Optimization of the apodization strength of linearly

chirped Bragg gratings for span dispersion compen-

sation

P . F E R N A N D E Z * , J . C . A G U A D O , J . B L A S , R . D U R A N , J . D U R A N ,

I . D E M I G U E L , R . L O R E N Z O A N D E . A B R I L

Department of Signal Theory, Communications and Telematic Engineering University of Valladolid,

Valladolid, Spain

(*author for correspondence: E.T.S.I de Telecomunicacion – Universidad de Valladolid, Campus Miguel

Delibes. 47011 Valladolid, Spain. E-mail: patfer@tel.uva.es)

Abstract. We present an optimization process for the apodization strength factor of fiber Bragg grat-

ing dispersion compensators. It will be demonstrated that a proper choice of the apodization strength

factor results in minimum deviation of the dispersion from the required level, maximum reflection

bandwidth and minimum group delay ripple impact, thus leading to a reduction of the Q-factor penalty of

the dispersion compensating system. We also discuss the influence of the fiber link length in the deter-

mination of the optimum apodization strength factor, taking into account that shorter links minimize the

group delay ripple amplitude, whereas the longer ones maximize the bandwidth of interest. It is determined

that a fiber link length of �80 km can balance the requirements of group delay ripple impact and

bandwidth. The results obtained through this analysis allow a faster modeling of the Bragg grating

parameters in order to achieve an improved spectral performance, as well as a cost-effective fabrication

process.

Key words: apodization, dispersion compensation, fiber bragg gratings, group delay ripple, modeling,

optical waveguides

1. Introduction

Apodized and linearly chirped fiber Bragg gratings have proved to be aneffective solution to compensate the chromatic dispersion of dense high bit-rate WDM optical communication systems (Gnauck et al. 1999). Theavailability of mass production techniques added to their inherent advanta-ges, such as polarisation insensitivity, fiber compatibility and the fact of beingpassive and low-loss devices, make them very attractive over competingtechnologies (Eggleton et al. 2001).

As a first step, it is well known that an accurate modeling of the spectralbehavior of the dispersion compensating gratings (DCGs) can improve theefficiency of the design process, as well as the quality of the resulting devices.In this sense, a particular effort has been made to evaluate the benefits of

Optical and Quantum Electronics 36: 57–66, 2004.

� 2004 Kluwer Academic Publishers. Printed in the Netherlands. 57

different apodization profiles in order to find the optimum one, that is, soas to ensure a top-flat reflectivity function with reduced side-lobe level, anda time delay response as linear as possible with minimized ripple impact inthe reflection bandwidth. This way, it has been reported (Pastor et al. 1996)that apodization functions like the sinc or the hyperbolic tangent, with ‘a wideflat region at the center and with non-abrupt decaying slopes toward theedges’ provide an optimum performance. However, the same kind of apo-dization function, as sinc, cosine, gaussian, etc., can be made smoother ortighter with adequate control parameters, obtaining a varying apodizationstrength factor (Ennser et al. 1998a), but preserving the same profile pro-perties. It would be helpful, then, to optimize the strength of those profiles,bearing in mind the linearity of the time delay response, the dispersiondeviation from the required level, the reflection bandwidth, etc., for adetermined optical link.

In this paper we demonstrate that an accurate selection of the apodizationstrength, even for those profiles defined as optimum in the literature, caneffectively improve the results obtained for bandwidth, dispersion and groupdelay ripple impact. We determine the optimum apodization strength range,and we also discuss the influence of the fiber link length in the DCG behaviorin order to establish the most suitable apodization strength in function of theoptical distance.

2. Model description

2.1. GRATING STRUCTURE

The apodized and linearly chirped Bragg grating can be described as follows(Pastor et al. 1996)

nðzÞ ¼ n0 þDn2T ðzÞ mþ cos

2pKB

zþ /ðzÞ� �� �

0 � z � L ð1Þ

where n0 is the refractive index of the unperturbed core, LB is the referenceBragg grating period at the center of the structure and Dn is the peak to peakamplitude of the index change modulation. The taper or apodization func-tion is T(z) and the non-zero dc-index change is represented by the mparameter, which will be considered zero, ideal fabrication case, in theanalysis that follows. The grating phase, /ðzÞ ¼ Fz2=2L2g (Litchinitser andPatterson 1997), describes the linear chirp along the length of the grating Lg,which is measured with the adimensional chirp parameter F.

