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Design and Modeling Tool for Chromatic Dispersion Compensation with Fiber Bragg Gratings

Design and Modeling Tool or Chromatic Dispersion Compensation with Fiber Bragg Gratings P. FERNÁNDEZ, J.C. AGUADO, J. BLAS, F. GONZÁLEZ, I. DE MIGUEL, J. DURÁN, R.M. LORENZO, E.J. ABRIL, M. LÓPEZ Dept.
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Design and Modeling Tool or Chromatic Dispersion Compensation with Fiber Bragg Gratings P. FERNÁNDEZ, J.C. AGUADO, J. BLAS, F. GONZÁLEZ, I. DE MIGUEL, J. DURÁN, R.M. LORENZO, E.J. ABRIL, M. LÓPEZ Dept. o Signal Theory, Communications and Telematic Engineering. University o Valladolid Campus Miguel Delibes, Cmno. del Cementerio s/n, 4711 SPAIN Abstract: - Given the great impact o iber Bragg gratings in DWDM communication networks we present a MatLab based graphical interace to design and model the FBG parameters to compensate the optical iber link chromatic dispersion. This tool allows the spectral analysis o Apodized and Linearly Chirped FBG s in general as well as the design o the grating device that can compensate or the chromatic dispersion o an speciied iber link. The spectral characteristics o the designed grating are shown as well as an estimation o the deviation rom the ideal theoretical compensator. The application is really cost-eective rom the computational point o view and inally, it can also be displayed the time domain original pulse to be transmitted, the dispersion broadened pulse and the recompressed one as well as the compression ratio, to check the behavior o the designed system. Key-Words: - Bragg gratings, dispersion compensation, pulse recompression, apodization, chirp, modeling. 1 Introduction The chromatic dispersion is one o the major limiting actors in long haul optical communication links, since transmission rates are constantly increasing. Several techniques have been proposed to or implementing dispersion compensation and pulse recompression as prechirped pulses 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 linearly chirped iber Bragg gratings because they are entirely passive and their size, cost and iber compatibility make them very attractive devices[1]. In this paper we present a graphic tool with three main interaces. In Section II we describe the Characterization view where it is possible to analyze the spectral response o apodized and linearly chirped iber Bragg gratings in general, computing relectivity, phase, time delay and dispersion responses. A determination o quality parameters as eective length or 3 db bandwidth are also shown. Section III introduces the Dispersion Compensation interace which allows the design o the Bragg grating device that can compensate or the chromatic dispersion o an optical iber link with a determined length and second-order dispersion parameter. Several kinds o apodization can be perormed to reduce the ripple level o the time delay response, and it is also computed the deviation range rom the ideal compensator. In section IV we analyze the Pulse Recompression view where a gaussian pulse o a determined duration can be transmitted along a selected optical link to observe its broadening due to chromatic dispersion and its recompression ater being relected rom the Bragg grating compensator designed. To conclude, in Section V we discuss the results obtained with the present design and modeling tool. Spectral Characterization To analyze the behavior o Fiber Bragg Gratings it is necessary to have inormation o its spectral characteristics as relectivity, phase, and time delay response[]. Given that the apodization technique is widely employed or both reducing the sidelobe level o the relectivity response and the time-delay ripple o the group-delay response, it would be helpul to include the possibility to apodize the grating with a wide variety o proiles. To allow a comparison between dierent schemes, it is computed an apodization actor, related to the eective length o the grating, to measure the tightness o the proile..1 Parameter Determination The application interace allows the user to enter the desired design parameters that deine the grating structure: The reractive index n, the index modulation depth n, the length o the grating L, the apodization unction a (, and the Bragg central wavelength λ. To compare the behavior between dierent apodization proiles, the apodization actor a e =L e /L will be computed. This parameter measures how tight is the apodization proile, the smaller the value, the tighter the apodization proile. When the parameter is close to the unit, the eect o the apodization is less signiicant, more similar to a uniorm non-apodized grating. In order to compute more realistic simulations it is possible to have into account the dc-averaged index change through the m parameter. In the ideal case m=, but more realistic simulations can use a positive value, m 1, to give a more accurate approximation. Although recent articles present new possible techniques that can create gratings o a much higher quality and small eature size than the mask process [3], the reerred parameter is an eicient way to simulate non-ideal cases. The general expression o the induced reractive index that includes the previous