Slow-light dispersion in coupled periodic waveguides

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Sukhorukov et al. Vol. 25, No. 12 /December 2008 /J. Opt. Soc. Am. B C65

Slow-light dispersion in coupled periodicwaveguides

Andrey A. Sukhorukov,1,* Andrei V. Lavrinenko,1,2 Dmitry N. Chigrin,3 Dmitry E. Pelinovsky,4 and Yuri S. Kivshar1

1Nonlinear Physics Centre and Centre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS),Research School of Physical Sciences and Engineering, Australian National University, Canberra,

ACT 0200, Australia2DTU Fotonik, Department of Photonics Engineering, NanoDTU, Technical University of Denmark, Building 345V,

DK-2800 Kongens Lyngby, Denmark3Physikalisches Institut, Universität Bonn, Nussallee 12, D-53115 Bonn, Germany

4Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1, Canada*Corresponding author: [email protected]

Received April 17, 2008; revised July 24, 2008; accepted July 25, 2008;posted July 28, 2008 (Doc. ID 95079); published September 18, 2008

Periodic waveguides bring a new twist to the typical waveguiding problems of the intermediate case betweenphotonic crystal waveguides and photonic wires or ridge waveguides. We develop an asymptotic theory appli-cable for a broad class of coupled periodic waveguide structures and use the analytical expressions to identifythe generic types of dispersion in the vicinity of a photonic band edge, where the group velocity of light isreduced. We show that the dispersion can be controlled by the longitudinal shift between the waveguides. Wealso demonstrate through finite-difference time-domain simulations examples of spatial and temporal pulsedynamics in association with different types of slow-light dispersion. © 2008 Optical Society of America

OCIS codes: 050.5298, 230.7370, 250.5300.

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. INTRODUCTIONhe physics and applications of slow light have been at-racting increasing attention in recent years. In particu-ar, the strong reduction of the pulse velocity by orders of

agnitude was demonstrated experimentally in periodi-ally modulated photonic structures [1–7]. Flexible con-rol over the light propagation in microscale structuresay lead to the realization of ultra-compact delay lines,ith the additional capability of flexible pulse shaping.

ndeed, the slow-light regime can be realized when theulse spectrum is tuned close to the edge of a photonicandgap, where the dispersion curves feature a turningoint and the group velocity is reduced to zero. On thether hand, in this spectral region the group-velocity dis-ersion becomes significant, defining the reshaping of theropagating pulse [8]. It was suggested that the slow-ight dispersion can be controlled in specially designedtructures [9–11], with the possibilities of obtaining eitheruartic band edges or split band edges where slow-lighttates feature nonvanishing phase velocity. Most recently,t was demonstrated, based on general symmetry analy-is, that the latter regime corresponds to the appearancef two distinct slow-light modes, for which group velocitys automatically matched in the same frequency region12]. The simultaneous excitation and beating of suchodes opens new possibilities for the spatiotemporal con-

rol of pulse propagation, including the operation of pho-onic crystal couplers for slow-light pulses [12].

In this work, we present a general analysis for the re-lization of different slow-light dispersion regimes by se-ecting a longitudinal shift between directionally coupled

0740-3224/08/120C65-10/$15.00 © 2

eriodic waveguides (as indicated by parameter z0 in thexamples of the structures sketched in Fig. 1). The keyossibilities of dispersion engineering with quadratic,uartic, or split band edges are illustrated in Fig. 2. Themportant effect of the photonic structure symmetry onhe mode dispersion and bandgap spectrum was recog-ized in previous studies of Bragg grating couplers13–17], photonic crystal waveguides [12,18], and coupledanopillar waveguides [19–22]. Our analytical expres-ions are valid for a broad range of photonic structures,ncovering the essential physics underlying the similari-ies of slow-light dispersion characteristics previously ob-ained in the framework of distinct analytical and nu-erical methods. We show that the most flexible control

ver the band-edge dispersion can be obtained when theongitudinal shift can be varied continuously. In two-imensional photonic crystals, the shift may be con-trained to particular values by the symmetry of the un-erlying periodic lattice. Whereas the shift is arbitrary inoupled Bragg grating waveguides, the weak refractivendex contrast imposes limits on the structure size [5,23],hich may be of the order of centimeters. On the otherand, the system of coupled nanopillar or one-imensional photonic crystal waveguides offers the com-ined advantages of unrestricted dispersion controlhrough nonconstrained positioning of individual nanopil-ars or holes, as illustrated in Fig. 1, and the capability ofwitching pulses in structures of micrometer dimensionsecause of the strong field confinement facilitated by thearge refractive index contrast. We present examples ofispersion dependencies in coupled nanopillar and photo-

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C66 J. Opt. Soc. Am. B/Vol. 25, No. 12 /December 2008 Sukhorukov et al.

ic wire waveguides, confirming that fundamentally dif-erent band-edge profiles can be obtained by varying justhe shift between the identical periodic waveguides.

