ultrashort pulse propagation in uniform and nonuniform waveguide long-period gratings

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M. Kulishov and J. Azaña Vol. 22, No. 7 /July 2005 /J. Opt. Soc. Am. A 1319

Ultrashort pulse propagation in uniform andnonuniform waveguide long-period gratings

Mykola Kulishov

Adtek Photomask Inc., 4950 Fisher Street, Montreal, Quebec H4T 1J6, Canada

José Azaña

Institut National de la Recherche Scientifique–Énergie, Matériaux et Télécommunications, 800 de la GauchetièreOuest, Suite 6900, Montréal, Québec H5A 1K6, Canada

Received October 29, 2004; accepted December 27, 2004

We conduct a detailed theoretical analysis of ultrashort pulse propagation through waveguide long-period grat-ing (LPG) structures operating in the linear regime. We first consider the case of uniform LPGs, and we alsoinvestigate the effect of the typical grating nonuniformities, e.g., grating profile apodization, grating periodchirping, and discrete phase shifts, on the spectral and temporal behavior of LPG structures. The two inter-acting modes are analyzed separately, and advanced representation tools, namely, space–wavelength andspace–time diagrams (where space refers to the longitudinal grating dimension), are used to provide a deeperinsight into the physics that determines the pulse evolution dynamics through the grating structures underanalysis. In addition to its intrinsic physical interest, our study reveals the strong potential of LPG-based de-vices for optical pulse reshaping operations in the subpicosecond regime. © 2005 Optical Society of America

OCIS codes: 050.2770, 130.3120, 250.5530, 320.5540.

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. INTRODUCTIONaveguide gratings have become critical components forany applications in fiber-optic communications. They

ave been widely used as filters, dispersion compensators,nd sensors. The operation principle of many optical de-ices is based on grating-assisted, resonant mode cou-ling either between two modes within a single wave-uide or between two mismatched guides. These opticalevices can be divided into two broad groups dependingn the direction of propagation of the coupled modes: (1)ontradirectional couplers if the interacting modes propa-ate in opposite directions and (2) codirectional couplers ifhey propagate in the same direction.

Short-period Bragg gratings (BGs) written along theore of optical fibers (by refractive-index perturbation) orithin integrated waveguides (by surface corrugation) arearticularly interesting examples of contradirectionalouplers.1 The interest in BGs as pulse-shaping devices isltimately related to their use for chromatic dispersionompensation in high-bit-rate optical communicationinks. The dispersion of transform-limited pulsesaunched at the input of uniform and nonuniform fiberGs (in both reflection and transmission) has been theo-etically and experimentally investigated.2–7 Typically, inhese applications the gratings are operated in the so-alled narrowband source regime where the pulse spec-ral bandwidth is narrower than that of the grating spec-rum.

Systematic investigations have been also conducted inhe so-called ultrashort propagation regime, where thepectral bandwidth of the incident pulse is substantiallyigher than the grating spectrum.8,9 In particular, theropagation of ultrashort pulses (e.g., reflection and

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ransmission) through weak and strong uniform BGs asell as through nonuniform gratings (with apodized or

hirped index profiles) has been the subject of intensiveesearch. These fundamental studies of the time-domainehavior of BGs have been essential to identify the strongotential of these devices for coherent manipulation andontrol of optical radiation(e.g., for optical pulse-shapingpplications). In fact, taking these fundamental works asstarting point, optical pulse shapers based on BGs have

een proposed and demonstrated.10 Other time-domainpplications of BGs include all-optical pulse repetitionate multiplication11 and optical code generation andecognition.8 Note that most of the mentioned time-omain applications are based on BGs operating withinhe linear regime. It is also worth noting that the BG ap-roach is specially suited for synthesizing temporal fea-ures in the range of a few tens of picoseconds. The syn-hesis of faster temporal features (e.g., in theubpicosecond regime) would require reducing the spatialcale of the BG profiles to an extremely demanding orven unpractical level (from a fabrication point of view).

Long-period (transmission) gratings (LPGs) provide ahase-matching condition in grating-assisted codirec-ional couplers (GACCs) (i) between two copropagatingodes in a fiber12 or in a planar waveguide,13,14 (ii) be-

ween two asymmetric planar waveguides,15 or (iii) be-ween two fiber cores in a mismatched twin-core fiber.16

he GACC is an attractive alternative for wavelength fil-ering applications as it offers the well-known advantagesf easy grating fabrication by low-resolution photolithog-aphy, a wide tuning range, and the potential for provid-ng separate input–output ports for add–drop multiplex-ng. Moreover, GACC filters have been realized by use of

005 Optical Society of America

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1320 J. Opt. Soc. Am. A/Vol. 22, No. 7 /July 2005 M. Kulishov and J. Azaña

emiconductors and electro-optic (EO) materials (e.g.,iNbO3, liquid crystals, and EO polymers), and their tun-

ng characteristics have been studied.17 Use of EO mate-ials in combination with interdigitated electrodes wouldllow one to control individually the physical grating pa-ameters such as grating length and coupling strength, asell as to introduce electronic phase shifts in any desiredosition along the grating.17,18