parameters can be expressed as: π n( = n 1+ n( m + cos + φ( (1) Λ Where n(= n a (/n is the slowly varying envelope o the induced index change that includes the eects o the index modulation depth, apodization unction and reractive index. The chirp actor F determines the phase term in (1) ollowing the well known relation φ(=fz /L. Finally, we must set the reerence Bragg wavelength o the grating, λ which is directly related to the grating period at the center o the structure as Λ = λ /(n ) as well as the range o wavelengths where we want to evaluate the spectral response o the iber grating. second one, known as Transer Matrix Method, because it is simple to implement, provides suicient accuracy and it is the astest method, what is a very important constraint in this type o modeling applications. This way, the grating can be considered as a number o uniorm sections [5], each one described by a x matrix that deines the propagation through it. Multiplying them we can obtain a single matrix that describes the whole grating. The number o sections M determines the level o accuracy but there exists an upper limit, given that CMT is only valid i the uniorm section length is much higher than the local period o the structure, this is z λ, that implies[6] M n L/λ () Once the parameters o the desired grating have been selected, the application tool can solve and calculate the spectral response. There are three possible levels o accuracy rom minimum to maximum, where the number o wavelengths to compute the response is increased to accommodate to dierent computational requirements. From the relection coeicient o the grating ρ(λ), it is showed the relectivity spectra R(λ)= ρ(λ) which can be displayed in both linear and logarithmic scale. There are two unctions obtained rom the phase o the relectivity coeicient θ ρ (λ) phase(ρ(λ)) that oer very important inormation about the grating response: the Group Delay τ ρ (λ) and the second-order dispersion coeicient d ρ (λ) : dθρ λ dθρ τ ρ = = (3) dω πc dλ dτ ρ πc d θ ρ d ρ = = (4) dλ λ dω At this point, with the calculation process completed, the application can represent any o the described spectral responses.. Algorithm We have chosen Coupled Mode Theory (CMT) or the analysis and modeling o iber Bragg gratings because it is straightorward, intuitive and one o the most important tools to understand the main optical properties o gratings. Besides, it gives quantitative inormation about diraction eiciency and spectral dependence o iber gratings and can model with a high level o accuracy the optical properties o interest [4]. The CMT theory leads to a pair o dierential coupled equations that deine the relectivity o the structure, that can be computed ollowing two dierent approaches, direct numeric integration or piecewise-uniorm techniques. We have chosen the 3 Dispersion Compensation Design Given recent advances in the ield o chromatic dispersion compensation with iber Bragg gratings [7],[8], we present a graphic interace dedicated to the design and modeling o this kind o devices. For a determined optical link, with an speciied length L 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 [9]: c λd L L = (5) 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, L is the required length or a uniorm grating, but or apodized gratings we should use a greater length to compensate the reduction o the coupling strength caused by the apodization proile at the grating ends [1]. The application calculates the real or eective length o the grating L e =a e L in unction o the apodization proile selected. Then, it only remains to calculate the linear chirp actor parameter F needed to compensate the dispersion o the iber link [11] that is computed according to 4πn Le F = (6) λ cd L B In case o designing a dispersion compensator there is an special interest in the linearity o the group delay response. To evaluate the quality o the grating, it would be helpul to compare the real group delay to the ideal linear response. One way to aord this study is to make a linear regression o the group delay along the 3 db bandwidth. This option can be selected and the linear regression will be showed in the same graphic view, as can be observed in Fig.1 Moreover, to complete this study, we can estimate the deviation between them calculating the mean error rate, whose value is showed in a numeric display. Where A is the peak amplitude, and T is the halwith at the 1/e intensity point. The parameter C ixes the linear requency chirp o the pulse, (in case o pre-chirped pulses). We consider the case where the carrier wavelength is ar away rom the zerodispersion wavelength so that the third-order dispersion is negligible, this way the amplitude o the transmitted pulse can be expressed as[1] AT A( z, t) = 1/ [ ( )] T iβ z 1 + ic (8) ( 1+ ic) t exp [ T i z( 1+ ic) ] β To compensate the transmitted pulse or the chromatic dispersion o the iber link, we can consider the classical setup where the broadened pulse is recompressed and back relected rom the Bragg grating and extracted with an optical circulator. 