Optical Fiber Systems with Dispersion Compensation

Ana Filipa Fazenda Cabete

Instituto Superior Tecnico, Technical University of Lisbon, Portugal.anafilipafcabete@gmail.com

Abstract

Pulse propagation in optical fibers, associated dispersion phenomenon and effective techniques that are able to solve thisdegrading effect are addressed.

The work begins with a brief description of the several types of dispersion, including these into the equation that governspulse propagation on linear regime for a single-mode fiber. Studies of dispersive effects (group velocity dispersion andhigher order dispersion) are carried out for various types of pulses.

The main techniques used to compensate the dispersive effects in the linear regime are analyzed. The first one is based ondispersion compensating fibers, which operation mode is described and, for several pulses, simulations are done consideringthe group velocity dispersion, higher order dispersion and both simultaneously. Another compensation approach is usingfiber Bragg gratings. In this topic, theoretical foundations essential for understanding the Bragg gratings are describedalong with uniform and non- uniform Bragg gratings parameters simulations.

Finally, optical fibers are analyzed as non-linear transmission media. As in the linear regime, the equation that governsthe pulse propagation in the nonlinear regime is derived. Dispersive effects of this type of regime are identified, namelythe group velocity dispersion and self-phase modulation, which equilibrium determines the propagation of a special typeof pulses, the Soliton. Simulations for this type of pulses are presented, as well as the study of one of the most commonnon-linear regime dispersion compensation technique: the dispersion decreasing fibers.

Keywords: Pulse Propagation in Optical Fibers, Group Velocity Dispersion, Higher Order Dispersion, Dispersion Compensation, DispersionCompensating Fibers, Fiber Bragg Gratings, Self-Phase Modulation, Solitons, Decreasing Dispersion Fibers.

1 INTRODUCTION

A communication system is a link between two pointsthrough which information is transmitted using a carrier.An optical communication system uses the electromagneticwaves from the optical spectrum, contained between the farinfrared (100µm) and the ultraviolet (0.05µm) to carry theinformation. The basic components of an optical commu-nication system are: an optical transmitter, that convertsthe electrical signal into optical signals sending them to theoptical fiber that represents the transmission media and anoptical receiver, that receives the optical signal converting itto an electrical signal [1].

The work developed by the physicists Daniel Colladon andJacques Babinet was essential to the development of opticalfiber based communication systems. In the decade of 1840,they were the first to demonstrate the possibility of redi-recting light through refraction, the fundamental principlefor light propagation in optical fibers. The experiment con-ducted by John Tyndall in 1854, gave him the credit for Col-ladon’s and Babinet’s ideas. In the second half of the twen-tieth century, optical fibers suffered major breakthroughs.In 1952, a partnership between Brian O’Brien and NarinderKapany resulted in the development of the first communica-tion system to ever use glass fibers, transmitting informationthrough pulses of light. In the 1960s, optical communicationsystems faced two major obstacles. First, there was a needfor a source capable of generating optical pulses. Secondly,

the inexistence of an adequate transmission media. The firstproblem was solved with the development of Laser, that al-lowed carrying up to 10000 times more information than thehighest radio frequencies used. Still, it wasn’t an adequatemedia for free space propagation due to the sensibility toenvironmental conditions. In 1966, Charles Kao and GeorgeHockman proposed optical fibers as the ideal light propa-gation media [2], as long as losses were of the order of 20dB/km. This goal was reached by Robert Maurer, DonaldKeck and Peter Schultz in 1970, making viable the use ofoptical fibers in communication systems.

In the last decades several generations of optical fiber com-munication systems were developed. The development of op-tical amplifiers allowed the amplification of signals withoutthe use of electronics. Erbium doped fiber amplifiers (ED-FAs) [2] were the ones that mattered the most, allowing theincrease in the distance between repeaters. Currently thefourth generation uses optical amplification to extend thedistance between amplifiers and wavelength division multi-plexing (WDM) technique that allows higher bit rates [2].The fifth generation is highly anticipated. Having solvedthe losses problems by using amplifying fibers, dispersionhas become the greater problem to be addressed. To elimi-nate this problem, several techniques have been developed,such as compensating dispersion systems, dispersion man-agement systems and soliton based systems [2]. All this solu-tions make use of optical amplification, WDM and dispersion

1

management.Optical fibers revolutionized the communication systems.

