10
Dispersion Compensating Fibers and Fiber Bragg Grattings Filipe Correia da Silva Matias Instituto Superior Técnico Technical University of Lisbon Lisbon, Portugal [email protected] Abstract This work aims to analyse the phenomenon of time dispersion inherent to pulse propagation in linear regime and study two techniques used to solve this degrading effect which is one of the most limiting factors in optical communication systems. The material dispersion and waveguide dispersion contributions in total dispersion single-mode optical fibers are studied. The pulse propagation equation in linear regime is derived which allows to verify the existence of dispersive effects caused by dispersion coefficients that governs the group velocity dispersion and higher-order dispersion. Propagation of gaussian pulses is simulated in Matlab in order to demonstrate and study the dispersive effects in their propagation. The two main techniques for dispersion compensation which are considered are dispersion compensating fibers and fiber Bragg gratings. The dispersion compensating fiber simulations demonstrated that is possible to fully recover the initial shape of the pulse when group velocity dispersion and higher-order dispersion are considered separately but not when both are considered simultaneously. The study of fiber Bragg gratings starts with a description of its operating principle that focuses on Fresnel’s reflection, on coupled-mode theory and on matrix theory. Finally, the uniform fiber Bragg gratings, apodized fiber Bragg gratings and chirped fiber Bragg gratings are presented and theirs main features and contributions to dispersion compensation are studied. Keywords: Pulse Propagation in Optical Fibers, Group Velocity Dispersion, Higher-order Dispersion, Dispersion Compensation, Dispersion Compensating Fibers, Fiber Bragg Gratings. 1. INTRODUCTION In the decade of 1840 were made the first experiments that demonstrated the operating principle of light propagation in optical fibers: the possibility of directing light through refraction [1]. However in the decade of 1950 the optical pulse communication still faced two major obstacles: the lack of sources capable of generating optical pulses and the inexistence of an adequate transmission medium [2]. The first obstacle was overcome with the invention of laser which operation was demonstrated by Theodore Maiman in 1960. In 1966 optical fibers were first proposed as propagation medium by Charles Kao and Charles Hockman as long as losses in order of could be achieved [3]. Ever since were made efforts to achieve better conditions for the use of optical fiber in communications systems which led to development of several generations of optical communication systems in recent decades. Each generation originated a considerable increase in transmission rates and distances achieved. The fifth generation of optical communication systems is in development and has led to great expectations. Once the problem related to losses have already been solved with the emergence of erbium-doped fiber amplifiers (EDFA's), the main hurdle to overcome with the fifth generation is dispersion. In this sense several techniques have been tested including compensating dispersion systems, dispersion management systems and soliton based systems [3]. The optical communication systems are at the forefront of telecommunications covering the needs of society: transmission of large information flows in an efficient and safe way at high speeds. However as mention before the nowadays optical communications systems still suffers dispersion which is one of the most limiting factors in these systems. In order to solve the dispersion problem several techniques of dispersion compensation have been studied including dispersion compensating fibers (DCFs) and fiber Bragg gratings (FBGs) which are two of the most used techniques. 2. PROPAGATION IN LINEAR REGIME A pulse suffers distortion during propagation in optical fiber due to attenuation and dispersion phenomena. These two phenomena impose limitations on distance and maximum transmission rate of an optical fiber. Dispersion defines the pulse broadening in time domain during the propagation in optical fiber and results from the fact that spectral components of pulse travels along the optical fiber with different group speeds. The pulse broadening causes pulses overlapping that sometimes make them indistinguishable in reception originating information loss. This effect is called Intersymbol interference (ISI). Dispersion in optical fibers results of two effects termed intermodal dispersion and group velocity dispersion (GVD). Single-mode fibers (SMFs) don’t suffer intermodal dispersion because the energy of the propagated pulse is transported only by one single mode and still present higher bit rates than multi-mode fibers and for that reason are more suitable for optical communication systems.

