Ultra-short pulse propagation in dispersion-managed birefringent optical fiber

Transcription

1 Chapter 3 Ultra-short pulse propagation in dispersion-managed birefringent optical fiber 3.1 Introduction This chapter deals with the real world physical systems, where the inhomogeneous parameters of the system have been taken into account. In the previous chapter, the dispersion parameters are considered to be constant under the assumption that the optical fiber is uniform. This would mean that there are no variations in the optical fiber. But in the real world, this may not be the case. So, the study of pulse propagation through inhomogeneous media demand special attention as it concerns to real physical systems. The presence of non-uniformity can very much influence the dynamics of soliton pulse propagation in the optical fiber. As a result, the transmitted soliton gets degraded by attenuation and dispersion, which are compensated by optical amplifier and DM systems, respectively [1, 78, 119]. In recent years, considerable research has been devoted to realize high-speed long-distance optical fiber communication systems employing DM for soliton transmission [1 13]. In a DM system, the fiber is made up of alternating sections of positive and negative GVD in such a manner so as to create a transmission line with high local and low average dispersion. 6

2 Such an arrangement of the link is certainly essential, as the dispersion must be compensated on average, which is provided by the alternation of positive and negative segments. Figure 3.1 represents the schematic diagram of DM fiber system. Let the positive and negative dispersion of the map have lengths and dispersions, L +,D + and L,D, respectively. Then, the path average dispersion, D av is [1,78,119]: Dz L / L / D + D av D L + L m z Figure 3.1: Sketch to explain dispersion management. The fiber line consists of alternating fiber segments with positive D + and negative D dispersion. The path average dispersion D av is then much lower than the local dispersion and close to zero. D av = D +L + +D L L m. 3.1 where, L m is the length of the dispersion map which is defined as: L m = L + +L. As in the case of conventional soliton, during the dispersion-managed soliton propagation, the dispersion and nonlinear effects cancel each other. The difference is that in the conventional case, this cancelation takes place continuously, whereas in the DM case, it takes place periodically with the periodof dispersion map length, L m. The strength of DM is characterized by a parameter, S, which is determined as [14 16] S = λ D+ D av L + D D av L, 3. πc T 61

3 where λ isthe wavelength ofthe incoming radiation, D + andd arepositive andnegative dispersion respectively, D av is the average dispersion and T is the input pulse width. The absolute values of the local dispersion are usually much greater than the path average dispersion: D +, D D av. The strength of the map is proportional to the number of the local dispersion lengths of the pulse in the map length: S L m /Z d where Z d is the dispersion length. Forysiak et al. [17] and Hasegawa et al. [18] have proposed adiabatic DM by changing dispersion profile in proportion to the soliton power in order to reduce dispersive wave radiation and collision induced frequency shift in WDM systems, respectively, at the amplifier. Mollenauer et al. [19] have succeeded in transmission of soliton based WDM at 1 Gb/s in seven channels by the use of the adiabatic dispersion map. Since the adiabatic dispersion map minimizes the imbalance between nonlinearity and dispersion, the transmitted pulse behaves almost as an ideal soliton in a fiber with the loss periodically compensated by amplifiers. In contrast, Suzuki et al. [13] used a non-adiabatic map to nullify the accumulated dispersion in a cross-oceanic soliton transmission experiment by periodically inserting dispersion compensating fibers. Smith et al. [131] proposed the use of a periodic map using both anomalous dispersion fiber and normal dispersion fiber alternatively. The nonlinear stationary pulse that propagates in such a fiber has a peak power larger than the soliton that propagates in a fiber with a constant dispersion whose value is given by the average value of the periodic map [14,15]. Such a nonlinear stationary pulse is commonly known as dispersion-managed soliton, and is found to exist even when the average dispersion is zero [13,133]. The dispersion-managed soliton is found to have enhanced energy compared to the energy of the soliton in a fiber with constant dispersion equal to the average dispersion. The power enhancement effect of dispersion-managed solitons is very important for practical applications. It provides an extra degree of freedom in the system design by giving the possibility to change the pulse energy while keeping the path-average fiber 6

