Pulse Propagation in Optical Fibers using the Moment Method

Pulse Proagation in Otical Fibers using the Moment Method Bruno Miguel Viçoso Gonçalves das Mercês, Instituto Suerior Técnico Abstract The scoe of this aer is to use the semianalytic technique of the Moment Method to study ulse roagation in otical fibers. It is made an overview of ulse roagation in both the linear and non-linear regime, exloring the effects of Grou Velocity Disersion (GVD) and Self-Phase Modulation (SPM) as well as an analysis of bit rate deendencies. Both the Gaussian ulse and the hyerbolic-secant ulse are evaluated in the linear and non-linear regimes, analysing the evolution of their arameters using the Moment Method. Disersion mas are also studied using the Moment Method so as to assess how different otical fiber characteristics influence the ulse arameters and roagation. Finally, the Moment Method the Moment Method for both the Gaussian and hyerbolic-secant ulses in the nonlinear regime is analysed, being comared against the Slit-Ste Fourier Method (SSFM) for the same roagation conditions in order to evaluate the Moment Method merits and shortcomings. Keywords Otical Fibers, Moment Method, semianalytic technique, GVD, SPM, chir, soliton, SSFM, linear regime, nonlinear regime, Gaussian ulse, hyerbolic-secant ulse, NLS, Disersion Mas. O I. INTRODUCTION tical fibers communication systems are systems that use otical fibers to transmit information. These systems have been being used on a global scale since 98 and have revolutionized the telecommunications field []. The main limitation factors in otical fibers are attenuation and GVD. However, another factor which condition ulse roagation is the develoment of non-linear effects, most notably, SPM [9]. The study of ulse roagation can have three distinct aroaches. We may consider the analytical aroach which results in recise and closed solutions but is not always ossible. Another aroach is a ure numerical one, which desite its recision requires a considerable comutational effort. Finally, we have the semi analytic aroach which can be variational or the Moment Method. Although the Moment Method cannot offer, by itself, a solution for ulse roagation it enables us to consider a arametric sace in order to gain hysical insight of the ulse through its arameters. A simle examle of the alication of the Moment Method can be considered in [][3]. It is my goal to show how one can use the Moment Method in different ulse shaes, roagation conditions and comare it with the SSFM in order to draw conclusions about its advantages and shortcomings as an alternative, versatile solution. Other alications of the Moment Method can be found in [][]. II. PULSE PROPAGATION IN OPTICAL FIBERS Considering the ulse roagation in the linear regime, according to [4] we have the equation that describes the ulse roagation in the linear regime as 3 A da d A d A i 3 A 3 () z dt dt 6 dt Where A is the enveloe of the ulse, t reresents time, z is the coordinate through which the ulse roagates in the fiber, β is the inverse of the grou velocity, β is the GVD coefficient, β 3 is the high order disersion coefficient and α is the attenuation coeffient. Taking into account the rocess to obtain the equation which describes the ulse roagation in the non-linear regime in [5] we can write the Non-Linear Schrödinger Equation (NLS) as u u i sgn( ) u u () as long as fiber losses and high order disersion effects are not taken into account and where u reresents the normalized amlitude u, NU, (3) Where is the normalized distance z LD (4) is the normalized time t z T T T (5) and N can be deduced from the exression LD T N LD P P L (6) NL Where L D reresents the disersion length, L NL reresents the non-linear length, T is the ulse width at the beginning of the fiber, is the non-linearity coefficient and P o reresents the inut ower. The Moment Method was first used, in 97, by Vlasov [6]. If we consider the NLS in the form du d U i U U (7) dz dt Where z ex[ ( z) dz] is non-linear arameter which contains both the non-linear effects and the fiber losses. The

