Pulse Propagation in Optical Fibers

Transcription

1 Pulse Propagation in Optical Fibers Arleth Manuela Gonçalves Departaento de Engenharia Electrotécnica e de Coputadores Instituto Superior Técnico Av. Rovisco Pais, 49- Lisboa, Portugal arlethgoncalves@hotail.co Abstrat This paper addresses the pulse propagation through a fiber optic syste, operating in the linear and nonlinear regies. After a brief introduction to optical fibers, we use the odal 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 phenoenon to which pulses are subject, while propagating in the fiber, in the linear regie. Since dispersion strongly liits the bandwidth of the transitted signal and ay cause interference between sybols, we address several ethods for reducing the effect of dispersion, which can be very efficient: the use of dispersion copensating fibers and the propagation of solitons. In non-linear syste, the non-linear Kerr effect explains the appearance of self-phase odulation (SPM) which has a contrary action to the phenoenon of dispersion in the anoalous dispersion region, allowing the appearance of solitons, which era pulses that conserve their shape along the propagation. The propagation of pulses in nonlinear regie is governed by the nonlinear Shrondinger (NLS) equation for which, their analytical solutions are calculated. An efficient ethod is used in the nuerical siulations under the nonlinear regie: the split-step fourier Method (SSFM). Finally, a brief study of the fundaental soliton is presented. Keywords Fibre optics, dispersion, pulse propagation, solitons. T I. INTRODUCTION he appearance of optical fibers drastically changed the paradig of data transission, because of two ain advantages that ake it particularly in relation to wires: The optical fibers are copletely iune to electroagnetic interference as they are ade of dielectric aterial capable of transitting pulses of light, which eans that data is not corrupted during transission. Optical fibers do not conduct electric current, so there will be no probles with electricity, as voltage instability probles or issues with spokes. The physical phenoenon known as total internal reflection is the responsible for light pulses to travel along the fiber by successive reflections []. Although since 87 it was known that light could describe a curved path within a aterial, only in 95 the Indian physicists Narinder Kapany could coplete its experients 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 transit inforation through the fiber [9]. There are two types of fiber: Multiode fibers with core diaeters greater than the single ode, allowing light to traverse the fiber through several paths, which are strongly affected by the effect of dispersion and, because of this, are ost often used for data traffic within walking distance. Single-ode fibers with core diaeters 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 ost often used for long distance counications [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 regies. II. MODAL THEORY Because of the difference between the refractive indices of the core and the cladding, it is possible to transit of light inside the fiber [3]. The core index of refraction of the core ust be always higher than the refractive index of the cladding. The odal theory can explain the existence of an angle of incidence that enables the phenoenon of total internal reflection. A. Fibers operated in the Linear Regie Optical fibers operated in the linear regie (onoodal regie), are the fibers that allow the use of only one light signal through the fiber. They have a saller radius and greater bandwidth due to lower dispersion. It is possible to define a axiu value for the angle of incidence with respect to the fiber axis, called acceptance angle axiu θ ax, iposing that only the rays entering the fiber at an angle of less than θ ax are propagated along it [3]. Let n be the refractive index in the core and n be the refractive index in the cladding. The paraeter which describes the ability to collect light in a fiber called a nuerical aperture (NA) which is related as θ ax follows