58 P. FERNANDEZ ET AL.

2.2. PARAMETER DETERMINATION

The minimum grating length Lming required to compensate the accumulated

dispersion of a determined optical fiber link can be calculated as follows(Ouellette 1991)

Lming ¼ 1

2

cn0

� �DkDfLf ð2Þ

where c represents the vacuum speed of light, Lf is the link length, Df is thedispersion parameter of the fiber and Dk is the compensating bandwidth.However, if the grating is apodized, the effective length decreases and it will berequired to increase the grating length in order to compensate the reduction ofthe coupling strength at the grating ends. The parameter that measures thestrength of the apodization profile, and consequently the reduction of theeffective length, is the apodization factor aeff (Ennser et al. 1998a)

aeff ¼R L0 zT ðzÞdzR L

0 zdzð3Þ

where L ¼ Lming is the calculated grating length in Equation (2). This way,

weak-apodized gratings, this is, almost rectangular shape, have aeff fi 1,whereas stronger or ‘tighter’ apodizations have aeff fi 0. Consequently, acorrected grating length determined by the inverse of the apodization factor,Lg ¼ Lmin

g =aeff, will be used. It only remains to determine the chirp factor, F,which is related to the link parameters as follows (Ouellette 1991)

F ¼8pn20L

2g

k2BcDfLfð4Þ

where kB ¼ 2n0LB represents the Bragg wavelength at the center of thestructure.

2.3. CASE OF STUDY

The aim of this study is to determine the optimum apodization strength for adetermined profile, the sinc function

T ðzÞ ¼ sincA2ðz� L=2Þ

L

��������B

!ð5Þ

OPTIMIZATION OF THE APODIZATION STRENGTH OF BRAGG GRATINGS 59

since it has been defined as optimum for dispersion compensation applica-tions (Pastor et al. 1996). The optimization process will be developed usingthe parameters A, B 2 ½1; 4� to acquire a series of different apodizationstrengths from �0.3 to �0.9, and comparing the results obtained whenapplying each one of the profiles to the DCGs. Fig. 1 illustrates the variationof the apodization strength of the sinc profile defined in (5) after the appli-cation of six different strength factors, i.e., for six different values of theparameters A and B.

The Bragg gratings under study will be modeled to compensate the dis-persion introduced by standard single-mode optical fiber links, with disper-sion parameter Df ¼ 17 ps/nm km, over a working bandwidth of 0.4 nmcentered at 1550 nm, since these are typical specifications for dispersioncompensation applications (Pastor et al. 1996; Benito et al. 1999). The linklength, Lf, will be varied from 70 to 120 km, every 10 km. A grating mod-ulation index depth of Dn ¼ 1.2 · 10)4 has been used in order to obtain stateof the art reflectivity levels of 75–95% (Mihailov et al. 1999). The requiredgrating length and chirp of the gratings will be adjusted to compensate thedispersion of each link following the modeling process described in theprevious section.

3. Results and discussion

Once the spectral results have been computed (Erdogan 1997), including timedelay and dispersion for each grating, it would be helpful to have a measure

0 L/2 L0

0.25

0.5

0.75

1

Apo

diza

tion

prof

ile T

(z)

Grating length

Fig. 1. Variation of the apodization profile, sinc function, for six different apodization strength values.

The apodization factors, aeff, correspond to 0.38 (tighter profile, shorter effective length), 0.47, 0.59, 0.68,

0.79 and 0.86 (almost rectangular shape, larger effective length).

60 P. FERNANDEZ ET AL.

of the group delay ripple impact and the deviation of the dispersion from therequired level in order to evaluate their spectral behavior. In this way, thetime delay ripple amplitude is computed with the aid of a linear regression ofthe group delay response, in a minimum mean square error fit sense, over thefull width at half maximum (FWHM) reflection bandwidth. The mean valueof the absolute differences between the actual group delay and its linear fitwill be considered representative of the group delay ripple amplitude. Sincethe ripple period is not constant, it will be also estimated as an averaged valueover the FWHM bandwidth. Regarding the dispersion results, we will esti-mate the deviation of the mean dispersion obtained with each grating fromthe required level, i.e., the accumulated dispersion of each link. All thespectral results have been computed using transfer matrix method algorithmsto solve the Coupled Mode Theory equations (Fernandez et al. 2002).

