Bandwidth-Efficient WDM Channel Allocation for Four-Wave Mixing-Effect Minimization Vrizlynn L. L. Thing, P. Shum, and M. K. Rao

Abstract—A novel channel-allocation method that allows reduction of the four-wave mixing (FWM) effect while maintaining bandwidth efficiency is presented. It is composed of a fractional bandwidth-allocation algorithm, taking into consideration the use of parameters with distinct differences. This proposed technique allows the computation of an optimal channel-allocation set, where degradation caused by interchannel interference and FWM is minimal. Simulation is carried out to show significant performance improvement, such as an average bit-error rate improvement factor of 1.336 for an eight-channel wavelength-division multiplexing system, without the requirement of increased bandwidth, unlike existing channel-allocation methods. Index Terms—Four-wave mixing (FWM), optical-fiber communications, unequally spaced channel allocation, wavelength-division multiplexing (WDM).



N THE attempt to reduce the four-wave mixing (FWM) [1]–[4] effect in wavelength-division multiplexing (WDM) systems, many unequally spaced channel-allocation methods [5]–[12] are proposed. However, they resulted in an increase of bandwidth requirement, compared with equally spaced channel allocation. This is due to the constraint of the minimum channel spacing between each channel, and that the difference in the channel spacing between any two channels must be assigned to be distinct. As the number of channels increases, the bandwidth for the unequally spaced channel-allocation methods increases in proportion. A novel method for channel allocation is presented here to reduce the FWM effect, so as to improve WDM system performance without inducing additional cost in terms of bandwidth. A fractional bandwidth-allocation algorithm is designed to allow the computation of a channel-allocation set to result in an optimal point where degradation caused by interchannel interference (ICI) and FWM is minimal. The method is simulated, and the results are analyzed to prove its effectiveness, such that the system performance has improved while maintaining bandwidth efficiency. II. FOUR-WAVE MIXING

For any three co-propagating optical signals with frequencies , and , the new frequencies generated by FWM are represented by (1) Paper approved by W. C. Kwong, the Editor for Optical Communications of the IEEE Communications Society. Manuscript received April 24, 2003; revised October 24, 2003; March 8, 2004; and March 9, 2004. The authors are with Nanyang Technological University, School of Electrical and Electronic Engineering, Singapore 119613 (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.838684

co-propagating Considering all the possible permutations, new optical sigoptical signals will give rise to original nals. Some of these new frequencies fall onto the channels, while others are found in other new frequency locations. Therefore, to prevent FWM signals from falling onto the channels, the frequency separation between any two channels . must be distinct, since The FWM signal power [13]–[15] is given by (2) where is the velocity of light in free space, is the wavelength, is the attenuation factor, is the refractive index of is the third-order nonlinear susceptibility, is the core, for for ), the degeneracy factor ( is the effective length, is the is the launched signal effective area of the guided mode, and peak power per channel under the assumption that all the channels have an equal optical input power. The mixing efficiency is a function of the phase mismatch between the signal waves involved, and is given by (3) is related to the fiber chromatic disperThe phase mismatch , and is given by sion (4) and are the frequency where is the dispersion slope. separations, and The ultimate factor which degrades system performance is the FWM power, rather than the number of FWM signals on each channel. However, it is generally assumed that FWM crosstalk is maximum for the center channels in an equally spaced channelallocation system, because the number of FWM signals at the center channels is maximum [14], [16]. The assumption that the FWM power is proportional to the number of FWM signals is used here. This case may not apply if the wavelengths in the WDM system are far from the zero dispersion wavelength and there is substantial difference in dispersion between the two different channels [17]. For example, if nondispersion shifted fibers are used, and the zero dispersion wavelength is at 1300 nm, and the operating wavelengths are around 1550 nm, the center channel does not suffer the most severe FWM crosstalk even though it has the largest number of FWM signals. This scenario is not considered here, as the impact of the FWM crosstalk would then be negligible.

