pedro miguel elói aleluia carrasco gonçalves · 1 super-gaussian pulse propagation in optical...

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SUPER-GAUSSIAN PULSE PROPAGATION IN OPTICAL FIBERS

Pedro Miguel Elói Aleluia Carrasco Gonçalves

Instituto Superior Técnico Av. Rovisco Pais, 1049-001 Lisboa, Portugal

E-mail: [email protected]

ABSTRACT

The principal idea of this dissertation is to study some pulses and compare them with the super-gaussian, in two different regimes in optical fiber: the linear and the non-linear. For both regimes, an analytical analysis was made, with the purpose of using some mathematical tools, FFT and SSFM, in order to illustrate with figures the analytical study done. In the linear regime, the dispersion effects for GVD and the high-order effects were studied, and perturbation known as Chirp parameter introduced. In the non-linear regime, two kinds of solitons were distinguished: the bright and the dark. The bright solitons are in fact the ones that interest in telecommunication, so a study was done introducing some perturbations and testing how they reacted during their propagation inside the optical fiber. The interaction between solitons and how these perturbations evolve in optical propagation was also described. Keywords: propagation, linear regime, non-linear regime, GVD, high-order effects, chirp’s parameter, pulse broadening, soliton

1. INTRODUCTION

A communication system is a link between two points through which information is transmitted using a carrier. An optical communication system uses the electromagnetic waves from the optical spectrum, contained between the far infrared (100 µm) and the ultraviolet (0.05 µm) to carry the information. The basic components of an optical communication system are: an optical transmitter, that converts the electrical signal into optical signals sending them to the optical fiber, an optical fiber and an optical receiver, that receives the optical signal converting it to an electrical signal [1]. The work developed by the physicists Daniel Collandon and Jacques Babinet was essential to the development of fiber optic based communication systems. In the decade of 1840, they were the first to demonstrate the possibility of redirecting light through refraction, the fundamental

principle for the propagation of light in optical fibers [2]. Credit for this discovery was, however, given to physicist John Tyndall in 1854.

Figure 1: Basic optical communication system.

In the second half of the twentieth century, optical fibers suffered major breakthroughs. In early 1950s, a partnership between Brian O’Brien and Narinder Kapany [2] resulted in the development of the first communication system to ever use glass fibers, transmitting information through pulses of light. In the 1960s, optical communication was faced with two obstacles. First, there was the need of a source capable of generating optical pulses. And second, the inexistence of an adequate transmission media. The first problem was solved with the development of Laser that allowed carrying up to 10000 times more information than the highest radio frequencies used. Still it was not an adequate media for free space propagation due to the sensibility to environmental conditions. In 1966, Charles Kao (who later received the Nobel Prize of Physics for his work in the field of optical fibers) and Charles Hockham, proposed optical fibers as the ideal light propagation media [2] as long as losses were of the order of 20 dB/km. This goal was reached in 1970 by Robert Maurer, Donald Keck and Peter Schultz making viable the use of opical fibers in communication systems. In the last decades several generations of optical fiber communication systems were developed. The development of optical amplifiers allowed the amplification of signals without the use of electronics. Erbium doped fiber amplifiers (EDFA’s) [1] where the more important of these allowing the increase of distance between repeaters. Currently the fourth generation uses optical amplification to increase distance between amplifiers and wavelength-division multiplexing (WDM) system [1] becoming the first

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photonic generation. The use of WDM techniques allows bit rates higher than 1 TB/s. The fifth generation was highly anticipated. Having solved the losses problem by using amplifying fibers, dispersion has become the greater problem to be addressed. To solve this problem, several techniques have been developed [1] such as pre and post compensating dispersion systems, dipersion management systems and soliton based systems [3]. All of these solutions use optical amplification, WDM and dispersion management. Optical fibers were a revolution in communication systems and, along with photonics, became the back bone of communication systems. The increasing need for high bandwidth digital networks with integrated services is leading to an increase of FTTC (fiber to the curb) and FTTH (fiber to the home) networks.

2. OPTICAL FIBERS: LINEAR REGIME

In this section light propagation in the linear regime in an optical fiber is described. An optical fiber is a cylindrical waveguide with the property of propagating light between two points. It is composed by a core with an index of refraction n1 and a cladding with an

index of refraction n2 .

