Cross-phase modulation effects in nonlinear fiber Bragg gratings

Broderick et al. Vol. 17, No. 3 /March 2000/J. Opt. Soc. Am. B 345

Cross-phase modulation effects in nonlinearfiber Bragg gratings

Neil G. R. Broderick, Domino Taverner, David J. Richardson, and Morten Ibsen

Optoelectronics Research Centre, University of Southampton, Southampton, SO17 1BJ, United Kingdom

Received June 14, 1999; revised manuscript received October 13, 1999

We report the results of a series of experiments examining cross-phase modulation effects in apodized fiberBragg gratings. All-optical switching and the optical pushbroom are observed, depending on the precisewavelength of the probe. The experimental results are then modeled with the coupled-mode equations.© 2000 Optical Society of America [S0740-3224(00)00203-4]

OCIS codes: 060.2340, 190.4370, 050.2770, 060.4370.

1. INTRODUCTIONOptical fiber Bragg gratings (FBG) possess many featuresthat make them attractive to the telecommunication in-dustry. Chief among these is the fact that they combinehigh reflectivity with a narrow bandwidth, making themideal for add/drop filters in a wavelength division multi-plexing (WDM) system. These properties also offer thepotential for the development of high-quality nonlineardevices such as optical switches. The narrow bandwidthof a FBG means that only a small change in the refractiveindex is necessary to detune light from inside the band-gap (where the reflection is high) to outside it, where com-plete transmission is possible. The strong reflectivity en-sures that the contrast ratio between the off and on statesof the grating is large. There are, of course, two ways toswitch a Bragg grating. The first is to use an opticalpulse, which is tuned to lie within the bandgap and whoseintensity is sufficient to detune itself from resonance,thus allowing propagation through the structure. In-deed, recently we demonstrated a nonlinear increase inthe transmission of a FBG from 2% to 40% using thismethod.1 This is similar to the pulse compression workdone by Eggleton et al.2 The second way is to use a high-intensity pump beam, tuned far from the Bragg reso-nance, to alter the propagation constant of a weak signalbeam whose wavelength is near or within the gratingbandgap. Such cross-phase modulation effects were firstseen by LaRochelle et al.,3, and Lauzon et al.,4 and, morerecently, ourselves.5,6

Although the first approach is perhaps more aestheti-cally pleasing, the second method offers considerable ad-vantages for a practical device. First, there is no restric-tion on the frequency separation between the pump andthe probe. This allows the pump and grating wave-lengths to be chosen separately. Second, as there is noenergy exchange between the pump and the probe, asingle pump can switch multiple channels in a WDM sys-tem. In addition, there are no requirements on the probeintensity. Last, we note that pulse compression usingcross-phase modulation is a proven technique and hasbeen used for many years in standard fiber.7,8

0740-3224/2000/030345-09$15.00 ©

It is for these reasons that the first nonlinear experi-ments in FBG’s used cross-phase modulation to switch aweak signal. LaRochelle et al.3 first demonstrated non-linear switching in 1990, whereas the first reports of self-switching in a FBG did not appear till much later.1 Thistime delay was due to the ease of observing cross-phasemodulation effects compared with self-phase modulationeffects in a FBG. In the above discussion we did not dis-tinguish between coupled-wave (CW) and pulse effects;however, the differences are important and lead to verydifferent behavior. In a Kerr medium the refractive in-dex n(x) can be modeled by

n~x ! 5 n0 1 n ~2 !I~x !, (1)

where n0 is the background index and I(x) is the local in-tensity. If we consider the effect of a strong CW pumpbeam on a low-intensity signal, then clearly the dominanteffect is that the signal ‘‘sees’’ a constant refractive indexthat is slightly different from that of the background me-dium. This slight change in the effective refractive in-dex, however, can still be sufficient to detune the signalfrom the Bragg grating (or any other resonant condition).If, however, the pump is a short intense pulse, thenthrough Eq. (1) it can be thought of as a moving wall ofrefractive index. The signal beam, upon encounteringthis moving wall, will be Doppler shifted, and thus its fre-quency will be altered. More precisely, it can be shownthat the probe’s frequency shift is proportional to the gra-dient of the pump’s intensity profile.9 In a medium witha positive Kerr nonlinearity @n (2) . 0# a positive inten-sity gradient (e.g., the leading edge of the pump pulse)causes a red shift of the probe’s frequency, and a negativeintensity gradient causes a blue shift of the probe’s fre-quency. If the pulse is asymmetric, then the frequencyshifts from the leading and trailing edges can be very dif-ferent, as we show below. Also, as demonstrated below,the difference between CW and pulse effects leads to verydifferent types of behavior in Bragg gratings. Finally, wenote that the effects that we have been describing dependonly on the intensity of the pump and not on its phase.If, however, the frequency difference between the pump

