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1 Pulse Propagation in Optical Fibers Arleth Manuela Gonçalves Departamento de Engenharia Electrotécnica e de Computadores Instituto Superior Técnico Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal [email protected] Abstrat — This paper addresses the pulse propagation through a fiber optic system, operating in the linear and nonlinear regimes. After a brief introduction to optical fibers, we use the modal theory approach to understand the operating principle for the pulses propagating in the fiber. Then, we try to understand the effect of dispersion, as the unavoidable phenomenon to which pulses are subject, while propagating in the fiber, in the linear regime. Since dispersion strongly limits the bandwidth of the transmitted signal and may cause interference between symbols, we address several methods for reducing the effect of dispersion, which can be very efficient: the use of dispersion compensating fibers and the propagation of solitons. In non-linear system, the non-linear Kerr effect explains the appearance of self-phase modulation (SPM) which has a contrary action to the phenomenon of dispersion in the anomalous dispersion region, allowing the appearance of solitons, which era pulses that conserve their shape along the propagation. The propagation of pulses in nonlinear regime is governed by the nonlinear Shrondinger (NLS) equation for which, their analytical solutions are calculated. An efficient method is used in the numerical simulations under the nonlinear regime: the split-step fourier Method (SSFM). Finally, a brief study of the fundamental soliton is presented. Keywords — Fibre optics, dispersion, pulse propagation, solitons. I. INTRODUCTION T he appearance of optical fibers drastically changed the paradigm of data transmission, because of two main advantages that make it particularly in relation to wires: The optical fibers are completely immune to electromagnetic interference as they are made of dielectric material capable of transmitting pulses of light, which means that data is not corrupted during transmission. Optical fibers do not conduct electric current, so there will be no problems with electricity, as voltage instability problems or issues with spokes. The physical phenomenon known as total internal reflection is the responsible for light pulses to travel along the fiber by successive reflections [10]. Although since 1870 it was known that light could describe a curved path within a material, only in 1952 the Indian physicists Narinder Kapany could complete its experiments leading to the invention the optical fibers. However, only with the appearance of devices capable of converting electronic pulses into pulses of light, it was possible to transmit information through the fiber [9]. There are two types of fiber: Multimode fibers with core diameters greater than the single mode, allowing light to traverse the fiber through several paths, which are strongly affected by the effect of dispersion and, because of this, are most often used for data traffic within walking distance. Single-mode fibers with core diameters of the order of less than one strand of hair and let light travel in the interior of fiber by a single path, and less sensitive to the dispersion effect and are therefore most often used for long distance communications [6], [8]. With this work, we hope to have contributed to better understanding of the effects to which light pulses propagate along the fiber are subject, both in the linear and nonlinear regimes. II. MODAL THEORY Because of the difference between the refractive indices of the core and the cladding, it is possible to transmit of light inside the fiber [3]. The core index of refraction of the core must be always higher than the refractive index of the cladding. The modal theory can explain the existence of an angle of incidence that enables the phenomenon of total internal reflection. A. Fibers operated in the Linear Regime Optical fibers operated in the linear regime (monomodal regime), are the fibers that allow the use of only one light signal through the fiber. They have a smaller radius and greater bandwidth due to lower dispersion. It is possible to define a maximum value for