Whereas different approaches to the control of band-dge dispersion have been discussed in the literature,here is limited research on pulse shaping under suchonditions. We perform finite-difference time-domainFDTD) simulations of pulse dynamics and identify novelegimes of spatiotemporal pulse shaping as the inputavelength it tuned close to different spectral regions. Inarticular, we observe transient spatial switching be-ween the waveguides due to the pulse splitting into slownd regular modes with different spatial profiles. We alsoegister asymmetric reshaping of slow-light pulses whenhe band-edge shape is selected close to a quartic profile.e anticipate that such tunable linear dynamics may

ead to new possibilities for the frequency conversion pro-esses and all-optical switching [5,17], since the nonlinearave mixing is enhanced in the slow-light regime [24,25].The paper is organized as follows. In Section 2, we de-

ive analytically the shapes of band-edge dispersion foroupled periodic waveguides. Then, in Section 3 we re-iew the basic properties of coupled nanopillaraveguides and demonstrate the engineering of slow-

ight dispersion, also presenting analysis of the spa-

ig. 1. Sketch of coupled periodic waveguides, consisting of (a)wo arrays of nanopillars or (b) two arrays of holes in coupledhotonic wire waveguides.

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(b)(a)ig. 2. (Color online) Characteristic dispersion dependencies (toand edge for (a) uncoupled periodic waveguides, (b) coupled peaveguides with the longitudinal offset by half a period �z =a /2
0
iotemporal dynamics of slow-light pulses in such struc-ures based on FDTD simulations. In Section 4 weemonstrate that similar types of slow-light dispersionan be realized in coupled photonic wire waveguides. Fi-ally, we summarize our key conclusions in Section 5.

. SLOW-LIGHT DISPERSION IN COUPLEDERIODIC WAVEGUIDESremarkable property of periodic waveguides is the pos-

ibility they afford for reducing the group velocity of opti-al pulses. The presence of the periodic refractive indexodulation along the waveguide leads to Bragg reflection

esonances and the appearance of photonic bandgaps forrange of frequencies where the light is reflected back

26–28]. In the vicinity of the band edges, the dispersionelation between the optical frequency �� and the Blochavenumber �k� is strongly modified, and it is commonlyxpressed as ��b+D2�k−kb�2. Here �b and kb denotehe values at the band edge, and D2 is the second-orderispersion coefficient. The group velocity Vg=d� /dk±2�D2��−�b��1/2 gradually reduces to zero as the fre-

uency is tuned toward the edge of a transmission band.In this section, we perform a general analysis of theodification of band-edge dispersion in photonic struc-

ures consisting of parallel waveguides. We consider two-imensional modulation of the optical refractive indexith the characteristic period a in the propagation direc-

ion �z�, with the dielectric constant profile written as

��x,z� = �p�x,z� + �w�x,z� + �w�− x + x0,− z + z0�, �1�

here x and z are the transverse and longitudinal coordi-ates, respectively; �w�x ,z� describes the profile of aingle waveguide; �x0 ,z0� is the relative offset of the sec-nd waveguide; and the function �p�x ,z� describes theackground refractive index. Then, the periodicity condi-ion is expressed as

�p�x,z� � �p�x,z + a�, �w�x,z� � �w�x,z + a�. �2�

e also require that the background index has exactlyhe same profile after the reflection and coordinate shift:

Mode 1Mode 2

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(c)and the corresponding group velocities (bottom row) close to thewaveguides with no longitudinal offset �z0=0�, and (c) coupled

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�p�x,z� � �p�− x + x0,− z + z0�. �3�

e note that these conditions are satisfied for a variety ofifferent photonic structures with coupled periodicaveguides, including the following situations:

(i) �p�x ,z�=constant, which is the case for coupledragg grating waveguides [17], arrays of nanopillars

19–22], or coupled one-dimensional photonic crystalaveguides shown schematically in Fig. 1.(ii) �p�x ,z� describes a hexagonal photonic crystal

tructure, and x0 and z0 correspond to the basis vectors ofhe photonic lattice [12].