Moment Method can be used to solve (7) as long as one can assume the ulse retains a secific shae during its roagation although its amlitude, width and chir can change in a continuous way. There are cases where this is valid, for examle, a Gaussian ulse retains its shae in a linear disersive medium as well as if the non-linear effects are of little significance. On the other hand, a ulse can also retain its shae even if the non-linear effects are strong as long as the disersive effects are weak. The concet behind the Moment Method is to address the otical ulse like a article in which its energy E, RMS width σ and chir C are related to U(z, T) as (8) E U dt T U dt E (9) * i * U U C T U U dt E T T () As the ulse roagates inside the fiber, these three moments change. To evaluate how they evolve with z we differentiate the equations with resect to z and use the NLS. After some algebra, detailed in [7], we find E () z C U 4 dt U dt z E T E () d dz C (3) If we consider a Gaussian ulse with chir, its ansatz may be described by T U z, T a ex ic iφ T (4) Where a reresents the amlitude of the ulse, reresents the chir of the ulse, T reresents the width of the ulse and reresents the hase of the ulse. All the four arameters are function of z. However since the hase does not affect the other arameters it can be ignored. a relates with the ulse energy through E a T (5) Since the energy C E does not change with z it can be relaced by its initial value E T. On the other hand, T. Using () and (3), the ulse width T and its chir may be described by the system of couled C differential equation, such as dt dz C (6) T dc T C P (7) dz T T On the other hand, if an hyerbolic secant ulse is to be considered, its ansatz takes the form T T U z, T a sech ex ic iφ T T (8) As with the Gaussian ulse all four arameters are a function of z and the hase may be ignored as it does not affect the other arameters. As the energy E remains the same throughout z it can be relaced by its initial value, E PT. Reeating the same rocess used for the Gaussian ulse, using (8) in () and (3), the width T and the chir C of the hyerbolic secant ulse are described by the following system of couled differential equations dt C (9) dz T dc 4 T C 4 P () dz T T The bit rate of an otical system may be defined as the number of bits that conveyed er unit of time. Bit rate is limited by inter symbols interference, which in turn is related to ulse broadening caused by disersion. As such, the study of bit rate is of high imortance in otical systems [4]. Pulse broadening is related to different arameters such as the RMS width of the source, the initial width of the ulse, the GVD and, sometimes, high order disersion. The RMS width of the ulse is given by [8] t t The broadening factor is defined in [8] as () LC L L 3 ( C ) 3 () 4 Considering T B the eriod of a bit slot, the bit rate is. In order to avoid inter symbol interference the B TB following rule must be uhold T B B B (3) 4 4B 4 Taking into consideration () and ignoring the high order disersion it can be written L CL ( C ) (4) The otimum value of to minimize ulse broadening can be calculated as d L C (5) d Substituting (5) in (4) we get L sgn( ) C C (6)

3 Alying (6) in rule (3), the maximum value of the bit rate is defined as B (7) 4 L C sgn( ) C However, analyzing equation (6) it can be concluded that for a given value of C,. This may be observed in the following icture. Fig. Evolution of and with chir Using equations (5) and (6) it s easy to conclude the intersection oint haens at C. From that oint on, 3 takes higher values than. As such, rule (3) should be rewritten as B B (8) 4 It is also interesting to analyse how the bit rate evolves in function of C, that behavior can be seen in the following icture The maximum bit rate haens at C, the intersection 3 oint between and. The bit rate is maximum at that oint because that is where the minimum of occurs. From that oint on rule (8) should be considered instead of rule (). As such, will take values higher than the minimum of and the bit rate will decrease. III. LINEAR REGIME SIMULATIONS USING THE MOMENT METHOD The first art of this chater will consider the Gaussian and hyerbolic secant ulses in constant disersion fibers, analyzing the evolution of the ulse arameters in the linear regime. Considering the linear regime,. As such, equations (6) and (7), which describe the evolution of the ulse width and chir for the Gaussian ulse, can be rewritten as dt C (9) dz T dc dz C (3) T If the ulse roagates in the anomalous region and that there is no chir in the beginning of the fiber C the evolution of the ulse chir, C, and the T broadening factor, T ictures, can be observed in the following Fig. 3 Chir evolution for the Gaussian ulse in the linear regime Fig. Bit rate evolution with chir