The GACC analysis (based on coupled-mode theory)as been generally limited to the spectral domain.1,19,20

his has similarly restricted use of these devices to con-entional filtering-type applications (bandpass and band-top filters, mode converters, spectrum equalizers). Time-omain analysis of GACCs has been considered only in aew studies. The pioneer works of Ouellette et al.21,22 andtegall and Erdogan23 suggested use of chirped GACCs

or dispersion compensation applications. It has beenhown that pulse recompression by factors of 10 or mores possible by use of chirped GACCs (for pulses of a fewens of picoseconds). Use of weak-coupling uniformACCs for encoding and decoding optical pulses was firstroposed by Syms et al.24 In more fundamental studies,25

he same authors developed optical path integrationethods to calculate the temporal impulse response of

niform GACCs. Kutz and co-workers have also investi-ated the temporal properties of GACCs operating in theonlinear regime for all-optical switching and passiveode-locking applications.26,27 Recently, fiber devices

ased on cascaded LPGs have been used for optical tem-oral pulse encoding and decoding28 and 23, 43, and 8

pulse rate multiplications of subpicosecond pulses.29

hese recent devices exploit the difference in the propa-ation speed between the core and cladding modes (thePGs are used to couple light from the core mode into theladding mode and vice versa).

Also, we recently investigated ultrashort pulse propa-ation through uniform GACCs in the linear regime.30

ur preliminary numerical results have demonstratedhe strong potential of GACCs for picosecond–ubpicosecond optical pulse processing and shaping. Inarticular, we have shown that, depending on the lengthnd coupling strength of the uniform perturbation in theACC, one can achieve different temporal shapes at the

utput of the device (for the energy receptor mode), in-luding square temporal waveforms as well as multipledouble and triple) pulse sequences. It should be men-ioned that squarelike temporal waveforms are highly de-ired for a range of nonlinear optical switching and fre-uency conversion applications,10 whereas the generationf multiple pulse sequences from a single input pulse cane exploited for repetition rate multiplication operationsn optical pulse trains.11 For comparison, although similaremporal waveforms can be also achieved with BGs, theireneration normally implies use of much more complexmplitude and phase grating profiles. Moreover, our re-ults have also shown that GACCs are more appropriateor the processing and shaping of temporal waveformsith features faster than a few picoseconds.Our previous studies were restricted to the case of uni-

orm transmission gratings. However, it is expected thathe range of achievable pulse-reshaping operations can beignificantly broadened by introducing suitable nonuni-

ormities along the grating profile (e.g., chirp or apodiza-ion profiles). The objective of this work is to complete ourreliminary studies on ultrashort pulse propagationhrough GACCs (linear regime) by considering the case ofonuniform grating perturbations as well. Specifically wenalyze here in greater detail the temporal and spectralesponses of uniform GACCs to ultrashort optical pulses,nd we extend this analysis to evaluate the effect of typi-al grating nonuniformities, namely, grating profilepodization, grating period chirping, and discrete phasehifts, on the GACC spectral and temporal behavior. Wenticipate that a proper understanding and interpreta-ion of the ultrashort pulse propagation dynamicshrough LPGs is less straightforward than for the case ofGs (where essentially the devices can be simply inter-reted as wavelength-selective mirrors), and conse-uently more complex analysis tools are required. In par-icular, in this work we make use of space–time andpace–wavelength diagrams (where space refers to theongitudinal grating dimension), thus providing a deepernsight into the physics that determines the pulse evolu-ion through the grating structures under analysis. Thevolution of the propagating radiation will be analyzed inetail for both the supplier and the receptor modes, and apecial emphasis will be placed on the analysis of thoseeatures with more immediate implications for imple-enting pulse-reshaping operations of practical interest.The remainder of this paper is structured as follows. In

ection 2 the theoretical principles of our analysis areutlined, and in particular the numerical procedures tobtain the space–wavelength and space–time diagramssed in our study are discussed. In Section 3 we analyzeltrashort pulse propagation through different LPGtructures using the numerical tools described in Section. Specifically, we consider the case of uniform, apodized,nd chirped LPG perturbations. Ultrashort pulse propa-ation through phase-shifted LPGs is also investigated.inally in Section 4 we summarize and conclude.

. PULSE EVOLUTION THROUGHRANSMISSION GRATINGSe assume that the wave packet (centered at frequency

c) has a Gaussian envelope function uhinstdu2 with a fullidth at half-maximum (FWHM) time width t0 and with-

ut initial chirp:

hinstd = expF−t2

2t022 lns2dGexpsi2pnctd. s1d

he Fourier transform Hinsnd of this pulse is

Hinsnd = S pt02

2 lns2dD1/2

expF−p2t0

2

2 lns2dsn − ncd2G , s2d

nd the corresponding FWHM bandwidth of its powerpectrum uHinsndu2 is

DnFWHM =2 lns2d1/2

pt0lnF pt0

2

lns2dG1/2

. s3d

he input pulse starts interacting with the grating as itnters into the perturbed region. We assume that the one-

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imensional (1D) refractive-index profile (along the z di-ection) through which the pulse propagates can be de-cribed as

dnszd = dn1szdcosF2p

Lz + fszdG , s4d

here dnI is the modulation peak amplitude, L is theominal grating period, and fszd describes a possiblerating chirp.