2 NA n sin n n n ax (.) where n is the refractive index of the outer ediu in which the fiber is inserted, usually the air, where n =. Δ is the dielectric contrast given by n n n (.) For better understanding the wave propagation inside a step index fiber with core radius a, one ay consider the schee depicted in Fig... Fig. Rays propagation in fiber []. The longitudinal coponent of the electric field in cylindrical coordinates has the following for Ez( r,, z, t) EF( r)exp( i )exp i( z t) (.3) where exp[i(βz-ωt)] is the tie and space dependence of onochroatic light traveling along the fiber axis, E is the aplitude of the field, is the aziuthal variation index, F (r) represents the radial odal eigenfunction, β is the longitudinal propagation constant, which is related to the free-space propagation constant as where k ( r) n ( r) k (.4) n for r a nr () (.5) n for r a and the propagation constant in vacuu is k c. 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 ust have q i. Introducing noralized (diensionless) wavenubers u ha and w a, and a noralized frequency v, their relation is given by v u w (.6) Moreover, introducing the noralized odal refractive index as u w ( k ) n b v v n n (.7) there is a iniu frequency of operation (noralized cutoff frequency v ) for which a certain ode starts to propagate in c the fiber. A special value is given by v. 448, which is the cutoff value for the second propagating ode, indicating whenever the fiber is operating below that value, the syste onoodal, or ultiodal if above, i.e., for singleode fiber, for B. Propagation Modes in Fiber v vc => ultiode fiber. c v v => Light, while propagating in the fiber, can be seen sa an electroagnetic phenoenon, and the whole propagating echanis, which can be described by the electroagnetic optical fields associated to it, is governed by Maxwell equations []. Generally speaking, odes can be classified into TE (Transverse Electric) odes, where there is no coponent of electric field in the direction of propagation (Ez = ), TM (Transverse Magnetic) odes, where there is no agnetic field coponent in the direction of propagation (Hz = ), and hybrid odes, for which there have both longitudinal field coponents, i.e., they have electric field (Ez ) and agnetic field (Hz ) coponents in the direction of propagation. These latter odes can be classified into HE n and EH n, where the paraeter refers to the aziuthal field variation and the paraeter n refers to the radial field variation []. In general, the surface waves guided by an optical fiber are hybrid odes, and their field coponents are governed by the Bessel functions of first (J) and second (K) species, satisfying the following equation odal where 4 u v R( u) S( u) v uv (.8) J ( u) K( w) R ( u) (.9a) uj ( u) wk ( w) J ( u) K( w) S( u) ( ) uj ( u) wk ( w) (.9b) The resolution of the Eq. (.8), while being very coplex, turns it to be necessary to use nuerical ethods. Only when the aziuthal index is zero ( = ) it is possible to propagate transversal TE n and TM n odes. According to the Gloge approxiation (i.e., for Δ << ), in which odes propagating in optical fibers are weakly guided, one ay take R(u)= S(u), which will cause the odal Eq.(.8) to be reduced to ( ) v R u uw (.) where the plus (+) sign corresponds to the odes EHn and the inus (-) sign corresponds to the odes HEn. Through the relationship between the Bessel functions and their derivatives under the Gloge approxiation, in which the c

3 3 aziuthal index zero, the Eq (.) for both odes (EHn HEn) reduces to J( u) K( w) uj ( u) wk ( w) (.) inferring that the odes EHn are actually TMn and odes HEn are odes TEn. Moreover for weakly guided fibers, these odes are alost linearly polarized, and therefore will be tered as LPpn. One should stress that the only ode capable of propagating in a single ode fiber syste (the only ode whose cutoff corresponds to v c =), called the fundaental ode, is the HE ode. Mosr propagating odes are degenerated odes, i.e., they have the sae cut-off frequency but with different field structure. In particular, HE n odes give rise to odes LP n Modes TE n, and TM n HE n are degenerate and give rise toodes LP n HE +,n odes and EH -,n with are also giving rise to degenerate odes LP n. Therefore, one has: EH n LP pn (with p=+); HE n LP pn (with p=-). Accordingly, one has the following odal for LP odes: J p( u) K p( w) u w J ( u) K ( w) (.) p For the fundaental ode (LP ), it reduces to uj ( u) K ( w) wj ( u) K ( w) p (.3) Figure. shows the variation of the noralized odal refractive index b as a function of noralized frequency v for the first six odes propagating in the fiber. Fig. - First six LP ode fiber: diagras b(v). One can observe a general increase in the noralized refractive index b odal with increasing noralized frequency v, and the greater the noralized frequency v, ore odes 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.3 Fig. 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 ay conclude that is possible to reduce effects of dispersion by using fibers with a saller contrast. III. PULSE PROPAGATION IN THE LINEAR REGIME Any pulse, while propagating in an optical fiber, especially in the linear regie, suffers the effect of tie dispersion, which causes its broadening and ay create interference between sybols, which can greatly liit the bandwidth of the signal to be transitted. Let one consider an optical pulse with a finite spectral width to be launched into the fiber. Each spectral coponent of the pulse, while traveling along the fiber, has a different group velocity which depends on its wavelength according to v c g with (3.) The dispersion due to the difference between the propagation velocities of different spectral coponents is called the Group Velocity Dispersion (GVD). There are two ain types of dispersion: the Interodal Dispersion (between the various odes of propagation) and intraodal dispersion (under the sae ode of propagation). Interodal dispersion is due to the various propagation odes travelling in the fiber. Each ode has a different group velocity for the sae wavelength and the pulse width at the output of the fiber will depend on the transission ties. The tie delay between the fastest (i.e., the fundaental ode) and the slowest ode (ode of higher order, depending on how any odes can propagate in this ultiode fiber) is responsible for the broadening of the pulse at the output of the fiber. In the case of the interodal dispersion or, as coonly referred to, the chroatic dispersion, the broadening of the pulse occurs within the sae ode, ainly because of the dependence of the refractive index of the aterial with the frequency. However, the chroatic dispersion is a consequence of cobined effect of two factors: the aterial dispersion and the waveguide dispersion [6], []. In the case of the aterial dispersion, the refractive index of the constituent aterials of the fiber has a non-linear variation with the wavelength dn ng n d (3.)