4 Fig. 4 Broadening factor evolution for the Gaussian ulse in the linear regime It can be seen the ulse broadens fast and develos considerable chir with a linear evolution. This haens because considering only the first term of the right side of equation (7) is taken into account, resulting in equation (3). As such, seeing the anomalous region is being considered and the terms in equation (5) would have oosite signals, there is no term contributing with the non-linear effects to even out the fast develoment evolution of the chir into negative values. This will influence the ulse broadening because considering the chir develos negative values and considering, then. So, considering (9), it is C easy to conclude the ulse will broaden fast and roortionally to the develoment of the chir. Considering now the hyerbolic secant ulse and the linear regime, equations (7) and (8) can be rewritten as dt C (3) dz T dc dz 4 C T (3) The evolution of the arameters for the hyerbolic secant ulse can be found in the next ictures. Fig. 6 Broadening factor evolution for the sech' ulse in the linear regime Relatively to Fig. 5 we can see the ulse acquires a negative chir. However, unlike the Gaussian ulse the evolution is not linear. In Fig. 6 there is a ulse broadening, consequence of the chir acquired by the ulse. However, since the chir values are smaller than in the Gaussian ulse also the ulse broadening is smaller than the one seen in Fig. 4. The second art of this chater will address fibers with variable disersion. This means fibers with different segments, each one with different characteristics. These are the so called disersion mas and are a owerful tool to overcome disersion in otical fibers. To start with we will consider different disersion mas for the Gaussian ulse. For that analysis it is imortant to consider the relationshi between the disersion coefficient and the GVD given by D (33) c For a disersion ma with two segments and an average GVD of zero which mean there is total comensation of the disersion we consider Fig. 7 Disersion ma for the Gaussian ulse with segments Fig. 5 Chir evolution for the sech ulse in the linear regime

5 The first segment of the ma is characterized by L 5km ; D 6 s ( km. nm) while the second segment is defined by L km ; D 8 s ( km. nm). Looking at Fig. 7 it easy to conclude this ma obeys to the exression D L D L. As such, there is total comensation of disersion which can be seen by the fact that after the 6km the ulse goes through it regains its initial width. For the hyerbolic secant ulse and considering a disersion ma of 3 segments instead of we can observe Fig. 9 Chir evolution for the Gaussian ulse in the non-linear regime Fig. 8 Disersion ma for the sech ulse with 3 segments The first and third segments are characterized by L km ; D,3 6 s ( km. nm) while the second segment,3 5 is carachterized by L 3km ; D 3 s ( km. nm). IV. NON-LINEAR REGIME: GAUSSIAN PULSE This chater will evaluate the Gaussian ulse roagation as well as its arameters in the non-linear regime. It will consider the roagation of the Gaussian ulse for different LD values of N, where L L NL. The results obtained NL P with the Moment Method will then be assessed against the SSFM to evaluate the Moment Method recision. Considering roagation in the anomalous zone, an unchired ulse at the beginning of the fiber C and equations (4) and (5) the evolution of the ulse width T and the chir for different values of N can be seen in the following ictures. It should be noted the ulse width is evaluated considering the broadening factor,. T T C Fig. Pulse broadening evolution for the Gaussian ulse in the non-linear regime It can be seen that as N increases (which mean the nonlinear effects are stronger) the ulse broadens less and less, eventually comressing for N.5. This is a consequence of the fact that for higher values of N the ulse gains less chir, going as far as to acquire ositive chir for N.5. This behavior would be exected by analyzing equations (4) and (5). In the non-linear regime the Self-Phase Modulation (SPM) should be considered. In equation (5) the SPM effects (reresented by the second term of the equation s right side) will comensate the disersive effects in chir evolution. This means that the bigger the non-linear comonent the bigger will the SPM is and as such the ulse will broaden less. Continuing analyzing equations (4) and (5) it can be inferred that for N it is exected that at some oint in the fiber the contributions of SPM and disersion will null each other with the chir taking on ositive values, which will result in a ulse comression. That can be observed in Fig. where for N, in 8 the chir takes ositive values which results in a ulse comression.