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Page 1: Dispersion Compensating Fibers and Fiber Bragg Grattings · techniques for dispersion compensation which are considered are dispersion compensating fibers and fiber Bragg gratings

Dispersion Compensating Fibers and Fiber Bragg

Grattings

Filipe Correia da Silva Matias

Instituto Superior Técnico

Technical University of Lisbon

Lisbon, Portugal

[email protected]

Abstract — This work aims to analyse the phenomenon of time

dispersion inherent to pulse propagation in linear regime and

study two techniques used to solve this degrading effect which is

one of the most limiting factors in optical communication

systems. The material dispersion and waveguide dispersion

contributions in total dispersion single-mode optical fibers are

studied. The pulse propagation equation in linear regime is

derived which allows to verify the existence of dispersive effects

caused by dispersion coefficients that governs the group velocity

dispersion and higher-order dispersion. Propagation of gaussian

pulses is simulated in Matlab in order to demonstrate and study

the dispersive effects in their propagation. The two main

techniques for dispersion compensation which are considered are

dispersion compensating fibers and fiber Bragg gratings. The

dispersion compensating fiber simulations demonstrated that is

possible to fully recover the initial shape of the pulse when group

velocity dispersion and higher-order dispersion are considered

separately but not when both are considered simultaneously. The

study of fiber Bragg gratings starts with a description of its

operating principle that focuses on Fresnel’s reflection, on

coupled-mode theory and on matrix theory. Finally, the uniform

fiber Bragg gratings, apodized fiber Bragg gratings and chirped

fiber Bragg gratings are presented and theirs main features and

contributions to dispersion compensation are studied.

Keywords: Pulse Propagation in Optical Fibers, Group Velocity

Dispersion, Higher-order Dispersion, Dispersion Compensation,

Dispersion Compensating Fibers, Fiber Bragg Gratings.

1. INTRODUCTION

In the decade of 1840 were made the first experiments that demonstrated the operating principle of light propagation in optical fibers: the possibility of directing light through refraction [1]. However in the decade of 1950 the optical pulse communication still faced two major obstacles: the lack of sources capable of generating optical pulses and the inexistence of an adequate transmission medium [2]. The first obstacle was overcome with the invention of laser which operation was demonstrated by Theodore Maiman in 1960. In 1966 optical fibers were first proposed as propagation medium by Charles Kao and Charles Hockman as long as losses in order of could be achieved [3]. Ever since were made efforts to achieve better conditions for the use of optical fiber in communications systems which led to development of

several generations of optical communication systems in recent decades. Each generation originated a considerable increase in transmission rates and distances achieved.

The fifth generation of optical communication systems is in development and has led to great expectations. Once the problem related to losses have already been solved with the emergence of erbium-doped fiber amplifiers (EDFA's), the main hurdle to overcome with the fifth generation is dispersion. In this sense several techniques have been tested including compensating dispersion systems, dispersion management systems and soliton based systems [3].

The optical communication systems are at the forefront of telecommunications covering the needs of society: transmission of large information flows in an efficient and safe way at high speeds. However as mention before the nowadays optical communications systems still suffers dispersion which is one of the most limiting factors in these systems. In order to solve the dispersion problem several techniques of dispersion compensation have been studied including dispersion compensating fibers (DCFs) and fiber Bragg gratings (FBGs) which are two of the most used techniques.

2. PROPAGATION IN LINEAR REGIME

A pulse suffers distortion during propagation in optical fiber due to attenuation and dispersion phenomena. These two phenomena impose limitations on distance and maximum transmission rate of an optical fiber.

Dispersion defines the pulse broadening in time domain during the propagation in optical fiber and results from the fact that spectral components of pulse travels along the optical fiber with different group speeds. The pulse broadening causes pulses overlapping that sometimes make them indistinguishable in reception originating information loss. This effect is called Intersymbol interference (ISI). Dispersion in optical fibers results of two effects termed intermodal dispersion and group velocity dispersion (GVD). Single-mode fibers (SMFs) don’t suffer intermodal dispersion because the energy of the propagated pulse is transported only by one single mode and still present higher bit rates than multi-mode fibers and for that reason are more suitable for optical communication systems.