Neglecting the inactive higher-order and radiationodes, the propagation field in the GACC can be ex-

ressed as a combination of the two orthogonal super-odes that are characteristics of the waveguide in the ab-

ence of the grating perturbation:31

Eszd = oj=1

2

AjszdEjsx,ydexps− jbjzd, s5d

here Aj, Ej, and bj represent the amplitudes of the inter-cting supermodes, the power-normalized transversaleld distribution of these supermodes, and the corre-ponding propagation constants along the propagation di-ection z, respectively. The presence of a grating pertur-ation induces a coupling between the two orthogonalupermodes, and this process can be analyzed by coupled-ode theory. For this purpose, Aj is assumed to be a

lowly varying function of z, and the orthonormality prop-rty of the supermodes Ejsx ,yd is also used. Consideringnly the fundamental Fourier component of the periodicndex perturbation, the coupled-wave equations can beritten as32

dA1szd

dz= − ikA2szdexpsj2szd,

dA2szd

dz= − ik*A1szdexps− j2szd, s6d

here 2s=b1snd−b2snd−2p /L represents the mismatch-ng condition for maximum energy transfer between su-ermodes. The coupling coefficient k can be derived as

k =v

4E E1

*sx,yde0De1E2sx,yddxdy, s7d

here e0 is the vacuum permittivity, De1 is the amplitudef the fundamental Fourier component of the periodic per-ittivity, De1<2ndn1, and v=2pn is the angular fre-

uency of the light. For isotropic index perturbation, theoupling occurs only between modes with the same polar-zation.

For simplicity and clarity, we limit the analysis of pulseropagation to the situation when the two guides of theACC are far from synchronism and weakly coupled, i.e.,

he only mode interaction mechanism is that induced byhe grating perturbation. Unlike for the general GACCase, where these equations are just a good approxima-ion for unsynchronized, weakly coupled modes, they areompletely accurate for in-fiber LPGs (fiber modes are al-ays orthogonal in an optical fiber).As the frequency deviates from the filter center, the

oupling efficiency drops and the filtering effect is clearly

elated to the dispersive properties of the coupling coeffi-ient k and the phase-mismatching factor s. In fact, be-ause of the slow variation of k with frequency, the filter-ng effect is mainly associated with the phase detuningactor s. We thus neglect the wavelength dependence ofhe coupling coefficient k in our following analysis.

The general solutions of the coupled equations of Eqs.6) can then be rewritten in the form of a transfer matrix

that relates the input amplitudes A1s0d and A2s0d at theoordinate z=0 with the amplitudes A1szd and A2szd at aiven arbitrary location z within the grating s0øzøLdRef. 32):

SA1sz,nd

A2sz,ndD = FT11sz,nd T12sz,nd

T21sz,nd T22sz,ndGSA1s0,nd

A2s0,ndD . s8d

ssuming further that only one guide or mode (supplier)s initially excited by the pulse launched at the input ofhe device [A1s0d=1; A2s0d=0], then the matrix element11sz ,nd will determine the pulse evolution through theupplier guide or mode, whereas pulse propagation in theeceptor guide or mode will be determined by the matrixlement T21sz ,nd. In this case, the temporal responses cor-esponding to the supplier and receptor guides or modesssz , td and hrsz , td can be calculated by taking the inverseourier transform sJ−1d of the result of multiplying the

nput pulse spectrum Hinsnd at z=0 by the GACC spectralransmission response corresponding to the supplierode T11sz ,nd and to the receptor mode T21sz ,nd, respec-

ively,

hssz,td = J−1fHinsndT11sz,ndg, s9ad

hrsz,td = J−1fHinsndT21sz,ndg s9bd

t any position z along the grating. In this way, space–ime diagrams (where space refers to the propagation dis-ance z) illustrating the pulse propagation dynamicshrough the transmission grating for both the suppliernd the receptor modes can be obtained. Similarly, theunctions T11sz ,nd and T21sz ,nd define the space–avelength (frequency) diagrams corresponding to the

upplier and receptor modes, respectively.

. Uniform Gratingor a uniform grating with constant period Lffszd=0gnd constant amplitude dn1, the transfer-matrix elementsave the following expressions:24

T11sz,nd = Fcossgzd + js

gsinsgzdGexpfjsb1 − sdzg,

T12sz,nd = jk

gsinsgzdexpfjsb1 − sdzg,

T21sz,nd = jk*

gsinsgzdexpfjsb2 + sdzg,

T22sz,nd = Fcossgzd − js

gsinsgzdGexpfjsb2 + sdzg, s10d

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. Uniform Grating with p Shiftor the pulse evolution calculation in supplier and recep-or guides, the elements T11

spdsz ,nd and T21spdsz ,nd of a

-shifted grating are simply determined by multiplicationf matrices of the corresponding uniform sections, i.e., theollowing matrix should be used in Eqs. (9):

Tspdsz,nd = HTsz,nd for 0 ø z ø L/2

TsL/2,ndIsp/2dTsz,nd for L/2 , z ø LJ ,

s11d

here Tsz ,nd is the matrix of a uniform grating describedn Eqs. (10) and the p-shift matrix is of the form

Isp/2d = Fexpsjp/2d 0

0 exps− jp/2dG . s12d

. Nonuniform (Chirped and Apodized) Gratingss shown in Fig. 1, any index profile dnszd can be ex-ressed as a sampled stair structure of uniform gratingieces Dzi, where L=SDzi. We used the piecewise uniformatrix approach32 to model the nonuniform gratings.his approach involves two main steps: (i) calculating theransfer matrix corresponding to each uniform sectionTisDzi ,nd for the uniform section Dzi] by fixing properlyhe values of the coupling coefficient ki or the grating pe-iod Li and using Eqs. (10), and (ii) multiplying all theatrices corresponding to the concatenated uniform sec-

ions in the appropriate order to obtain a final 232 trans-er matrix TS that characterizes the whole grating sec-ion:

TSsz,nd = Fpi=1

n+1

TisDzi,ndG for z P Dzn+1. s13d

he elements of this final matrix are used to evaluate theulse dynamics in the device following the indicationsiven above [see Eqs. (9)].