The need to increase the traffic capacity, results from thegeneralization of new technologies. Given all the advantagesand capabilities of optical fibers, it is possible to assess thatthe use of these devices as a mean of transmission (FTTxnetworks), is the most appropriate way to meet these re-quirements.

2 PULSE PROPAGATION IN A OPTICALFIBER

Besides fiber losses (α), that reduce the available opticalpower increasing the bit error rate (BER) in the receptionlimiting the maximum distance between transmitter and re-ceiver, the dispersion is another major degrading phenomenathat affects pulse propagation. The time dispersion limitsthe bandwidth because it causes broadening of the opticalpulse. In long distance communication systems it can causepulses to interfere with each other (intersymbolic interfer-ence (ISI)) [2] and consequently information loss. The mainadvantage of single mode fibers (SMF), the ones that are rel-evant for this work, is that intermodal dispersion is absentbecause the energy of the propagated pulse is transportedonly by one single mode. However, in this type of fibersthere is still a dispersion source entitled group velocity dis-persion (GVD).

2.1 Time Dispersion in SMF

For a monomodal fiber, the group velocity relates with thegroup index ng by

vg =c

ng. (1)

Considering the fact that the frequency dependence causesa time delay, one has

D = −2πc

λ2β2, (2)

where D is the dispersion coefficient and β2 is the GVDcoefficient responsible for the pulse broadening inside thefiber. It is given by

β2 =d2β

dω2. (3)

The GVD is the result of different spectral componentsof the pulse traveling at a slightly different group veloci-ties due to the variation of the refractive index of the fibercore and cladding with the frequency. The dispersion coef-ficient includes both the material dispersion (DM ) and thewaveguide dispersion (DW ). The first one is a consequenceof existing changes in the cladding’s refractive index withthe wavelength and the waveguide dispersion occurs becauseof the light-confinement problem in the fiber core. Thesecontributions are given by

DM = M2 =1

c

∂N2

∂λ(4)

DW = M2∆d(νb)

dν− N2

2 ∆

n2λc

[νd2(νb)

dν2

]. (5)

where c is the light velocity, N2 is the group index of thecladding, ν is the normalized frequency and b is the normal-ized propagation constant.

Figure 1: Total dispersion D, material dispersion DM and waveguide dis-persion DW , in a conventional SMF [1].

DM has a positive slope and DW a negative slope. D ≈ 0when the fiber is operating near the zero dispersion wave-lenghth λZD. However, the dispersion effects do not disap-pear completely. In this case and when the pulse is ultra-short, one has to consider the effects of the higher orderdispersion (HOD), which is governed by the dispersion slopeS = ∂D

∂λ which equals

S =4πc

λ3β2 +

(2πc

λ2

)2

β3 =SD4

[1 + 3

(λZDλ

)4]. (6)

When λ = λZD and β2 = 0, S is proportional to the higherorder dispersion coefficient β3, which equals

β3 =∂β2

∂ω. (7)

2.2 Pulse Propagation Equation in the LinearRegime

The pulse propagation equation in the linear regime isderived in order to determine the shape of the pulse at theend of the link.

Assuming that the refractive index is invariant and con-sidering one pulse A(0, t) at the input of the fiber with thecarrier frequency of ω0, the field equations are

E(x, y, 0, t) = xF (x, y)B(0, t), (8)

B(0, t) = A(0, t)exp(−iω0t). (9)

Applying the properties of Fourier transform, the ampli-tude can be written as

B(z, t) = A(z, t)exp[i(β0z − ω0t)]. (10)

2

To find the basic the propagation equation it is useful todetermine A(z, t) in terms of A(0, t). Considering losses (at-tenuation coefficient α), it is obtained the following equation

∂A

∂z+

∞∑m=1

im−1

m!βm

∂mA

∂tm+α

2A = 0. (11)

Ignoring the fourth and greater propagation terms (m ≥ 4)and α, it is reached

∂A

∂z+ β1

∂A

∂t+ i

1

2β2∂2A

∂t2− 1

6β3∂3A

∂t3= 0. (12)

This differential equation can be rewritten applying thefollowing normalized variables

ζ =z

LD, τ =

t− β1z

τ0, LD =

τ20

|β2|. (13)

Then it is obtained

∂A

∂ζ+ i

1

2β2LDτ20

∂2A

∂τ2− 1

6β3LDτ3o

∂3A

∂τ3= 0. (14)

2.3 Merit Figure

The pulse broadening caused by dispersion, has influenceon the intersymbolic interference. As a consequence, the bitrate is affected. To study the repercussions in this parameter,it is essencial to determine the effective width of the pulse,given by

σ(z) =√〈t2〉 − 〈t〉2, (15)

where the angle brackets refer to averaging with respect tointensity.