First of all, it is worth remembering that the stronger the apodization, thelonger the dispersion compensating grating should be, as Fig. 2 shows. Sincegrating length is a major constraint in terms of mask length availability andexposure times (Ennser et al. 1998a), optimum values of aeff should be aslarge as possible, thus avoiding excessively strong profiles. Fig. 3 shows thedeviation of the mean dispersion from the required amount to compensatethe dispersion of each link length, Df · Lf (ps/nm), in function of the apo-dization strength factor. As it can be observed, the mean dispersion valueequals the desired level, that is, zero deviation, for apodization factors be-tween �0.7 and �0.8, becoming closer to 0.8 for increasing link lengths.

With regard to the group delay ripple (GDR) impact, it is shown in Fig. 4that there always exists a determined apodization strength value for each link

0.3 0.4 0.5 0.6 0.7 0.8 0.9 15

10

15

20

25

30

Gra

ting

leng

th (

cm)

Apodization factor

70 km80 km90 km100 km110 km

120 km

Fig. 2. Grating length versus apodization strength for each DCG.

OPTIMIZATION OF THE APODIZATION STRENGTH OF BRAGG GRATINGS 61

length, which minimizes the GDR amplitude, and this value is located be-tween 0.7 and 0.8. Since it is known that the ripple amplitude degrades theperformance of the system (Ennser et al. 1998b), it is important to maintainthis parameter as low as possible. It can also be deduced, that the longer thelink, the more severe the GDR amplitude impact becomes, with anapproximate increase of 0.5 ps for each 10 km of optical fiber.

0.3 0.4 0.5 0.6 0.7 0.8 0.9-50

-40

-30

-20

-10

0

10

20

30

40

50

60

Mea

n D

ispe

rsio

n D

evia

tion

(ps/

nm)

Apodization factor

70 km80 km90 km100 km110 km

120 km

Fig. 3. Deviation of the mean dispersion from the ideal value Df · Lf (ps/nm) versus apodization strength

for each DCG.

0.3 0.4 0.5 0.6 0.7 0.8 0.91

1.5

2

2.5

3

3.5

4

4.5

Mea

n G

DR

am

plitu

de (

ps)

70 km80 km90 km100 km110 km

120 km

Apodization factor

Fig. 4. Mean group delay ripple amplitude versus apodization strength for each DCG.

62 P. FERNANDEZ ET AL.

The mean group delay ripple period variation is analyzed in Fig. 5 forstrength values between 0.7 and 0.85, due to the fact that the period decreasesas the apodization factor increases, and also to focus on the optimum rangethat minimizes the dispersion deviation and the group delay ripple amplitude.It has been reported (Eggleton et al. 2000) that the worst group delay rippleimpact occurs when its period equals the bit rate: 80 pm for 10 Gb/s, 160 pmfor 20 Gb/s, 320 pm for 40 Gb/s, etc. It has also been concluded (Eggletonet al. 2000) that the faster the GDR variation, the minor impact on thesystem, with independence of the modulation format. In this way, we cannotice in Fig. 5 that the GDR period experiences a great decreasing towardshigher apodization strengths; for aeff > 0.8 the period is kept under 20 pm,which can be considered much lower than 80 pm, that is, a GDR variationfast enough to cause minor system impact for any modulation format.

Another interesting result is the FWHM reflection bandwidth. These dis-persion compensating gratings have been modeled to cover a range of 0.4 nmalthough the apodization strength applied can improve or degrade that value.As Fig. 6 shows, there exists a determined apodization factor between 0.7and 0.8 that maximizes the bandwidth for each one of the link lengthsconsidered. Despite the fact that all the designed gratings describe the sametendency with regard to the apodization strength, it is also shown that thelonger the link, the maximum bandwidth acquired, since longer distancesrequire longer grating lengths, resulting in more square-shaped filters withimproved bandwidths. However, it must be noted that the design for 70 kmfiber link achieves a maximum bandwidth of 0.396 approx., less than

0.7 0.75 0.8 0.850

5

10

15

20

25

30

35

40

45

5070 km80 km90 km100 km110 km

120 km

Mea

n G

DR

per

iod

(pm

)

Apodization factor

Fig. 5. Mean group delay ripple period for each dispersion compensating grating in the apodization

strength range of interest 0.7 < aeff < 0.85.