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1.1. Time Dispersion in Single-Mode Fibers

Total dispersion in SMF is given by

(2.1)

where is the GVD coefficient responsible for the pulse broadening. This parameter can be written as the sum of two terms such that

(2.2)

where represents material dispersion and represents waveguide dispersion given by the expressions

(2.3)

[

] (2.4)

where is the velocity of light, is the refractive index of the cladding material, is the group index of the cladding, is the normalized frequency parameter and is the normalized propagation constant.

Material dispersion is caused due to the non-linear variation of refractive index of the cladding material with wavelength. Since group velocity of a mode varies depending on the refractive index, the several spectral components propagate with different group velocities introducing a delay between each other, which leads to temporal broadening of the transmitted pulses.

The waveguide dispersion occurs because the optical power propagating in the fiber is not entirely confined in fiber core having also propagation in cladding. Since the refractive index of cladding is lower than the refractive index of core, the pulses propagate with higher velocity in cladding causing pulse broadening.

Figure 2.1. Total dispersion, material dispersion and waveguide

dispersion in a conventional SMF.

Figure 2.1 shows that total dispersion is zero for a wavelength near which is known as zero dispersion

wavelength . It’s possibly to design optical fibers with zero dispersion for wavelength of where is the minimum attenuation. These fibers are called dispersion-shifted fibers (DSFs).

DVG’s effects are predominant in dispersion but when the carrier is in near to or when the pulse is ultra-short the higher-order dispersion (HOD) must be considered [4]. HOD is given by

(

)

[ (

)

] (2.5)

where is known as the higher-order dispersion coefficient.

1.2. Pulse Propagation Equation

The pulse propagation equation in linear regime is derived in order to observe the influence of the dispersion coefficients in the shape of the pulse at the optical fiber output.

Representing a pulse that propagates through optical fiber by , the same pulse at the fiber entrance can be represented by . Assuming that the pulse modulates a carrier with angular frequency and the electric field is polarized linearly with x axis, the field can be written by

(2.6)

with

(2.7)

Applying the properties of Fourier transform, the amplitude can be written as

[ ] (2.8)

with

∫ [ ]

(2.9)

where

(2.10)

(2.11)

Using equation (2.9) is possible to calculate in terms of but it’s useful to define first

(2.12)

Now from equation (2.9) and considering losses it’s obtained the differential equation which allows determining from

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(2.13)

Ignoring the fourth and greater propagation terms and the attenuation constant it’s reached

(2.14)

Considering the following normalized variables

(2.15)

(2.16)

| | (2.17)

equation (2.14) can be rewritten as

(2.18)

This equation can be simplified using

(2.19)

| |

(2.20)

and finally it’s obtained

(2.21)

Introducing a new variable designated normalized frequency, so that it’s reached

[

] (2.22)

Now, it’s easy to determine the spectral pulse value, at any position in the fiber, using the FFT (Fast Fourier Transform) and the IFFT (Inverse Fast Fourier Transform) and following the next steps [3]

(i) [ ]

(ii) [

]

(iii) [ ].

1.3. Gaussian Pulse

A Gaussian pulse is described by the following equation

( )

(2.23)

The Gaussian pulse broadening caused by dispersion during the propagation in an optical fiber is represented by the broadening factor that is given by

√(

)

(

)

(2.24)

when considering the normalized spectral width which is a reasonable approach for an monomodal laser with a small spectral width and ignoring the HOD. In this equation L is the fiber length and C is the chirp parameter.

Figure 2.2 shows the influence of the chirp parameter in the broadening factor. For the anomalous region a negative chirp causes a faster broadening of the pulse caused by the GVD effect and for a positive chirp it’s observed an initial pulse contraction but at some point is this effect is overruled by the GVD effect.

Figure 2.2. Pulse broadening evolution with the normalized distance

for three values of C.

The following subsections are focused on the study

separately of the two types of dispersive effects (GVD and HOD) caused in the propagation of Gaussian pulses in optical fiber.

1.3.1. Group Velocity Dispersion

In order to identify the GVD effects in Gaussian pulses, the simulations done for three different chirp values are presented.