To be specific, in the following we consider the TModes (electric field polarized in the y axis), and similar

nalysis can be performed for TE-polarized waves. Then,he dispersion of guided modes can be determined basedn the scalar Helmholtz equation for the complex electriceld envelopes E�x ,z�,

�E + ��x,z��2c−2E = 0, �4�

here � is the angular frequency and c is the speed ofight in vacuum. First, we define the modes of individualaveguides with �1�x ,z�=�p�x ,z�+�w�x ,z� or �2�x ,z��p�x ,z�+�w�−x+x0 ,−z+z0�. Due to the periodicity condi-

ion (2) and additional symmetry (3), the waveguides sup-ort equivalent sets of modes, which satisfy the Blochondition:

E1�x,z;�1,k� = �1�x,z;�1,k�exp�ikz�,

E2�x,z;�2,k� = �2�x,z;�2,k�exp�ikz�, �5�

here the subscripts 1 and 2 refer to the correspondingaveguides, � are the eigenfunctions that are localized inand periodic in z, and k is the Bloch wavenumber. The

ispersion dependence �1�k� and �2�k� can be determinedy solving the wave equation (3), using well-establishedpproximate analytical or exact numerical methods26–28]. We note that in dielectric structures �j�k�

�j�−k� and �j�x ,z ;�1 ,k��j*�x ,z ;�1 ,−k�, where � stands

or complex conjugation. On the other hand, due to theymmetry conditions, �1�k��2�−k�, and we can alwayshoose the absolute phase to satisfy �2�x ,z ;�1 ,k��1�−x+x0 ,−z+z0 ;�1 ,−k�. Then, we conclude that

�1�±k� � �2�±k�,

�2�x,z;�1,k� � �1*�− x + x0,− z + z0;�1,k�. �6�

Now we seek solutions for the coupled-waveguide struc-ure defined by Eq. (1) as a combination of individualodes with the amplitudes A1 and A2:

E�x,z;�,k� = A1E1�x,z;�1,k� + A2E2�x,z;�1,k�. �7�

e substitute Eq. (7) into Eq. (4), multiply the resultingquation by E1

* or E2*, and integrate over the area (−�

x� +�, 0�z�a). Then, we obtain a set of two equationsor the amplitudes A1 and A2, which have a nontrivial so-ution when the following condition is satisfied:

��b1 + �12b2�2 = ��b3 + �1

2b4�2. �8�

ere �=�2−�12 defines the frequency detuning due to

aveguide coupling, and the coupling coefficients are

b1 = ��p�x,z� + �w�x,z� + �w�− x + x0,− z + z0��

��1�x,z;�1,k��2,

b2 = ��w�− x + x0,− z + z0��1�x,z;�1,k��2,

b3 = ��p�x,z� + �w�x,z� + �w�− x + x0,− z

+ z0��1�x,z;�1,k�

�1�− x + x0,− z + z0;�1,k�,

b4 = ��w�− x + x0,− z + z0��1�x,z;�1,k�

�1�− x + x0,− z + z0;�1,k�, �9�

here the brackets � denote −�+�dx0

adz� �.We consider the case of weakly coupled waveguides.

hen, we estimate that b1�O�1�, b2,3,4�O��, and �O��, where is a small parameter. By keeping the

owest-order terms in Eq. (8), we obtain the followingsymptotic relation:

��b1 + �12b2�2 � �1

4�b4�2. �10�

hen we determine the dispersion relation analytically:

� = �2 − �12 � �1

2�− b2/b1 ± �b4�/b1�. �11�

We assume that the individual waveguides have theommonly occurring parabolic shape of the dispersionurves at the edge of the Brillouin zone,

�1�k� � �1b + D2�k − k1b�2, �12�

here k1b=� /a. We introduce the notation �=k−k1b andxpand the coupling coefficients in �:

b1 � b1b + b1b2�2,

b2 � b2b + b2b2�2,

b4 � b4b + ib4b1� + b4b2�2. �13�

We substitute Eqs. (12) and (13) into Eq. (11), take intoccount that �2−�1

2�2��−�1��1, and obtain the final ex-ression for mode dispersion in coupled waveguides closeo the band edge, in the regime of slow-light propagation:

� � ��1b + D2�2�

�1 −b2b + b2b2�2 ± ��b4b + b4b2�2�2 + �2b4b1

2 �1/2

2�b1b + b1b2�2� � .