2.1. Modal theory To obtain a precise model of light propagation within an optical fiber one must recur to the electromagnetic theory, namely, Maxwell’s equations [12]

∇×E = − ∂B

∂t (1)

∇× H = J + ∂D

∂t (2)

∇⋅D = ρ (3) ∇⋅B = 0 (4) and the following constitutive relations D = ε0E+ P (5)

H = 1

µo

B − M. (6)

The polarization P should be given by [1]

P r,t( ) = PL r,t( ) + PNL r,t( ). (7)

These equations allow reaching to the wave equation

∇2E+ n2 ω( )ω

2

c2E. (8)

Equation (8) describes wave propagation inside an optical fiber according to the electromagnetic theory and each solution corresponds to a mode. Both fields E and H satisfy Maxwell’s equations. Given the cylindrical symmetry of an optical fiber and assuming

Ez as the supporting component one as

Ez r,ω( ) = A ω( )F r( )exp imφ( )exp −iβz( ) (9)

and applying it to Equation (8) one gets

d 2F r( )dr 2 + 1

rdF r( )

dr+ k0

2n2 − β 2 − m2

r 2

⎡

⎣⎢

⎤

⎦⎥F r( ) = 0 (10)

The solutions to this equation are the Bessel functions

F r( ) =BJm u r

a⎛⎝⎜

⎞⎠⎟

, r ≤ a

CKm w ra

⎛⎝⎜

⎞⎠⎟

, r > a

⎧

⎨⎪⎪

⎩⎪⎪

(11)

where B and C are arbitrary constants and u and w are normalized variables given by

u = a k02n1

2 − β 2 (12)

w = a β 2 − k02n2

2 . (13) An optical fiber supports hybrid modes which are characterized by having both electric and magnetic components in the propagation direction. For guided modes one has k0n2 < β < k0n1. (14) It is common to define the normalized propagation constant b and the normalized frequency V as

b =

n 2 − n22

n12 − n2

2 (15)

V = k0a n1

2 − n22( )1 2

≈ 2πλ

⎛⎝⎜

⎞⎠⎟

an1 2Δ (16)

Since practical optical fibers are characterized by a low dielectric contrast their modes are linearly polarized (LP) [13]. Figure 2 shows the first six modes in an optical fiber, as well as it gives the maximum value for V in order to have a single mode fiber.

Figure 2: LP modes in an optical fiber: Dispersion diagram.

2.2. Fiber optics: transmission characteristics Two of principle phenomena that affect pulse propagation in an optical fiber are fiber losses and dispersion. Fiber losses (α ) reduce the available optical power increasing the bit error rate (BER) in the reception and

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

v

b (v

)

LP01LP11LP21LP02LP31LP12V = 2.4048

3

limiting maximum distance between transmitter and receiver. The dispersion on the other hand limits the bandwidth because it causes the broadening of the optical pulse. In long distance communication systems it can cause pulses to interfere with each other (Intersymbol Interference - ISI) [1] causing information loss. There are two kinds of dispersion intramodal and intermodal dispersion. Intramodal dispersion which results from the fact that the group velocity is a function of the wavelength, is the only one relevant for this work since the object of study is single mode fibers. For a monomodal fiber the group velocity is given by

vg =

dβdω

⎛⎝⎜

⎞⎠⎟

−1

(17)

and it relates with the group index ng , by

vg =

cng

. (18)

Considering the fact that the frequency dependence causes a time delay one has

D = − 2πc

λ 2 β2 (19)

where D is the dispersion coefficient and β2 is the group velocity dispersion (GVD) coefficient given by

β2 =

d 2βdω 2 . (20)

The dispersion coefficient includes both the material dispersion ( DM ) and the waveguide dispersion ( DW ). When D 0 the fiber is operating near the zero-dispersion wavelength λD , however the dispersion effects do not disappear. In this case, one has to consider the effects of the high order dispersion which are governed by the dispersion slope S = ∂D ∂λ which equals

S = 2πc

λD2

⎛

⎝⎜⎞

⎠⎟

2

β3 (21)

where β3 is the high order dispersion coefficient.