As a general rule, the required number of sections inhe piecewise uniform approach is mainly determined byhe desired accuracy. For most of apodized or chirpedragg gratings, where the number of grating periods nor-ally exceeds 104, M,100 sections can be sufficient.32 In

his case, it is generally accepted that M cannot be maderbitrarily large since the coupled-mode theory approxi-ations that lead to Eqs. (6) are considered not valid

ig. 1. Envelope of refractive-index distribution along fiber axisand its piecewise approximation by uniform sections Dzi with aulse propagating through the Dz section.
n+1
hen a uniform grating section is only a few grating pe-iods long.32 However, in the case of transmission LPGs,here the number of grating periods is more than 1 order

f magnitude smaller, we divide the apodized and chirpedransmission gratings in N elementary cells, each one of aength equal to the grating period L. Our tests show thathis simplification is not too restrictive in practice, and inact it yields results in good agreement with the analyticormulas. This numerical treatment, already used forcousto-optic and EO filter designs,33,34 allows us to simu-ate the pulse evolution along the grating length with areat level of detail and an unparalleled precision.

. NUMERICAL RESULTS ANDISCUSSIONShe space–time and space–wavelength diagrams that wese in our analysis here provide useful information re-arding the evolution of the input pulse as it propagateshrough the grating structure. In other words, these dia-rams offer a deeper insight into the physics that deter-ine the observed pulse-reshaping process along the

rating length. This is especially true if the informationrovided by the two diagrams is conveniently correlated.It is important to note that the wavelength and tempo-

al responses of a given GACC evaluated at a grating dis-ance z are the same as those of an identical GACC (iden-ical grating parameters) with a length L=z (this isssociated with the fact that a GACC is a transmissiveevice). Consequently, the space–time diagram of a givenACC (e.g., uniform, apodized, chirped) also shows all theifferent temporal shapes that can be achieved with thatpecific kind of GACC (e.g., by simply fixing a differentrating length).

A general aspect of the obtained diagrams is that thepace–wavelength diagrams corresponding to the sup-lier and receptor modes (for the same grating structure)re simply complementary images. This is due to poweronsiderations, uHrsldu=1− uHssldu. The same is not validor the space–time diagrams. In general, these diagramsre different for the supplier and receptor modes. In otherords, the temporal pulse evolves differently in the two

nteracting modes, and in fact the temporal evolution inne of the interacting modes cannot be inferred from theemporal evolution in the other mode. Note that this in-eresting feature has its origin in the different spectralhase behavior of the supplier and receptor modes.

. Uniform Gratingigures 2 and 3 show the results corresponding to aACC with a uniform grating perturbation of period L40 mm. Specifically, the grating length is L=15 mm and

he coupling strength k is fixed so that kL=p. In theseimulations, the effective index of the energy supplierode (or supplier guide) was set to ns=2 and that of the

nergy receptor mode was set to nr=1.96115. This en-ures that the phase-matching condition between thesewo modes is exactly satisfied at a wavelength of 1555m. The indices were fixed to the given values targeting

ntegrated tunable GACC designs based on EO materials

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e.g., LiNbO3). In all the simulations presented here, thenput pulse was assumed to be a transform-limitedaussian pulse centered at the resonance wavelength of

he GACC filter (1555 nm), with a peak intensity of 100arbitrary units) and a time width depending on the grat-ng under analysis (in all cases, the input pulse band-idth was fixed to be slightly broader than the GACC

pectral response in the receptor mode so as to analyzehe so-called ultrashort pulse propagation regime). In thease of the uniform GACC, the FWHM time width of thenput pulse was fixed to 500 fs.

Figure 2 shows the results for the supplier mode, andig. 3 shows the results for the receptor mode. In bothases the corresponding space–time and space–avelength two-dimensional (2D) diagrams are pre-

ented. These diagrams were computed following theathematical procedure described in Section 2. In the

ig. 2. Space–time (left) and space–wavelength (right) 2D diagrith a uniform grating. The corresponding temporal (left) and sp

he grating length are also shown in the 1D plots (solid curves). Fdashed curves).

iagrams, different intensity levels are represented byifferent colors, according to the color maps shown in thedjacent bars. It is worth mentioning that for representa-ion purposes, the temporal axis in the 2D space–timeiagrams was reframed as follows: trepresentation= tphysicalsz /n0d, where the parameter n0 was fixed in each case tonsure that the information is represented in a compactormat while showing all the relevant details of the pulseropagation process.In addition, the temporal and spectral responses at

hree specific, representative locations along the gratingength are also shown in each case (solid curves). For com-arison, the input pulse is shown in these 1D plots as welldashed curves). Notice that the optical temporal intensi-ies are represented in normalized units, but one has toeep in mind that the output pulses have N times lowermplitude than the input pulse, where the respective val-

optical pulse propagation through the supplier mode of a GACC(right) responses at three specific, representative locations alongparison, the input pulse is represented in these 1D plots as well

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es of N are given in the corresponding 1D plots (note fur-her that the 1D temporal waveforms are represented asfunction of the physical time).As a result of the continuous interaction between the

eceptor and supplier modes during the pulse propagationlong the GACC, the corresponding temporal responsesndergo a strong reshaping process. This pulse reshaping

s essentially determined by the difference in group veloc-ty between the optical radiation propagating within theupplier mode and that propagating within the receptorode (slower and faster modes, respectively, in our ex-

mple).Initially when the input pulse enters the grating, the

nergy in the pulse is coupled from the supplier mode intohe receptor mode. At the resonance wavelength (1555m) the supplier mode power is fully coupled into the re-eptor mode over the first half of the grating length L /2

ig. 3. Space–time (left) and space–wavelength (right) 2D diagrith a uniform grating. The corresponding temporal (left) and sp

he grating length are also shown in the 1D plots (solid curves). Fdashed curves).