4 dispersion constant. Single-channel high-bit-rate dispersion-managed soliton transmission over long distances with weak guiding filters and without guiding filters was experimentally demonstrated [14, 16]. DM considerably modifies the dynamics of pulse evolution because of the periodic variation of GVD. The alternating GVD sign within one period brings about large pulse width oscillation. This leads to strong overlap of neighboring bits in a pulse train, resulting in significant nonlinear interactions [135,136]. DM also alters the sign of the velocity of the pulse within each map period, and thus the trajectory of the central pulse position draws large zigzags. This zigzag motion leads to repeated collisions among pulses in different wavelength channels in WDM transmission [137, 138]. In dispersion-managed soliton transmission, WDM interactions due to the repeated collisions is found to be a major transmission penalty [139, 14]. In this chapter, we are concerned with solitons propagation in DM birefringent fibers where the magnitude and sign of the GVD is deliberately varied along the distance to improve the transition performance. The use of DM technique has dramatically improved the pulse robustness, and the possibility of transmitting information at higher data rates over the installed standard fiber network. It is therefore obvious to raise the fundamental question as whether or not the interesting robustness properties of dispersion-managed solitons can also exist in more elaborate systems. As a first step to address this question, in this work, we have to analyze the pulse propagation in a dispersion-managed two-modes system, in which the XPM is acting as a strong perturbation, and the optical pulses can also propagate over relatively long distances. Subsequently, we will also demonstrate the ability of short pulses to execute a highly stable propagation in a special system made of concatenated pieces of high-birefringence fibers with alternately positive and negative signs of the dispersion. Each piece of fiber is a high birefringence fiber, whose intrinsic birefringence is much higher than the random birefringence and even much higher than the intrinsic 63

5 birefringence that one could obtain by winding or spooling up the fiber. The main interest of a such setup lies in the demonstration that the technology of DM is a quite robust technique, which can ensure highly stable propagation of ultra-short pulses in perturbation conditions much stronger than those of a standard telecommunication fiber. In this chapter, we obtain the explicit form of the CV equations for the coupled HNLS equations using the generalized projection operator method. We illustrate the effectiveness of our CV equations by applying them to the characterization of light pulses in a birefringent DM fiber, in which higher-order linear and nonlinear effects are acting as strong perturbation effects such as the TOD, XPM, SRS and SS. This chapter is organized as follows. In section 3., we introduce the theoretical model to describe the propagation of ultra-short pulses in a two-mode fiber system. In section 3.3, we evaluate the pulse parameters such as amplitude, temporal position, width, chirp, frequency and phase through the projection operator method. Section 3.4 illustrates the evolution of pulse parameters against propagation distance. In this case, we show that the SRS has a strong impact on the pulse dynamics. Finally, in section 3.5 we give some concluding remarks. 3. Theoretical Model In this section, we will review the derivation of coupled HNLS equations which describe simultaneous propagation of two optical pulses in DM birefringent fiber, presenting it from a more physical perspective. The dispersion terms of the coupled HNLS equations can be derived in the same way as discussed in the previous chapter. In this chapter, we have examined the ultra-short pulse propagation in the DM birefringent fiber with linear attenuation. For this purpose, equations.13a.13b can be modified as i Q 1 z = γ i NL R1 +iαq 1, ω 1 t a

6 i Q z = γ 1 1+ i NL R +iαq, ω t 3.3b where α is the linear attenuation. Here, equations.14a-.15b can be rewritten in the following form NL R1 = NL R = Rt t 1 Q 1 z,t +cosθ Q 1 z,t t 1 ++sinθ Q z,t t 1 dt 1, 3.4a Rt t 1 Q z,t +cosθ Q z,t t 1 ++sinθ Q 1 z,t t 1 dt 1, 3.4b NL R1 = +cosθ Q 1 ++sinθ Q Q 1 Q 1 t 1 +cosθ Q 1 ++sinθ Q, 3.4c t NL R = +cosθ Q ++sinθ Q 1 Q Q t 1 t +cosθ Q ++sinθ Q d Equations 3.4c and 3.4d are valid if the pulse envelope evolves slowly along the fiber. Defining the first moment of the nonlinear response function as [1] T R = trtdt = th R tdt = f R dim h R dδω 3.5 Δω= and using Rtdt = 1 along with equations 3.4c and 3.4d into equations.13a and.13b and following the same way as we did in the previous chapter, we obtain coupled HNLS equations which describe two mode propagation in conventional birefringent optical fiber. In this chapter, we intend to investigate the dynamics of the ultra-short optical pulses propagation in DM birefringent optical fiber. For this purpose, coupled HNLS equations can be expressed as 65