2000 Optical Society of America

346 J. Opt. Soc. Am. B/Vol. 17, No. 3 /March 2000 Broderick et al.

and the probe is sufficiently small, then parametric am-plification of the probe can take place.10 For the pur-poses of this paper we restrict ourselves to the non-phase-matched regime, where any parametric amplification isnegligible.

Earlier we reported two sets of experiments that exam-ined cross-phase modulation in FBG.5,6 Each of these ex-periments was performed at a single frequency andlooked at either the transmitted or the reflected light butnot both. Clearly what was lacking in these earlier re-ports was a description of how cross-phase modulation ef-fects varied with frequency, and this is rectified in thispaper. The outline of the paper is as follows: In Section2 we present a theoretical model of our system, and inSection 3 we describe the experimental setup. In Sec-tions 4 and 5 we present our results along with a compari-son of the results from our numerical model. Finally wediscuss these results in Section 6.

2. THEORETICAL MODELIn a fiber Bragg grating the linear refractive index variesperiodically with a period d and can be approximated as11

n~z ! 5 n0 1 Dn~z !cos~2k0z !, (2)

where n0 is the background refractive index and Dn(z) isthe modulation depth of the grating. The wave vectork0 5 p/d and corresponds to the Bragg frequency v0around which the grating is highly reflective. We takeEq. (2) to refer to the effective index of the fiber mode un-der consideration.

The effect of the grating is to couple forward- andbackward-propagating light at frequencies v close to v0 .Thus in our model we can assume that there are only twofrequencies of interest, the Bragg frequency v0 and thepump frequency vp . Also, we assume that uvp 2 v0u issufficiently large that the pump is unaffected by the grat-ing, a condition satisfied in our experiments. Underthese conditions we can write the electric field as

E~z, t ! 5 $ f1~z, t !exp@i~k0z 2 v0t !#

1 f2~z, t !exp@2i~k0z 1 v0t !#x

1 P~z, t !exp@i~kpz 2 vpt !#y 1 c.c., (3)

where f1, and f2, are the slowly varying envelopes of theforward- and backward-propagating waves at the Braggfrequency. The pump envelope is given by P(z, t),where z is the propagation direction. Note that we havetaken the pump and the probe to be orthogonally polar-ized, which is not necessary but represents the experi-mental conditions. If we make the usual slowly varyingassumptions, the coupled-mode equations (CMEs) for f1

and f2 can be written as12

i]f1

]x1

i

vg

]f1

]t1 k f2 1 d f1 1

2

3GuP~z, t !u2f1 5 0,

(4a)

2i]f2

]x1

i

vg

]f2

]t1 k f1 1 d f2 1

2

3GuP~z, t !u2f2 5 0.

(4b)

Note that we have assumed that the pump propagates un-changed throughout the fiber; i.e., P(z, t) 5 P(z 2 vgt).For an optical fiber,11,13

d 5v 2 v0

vg

, k~z ! 5pDn~z !

l, G 5

4pn0

lZn ~2 !, (5)

vg is the group velocity in the absence of a grating, l isthe free-space Bragg wavelength, and Z is the vacuum im-pedance. The parameter k measures the strength of thecoupling between f1 and f2 . In addition to Eq. (4) we as-sume that no light is incident on the grating from theright, i.e., backwards, direction and that initially thefields in the grating are in steady state.