the angle of incidence with respect to the fiber axis, called acceptance angle maximum θ0max, imposing that only the rays entering the fiber at an angle of less than θ0max are propagated along it [13]. Let n1 be the refractive index in the core and n2 be the refractive index in the cladding. The parameter which describes the ability to collect light in a fiber called a numerical aperture (NA) which is related as θ0max follows 2 NA n0 sin 0max n12 n22 n1 2 there is a minimum frequency of operation (normalized cutoff (2.1) frequency where n0 is the refractive index of the outer medium in which the fiber is inserted, usually the air, where n0=1. Δ is the dielectric contrast given by n12 n22 (2.2) 2n12 For better understanding the wave propagation inside a step index fiber with core radius a, one may consider the scheme depicted in Fig. 2.1. Fig 2. 1 – Rays propagation in fiber [10]. The longitudinal component of the electric field in cylindrical coordinates has the following form Ez (r, , z, t ) E0 F (r )exp(im )exp i( z t ) (2.3) where exp[i(βz-ωt)] is the time and space dependence of monochromatic light traveling along the fiber axis, E0 is the amplitude of the field, m is the azimuthal variation index, F (r) represents the radial modal eigenfunction, β is the longitudinal propagation constant, which is related to the free-space propagation constant as k 2 ( r) n2 ( r)k02 2 vc 2.4048 , which is the cutoff value for the second propagating mode, indicating whenever the fiber is operating below that value, the system monomodal, or multimodal if above, i.e., for v vc => singlemode fiber, for v vc => multimode fiber. B. Propagation Modes in Fiber Light, while propagating in the fiber, can be seen sa an electromagnetic phenomenon, and the whole propagating mechanism, which can be described by the electromagnetic optical fields associated to it, is governed by Maxwell equations [1]. Generally speaking, modes can be classified into TE (Transverse Electric) modes, where there is no component of electric field in the direction of propagation (Ez = 0), TM (Transverse Magnetic) modes, where there is no magnetic field component in the direction of propagation (Hz = 0), and hybrid modes, for which there have both longitudinal field components, i.e., they have electric field (Ez ≠ 0) and magnetic field (Hz ≠ 0) components in the direction of propagation. These latter modes can be classified into HEmn and EHmn, where the parameter m refers to the azimuthal field variation and the parameter n refers to the radial field variation [10]. In general, the surface waves guided by an optical fiber are hybrid modes, and their field components are governed by the Bessel functions of first (Jm) and second (Km) species, satisfying the following equation modal u2 v Rm (u) Sm (u) m2 1 2 2 v uv 4 (2.8) where (2.5) in vacuum is Let k q in the cladding and k h in core. Since there is a surface wave guided by the cladding sheath, suffering total internal reflection at the core-cladding interface, under these conditions, one must have q i . Introducing normalized (dimensionless) wavenumbers u ha and w a , and a normalized frequency v , their relation is given by v 2 u2 w2 the fiber. A special value is given by (2.4) where n1 for r a n( r ) n2 for r a and the propagation constant k0 c 2 . vc ) for which a certain mode starts to propagate in (2.6) Moreover, introducing the normalized modal refractive index as u 2 w2 ( k0 )2 n22 b 1 2 2 (2.7) v v n12 n22 Rm (u) Sm ( u ) J m' (u) K ' ( w) m uJ m (u) wKm ( w) J m' (u) K ' ( w) (1 2) m uJ m (u) wKm ( w) (2.9a) (2.9b) The resolution of the Eq. (2.8), while being very complex, turns it to be necessary to use numerical methods. Only when the azimuthal index is zero (m = 0) it is possible to propagate transversal TE0n and TM0n modes. According to the Gloge approximation (i.e., for Δ << 1), in which modes propagating in optical fibers are weakly guided, one may take Rm(u)= Sm(u), which will cause the modal Eq.