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Figure 2.3. Gaussian pulse evolution along the optical fiber for

.

Figure 2.4. Gaussian pulse evolution along the optical fiber for

.

Figure 2.5. Gaussian pulse evolution along the optical fiber for

.

Observing Figure 2.3, Figure 2.4 and Figure 2.5 it’s

possible to conclude that for all the three cases there is a reduction in the pulse amplitude along the optical fiber which is caused by the pulse broadening that is inflicted by the GVD effect. Therefore the greater the link length the most significant the dispersive effects will be.

When comparing the three figures it’s possible the observe that when chirp is considered the pulse broadening is faster than in the absence of chirp . For it is concluded that the presence of positive chirp introduces in an initial period a compensation of GVD opposing the pulse broadening but at a certain point the GVD becomes dominant.

1.3.2. Higher-Order Dispersion

After the study of the GVD effects it’s intended now to analyse the HOD contribution in Gaussian pulses propagation.

Figure 2.6 shows that as the pulse propagate in the optical fiber the larger are the deformations relative to the initial pulse becoming asymmetric with more and larger oscillations on the pulse front.

Figure 2.6. Gaussian pulse evolution along the optical fiber.

In order to observe the influence of the chirp parameter in

the HOD effect was simulated the propagation of chirped Gaussian pulses for three values of the chirp parameter.

Figure 2.7 shows that the chirp parameter causes the deterioration of the pulse as observed for GVD. Since is affected by a factor were only considered positive values because for asymmetric values the evolution of the pulse would be similar. It can also be concluded that the greater the absolute value of C the more significant the HOD effects are.

Figure 2.7. Higher-order dispersion effects on chirped Gaussian

pulses for three values of C.

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3. DISPERSION COMPENSATING FIBERS

This section introduces a technique for dispersion compensation in linear regime: dispersion compensation using DCFs which allows full dispersion compensation as long as the average power of the optical signal is low enough to consider the nonlinear effects negligible. This technique uses a DCF segment with length after a SMF segment with length to eliminate the dispersion effect taking advantage of the linear nature of the pulse propagation equation in optical fibers given by equation (2.14). This way it is intended to reduce the average dispersion of the link to zero by combining sections of optical fiber with dispersion coefficients of opposite sign.

Considering the pulse propagation along two segments of fiber (SMF and DCF), the output pulse is given by

[

]

(3.1)

where the total length link is given by and

and

correspond to GVD and DOS coefficients,

respectively, for the section of fiber with length with

.

In order to recover the initial pulse shape the DCF must have parameters that allows to eliminate the terms and . To satisfy this condition

(3.2)

Evaluating equation (3.2) it’s possible to conclude that and with must have opposite signs and the link to be economically viable must have a high absolute value to allow a DCF length as lower as possible.

However this technique has some disadvantages, including the fact that potentiate the nonlinear effects, the high manufacturing costs, high losses caused by the reduced value of the normalized frequency compared to losses exhibited by SMF (α = 0.2 dB / km), high insertion losses (typically greater than 5dB) and a reduced effective area [2].

The study of dispersion compensating based on DCF for two cases is presented in the following subsections. In the first case it’s only considered the GVD coefficient neglecting the DOS coefficient and the second case it’s the opposite study where is compensated and .

3.1. Group Velocity Dispersion Compensation

When considering a SMF with the coefficient can be neglected obtaining from equation (3.1)

∫ [

]

(3.3)

which has as a condition for perfect compensation DVG

(3.4)

Since the fibers typically used in telecommunications have positive values for the GVD coefficient DVG ( it is inferred from equation (3.4) that the DCF fiber should have negative values of DVG ( . Thus the DCF length used for perfect compensation is given by

(3.5)

The results of the GVD compensation of a Gaussian pulse are presented in Figure 3.1 and Figure 3.2.

Figure 3.1. Gaussian pulse evolution in SMF for GVD compensation.

Figure 3.2. Gaussian pulse evolution in DCF for GVD compensation.