Equations (4) for the optical pushbroom are identical tothose derived by de Sterke12 except for two minorchanges. First, since we have an orthogonally polarizedpump and probe, the nonlinearity is reduced by a factor of3. Second, we have assumed a nonuniform grating sothat k is a function of position. In deriving Eq. (4) wemade two important assumptions. First we assumedthat the wavelength difference between the pump and theprobe is sufficiently large that there is no parametric am-plification of the probe beam. Second, we assumed thatthe probe beam is sufficiently weak that by itself it doesnot experience any nonlinear effects. Both of these ef-fects have been included theoretically by others,14–16 butsince such effects were not present in our experiments, wedo not include them in our model.

A. Linear Properties of a Fiber Bragg GratingTo understand the nonlinear behavior of a FBG, first re-call the linear properties. In the absence of a pump andfor a CW input, the coupled-mode equations can be writ-ten as

]

]xS f1~x !

f2~x ! D 5 F id ik~x !

2ik~x ! 2id G S f1~x !

f2~x ! D . (6)

For a uniform grating @k(x) 5 k# we can solve Eqs. (6) byexponentiating, yielding the transfer matrix M:

M 51

bF id sin bx 1 b cos bx ik sin bx

2ik sin bx b cos bx 2 id sin bxG ,(7)

where b2 5 d 2 2 k2. Equation (7) is still valid when bis complex, which occurs when udu , k. In terms of M,the solutions to the coupled-mode equations are

S f1~x !

f2~x ! D 5 MS f1~0 !

f2~0 ! D . (8)

From Eq. (8) it can easily be shown that for a finite struc-ture the reflection coefficient is given by r 5 2M11 /M21and that there are discrete frequencies where the reflec-tivity drops to zero.17 These zeros are important for thediscussion of the optical pushbroom in Subsection 2.C.

For a uniform grating the bandgap extends from 2k, d , k, and for frequencies within this region the in-tensity decreases exponentially along the length of thegrating. Outside this region there are plane-wave solu-tions to Eq. (6) that propagate unchanged through thegrating; they can be written as


S f1

f2D 5 S k

q7Aq2 1 k2D exp@i~qx 2 dt !#, (9)

where

q 5 6Ad 2 2 k2. (10)

The dispersion relationship given by Eq. (10) is illus-trated in Fig. 1. The group velocity V is

V 5dd

d q5

1

dAd 2 2 k2. (11)

Note that V 5 0 at the band edge and asymptotically ap-proaches unity as d → `, indicating that the dispersionrelationship approaches that of the background mediumfar from the grating, as one would expect. In addition,the group velocity dispersion is given by the curvature ofthe dispersion relationship. From Fig. 1 it can be seenthat this is anomalous above the bandgap and normal be-low it. For typical grating parameters the dispersionnear the bandgap can be six orders of magnitude greaterthan the bare fiber dispersion, which is why we havetreated the background material as dispersionless in Eqs.(4). For an apodized grating such as the one used inthese experiments, the group velocity of a single fre-quency depends on its position; in that case, V and k inEq. (11) should be replaced by V(x) and k(x).

B. Coupled-Wave Switching of a Fiber Bragg GratingBy far the simplest application of the CME’s [Eq. (4)] isthe modeling of the switching of a uniform grating by astrong CW pump beam. In this case both k and P areconstants and the CME’s can then be solved exactly. Theonly effect the pump has on the solution is to introduce anadditional detuning of 2/3GuPu2. This has the effect ofuniformly shifting the entire reflection spectrum to lowerdetunings by 2/3GuPu2. For a uniform grating the fre-quency difference between the center of the bandgap andthe position of the first minima in the reflection spectrumis 'k. Thus the intensity needed to switch a probe thatis centered at the Bragg frequency from being reflected tobeing totally transmitted is3

Fig. 1. Dispersion relationship for a uniform grating with thesame parameters as the grating in Fig. 8. Note that no solu-tions exist in the bandgap. The dashed line indicates the back-ground dispersion relationship.

uPu2 '3

2Gk. (12)

This effect was first seen by LaRochelle et al.,3 who ob-served an increase in the average power transmissionfrom 4% to 6%. The degree of switching that they wereable to observe was limited by the temporal resolution oftheir detector, and theoretical estimates suggest thatcomplete switching was actually occurring in theirsystem.3 In LaRochelle’s experiments the pump beamwas in fact a 100-ps long pulse. However, as their grat-ing was only 3.5 cm long, the pulse was longer than thegrating, justifying the CW nature of the experiment. Weshall return to the question of what constitutes a CWpump later in the paper.