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III. UNEQUALLY SPACED CHANNEL ALLOCATION In [5], three methods were proposed to obtain the channel allocation based on algebraic approach. They are: constructions based on modified extended quadratic congruence (EQC) sequences; constructions based on searching; and constructions based on disjoint difference sets. These three methods are able to achieve optimal unequal channel allocation in that no FWM signals will fall onto the channel signals. However, the application of the algorithms is limited to prime powers, and the bandwidth to be assigned to the system is much larger than that for the bandwidth for equally spaced channel allocations. For example, the increase in the bandwidth is 16% for an eight-channel allocation, compared with equally spaced channel allocation. It is also shown that the increase in bandwidth will correspond to the increase in the number of channels. In [11] and [12], a simple construction algorithm is proposed. It aims to obtain close-to-minimum operating bandwidth and to result in a close-to-minimum number of FWM signals in the carrier channels. The objective is to redistribute the FWM signals in such a way that the number of FWM signals falling onto a specific carrier channel gets reduced substantially. Using this method does reduce the FWM effect, compared with the equally spaced channel allocation. However, the FWM effect is not eliminated, as a small number of FWM signals will still fall on the carrier channels. Compared with the above three methods, the bandwidth used is reduced, as this algorithm does not provide an optimum allocation. However, the bandwidth used here is still much larger, compared with the equally spaced channel allocation. These papers [8]–[10] cover investigation of unequally spaced channel allocation versus input power. An algorithm for finding a Golomb ruler, whereby the spacing must be greater than or equal to the pulse separation slot, is described. The Golomb ruler found is then used to allocate the channel frequencies. However, the operating bandwidth has to be expanded to accommodate the channels for this allocation. IV. FRACTIONAL-BASED WDM CHANNEL ALLOCATION In WDM systems, the slot width must be large enough to avoid appreciable overlap of channel and the FWM spectra, even with some instability in the channel frequencies. Since the root mean square (rms) frequency jitter for an FWM wave is three times that of a channel, the superimposition of the spectra is negligible when the channel-frequency stability is on the order of slot width/10. This is provided that the slot width is greater than or equal to 2 bit rate. In order to provide an adequate amount of rejection without distorting the desired channel, a minimum channel separation of greater than or equal to 10 bit rate should be provided [8]. As mentioned in Section II, the number of FWM signals could be a useful measure of FWM effect. In the equally spaced channel allocation, it is generally assumed that the FWM crosstalk is maximal for the center channel, as the number of FWM signals at the center channel is maximum. The assumption that the FWM power is proportional to the number of FWM signals is used here as well.

Fig. 1.

Operating bandwidth allocation.

In this scheme, we aim to achieve reduction in FWM effect with the WDM system using the same operating bandwidth as the one for equally spaced channel allocation. An algorithm was designed to allocate the channels so as to restrict the expansion of the bandwidth. We also propose to use a set of marks with distinct differences as parameters in the algorithm. Since the difference between any two numbers is distinct, the new FWM frequencies generated would not fall into the one already assigned for the carrier channels. In this case, the FWM signals would be spread out and not accumulate in strength at a certain location. Since the same operating bandwidth as the equally spaced channel allocation will be used, some channels would end up nearer to other channels, due to the distinct difference in spacing required. With this reduction in frequency spacing between certain channels, the ICI would therefore be increased. However, this algorithm will be used to find an optimal point where the distortion caused by both the ICI and FWM will be minimal by means of iteration, where the channels closely spaced will spread out after each iteration. Applying this algorithm, a systematic approach to the channel allocation could be achieved whereby no FWM signals fall exactly on the carrier channel, and the operating bandwidth would not need to be expanded. The algorithm is presented as follows. channels, select distinct difference marks • For . (DDM) and • Split the operating bandwidth to pre-allocate post-allocate sections (see Fig. 1). is the frequency allocated for the first channel. • . • Initial channel spacing • Remove first element of the DDM to get the DDM vector . • Rearrange the elements so that channel spacing of channels nearer to the center frequency will be wider to get the modified DDM vector, . For odd . For even . . • We label the elements of as • Let the number of iterations processed (for locating optimal allocation set) be , which has a starting value of 0. • Let the incrementation factor be . • Frequency of the th channel is then allocated by