Being V the normalized spectral width, considering V <<1 a reasonable approach for an monomodal laser with a smallspectral width and ignoring the HOD, for a gaussian pulsethe following equation represents the broadening factor(

σ

σ0

)2

=

(1 + C

β2L

2σ20

)2

+

(β2L

2σ20

)2

, (16)

where C is the chirp parameter and L the fiber length. As-suming a broadening coefficient η = σ

σ0, the correspondent

bit rate is B.Figure 2 shows the effects of the chirp parameter on the

broadening coefficient of the pulse along the fiber. For theanomalous region (β2 < 0), a negative chirp causes a fasterbroadening of the pulse. However, for a positive chirp onecan observe a pulse contraction in its initial stage. Later on,the effect of the chirp is overruled by the GVD effect.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

ζ = z / LD

η

C = − 2

C = 0

C = 2

Figure 2: Pulse broadening evolution with the normalized distance ζ.

Being

x =|β2|L2σ2

0

= 2γ20 |β2|(B2L), (17)

and applying it to the equation (16), it is reached the finalequation that represents the figure of merit

B2L =−C + sgn(β2)

√η2(1 + C2)− 1

2γ20β2(1 + C2)

. (18)

This product is dimensionless and given two or more links,this factor allows to verify which one is the best (the biggerthis factor is, the better is the link). It is possible to adjustthe values of B2 and L in order to have the best link possible.

−6 −4 −2 0 2 4 60

2

4

6

8

10

12

14

16

18x 10

24

C

B02 L

β

2 = − 20 ps

2/km

β2 = 20 ps

2/km

Figure 3: Influence of C parameter in the B2L product.

2.4 Group Velocity Dispersion Effect

This section focuses on the study of the GVD effect onpulses, namely chirped gaussian ones.

2.4.1 Gaussian pulse

A chirped gaussian pulse is described by the followingequation

A(0, t) = exp

[− 1 + iC

2

(t

t0

)2]. (19)

In order to identify the GVD effects in gaussian pulses,the simulations done for three different chirp values are pre-sented.

3

Figure 4: Gaussian pulse evolution along the fiber length for C = 0.

Figure 5: Gaussian pulse evolution along the fiber length for C = 2.

Figure 6: Gaussian pulse evolution along the fiber length for C = −2.

Observing the above images, it is possible to conclude thatthere is a substantial reduction in the pulse amplitude alongthe fiber. So, for distances bigger than the ones used, theresults will be even more degraded. This happens becauseof the broadening phenomenon that comes with the GVDinfluence. For C < 0 the chirped gaussian pulse broadensfaster than the same type of pulse in the absence of frequencychirp (C = 0). In the C > 0 situation, it is verified that thechirp initially compensates the GVD effects that later in thepropagation become dominant.

2.5 Higher Order Dispersion Effect

This section focuses on the study of the HOD effect onpulses, namely gaussian ones.

2.5.1 Gaussian pulse

Although the contribution of GVD dominates in mostcases of practical interest, it’s sometimes necessary to in-clude the higher order dispersion, governed by β3 [3]. Thisparameter can’t be ignored in situations where ultra-shortpulses are considered (characteristic width < 5ps) and incases where the pulse wavelength is very close to the zerodispersion wavelength λZD (β2 = 0) [1].

For this case, it’s considered L′D = τ0|β3| . For one link

distance z = 5L′D and width τ0 = 1ps, Figure 7 representsgaussian pulse shapes for three distinct cases: initial pulse,L′D = LD and β2 = 0.

Figure 7: HOD effects in a gaussian pulse: initial pulse, L′D = LD and

β2 = 0.

The presence of HOD effects changes the pulse shape,causing deformations. For β2 = 0, the pulse becomes asym-metric when comparing to the initial pulse presenting strongoscillations in its extreme. When considering L′D = LD, thepulse is still distorted and remains asymmetric.