For our experiments we used a nonuniform grating.In the CW regime this makes no difference to the physics;however, the switching power may be increased or de-creased depending on whether the bandgap is larger orsmaller than that of a uniform grating [i.e., k in Eq. (12)should be replaced by kmax].

C. Optical PushbroomIn the regime where the pump pulse is shorter than thegrating, the interaction can no longer be treated asthough it were CW. In this regime the dominant effect isthe frequency shift of the probe rather than the nonlin-early induced shift in the Bragg resonance. This fre-quency shift can result in the compression of the probe—the so called optical pushbroom.12 This works as follows.

From Eq. (11) it can be seen that the farther away apulse is from the bandgap, the greater its group velocity.Clearly, then, a FBG could be used to efficiently compressan appropriately chirped pulse as it propagates throughthe grating. The basics of the optical pushbroom are thatthe intense pump introduces a chirp on the probe pulsethat is then compressed by the FBG. More precisely,consider a CW beam centered at the first transmissionresonance of the FBG. The intensity profile of the lightinside the grating is shown in Fig. 2. Note that a signifi-cant amount of energy is stored inside the grating at thisdetuning. When the pump enters the grating, it first en-counters the back of the probe resonance. Through cross-phase modulation it lowers the frequency of the back ofthe probe, causing it to speed up. This increase in veloc-ity allows the back of the probe to sit on the leading edgeof the pump, where it experiences still further cross-phasemodulation. This process continues through the gratingwith more and more of the probe’s energy being swept uponto the leading edge of the pump.

In transmission the hallmark of the optical pushbroomis thus a sharp spike (temporally coincident with thepump) followed by a relatively longer dip in the transmis-sion. The decrease in transmission is due to the fact thatthe pump has swept up all the stored energy in the grat-ing and the transmission resonance needs time to rees-tablish itself. The effects of the optical pushbroom can beseen in Fig. 3, which shows a theoretical trace of thetransmitted light obtained by solving Eqs. (4) numeri-cally. In the simulation the various parameters werechosen to correspond to the measured experimental val-ues. In particular,


k~x ! 5 0.45 sin~px/ll !cm21, (13)

where ll is the length of the grating and we used a detun-ing of 20.67 cm21 for the probe beam.

In its original guise the optical pushbroom was shownto work in a uniform Bragg grating, where it swept outthe energy associated with one of the linear resonances.12

The same physical process was soon shown to compressoptical pulses that were co-propagating with the pumpthrough the grating.10,18 However, as Fig. 3 shows, thesame effect can be seen in apodized gratings, which do nothave as strong resonances as uniform Bragg gratings.The optical pushbroom works in this case since for fre-quencies close to the edge of the bandgap there is still anappreciable amount of energy stored in the grating thatcan be swept out by means of cross-phase modulation (seeFig. 2). The energy stored in our apodized grating near

Fig. 2. Field profile inside the grating at the first transmissionresonance. Solid curves, total intensity u f1u2 1 u f2u2; short-dashed curve, u f1u2; long-dashed curve, u f2u2. Note that thefield structure is single peaked, with the maximum intensity be-ing significantly higher than the input intensity. This illus-trates the energy storage capability of a FBG.

Fig. 3. Theoretical trace of the optical pushbroom. Solid curve,transmitted probe intensity; dashed curve, pump profile. Theinset is a blowup of the front spike in the transmission. The pa-rameters chosen match those used in the actual experiment.

resonance is at most 82% of what can be stored in a uni-form grating with the same peak reflectivity and length.This is still 40% more than for a uniform medium. Infact, the use of an apodized grating relaxes considerablythe experimental requirements for seeing the opticalpushbroom. Because of the lack of any well-defined reso-nances in the transmission spectrum, the detuning of theprobe does not need to be as critically tuned as it wouldneed to be if the probe had to lie exactly on a transmissionresonance. Thus, as we show below for an apodized grat-ing, there are a wide range of frequencies where the opti-cal pushbroom can be observed. In fact, any apodizedgrating will store energy at frequencies near the bandgap.This is because the closer the frequency of light is to theedge of the local bandgap, the more slowly it will propa-gate. Since Bragg gratings are lossless media reducedvelocity must be associated with energy storage. Theamount of energy at a frequency v0 that is stored will de-pend on the average velocity that frequency sees duringpropagation through the grating. In addition, the pres-ence of any transmission resonances will strongly affectthe amount of energy stored.