�14�

We now analyze particular cases, considering symmet-ic waveguides with �p,w�x ,z��p,w�−x ,z��p,w�x ,−z�:

(i) If there is no shift between the waveguides, z0=0,hen b =0, and
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� � ��1b + D2�2��1 −b2b + b2b2�2 ± �b4b + b4b2�2�

2�b1b + b1b2�2� � .

�15�

n this case, the dispersion branches are split vertically,s illustrated in Fig. 2(b). At the band edge, there appears

single slow-light mode, featuring conventional qua-ratic dispersion dependence.(ii) If the waveguides are shifted by half of the modu-

ation period, z0=a /2, then b2b=b4b=0, and

� � ��1b + D2�2��1 −b2b2�2 ± ��b4b1�

2�b1b + b1b2�2� � . �16�

n this case, the dispersion branches predominantly splitorizontally, as illustrated in Fig. 2(c). Accordingly, the so-alled split band edge appears, which is associated withhe coexistence of two distinct slow-light modes in theame frequency region as the photonic band edge.

(iii) By choosing the intermediate shifts between theaveguides, it becomes possible to tailor the dispersionependencies, as they are transformed between the casesi) and (ii). We elaborate more on this possibility in Sec-ion 3.

. SLOW-LIGHT DYNAMICS INISPERSION-ENGINEERED COUPLEDANOPILLAR WAVEGUIDESe now demonstrate how the band-edge dispersion can

e engineered in coupled nanopillar waveguides, shownchematically in Fig. 1(a). The potential for practical re-lization of nanopillar waveguide structures was demon-

0.35 0.4 0.45 0.5 0.55 0.6

0.32

0.34

0.36

0.38

0.4

0.42

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Mode 1Mode 2

0.35 0.4 0.45 0.5 0.55 0.6 0Wavenumber

-0.5

0

0.5

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(b)

(c)

z

xRods offset 0

(a)

ig. 3. (Color online) (a) Transverse refractive index profile of coower and upper rows of nanopillars �z0=0�. (b) Dispersion depenwo fundamental guided modes, shown with solid (mode 1) andpectrum used in FDTD numerical simulations. (c) Dependence oumber. (d), (e) Characteristic amplitude and phase profiles of theloch wavenumber k=0.475.

trated in experimental studies [29–32]. For the numeri-al simulations presented below, we solve Maxwell’squations in a two-dimensional geometry for TM-olarized electromagnetic waves (with the electric fieldolarized along the y axis). We choose the nanopillar di-meter 0.29 a, the material refractive index 2.5, and theransverse separation between the coupled waveguides0=a. For convenience, we normalize the time to =a /cnd the Bloch wavenumber to 2� /a, and the frequency iseasured in multiples of a /�, where � is the wavelength

n vacuum. In FDTD simulations, we use pulses with theull-width at half-maximum of the frequency spectrumqual to 0.5% of the central frequency. We estimate that,o observe the slow-light propagation at 1.55 �m, one canse a structure with a period of a�0.62 �m, a nanopillariameter of 180 nm, a time scale of �2.07 fs, and a pulseuration of 655 fs.

. Symmetrically Coupled Waveguides with Noongitudinal Shifte start by considering the case of zero longitudinal off-

et �z0=0�, when the photonic structure is symmetric ashown in Fig. 3(a). Calculations of the dispersion depen-encies were performed using the freely available MIThotonic Bands (MPB) software package utilizing thelane-wave expansion method [33]. In agreement withhe general analysis presented in Section 2, the disper-ion branches are split vertically [see Fig. 3(b)]. Accord-ngly, the slow-light regime is realized for both modes athe edge of the Brillouin zone, corresponding to the samealue of the normalized wavenumber �k=0.5� [see Fig.(c)]. We note that the first mode has an odd symmetrysee Fig. 3(d)], such that the electric field has a � phasehift between the waveguides. On the other hand, the sec-

Mode 1 − Amplitude Phase


nanopillar waveguides, with zero longitudinal offset between thes of the normalized frequency on the Bloch wavenumbers for thed (mode 2) lines, respectively. Gray shading indicates the pulseormalized group velocity of the guided modes on the Bloch wave-s, corresponding to the marked points in plots (b) and (c) with the

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nd mode has an even symmetry [see Fig. 3(e)], and theoupled waveguides are in phase. This situation is similaro the mode structure in a conventional directional cou-ler [34–36], indicating the possibility of light switchingetween the parallel waveguides. However, the mode cut-ffs and the associated reduction of their group velocitiesppear at different frequencies, and we perform FDTDimulation to investigate the pulse dynamics in the sys-em.