It is also important to examine the HOD effects in achirped gaussian pulse. The next figure, represents the de-terioration of a gaussian pulse with the chirp parameter.

Figure 8: Deterioration of a gaussian pulse when β2 = 0 and τ0 = 1ps, forthree values of C.

4

The β3 parameter is affected by a C2 factor, so for sym-metric values of C the evolution of chirp gaussian pulses issimilar [4]. Considering Figure 8, it is possible to concludethat the greater C value is, more significant the effects ofHOD will be.

3 DISPERSION COMPENSATION IN LINEARREGIME

In this section two of the techniques used for compensatingthe dispersion phenomenon are presented: Dispersion Com-pensating Fibers (DCF) and Fiber Bragg Gratings (FBG).

3.1 Dispersion Compensating Fibers

The broadening of pulses that occur due to GVD, can becompensated using DCF. This technique allows full disper-sion compensation as long as the average optical power ofthe signal is low enough so that nonlinear effects can be ne-glected. This technique combines segments of optical fiberwith different characteristics in order to reduce the averagedispersion of the entire fiber link to zero.

Considering the pulse propagation along two segments offiber, the correspondent pulse envelope is given by

A(L, t) =1

2π

∫ ∞−∞

A(0, ω)exp

[i

2ω2(β21L1+β22L2)−iωt

]dω,

(20)where L = L1 + L2 and β2j is the dispersion coefficient ofthe fiber segment Lj(j = 1, 2). The DCF is such that thefactor in ω2 is cancelled, meaning that

β21L1 + β22L2 = 0. (21)

The results obtained for a gaussian pulse are presented inthe Figures 9,10 and 11.

−30 −20 −10 0 10 20 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

Am

plit

ud

e

SMF entrance

SMF output

DCF output

Figure 9: Gaussian pulse evolution along the optical fiber and DCF.

As expected the pulse recovers its original shape showingthat full dispersion compensation is possible. As in this case,when considering only the HOD effects, the pulse also recov-ers its original shape. However, when GVD and HOD areconsidered simultaneously, it is not possible to compensateboth degrading effects.

Figure 10: Evolution of a gaussian pulse since the SMF entrance to itsexit for C = 0.

Figure 11: Evolution of a gaussian pulse since the DCF entrance to itsexit for C = 0.

3.2 Fiber Bragg Gratings

Due to recent studies, it is possible to change the refractionindex inside the fiber core through ultraviolet light absorp-tion. This photosensitivity allows the fabrication of phasestructures inside the fiber core. This enables the formationof phase structures, the Bragg gratings [5].

Bragg gratings are a periodic disturbance of the refractionindex along the fiber length. They are formed by a set of el-ements spaced a certain distance. These segments of opticalfiber reflect only the wavelegths that satisfy the Bragg con-dition and transmit all the others. This way, a FBG acts likea reflective optical filter [1]. This because of the existence ofa stop band, the frequency region where the most part of theincident light is reflected back [1]. The Bragg wavelength,where the stop band is centered, is given by

λB = 2nΛ, (22)

where Λ is the grating period and n is the average modeindex. For λB , maximum reflectivity occurs. To analyze thebehavior of these structures, it is used the couple-mode equa-tions that describe the coupling between the waves that aretransmitted (forward waves) and the ones that are reflectedbackwards (backward waves).

5

3.2.1 Uniform FBG

A FBG is considered uniform if all its spacial propertiesremain constant throughout its length. Along the uniformeFBG, the two waves referred are given by

∂Af∂z

= iδAf + iκgAb (23)

−∂Ab∂z

= iδAb + iκgAf (24)

where Ab and Af are the spectral amplitudes of the twowaves, δ is the detuning from the Bragg wavelength and κgis the coupling coefficient. Solving analytically the couple-mode equations, the expressions obtained for the reflectioncoefficient rg and its phase φg are

rg =Ab(0)

Af (0)=

iκgsin(qgLg)

qgcos(qgLg)− iδsin(qgLg)(25)

φg = −arctan[Im(rg)

Re(rg)

], (26)

where q2g = δ2 − κ2

g and Lg is the FBG length.

−10 −8 −6 −4 −2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

δ Lg

|rg|2

k

g L

g=2

kg L

g=4

Figure 12: Reflectivity in function of δLg product.

