponding OSNR of L=40 and 50 km in Fig. 4 versus wavelength.

7. Conclusions

A HGA assisted by stochastic perturbation and the adaptive technique has been constructed in this paper. In comparison with the old version of our HGA [10

10. X. M. Liu and B. Lee, “Optimal design for ultrabroadband amplifier,” J. Lightwave Technol. 21, 3446–3455 (2003).. [CrossRef]

12

12. X. M. Liu and Y. H. Li, “Optimizing the bandwidth and noise performance of distributed multi-pump Raman amplifiers,” Opt. Commun. 230, 425–431 (2004). [CrossRef]

], the new HGA can exploit the better solutions (e.g., it can in parallel offer more than seven global maxima that are shown in Tables 1 and 2, but the old HGA offers only part of the global maxima accompanied with some local maxima) and shortens the amount of run time by more than half in our simulations. An example of a four-pump FRA shows that the design of multipump Raman amplifiers involves the multimodal function optimization problems with multiple variables. From the new HGA, relationships of the optimal signal bandwidth Δλ with L and G ON-OFF are obtained, and Δλ approximately linearly decreases with the increase of L and G ON-OFF; i.e., Δλ=-0.88322×L+114.59 nm and Δλ=-7.09127×G ON-OFF +71.35 nm in our optimizations, respectively. A movie based on an optimal example demonstrates that there are strong interactions of pump-to-signal and signal-to-signal, and the SRS effect makes the power of higher-frequency waves transfer into that of lower-frequency waves. The corresponding OSNR of optimal results of L=40 and 50 km exhibits that (1) the noise performance deteriorates with increasing the span length L and (2) shorter-wavelength signals have less OSNR than higher-wavelength signals.

Acknowledgments

This research has been supported in part by the National Natural Science Foundation of China (60132020). The authors thank Xin-Jun Liu and Xinjie Yu, Tsinghua University, respectively, for helping to make three movies and for fruitful discussions on the GA.

Xueming Liu is currently engaged in research at the School of Electrical Engineering, Seoul National University, Korea.

References and links

1.

P. Parolari, L. Marazzi, L. Bernardini, and M. Martinelli, “Double Rayleigh scattering noise in lumped and distributed Raman amplifiers,” J. Lightwave Technol. 21, 2224–2228 (2003). [CrossRef]

2.

X. Zhou and M. Birk, “Performance limitation due to statistical Raman crosstalk in a WDM system with multiple- wavelength bidirectionally pumped Raman amplification,” J. Lightwave Technol. 21, 2194–2202 (2003). [CrossRef]

3.

S. Faralli and E. Di Pasquale, “Impact of double Rayleigh scattering noise in distributed higher order Raman pumping schemes,” IEEE Photon. Technol. Lett. 15, 804–806 (2003). [CrossRef]

4.

Y. Hadjaret al., “Enhanced double Rayleigh backscattering in second order Raman amplification and system performance implications,” Opt. Commun. 229, 419–423 (2004). [CrossRef]

5.

M. Tang, P. Shum, and Y. D. Gong, “Design of double-pass discrete Raman amplifier and the impairments induced by Rayleigh backscattering,” Opt. Express 11, 1887–1893 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-16-1887. [CrossRef] [PubMed]

6.

P. B. Hansen, L. Eskilden, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiBiovanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” IEEE Photon. Technol. Lett. 10, 159–161 (1998). [CrossRef]

7.

P. C. Xiao, Q. J. Zeng, J. Huang, and J. M. Liu, “A new optimal algorithm for multipump sources of distributed fiber Raman amplifier,” IEEE Photon. Technol. Lett. 15, 206–208 (2003). [CrossRef]

8.

M. Yanet al., “Automatic design scheme for optical-fiber Raman amplifiers backward-pumped with multiple laser diode pumps,” IEEE Photon. Technol. Lett. 13, 948–950 (2001). [CrossRef]

9.

X. Zhou, C. Lu, P. Shum, and T. H. Cheng, “A simplified model and optimal design of a multiwavelength backward-pumped Raman amplifier,” IEEE Photon. Technol. Lett. 13, 945–947 (2001). [CrossRef]

10.

X. M. Liu and B. Lee, “Optimal design for ultrabroadband amplifier,” J. Lightwave Technol. 21, 3446–3455 (2003).. [CrossRef]

11.

X. M. Liu and B. Lee, “Optimal design of fiber Raman amplifier based on hybrid genetic algorithm,” IEEE Photon. Technol. Lett.16, (to be published).

12.

X. M. Liu and Y. H. Li, “Optimizing the bandwidth and noise performance of distributed multi-pump Raman amplifiers,” Opt. Commun. 230, 425–431 (2004). [CrossRef]

13.

S. W. Mahfoud, “Niching methods for genetic algorithms,” Ph.D. dissertation (University of Illinois, Urbana, Ill., 1995).

14.

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, New York, 1989).

15.

A. A. B. Tio, P. Shum, and Y. D. Gong, “Wide bandwidth flat gain Raman amplifier by using polarization-independent interferometric filter,” Opt. Express 11, 2991–2996 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-2991. [CrossRef] [PubMed]

16.

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photon. Technol. Lett. 11, 530–532 (1999). [CrossRef]

17.

X. M. Liu, “Powerful solution for simulating nonlinear coupled equations describing bidirectionally pumped broadband Raman amplifiers,” Opt. Express. (this issue).

18.

X. M. Liu and B. Lee, “A fast and stable method for Raman amplifier propagation equations,” Opt. Express 11, 2163–2176 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2163. [CrossRef] [PubMed]

19.

X. Liu and Y. Li, “Optimal design of DFG-based wavelength conversion based on hybrid genetic algorithm,” Opt. Express 11, 1677–1688 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-14-1677. [CrossRef] [PubMed]

20.

X. M. Liu and B. Lee, “Effective shooting algorithm and its application to fiber amplifiers,” Opt. Express 11, 1452–1461 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1452. [CrossRef] [PubMed]

21.

D. Dahan and G. Eisenstein, “Numerical comparison between distributed and discrete amplification in a point-to-point 40 Gbit/s 40-WDM- based transmission system with three different modulation formats,” J. Lightwave Technol. 20, 379–388 (2002). [CrossRef]

OCIS Codes
(000.3860) General : Mathematical methods in physics
(000.4430) General : Numerical approximation and analysis
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

ToC Category:
Research Papers

History
Original Manuscript: January 5, 2004
Revised Manuscript: February 3, 2004
Published: February 23, 2004

Citation
Xueming Liu and Yanhe Li, "Efficient algorithm and optimization for broadband Raman amplifiers," Opt. Express 12, 564-573 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-4-564


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References

  1. P. Parolari, L. Marazzi, L. Bernardini, and M. Martinelli, �??Double Rayleigh scattering noise in lumped and distributed Raman amplifiers,�?? J. Lightwave Technol. 21, 2224-2228 (2003). [CrossRef]
  2. X. Zhou and M. Birk, �??Performance limitation due to statistical Raman crosstalk in a WDM system with multiple- wavelength bidirectionally pumped Raman amplification,�?? J. Lightwave Technol. 21, 2194-2202 (2003). [CrossRef]
  3. S. Faralli and E. Di Pasquale, �??Impact of double Rayleigh scattering noise in distributed higher order Raman pumping schemes,�?? IEEE Photon. Technol. Lett. 15, 804-806 (2003). [CrossRef]
  4. Y. Hadjar et al., �??Enhanced double Rayleigh backscattering in second order Raman amplification and system performance implications,�?? Opt. Commun. 229, 419-423 (2004). [CrossRef]
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