p / s2kd. This same energy is fully coupled back into theupplier mode over the second half of the grating length.n contrast, the energy of light at wavelengths outsideesonance is exchanged between the supplier and the re-eptor modes only partially and with a shorter periodicityalong the grating length). For example, only 12% of theptical power at 1543 and 1565 nm (maxima of the firstidelobes) is exchanged between the supplier and receptorodes, and three full cycles of energy exchange are com-

leted over the whole grating length L=p /k.In the first part of the device, the energy in the optical

ulse is being coupled from the supplier mode (slowerode) into the receptor mode (faster mode) in such a way

hat the optical radiation coming from the supplier moderrives into the receptor mode with a temporal delay withespect to the radiation already propagating through theeceptor mode. This process is essentially responsible for

optical pulse propagation through the receptor mode of a GACC(right) responses at three specific, representative locations alongparison, the input pulse is represented in these 1D plots as well

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he slight temporal broadening observed in the receptoremporal pulse along the initial section of the grating.rom a given location within the grating, some of the en-rgy outside the resonance region in the receptor modetarts to be coupled back gradually into the supplierode. Because of the velocity difference between theodes, the recoupled radiation from the receptor mode

nticipates temporally to the radiation already in the sup-lier mode, thus giving rise to the double-pulse structurebserved in the supplier mode. Initially, the pulse associ-ted with radiation recoupled from the receptor mode (theeading pulse in the sequence) is obviously less intensehan the pulse associated with radiation already in theupplier mode (the trailing pulse in the sequence). How-ver, as the light continues propagating through the de-ice, more energy is coupled back from the receptor modento the supplier mode, thus feeding the leading pulse inhe supplier temporal sequence. Simultaneously, an im-ortant part of the energy in the trailing supplier pulse isoupled into the receptor mode. The combination of thesewo processes leads to the observed intensity decrease (in-rease) of the trailing (leading) pulse in the sequence. Inhis way, in the final section of the device, the intensity ofhe leading pulse becomes higher than that of the trailingulse (there is a point within the device where both pulsesre equalized in intensity).The described energy transference dynamics is also re-

ponsible for the observed evolution of the temporal shapen the receptor mode. As detailed above, the recoupled en-rgy from the receptor mode into the supplier modeainly comes from the edges of the main spectral lobe of

he radiation in the receptor mode (this recoupled energypproaches the phase-matching wavelength as the pulseontinues propagating through the grating). This is evi-enced by the space–wavelength diagram, where it can bebserved that the main spectral lobe in the receptor modeecomes narrower with the grating distance. This spec-ral narrowing translates into the observed smoothingnd duration increase of the associated temporal wave-orm. In particular, as the pulse propagates through therating, the receptor mode first evolves into a nearlyquarelike temporal shape (with a more or less significantipple on top depending on the observation distance). Asnticipated, after a certain point within the device szL /2d, most of the recoupled energy from the receptorode into the supplier mode comes from the optical fre-

uencies around the phase-matching (or resonance) wave-ength, and these frequencies mainly lie in the centralortion of the associated temporal waveform. The con-inuation of this recoupling process causes a depletion ofnergy in the central portion of the receptor temporalaveform is such a way that this waveform evolves sub-

equently into a symmetric double-pulse temporal struc-ure (with two separated optical pulses). In the secondalf of the device, some energy is again coupled from theupplier mode into the receptor mode. As mentionedbove, most of this energy lies in spectral sidelobes farrom the phase-matching wavelength. Thus an importantart of this coupled energy has first traveled through bothhe receptor and the supplier modes so that the averagepeed of this coupled energy is in between the speeds ofhe interacting modes. As a result, this energy, when re-

oupled into the receptor mode, essentially contributes tohe central part of the corresponding temporal waveform.his process is then responsible for the evolution of theeceptor temporal waveform into the three-pulse se-uence observed in the final part of the GACC device.It is also important to note that the described temporal

volution in the supplier and receptor modes is generalor any arbitrary uniform GACC, and the whole cyclee.g., square waveform, double pulse, triple pulse) is com-leted along each grating section of length L so that kLp (this would also justify our parameter choice in the ex-mples presented here). We emphasize that the temporalaveform in the receptor mode is temporally symmetrict any position along the grating length. This generalymmetry property of uniform GACCs has its origin inwo main factors: (i) all input power is launched into theupplier mode (i.e., initial absence of the radiation in theeceptor mode); and (ii) in the presence of a uniform grat-ng, the receptor mode exhibits a constant average groupelay along the whole pulse spectrum. This is true at anyosition along the grating length. It should be mentionedhat this is an intrinsic property of the receptor mode in aniform GACC, which can be inferred from the numericalnalysis of the complex field amplitude in the receptorode [T21sz ,nd in Eqs. (10)]. In contrast, the average

roup delay in the supplier mode varies strongly with fre-uency.Another important observation from our numerical re-

ults is that, in addition to the differences in the temporalhapes, the optical pulses generated from a GACC exhibitmuch shorter duration (generally more than 1 order ofagnitude) than those generated from an equivalent BG

with the same grating length). The discordance in theemporal scales that can be obtained with the BG andACC approaches (for the same grating length) is mainlyssociated with the difference in their respectiveefractive-index contrasts. The refractive-index contrasts generally more than 1 order of magnitude larger for aG sns+nr=2neffd than for a GACC sns−nrd. Note that theefractive-index contrast determines the speed differenceetween the two coupled modes, which in turn fixes theemporal resolution associated with a given gratingength. As an example, a uniform GACC similar to as thene considered here (15 mm length with a grating periodf 40 mm) could be designed (by properly fixing the cou-ling coefficient) to reshape an input Gaussian pulse intosquare pulse of <2 ps duration (see Fig. 3). For this

ame purpose, a 200 mm BG (grating period <550 nm)ould be required. This simple example clearly shows the

ignificant difference in precision requirements in the re-pective fabrication processes of LPGs and BGs for opticalulse-shaping operations.