As expected the pulse recovers its initial shape at the DCF output and thus it is demonstrated that using the DCF the perfect compensation of DVG can be achieved.

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3.2. Higher-Order Dispersion Compensation

When using bit rates higher than ultra-short pulses must be used however in this case the DOS effects can’t be neglected and becomes difficult compensate DVG. Thus, for the DOS compensation study the GVD effects are considered negligible using the then the equation. Then it is obtained from equation (3.1)

∫ [

]

(3.6)

The condition for perfect compensation of DOS is given by

(3.7)

So the DCF length must be

(3.8)

The results of the DOS compensation of a Gaussian pulse are presented in Figure 3.3 and Figure 3.4.

Figure 3.3. Gaussian pulse evolution in SMF for DOS compensation.

Figure 3.4. Gaussian pulse evolution in DCF for DOS compensation.

Figure 3.3 and Figure 3.4 shows that the use of DCF eliminates fluctuations in the pulse front and reduces its broadening (effects caused by DOS) recovering the initial shape of the pulse.

When GVD and HOD are considered simultaneously, it is not possible to compensate both degrading effects.

4. FIBER BRAGG GRATINGS

Diffraction gratings are optical components with periodical structures that allow the spatial separation of polychromatic light at its wavelengths. The structures origin diffraction effects and cause mutual interference in each wavelength of the incident light being transmitted or reflected in discrete directions designated by orders [5].

The FBGs arise from diffraction gratings and have varied applications in telecommunications. In this work the study of FBGs it’s focused in dispersion compensation.

4.1. Operation Principle

FBGs act by filtering through reflection certain spectrum wavelengths of the incident light allowing lossless transmission of the other wavelengths. This behavior is achieved by varying at periodic or aperiodic intervals the refractive index of the fiber core forming a dielectric mirror onto a specific wavelength acting as an optical filter reflector.

The Bragg condition which requires that the reflection spectrum is centered at the Bragg wavelength, , is given by

(4.1)

where is the perturbation period of the refractive index and is the average modal refractive index of the core of optical fiber.

The fundamental operation principle used by FBGs is the Fresnel reflection which announces that the light propagating between mediums with different refractive indices can be reflected and refracted simultaneously [5].

The theory of coupled modes allows the analysis of the FBGs behavior in the fiber as it allows the definition of the spectral response of uniform FBGs with periodic modulation of the refractive index. The basic idea of this theory consists in calculating the modes of undisturbed or uncoupled structures since these are previously defined and for more complex structures, with disturbances, the solution can been found from a linear combination of these modes. However, this type of analysis becomes complicated when considering situations with aperiodic disturbances and/or with variable amplitude and for that reason the matrix theory emerge. This theory consists in an approximation of the whole structure to a set of contiguous sections of uniform and periodic modulation wherein each region is represented by a corresponding matrix. Thus, the overall behavior of the whole structure can be

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obtained by multiplying the matrices of each section since phase continuity between elements is ensured [6].

4.2. Classification

The spectral behavior of FBGs depend on the refractive index modulation profile that can be represented generally by

(4.2)

where represents the modulation introduced in the optical fiber core. According to this parameter it is possible to distinguish different types of FBGs. In this work is studied the uniform fiber Bragg gratings, the apodized fiber Bragg gratings and the chirped fiber Bragg gratings (CFBG).

4.2.1. Uniform Fiber Bragg Gratings

The uniform FBGs are the simplest type of FBGs, involve a process of manufacturing more common and are the lesser specific in terms of application. Due to their ease manufacturing are widely used in telecommunications area and sensors area. The spectral properties of uniform FBGs remain constant along the propagation axis with a uniform and periodical variation of the refractive index according to the equation

(

) (4.3)

The equations that describe the coupled modes which represent the coupling between waves propagating in the positive direction (without reflection) called forward waves, and waves propagating in the negative direction (with reflection), denoted by backward waves are given by

(4.4)

(4.5)

where and are the spectral amplitudes of two waves,

is the detuning from the Bragg wavelength and is the

coupling coefficient. Solving analytically the coupled-mode equations the reflection coefficient and its phase are respectively given by

( ) (4.6)

[

] (4.7)

where is the FBG length and √ . For values

contained in the range of ,

becomes pure

imaginary which implies that most of the incident field on the

FBG is reflected demonstrating the existence of a stop band region which is one of the majors characteristics of FBGs and is shown in Figure 4.1.