The wide range of possible behaviors that can be ob-served in a nonuniform Bragg gratings are shown in Figs.4 and 5. These graphs show the transmitted intensities(dashed curves) and the reflected intensities (solid curves)for a CW beam at various detunings. The parametersused in these simulations match as closely as possible theexperimental parameters described below. In Fig. 4 thedetuning is above the center of the bandgap, and simpleCW switching can be seen particularly in Figs. 4(b) and4(c). In Fig. 5 the detuning is below the center of thebandgap, and, as expected, optical pushbroom effects canbe seen. Note that because of the length of the pumppulse, we can assume that the field in the grating evolvesadiabatically (except for the first 100 ps resulting fromthe sharp leading edge of the pump), and thus at mosttimes the reflection and the transmission should sum tounity, as can be seen in the theoretical traces.

Fig. 4. Theoretical transmitted (dashed curves) and reflected(solid curves) intensity profiles of the probe beam as a function ofthe time in nanoseconds. The pump is incident on the grating att 5 0. The detunings used were 0.662 cm21, 0.466 cm21, 0.049cm21, and 20.197 cm21 for (a), (b), (c), and (d), respectively.


3. EXPERIMENTAL SETUPOur experimental setup is shown in Fig. 6. High-powerpump pulses at 1550 nm are used to switch a low-power(1-mW), narrow-linewidth (,10-mHz) probe that couldbe temperature tuned right across the grating’s bandgap.The pump pulses, derived from a directly modulateddistributed-feedback laser, were amplified to a highpower (.10 kW) in an erbium-doped-fiber amplifier cas-cade based on large-mode-area erbium-doped fiber andhad a repetition frequency of 4 kHz. Figure 7 shows theintensity profile of the pump pulse. Its shape is asym-metric owing to gain-saturation effects within the ampli-fier chain and has a 30-ps rise time and a 3-ns half-width.The spectral half-width of the pulses at the grating inputwas measured to be 1.2 gHz, as defined by the chirp onthe input seed pulses.

The pump and probe were polarization coupled into theFBG and were thus orthogonally polarized within theFBG. We note that although the fiber used was nomi-nally circular, a small residual birefringence of 4 3 1026

was measured, and hence it was necessary to align the in-put polarizations along the principle axes of the grating.A half-wave plate was included within the system beforethe FBG to achieve this. Both the reflected and thetransmitted probe signals could be measured in our ex-perimental system by use of a fiberized detection systembased on a tunable, narrow-band (,1 nm) optical filterwith .80-dB differential loss between pump and probe(sufficient to extinguish the high-intensity pump signal),a low-noise preamplifier, a fast optical detector, and sam-pling scope. The temporal resolution of our probe beammeasurements was '50 ps.

The FBG was centered at 1536 nm and was 8 cm longwith a sine-squared apodization profile @k(x)5 k0 sin2(2px/l)#, resulting in almost complete suppres-sion of the sidelobes. The grating had a peak reflectivityof 98% and a measured width of less than 4 GHz. Themeasured reflection spectrum is shown in Fig. 8 (solid

Fig. 5. Theoretical transmitted (dashed curves) and reflected(solid curves) intensity profiles of the probe beam as a function ofthe time in nanoseconds. The pump is incident on the grating att 5 0. The detunings used were 20.444 cm21, 20.592 cm21,20.643 cm21, and 20.891 cm21, for (a), (b), (c), and (d), respec-tively.

curve), along with a theoretical reflection spectrum for anidealized grating with identical parameters. In Fig. 8the wavelengths along the x axis are given in terms of thedifference from the center wavelength of 1535.930 nm.The grating was mounted in a section of capillary tube,angle polished at both ends so as to eliminate reflectionsfrom the grating end faces, and was appropriately coatedto strip cladding modes.