7 q 1 3 z + i l 1β lz l q i γ l! τ l 1 q 1 +γ q q 1 + γ 3 q 1 q 1 +γ 4 q q 1 τ l= +iq 1 γ 5 q 1 +γ 6 q + α τ q 1 =, 3.6a q 3 z + i l 1β lz l q i γ l! τ l 1 q +γ q 1 q + γ 3 q q +γ 4 q 1 q τ l= +iq τ γ 5 q +γ 6 q 1 + α q =, 3.6b where β l z represent the l th order dispersion varies with z, γ 5 = T R γ 1 and γ 6 = T R γ represent the SRS and α is the linear attenuation. In practice, equations 3.6a and 3.6b describe a highly birefringent fiber system that makes use of two waves of different frequencies as the carrier waves. The two carrier waves are orthogonally polarized parallel to the fast and the slow axes of the fiber. The frequency difference between the two carrier waves can be varied continually in such a way that the group velocity mismatch becomes zero. As the system of equations 3.6a and 3.6b describe the propagation of ultra-short pulses in DM birefringent fibers, the dynamics of the pulse parameters in such a system will be very useful to precisely characterize the light pulses in many of the above mentioned two-mode fiber systems. Several analytical, approximation and perturbation methods have been developed to study the pulse dynamics in DM fiber systems [141 15]. Those methods include the Lagrangian variational method [141], the Hamiltonian method [14], the projection operator method [ ], the non-lagrangian collective variable approach [148], the collective variable CV technique [ ] and the moment method [151, 15]. Among these methods, we choose the generalized projection operator method to investigate the dynamics of ultra-short pulses in birefringent fiber. The major advantage of the projection-operator method over other existing methods is that it does not require one to go through the complex procedure of derivation of the Lagrangian. 66

8 3.3 Projection operator method In 1986, Boesch et al., proposed the projection operator method scheme for the Klein- Gordon equation to derive the ordinary differential equations which describe the pulse dynamics ofthat system [143]. Tchofo Dinda et al.,[151] proposedacvtheorywhich was claimed to be equivalent to the Lagrangian variational method. Nakkeeran and Wai [145] proposed the generalized projection operator method for complex nonlinear partial differential equations from which one can derive the ordinary differential equations that could be derived either by the Lagrangian variational method or the bare approximation of the CV theory. Wai and Nakkeeran [144] showed that the resultant CV equations of motion derived through the Lagrangian variational method and the bare approximation of the CV theory are unique for the Gaussian ansatz. In the presence of SRS, the system becomes dissipative and writing a Lagrangian density for such a system is not possible. Hence the variational method cannot be used to study the effects of SRS on pulse propagation in a fiber. To study the SRS in solitons, perturbation theory can be used [151]. In effect, although the variational approach works well for conservative systems, it fails in the presence of SRS in the fiber. Hence a more generalized theory that will work for both dissipative and nondissipative systems has to be used in order to study the pulse propagation in the case of systems using ultra-short pulses. In this chapter, we use the projection operator method to develop a general theory to study the ultra-short pulse propagation in birefringent DM optical fibers. In all the studies mentioned above, a Gaussian profile for the pulse propagating in the DM fiber transmission line was assumed. In fact, it is its analytical tractability that makes the Gaussian ansatz attractive, especially in the variational analysis of DM fiber systems. We assume for the pulse in the q 1 mode and the pulse in the q mode as a Gaussian shape of the following form q 1 z,τ = f[x l z,τ], 3.7a 67

9 q z,τ = g[y l z,τ], 3.7b where f = x 1 exp [ τ x x 3 [ τ y g = y 1 exp y 3 +ix 4 τ x +iy 4 τ y ] +ix 5 τ x +ix 6, 3.8a ] +iy 5 τ y +iy 6, 3.8b where x l s designate the pulse parameters in the q 1 mode, y l s are the pulse parameters in the q mode and l = 1,,3,4,5,6. The parameters x 1,y 1, x,y, logx3,y 3, x 4,y 4 /π, x 5,y 5, and x 6,y 6, represent the pulse amplitude, temporal position, FWHM Full Width at Half Maximum of peak power, chirp, frequency, and constant phase, respectively. Note that the parameters x 5,y 5 that we call frequency correspond in fact to the frequency shift of the pulse with respect to the carrier frequencies. To proceed further, we substitute equations 3.7a and 3.7b into equation 3.6a and multiply by the projection operator P q1 = f x l e iθ, where θ is a phase constant and f x l represents complex conjugate of f xl, and integrate over τ. Collecting real part, we get Rf z fx l e iθ dτ β z + γ 1 f +γ g Iff x l e iθ dτ + If ττ fx l e iθ dτ β 3z 6 γ 5 f +γ 6 g τ Iff x l e iθ dτ + α Rf τττ f x l e iθ dτ R[γ 3 f f +γ 4 g f τ f x l e iθ ]dτ Rff x l e iθ dτ =. 3.9 On substituting equations 3.8a and 3.8b into equation 3.9 and choosing any value for θ between π/ to π/, we obtain the CV equations of motion of the field q 1 z,τ as follows x 1 = 1 αx β zx 1 x 4 1 β 3zx 1 x 4 x 5 +S 1 S 4 γ 4 x 1 y1 y 3x y x 3 7x 3 4x y +11y3 +4y4 3, 3.1a 68