2.3. Pulse propagation equation in the linear regime In the linear regime the pulse spectrum shape is invariant along the fiber. As such, one can neglect the term PNL from Equation (7). So, if one considers the field equations [10]

E x, y,o,t( ) = x̂F x, y( )B 0,t( )B 0,t( ) = A 0,t( )exp −iω0t( ) (22)

one can obtain the pulse propagation equation for the linear regime given by

∂A∂z

+ β1

∂A∂t

+ i 12β2

∂2 A∂t2 − 1

6β3

∂3 A∂t3 + α

2A = 0. (23)

For the following analysis high order dispersion and fiber losses α are neglected. Considering the case of a gaussian pulse

A 0,t( ) = exp − t2

2τ 02

⎛

⎝⎜⎞

⎠⎟ (24)

where τ 0 is the initial width of the pulse. Applying Equation (24) to Equation (23) one gets

A z,t( ) = τ 0

τ 0 − iβ2z( )1 2 exp − t2

2 τ 02 − iβ2z( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥. (25)

Attaining to the dispersion length given by LD = τ 0

2 β2 one gets the expression for the pulse width along the fiber

τ z( ) = τ 0 1+ z

LD

⎛

⎝⎜⎞

⎠⎟

2

. (26)

As Figure 3 shows GVD broadens the pulse’s width.

Figure 3: Pulse width evolution.

Also from Equation (25) one realizes that the pulse acquires chirp although it did not existed initially. To show it clearly one rewrites the pulse equation in the form

A z,t( ) = A z,t( ) exp iφ z,t( )⎡⎣ ⎤⎦ (27)

where

φ z,t( ) = −sgn β2( ) z LD( )

1+ z LD( )2

t2

2τ 02 tan−1 z

LD

⎛

⎝⎜⎞

⎠⎟. (28)

The time dependence of φ z,t( ) implies that the instant

frequency differs from the central frequency ω0 along the pulse. The difference is given by

δω t( ) = − ∂φ∂t

=sgn β2( ) z LD( )

1+ z LD( )2

tτ 0

2 (29)

and is called chirp.

2.3.1. Gaussian pulse Through Equation (23) one can simulate the propagation of a pulse along the optical fiber in the linear regime, using the normalized variables

4

ζ = z

LD

(30)

τ =

t − β1zτ 0

(31)

and applying the algorithm

i( ) A 0,ξ( ) = FFT A 0,τ( ){ }ii( ) A ζ ,ξ( ) = A 0,ξ( )exp i 1

2sgn β2( )ξ 2 −ξ 3⎛

⎝⎜⎞⎠⎟

iii( )A ζ ,ξ( ) = FFT −1 A ζ ,ξ( ){ }.

2.3.2. Super-gaussian pulse In this section, the case study is a super-gaussian pulse [10] given by

A 0,t( ) = exp −1+ iC2

tt0

⎛

⎝⎜⎞

⎠⎟

6⎡

⎣⎢⎢

⎤

⎦⎥⎥

(32)

where C = 0, ± 2 . For C = 0 one gets Figure 4.

Figure 4: Super-gaussian pulse at begin and end of the optical fiber

C = 0.

For C = −2 the result is Figure 5.


C = −2.

For C = 2 one obtains Figure 6.


C = 2.

The broadening of the pulse due to dispersion is clear in the situation C = 0 . A decrease of pulse amplitude is caused by the conservation of energy. From the cases C = ±2 it is possible to observe that chirp increases the broadening effect caused by the GVD.

2.4. Bit rate The bit rate is an important merit figure in an optical communication system as it defines the amount of information transmitted per time unit. It is dependent of the effects of dispersion [13]. The number of pulses to be transmitted per time unit depends on the RMS width of the pulse

σ 2 = t2 − t

2⎡⎣⎢

⎤⎦⎥

(33)

where the angle brackets refer to averaging with respect to intensity

tm =tm A z,t( ) 2

dt−∞

∞

∫A z,t( ) 2

dt−∞

∞

∫. (34)

The broadening factor is given by [1]