As a characteristic example demonstrating the effect ofeating between the modes on the pulse dynamics, weune the pulse frequency close to the cutoff of the secondode. The pulse spectrum is shown with a gray-shaded

tripe in Fig. 3(b), at the frequency around ��0.3784. Weee that, in this frequency region, both modes can propa-ate in the structure. However, the two modes have sig-ificantly different group velocities, since the slow-lightropagation is realized only for the second mode. Indeed,ur simulations demonstrate that the leading edge of theulse has an odd symmetry, associated with the faster-ropagating first mode [see Fig. 4(a)]. At a later time [Fig.(b)], periodic switching of light is observed along theaveguides, due to the transient overlap between the

lowly propagating even mode and the odd-mode state.inally, at longer times only the slow even mode remains

nside the structure [Fig. 4(c)], and the spatial switchinggain disappears.

. Coupled Waveguides Shifted by Half a Periode now consider the light dynamics in the case when theaveguides are shifted longitudinally by half of the pe-

iod, as shown in Fig. 5(a). In agreement with the generalredictions of Section 2, now the dispersion curves splitorizontally [21], resulting in the appearance of the splitand edge [see Figs. 5(b) and 5(c)]. In this case, there ap-ear two band edges with different wavenumbers, equallyeparated from the Brillouin zone edge. However, the cut-ff frequencies for the two modes now coincide. In agree-

z

x

Rods offset 0; Normalized frequ

(a)

(c)

(b)

ig. 4. (Color online) Pulse dynamics in the coupled nanopill0.3784, and the pulse spectrum truncated at full-width at hal

uency region in Fig. 3(b). Shown are the snapshots of the electr

ent with the previous results for other types of couplederiodic waveguides [12,17], we observe that this type ofispersion also facilitates switching of slow-light pulses inoupled nanopillar waveguides. Although the modal pro-les are now neither even nor odd [see Figs. 5(d) and 5(e)],heir beating still results in the periodic tunneling of lightetween the waveguides, as confirmed by the numericalimulations presented in Fig. 6. We see that the pulse fol-ows a sinusoidal trajectory, as it periodically switches be-ween the waveguides during the propagation along thetructure. Such a characteristic pulse trajectory is pre-erved at all time steps, and this is only possible becauseoth modes propagate in the slow-light regime with theame group velocities.

. Coupled Waveguides with Intermediate Longitudinalhiftinally, we consider the case of intermediate longitudinalhifts between the coupled periodic waveguides. By con-inuously varying this structural parameter, we can flex-bly control the band-edge dispersion. In particular, forhe structure shown in Fig. 7(a), we realize the situationhen the first mode dispersion becomes almost flat close

o the band edge [see Fig. 7(b)]. As a result, two zero-roup-velocity regions appear at frequencies in the vicin-ty of the band edge [see Figs. 7(c)]. Since the coupledaveguide structure is no longer symmetric or antisym-etric, the modes feature skewed amplitude and phase

rofiles [see Figs. 7(d) and 7(e)].We simulate the pulse dynamics when its central fre-

uency is tuned right at the band edge ��0.4029�, asarked with the gray-shaded stripe in Fig. 7(b). This fre-

uency is above the cutoff of the second mode; however,e observe the periodic light tunneling between theaveguides, which is characteristic of two-mode beating

see Fig. 8]. Indeed, for frequencies close to the band edge,here exist two sections of the dispersion curve above andelow the Brillouin zone edge, where the group velocities

0.3784

= 124

= 446

e = 294

veguides shown in Fig. 3(a). The central pulse frequency is �mum is shown with the gray shading at the corresponding fre-component E at different times, as indicated by labels.

ency

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or the first mode are small and positive. It is the beatingetween these states that results in the tunneling of lightetween the waveguides. In contrast to the case of longi-udinal shift by half a period discussed above in Subsec-ion 3.B, the switching between the waveguides is notymmetric. Indeed, the switching lengths from the uppero the lower waveguides and back are different, due to thesymmetry of the underlying photonic structure. Never-heless, we observe that the switching is almost completet certain propagation distances. Therefore, such asym-

0.35 0.4 0.45 0.5 0.55 0.6

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0.35 0.4 0.45 0.5 0.55 0.6 0Wavenumber

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z

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ig. 5. (Color online) Dispersion and modal characteristics for chifted by half of the longitudinal periods �z0=0.5 a�. Notations a=0.455.

z

etric structures can operate as slow-light couplers, withhe possibility for fine-tuning of the band-edge dispersion.