4 4 Waveguide dispersion follows fro the fact that, for a given ode, the energy distribution between the core and the cladding is a function of wavelength. Generally, 8% of optical power propagating in the fiber reains confined to the core while the reaining % 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 coponents at different wavelengths have alost the sae propagation velocity, in other words, propagation delay is quite constant at different wavelengths. Fig 3. - Zero dispersion [5]. Discarding higher order dispersion effects, the paraeter which describes the fiber dispersion is given by c D (3.3) with the DGV coefficient β is given by vg v ( ) (3.4) g Let one take into consideration the electroagnetic spectru as depicted in Figure 3.. Fig 3. - Electroagnetic spectru. In the anoalous dispersion region, one has D λ > => β <. The group velocity increases with the frequency, i.e., the blue spectral coponents travel faster than those red (see Figure 3.3.). In noral dispersion region, one has D λ < => β >. The group velocity decreases with the frequency, i.e., in the red spectral coponents travel faster than the blue ones. Fig Noral dispersion [5]. For a standard silicon optical fiber, the zero dispersion wavelength occurs around 3 n. Around this value, there is no pulse broadening. For this reason, the ost current optical counication systes have been developed to take advantage is this characteristic []. A. Pulse Propagation Equation in linear Regie In this Section, the differential equation that governs the propagation of pulses in the linear regie along a single ode optical fiber with sall contrast (Δ << ), is derived ignoring the effect of higher order dispersion. Let A(,t) be the pulse at the fiber input at z =. Assuing that the pulse odulates a carrier, with an angular frequency ω, and that the electric field is linearly polarized along the x, one ay write E( x, y,, t) xe ˆ F( x, y) A(, t)exp( i t) (3.5) where F (x, y) is the odal function representing the transversal variation of the fields of the LP ode. With the help of the Fourier transfor (nuerically coputed through the FFT-Fast Fourier Transfor and Inverse Fast Fourier Transfor-IFFT), the pulse envelope at a generic point of the fiber is given by A ( z, t ) (, )exp( ) A z i t d (3.6) Thus the electric field in a generic point of the fiber is given by E( x, y, z, t) xe ˆ F( x, y) A( z, t)exp i( z t) (3.7) where ( ) (3.7a) In order to obtain A(z,t)) fro A(,t), we introduce the following GVD coefficients Fig Anoalous dispersion [5]. (3.8a)

5 5 where v g ( ) vg ( ) vg (3.8b) (3.8c) The frequency deviation over the carrier is given by (3.9) As Ω <<ω, it is reasonable to disregard the dispersion coefficients above β. Moreover, discarding the losses in the fiber, the linear propagation equation, which allows calculating A(z,t) fro A(,t), can be written by A( z, t) A A( z, t) i z t t (3.) This equation can be solved using a siple algorith which allows the coputation of pulse propagation along the fiber. This algorith is herein designated RIMF algorith and is applied in three steps: First step: To copute with A(, ) FFT A(, t) A(, t)exp( it) dt Second step: Then copute A( z, ) A(, )exp i ( ) z Third step: Finally A( z, t) IFFT A( z, ) A( z, )exp( it) d ( ) (3.) Because the pulses are usually narrowband Ω <<ω, A(z,t) is a slowly varying function in tie and oscillates with exp(-iωt). In the absence of the dispersive effects, the pulse propagate without distortion with a group delay For the nuerical solution of Eq. (3.) is useful to introduce the noralized (diensionless) space and tie variables z (3.4a) L D t z So equation (3.) can be rewritten as and the Fourier pair as (3.4b) A(, ) i A(, ) sgn( ) (3.5) A(, ) A(, )exp( i ) d (3.6a) A (, ) (, )exp( ) A i d (3.6b) where ψ is a noralized frequency given by ( ) (3.7) To apply the RIMF algorith to Eq. (3.5), one has First step: To calculate A(, ) FFT A(, ) Second step: Then i A(, ) (, )exp sgn( ) A 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 A ( ) A (, ) sec h ( ) (3.8) z (3.) g Since it is reasonable to neglect the influence of the higher order dispersion (β 3 =), it is possible to isolate the effect of the GDV in the pulse propagation. Defining τ as the characteristic pulse tie width, the dispersion length is introduced as L D (3.3) The dispersive effects in a optical link of length L are negligible only if L D >L. Fig Coparison of the pulse input and output fiber.