(5) for from 1 to . The notion here is that since the difference for any two elements in the DDM vector is distinct, incrementing each element



by a same value would still result in a distinct difference. The incrementation factor will also have an impact on the allocation, in that it will be in increasing order of significance from the outer to the inner elements. After which, a near-equally spaced channel-allocation situation would be reached following multiple iterations of the algorithm. The performance of the WDM system is then observed for each iteration to locate the allocation where the performance is optimal. This channel-allocation set will be used for the WDM system design eventually. Some considerations here are as follows. 1) Question 1: Why not map the set of DDM directly on the total bandwidth? - Answer: If the total bandwidth is used for mapping, the channel spacing for some channels will be very much smaller, compared with the rest. For example, if there are eight channels and the DDM vector is [1, 4, 9, 15, 22, 32, 34], the spacing for the first channel is , which is about 1% of the total bandwidth, while the largest , which is about 30% of channel spacing will be the bandwidth. If there are 16 channels and the DDM vector is [1, 4, 11, 26, 32, 56, 68, 76, 115, 117, 134, 150, 163, 168, 177], the sum of the elements will be 1298. Therefore, the spacing for the first channel is 0.08%, and the largest spacing is about 14% of the bandwidth. This will lead to a very high number of iterations required before the optimal unequal spacing allocation set can be found. 2) Question 2: What percentage should be chosen for “Preallocate” and “Post-allocate”? - Answer: If a high percentage is chosen for pre-allocate, there will be little bandwidth left for distribution. The channel allocation will then be very close to the equally spaced channel allocation. However, if the pre-allocate section is set to be very small, the post-allocate section will then be very large, and it will result in the same situation as mapping the set of DDM directly on the total bandwidth. Therefore, a choice of 50%–90% of the bandwidth for the pre-allocate section is recommended. 3) Question 3: What should be the value chosen for the incrementation factor? - Answer: A value too high will result in fewer iterations. However, precise simulation is required for arriving at the optimal allocation set discovered. In this case, the incrementation factor is set to be similar to the first element of the DDM vector.

V. SIMULATION The eight-channel x 10 Gb/s, 100 GHz spacing for an equal channel-spacing allocation WDM system to be simulated in VPI Transmission Maker was designed. The 200-km fiber has an erbium-doped fiber amplifier (EDFA) after each 100-km span to compensate for the attenuation loss of 0.2 dB/km. The average power of each laser was set to 1 mW. Dispersion was set to 3 ps/nm.km, and the pre-allocated bandwidth was set at 50%. The DDM vector used was [1, 4, 9, 15, 22, 32, 34]. The simulation flowchart is shown in Fig. 2.

Fig. 2.

Simulation flowchart.

During simulations, the first iteration applied the frequency set for equally spaced channel allocation. The next 50 iterations applied the proposed method, whereby the channel spacing between any two channels is no longer equal. The bit-error rate (BER) improvement should be computed as follows: BER improvement BER decrease BER BER BER BER BER BER However, since the improvement is normally in the terms of 1E2 to 1E5 and will result in a very high BER improveE ment throughout the computation (e.g., BER and BER E , BER improvement will be 99.9%), therefore, a better way of computing the improvement has to be used. The BER improvement factor aims to find the ratio of improvement of the new allocation set over the equally spaced channel allocation. A logarithm is used to prevent presenting an improvement of one when the old and new BER are the same. Therefore, a BER improvement factor of zero will indicate no improvement, a negative value will indicate degradation, and a positive value will indicate improvement. The BER improvement factor was computed using the following formulas: BER improvement factor




Using the above formulas, the average BER improvement factor of the eight channels for each channel-allocation set was computed and analyzed. In the simulator, the BER is calculated as follows [18]: BER


where and denote the mean and variance of bit number , is the decision threshold, and erfc is the complementary error function. They are defined as (8)


Fig. 3.

Fig. 4.

Simulation results of 50 unequally spaced allocation sets.


Simulations for ICI consideration.


(10) where the noise contributions are (from left to right): signal-amplified spontaneous emission (ASE) beat noise, ASE-ASE beat noise, intraband crosstalk from parameterized signals, interband crosstalk from parameterized signals, thermal noise, and shot noise. The threshold is found by sweeping the decision threshold from 1% above the worst-case zero-bit level, to 1% below the worst-case one-bit level, while monitoring the BER. The threshold that gives the lowest BER is the threshold used.

