PHYSICAL REVIEW B 15 MAY 1999-IIVOLUME 59, NUMBER 20

Defect modes in one-dimensional comblike photonic waveguides

J. O. Vasseur,* B. Djafari-Rouhani, L. Dobrzynski, and A. AkjoujLaboratoire de Dynamique et Structures des Mate´riaux Moleculaires, UPRESA CNRS 8024, UFR de Physique,

Universitede Lille I, 59655 Villeneuve d’Ascq Ce´dex, France

J. ZemmouriLaboratoire de Physique des Lasers, Atomes et Mole´cules, Centre d’Etudes et de Recherches Lasers et Applications,

UMR CNRS 8523, Universite´ de Lille I, 59655 Villeneuve d’Ascq Ce´dex, France~Received 9 October 1998!

We investigate the existence of defect modes in the photonic band structure of a comblike waveguidegeometry made of dangling side branches grafted periodically along an infinite monomode waveguide. Thisone-dimensional photonic crystal exhibits forbidden bands that originate both from the periodicity of thesystem and the resonance states of the grafted branches. The defect modes result from the presence of defectivebranches in the comb and may occur in these stop bands. The localized states appear as very narrow peaks inthe transmission spectrum of finite comblike waveguides composed of a finite number of grafted branches. Thebehavior of the localized states is analyzed as a function of the length, of the position, and of the number of thedefective branches. These theoretical results are confirmed by experiments using coaxial cables in the fre-quency range of a few hundreds of MHz.@S0163-1829~99!04519-1#

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I. INTRODUCTION

For the last ten years, numerous theoretical and expmental investigations have focused on the existence of gin the electromagnetic band structure of thephotonic crys-tals, i.e., artificial structures exhibiting a spatially dependedielectric constant. Various one-dimensional~1D!, 2D, and3D structures of periodic photonic crystals were studie1

Gaps were also observed in slightly disordered2,3 as well asin quasiperiodic4 2D photonic crystals. In these forbiddebands, electromagnetic modes, spontaneous emission,zero-point fluctuations are all absent.5 These properties become more pronounced when the band gap is made laSome studies have also addressed the problem of the egence of localized states in the photonic band gaps~PBG! byintroducing defects in the periodic dielectric structure. In 1photonic crystals, i.e., classic Bragg mirrors,6,7 defects maybe layers of different nature than the other slabs. In 2D3D photonic crystals made of a periodic distribution of delectric cylindrical or spherical inclusions embedded in aelectric matrix, defect modes were observed by removingadding some inclusions8–10or by changing the characteristic~material or diameter! of several inclusions.11 These proper-ties started also to be investigated in quasi-one-dimensiphotonic crystals.12 Photonic crystals possessing localizmodes in the forbidden bands should have several apptions to optical devices such as selective frequency filtervery efficient waveguides.

In two previous papers,13,14 we proposed a model ofone-dimensional photonic crystal exhibiting very narropass bands separated by large forbidden bands. This syis composed of an infinite one-dimensional waveguide~thebackbone! along which stars ofN8 finite side branches argrafted atN equidistant sites,N and N8 being integers~seeFig. 1!. This star waveguideis described by two structuraand two compositional parameters, namely the periodi

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d1, the lengthd2 of each side branch and the relative dieletric permittivity « i of each medium withi 51 for the back-bone andi 52 for the side branches. The one-dimensionnature of the model requires that the lengthsd1 andd2 mustbe very much greater than the diameter of the backbonethe side branches. This retains its validity to monomopropagation of electromagnetic waves. The stop bands onate both from the periodicity of the system and the renance states of the grafted branches, which play the rolresonators. Wide gaps/narrow bands can be obtained bappropriate choice of the parameters, in particular the rbetween the two characteristic lengthsd1 andd2. Moreover,by increasing the numberN8 of side branches grafted oeach node, the forbidden bands can be strongly enlargedus stress that, unlike in the usual photonic crystals,1 rela-tively large gaps still remain for homogeneous systewhere the branches and the backbone are constituted osame material. We also proposed another way of makvery large stop bands,15 i.e., a tandem structure composedtwo or several succesive comblike waveguides, which difby their physical characteristics. Very large gaps in this stem result from the superposition of the band gaps inindividual combs.