(2.8) to be reduced to mv 2 Rm (u) 2 2 (2.10) uw where the plus (+) sign corresponds to the modes EHmn and the minus (-) sign corresponds to the modes HEmn. Through the relationship between the Bessel functions and their derivatives under the Gloge approximation, in which the 3 azimuthal index zero, the Eq (2.10) for both modes (EHmn HEmn) reduces to J 1 (u ) K ( w) 1 0 (2.11) uJ 0 (u) wK0 ( w) inferring that the modes EH0n are actually TM0n and modes HE0n are modes TE0n . Moreover for weakly guided fibers, these modes are almost linearly polarized, and therefore will be termed as LPpn. One should stress that the only mode capable of propagating in a single mode fiber system (the only mode whose cutoff corresponds to vc=0), called the fundamental mode, is the HE11 mode. Mosr propagating modes are degenerated modes, i.e., they have the same cut-off frequency but with different field structure. In particular, HE1n modes give rise to modes LP0n Modes TE0n, and TM0n HE2n are degenerate and give rise tomodes LP1n HEm+1,n modes and EHm-1,n with m ≥ 2 are also giving rise to degenerate modes LPmn. Therefore, one has: EHmn→LPpn (with p=m+1); HEmn→LPpn (with p=m-1). Accordingly, one has the following modal for LP modes: J (u ) K ( w) u p 1 w p 1 0 (2.12) J p (u ) K p ( w) For the fundamental mode (LP01), it reduces to uJ1 (u) K0 ( w) wJ 0 (u) K1 ( w) (2.13) Figure 2.2 shows the variation of the normalized modal refractive index b as a function of normalized frequency v for the first six modes propagating in the fiber. Fig 2. 2 - First six LP mode fiber: diagrams b(v). One can observe a general increase in the normalized refractive index b modal with increasing normalized frequency v, and the greater the normalized frequency v, more modes propagating in the fiber. It is also interesting to analyze the influence of the dielectric contrast on the dispersion curves b(v). as in Figure 2.3 Fig 2. 3 - Influence of dielectric contrast on the dispersion curve b (v). Noting that an increase in the dielectric contrast Δ causes an increase in dispersion curves b(v), one may conclude that is possible to reduce effects of dispersion by using fibers with a smaller contrast. III. PULSE PROPAGATION IN THE LINEAR REGIME Any pulse, while propagating in an optical fiber, especially in the linear regime, suffers the effect of time dispersion, which causes its broadening and may create interference between symbols, which can greatly limit the bandwidth of the signal to be transmitted. Let one consider an optical pulse with a finite spectral width to be launched into the fiber. Each spectral component of the pulse, while traveling along the fiber, has a different group velocity which depends on its wavelength according to 1 2 c vg (3.1) with The dispersion due to the difference between the propagation velocities of different spectral components is called the Group Velocity Dispersion (GVD). There are two main types of dispersion: the Intermodal Dispersion (between the various modes of propagation) and intramodal dispersion (under the same mode of propagation). Intermodal dispersion is due to the various propagation modes travelling in the fiber. Each mode has a different group velocity for the same wavelength and the pulse width at the output of the fiber will depend on the transmission times. The time delay between the fastest (i.e., the fundamental mode) and the slowest mode (mode of higher order, depending on how many modes can propagate in this multimode fiber) is responsible for the broadening of the pulse at the output of the fiber. In the case of the intermodal dispersion or, as commonly referred to, the chromatic dispersion, the broadening of the pulse occurs within the same mode, mainly because of the dependence of the refractive index of the material with the frequency. However, the chromatic dispersion is a consequence of combined effect of two factors: the material dispersion and the waveguide dispersion [6], [11]. In the case of the material dispersion, the refractive index of the constituent materials of the fiber has a non-linear variation with the wavelength dn ng n (3.2) d 4 Waveguide dispersion follows from the fact that, for a given mode, the energy distribution between the core and the cladding is a function of wavelength. Generally, 80% of optical power propagating in the fiber remains confined to the core while the remaining 20% propagate along the cladding at a speed greater than the core causing the broadening of the pulse at the fiber output [6]. In the neighborhood of the zero dispersion wavelength (ideal frequency band) the spectral components at different wavelengths have almost the same propagation velocity, in other words, propagation delay is quite constant at different wavelengths. In normal dispersion region, one has Dλ <0 => β2> 0. The group velocity decreases with the frequency, i.e., in the red spectral components travel faster than the blue ones. Fig 3. 