10 x = β zx 5 + β 3z 4+x 4 8x 3 x 4 +4x 3 x 5 + 3γ 3x 1 3 +S 1 S 3 γ 4 y 1y 3 x 3x 3 x y +3x 3y 3 +y 4 3, 3.1b x 3 = β zx 3 x 4 +β 3 zx 3 x 4 x 5 S 1 S 4 γ 4 x 3 y1 y 3x y x 3 5x 3 4x y +7y3 +y4 3, 3.1c x 4 = β z β 3 zx 5 x 4 4 x 1 γ1 γ x 4 3 x 3 x 5 4S1 S 3 γ y1 y 3 3 x 3 4x y +y3 +4S 1 S 4 γ 4 y 1y 3 x 3x 5 x 4 x y x 3 4x y + 4x x 5 4x 5 y +x 3 x 5 x 4 y +x x 3x 4 +8x 5 y y3 +x 5 +x 4 x y y S 1 S 4 γ 6 y1y 3 x y 4x y 3x 3 +y3, 3.1d x 5 = γ 3x 1x 4 γ5 x 1 + 4S x 1 S γ y1 x y y 3 +4S 1 S 3 γ 6 y1 y 3 3 x 3 4x y +y3 +S 1 S 3 γ 4 y 1y 3 x 3x 3x 4 4x x 4 y x 5 +x 4 y +x x 5 +4x 4 y +x 3 x 4 +x 5 x y y3, 3.1e x 6 = β z x x 3 x 5 + β 3z 3x 4 3 4x 3 x 4 x 5 1x 5 +8x 3 x x γ 1 +γ 3 x 5 + S 1S 3 γ y1 y 3 y3 4 +5y 3 4x y +3x 3 x 3 +S 1 S 4 γ 6 y1 y 3x y y 43 +x3 5x3 4x y +7y S 1S 4 γ 4 x 3y 1y 3 y x x 34x y 3x 3 +y 3x 4 +x 3 +y 3y 3 4x y +x 3x 5, 3.1f where [ x y S 1 = exp x 3 +y 3 S 3 = x 3 +y3 ] ; S = x 3 +y 3 5 ; S 4 = x 3 +y 3 3, 7. 69

11 The overdot denotes differentiation with respect to z. We substitute equations 3.7a and 3.7b in equation 3.6b and multiply by P q = gy l e iφ where φ is a phase constant and gy l represents complex conjugate of g yl and integrate over τ. We get the following equation Rg z gy l e iφ dτ β z + γ 1 g +γ f Igg y l e iφ dτ + Ig ττ gy l e iφ dτ β 3z 6 γ 5 g +γ 6 f τ Igg y l e iφ dτ + α Rg τττ g y l e iφ dτ Rγ 3 g g +γ 4 f g τ g y l e iφ dτ Rgg y l e iφ dτ = On substituting equations 3.8a and 3.8b into equation 3.11 and choosing any value for φ between π/ to π/, we obtain the real part of CV equations of motion of the field q as follow y 1 = 1 αy β zy 1 y 4 1 β 3zy 1 y 4 y 5 +S 1 S 4 γ 4 x 1y 1 x 3 x y y3 7y 3 +4x y 11x 3 4x 4 3 y = β zy 5 + β 3z 4+y 4 8y3 3 y 4 +4y 3 y γ 3y1 +S 1 S 3 γ 4 x 1 x 3 y3 y 3 4x y +3x 3 +x4 3, 3.1a, 3.1b y 3 = β zy 3 y 4 +β 3 zy 3 y 4 y 5 +S 1 S 4 γ 4 y 3 x 1 x 3 x y y3 5y 3 4x y +7x 3 +x4 3, 3.1c y 4 = β z β 3 zy 5 y4 4 x 1 γ y3 4 x 1 γ 3 x 5 4S 1 S 3 γ x 1 x 3 3 y3 4x y +y3 +4S 1 S 4 γ 4 x 1x 3 y x x 4 3 +y3 x 3 y 3 +4x y y 4 +x 3 +y 3 x 3 4x y +y3y 5 +16S 1 S 4 γ 6 x 1x 3 x y 3x 3 +y 3 4x y, 3.1d 7