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, and then reconverted to the time domain. The time axis is displayed assuming a reerence rame moving with the pulse t =t-β 1 z where β 1 =1/v g. 4 Pulse recompression As we have mentioned in the previous section, there is not a single Bragg grating device that can compensate or the chromatic dispersion o an optical iber link. Some characteristics, as the time delay ripple, or the relectivity level, will have a very important inluence in the recompression o the optical pulses transmitted, and can be controlled by the design requirements. Consequently, why we have developed a new interace to observe the eect o the Bragg grating compensator on the time domain pulses allowing the study o the optimum behavior according to dierent criteria (peak power, RMS width, etc.) We consider the propagation o gaussian input pulses, whose initial amplitud can be expressed as 1+ ic t A (, t) = A exp (7) T Fig 1(a) Group delay or a uniorm grating (Grating length L=3.5 cm, chirp actor F=54.17) and linear regression computed or the 3 db bandwidth. Fig 1(b) Group delay or a raised cosine-apodized grating (Grating length L=6 cm, chirp actor F=154.56, apodization actor a e =.59) and linear regression computed or the 3 db bandwidth. To observe the eect o the recompression o pulses we can analyze an standard iber link o 5 km long, with second order dispersion parameter β = ps /km. I gaussian pulses with a ull width at hal maximum duration τ FWHM =4 ps are transmitted, they will be broadened to approx. 136 ps. At that moment we can recompress the pulses with a linearly chirped Bragg grating ollowing the dispersion compensation process o the previous section. I we do not apodize the grating, we will beneit o a short grating, with only 35 mm long but, on the other hand, the time delay response will suer o a high ripple level as can be observed in Fig. 1(a). Nevertheless, we can select a raised cosine apodization, obtaining an smoother time delay response as can be seen in Fig.1(b). Now, we can compare the results obtained in Fig.. In the irst case, the uniorm grating, we have recompressed the pulse to 48 ps, this is, a compression ratio o.83. In the second case, with the apodized grating, the pulse have been recompressed to 44 ps reaching a compression ratio o.9, which is good enough or signal regeneration. In terms o amplitude, it has a better behavior the irst dispersion compensator, due to the act that uniorm gratings reach higher levels o maximum relectivity but with the disadvantage o appearing a noticeable sidelobe level. The mean error rate rom the dispersion achieved to the ideal linear response is ten times higher in the case o the non-apodized design Fig.(a) Initial pulse (solid line), transmitted pulse (dashed line) and recompressed pulse (dotted) Grating length L=6 cm, chirp actor F= Fig.(b) Initial pulse (solid line), transmitted pulse (dashed line) and recompressed pulse (dotted) Grating length L=3.5 cm, chirp actor F= Summary We have presented a design and modeling tool that allows pulse recompression with a iber Bragg grating, one o the most versatile components in optical communications. In order to cover all the design steps, including an exhaustive analysis o Bragg gratings spectral responses, we have developed a three interace system with an intuitive modeling process. It allows a detailed study, analysis and design o iber Bragg gratings, specially as dispersion compensators, with a high level o accuracy but nonrestrictive rom the computational point o view. The pulse recompression eature adds inormation to the dispersion compensation design oering graphical time-domain results and quality parameters as compression ratio and sidelobe level. 6. Acknoledgments 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 RETECAL. Reerences: [1] B.J.Eggleton, K.A. Ahmed, F.Ouellette, P.A.Krug, H.F.Liu Recompression o pulses broadened by transmission through 1 km o nondispersion shited iber at 1.55 ìm using 4-mmlong optical iber bragg gratings with tunable chirp and central wavelength. IEEE Photonics.Techn.Lett. Vol.7,no5,1995 [] F. Alonso. Study o Fiber Bragg Gratings as Dispersion Compensators. Final Career Project, University o Valladolid 1 [3] Britain s Payne May have Key to All-Optical Routers IEEE Spectrum, Novembre 1 [4] H. Kogelnik. Theory o Optical waveguides Guided-Wave Optoelectronics. T. Tamir. Ed. New York, Springer-Verlag, 199 [5] M.Yamada, K.Sakuda, Analysis o almostperiodic distributed eedback slab waveguides via a undamental matrix approach Applied Optics, vol. 6, no. 16. [6] T.Erdogan, Fiber grating spectra J.Lightwave Tech. vol.15, no.8, 1997 [7] J. E.Sipe, C. M. de Sterke. Dispersion o Optical ibers with Far-O resonance gratings. J.Lightwave Tech. vol.19, no.1, 1 [8] Jamal, J.C. Cartledge. Variation o the perormance o multispan 1 Gb/s systems due to the Group Delay Ripple o Dispersion Compensating Fiber Bragg Gratings.J.Lightwave Tech. vol., no.1, [9] K. Ennser, M.N.Zervas, R.I.Laming Optimization o Apodized Linearly Chirped Fiber Gratings or Optical Communications J. o Quantum Electronics. Vol.34, no. 5, 1998 S. [1] 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, 1999 [11] F. Ouellette. All iber ilter or eicient dispersion compensation. Opt. Letters. Vol.16, no 5, [1] G.P.Agrawal. Fiber-Optic Communication Systems Wiley Series Editors, USA 199
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