. PHOTONIC WIRE WAVEGUIDESe now show that the band-edge dispersion can also be

ontrolled in other types of periodic structures, consider-ng the case of coupled photonic wire waveguides shownchematically in Fig. 1(b). Such waveguides have been ex-ensively investigated in recent years, and we choose the



waveguides, where the lower and upper rows of nanopillars aresame as in Fig. 3, and the mode profiles in (d) (e) are shown for

z

x

Rods offset 0.5; Normalized frequency 0.4035

(a) Time = 134

(c) Time = 458

(b) Time = 296

ig. 6. (Color online) Pulse dynamics in the coupled nanopillar waveguides shown in Fig. 5(a). The central pulse frequency is �0.4035, and the pulse spectrum truncated at full-width at half-maximum is shown with the gray shading in Fig. 5(b). Shown are the

napshots of the electric field component E at different times, as indicated by labels.

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arameters based on the characteristics of previously fab-icated structures [37,38]. Specifically, we solve Maxwell’squations in a two-dimensional geometry for TE-olarized electromagnetic waves (with the magnetic fieldolarized along the y axis). We choose the width of indi-idual waveguides equal to the structure period a, theole diameter 0.4 a, the effective material refractive in-ex 2.798, and the transverse separation between theoupled waveguide centers x0=1.2 a. Similar to the re-ults presented for coupled nanopillar waveguides in Sec-ion 3, we normalize the Bloch wavenumber to 2� /a, andhe frequency is measured in multiples of a /�, where � ishe wavelength in vacuum. According to the results of

0.35 0.4 0.45 0.5 0.55 0.6

0.32

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0.35 0.4 0.45 0.5 0.55 0.6 0Wavenumber

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0.5

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(b)

(c)

z

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(a)

ig. 7. (Color online) Dispersion and modal characteristics for chifted by z0=0.35 a. Notations are the same as in Fig. 3, and th

z

x

Rods offset 0.35; Normalized fr

(a)

(c)

(b)

ig. 8. (Color online) Pulse dynamics in the coupled nanopill0.4029, and the pulse spectrum truncated at full-width at half

uency region in Fig. 7(b). Shown are the snapshots of the electr

37,38], the slow-light propagation at 1.5 �m can be real-zed when the period is of the order of a�0.35 �m.

Whereas the analytical analysis in Section 2 is pre-ented for TM-polarized waves, numerical simulations in-icate that the dispersion of TE-polarized modes inoupled periodic waveguides follows the same generalrends as sketched in Figs. 2(a)–2(c). Specifically, we findhat the slow-light regime appears at separate frequencyutoffs for even and odd modes when the holes in adjacentaveguides are aligned (Fig. 9). Then, quartic band-edgeispersion can be realized for a particular value of theongitudinal shift between the holes [see Fig. 10]. Finally,he split band edge is observed when the holes are shifted



waveguides, where the lower and upper rows of nanopillars aree profiles in (d) (e) are shown for k=0.475.

ncy 0.4029

= 134

= 459

e = 297

veguides shown in Fig. 7(a). The central pulse frequency is �um is marked with the gray shading at the corresponding fre-component E at different times, as indicated by labels.

.65 (e)

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Fto(p

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y half a period [see Fig. 11]. Due to the similarity of dis-ersion characteristics and the symmetries of mode pro-les with those of nanopillar waveguides, we expect thathe pulse propagation in photonic wires would mimiclosely the key features of the spatial–temporal dynamicsiscussed in Section 3.