6 6 A(, ) FFT A(, t) A exp ic ( ic) (3.4) The pulse spectral intensity will be B. Chirp Fig Evolution of the pulse. Since the optical signals eitted by a laser source suffer fro chirp, it is useful to introduce a chirp paraeter C which quantifies the variation in the carrier frequency C c (3.9) where β c is Henry factor, responsible for the enlargeent of the spectral line. When considering the linear propagation of a Gaussian pulse with the Chirp [3], the initial pulse can be written as A(, ) A exp C ( C ) The spectral width Δω at /e axiu will be (3.5) C (3.6) One ay conclude that the larger the value of C, the larger the pulse spectral width. ic t A(, t) A exp where A represents the pulse aplitude. (3.) Fig Evolution of Gaussian pulse with C=. Fig Initial Gaussian pulse for C=. The frequency shift δω(z,t) caused by the existence of the chirp is z t z ( z, t) C ( C ) (3.) ( z) with ( z) ( z) (3.) and η(z) is the pulse broadening factor Fig Evolution of Gaussian pulse with C=. The pulse broadening factor η(z), introduced in Eq. (3.3), also shows that a pulse ay also suffer a tie copression as it propagates, as long as β C< [3]. Figure 3. shows the variation of the broadening factor with the travelled distance for a Gaussian pulse in the anoalous dispersion region (β <) z z ( z) sgn( ) C L D L D (3.3) When applying the first step of the RIMF algorith to the initial ipulse Gaussian, one has Fig 3. - Spatial evolution of spectral width pulse for different values of C.

7 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 oentu is given by A(, t) A exp ic t (3.7) Fig Evolution of Super-Gaussian pulse with C=. This expression is quite identical to the Gaussian pulse, differing only in the value of the paraeter. For a Gaussian pulse is equal to. The edges of the pulses becoe increasingly steep as increases, as shown in Figure 3., when copared with figure 3.7. Fig 3. - Initial Super-Gaussian pulse for C=, to =3. The paraeter is related to the duration t r for which the intensity of the pulse increases fro % to 9% of its peak value t r (3.8) Eq. (3.8) shows that a pulse with a saller rise tie increases faster []. 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 =3. Fig 3. - Super-Gaussian coparison pulse input and output fiber. C. Dispersion Copensation A technique used to copensate for or iniize the proble of dispersion is the use of dispersion copensating fibers (DCF Dispersion Copensation Fiber). This technique is to cobine optical fibers with different characteristics, such that the average GDV of the link becoes quite sall, whereas the GDV of each section ay be large [7]. In each two consecutive sections, there are two types of fiber. A section of greater length L, operating in the anoalous dispersion region (with β <), and a shorter section of length L, operating in the noral region (with β >). Nevertheless, the two GDV coefficients are quite different ( β β ). This technique takes advantage of the linear nature of the syste when considering the propagating optical pulse into two sections of the fiber, whose ipulse propagation equation is given by i A( L, ) A(, )exp ( L L ) i d (3.9) where L L L. The length of dispersion copensating fiber L is chosen so that L L L L (3.3) Pick-up β << β so that L is uch lower than L. When using this technique it is coon for L to be in the order of - k, while L is in the order of k. Provided that they fulfill the condition β L +β L =, A(L,τ)=A(,τ) thus recovering its original shape once every two consecutive sections, although pulse width ay change significantly in each section. As an exaple of application, we consider L = k, β = -, L = k e β = and the pulse input of the first fiber is given by A ( ) A (, ) sec h ( ) (3.3) Fig Evolution of Super-Gaussian pulse with C=. Fig 3. 5 Coparison input and output pulse in the st fiber.