. Apodized Gratingigures 4 and 5 show the results corresponding to anpodized uniform-period GACC. Again, space–time andpace–wavelength diagrams are used to represent theulse evolution in both the supplier (Fig. 4) and the recep-or (Fig. 5) modes. In all cases, the temporal and spectralesponses in three specific locations within the GACCength are also represented. The features of the simulatedrating are identical to those of the uniform grating in

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ubsection 3.A, except for the fact that a Gaussianpodization profile is now introduced along the gratingength. We assume the following coupling coefficient de-endence: kszd=1.73p expf−0.938sz−L /2d2g in cm−1,here the coordinate and the grating length are in centi-eters. Such an apodized grating still provides an aver-

ge value of the grating strength equal to p. The inputulse into the GACC now exhibits a FWHM time width of00 fs.The temporal pulse evolution in the supplier mode of

he apodized GACC is very similar to that of the un-podized uniform GACC (compare the results in Figs. 2nd 4). The descriptions given in Subsection 3.A are alsoalid here to explain this similar pulse evolution in theupplier mode. The pulse evolution in the receptor modef the apodized GACC, however, is different from that ofhe unapodized uniform GACC (compare the results inigs. 3 and 5). Two main differences can be pointed out:

ig. 4. Space–time (left) and space–wavelength (right) 2D diagrith an apodized grating. The corresponding temporal (left) anlong the grating length are also shown in the 1D plots.

(i) Although in a uniform GACC the temporal wave-orm corresponding to the receptor mode is always tempo-ally symmetric (along all the grating length), this is notalid anymore when the grating profile is apodized (evenf this apodization is spatially symmetric). In general, wenticipate that the temporal symmetry property in the re-eptor mode is broken when any nonuniformity is intro-uced along the grating profile (e.g., period chirp, phasehifts).

(ii) The grating apodization process precludes the evo-ution of the propagating optical pulse into the interestingemporal waveforms observed in the case of an un-podized uniform GACC (e.g., squarelike waveforms andultiple pulse sequences). Basically, in the apodizedACC, the receptor pulse is simply a slightly broadenedersion of the input pulse. This pulse is strongly reshapednly along the final part of the device (see, for instance,

optical pulse propagation through the supplier mode of a GACCtral (right) responses at three specific, representative locations

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he pulse shapes at z=12 and 14.4 mm, where temporalscillations can be observed). However, the remaining en-rgy in the receptor mode along this final section of theevice may be too low to be of interest from a practicalerspective.

In view of these results, we can conclude that the inter-sting pulse-reshaping operations observed along the re-eptor mode in a uniform GACC are mainly due to the es-ential contribution of the coupled radiation lying inpectral sidelobes far apart from the phase-matchingavelength, a contribution that does not take place in the

ase of apodization (as observed in our example in Figs. 4nd 5, the main effect of apodizing the profile of a uniformACC is precisely that of eliminating the coupling pro-

ess in the spectral sidelobes35). This is in good agreement

ig. 5. Space–time (left) and space–wavelength (right) 2D diagrith an apodized grating. The corresponding temporal (left) anlong the grating length are also shown in the 1D plots.

ith our discussions in Subsection 3. A. Thus the apodiza-ion of the grating profile drastically limits the temporaleshaping operations that can be achieved with a GACC,nd in particular it precludes the formation of the inter-sting temporal shapes observed along the receptor moden an ideal uniform GACC.

. Chirped Gratingor the chirped gratings, a linear chirp in the grating spa-ial period is permitted, and in particular, Lszd=L0+CszL /2d, where L0=40 mm is the grating period at the cen-

er of the device and C=250 nm/cm is the grating chirp.gain the grating is assumed to have a length L15 mm and the coupling coefficient is fixed so that kLp. The simulation results corresponding to this chirped

optical pulse propagation through the receptor mode of a GACCtral (right) responses at three specific, representative locations

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ACC are shown in Figs. 6 and 7. For these simulations,n input pulse of 300 fs (FWHM) has been assumed.The linear chirp in the grating period translates into an

nduced linear variation of the phase-matching wave-ength along the GACC length. In particular, the coupledpectral band shifts toward longer wavelengths as theulse propagates along the grating. This can be clearlybserved in the space–wavelength diagrams in Figs. 6nd 7. As in a uniform GACC, the propagating pulse inhe supplier and receptor modes also undergoes a strongemporal reshaping process, which is in fact a conse-uence of the continuous interaction between these twoodes as the pulse propagates along the GACC. The men-

ioned wavelength shifting determines, however, a differ-nt temporal evolution of the propagating pulse as com-ared with the uniform case. Our most significantbservations are as follows.

ig. 6. Space–time (left) and space–wavelength (right) 2D diagrith a chirped grating. The corresponding temporal (left) and sp

he grating length are also shown in the 1D plots.

