Optimization of the apodization strength for linearly chirped Bragg grating dispersion compensators in optical fiber communications links

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1 Optimization o the apodization strength or linearly chirped Bragg grating dispersion compensators in optical iber communications links P. FERNÁNDEZ, J.C. AGUADO, J. BAS, F. GONZÁEZ, I. DE MIGUE, J. DURÁN, R.M. ORENZO, E.J. ABRI, M. ÓPEZ Dept. o Signal Theory, Communications and Telematic Engineering. University o Valladolid Campus Miguel Delibes, Cmno. del Cementerio s/n, 47 SPAIN Abstract: - The optimization o apodized linearly chirped iber Bragg gratings in the ield o chromatic dispersion compensation is discussed in terms o group delay response and pulse recompression. To develop this study two prototype scenarios are considered to show the dierent behavior o the compensating devices in unction o the link length. Simulation results or ps gaussian pulses transmitted and recompressed with dierent grating designs over the both links are presented. The maximum deviation error and the regression slope parameters are introduced to compare the quality o the group delay response to establish an optimum design o the apodization strength. Key-Words: - Bragg gratings, dispersion compensation, pulse recompression, apodization, chirp, group delay ripple, compression ratio. Introduction Degradation o transmitted signals due to chromatic dispersion is one o the major limiting actors in long haul optical communication links, since transmission rates are constantly increasing and the loss o optical iber becomes lower. Several techniques have been proposed to achieve dispersion compensation and pulse recompression as prechirped pulse transmission or dispersion shited ibers. However, the irst one does not cancel the dispersion completely, and the second one requires modiying existing iber links. In recent years, there has been increasing interest in dispersion compensating iber Bragg gratings because they are entirely passive and their size, cost and iber compatibility make them very attractive devices[]. Given that the group delay response plays a decisive role in the dispersion compensation behavior [], a detailed study o the design parameters that inluence this delay unction would be helpul to achieve optimum results. In this paper we present a study about the eect o the apodization strength in unction o the total amount o dispersion to be compensated, this is, the iber link length. Two general prototypes has been considered, and we have computed the simulated results or gaussian pulses transmitted and recompressed with the dierent ACFBG s designs. In Section II we develop the dispersion compensation design and in Section III we present and discuss the computed results or the dierent apodization strength proiles in the two scenarios. Finally, In Section IV the main conclusions are presented. Dispersion compensation design For a determined optical link, with an speciied length and dispersion parameter D, we can design a Bragg grating that can achieve the opposite dispersion level in order to cancel this undesirable eect. Some o the parameters as the linear chirp will be determined by the time delay slope required, but others as apodization unction and modulation depth open a wide variety o possibilities to improve the response o the device. The minimum length required to compensate the dispersion introduced by the iber link is [3]: c λd = () n Where c is vacuum speed, n is the reractive index o the iber, and λ is the bandwidth to compensate or chromatic dispersion. In act, is the required length or a uniorm grating. Nevertheless, we can apodize the Bragg grating, but in that case we should use a greater length to compensate the reduction o the coupling strength caused by the apodization proile at the grating ends [4]. In order to establish an optimum proile to acquire the best dispersion compensation, several parameters have been reported in previous works, as the perormance versus the apodization actor or the grating length. [5]. However, the