Figure 4.1. Stop band.

Figure 4.2 shows the reflectivity spectrum for two values

of in which it’s observed that in the stop band region the

reflectivity gets closer to as the value increases

presenting a filter characteristic. However, it is also possible to observe the presence of secondary lateral lobes of high amplitudes caused by reflections at the FBG extremes, where there is no variation of the refractive index, which contribute to the existence of crosstalk between very close channels and this is one disadvantage of uniform FBGs.

Figure 4.2. Reflectivity spectrum for and .

From the phase of the signal reflected, the expression for the group delay given by

(4.8)

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Figure 4.3. Group delay for and .

The uniform FBG dispersion is obtained from the derivative of the group delay in order to wavelength

(4.9)

where is the FBG group velocity dispersion coefficient.

Figure 4.4. FBG dispersion for and .

When analysing Figure 4.3 and Figure 4.4 it’s possible to

conclude that the higher dispersion values are given out of the stop band region and are justified by the high group delay presented near this region which is caused by successive reflections at the FBG ends.

Another important parameter is the grating bandwidth given by

√( )

(4.10)

Figure 4.5. Bandwidth in function of the grating length.

Observing Figure 4.5 it’s possible to conclude that shorter gratings lengths lead to higher bandwidths but the value of the maximum reflectivity is also lower concluding that uniform FBG present reduced bandwidth not allowing the use of high bit rates [2].

4.2.2. Apodized Fiber Bragg Gratings

To suppress the unwanted secondary lobes displayed in the uniform FBG reflectivity spectrum techniques of apodization are used. Such techniques results of varying the amplitude of the modulation coefficient of the refractive index along the grating length.

The spectral properties of apodized FBGs are given by

(

) (4.11)

where is the apodization function.

Figure 4.6 shows the reflectivity spectrum for a uniform FBG and Figure 4.7 shows the result of two apodization techniques implemented in this FBG. The use of apodization techniques allows the elimination of the lateral lobes bridging one of the disadvantages of uniform FBGs.

Figure 4.6. Reflectivity spectrum of uniform FBG.

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Figure 4.7 Reflectivity spectrum of the uniform FBG with raised sine

and Blackman apodization.

4.2.3. Chirped Fiber Bragg Gratings

CFBGs were developed in order to solve one of the aforementioned disadvantages of uniform FBGs: the reduced bit rate. CFBGs allow dispersion compensation for high bit rates by varying the Bragg condition throughout its length or in other words by the progressive variation of the stop band center. This behavior is achieved by varying the spatial modulation period or the average amplitude of the refractive index of the grating or both simultaneously.

Considering a linear grating period variation, given by

(4.12)

where represents the grating spatial period in one of its extremes and is the aperiodicity coefficient [7].

The linear aperiodicity causes the increase of the Bragg wavelength and a consequent translation of the stop band to lower frequencies. Due to the variation of the Bragg wavelength the multiple signal wavelengths matching the various Bragg wavelengths are reflected at different positions of the CFBG. Thus, the slower spectral components (higher frequencies) are first reflected and faster spectral components (lower frequencies) travel a longer distance until reflected at the CFBG. This leads to a group delay dependent of the wavelength and for a linear aperiodicity the group delay will be also linear. This property makes CFBs attractive for dispersion compensation in communication systems through optical fiber again because make possible the arrival of the spectral components practically at same time [5].

CFBGs have a wider bandwidth compared with the uniform FBGs that result from the fact that the Bragg condition is verified by a larger number of spectral components [7].

Having as reference the spectral component at the grating initial extreme, , it is observed that the reflected spectral component at the opposite end has a group delay given by

(4.13)

Figure 4.8. Reflectivity and group delay of a CFBG with a 2.5 cm

length and a linear aperiodicity coefficient of 0.8 nm/cm.