-15 -10 -5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

Ampli

tude

Initial PulseFinal Pulse

-15 -10 -5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

Ampli

tude


-15 -10 -5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

Ampli

tude


5

σσ 0

⎛

⎝⎜⎞

⎠⎟

2

= 1+LCβ2

2σ 02

⎛

⎝⎜⎞

⎠⎟

2

+β2L2σ 0

2

⎛

⎝⎜⎞

⎠⎟+ 1+C 2( )2 Lβ3

4 2σ 03

⎛

⎝⎜

⎞

⎠⎟

2

(35) where L is the fiber length and C is the chirp parameter. Through some manipulation and considering the normalized variables

η = σ

σ⎛⎝⎜

⎞⎠⎟

2

, ζ = zLD

(36)

Equation (35) can be written as

η = 1+ sgn β2( )Cζ( )2

+ζ 2. (37) Figure 7 shows the effects of the chirp parameter on the broadening of the pulse along the fiber. For the anomalous region ( β2 < 0 ) a negative chirp causes a faster broadening of the pulse. However if the chirp is positive one can observe a pulse contraction at least in its the initial stage. Later the effect of the chirp is overruled by the GVD effect.

Figure 7: Pulse width evolution in function of C for β2 < 0

2.5. Higher-order dispersion

When β2 = 0 , situations in which the carrier is in the

neighboring of the considered wavelength, λD , or, the signal has a high spectral width, it is necessary to consider the higher order dispersion. For that, it is primary to define the dispersion slope as

S = ∂D

∂λ (38)

Which it is obtained

SD = 2πc2

λD2

⎛

⎝⎜⎞

⎠⎟β3 λD( ) (39)

2.5.1. LD = L’D In this case, it has to be considered both effects, although, in this case as we are about to see, the effect of β3 prevails

A ζ ,ξ( ) = A 0,ξ( )exp −i

12

sgn β2( )ξ 2ζ − iκξ 3ζ

⎛⎝⎜

⎞⎠⎟

(40)

With that equation it was possible to achieve the following figures

Figure 8: Input and output of a gaussian pulse with the high order effect

and C=0.

Figure 9: Input and output of a super-gaussian pulse, m=3, with the high

order effect and C=0.

According to these results, it can be concluded that the higher-order effect causes a tail in the output pulse, due to its odd effect. It is also shown that when the pulse has high spectral width the high order effect prevails for a longer period of time. 2.5.2. LD >> L’D

Considering

LD

L 'D= 10 , we have:

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6Evolução da largura dos impulsos com a distância: sgn (β2) = - 1

ζ = z / LD

η

C = - 2C = 0C = 2zmin

6


and C=0.



By observing the figures, where, once again, it has to be considered both effects, we are able to see that the effect of

β3 prevails. 2.5.3. LD << L’D

Now, we considered

LD

L 'D= 0.1 .


and C=0.



In this cases are able to see the the GVD is the main effect. Anyway, we can also see that in the super-gaussian pulse theres a little of the β3 effect, but the GVD is the one that prevails.

3. OPTICAL FIBERS: NON-LINEAR REGIME

In this section light propagation in the non-linear regime in an optical fiber is described.

3.1. Self-phase modulation Optical fibers are also affected by nonlinear effects, such is the case of the self-phase modulation (SPM) effect. When the electromagnetic field intensity increases there is a consequent change of the index of refraction. This nonlinear effect is known as the Kerr [17] effect and is described by [14]

′nj = nj + n2 P Aeff( ), j = 1,2, (41)

where n2 nonlinear index coefficient, P is the optical

power and Aeff is the effective area.

The propagation constant becomes a function of the optical power ′β = β + γ P (42)

where γ = 2πn2 Aeff λ .

The nonlinear phase is given by

φNL = γ Pin t( )Leff (43)

where Leff is the effective length. Self-phase modulation

causes chirp.

3.2. Pulse propagation equation in the non-linear regime

In this situation the term PNL from Equation (7) is no longer

neglected. This term originate the component iγ U

2U that

accounts for the nonlinear effects.