6 Rebuilding the ansatz for the Gaussian ulse described in () for N, using the Moment Method and using the SSFM it can be obtained Fig. 3 Gaussian ulse- Moment Method, N.5 Fig. Gaussian ulse - Moment Method, N Fig. 4 Gaussian ulse - SSFM, N.5 Fig. Gaussian ulse - SSFM, N Finally, for N.5 the confrontation between the Moment Method and the SSFM can be found in the below ictures Comaring both figures it can be concluded that for N the Moment Method is a valid technique to study the roagation of a Gaussian ulse. The ulse shae is similar in both techniques and the amlitude value at the end of the fiber is of a.8433 for the Moment Method and a.8894 for the SSFM which is a good indicator of the validity of the Moment Method. Comaring this situation with the linear case there is less broadening of the ulse thanks to the non-linear resent in the fiber. For N,5, the Moment Method is also a valid technique to study the Gaussian ulse. The amlitude values at the end of the fiber are quite similar for both techniques and the ulse shae throughout the fiber is identical as it can be seen in the following ictures. Fig. 5 Gaussian ulse - Moment Method, N.5

7 Fig. 6 Gaussian ulse - SSFM, N.5 Fig. 7 Chir evolution for the 'sech' ulse in the non-linear regime In this situation the Moment Method roves even more reliable than in the revious situations with final value of amlitude being very similar between both techniques, with a.44 for the Moment Method and a.9 for the SSFM. The different situations can be organized in the following table Amlitude, a, at the end of the fiber (ζ = ) Moment Method SSFM N =,6686,6684 N =.5,736,7554 N =,8433,8894 N =.5,44,9 Table. Moment Method and SSFM comarison for the Gaussian ulse From the above table it can be concluded that the Moment Method is a valid technique for realistic situations where the non-linear effects are not that intense but it starts to lose validity when they become more significant. This is because the Moment Method assumes the ulse shae doesn t change during the ulse roagation, something the non-linear effects cause to haen. As such, the Moment Method does not realistically reresent a ulse in those conditions because it acts like a straitjacket for the ulse. V. NON-LINEAR REGIME: HYPERBOLIC SECANT PULSE This chater will evaluate the Gaussian ulse roagation as well as its arameters in the non-linear regime, considering different values of N and assessing the results of the Moment Method against those of the SSFM. Considering roagation in the anomalous zone, an unchired ulse at the beginning of the fiber C and equations (7) and (8) the evolution of the ulse width T and the chir C can be seen in the following ictures. It should be noted the ulse width is evaluated considering the broadening factor. Fig. 8 Pulse broadening evolution for the sech ulse in the non-linear regime For N and N.5 the chir and broadening factor evolution is similar to the one observed for the Gaussian ulse although the chir takes smaller values which result in less broadening of the ulse. However for N bigger changes are seen in the ulse behavior when comared to the Gaussian ulse. For N the chir remains throughout the fiber. Since there is no chir variation there is also no variation in the ulse width and as such the broadening factor remains. This behavior is similar to the behavior of the fundamental soliton [5]. For N.5 the chir and broadening factor behavior is similar to the one in the Gaussian ulse, however with some changes. The hyerbolic secant ulse develos a higher ositive chir value than the Gaussian ulse which will result in a bigger ulse comression. This behavior is relevant because it is oosed to what is erceived for N where the chir values and broadening factors are smaller than in the Gaussian ulse. The exlanation for this is in the comarison between equations (5) and (8). When the coefficients of both equations are taken into account there is a bigger