Considering the spectral components reflected at the ends is obtained from the derivation of group delay in order to wavelength the grating dispersion

(4.14)

where represents the difference between the

spectral components reflected at CFBG extremes. Then it’s obtained

(4.15)

from which it is concluded that the dispersion at CFBGs are independent of its length having only dependence of the aperiodicity coefficient. Figure 4.9 shows the variation of dispersion with aperiodicity coefficient.

Figure 4.9. Dispersion in a CFBG with a 2.5 cm length as function of

the aperiodicity coefficient.

Thus using CFBGs with lengths on the order of dozen

centimeters is possible to compensate the GVD imposed by a SMF with length in the order of hundreds of kilometers [2].

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5. CONCLUSION

This paper presents a study on pulse propagation in optical fibers on linear regime to assess one of the most limiting factors of optical communication systems: time dispersion. Once analysed the phenomena inherent to dispersion two techniques used to overcome this problem were addressed.

Pulse propagation equation confirmed the influence of dispersive mechanisms in the propagation represented by , called second order coefficient and governs GVD and called of third order coefficient, which represents DOS. GVD is responsible for the temporal broadening and decreased amplitude of pulses causing interference between signals, called ISI, and consequently limits the bit rate and link length. Gaussian pulses are more affected by such phenomena with the advance along the fiber concluding that the greater the link length the worse the signal condition at reception. When considering Gaussian pulses with chirp it was verified a higher pulse broadening. It was observed that a negative chirp parameter is the one that provokes the most degrading effects. When considering ultra-short pulses or when GVD is zero the DOS effects can’t be disregarded. In the propagation of Gaussian pulses was observed that term causes enlargement and asymmetry with oscillations at the pulse front. It was also observed that the greater the absolute value of chirp the most significant DOS effects are.

Regarding DCFs was demonstrated that is possible to recover the initial shape of the pulse when considering the DVG and DOS effects. The FBGS study was initiated with uniform FBGs which present the existence of a stop band where the maximum reflection occurs showing a behavior similar to an optical filter and where the dispersion is zero. However, on the stop band border signal suffers distortion caused by multiple reflections at the grating extremes making impossible the dispersion compensation for the wavelengths of this area. These reflections cause the secondary lateral lobes observed in reflectivity spectrum which is one of the disadvantages of uniform FBGs. However these lobes can be eliminated using apodization techniques as shown for raised sine and Blackman profile. It was verified that the smaller the grating length the wider is the bandwidth but consequently reflectivity will be lower. The reduced bandwidth and high signal distortion at the stop band borders presented by the uniform FBGs led to the development of CFBGs. These by varying the spatial modulation period or the average amplitude of the refractive index of the grating or both simultaneously allows the reflection of the multiple signal wavelengths at different positions of the grating. It was concluded that is possible to compensate the dispersion caused by the propagation in an optical fiber with hundreds of kilometers with a CFBG with length on the order of only dozens of centimeters.

BIBLIOGRAPHY

[1] J. Hecht, City of Light: The Story of Fiber Optics, Oxford

University Press, 2004.

[2] G. P. Agrawal, Fiber-Optic Communications Systems,

Third Edition, John Wiley & Sons, 2002.

[3] C. R. Paiva, Fotónica - Fibras Ópticas, Instituto Superior

Técnico, Abril de 2008.

[4] G. P. Agrawal, Nonlinear Fiber Optics, Third Edition,

Academic Press, 2007.

[5] C. A. F. Marques, Gravação de redes de Bragg avançadas

em fibra óptica, Universidade de Aveiro, Departamento de

Física, 2008.

[6] L. M. P. Marques, Optimização de Processos de Produção

de Sensores de Bragg em Fibra Óptica, Faculdade de

Engenharia da Universidade do Porto, 2008.

[7] B. M. B. Neto, Redes de Bragg Dinamicamente

Reconfiguráveis para Compensação da Dispersão

Cromática, Universidade de Aveiro, Departamento de

Física, 2005.