In the first part of the device, the energy in the opticalulse is coupled from the supplier mode into the receptorode, and this coupled radiation arrives into the receptorode with a temporal delay with respect to the radiation

lready in the receptor mode. This process is essentiallyesponsible for the temporal broadening observed in theeceptor temporal pulse along the initial section of theACC. At the very beginning of the device the coupled en-

rgy proceeds from the whole input spectral band. How-ver, after this initial section and as the pulse propagateslong the grating, a different spectral band is coupledrom the supplier mode into the receptor mode at each dif-erent location within the LPG (as determined by therating period chirp). Since the energy proceeds from dif-erent spectral regions of the input pulse, basically theame amount of energy is coupled at every position alonghe GACC. This coupled energy arrives, however, with a

optical pulse propagation through the supplier mode of a GACCright) responses at three specific, representative locations along

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ertain delay with respect to the energy already propagat-ng through the receptor mode. This basic process is re-ponsible for the observed tendency toward intensityqualization along the waveform duration. In this way,he original pulse evolves into a nearly squarelike tempo-al shape during propagation through the initial sectionf the device. However, as the pulse continues propagat-ng through the device, this squarelike temporal shape isroadened and, more importantly, it is also slightly dis-orted. In fact, this distortion is especially significantlong the final section of the device, where the centralortion of the receptor pulse seems to accumulate aigher amount of energy. This effect is again associatedith the continuous energy exchange between the sup-lier and the receptor modes and will become clearer inur following discussions.

As in a uniform GACC, part of the energy coupled intohe receptor mode is subsequently recoupled back into the

ig. 7. Space–time (left) and space–wavelength (right) 2D diagrith a chirped grating. The corresponding temporal (left) and sp

he grating length are also shown in the 1D plots.

upplier mode. This is a process that takes place alongost of the device length. The recoupled radiation from

he receptor mode (faster mode) anticipates the radiationlready propagating in the supplier mode and is respon-ible for the formation of the observed double-pulse struc-ure. In contrast to the uniform GACC case, the twoulses in this temporal structure become highly dissimi-ar as they evolve along the grating device. Specifically,he first pulse in the sequence develops a prominent lead-ng edge that subsequently breaks into an oscillatory tem-oral structure (with a narrow, high-intensity trailingulse) as the signal approaches toward the output end ofhe grating. The origin of this different behavior (as com-ared with a uniform grating) is again associated with thehase-matching wavelength shifting along the GACCength. As mentioned above, the leading pulse in the origi-al double-pulse sequence is associated with energy,hich is being recoupled into the supplier mode from

optical pulse propagation through the receptor mode of a GACCright) responses at three specific, representative locations along

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pectral regions that were first coupled into the receptorode along the immediate previous sections of the device.

n contrast, the trailing pulse in this same sequence isostly associated with energy still to be coupled into the

eceptor mode. Thus, as in a uniform grating, the leadingtrailing) pulse increases (decreases) as the signal propa-ates along the device. Because of the very limited band-idth of the radiation being coupled between the suppliernd receptor modes (and vice versa), this energy exchangeccurs, however, much more slowly than in the uniformase. In the final section of the device, part of the energyn the leading pulse of the supplier waveform is recoupledgain into the receptor mode (this is energy that now liesn the edges of the main coupled spectral band), thus giv-ng rise to the mentioned temporal oscillatory structure inhe supplier mode. The energy recoupled back into the re-

ig. 8. Space–time (left) and space–wavelength (right) 2D diagrith a uniform grating having a p shift in its center. The corres

epresentative locations along the grating length are also shown

eptor mode along the final section of the device mostlyccumulates around the central portion of the receptoremporal waveform.

. Phase-Shifted Gratingigures 8 and 9 show the results corresponding to ahase-shifted GACC. The features of the simulated grat-ng are identical to those of the uniform LPG described inubsection 3.A, except for the fact that a discrete p phasehift is now introduced at the center of the grating (i.e., at=L /2). The input pulse into the GACC now exhibits aWHM time width of 500 fs. Again, space–time andpace–wavelength diagrams are used to represent theulse evolution in both the supplier (Fig. 8) and the recep-or (Fig. 9) modes. The temporal and spectral responses

optical pulse propagation through the supplier mode of a GACCg temporal (left) and spectral (right) responses at three specific,1D plots.

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orresponding to three specific locations within the GACCre also shown.The temporal pulse evolution along the first half of the

hase-shifted GACC is identical to that of a uniformACC. As a result, the descriptions given in Subsection.A are also valid here to explain the pulse propagationynamics along the first half of the device (for both theupplier and the receptor modes). However, as expected,he presence of a p shift at the center of the LPG affectsignificantly the pulse propagation dynamics along theecond half of the device, this dynamics being in fact veryifferent from that observed for a uniform GACC.As for the case of continuous nonuniformities (e.g.,

rating apodization or period chirping), the introductionf a discrete nonuniformity (p phase shift) in the gratingrofile also breaks the temporal symmetry property ob-erved for an ideal uniform LPG (in the receptor mode).ore specifically, the temporal waveform in the receptor

ig. 9. Space–time (left) and space–wavelength (right) 2D diagrith a uniform grating having a p shift in its center. The corres

epresentative locations along the grating length are also shown

ode evolves now into two separated, consecutive opticalulses of different intensities, where the intensity ratioetween the two pulses changes along the LPG length.his behavior resembles that observed in the supplierode of a uniform LPG, and its origin is again associatedith the continuous energy exchange between the sup-lier and the receptor modes (in combination with theell-known difference in group delay between these two

nteracting modes). For instance, in the final section ofhe device, we observe an intensity increase in the leadingulse as it is fed by radiation lying in the spectral side-obes far from the phase-matching wavelength (secondarypectral sidelobes). Obviously, this radiation is recoupledgain from the supplier mode (most of this radiation hadlready propagated within the receptor mode along theery first section of the LPG). In contrast, the energy inhe trailing pulse mainly lies around the phase-matchingavelength and the first spectral sidelobes (edges of the

optical pulse propagation through the receptor mode of a GACCg temporal (left) and spectral (right) responses at three specific,1D plots.