By observing the above figure, it’s possible to concludethat in the stop band region the greater the κgLg product,the more reflectivity approaches its maximum value 100%.However, it is verified the presence of secondary maximumsthat are justified by the existence of multiple reflections inthe FBG extremes. To solve this problem apodisation tech-niques are used.

From the reflected signal phase, the group delay τg is givenby

τg =∂φg2πc

= − λ2

2πc

∂φg∂λ

, (27)

and the grating induced dispersion is

Dg =∂τg∂λ

= −2πc

λ2

∂2φg∂ω2

= −2πc

λ2β2g, (28)

where β2g is the dispersion coefficient related to the groupvelocity in FBG.

−10 −8 −6 −4 −2 0 2 4 6 8 10−25

−20

−15

−10

−5

0

5

δ Lg

φ [

rad

]

k

g L

g=2

kg L

g=4

Figure 13: Reflectivity phase φg in function of δLg product.

Figure 14: Group delay τg in function of wavelength λ.

Figure 15: Dispersion Dg in function of wavelength λ.

In Figure 13, it is observed that in the stop band the phasevariation is almost linear. So, this region will correspondto a minimum group delay value as can be seen in Figure14. Consequently, the dispersion value is lower (Figure 15).From this, it’s possible to state that the grating induceddispersion exists only outside the stop band and the biggerthe κgLg product is, the higher the dispersion value will be.

Another important parameter is the bandwidth of thegrating, given by [6]

∆λ =2λB

2

2nLgπ

√(κgLg)2 + (π)2. (29)

The smaller the grating length Lg, the bigger the band-

6

Figure 16: Bandwidth in function of the grating length Lg.

width will be. However, this fact leads to an undesirablesituation related to the fact that the value of the maximumreflectivity decreases. Although FBG are used for disper-sion compensation, they have a very narrow stop band widthwitch forbids its use for very high bit rates [1].

3.2.2 Chirped FBG

Chirped FBG (CFBG) are used for dispersion compensa-tion when considering high bit rates. This type of device en-ables variation of the Bragg condition throughout its lengthby varying the physical grating period Λ(z) or by changingthe effective mode index n. The grating period variation hasa linear characteristic, given by the expression[7]

Λ(z) = Λ(0) + CΛz, (30)

where Λ(0) represents the grating spacial period in one of itsextremes and CΛ the aperiodicity coefficient.

Hence, it is possible to obtain a linear aperiodicity whichis caused by an increase in the Bragg wavelength. Conse-quently, a shift on the stop band center occurs, changing itto progressively lower frequencies as the spacial period in-creases. Different frequency components of an incident op-tical pulse are reflected at different points, depending onwhere the Bragg condition is satisfied. This way, for a situa-tion that corresponds to anomalous GVD, the high frequencycomponents of the pulse are the first to be reflected and thelower frequency ones are reflected later [1]. The remainingwavelengths are reflected normally.

A CFBG has a wider bandwidth than a uniform FBG.This fact is related to the fact that the Bragg condition isverified, in CFBG, for a bigger number of spectral compo-nents.

The group delay τg in CFBG is given by[1]

τg =2nLgc

. (31)

The dispersion in a CFBG, where the optical period varieslinearly along its length, is given by

Dg =2nLgc∆λ

, (32)

Figure 17: Reflectivity (left) and group delay τg (right) of a CFBG withLg = 2.5cm.

where ∆λ = 2nLgCΛ represents the difference between thespectral components reflected at the CFBG extremes. Then,it is obtained

Dg =1

cCΛ. (33)

Figure 18: Dispersion in a CFBG with CΛ = 1nm/cm, in function of thegrating length Lg .

If the grading is long enough, the CFBG dispersion will beindependent from its length (Figure 18), varying only with

Figure 19: Dispersion in a CFBG with Lg = 2.5cm, in function of theaperiodicity coefficient CΛ.

7

the aperiodicity coefficient (Figure 19). This way, it is possi-ble to use a CFBG with length in the order of centimeters tocompensate the GVD effects in a SMF with approximatelyhundreds of kilometers.