8 8 The output pulse of the first fiber is the ipulse input of the second fiber (DCF). So, taking into account the shape of the pulse to propagate Fig 4. - Power of a Gaussian pulse [7]. Fig Coparison input and output pulse in the nd fiber. IV. PULSE PROPAGATION IN NONLINEAR REGIME Pulse propagation in nonlinear regie is affected by the optical Kerr effect. The propagation of ipulses in Nonlinear Dispersive Regie (NLDR) is governed by the SPM and the GDV siultaneously [4]. To disrupt the relative dielectric constant ( xy, ) (4.) where n( x, y) n (4.) Whatever the process that led to that disturbance, the new longitudinal propagation constant is given by (4.3) Introducing the paraeter nk n n (4.4) ref wherein is the effective area and r ef is the effective radius. The optical Kerr effect provides that for certain values of P( z, t) (4.5) n 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 P( z, t) P ( t)exp( z) (4.6) in The nonlinear phase generated by Kerr effect will be L L L ( ) ( ) NL (, ) (4.7) t dz dz P z t dz And because the effective length is exp( L ) (4.7) Coes to (t NL ) ( t) P ( t) (4.8) NL in Noting then that the nonlinear phase depends only on P in (t), hence the nae of Self-Phase Modulation. The instantaneous frequency deviation of the local (over carrier) caused by the SPM pulse is dnl dpin () () t t (4.9) dt dt It has in front of the ipulse dp dt ( t), yielding a redshift. Siilarly the tail of pulse dp dt ( t), causes a blueshift. Since this is an in effect contrary to what happens in GDV to the anoalous dispersion, thus allows the propagation of pulses that retain their shape along the propagation [4]. A. Non-Linear Equation Shrodinger In non-linear regie, the equation of propagation of pulses is governed by NLS equation, that neglecting effects of higher order and losses is given by u u i sgn( ) u u (4.) If incident peak power P, observe the relationship P (4.) nl The propagation of pulses in a fiber of length L in siplistic ters [4], is governed by the nonlinear regie for L>L NL and the linear regie for L<L NL, for a grossly way, is possible to distinguish four regies of propagation in the fiber: Non-dispersive linear regie (NDLR), when L<L NL and L<L D ( disregard GDV and APM effects) Dispersive linear regie (DLR), when L<L NL and L>L D (disregard APM, only acts GDV) Non-dispersive nonlinear regie (NDNLR), when L>L NL and L<L D (disregard GDV, only acts APM) Non-dispersive nonlinear regie (NDNLR), when L>L NL and L>L D (acts GDV and APM siultaneously, thus allowing the propagation of solitons). B. Analytical Solutions of NLS Equation By liiting the analysis to just in case the solitons as they occur (anoalous dispersion zone, where sgn(β ) = -), coes to NLS equation u u i u u (4.) Which has the analytical solution for solitary waves u(, ) sec h( q) exp i (4.3) Where is the paraeter which sets both the aplitude and pulse width, q is pulse s center in relation and is initial phase (in ). in