OPTIMIZATION OF THE APODIZATION STRENGTH OF BRAGG GRATINGS 63

0.003 nm below the required value. The next one, the design for 80 km,achieves the required bandwidth of 0.4 nm for all the apodization factorsinside the range 0.7 < aeff < 0.8, and the following ones such as 90, 100, 110and 120 km, achieve improved values. However, since the initial conditionwas a FWHM bandwidth of 0.4 nm, and given that the results obtained formean group delay ripple amplitude recommend shorter links, 80 km canmeet the requirements for both parameters. This choice is reinforced by thefact that shorter link lengths require shorter gratings, reducing the cost of thefabrication process.

Owing to the previous analysis, we can state that, in general, apodizationstrength factors between �0.7 (A ¼ 1, B ¼ 1.75) and �0.8 (A ¼ 1, B ¼ 3)optimize the behavior of apodized and linearly chirped dispersion compen-sating gratings with independence of the link length. Comparing the relativebenefits of the different link lengths considered, it has been shown thatshorter links designs minimize the GDR amplitude impact but the longerones maximize the bandwidth of interest.

Finally, to justify the results, the Q-factor penalty of the system due todispersion is evaluated as a function of the apodization strength by means ofsimulation (using OptSim 3.3). We have simulated a 20 Gb/s system withnon-return-to-zero (NRZ) pseudorandom data and external modulation,injecting an average power of )5 dBm in the optical fiber. The dispersionparameter of the fiber link is 17 ps/nm km, and non-linear effects were notsimulated in order to focus on dispersion penalty. The signal is received by aPIN photodetector and electrically filtered by a 11 GHz bandwidth third-order Bessel filter. To carry out the analysis of the Q-factor penalty due to

0.3 0.4 0.5 0.6 0.7 0.8 0.90.35

0.36

0.37

0.38

0.39

0.4

0.41

0.42

0.43

FW

HM

Ban

dwid

th (

nm)

70 km80 km90 km100 km110 km

120 km

Apodization factor

Fig. 6. FWHM Bandwidth for each dispersion compensating grating as a function of the apodization

strength.

64 P. FERNANDEZ ET AL.

dispersion, we first simulated a system consisting of a span of optical fiberfollowed by a dispersion compensating grating, and measured the Q-factor.Next, we obtained the Q-factor in a system where the dispersion compensatorwas extracted and the fiber was substituted by an ideal attenuator with thesame loss of the link analyzed. Hence, dispersion effects were not presentwhen making this measurement. The difference between the two measuredvalues is the Q-factor penalty due to dispersion. Links of 80, 100 and 120 kmwere evaluated when using dispersion compensating gratings with aeff in therange from 0.35 to 0.86. As shown in Fig. 7, the minimum Q-factor penaltywas obtained for compensating gratings with apodization factors in theinterval 0.7 < aeff < 0.8. These results are therefore consistent with thespectral analysis previously performed, confirming that an apodization factorbetween 0.7 and 0.8 effectively maximizes the equalization performance ofthese devices.

4. Summary

This paper has demonstrated that a proper selection of the apodizationstrength for DCGs can improve their spectral behavior obtaining minimizedgroup delay ripple impact, and minimum deviation of the mean dispersionfrom the required value. A range of optimum apodization strengths,0.7 < aeff < 0.8, has been determined for standard optical links between 70and 120 km. It has also been shown that there exists a compromise betweenGDR amplitude impact and the reflection bandwidth for this kind of devices,

0.3 0.4 0.5 0.6 0.7 0.8 0.91.5

2

2.5

3

3.5

4

4.5

5

Q-F

acto

r P

enal

ty (

dB)

80 km

100 km

120 km

Apodization factor

Fig. 7. Q-factor penalty obtained as a function of the apodization strength of the dispersion compensating

gratings.

OPTIMIZATION OF THE APODIZATION STRENGTH OF BRAGG GRATINGS 65

since longer gratings achieve improved results for bandwidth but at the ex-pense of a more serious GDR amplitude impact. In this way, it has beenconcluded that, a span length of 80 km can meet the requirements forbandwidth and GDR amplitude impact in standard optical links. TheQ-factor penalty analysis confirms that an apodization factor between 0.7and 0.8 maximizes the benefits of the dispersion compensation grating.

Acknowledgements

This work is supported by the Spanish Ministry of Science and Technology(Ministerio de Ciencia y Tecnologıa) under Grant TIC2000-0265-P4-02 andhas been developed in collaboration with RETECAL.

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