8 discreancy between the SPM and the GVD contributions in the Gaussian ulse than in the hyerbolic secant ulse where the contributions of both henomenon are equal as it can be seen by the ulse behavior for N. For the Gaussian ulse the term which translates the GVD has more significant contribution than the one which translates the SPM. This results in a bigger broadening of the ulse for N and a smaller comression of the ulses for N as it can be seen when comaring Fig. and 8. This way, for the sech ulse the GVD will not influence the imulse in such an exressive way as in the Gaussian ulse, as such the ulse will develo less chir and the ulse will broaden less. On the other hand, for N the oosite haens. As GVD as a smaller contribution it won t comensate the influence of the SPM, resulting of the non-linear effects which in turn will result in higher chir values and bigger ulse comression. Physically, this behavior is in agreement with the exected behavior of the sech ulse which for higher values of non-linearity it will aroach solitons of higher order. Next, similar to chater IV an assessment of the Moment Method against the SSFM will be made for the hyerbolic secant ulses in different non-linear situations. Starting with N, For this situation the Moment Method is a erfect solution as it comletely relicates the ulse as it is obtained using the SSFM. The ulse behaves as a fundamental soliton, keeing its amlitude, shae and width throughout the fiber. For N.5 Fig. Sech ulse - Moment Method, N.5 Fig. 9 Sech ulse - Moment Method, N Fig. Sech ulse - SSFM, N.5 It can be seen that for this case the Moment Method also rovides a valid solution as the ulse evolution is quite similar when using both the Moment Method and the SSFM. Finally, for N.5 it is to be exected to observe a ulse comression as well as a eriodic behavior as er Fig. 7. As such, in order to observe the eriodic behavior the roagation distance of the simulation will be longer than the one used in the revious situations. Fig. Sech ulse - SSFM, N

9 Fig. Sech ulse - Moment Method, N.5 Fig. 4 Sech ulse - Moment Method, N 4 Fig. 3 Sech ulse - SSFM, N.5 Fig. 5 Sech ulse - SSFM, N 4 Analysing Fig. the eriodic behavior observed in Fig. 7 can be confirmed, with the eaks of Fig., where the is a ulse comression corresond to the minimums of Fig. 7. This behavior is also starting to aroach that of the second order soliton. It is to be noted that the roagation distance in both simulations is different. This haens because using a roagation distance of 6 with the SSFM is not realistic as it takes too much time to run. Even with 8 the comutation time was too high and unractical. This situation is a ractical examle where the Moment Method rovides advantages against the SSFM as it takes less comuting effort. However some discreancies between the two methods start to show as even though the eaks of the ulse occur in the same laces of the fiber, their amlitude values are somewhat different. In order to better exlore the discreancies that are starting to show between the Moment Method and the SSFM simulations using both methods for N 4 were run, in order to simulate a second order soliton. Although the hyerbolic secant ulse evolution using the Moment Method has a similar behavior to that of a second order soliton the fact is both the eriod and eak amlitude of both ulses are quite different. This agrees with the conclusion of chater IV that for high values of non-linearity the Moment Method does not hold true, losing recision as the nonlinearity increases. The comarison between the Moment Method and the SSFM is best illustrated in the below table Amlitude, a, at the end of the fiber (ζ = ) Moment Method SSFM N =,775,7855 N =.5,8668,843 N = N =.5,65,78 Table. Moment Method and SSFM comarison for the sech ulse

Analysing the table it can be verified that the Moment Method, in general, agrees with the SSFM. An interesting inference is to observe that when the non-linear effects are less significant the agreement between both techniques is smaller, increasing as the non-linear effects increase. As such, for values close to N the Moment Method is a very valid solution, however for N the Moment Method loses its validity as the non-linear effects are too intense. Comaring with Table. it can be concluded that when the non-linear effects are weaker and the ulse is roagating near the linear regime the Gaussian ansatz rovides more accurate solutions. However, when transitioning to the non-linear regime the ansatz of the hyerbolic secant ulse becomes more accurate than the Gaussian one. This henomenon is related to the aearance of solitons in otical fibers which are better described by an hyerbolic secant shaed ulse. VI. CONCLUSION The main goal of this aer was to study ulse roagation in otical fibers using the Moment Method, addressing both its advantages and disadvantages, comaring it with analytical and numerical solution, like the Slit-Ste Fourier Method. The main conclusion drawn from this aer is that the main advantage of the Moment Method against a numerical solution is that we can observe which secific arameter is affecting the ulse behavior. As it was