VI. RESULTS AND ANALYSIS Fig. 3 shows the average BER improvement factor of each allocation set over the equally spaced channel allocation. The average was computed from the eight channels. With the exception of allocation set 7, which showed deterioration of BER by a factor of 0.0068, the rest of the allocation set showed improvement over the equally spaced channel allocation. Table I shows the allocation sets with the top 10 average BER improvement factor (sorting in this order). Improvements or deteriorations at each channel were also presented. Although Set 2 results in the highest average BER improvement, it should not be chosen for deployment, as channel six will suffer from a severe deterioration of 2.5604 in BER. Instead, an appropriate choice would be Set 12, as all channels will enjoy the benefits of this allocation scheme. We can also see that channels one and eight were the least affected (minimal improvement), as being “edge” channels, they were less likely than the rest to suffer from severe distortion due to the FWM effect. The results presented above were derived from simulations involving all possible distortions or degradations resulting from ICI and nonlinear effects such as FWM, cross-phase modulation, stimulated Raman scattering, etc. Separate simulations were also conducted for ICI and nonlinearity effects, and the results were as follows. For simplification, we will refer them to

Fig. 5. Simulations for FWM consideration.

ICI or FWM effects’ consideration. Figs. 4 and 5 show the average BER improvement factor of each allocation set over the equally spaced channel allocation for ICI and FWM consideration, respectively. It can be seen that Set 2 resulted in the best improvement of 1.9416 and 3.5788 during both the ICI and FWM considerations, respectively. Therefore, the overall effect’s simulation results in Fig. 3 reflected this, too. As shown in Fig. 4, Set 7 was the only allocation set which degraded in BER by 0.036 using this allocation scheme. Whereas, as shown in Fig. 5, Sets 6, 7, 46, 47, 48, and 49 suffered deterioration in BER by 0.1296, 0.6404, 0.0024, 0.4247, 0.3439, and 0.0055, respectively. Tables II and III present the top 10 allocation sets (for ICI and FWM consideration, respectively), corresponding to those top 10 for the overall effect’s simulation. It can be observed that for Set 2, channel six suffered BER degradation from both ICI and FWM effects. Therefore, the overall performance of channel six was not favorable. Channels one and eight also suffered minor degradation from the ICI effect, but were sufficiently compensated by improvements from FWM effect reduction. As for Set 12, which has overall improvement in all eight channels, separate simulations showed deterioration in channel eight by a factor of 0.0042 from ICI, but was compensated by an improvement of 0.3586 from FWM effect reduction. On the other hand, channel five, which showed a drop of 0.9131 from the FWM effect, was compensated by ICI reduction by a BER improvement of 0.7487.






VII. CONCLUSION The fractional bandwidth-allocation algorithm, combined with the use of parameters with distinct differences, was presented in this paper, intending to bring about FWM effect reduction while, at the same time, maintaining operating bandwidth efficiency. The scheme was simulated and results showed improvement in the performance of the WDM system. Although the best improvement in average BER of eight channels by a factor of 2.1059 was achievable, the objective of improving performance for all channels was not met. This

was due to deterioration in one of the channels by a factor of 2.5604. Instead, an allocation set resulting in improvement by a factor of 1.336 in average BER was found to be a better choice, as the benefit of the scheme was evidenced from the BER improvements shown across all channels. Simulations by isolating ICI and FWM (nonlinear, to be exact) effects were also presented to show the consequences brought about by the scheme in the two areas. Although this scheme does not allow total elimination of FWM effects, as in other existing optimum unequally spaced channel-allocation methods (which requires a 1.6 times expansion of the bandwidth for an