12 y 5 = y 6 = β z y 3 γ5 y1 + γ 3y1 y 4 +4S y3 1 S γ x 1 x y x 3 +4S 1 S 3 γ 6 x 1 x 3 x 3 4x y +y3 +S 1 S 3 γ 4 x 1 x 3 y3 x 3 4x y +y3 y 4 x y x 3 +y 3 y 5, 3.1e y3y 5 3y 3y 4 4y 5 1y 5 +8y3y β 3z 4y 3 + 5y γ 1 +γ 3 y 5 + S 1S 3 γ x 1 x 3 x x 3 4x y +3y3 y 3 S 1 S 4 γ 6 x 1 x 3x y x 43 +y3 5y3 4x y +7x S 1S 4 γ 4 y3 x 1 x 3 x y y3 4x y 3x 3 +y 3 y 4 +x 3 +y 3 x 3 4x y +y3 y f Equations 3.1a - 3.1f represent the CV equation of motion of q mode. 3.4 Results and Discussions To evaluate the correctness of our variational equations, the corresponding solutions should be compared with the CVs obtained directly from the pulse profile equation 3.8a, by use of the following formulae that correspond to the moment method [151,15]: x 11 = max q 1, τ q 1 dτ x 1 = q 1 dτ x 13 = 1 L 1 x 1 N 1 = N a, 3.13b q 1 dτ, L 1 = τ q 1 dτ, 3.13c 71

13 x 14 = x 15 = i τ x 1 q 1 q 1τ u q 1τ dτ w q 1 dw q 1 dw x 16 = q 1τ = q 1 τ =, τ x 1 q 1 dτ, 3.13d, 3.13e 3.13f In equation 3.13e, q 1 represents the spectral Fourier transform of q 1. Our variational equations are basically relevant for the characterization of the phase and intensity profile of ultra-short pulses only in situations where the exact solutions of the coupled HNLS equations are not too far from a Gaussian-shaped profile as those considered in the ansätz 3.8a. Such a situation often exists in DM fiber systems, in which highly robust pulses, which are the fixed points of the system, display an intensity profile which is in general very close to a Gaussian profile [ ]. Here also, we will consider a DM fiber system, which is made of a juxtaposition of fiber sections with alternately positive and negative dispersion. To obtain the evolution of the pulse parameters, we have carried out numerical simulations of the pulse propagation in a typical DM fiber system, which consists of a juxtaposition of identical dispersion maps L /, L +, L / made of pieces of nonzero dispersion-shifted fibers, with the following typical parameters: second-order dispersion β ± = ±6.8ps /km, TOD β 3± = ±.6ps 3 /km higher-order dispersion compensation. The fiber lengths are determined by means of an analytical design procedure for DM fiber systems [153] with the following input data: Input pulse width FWHM 11.8ps, energy E =.5pJ. Hence we obtain: L = 1.36km, L + = km. The nonlinear coefficients are chosen as follows: γ 1 = W 1 m 1, γ = γ 1 /3, γ 3 = λ cπ γ 1, γ 4 = γ 3 /3, γ 5 = T R γ 1, γ 6 = γ 5 /3, λ = 1.55μm and T R = ps. 7

14 Amplitude W 1/.18 a Temp. Pos. ps b.1 Width ps 5 15 c Chirp THz/ps 4 x 1 3 d 1 Frequency THz x 1 5 e Phase Rad 1 f z km z km Figure 3.: Evolution of the CVs without the effect of SRS. Solid curves and asterisks respectively represent the result obtained from our CV equations and full numerics. The effect of fiber losses is known to cause an exponential attenuation of the pulse amplitude in all fiber systems. As this effect generally dominates all the other effects, wehavesetthelosscoefficientαtozeroinordertoshedmorelightontheothereffects. We have found from numerical simulations not presented here, that in the situation of zero group-velocity mismatch under consideration, the pulse dynamics in the q 1 -mode and q -mode are qualitatively the same. Consequently only the dynamical 73