4 - Normal dispersion [5]. For a standard silicon optical fiber, the zero dispersion wavelength occurs around 1300 nm. Around this value, there is no pulse broadening. For this reason, the most current optical communication systems have been developed to take advantage is this characteristic [12]. Fig 3. 1 - Zero dispersion [5]. Discarding higher order dispersion effects, the parameter which describes the fiber dispersion is given by 2 c D 2 2 (3.3) with the DGV coefficient β2 is given by 1 vg 2 2 (3.4) vg (0 ) Let one take into consideration the electromagnetic spectrum as depicted in Figure 3.2. A. Pulse Propagation Equation in linear Regime In this Section, the differential equation that governs the propagation of pulses in the linear regime along a single mode optical fiber with small contrast (Δ << 1), is derived ignoring the effect of higher order dispersion. Let A(0,t) be the pulse at the fiber input at z = 0. Assuming that the pulse modulates a carrier, with an angular frequency ω0, and that the electric field is linearly polarized along the x, one may write ˆ 0 F ( x, y) A(0, t )exp( i0t ) E( x, y,0, t ) xE where F (x, y) is the modal function representing the transversal variation of the fields of the LP01 mode. With the help of the Fourier transform (numerically computed through the FFT-Fast Fourier Transform and Inverse Fast Fourier Transform-IFFT), the pulse envelope at a generic point of the fiber is given by Fig 3. 2 - Electromagnetic spectrum. In the anomalous dispersion region, one has Dλ> 0 => β2 <0. The group velocity increases with the frequency, i.e., the blue spectral components travel faster than those red (see Figure 3.3.). (3.5) A( z, t ) 1 2 A( z, )exp(it )d (3.6) Thus the electric field in a generic point of the fiber is given by ˆ 0 F ( x, y ) A( z, t )exp i( 0 z 0t ) E ( x, y, z, t ) xE (3.7) where 0 (0 ) (3.7a) In order to obtain A(z,t)) from A(0,t), we introduce the following GVD coefficients Fig 3. 3 - Anomalous dispersion [5]. m m m (3.8a) 5 where 1 2 1 vg (0 ) 2 1 1 vg 2 2 vg (0 ) (3.8b) (3.8c) For the numerical solution of Eq. (3.10) is useful to introduce the normalized (dimensionless) space and time variables z (3.14a) LD The frequency deviation over the carrier is given by 0 (3.9) As |Ω|<<ω0, it is reasonable to disregard the dispersion coefficients above β2. Moreover, discarding the losses in the fiber, the linear propagation equation, which allows calculating A(z,t) from A(0,t), can be written by t 1 z 0 (3.14b) So equation (3.10) can be rewritten as A( , ) i 2 A( , ) sgn( 2 ) 0 2 2 (3.15) and the Fourier pair as A( z, t ) A 2 A( z, t ) 1 i 0 z t 2 t 2 2 (3.10) A( , ) A( , ) exp(i )d (3.16a) This equation can be solved using a simple algorithm which allows the computation of pulse propagation along the fiber. This algorithm is herein designated RIMF algorithm and is applied in three steps: First step: To compute A( , ) 1 2 A( , ) exp(i )d (3.16b) where ψ is a normalized frequency given by A(0, ) FFT A(0, t ) A(0, t ) exp(it )dt 0 ( 0 ) 0 (3.17) Second step: Then compute To apply the RIMF algorithm to Eq. (3.15), one has A( z, ) A(0, )exp i() z First step: To calculate A(0, ) FFT A(0, ) Third step: Finally 1 A( z, t ) IFFT A( z, ) 2 with () 1 2 2 Second step: Then A( z, ) exp(it )d 2 (3.11) Because the pulses are usually narrowband |Ω|<<ω0, A(z,t) is a slowly varying function in time and oscillates with exp(-iΩt). In the absence of the dispersive effects, the pulse propagate without distortion with a group delay g 1 z i A( , ) A(0, ) exp sgn( 2 ) 2 2 Third step: Find A( , ) IFFT A( , ) Figure 3.5 illustrates the effect of the GVD on a secanthyperbolic shaped pulse, i.e., one whose initial shape is A0 ( ) A(0, ) sec h( ) (3.18) (3.12) Since it is reasonable to neglect the influence of the higher order dispersion (β3=0), it is possible to isolate the effect of the GDV in the pulse propagation. Defining τ0 as the characteristic pulse time width, the dispersion length is introduced as LD 02 2 (3.13) The dispersive effects in a optical link of length L are negligible only if LD>L. Fig 3. 