ointed out throughout the aer a ulse is described by different arameters. While some have no influence on the way the ulse roagates some may be resonsible for the ulse broadening or comression, for its change in amlitude or even its eriod. Using the Moment Method there is the ossibility to only maniulate one of those arameters at a time and analyse what effect it has on the ulse. If we think of the ulse as being controlled by a control anel with different buttons, each arameter would be a button. Switching different buttons, different asects of the ulse may be controlled which rovides a rofound qualitative analysis of the ulse, achieving a better hysical ercetion of the ulse as oosed to the brute force method of a numerical solution. Besides, it was ossible to conclude that a numerical method is vastly more demanding in regards to comuting time, taking significantly more time to run than the Moment Method, while roducing similar oututs. So, even though the SSFM is more recise there are situations where its use is not justified as the Moment Method rovides accurate and faster solutions. It can also be concluded that the GVD and the non-linear effects have a big influence on ulse behavior. If in the linear regime the GVD influence is absolute SPM does not exist as the non-linear effects raise their significance there starts to exist a balance between the SPM and the GVD. As such, while the GVD causes the ulses to broaden the SPM challenges this effect causing ulse comression. This dynamic is intimately connected with the develoment of chir and its analysis and maintenance is of high imortance in otical communication systems. After extensive analysis of the Gaussian and hyerbolic secant ulse it can be concluded both ulses translate a fair reresentation of a ulse roagating in an otical fiber, however each ulse shae better reresents this in different roagation conditions. While in the linear regime the Gaussian ulse translates a ulse in an otical fiber in a fairly accurate fashion when the non-linear effects start to aear and become more significant, accumulating over distance the Gaussian ulse evolves towards an hyerbolic secant shae, a behavior linked with the aearance of solitons. As such, in non-linear roagation conditions, the hyerbolic secant ulse better translates an otical ulse. These statements are suorted by the comarison between the Moment Method and the SSFM as when considering the Gaussian ulse there is a bigger agreement between the Moment Method and the SSFM in the linear regime while when considering the hyerbolic secant ulse the biggest agreement aears for N. Finally, it can be conclude that for very high non-linear effects the Moment Method loses recision. This haens because when using the Moment Method a ulse shae has to be assumed and that ulse shae does not change during roagation. However the non-linear effects cause changes in the ulse shae that when the non-linear effects are very high cause the Moment Method to lose validity. This means the Moment Method is a good solution when the non-linear effects do not dominate the ulse roagation in an otical fiber. However this is not exactly a disadvantage as in real roagation conditions the non-linear effects are never that significant nor do they surass the limits in which the Moment Method is valid, making it a valid and versatile solution in most cases. REFERENCES: [] G.P. Agrawal, Introduction in Fiber-Otic Communication Systems, 4 th ed. New York: Willey,,. -8. [] K.C. Kao e G. A. Hockham, Proc. IEE 3, 5 (966) [3] F.P. Karon, D. B. Keck, e R. D. Maurer, Al. Phys. Lett. 7, 43 (97) [4] C. R. Paiva, Fibras Óticas. IST, 8 [5] C. R. Paiva, Solitões em Fibras Óticas. IST, 8 [6] S. N. Vlasov, V. A. Petrishchev, e V. I. Talanov, Radiohys. Quantum Electron. 4, 6 (97) [7] J. Santhanam, Alications of the Moment Method to Otical Communications Systems: Amlifier Noise and Timing Jitter, Ph. D. dissertation, Det. Phys and Astron., University of Rochester, Rochester, NY, 4 [8] G.P. Agrawal, Control of Nonlinear Effects in Fiber-Otic Communication Systems, 4 th ed. New York: Willey,,. 48-43 [9] G.P. Agrawal, Nonlinear fiber otics: its history and recent rogress, J. Ot. Soc. Am. B, vol. 8, no., Dec. [] J. Santhanam and G. P. Agrawal.(July, 3). Raman-induced sectral shifts in otical fibers: general theory based on the moment Method. Otics Communication.[Online]... 43-4 [] S. Lefrancois, C. Husko, A. Blanco-Redondo and B. J. Eggleton, Nonlinear silicon hotonics analyzed with the moment method, J. Ot. Soc. Am. B, vol. 3, no., Feb. 5