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The pulse propagation is now given by [14]

∂U∂ζ

+ iβ2

2∂2U∂τ 2 = −α

2U + iγ U

2U (44)

and it is called the nonlinear Schrödinger (NLS) equation. The equation can be rewritten as

i ∂U∂ζ

=sgn β2( )

2LD

∂2U∂τ 2 −

exp α z 2( )LNL

U2U (45)

where LNL = 1 γ P0 is the nonlinear length. This equation allows to classify the propagation regime in four categories by comparing L , LD and LNL . When

L LNL and L LD the regime is linear and non-

dispersive. If L LNL and L ≈ LD the regime is linear and

dispersive. For L ≈ LD and L LD the regime is nonlinear

and non-dispersive. And if L ≈ LNL and L ≈ LD the regime is nonlinear and dispersive. The latter is the most frequent situation in an optical fiber. For the anomalous region and when the effects of GVD and SPM are balanced it is possible to propagate solitons. In this case and if losses are neglected, Equation (44) can be written as

i ∂U∂ζ

+ 12∂2U∂τ 2 + N 2 U

2U = 0. (46)

When N = 1 the solution of this equation is the fundamental soliton [14] and is given by

U ζ ,τ( ) = sech τ( )exp iζ

2⎛⎝⎜

⎞⎠⎟

. (47)

The fundamental soliton maintains its shape along the fiber although it suffers a phase shift of ζ 2 .

3.3. Bright solitons In order to obtain this kind of solitons it is necessary to operate in the anomalous region, i.e., setting

sgn β2( ) = −1 .

With this consideration and using normalized amplitude, i.e., u = NU , Equation (46) will become

i∂u∂ξ

−12∂2 u∂τ 2

+ u2u = 0 (48)

3.3.1. Fundamental soliton

The relation between LD and LNL is given by

N 2 =

LD

LNL

(49)

and for the fundamental soliton one has N = 1 . The usual form for the fundamental soliton is obtained by doing

u 0,0( ) which implies that η = 1 [14]. So,

u ξ ,τ( ) = sech τ( )exp iξ 2( ) (50)

And it was possible to obtain the following figure

Figure 14: Fundamental soliton evolution along the optical fiber.

As it can be seen, the soliton will propagate undistorted which is very important for communication systems. This is a result of the perfect compensation between fiber nonlinearity and the GVD.

3.3.2. Gaussian pulse In this section, the case study is a Gaussian pulse propagating along the optical fiber given by

u0 τ( ) = exp

τ 2

2⎛⎝⎜

⎞⎠⎟

. (51)

The results are in the next figure:

Figure 15: Gaussian pulse evolution along the optical fiber.

-15 -10 -5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time τ

Ampli

tude

η


η = 0.74235

8

Figure 16: Gaussian pulse evolution along the optical fiber.

In this case, there is broadening of the pulse and consequent amplitude is reduction due to the conservation of energy. Also from this simulation one can observe that even if a pulse is not a soliton it will tend to it as it propagates along the fiber. In this case it stabilizes at η = 0.74235 . 3.3.3. Super-gaussian Comparing a super-gaussian pulse propagating along the optical fiber with different m values gives us the next two Figures:

Figure 17: Super-gaussian pulse evolition with m=2, along the optical

fiber.

Figure 18: Super-gaussian pulse evolution with m=3, along the optical

fiber.

By the analysis of the figures, once again, there is broadening of the pulse and consequent amplitude is reduction due to the conservation of energy.


fiber.


fiber.

Also from this simulations we can observe that even super-gaussion pulses will tend to solitons as they propagates along the fiber.

3.4. Soliton Interaction The first conclusion which is important to emphasize is that the parameter that will “control” this effect is the time interval between two consecutive solitons, TB . This interval influences drastically the bit-rate of the communication as

B = 1 TB . The equation used to simulate this interaction was [17]

i∂u

1

∂ξ+

12∂2 u

1

∂τ 2+ u

1

2u

1= −2 u

1

2u

1− u

12u

2* (52)

Where its solution u0

τ( ) = sech τ − q0( ) + r sech r τ + q

0( )⎡⎣ ⎤⎦exp iθ( ) (53) With that it was possible to do the following simulations

9

Figure 21: Soliton interaction when q0

= 5.51

It is possible to visualize that for high values of q0 there are no soliton interactions, which means that there is not any problem with the information they carry. The next figure shows the relation between q0 and initial phase, φ .