15 behaviour in the q 1 -mode will be presented below. Figures 3. and 3.3 represent the pulse dynamics obtained without and with the effect of SRS, respectively. The Amplitude W 1/.18 a Temp. Pos. ps b.1 Width ps 5 15 c Chirp THz/ps 4 x 1 3 d 1 Frequency THz x 1 5 e Phase Rad 1 f z km z km Figure 3.3: Evolution of the CVs with the effect of SRS. Solid curves and asterisks respectively represent the result obtained from our CV equations and full numerics. solid curves represent the solution of our variational equations. The asterisks show the results obtained from the moment method based on the pulse profile equations 3.13a-3.13f. Figures 3. and 3.3 exhibit the following general features: i The CVs obtained from the moment method agree extremely well with the solution of our variational equations. This agreement provides strong evidence of the reliability of 74

16 our equations 3.1a-3.1f for the characterization of ultra-short light pulses. This agreement indicates that, in the parameter region under consideration, the exact pulse profile is close to a Gaussian profile. There are two types of oscillatory dynamics in dispersion-managed systems: the fast oscillationsas those represented in figures 3., and the slow dynamics which are only partially visible in figures 3.. In fact, only a single period of the slow dynamics is clearly visible in figures 3.a or 3.c. The largest oscillations observed in figures 3.a or 3.d at nearly km correspond to the maximum of the slow dynamics. ii Figures 3.a, 3.c and 3.d show that the pulse amplitude, width and chirp, execute large-amplitude oscillations, which result from the periodic alternation of fiber segments with positive and negative dispersion. On the other hand, figures 3.e and 3.b show that no frequency shift and no temporal shift occur when the system modelling does not include SRS. In contrast, figure 3.3e shows that SRS causes the pulse to execute a frequency x 5m shift, which is converted into temporal shift as the first and second terms in the r.h.s of equation 3.1b show. The temporal shift due to the second-order dispersion is linear as the first term in the r.h.s of equation 3.1b shows. One can see in figure 3.3b that the temporal shift due to SRS is indeed linear the zigzags are simply due to the change of sign of the second-order dispersion. Each piece of the zigzag corresponds to a linear temporal shift. The effect of TOD, which induces a temporal shift that is rather quadratic x 5m, is much smaller than the effects of the second-order dispersion. 3.5 Conclusion We have examined the dynamics of ultra-short pulses in a two-mode fiber system, by use of a set of variational equations obtained from the generalized projection operator method with Gaussian ansatz. Although we have illustrated this variational approach by applying it to the dynamics of two light pulses polarized respectively along the 75

17 two birefringence axes of a lossless fiber with strong intrinsic birefringence and zero group-velocity mismatch, the method of the present chapter can easily be applied to many other nonlinear two-mode fiber systems that support localized structures with relatively long lifetime, such as LP 1 and LP 11 modes of a standard step-index fiber [154]. The use of this technique of DM has dramatically improved the pulse robustness, and the possibility of transmitting information at higher data rates over the installed standard fiber network. It is therefore natural to raise the fundamental question as whether or not the interesting robustness properties of dispersion-managed solitons can also exist in more elaborate systems. As a first step to answering this question, in this work we have analyzed the pulse propagation in a DM two-mode system, in which the XPM is acting as a strong perturbation, and we have shown that pulses can also propagate over relatively long distances, as can be seen in figures 3. and 3.3. The effects of SRS and SS have no significant impact on the input pulse width greater than 5 ps. In this chapter, at the input pulse width of 11.8 ps, we have shown that SRS has a strong impact on the pulse dynamics which is quite contrary to the established results. Another important point to be noted is that the spatial dynamical behavior of ultra-short pulses in the birefringent DM fiber in terms of the above studied pulse parameters have not been investigated so far. Finally, our study demonstrates the ability of short pulses to execute a highly stable propagation in a special system made of concatenated pieces of high-birefringence fibers with alternately positive and negative signs of the second-order dispersion, which includes several perturbing effects such as the TOD, SRS, or XPM. The main virtue of our variational equations is that they show explicitly the mode of action of each physical effect on the pulse parameters, thus giving a deeper insight into the dynamical behaviour of the localized structure. It would have been practically impossible to obtain comparable information through a theoretical analysis that does not use a CV approach. 76