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pectral mainlobe) of the propagating radiation. Whereashe energy around the phase-matching wavelength re-urns to the supplier mode along the second half of the de-ice, the spectral sidelobes are first fully coupled from theupplier mode (100% coupling) and partially recoupledack into this same mode along the final section of theACC. Interestingly, as a result of the described energy

xchange, the two pulses in the receptor mode are nearlyqualized in intensity at the output end of the GACC.

The complexity of the mode interactions and energy ex-hanges induced by a simple discrete p phase shift areurther evidenced when the pulse propagation dynamicsre analyzed in the supplier mode. The two-pulse struc-ure typical of a uniform grating is again formed along therst half of the device, but this structure undergoes alightly different evolution along the second half of therating. In particular, after the p phase shift, the leadingulse in the supplier mode is practically depleted. This isue to the fact that this leading intensity peak is associ-ted with radiation lying in the edges of the spectralainlobe, and this radiation is fully coupled back into the

eceptor mode just after the p phase shift. However, inhe final section of the device, the radiation around thehase-matching wavelength is fully recoupled back fromhe receptor mode, thus feeding again the leading pulse inhe supplier mode and giving rise to the observed energyncreasing in this leading pulse (this radiation has trav-led mostly along the receptor mode and this is why it isuch anticipated to the rest of the radiation in the sup-

lier mode). On the other hand, the trailing pulse in theupplier mode is associated with radiation that lies in thepectral sidelobes (and edges of the spectral mainlobe). Asentioned above, this radiation is coupled into the recep-

or mode after the p phase shift, leading to the corre-ponding intensity depletion in this trailing pulse alonghe second half of the grating device.

. CONCLUSIONSn this paper, ultrashort pulse propagation through uni-orm and nonuniform waveguide LPGs (i.e., GACCs) haseen analyzed in detail. For this purpose, we have madese of advanced representation tools, namely, space–timend space–wavelength diagrams; and by correlating thenformation provided by these two diagrams, we have pro-ided a deep and clear insight into the physics that deter-ines the optical pulse evolution in the analyzed grating

evices. In general, we have observed that as the pulseropagates through the LPG structure, it undergoes atrong temporal reshaping. This is true for both the sup-lier and the receptor modes, but we have shown thathese two modes exhibit, in fact, a very different behaviorn the temporal domain. As evidenced by our numericalesults and representations, the evolution of the originalemporal pulse into the described different temporalhapes has its origin in the continuous interaction (en-rgy coupling) between the supplier and the receptorodes and is mostly associated with the difference in

roup velocity between these two modes.For instance, we have demonstrated that the pulse

ropagating through an ideal uniform LPG evolves intoifferent temporal shapes of practical interest, e.g., tem-

oral reshaping into square temporal waveforms and se-uences of equalized multiple pulses can be achieved inhe receptor mode. An outstanding property of the tempo-al waveforms in the receptor mode of a uniform LPG ishat they are always temporally symmetric at any posi-ion along the device length. The introduction of anpodization profile in the LPG profile will break this tem-oral symmetry in the receptor mode while precluding thebservation of the mentioned temporal shapes of practicalnterest. In either uniform or apodized cases, the tempo-al waveform in the supplier mode evolves into two sepa-ated pulses that appear equalized in intensity only at apecific location within the LPG. As expected, chirping therating period also affects the temporal pulse evolution inhe supplier and receptor modes. One outstanding featuref a chirped LPG is that the pulse can be reshaped into aearly squarelike temporal waveform in both the suppliernd the receptor modes. However, the formation of equal-zed multiple pulse sequences is generally not possiblehen a chirped LPG is used. The introduction of a simpleiscrete p phase shift in the grating structure also affectsrastically the pulse propagation dynamics along theACC device in both the supplier and the the receptorodes. For instance, in the case of a p phase shift located

ust at the center of the grating, the temporal waveformn the receptor mode evolves into two separated pulseslong the second half of the device. These two pulses areearly equalized in intensity at the output end of theACC.The results presented here are not only interesting

rom a physical perspective but can also have importantractical implications. For instance, our results clearlyhow the strong potential of LPG-based devices for opticalulse manipulation applications and point at the LPG ap-roach as an interesting alternative to implement inte-rated (or in-fiber) optical pulse shapers. We emphasizehat the time duration associated with a given LPGength is generally much shorter (more than 1 order of

agnitude) than the time duration corresponding to thequivalent BG (with the same length). This would makehe LPG approach a much more suitable solution for syn-hesizing temporal shapes in the picosecond–ubpicosecond regime (faster than a few picoseconds), aemporal regime that cannot be easily resolved with BGechnology.

CKNOWLEDGMENThis research was supported by the Natural Sciences andngineering Research Council of Canada (NSERC).

The e-mail address for M. Kulishov [email protected]. The e-mail address for

. Azaña is [email protected].

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ultrashort pulse propagation in uniform and nonuniform waveguide long-period gratings

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