. CONCLUSIONSe have shown that dispersion characteristics of slow-

ight at photonic band edges can be flexibly controlled in

0.35 0.4 0.45 0.5 0.55 0.60.19

0.2

0.21

0.22

0.23

0.24

0.25Fr

eque

ncy

Mode 1Mode 2

0.35 0.4 0.45 0.5 0.55 0.6 0Wavenumber

-0.4

-0.2

0

0.2

0.4

Gro

upve

loci

ty

(b)

(c)

z

xHoles offset 0

(a)

ig. 9. (Color online) Dispersion and modal characteristics for che lower and upper rows of holes �z0=0�. (a) Transverse refractivn the Bloch wavenumbers for the two fundamental guided modec) Dependence of the normalized group velocity of the guided mhase profiles of the modes, corresponding to the marked points

0.35 0.4 0.45 0.5 0.55 0.60.19

0.2

0.21

0.22

0.23

0.24

0.25

Freq

uenc

y

Mode 1Mode 2

0.35 0.4 0.45 0.5 0.55 0.6 0Wavenumber

-0.4

-0.2

0

0.2

0.4

Gro

upve

loci

ty

(b)

(c)

z

xHoles offset 0.28

(a)

ig. 10. (Color online) Dispersion and modal characteristics foroles are shifted by z =0.28 a. Notations are the same as in Fig
0
oupled periodic waveguides, simply by changing the lon-itudinal offset. Our general theoretical analysis provides

link between the variety of previously consideredoupled guiding structures, formulating the generic basisor dispersion engineering. We further demonstrate howarious dispersion dependencies can be realized for a sys-em of coupled nanopillar waveguides comprising periodicows of dielectric rods and photonic wire waveguides withperiodic set of holes. In such structures, the longitudinal

ffset between the rods or holes can be freely varied, of-ering full flexibility in tailoring the dispersion properties.



photonic wire waveguides with zero longitudinal offset betweenprofile. (b) Dispersion dependencies of the normalized frequencyn with solid (mode 1) and dashed (mode 2) curves, respectively.

on the Bloch wavenumber. (d) (e) Characteristic amplitude ands (b) and (c) with the Bloch wavenumber k=0.475.



d photonic wire waveguides, where the lower and upper rows of

.65 (e)

(d)

ouplede indexs, showodes

in plot

.65 (e)

(d)

couple. 9.

Ofsis

sgsddatnpnepamsm

AWMmsaCwMAem

LC0

R

1

Fh otatio


ur numerical FDTD simulations illustrate distinctly dif-erent regimes of spatio-temporal pulse dynamics, demon-trating nontrivial dependencies on the frequency detun-ng from the band edge and high sensitivity to thetructure symmetry.

These results suggest several directions for furthertudies. We note that by optimizing the value of the lon-itudinal shift between the periodic waveguides, it is pos-ible to obtain quartic band-edge dispersion. It was pre-icted that such flat band edges can lead to an increasedensity of states at frequencies near the band edge [9,10],nd this may offer a potential to enhance the characteris-ics of slow-light resonators. Another parameter thateeds to be carefully addressed is the distance betweenhotonic wires or transverse offset between two rows ofanopillars providing the optimal coupling strength forffective interactions. The demonstrated control over thehase velocities and spatial profiles of slow-light modeslso suggest the potential for flexible engineering of wave-ixing processes and all-optical switching with discrete

olitons [39], taking advantage of nonlinearity enhance-ent in the slow-light regime.

CKNOWLEDGMENTSe thank S. Ha, M. de Sterke, L. Botten, K. Dossou, C.onat, and B. J. Eggleton for useful discussions and com-ents. This work was supported by the Australian Re-

earch Council under the Centres of Excellence Programnd through the Australian Partnership for Advancedomputing National Facility (APAC). Part of this projectas completed during the visit of A. A. Sukhorukov to Mc-aster University (Canada) supported by the Australiancademy of Science travel grant. D. N. Chigrin acknowl-dges partial support by the Deutsche Forschungsge-einschaft (DFG) through the project FOR 557. A. V.

0.35 0.4 0.45 0.5 0.55 0.60.19

0.2

0.21

0.22

0.23

0.24

0.25Fr

eque

ncy

Mode 1Mode 2

0.35 0.4 0.45 0.5 0.55 0.6 0Wavenumber

-0.4

-0.2

0

0.2

0.4

Gro

upve

loci

ty

(b)

(c)

z

xHoles offset 0.5

(a)

ig. 11. (Color online) Dispersion and modal characteristics foroles are shifted by half of the longitudinal periods �z0=0.5 a�. N

aurinenko acknowledges partial support from Europeanommission FP6, project NewTon, NMP4-CT-2005-17160.

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Slow-light dispersion in coupled periodic waveguides