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Design of an ultrafast all-optical differentiatorbased on a fiber Bragg grating in transmission

Miguel A. Preciado* and Miguel A. Muriel

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

Received July 23, 2008; revised September 4, 2008; accepted September 5, 2008;posted September 30, 2008 (Doc. ID 99330); published October 21, 2008

We propose and analyze a first-order optical differentiator based on a fiber Bragg grating (FBG) in trans-mission. It is shown in the examples that a simple uniform-period FBG in a very strong coupling regime(maximum reflectivity very close to 100%) can perform close to ideal temporal differentiation of the complexenvelope of an arbitrary-input optical signal. 2008 Optical Society of America

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

A first-order optical temporal differentiator is a de- vice that provides the first-order derivative of thecomplex envelope of an arbitrary input optical signal.This operation is performed on optical devices at op-eration speeds several orders of magnitude over elec-

tronics. These devices may find important applica-tions as basic building blocks in ultra-high-speed all-optical analogdigital signal-processing circuits [1].Moreover, optical differentiators are of immediate in-terest for the generation of optical monocycle pulsesfrom input-optical Gaussian pulses for ultrawide-band systems, recently emerging as a solution for fu-ture wideband personal access networks [24], andgeneration of a HermiteGaussian temporal wave-form from an input Gaussian pulse to synthesize anytemporal shape by superposition [5]. Several schemeshave been previously proposed based on integrated-optic transversal filter [1], long-period fiber gratings[6], fiber Bragg gratings (FBGs) [710], and two-arm

interferometers [11].In this Letter, we propose and analyze a first-orderoptical differentiator based on an FBG operating intransmission, with its inherent advantages (all-fiberapproach, low insertion loss, and the potential for lowcost). Regarding other in-fiber differentiators [610],our approach requires only a single FBG without anyadditional elements (optical circulator, coupler, FBG,or claddingcore mode converter). It is worth notingthat, although FBGs are typically used in reflection,using them in transmission offers interesting proper-ties. First, since the use of a coupler or circulator isnot required, the energy efficiency is increased, andthe cost and complexity of the system are reduced.

Second, we have less sensitivity of the phase re-sponse to grating-fabrication errors in transmissionthan in reflection mode [12,13].

In the remainder of this Letter we explain the the-oretical basis of this method, and, as example, a FBGdifferentiator is designed, numerically simulated,and applied over several input signals. Moreover, wealso compare the accuracy and length of several FBGdifferentiators. Finally, we summarize and concludeour work.

The temporal operation of a first-order differentia-tor can be expressed as foutt=dfint/ dt, where fintand foutt are the complex envelopes of the input and

the output of the system, respectively, and t is thetime variable. We can also express this in frequencydomain as Fout=jFin, where Fin and Foutare the spectral functions offint and foutt, respec-tively, is the base-band angular pulsation i.e.,

=opt0, opt is the optical angular pulsation, 0 isthe central angular frequency of the signals, and j=11/2 is the imaginary unit. Thus, the spectral re-sponse (SR) of the ideal first-order differentiator is

Hdiff=Fout/Fin=j, the phase of which pre-sents a -phase shift at =0, and the magnitude ofwhich is Hdiff= .

On the other hand, it is well known that the SRmagnitude and phase of an FBG in transmission arerelated by means of the Hilbert transform [12,14]:

argHT = HTlnHT, lnHT = C0

+ HT1argHT , 1

where HT{} stands for the Hilbert transform, ln de-notes the natural logarithmic function, and C0 is anarbitrary real number. Let us suppose a local -phaseshift in HT at =0 in a certain interval Wwhere we can approximate argHT / 2sign, with W0 the radius of the inter-val. We can estimate lnHT locally for fromEq.(1), applying the Hilbert transform integral [15]:

lnHT = C0 +1

argHT

d = C0

+1

argHT +

d C0

+1

I

W

W /2sign +

d = C0

+I

ln/W = lnC1 , 2

where is a sufficiently small value in the sense thatwe can approximate W

W argHT+/dW

W / 2sign+/d and W argHT

+/d+W argHT+/dI, with I

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=W argHT/d+W

argHT/d; C1 is apositive real number (arbitrary, since it contains thearbitrary C0), and it is implicitly supposed that theCauchy principal value of the improper integralsmust be taken. From the positive solution of Eq. (2),we finally deduce HTC1. Thus, sinceargHT is uniquely related to lnHT [12], wecan expect that if HT is locally satisfied, wealso obtain the desired -phase shift at =0, andtherefore the entire phase (neglecting constant andlinear terms) and proportional amplitude of the first-order differentiator SR in a certain bandwidth. It isworth noting that taking the negative solution fromEq. (2) leads to HT

1, which agrees with thespectral -phase shift observed in [16] when approxi-mating the integrator SR magnitude with a phase-shifted FBG in transmission.

Since the SRs in reflection and transmission are re-lated by HR=1 HT2, the objective SR of theFBG in reflection must locally satisfy HR=1 C12 (a semicircumference function). Severalapproximations have to be done in order to achieve

HR with a feasible FBG. First, we have a finitebandwidth in a real system. Moreover, in order to re-duce the complexity of the grating profile required,we propose Gaussian and Lorentzian functions [14]as approximation functions of HR in the limitedbandwidth. In Fig. 1, the resulting approximationand ideal bandwidth-limited functions are repre-sented. Moreover, we have to take into account that aperfect zero in transmission is impossible with anFBG, but a transmission dip can be imposed. Finally,in the grating design, we have to apply an inversescattering algorithm [17] to obtain the grating pro-file.