Fig. 6. Schematic diagram of the experimental setup. PBS, po-larization beam splitter; BPF, bandpass filter with a width of ,1nm; LD, laser diode; LA-EDFA, large-mode-area erbium fiberamplifier. The polarizer, POL, is set to minimize the pump.See the text for more details.

Fig. 7. Measured intensity profile of the pump pulse used in theexperiments.

Fig. 8. Measured reflection spectrum of the grating used in ourexperiments (solid curve). The dashed curve is a theoreticaltrace for a grating with the same parameters. The center of thegrating is at 1535.9290 nm, and the horizontal scale gives the de-tuning from the center wavelength. The effect of the apodiza-tion can be seen clearly in the lack of sidelobes in the spectrum.


It should be noted that the pump-pulse shape require-ments for CW switching and for the optical pushbroomare somewhat incompatible. The pushbroom requirespulses with a large intensity gradient, whereas for CW ef-fects to be seen, the intensity gradient should be zero.However with our pulse (shown in Fig. 7) we are able tohave our cake and eat it, too. The exceedingly rapid risetime of the pulse allows us to see the optical pushbroom,yet the fact that the pulse is longer than the grating al-lows CW effects to be seen. As we discuss in detail be-low, the frequency of the probe pulse determines whetherwe are in a pushbroom or a CW regime. In some cases,however, we see a combination of both effects.

We were able to tune the wavelength of our probe beamby changing the temperature of the laser diode. The fol-lowing empirical relationship was found between thewavelength and the resistance of the heating element:

l~T ! 5 1538.00 2 0.149639T 2 0.00199972T2, (14)

where T is measured in kilohms and l in nanometers.We were able to control the temperature to within 0.01kV corresponding to a wavelength tunability of 1 pm.The actual center wavelength of the grating was con-stantly shifting, owing to small temperature changes ofthe laboratory; the Bragg wavelength shifted by approxi-mately 10 pm/C°, which is an extremely large shift giventhat the bandgap is ;30 pm. This made it difficult to de-termine in real time the actual detuning of the probe fromthe Bragg wavelength. For this paper we estimated thedetuning for a given data set by measuring the amount ofreflected or transmitted light in the linear regime andcomparing it with that measured at large detunings(where the transmission is assumed to be unity). From aprevious high-resolution scan of the grating’s response wecould then determine the detuning to good accuracy.

We performed a series of measurements, looking atboth the transmitted and the reflected light as a functionof both the probe wavelength and the pump power. Forclarity, these results are split into the reflected and thetransmitted outputs. We will first discuss thetransmitted-light case.

4. FORWARD-PROPAGATING CASEAs discussed in Section 2, the transmitted probe shape isexpected to be a strong function of its detuning from thegrating’s bandgap. This can be seen in Figs. 9 and 10,which show the averaged probe’s waveform as we tuneacross the bandgap from the short-wavelength side to thelong-wavelength side while keeping the pump power con-stant. In these graphs the transmitted intensity hasbeen scaled so that the normalized linear intensity at thecenter of the bandgap is 0.04 and the normalized inten-sity at wavelengths well outside the bandgap is unity.This in fact slightly underestimates the true transmissionfor long wavelengths, owing to a slight wavelength depen-dence of the detection process. In all the traces the ori-gin of the time is set to coincide with the start of thetransmitted pump pulse. These experimental tracesshould be compared with the theoretical traces in Fig. 4.

Examining the traces in Fig. 9 we see that the commonfeature is that the transmission increases in the presence

of the pump. This effect is due to the nonlinear indexchange induced by the pump, which shifts the Braggwavelength to lower frequencies, and thus the probe is ef-fectively further from the Bragg resonance and hence itstransmission increases. In this regime our results aresimilar to those of La Rochelle et al.,3 the most importantdifference being that we are able to resolve the temporalshape of the transmitted light. In addition we have acleaner source and a better grating, giving significantly

Fig. 9. Measured transmitted intensity profiles of the probepulses as a function of the time in nanoseconds. The derived de-tunings are 0.662 cm21, 0.466 cm21, 0.049 cm21, and20.197 cm21 for (a), (b), (c), and (d), respectively. The probe’sintensity has been normalized to the peak of the transmission atwavelengths far from the grating.