2 problem about the grating length required to compensate the chromatic dispersion can be directly solved increasing the minimum length required proportionally to the apodization actor <a e <[3]: a e z( z) dz e = = zdz () The smaller the actor, the tighter the apodization proile. This way the equivalent length o the apodized grating is eq = /a e which increases or tight apodization unctions (z), and tends to the minimum length or proiles more similar to nonapodized or square proiles. From the point o view o the kind o the apodization unction several proposals have been made. In [5] it has been concluded that broad lat center and smoothly decaying wings give better grating perormance. On the other hand, in [3] is showed that optimum apodization proiles have not only a lat center region but also edges with continuously decaying slopes. However the optimum perormance will strongly depend on the length o the link, this is, the total amount o dispersion to be compensated. This way, we are not going to compare dierent apodization unctions as sinc, hyperbolic tangent, Blackman, etc., since their characteristics have been studied in the literature. In act, we have chosen a raised cosine proile as expressed in (3) because it is in good a z π ( ) = cos z (3) eq agreement with the previous statements, continuously decaying edges, and we have studied the inluence o the strength o this proile in unction o the total amount o dispersion to be compensated. Each time we change the tightness o the proile the chirp actor will be computed depending o the equivalent length accordingly to [6] 4πn eq F = (4) λ cd B 3 Pulse recompression We are going to analyze two scenarios, a short link o 3 km, and a longer link o km o standard singlemode iber with a second-order dispersion parameter o 7 ps/nm km. The pulses under consideration will be gaussian pulses, deined as: + ic t A (, t) = A exp (5) T Where T is the hal-width at /e intensity point, ixed to ps, and the peak amplitude has been set to A =. It is also possible to analyze prechirped pulses, with a determined value or the chirp parameter C, although or the present work we have limited our study to non-prechirped pulses. We consider the case where the carrier wavelength is ar away rom the zero-dispersion wavelength so that the third-order dispersion is negligible, this way the amplitude o the transmitted pulse can be expressed as [7]: AT A( z, t) = / [ ( )] T iβ z + ic (6) ( + ic) t exp [ T i z( + ic) ] β To show the results or the transmitted pulse the time axis in Fig. and Fig.4 is displayed assuming a reerence rame moving with the pulse t =t-β z where β =/v g. In order to compensate the transmitted pulse or the chromatic dispersion o the iber link, we can consider one o the classical setups where the broadened pulse is recompressed and back relected rom a chirped Bragg grating and extracted with an optical circulator. We have considered or the Bragg grating ê e product a value o 7.8, which is inside the range o maximum restoration, to guarantee a good peak power result. The recompressed pulse is computed in the requency domain as the product o the Fourier transorm o the transmitted pulse amplitude and the relection coeicient o the Bragg grating, obtained through Coupled Mode Theory and computed with a transer matrix method [8]. Finally, the inverse Fourier transorm is calculated to show the proile o the recompressed pulse in the time domain. 3. Short links We start our study with a 3 km standard non-shited iber with a total amount o 5 ps/nm chromatic dispersion as a prototype o relatively short link. To study the eect o the apodization strength in pulse recompression we have compared the non-apodized design with the perormance o three dierent raisedcosine proiles o increasing strength shown in Fig. The computed numerical results are summarized in Table. Obviously, the uniorm grating achieves the shorter length, even reaching the same compression ratio o the apodized ones, C ô =.95. But given the high group delay ripple (mean ripple=3.84 ps in the 3 db band) the recompressed pulse will suer o a

3 noticeable sidelobe level, as can be observed in Fig.(b) Fig. Raised cosine apodization proiles o increasing strength a=.4 (solid, a e =.7), a=(dotted, a e =.6), a=4 (dashed, a e =.37) In Fig.(a) is showed the group delay response as well as the irst degree polynomial that best its the response in a least-squares sense. In order to justiy the not so bad results obtained or a non-apodized grating it is important to note that although the mean dispersion o this device is ps/nm ar away rom the ideal 5 ps/nm o the link, the slope o the linear regression is 58 ps/nm. In other words, the quite high ripple oscillates around a linear response very close to the ideal slope. When we apodize the grating with a raised cosine proile with parameter a=.4, it is obtained that the compression ratio has increased to C ô =, the mean dispersion acquired is 439 ps/nm and the regression slope is 57 ps/nm, not really ar rom the uniorm case, but now the mean ripple is only.47 ps, as can be seen in Fig.3. Besides, the grating length is only one centimeter longer than the uniorm case. Following the study or tighter apodization proiles, some important conclusions can be extracted. First o all, we can not airm that the stronger the apodization, the better pulse recompression. As we can observe in Table, when the apodization actor is increased, the regression slope D r eectively is more closer to the ideal 5 ps/nm, but the mean error is not constantly decreasing, and the Compression ratio does not improve. However, as expected, the grating length increases, which is a very important constraint or the mask process or exposure times. This behavior can be explained since or small apodization actors the ripple is high but with a linear slope, very close to the ideal one Fig.(a) Group delay response (rippled) and linear regression or a 5 km, D =7 ps/nm/km link, nonapodized grating. x x 4 Fig.3.Group delay response (rippled) and linear regression or a 5 km, D =7 ps/nm/km link, raised cosine-apodized grating (a=.4) Fig.(b) Initial, transmitted and recompressed pulse or a 5 km, D =7 ps/nm/km link, non-apodized grating. Uniorm r.c(a=.4) r.c(a=) r.c(a=4) eq 4. cm 5.3 cm 6.64 cm.7 cm R max D r å m C ô Table. Equivalent length, maximum Relectivity, regression slope, mean error rate and compression ratio or uniorm, and raised cosine apodizations (a parameter values:.4,,4)