4 DISPERSION COMPENSATION IN NONLIN-EAR REGIME

All the former studies were made considering an opticalfiber as a linear propagation medium. It happens that notalways this devices have this type of behavior. In fact, whenconsidering high optical powers or long transmission dis-tances, nonlinear effects appear and cannot be neglected.When the electromagnetic field intensity increases, there isa consequent change of the optical fiber refraction index [8].This nonlinear effect is known as the Kerr effect. In this, itis noticed a nonlinear phase deviation φNL which is calledself-phase modulation (SPM). So, in this regime, SPM phe-nomenon and GVD limit the system performance by caus-ing spectral broadening of optical pulses. However, for theanomalous region a fascinating manifestation occurs whenthere is a balance between both GVD and SFM: the appear-ance of a special type of pulse, the soliton. This sectionfocuses on pulse propagation in the nonlinear regime, iden-tifying the behavior of systems based on optical solitons andpresenting one of the most common solutions in dispersioncompensation for this regime, the dispersion decreasin fibers(DDF).

4.1 Optical Solitons

Using normalized variables, the expression that rules thepulse propagation in nonlinear dispersive regime is given by

i∂u

∂ζ− 1

2sgn(β2)

∂2u

∂τ2+ |u|2u = −iΓ

2u+ ik

∂3u

∂τ3. (34)

Neglecting the losses (Γ = 0) and the HOD (k = 0), equa-tion (34) is reduced to

i∂u

∂ζ− 1

2sgn(β2)

∂2u

∂τ2+ |u|2u = 0, (35)

which represents the nonlinear Schrodinger (NLS) equation.Through the inverse scattering transform (IST) method,

it’s possible to prove that the NLS equation accepts solutionslike [9]

µ(ζ, τ) = sech(τ)exp

[iζ

2

]. (36)

Any incident pulse has the following shape

µ(0, τ) = Nsech(τ), (37)

where N = LD/LNL, LD is the dispersion length and LNLis the nonlinear length. The first order solution (N = 1) cor-responds to the fundamental soliton. This type of pulse hasa special property related to the fact that its shape remains

constant along propagation (Figure 20). Unlike of what hap-pens with the fundamental soliton, the second and third or-der solitons (N=2 and N=3, respectively) change its shapealong propagation (Figure 21). Nevertheless, a periodic evo-lution is observed recovering totally its original shape in asoliton period ζ0 = π

2 .

Figure 20: Fundamental soliton evolution along the optical fiber.

Figure 21: Third order soliton evolution along the optical fiber.

In the above case, it can be seen that the pulse width con-tracts until a maximum value, where the SPM effects aredominant on GVD ones. Next, the situation is reversed,having the dispersion effects ruling on the SPM ones. Con-sequently, broadening and a decrease in the pulse amplitudeis verified. Finally, the several pulse components join and re-cover completely the initial pulse shape. This happens whenthe propagation distance equals the soliton period.

Although the higher order solitons can be used for com-pressing pulses, the fundamental soliton is the most interest-ing one for current communications systems, because it doesnot suffer distortion during its propagation.

4.2 Decreasing Dispersion Fibers

The existence of solitons in an optical fiber depends onthe balance between GVD and SPM. This same balance,however, can be broken by fiber loss presence. To solve thisproblem, a dispersion decreasing fiber (DDF) can be used[3].

8

It is known that the relation between LD and LNL is givenby

N2 =LDLNL

, (38)

and for the fundamental soliton one has N = 1. Therefore,

β2 = τ20 γP, (39)

where P = P0exp(−αz). Then, it is possible to concludethat

β2 = |β2|exp(−αz), (40)

with β2 = τ20 γP0 and γ being the nonlinear coefficient. Equa-

tion (40) describes the ideal profile for β2 along the fiber inorder to solve this problem. However, in practical terms thissolution is not easy to implement and so it is common toappply a staircase approximation to equation (40).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ζ

β2

Ideal DDF

Step approximation DDF

Figure 22: Step approximation of the dispersion profile β2 with 4 steplevels.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ζ

β2

Ideal DDF

Step approximation DDF

Figure 23: Step approximation of the dispersion profile β2 with 6 steplevels.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ζ

β2

Ideal DDF

step approximation DDF

Figure 24: Step approximation of the dispersion profile β2 with 300 steplevels.

Figure 25: Fundamental soliton propagation along a DDF with 4 steplevels.

Figure 26: Fundamental soliton propagation along a DDF with 6 steplevels.