eight-channel WDM system, compared with equally spaced channel allocation), it allows significant improvement in the system performance at no additional cost. REFERENCES [1] K. Inoue, “Experimental study on channel crosstalk due to fiber four-wave mixing around the zero-dispersion wavelength,” J. Lightwave Technol., vol. 12, pp. 1023–1028, June 1994. [2] S. Song, “The number of four-wave mixing (FWM) waves in WDM systems and its applications,” in Proc. 14th Annu. Meeting IEEE Lasers and Electro-Optics Society, vol. 1, 2001, pp. 283–284. [3] W. C. Kwong, G.-C. Yang, and K.-D. Chang, “Locating FWM crosstalks in high-capacity WDM lightwave systems,” in Proc. IEEE Int. Conf. Communications, vol. 3, 2001, pp. 726–730. [4] R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol., vol. 13, pp. 841–849, May 1995. [5] W. C. Kwong and G.-C. Yang, “An algebraic approach to the unequalspaced channel-allocation problem in WDM lightwave systems,” IEEE Trans. Commun., vol. 45, pp. 352–359, Mar. 1997. [6] H. P. Sardesai, “A simple channel plan to reduce effects of nonlinearities in dense WDM systems,” in Proc. Conf. Lasers, Electro-Optics, 1999, pp. 183–184. [7] W. C. Kwong and G.-C. Yang, “Allocation of unequal-spaced channels in WDM lightwave systems,” Electron. Lett., vol. 31, no. 11, pp. 898–899, May 1995. [8] F. Forghieri, R. W. Tkach, A. R. Chraplyvy, and D. Marcuse, “Reduction of four-wave mixing crosstalk in WDM systems using unequally spaced channels,” IEEE Photon. Technol. Lett., vol. 6, pp. 754–756, June 1994. [9] F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, “WDM systems with unequally spaced channels,” J. Lightwave Technol., vol. 13, pp. 889–897, May 1995. , “Performance of WDM systems with unequal channel spacing to [10] suppress four-wave mixing,” in Proc. Eur. Conf. Optical Communication, Sept. 1994, pp. 741–744. [11] B. Hwang and O. K. Tonguz, “A generalized suboptimum unequally spaced channel allocation technique—Part I: In IM/DD WDM systems,” IEEE Trans. Commun., vol. 46, pp. 1027–1037, Aug. 1998. [12] O. K. Tonguz and B. Hwang, “A generalized suboptimum unequally spaced channel allocation technique—Part II: In coherent WDM systems,” IEEE Trans. Commun., vol. 46, pp. 1186–1193, Sept. 1998. [13] K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode fibers,” J. Appl. Phys., vol. 49, no. 10, pp. 5098–5106, Oct. 1978. [14] M. W. Maeda, W. B. Sessa, W. I. Way, A. Yi-Yan, L. Curtis, R. Spicer, and R. I. Laming, “The effect of four-wave mixing in fibers on optical frequency-division multiplexed systems,” J. Lightwave Technol., vol. 8, pp. 1402–1408, Sept. 1990. [15] N. Shibata, K. Nosu, K. Iwashita, and Y. Azuma, “Transmission limitations due to fiber nonlinearities in optical FDM systems,” IEEE J. Select. Areas Commun., vol. 8, pp. 1068–1077, Aug. 1990.


[16] K. Inoue, K. Nakanishi, K. Oda, and H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” J. Lightwave Technol., vol. 12, pp. 1423–1439, Aug. 1994. [17] A. Yu and M. J. O’Mahony, “Optimization of wavelength spacing in a WDM transmission system in the presence of fiber nonlinearities,” Proc. IEE OptoElectronics, vol. 142, no. 4, pp. 190–196, Aug. 1995. [18] VPI Photonics, Holmdel, NJ, “Transmission Maker,”.

Vrizlynn L. L. Thing received the B.Eng. and M.Eng. degrees in electrical and electronics engineering from Nanyang Technological University, Singapore, in 2000 and 2003, respectively. She is currently working toward the Ph.D. degree in the Department of Computing, Imperial College London, London, U.K. Her research work was on optical fiber communications, focusing on the channel-allocation problem in WDM systems. She joined the Institute for Infocomm Research, Nanyang Technological University, Singapore, formerly known as Kent Ridge Digital Labs, in 2000, and worked on research and developement in the areas of IP mobility (e.g., Mobile IPv4 and IPv6 protocols, Localized Mobility Management) and internet security (specifically, in denial-of-service (DoS), distributed DoS attacks, IP traceback, and resilient networks).

P. Shum received the B. Eng. and Ph.D. degrees in electronic and electrical engineering from the University of Birmingham, Birmingham, U.K., in 1991 and 1995, respectively. After graduation, he stayed on in the same university as an honorary Postdoctoral Research Fellow. In 1999, he joined the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. Since 2002, he has been the Director of the Network Technology Research Center. He has published more than 150 international journal and conference papers. His research interests are concerned with optical communications, nonlinear waveguide modelling, fiber gratings, and WDM communication systems. Dr. Shum received the Singapore National Young Scientist Award in 2002 for his contribution to next-generation optical communication technologies.

M. K. Rao Photograph and biography not available at time of publication.

Bandwidth-Efficient WDM Channel Allocation for Four ... - IEEE Xplore

52, NO. 12, DECEMBER 2004. Bandwidth-Efficient WDM Channel Allocation for. Four-Wave Mixing-Effect Minimization. Vrizlynn L. L. Thing, P. Shum, and M. K. ...

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