Fig. 10. More transmitted intensity profiles for frequencies be-low the center of the bandgap. The derived detunings are20.444 cm21, 20.592 cm21, 20.643 cm21, and 20.891 cm21

for (a), (b), (c), and (d), respectively.


Fig. 11. Experimental traces of the optical pushbroom. (a) Result of optimizing the wavelength for maximum energy storage in theBragg grating, (b) effect on increasing the pump power.

clearer results. Note that in Figs. 9(a) and 9(b) thetransmission follows very closely the pump profile of Fig.7 owing to the fact that the grating’s slope is almost linearnear the short wavelength edge. However, as we movecloser toward the center of the Bragg grating, only thepeak of the pulse is sufficiently intense to switch theprobe. Hence, instead of seeing the broad switchedpulses, we see only a narrow pulse corresponding in timeto the peak of the pump pulse. This effect can also beclearly seen in Fig. 4(d).

In Fig. 10 the traces for frequencies below the Braggwavelength (negative detunings) are shown, where againwe see a strong dependence on the probe’s wavelength.As we are now below the Bragg resonance, the effect ofthe pump is to move the probe’s wavelength closer to thecenter of the bandgap, thereby decreasing the transmis-sion. This is the cause of the long dips in the transmis-sion that can be seen in all the traces. In addition to thedips in the transmission, it can also be seen that thetransmission initially increases owing to the presence ofthe pump. This is the optical pushbroom effect describedabove. As expected, it can be seen only for frequenciesclose to the long-wavelength edge of the grating. How-ever, as discussed in Subsection 2.C, this wavelengthrange is much broader than that of a comparable uniformgrating. Again these features can be seen in the theoret-ical traces. However, in the theoretical case the degreeof switching is typically larger than that seen in the ex-periments. This is due largely to the difference in pulseshapes between the theory and experiments. The theo-retical pulse shape decays more smoothly than the experi-mental one. Note, however, that the overall agreementis excellent with small features such as the increase inthe transmitted light at t 5 3 ns in Fig. 10(d), which alsoappears in the corresponding theoretical trace [Fig. 5(d)].

Optimizing the wavelength for the optical pushbroomresults in the trace shown in Fig. 11(a). In this case theprobe was detuned by 0.02 nm from the Bragg resonance,which is right on the very edge of the grating resonancewhere the transmission is near unity in the absence of theprobe. Note that the parameters used in the theoreticaltrace (Fig. 3) correspond as closely as possible to the ex-

perimental parameters of Fig. 11(a). Note also thatthere is an excellent agreement between the theoreticaland the experimental trace that is matched by the agree-ment at other detunings. This allows us to be confidentthat we are observing the optical pushbroom and notsome other nonlinear effect. Once again the main dis-agreement between the two comes from the difference inpulse shapes between the theoretical model, which as-sumed a triangular pulse, and the actual pump profile,which is more complicated (see Fig. 7).

Finally, we examined the effects on the transmittedprofiles of changing the pump power. These results areshown in Fig. 11(b) and basically followed the expectedtrend. Since we are examining a nonlinear effect, itshould die away with decreasing pump power, as indeedit does. We now turn to the traces of the reflected lightthat have not been treated before either analytically orexperimentally in any detail to our knowledge.

5. BACKWARD-PROPAGATING CASEIn Figs. 12 and 13 are shown the reflected traces of theprobe as a function of time. These traces are normalizedso that the peak reflection in the linear regime corre-sponds to a value of 0.96. This normalization again un-derestimates slightly the actual reflectivity for frequen-cies near the edge of the bandgap, owing to the unevengain from the amplifier before the detector. We note thatowing to a thermal shift in the Bragg resonance of thegrating, the actual detunings in Figs. 12 and 13 are dif-ferent from those in Figs. 9 and 10 even though the actualtemperature of the diode was the same in each case.