Through the results, it can be verified that between 4 and6 step levels there is an improvement on the amplitude oscil-lation of the fundamental soliton. For a considerable numberof step levels, the approximation becomes almost linear over-lapping the ideal function of β2. Thus, it may be concludedthat the larger the number of step levels used, the betterthe approximation to the real function and hence the moreefficient compensation will be. However, the fact of increas-ing the number of step levels to improve the approximationrequires an increased processing capability [10].

9

Figure 27: Fundamental soliton propagation along a DDF with 300 steplevels.

5 CONCLUSION

In this paper the optical fiber dispersion phenomenon,its degrading influences and a few compensating techniqueswere addressed.

For a SMF, β2 (GVD) is responsible for the pulse broad-ening and ISI that limits the bit rate. Regarding the GVDeffects on pulse propagation, a decrease in the pulse ampli-tude and an increase in its width are expected. Moreoverwhen considering gaussian pulses with chirp parameter C, itcan be verified that there is an additional broadening effect.However, when considering ultra short pulses or when theGVD coefficient is null, the HOD effects ruled by β3 can notbe neglected. For this case, a gaussian pulse shows changesin its shape, becoming asymmetric when comparing to theinitial pulse and oscillations in the extremes. Chirp param-eter in this case also contributes for the pulse deterioration.

Regarding dispersion compensation techniques, the DCFwas the first to be addressed. It was proven that it is pos-sible to reduce to zero the average value of dispersion asthe pulse initial shape was fully recovered when consideringGVD and HOD effects separately. FBG is another techniquecommonly used. Uniform Bragg gratings act like an opticalfilter, reflecting only the wavelengths that satisfy the Braggcondition. The most important parameters were studied.This type of Bragg gratings are not efficient for very high bitrates and consequently, CFBG are used. In these, throughrefractive index changes or by grating period variation it ispossible to reflect different wavelengths at different places inthe grating. The main parameters were studied, concludingthat a CFBG with length in the order of centimeters is ableto compensate the GVD effects in a SMF with approximatelyhundreds of kilometers.

In the nonlinear regime, high electromagnetic field inten-sities cause a refraction index variation (nonlinear opticalKerr effect). This effect allows the appearance of the SPMphenomenon. Under certain circumstances, it is possible tohave an balance between GVD and SPM, allowing the prop-agation of a special type of pulses, the solitons. These pulses

maintain their shape throughout the fiber propagation, witchmakes them very desirable when considering its applicationin optical systems. When GVD and SPM effects are notbalanced due to fiber losses, it is required a compensationtechnique. For this goal, the DDF technique was studied.Using a staircase approximation for the ideal dispersion pro-file, it was shown that the compensation was efficient.

From this study, it was shown that dispersion phenomenonhas a deep influence on optical systems and must be consid-ered all times.

6 REFERENCES

[1] Agrawal, Govind P., Fiber-Optic Communications Sys-tems Third Edition, John Wiley & Sons, 2002.

[2] Paiva, Carlos R., Fotonica - Fibras Opticas, InstitutoSuperior Tecnico, Abril de 2008.

[3] Agrawal, Govind P., Nonlinear Fiber Optics FourthEdition, Academic Press, 2007.

[4] Matos, S. A., Dissertacao sobre Gestao De DispersaoEm Sistemas de Comunicacao Convencionais - Regime Lin-ear, Junho 2004.

[5] Othonos, Andreas, Review Article - Fiber Bragg Grat-ings, Departmente of Natural Science, Physics, University ofCyprus, 12 September 1997.

[6] Nogueira, Rogerio N., Dissertacao sobre Redes deBragg em Fibra Optica, Universidade de Aveiro, Departa-mento de Fısica, 2005.

[7] Neto, B. M. B., Dissertacao sobre Redes de BraggDinamicamente Reconfiguraveis para Compensacao da Dis-persao Cromatica, Universidade de Aveiro, Departamentode Fısica, 2005.

[8] Paiva, Carlos R., Fotonica - Solitoes em Fibras Opticas,Instituto Superior Tecnico, Abril de 2008.

[9] A. Hasegawa, M. Matsumoto, Solitons in Optical Cm-munication Third Edition, Springe, New York 2003.

[10] Santos, N. M. V-D. N., Dissertacao sobre Metodosvariacionais aplicados ao estudo das fibras opticas e tecnicasde compensacao de dispersao, Setembro 2011.

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