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Ultrafast all-optical integrator based on a fiberBragg grating: proposal and design

Miguel A. Preciado* and Miguel A. Muriel

ETSI Telecomunicacion, Universidad Politecnica de Madrid (UPM), 28040 Madrid, Spain*Corresponding author: ma.preciado@upm.es

Received April 4, 2008; revised April 29, 2008; accepted May 1, 2008;posted May 8, 2008 (Doc. ID 94707); published June 13, 2008

We demonstrate a simple technique for the implementation of an all-optical integrator based on a uniform-period fiber Bragg grating (FBG) in reflection that is designed to present a decreasing exponential impulseresponse. The proposed FBG integrator is readily feasible and can perform close to ideal integration of few-picosecond and subpicosecond pulses. 2008 Optical Society of America

OCIS codes: 060.2340, 060.3735, 200.4740, 230.1150, 320.5540, 320.7080.

An optical integrator is a recent concept proposed asa basic building block of the ultrafast all-optical sig-nal processing system. It performs the time integralof the intensity (incoherent integrator [1]) or the com-plex envelope (coherent integrator [27]) of an arbi-trary optical input signal, finding concrete applica-

tions as a dark-soliton detector and pulse shaper[24]. Passive optical integrators [57] do not requireany gain element (active medium) and can performclose to ideal integration, but they are not suited forall signals and applications.

A fiber Bragg grating (FBG) is a low-cost, all-fiber,passive optical device and has been proposed as anoptical integrator [6,7]. In [6] it has been demon-strated that a transmission phase-shifted FBG in a very strong coupling regime (reflectivity extremelyclose to 100%) can approximate the ideal integratorspectral response over a certain limited bandwidth(below a few hundreds of gigahertz). In [7] a weakcoupling uniform FBG (approximately rectangular

impulse response) exploits the property that the tem-poral convolution of an arbitrary input signal with arectangle signal can be considered as the temporalintegration of the input signal over a limited timewindow. Unfortunately, the output signal inevitablypresents not only a single integrated pulse, but also asecond integrated pulse after the temporal window ofvalidity.

In this Letter we propose an optical coherent pas-sive integrator based on a uniform period FBG, de-signed to present a decreasing exponential impulseresponse in reflection. The proposed method is not re-stricted to a weak or strong coupling regime and per-forms a close to ideal integration of few-picosecond

and subpicosecond pulses. In the remainder of thisLetter we explain the theoretical basis of this methodand, as an example, a FBG integrator is designed,numerically simulated, and applied over several in-put signals. Moreover, we also compare the accuracyand energy efficiency of several FBGs of differentlengths. Finally, we summarize and conclude ourwork.

In the following, temporal signals are representedas complex envelopes with 0 as the central carrierangular pulsation and spectral signals are repre-sented in the base-band angular pulsation =opt0, where opt is the optical angular pulsation. The

temporal operation of an ideal optical coherent tem-poral integrator can be expressed as foutt=

t find, where fint and foutt are the complexenvelopes of the input and output of the system, re-spectively, and t is the time variable. We can also ex-press this in the frequency domain as Fout

=Fin/j, where Fin and Fout are the spec-tral functions of fint and foutt, respectively, and j=11/2 is the imaginary unit. Thus, the spectraltransfer and impulse responses of the ideal integra-tor are, respectively,

Hideal =Fout/Fin = 1/j , 1

hidealt = ut, 2

where ut is the unit step function. Equation (1) im-plies infinite gain at =0 and finite Fout is only ob-tained if Fin= 0=0. Therefore, a real system can-not perform this ideal response, even in case of an

active device. However, Hideal can be accuratelyapproximated with a complex Lorentzian function[8], which corresponds to a decreasing exponentialimpulse response in the temporal domain,

Haprx =A/j+ 1 , 3

haprxt =A exp t/ut, 4

where A and are amplitude and time constants, re-spectively. It is worth noting that this approximationis commonly used in electronics passive RC integra-tors. From Eqs. (1) and (3), we can observe that highvalues imply an accurate integration operation. In-deed, t must be satisfied for accurate operation,where t is the input pulse width. However, since wehave a passive device, Haprx=0 =A1A1 /,so high implies low A, and from Eq. (4) it implieslow haprxt intensity. From foutt=finthaprxt, thisfinally leads to a low output intensity, where the sym-bol indicates the temporal convolution operator.Hence, it is necessary to reach a trade-off solution be-tween accuracy and energy efficiency.

A FBG can be designed to exhibit a decreasing ex-ponential impulse response. Under the weak cou-pling regime, the grating profile can be easily de-duced as a spatial decreasing negative exponential

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function (Born approximation). For the strong cou-pling regime, an inverse scattering algorithm [9,10]must be applied to synthesize the corresponding grat-ing profile. It is worth noting that, in the weak cou-pling regime, the grating length L is of in the order of6c/2nav, where the exponential function has beensupposed limited to a duration of 6, c is the speed oflight in a vacuum, and nav is the average refractiveindex of the grating. Thus, from t we deducethat L6t c/ 2nav must be satisfied for accurate op-eration in the weak coupling regime. Regarding thestrong coupling regime, we have to take into accountthat the required grating obtained from inverse scat-tering is usually several times longer than in theweak coupling regime, depending on the maximumreflectivity.