5 - Comparison of the pulse input and output fiber. 6 A(0, ) FFT A(0, t ) A0 0 2 02 2 exp 1 iC 2(1 iC ) (3.24) The pulse spectral intensity will be Fig 3. 6 - Evolution of the pulse. 2 B. Chirp Since the optical signals emitted by a laser source suffer from chirp, it is useful to introduce a chirp parameter C which quantifies the variation in the carrier frequency C c (3.19) where βc is Henry factor, responsible for the enlargement of the spectral line. When considering the linear propagation of a Gaussian pulse with the Chirp [3], the initial pulse can be written as 1 iC t 2 A(0, t ) A0 exp 2 0 where A0 represents the pulse amplitude. A(0, ) A02 02 2 02 exp 2 1 C2 (1 C ) 2 (3.25) The spectral width Δω at 1/e maximum will be 1 C2 0 (3.26) One may conclude that the larger the value of C, the larger the pulse spectral width. (3.20) Fig 3. 8 - Evolution of Gaussian pulse with C=0. Fig 3. 7 - Initial Gaussian pulse for C=0. The frequency shift δω(z,t) caused by the existence of the chirp is z t z (3.21) ( z, t ) C (1 C 2 ) 22 2 1 0 1 ( z) with (3.22) 1 ( z ) 0 ( z) and η(z) is the pulse broadening factor 2 z z ( z ) 1 sgn( 2 )C LD LD Fig 3. 9 - Evolution of Gaussian pulse with C=2. The pulse broadening factor η(z), introduced in Eq. (3.23), also shows that a pulse may also suffer a time compression as it propagates, as long as β2C<0 [3]. Figure 3.10 shows the variation of the broadening factor with the travelled distance for a Gaussian pulse in the anomalous dispersion region (β2<0) 2 (3.23) When applying the first step of the RIMF algorithm to the initial impulse Gaussian, one has Fig 3. 10 - Spatial evolution of spectral width pulse for different values of C. 7 The pulse broadening due to the dispersion is sensitive to the sharpness of the pulse. When considering pulses with sharper steep edges, the broadening is generally greater, as is the case Super-Gaussian pulse whose initial momentum is given by 1 iC t 2 m A(0, t ) A0 exp 2 0 (3.27) Fig 3. 14 - Evolution of Super-Gaussian pulse with C=2. This expression is quite identical to the Gaussian pulse, differing only in the value of the parameter m. For a Gaussian pulse m is equal to 1. The edges of the pulses become increasingly steep as m increases, as shown in Figure 3.11, when compared with figure 3.7. Fig 3. 11 - Initial Super-Gaussian pulse for C=0, to m=3. The parameter m is related to the duration tr for which the intensity of the pulse increases from 10% to 90% of its peak value tr 0 m (3.28) Eq. (3.28) shows that a pulse with a smaller rise time increases faster [2]. For this reason the super-Gaussian pulses as well as expand faster than the Gaussian, also strongly distort its original shape, as shown below, when considering m=3. Fig 3. 12 - Super-Gaussian comparison pulse input and output fiber. Fig 3. 13 - Evolution of Super-Gaussian pulse with C=0. C. Dispersion Compensation A technique used to compensate for or minimize the problem of dispersion is the use of dispersion compensating fibers (DCF –Dispersion Compensation Fiber). This technique is to combine optical fibers with different characteristics, such that the average GDV of the link becomes quite small, whereas the GDV of each section may be large [7]. In each two consecutive sections, there are two types of fiber. A section of greater length L1, operating in the anomalous dispersion region (with β21<0), and a shorter section of length L2, operating in the normal region (with β22>0). Nevertheless, the two GDV coefficients are quite different (|β21|≠|β22|). This technique takes advantage of the linear nature of the system when considering the propagating optical pulse into two sections of the fiber, whose impulse propagation equation is given by 1 i A( L, ) A(0, ) exp 2 ( 21L1 22 L2 ) i d 2 2 (3.29) where L L1 L2 . The length of dispersion compensating fiber L2 is chosen so that 21L1 22 L2 0 L2 21 L 22 1 (3.30) Pick-up |β21|<<|β22| so that L2 is much lower than L1. When using this technique it is common for L1 to be in the order of 100-1000 km, while L2 is in the order of 1 km. Provided that they fulfill the condition β21L1+β22L2=0, A(L,τ)=A(0,τ) thus recovering its original shape once every two consecutive sections, although pulse width may change significantly in each section. As an example of application, we consider L1=20 km, β21= -1, L1=20 km e β21=1 and the pulse input of the first fiber is given by A0 ( ) A(0, ) sec h( ) (3.31) Fig 3. 