As example we design an uniform-period FBG in

transmission on the ideas introduced above, wherewe assume a central frequency 0/ 2=193 THz;we use the Gaussian approximation function inHR with a FWHM of 0.56 THz, and we impose atransmission dip 20 logHT= 0 =60 dB (maxi-mum reflectivity of 99.9999%). By applying an in-verse scattering algorithm, we obtain the coupling co-efficient z, which is represented in Fig. 2, in three

different scales for a clear visualization. Note thatthe maximum coupling coefficient maxz=6555.1 m1, is a very high coupling value but iswithin current technology. As can be seen, z con-sists of a strong peak at the beginning and anasymptotic decay at the back, the end value of whichis 16.67 m1 (0.254% of the maximum value). It isworth noting that z must not be excessively spa-tially limited at the back in order to obtain the de-sired transmission dip in =0. Thus, the resulting

uniform-period FBG has a length ofL =10 cm, an av-erage refraction index nav=1.452, and a grating pe-riod of0 =534.888 nm. Figure 3 shows the magni-tude and phase of the FBG SR in transmission,

HT, obtained from numerical simulation. As it canbe seen, the phase of HT presents the desired-phase shift in =0. In the operation bandwidth,which can be estimated as 400 GHz, the magnitudeof HT is approximately proportional to Hdiff,and the phase is approximately linear (nondistorting,pure delay).

Figure 4 shows the numerically obtained outputtemporal waveforms for different input temporalwaveforms, comparing ideal and designed FBG dif-

ferentiator, where all the considered input signalsare spectrally centered at 0. In Figs. 4(a) and 4(b),we consider the input waveforms of a 7 ps transform-limited Gaussian optical pulse (with a correspondingspectral width of 126.08 GHz) and its first-order de-rivative, respectively, where both spectral and tempo-ral widths are expressed as FWHM. As can be ob-served, the FBG differentiator results are in verygood agreement with the ideal differentiator results.The energy efficiencies, calculated as the ratio of theoutput signal energy to the input signal energy indecibels, are 37.065 and 26.328 dB, respectively,for each input signal.

Fig. 1. FBG differentiator SR amplitude in (a) reflectionand (b) transmission, corresponding to ideal bandwidth-limited (solid curve), Gaussian approximation (dottedcurve), and Lorentzian approximation (dashed curve)functions.

Fig. 2. Grating profile obtained by inverse scattering indifferent scales.

Fig. 3. Spectral response in transmission of the designedFBG (solid curve) and the ideal differentiator (dottedcurve).

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Finally, in Fig. 5, we show the dependence of thedifferentiator accuracy and length on the FBG trans-mission dip. We calculate the output pulse consider-ing a 7 ps Gaussian pulse as input waveform for sev-eral FBG differentiators in transmission, designedwith the same approximation function for HR(Gaussian with a FWHM of 0.56 THz), with thetransmission dip varying from 20 to 80 dB. The accu-racy of the differentiator has been calculated as thedegree of similarity between the FBG and the idealdifferentiator output signals, which can be estimatedwith the normalized cross-correlation coefficient,Corr [14]:

Corr = max

fout,FBGtfout,ideal* tdt

fout,FBGt2dt

fout,idealt2dt

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where fout,FBGt and fout,idealt are the complex enve-lopes of the output signals corresponding to FBG andthe ideal differentiator, respectively, and * denotesthe complex conjugate. The length has been obtainedfrom properly limited grating obtained by the inversescattering algorithm. As it can be seen in Fig. 5,lower minimum in transmission (higher maximum

reflectivity) implies higher accuracy but also longergrating (ideal zero in transmission is achieved onlyby a hypothetical infinite-length FBG). Moreover,there are many factors that can decrease the trans-mission dip magnitude, for instance, spatial nonuni-formity of the fiber, which may restrict the feasibletransmission dip values in practice.

In conclusion, in this Letter we have presented asimple approach based on an FBG in transmission as

an ultrafast all-optical differentiator. A key aspect ofthis method is that, because of the logarithmic Hil-bert transform relations between SR amplitude andphase of an FBG in transmission, the required-phase shift appears in the SR phase when the cor-responding amplitude approximates the first-orderdifferentiator SR amplitude. We want to emphasizethe high energetic efficiency and simplicity of the re-sulting scheme compared to other implementations,since only a single FBG working in transmission isrequired to obtain the first-order differentiator, with-out any additional element (optical circulator, FBG,coupler, or claddingcore mode converter). On theother hand, depending on the application require-

ments and the design, the required length, couplingcoefficient, and transmission dip may be a fabricationchallenge.

This work was supported by the Spanish Ministe-rio de Educacion y Ciencia under project Plan Nacio-nal de I D I TEC2007-68065-C03-02.

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Fig. 4. Temporal waveforms of the input pulse (dashedcurve) and output pulse corresponding to FBG (solid curve)and the ideal differentiator (dotted curve), which arehardly distinguishable in both plots. Input pulses of plots(a) and (b) respectively are a 7 ps Gaussian pulse and thefirst-order derivative of a 7 ps Gaussian pulse.

Fig. 5. Grating length (squares, dotted curve) and cross-correlation coefficient, Corr (circles, dashed curve), whichrepresents the operation accuracy. Thirteen FBG differen-tiators are designed assuming the same operation band-width to obtain transmission dip values from 20 to 80 dBand are applied to a 7 ps Gaussian pulse.

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