As an example we design a uniform period FBGwith a maximum reflectivity of 80%, for which theimpulse response is a negative exponential, as de-scribed in Eq. (4), with a decay time of=200 ps anda central frequency of0/2=193 THz. The FBG pre-sents an effective refractive index of the mode in the

unperturbed fiber neff=1.452, L = 24 cm, and a gratingperiod of 0 =534.88 nm, which corresponds to aBragg frequency of 193 THz. By applying an inversescattering algorithm we obtain a readily feasible cou-pling coefficient, z, shown in Fig. 1, with a maxi-mum coupling coefficient of max=43.225 m

1. Thespectral response in reflection of the FBG is numeri-cally obtained and represented in Fig. 2. It presents agood agreement with the ideal response, except forcentral frequencies, where the ideal integrator re-sponse tends to infinity.

Figure 3 shows the numerically obtained outputtemporal waveforms for different input temporalwaveforms, comparing ideal and designed FBG inte-

grator, where all the considered input signals arespectrally centered at 0. In Figs. 3(a) and 3(b) weconsider the input waveforms of the first-time deriva-tive of a 1 ps transform-limited Gaussian opticalpulse and a 1 ps transform-limited Gaussian opticalpulse respectively, where width is expressed as theFWHM. As it can be observed, the FBG integrator re-sult is in good agreement with the ideal integrator re-sult. In the example of Fig. 3(b), the ideal integratedsignal presents a not-null constant value in t. How-ever, it is obvious that the output intensity of thispassive device cannot be a not-null constant in t,and the FBG output signal presents a decay time

equal to (200 ps in this example). Fig. 3(c) shows anexample of the first-time derivative of a 100 psGaussian optical pulse as the input temporal wave-form, which is too long to be correctly processed bythe FBG integrator, since the accurate operation con-dition t is not satisfied.

Finally, in Fig. 4, we compare the accuracy and en-ergy efficiency of eight FBG integrators with lengths

of 24, 12, 6, 3, 1.5, 0.75, 0.375, and 0.1875 cm, all ofthem with the same maximum reflectivity of 80%when applied over the first derivative of a 10 psGaussian pulse. The energy efficiency of the integra-tor is represented as the ratio of the output signal en-ergy to the input signal energy in decibels, and theaccuracy of the integrator has been calculated as thedegree of similarity between the FBG and the idealintegrator output signals, which can be estimatedwith the normalized cross-correlation coefficient C as[8]

C = max

fout,FBGtfout,ideal* tdt

fout,FBGt2dt

fout,idealt2dt1/2

,5

where fout,FBGt, and fout,idealt are the complex enve-lopes of the output signals corresponding to the FBGand ideal integrator, respectively, and * denotes thecomplex conjugate operator. As can be seen in Fig. 4,longer lengths imply higher accuracy but also lowerenergy efficiency, so a trade-off solution must beadopted for each case.

Fig. 1. Coupling coefficient function of the FBG integrator,obtained by an inverse scattering algorithm.

Fig. 2. Magnitude and phase of the spectral response cor-responding to the designed FBG (solid curve), and to anideal integrator (dotted line).

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In conclusion, in this Letter we have presented a very simple optical integrator based on a uniform-period FBG, designed to present a negative exponen-tial impulse response in reflection. As an example wehave designed a FBG integrator with 80% of maxi-mum reflectivity and have applied it over several in-put signals. We have also compared the accuracy andenergy efficiency of several lengths of FBG integra-tors and conclude that a trade-off grating lengthmust be selected, depending on the concrete applica-tion. Some of the lengths of the FBGs analyzed arelonger than typical (approximately millimeters), but

they are within current technology [11,12]. The pro-posed method is readily feasible, presents the inher-ent advantages of FBGs without the requirement of astrong or weak coupling regime to work, and is spe-

cially suited to few-picosecond and subpicosecondpulses. As main drawbacks, this approach may re-quire very long FBGs for accurate integration of longpulses (100 ps or longer); it also presents the intrinsiclimitations of passive integrators.

This work was supported by the Spanish Ministe-rio de Educacion y Ciencia under Plan Nacional deIDI TEC2007-68065-C03-02.

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Fig. 3. Numerical simulations results, where input pulsesof plots (a)(c), respectively, are the first-time derivative ofa 1 ps Gaussian pulse, a 1 ps Gaussian pulse, and the first-time derivative of a 100 ps Gaussian pulse (too long to becorrectly processed). Plots show the temporal waveforms ofthe input pulse (dashed curves), and output pulse corre-sponding to FBG (solid curves) and ideal (dotted curves) in-tegrators, which are hardly distinguishable in plots (a) and

(b).

Fig. 4. Measures of energy efficiency (squares, dottedcurve) and normalized cross correlation coefficient (circles,dashed curve) that estimate the similarity between the out-put pulse and the ideal integrated pulse and represent theintegration accuracy. Eight FBG integrators applied to thefirst derivative of a 10 s Gaussian pulse are represented,with lengths of 24, 12, 6, 3, 1.5, 0.75, 0.375, and 0.1875 cm

and a maximum reflectivity of 80%.

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