15 – Comparison input and output pulse in the 1st fiber. 8 The output pulse of the first fiber is the impulse input of the second fiber (DCF). So, taking into account the shape of the pulse to propagate Fig 4. 1 - Power of a Gaussian pulse [7]. Fig 3. 16 - Comparison input and output pulse in the 2nd fiber. IV. PULSE PROPAGATION IN NONLINEAR REGIME Pulse propagation in nonlinear regime is affected by the optical Kerr effect. The propagation of impulses in Nonlinear Dispersive Regime (NLDR) is governed by the SPM and the GDV simultaneously [4]. To disrupt the relative dielectric constant (4.1) ' ( x, y ) where (4.2) 2n( x, y )n Whatever the process that led to that disturbance, the new longitudinal propagation constant is given by (4.3) ' Introducing the parameter wherein n2' k0 n2' 2 ref2 (4.4) is the effective area and ref is the effective radius. The optical Kerr effect provides that for certain values of n2' (4.5) P( z, t ) P( z, t ) is the transported power, that is related to the power of the internal fiber Pin (t ) and attenuation coefficient of the fiber as follows (4.6) P( z, t ) Pin (t )exp( z) The nonlinear phase generated by Kerr effect will be L L L NL (t ) ( ' )dz dz P( z, t )dz 0 0 And because the effective length is 1 1 exp( L) Comes to NL (t ) A. Non-Linear Equation Shrodinger In non-linear regime, the equation of propagation of pulses is governed by NLS equation, that neglecting effects of higher order and losses is given by u 1 2u 2 (4.10) i sgn( 2 ) 2 u u 0 2 If incident peak power P0, observe the relationship P0 n2' 2 It has in front of the impulse dPin dt 0 (t ) 0 , yielding a redshift. Similarly the tail of pulse dPin dt 0 (t ) 0 , causes a blueshift. Since this is an effect contrary to what happens in GDV to the anomalous dispersion, thus allows the propagation of pulses that retain their shape along the propagation [4]. NL (t ) Pin (t ) (4.7) 0 (4.7) (4.8) Noting then that the nonlinear phase depends only on Pin(t), hence the name of Self-Phase Modulation. The instantaneous frequency deviation of the local (over carrier) caused by the SPM pulse is d dP (t ) (t ) NL in (4.9) dt dt 2 n2' L (4.11) The propagation of pulses in a fiber of length L in simplistic terms [4], is governed by the nonlinear regime for L>LNL and the linear regime for L<LNL, for a grossly way, is possible to distinguish four regimes of propagation in the fiber: Non-dispersive linear regime (NDLR), when L<LNL and L<LD ( disregard GDV and APM effects) Dispersive linear regime (DLR), when L<LNL and L>LD (disregard APM , only acts GDV) Non-dispersive nonlinear regime (NDNLR), when L>LNL and L<LD (disregard GDV, only acts APM) Non-dispersive nonlinear regime (NDNLR), when L>LNL and L>LD (acts GDV and APM simultaneously, thus allowing the propagation of solitons). B. Analytical Solutions of NLS Equation By limiting the analysis to just in case the solitons as they occur (anomalous dispersion zone, where sgn(β2) = -1), comes to NLS equation u 1 2u 2 i u u0 (4.12) 2 2 Which has the analytical solution for solitary waves u( , ) sec h ( q0 ) exp i 2 0 (4.13) 2 Where is the parameter which sets both the amplitude and pulse width, q0 is pulse’s center in relation 0 and 0 is initial phase (in 0 ). 9 Treating a wave with localized surrounding ( u( , ) is independent of ), apart from that when tends to , u( , ) approaches 0. When considering the loss ( LD 0 ) and 3rd dispersion order ( 3 6 2 0 ), variables represents Give name of fundamental soliton the canonical form of Eq. (4.13) by making 1 and q0 0 to phase zero. u( , ) sec h( ) exp i 2 (4.14) N which non-linearity and D which represents the dispersion, are given as a function of 2 (4.18a) N i u 2 i 2 3 (4.18b) D sgn( 2 ) 2 3 2 What is not entirely true, but it is considered the variation of u with negligible . When considering the incident pulse u0 ( ) u( 0, ) type, Eq.’s (4.17) solution will be u( , ) exp ( N D ) u0 ( ) Making u( h, ) exp h( N D ) u( , ) Fig 4. 