9 9 Treating a wave with localized surrounding ( u(, ) is independent of ), apart fro that when tends to, u(, ) approaches. Give nae of fundaental soliton the canonical for of Eq. (4.3) by aking and q to phase zero. u(, ) sec h( )exp i (4.4) When considering the loss ( L D ) and 3rd dispersion order ( 3 6 ), variables N which represents non-linearity and D which represents the dispersion, are given as a function of N i u (4.8a) 3 i D sgn( ) (4.8b) 3 What is not entirely true, but it is considered the variation of u with negligible. When considering the incident pulse u ( ) u(, ) type, Eq. s (4.7) solution will be u(, ) exp ( N D ) u ( ) (4.9) Fig 4. - Evolution of fundaental soliton. In case of periodic waves, ust be taken into account that using the inverse of the dispersion or IST (inverse scattering transfor), shows that any incident pulse shape u ( ) u (, ) N sec h ( ) (4.5) Where N represents the soliton order. For soliton order N, unlike the fundaental soliton, pulses do not retain their shape along the propagation, shows instead an evolution periodic with period in real units represents (4.6) z L D Fig Evolution solution of the 3rd order. C. Nuerical Siulation NLS Equation: Split-Step Fourier Method The disclosed ethod for solving nonlinear equations propagating pulse is SSFM, which the optical apply the nuerical [4], is to separate the equation (4.), non-linear of dispersion. Equation (4.) can be written in a ore copact for u(, ) ( D N) u(, ) (4.7) Making u( h, ) exp h( N D ) u(, ) (4.) Eq. (4.) sets up an iterative schee of longitudinal step h, which enables the start ( ) at the end of the fiber ( L LLD L ). Trying to divide the total space propagation in sall sections eleentary length h. SSFM consists of two consecutive procedures, dividing the Eq. (4.) in v(, ) exp( hn ) u(, ) (4.a) u( h, ) exp( hd ) v(, ) (4.b) According to Eq. (4.3a), v (, ) is given by h v(, ) exp exp ih u(, ) u(, ) (4.) Using the Fourier transfors (st step RIMF algorith), then defines v(, ) FFT v(, ) v(, )exp( i ) d (4.3) And these conditions D, according to Eq. (4.8b) has been D becoes na algebraic operator i 3 D sgn( ) i (4.4) By taking into account Eq. (4.b) - nd step of the RIMF algorith u( h, ) exp( hd ) v(, ) h 3 exp i sgn( ) exp( ih ) v(, ) (4.5)

10 Each iteration to finally, apply 3rd step RIMF s algorith u( h, ) IFFT u( h, ) u( h, )exp( i ) d (4.6) Thus SSMF suarizes the application of RIMF algorith in case of non-linear equations, using an iterative schee longitudinal h, showing such a ethod since it is very effective to use longitudinal steps quite sall, because the saller step size, greater iterations nuber and increasing the nuber of iterations, the ore efficient it becoes ethod. Enhancing here that for siulation of all pulses in nonlinear regies, used SSFM (as was case in Figures and ). D. Characteristics of Fundaental Soliton Area of soliton does not depend on any paraeter characteristic fiber [4] A (4.7) Energy of the soliton is given by E s (4.8) Shown to be inversely proportional to the teporal width Power spectral density is directly related to energy, as follows Es S( ) d (4.9) [6] Andrade, M. A. (9). Modelização da propagação e sisteas de counicação óptica baseados na tecnica de ultiplexage por divisão no copriento de onda. Vila Real: Universidade de Trás-os-Montes e Alto Douro. [7] Dos Santos, N. M.-D. (). Métodos variacionais aplicados ao estudo das fibras ópticas e técnicas de copensação da dispersão. Lisboa: Instituto Superior Técnico. [8] fibra.no.sapo. (5 de Outubro de 4). Obtido e Janeiro de, de [9] Hardware, C. d. (s.d.). Clube do Hardware.co.br. Obtido e Novebro de, de [] paginas.fe.up.pt. (s.d.). Obtido e Setebro de, de fibrasopticas.pdf [] PUC - Rio, C. D. (s.d.). dbd.puc-rio.br. Obtido e Novebro de, de [] RNP, R. N. ( de Abril de ). rnp.br. Obtido e Janeiro de, de [3] UFRJ, G. d.-g. (s.d.). gta.ufrj.br. Obtido e Novebro de, de CoceitosCoposio.htl 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 liiting factor for the bandwidth of the transitted signal. In linear regie, the technique of using the dispersion copensating fibers every two consecutive sections, shows to be very efficient and to solve the effect of the dispersion. In non-linear regie, pulses are subject to AMF (caused by the optical Kerr effect). In the anoalous 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 siulation pulse schee is nonlinear regie, we have used the SSFM, which has proved to be as efficient as higher the nuber of iterations used in the ethod. REFERENCES [] Agraval, G. P. (7). Maxwell s Equations. In Nonlinear Fiber Optics, 4ª edição. Burlington, USA: Acadeic Press. [] Agrawal, G. P. (7). Super-Gaussian Pulses. In Non Fiber Optics, 4ª Edição. Burlington, USA: Acadeic Press. [3] Paiva, C. (). Fotónica. Fibras Ópticas. [4] Paiva, C. R. (). Fotónica. Solitões E Fibras Ópticas. [5] Cartaxo, A. (). Sisteas de Telecounicações. Counicações Ópticas.