Figure 22: Soliton interaction considering q0

= 3.22 and different values

of φ

It is interesting that although it was considered a low value of q0 , the influence of φ will prevail. For lower values ofφ , there are some collisions between the two solitons, but for higher values of φ the effect of q0 will tend to vanish. In order to understand how solitons interact between them another study was made

Figure 23: Interaction between two solitons when θ = π

4and r = 1

Figure 24: Interaction between two solitons when θ = 0 and r = 1.1

If we look attentively to Figure 23 it is possible to see that the two solitons at the beginning of the input are attracted but when they continue propagating inside the fiber, they start to separate themselves. In Figure 24 another important parameter was: the relative amplitude. As it was considered in the situation of the two in-phase solitons, the major difference is that they oscillate periodically although they never collide.

4. CONCLUSION

To have an accurate description of light propagation in an optical fiber one needs to study the modal theory. This theory shows that for low dielectric contrast optical fibers the modes that propagate are linearly polarized. Also one concludes that to have single mode fibers the normalized frequency needs to be lower than 2.4048. The study of pulse propagation in the linear regime shows that the pulse broadens due to GVD that leads to the need of reducing the system’s bit rate in order to prevent ISI. The propagation of the pulse also leads to the appearance of chirp. It is shown that the nonlinear regime is governed by the NLS equation. Also in the nonlinear regime, there is the

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appearance of new frequency components due to the Kerr effect, leading to SPM effect. The relation between L , LD and LNL also gives insight on the behavior of the regime the pulse is propagating in. The numerical analysis of the super-gaussian pulse propagating in the linear regime confirmed the broadening of the pulse and the chirp’s effect predicted in the theoretical analysis. It was also shown that the broadening of the pulse leads to a reduction of the pulse amplitude due to the conservation of energy. The simulation of the propagation of the fundamental soliton showed that, when the GVD and SPM effects are balanced, it does propagate along an optical fiber without suffering changes in its shape. The simulation of a gaussian pulse in this situation shows how robust the system is, as the pulse reshapes into a soliton along the propagation. When GVD and SPM effects are not balanced due to fiber losses, soliton propagation is not possible.

5. REFERENCES

[1] G. P. Agrawal, Fiber-Optic Communication Systems, 3rd Ed. New York: Wiley, 2002.

[2] J. Hecht, City of Light: The Story of Fiber Optics. New York: Oxford University Press, Inc, 2004.

[3] G. P. Agrawal and Y. S. Kivshar, Optical Solitons: From Fibers to Photonic Crystals. New York: Academic Press, 2003.

[4] B. Dacorogna, Introduction to the Calculus of Variations, 2nd Ed. London: Imperial College Press, 2004.

[5] C. R. Paiva, Introdução aos Métodos Variacionais, IST, 2008. [6] —, Problemas, Fotónica. IST, 2011. [7] R. K. Luneburg, Mathematical Theory of Optics, Providence,

Rhode Island: Brown University, 1994, pp. 189-213. [8] R. Chatterjee, Antenna Theory and Pratice, 2nd Ed. New Age

International, 1996, pp. 198-199. [9] S. P. Morgan, “General Solution of the Luneberg Lens

Problem”, Journal of Applied Physics, pp. 1358-1368, 1958. [10] C. R. Paiva, Projecto, Fotónica. IST, 2011. [11] J. Evans and M. Rosequist, “ F=m a Optics”, American

Journal of Physics, pp. 876-883, 1986. [12] J. C. Maxwell, A Treatise on Electricity & Magnetism, New

York: Dover Publications, 1954. [13] C. R. Paiva, Fibras Ópticas, IST, 2008. [14] G. P. Agrawal, Nonlinear Fiber Optics, 4th Ed. Boston:

Academic Press, 2007. [15] A. Biswas, “Dispersion-managed solitons in optical fibers”,

Journal of Optics.1, Vol. 4, 2002. [16] V. S. Grigoryan and C. R. Menyuk, “Dispersion-managed

solitons at average dispersion”, Optical Letters. 8, Vol. 23, 1998.

[17] Carlos R. Paiva, Solitões em Fibras Ópticas, IST Press, 2008.

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