2 - Evolution of fundamental soliton. In case of periodic waves, must be taken into account that using the inverse of the dispersion or IST (inverse scattering transform), shows that any incident pulse shape (4.15) u0 ( ) u( 0, ) N sec h( ) Where N represents the soliton order. For soliton order N≥2, unlike the fundamental soliton, pulses do not retain their shape along the propagation, shows instead an evolution periodic with period 0 2 in real units represents z0 2 LD 02 2 2 (4.16) (4.19) (4.20) Eq. (4.20) sets up an iterative scheme of longitudinal step h , which enables the start ( 0 ) at the end of the fiber ( L L LD ). Trying to divide the total space propagation 0 L in small sections elementary length h . SSFM consists of two consecutive procedures, dividing the Eq. (4.20) in v( , ) exp(hN )u( , ) u( h, ) exp(hD )v( , ) According to Eq. (4.23a), (4.21a) (4.21b) v( , ) is given by 2 h v( , ) exp exp ih u( , ) u( , ) (4.22) 2 Using the Fourier transforms (1st step RIMF algorithm), then defines Fig 4. 3 - Evolution solution of the 3rd order. v ( , ) FFT v( , ) v( , ) exp(i )d (4.23) C. Numerical Simulation NLS Equation: Split-Step Fourier Method The disclosed method for solving nonlinear equations propagating pulse is SSFM, which the optical apply the numerical [4], is to separate the equation (4.10), non-linear of dispersion. Equation (4.10) can be written in a more compact form u( , ) ( D N )u( , ) 0 (4.17) And these conditions D becomes na algebraic operator D , according to Eq. (4.18b) has been i sgn( 2 ) 2 i 3 (4.24) 2 By taking into account Eq. (4.21b) - 2nd step of the RIMF algorithm u( h, ) exp(hD )v ( , ) D h exp i sgn( 2 ) 2 exp(ih 3 )v ( , ) 2 (4.25) 10 Each iteration to finally, apply 3rd step RIMF’s algorithm u( h, ) IFFT u( h, ) 1 2 u( h, ) exp(i )d (4.26) Thus SSMF summarizes the application of RIMF algorithm in case of non-linear equations, using an iterative scheme longitudinal h , showing such a method since it is very effective to use longitudinal steps quite small, because the smaller step size, greater iterations number and increasing the number of iterations, the more efficient it becomes method. Enhancing here that for simulation of all pulses in nonlinear regimes, used SSFM (as was case in Figures 21 and 22). D. Characteristics of Fundamental Soliton Area of soliton does not depend on any parameter characteristic fiber [4] (4.27) A 2 Energy of the soliton is given by 2 2 (4.28) Es 0 Shown to be inversely proportional to the temporal width Power spectral density is directly related to energy, as follows 1 (4.29) Es S ( ) d 2 V. CONCLUSIONS In this paper we have shown that an optical pulse propagating along an optical fiber, suffers the effect of dispersion which broads the pulse at the output of the fiber. This is a limiting factor for the bandwidth of the transmitted signal. In linear regime, the technique of using the dispersion compensating fibers every two consecutive sections, shows to be very efficient and to solve the effect of the dispersion. In non-linear regime, pulses are subject to AMF (caused by the optical Kerr effect). In the anomalous dispersion region, this effect is contrary to the DGV, thereby allowing the propagation of pulses very interesting because it does not changes its shape over propagation in the fiber. These pulses are called soliton. For the simulation pulse scheme is nonlinear regime, we have used the SSFM, which has proved to be as efficient as higher the number of iterations used in the method. REFERENCES [1] [2] [3] [4] [5] Agraval, G. P. (2007). Maxwell´s Equations. In Nonlinear Fiber Optics, 4ª edição. Burlington, USA: Academic Press. Agrawal, G. P. (2007). Super-Gaussian Pulses. In Non Fiber Optics, 4ª Edição. Burlington, USA: Academic Press. Paiva, C. (2010). Fotónica. Fibras Ópticas . Paiva, C. R. (2010). Fotónica. Solitões Em Fibras Ópticas . Cartaxo, A. (2011). Sistemas de Telecomunicações. Comunicações Ópticas . [6] [7] [8] [9] [10] [11] [12] [13] Andrade, M. A. (2009). 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Obtido em Novembro de 2011, de http://www.gta.ufrj.br/grad/08_1/wdm1/FibraspticasComceitosComposio.html