When we look at the traces in Figs. 12 and 13 a numberof features are apparent. First, as expected, the reflec-tivity decreases in the presence of the pump above thebandgap and increases below the bandgap. This is due tothe simple effect of cross-phase modulation as discussedin Section 4 whereby the effective frequency of the centerof bandgap decreases owing to the presence of the pump.However, compared with the transmitted cases, the ef-fects of cross-phase modulation are not as apparent [e.g.,compare Fig. 9(b) with Fig. 12(b)]. We are not fully con-


fident of the reason for this, but it is most likely due to adrop in the pump power during the course of the measure-ments.

The other main feature of interest in the reflectedtraces can be seen in Fig. 12(a). Note that initially thereflected intensity increases in a manner similar to thatof the traces of the optical pushbroom. This is a new ef-fect caused by the apodization profile of the grating,which could not have been seen if one used a uniformgrating.6 Note that this effect can also be seen clearly inFig. 4(a). The reason behind this peak is very similar tothe explanation for the optical pushbroom in that a non-

Fig. 12. Measured reflected intensity profiles of the probepulses as a function of the time in nanoseconds. The detuning ofthe probe is 0.588 cm21, 0.490 cm21, 0.246 cm21, and 0.000 cm21

for traces (a), (b), (c), and (d), respectively. The probe’s intensityhas been normalized so that the peak reflection in the linear re-gime corresponds to a value of 0.96.

Fig. 13. More reflected intensity profiles for frequencies be-low the center of the bandgap. The detuning of the probeis 20.247 cm21, 20.494 cm21, 20.593 cm21, and 20.643cm21, for traces (a), (b), (c), and (d), respectively. The graphsare normalized as in Fig. 12.

linear shift in frequency is responsible for the appearanceof the peak. In our situation, as a result of the apodiza-tion profile, light propagates through almost half thegrating before being reflected. This means that com-pared with a uniform grating, significantly more energy isstored in the grating at frequencies within its bandgap.The pump pulse acts on this stored energy by lowering itsfrequency through cross phase modulation. The apodiza-tion profile then ensures that lower frequencies are re-flected earlier in the grating and thus creates the rightamount of dispersion to compress the reflected pulse.

Unlike the forward pushbroom, this effect can be seenonly in apodized gratings, and it takes place over a nar-rower frequency range, as can be seen from the experi-mental traces. We have performed numerical simula-tions that show that it is a robust phenomenon thatoccurs in a wide variety of nonuniform gratings, includingboth linearly chirped gratings and gratings with a rampin k. It is also possible to increase the size of the effect byappropriately designing the grating; however, it will al-ways remain a relatively small effect compared with thatof the optical pushbroom, as less energy is stored in thegrating at frequencies within the bandgap compared withthat for frequencies outside the bandgap.

6. CONCLUSIONWe have presented, for the first time to our knowledge, acomplete investigation of the effects of cross-phase modu-lation that shows the effects on both the reflected and thetransmitted light of varying the pump power and probefrequency. These experimental results can be accuratelymodeled with use of the standard coupled-mode equa-tions, which indicates that no other effects are present inour experiments.

The results here show clearly how a number of differ-ent phenomena such as the optical pushbroom and CWswitching of gratings, which were previously consideredseparately, need to be considered together to obtain acomplete understanding of our measurements. In addi-tion, these results show the advantage of using apodizedgratings for such experiments. Apodized gratings areimportant because they allow light to penetrate fartherinto the grating at frequencies within the bandgap in thelinear regime, making it easier to observe nonlineareffects.5 Using apodized gratings allows novel effects tobe seen that were not previously predicted owing to thetheoretical focus on pulse dynamics in uniform gratings.These effects show clearly that it is possible to obtain sig-nificant switching with all-fiberized sources and gratings.Finally, the agreement between the theoretical modelingand the experimental results gives us faith in our numeri-cal model and should allow the development of improvedand functional devices based on fiber Bragg gratings.

The authors’ phone number is 44-0-1703-593144; fax,44-0-1703-593142; e-mail: [email protected].

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Cross-phase modulation effects in nonlinear fiber Bragg gratings

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