4 However, or higher apodization actors the group delay is smoother, but not so linear, leading to not so good compression ratios, as happened or the raised cosine with a=. I we extreme the conditions reaching to a very tight proile (a=4), we will obtain again good results, with a regression slope o 54 ps/nm. The prize paid in terms o grating length, however, can be very expensive. The improvement o the behavior rom the a=.4 raised cosine apodization to the a=4 proile, does not justiy a double length o the Bragg grating, at least or relatively short links, specially since the compression ratio with the irst one is as good as the second one. 3. ong links The results o dispersion compensation or longer links can give an extra amount o inormation to design the optimum grating device. For km o standard singlemode optical iber, with the same second-order dispersion parameter 7 ps/nm/km, this is, a total dispersion o 7 ps/nm, we have compared the same our cases o the previous section. Now, ollowing the Bragg grating design steps showed in (), () we have a chirped Bragg grating with ê e =.8. Uniorm r.c(a=.4) r.c(a=) r.c(a=4) eq 7 cm 8.83 cm.6 cm 8.78 cm R max D r å m å d C ô Table. Equivalent length, maximum Relectivity, regression slope, mean error rate, maximum deviation error, and compression ratio or uniorm, and raised cosine apodizations (a parameter values:.4,,4) In this case, we can observe rom Table. that the compression ratio does not improve with the apodization strength, but there are other parameters that predict a higher amount o sidelobe level in the time domain recompressed pulse, as the mean error rate, or the deviation o the regression slope rom the ideal case o 7 ps/nm. The uniorm proile can be directly discarded given the ps o mean error rate, what will degrade the recompressed pulse as can be seen in Fig.4. To choose between lower or higher apodization actors now we have to compare the mean error, and the regression slope. This time, the stronger the apodization, the better results, given that the total amount o dispersion to compensate is higher Fig 4. Initial, transmitted and recompressed pulses or a km, D =7 ps/nm/km link, non-apodized grating. But we have introduced another parameter to establish a better comparison, the maximum deviation error, computed as the maximum deviation rom the delay response to the regression slope in the 3 db bandwidth. This shows us that the a=4 proile has more deviation error (å d =4.47) than the a= (å d =3.85), although the rest o parameters are better. It can be seen in Fig.5(b) and Fig.5(c) how the delay response losses linearity in the second case. Consequently, it is not worth to increase arbitrarily the apodization strength, and consequently the grating length, in order to provide a better perormance, because with middle apodization actors (about.6) both the compression ratio and the sidelobe level give good enough results or signal regeneration [Ou]. The graphic representation o the group delay or the three previous apodization proiles are presented in Fig.5 where it can be observed how the delay is smoother or stronger apodization proiles but begins to loss linearity x 4 Fig.5(a) Group delay response (rippled) and linear regression or a km, D =7 ps/nm/km link, raised cosine-apodized grating (a=.4)

5 9 8 7 unction (a e =.6). In any case, really strong unctions with low apodization parameters (a e <.4) are not justiied in terms o compression ratio or sidelobe level, taking into account the increasing diiculty o the mask process x 4 Fig.5(b) Group delay response (rippled) and linear regression or a 5 km, D =7 ps/nm/km link, raised cosine-apodized grating (a=) x 4 Fig 5(c) Group delay response (rippled) and linear regression or a 5 km, D =7 ps/nm/km link, raised cosine-apodized grating (a=4) 4 Summary We have presented a review o the state-o-the art criteria or the design o optimum apodized linearly chirped iber Bragg gratings or dispersion compensation and extended the study to short distance and long distance scenarios. We have introduced new parameters as maximum deviation error and regression slope to study the behavior o the time delay response that can give extra inormation to select the proper design in unction o dierent requirements. The analysis o two prototype o standard iber links suggests that there is not a general optimum apodization actor or chromatic dispersion compensation, but we can observe that or short links it will be enough with proiles like raised cosine with a=.4 (a e =.79) while or long links it is better to use stronger unctions like the a= 5 Acknowledgments This work is supported by the Spanish Ministry o Science and Technology (Ministerio de Ciencia y Tecnología) under grant TIC-65-P4- and has been developed in collaboration with RETECA. Reerences: [] B.J.Eggleton, K.A. Ahmed, F.Ouellette, P.A.Krug, H.F.iu Recompression o pulses broadened by transmission through km o nondispersion shited iber at.55 ìm using 4 -mmlong optical iber bragg gratings with tunable chirp and central wavelength. IEEE Photonics.Techn.ett. Vol.7,no5,995 [] Jamal, J.C. Cartledge. Variation o the perormance o multispan Gb/s systems due to the Group Delay Ripple o Dispersion Compensating Fiber Bragg Gratings.J.ightwave Tech. vol., no., [3] K. Ennser, M.N.Zervas, R.I.aming "Optimization o Apodized inearly Chirped Fiber Gratings or Optical Communications J. o Quantum Electronics. Vol.34, no. 5, 998 S. [4] D.Benito, M.J.Erro, M.A.Gomez, M.J.Garde, M.A.Muriel, "Emulated single-mode iber optic link by use o a linearly chirped iber bragg grating" J.Selected Topics in Quantum Elec., vol. 5, no.5, 999 [5] D.Pastor, J.Capmany, D.Ortega, V.Tatay, J.Martí "Design o apodized linearly chirped Bragg gratings or dispersion compensation" J.ightwave Tech. Vol.4, no., 996 [6] F. Ouellette., All-iber ilter or eicient dispersion compensation, Opt.ett., vol6, no. 5, pp , 99. [7] G.P.Agrawal. Fiber-Optic Communication Systems Wiley Series Editors, USA 99 [8] M.Yamada, K.Sakuda, "Analysis o almostperiodic distributed eedback slab waveguides via a undamental matrix approach" Applied Optics, vol. 6, no. 6.