In this paper, we focus on the existence of localizmodes inside gaps when defective branches of differlength and constituted of a different material are inserted

FIG. 1. Periodic waveguide withN stars ofN8 (N856, here!grafted branches.

13 446 ©1999 The American Physical Society

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PRB 59 13 447DEFECT MODES IN ONE-DIMENSIONAL COMBLIKE . . .

the star waveguide. We show that the localized states apas very narrow peaks in the transmission spectrum of colike waveguides composed of a finite number of grafbranches. We analyze the behavior of the localized statea function of the length, of the position, and of the numberthe defective branches. The theoretical results are confirby experiments using coaxial cables in the frequency raof a few hundreds of MHz.

This paper is organized as follows. In Sec. II, we give tanalytic expressions for the band structure of the periostar waveguide and for the transmission coefficient througfinite comb with and without defects. Section III is devotto a numerical discussion of these expressions as well ascomparison of the theoretical results with experimental msurements. Finally, conclusions of this work are reportedSec. IV.

II. MODEL

The pass and stop bands of the star waveguide depicteFig. 1 can be displayed in the dispersion relation for anfinite number of sites (N→`) or in the transmission coefficient T through the waveguide when the side branchesgrafted at a finite numberN of nodes. Both the dispersiorelation and the transmission factorT were obtained in aclosed form by using the ‘‘interface response theor~IRT!.13,16 They are given as

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wherek is the modulus of the wave vector for propagationelectromagnetic waves in the star waveguide andt5exp(jkd1). In Eqs.~1! and ~2!, the quantitiesCi , Si , andFi ( i 51,2) are defined asCi5cosh(aidi), Si5sinh(aidi), andFi5a i with a i5 j vA« i /c wherev is the wave circular fre-quency,c the speed of light in vacuum, andj 5A21. Thedispersion relations~1! and~2! were obtained for two differ-ent choices of the boundary conditions, namely, the vaning of either the electric field (E50) or the magnetic field(H50) at the free extremities of the side branches. Tmodulus of the wave vectork is then given by either Eqs.~1!or ~2!. In Eq. ~3!, A andB are defined as

A512t~C12S1! ~4!

and

B5t2~C12S1!. ~5!

On the other hand, the localized states in the band strucare obtained by removing, on one site of the photonic crydepicted in Fig. 1,N28 resonators and replacing them byN38

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branches of lengthd3 (Þd2), constituted of a material orelative dielectric permittivity«3 (Þ«2). Using the IRT, onecan calculate analytically the frequencies of the localizstates in the case of an infinite comb, i.e., with an infinnumber of nodes (N→`). They are given by the relation

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In the latter equation, the parameterx characterizes theperturbation introduced in the comb by the defective resotors and is defined as

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when the electric fieldE50 vanishes at the free ends of athe resonators and as

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when the boundary conditionH50 has been taken into account. In Eqs.~7! and ~8!, the symbolsF3 , C3, andS3 as-sociated with the defects have the same meaning as abThe transmission coefficient through a defective star waguide with a finite numberN of nodes can be obtained as

T5U 2S1~ t221!2tN

~A22B2t2N!~ t221!1xcU2

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where

c5$t~A21B2t2N!2AB@ t2(N2n11)1t2n#%. ~10!

In Eq. ~10!, the integern, such as 1<n<N, stands for thelocation of the defects in the finite comb. One can eascheck that forx50 ~i.e.,N285N3850), which corresponds tothe comb without defect, Eq.~10! leads to Eq.~3!. One canalso readily verify that the same transmission coefficienobtained for two symmetrical locations of the defects wrespect to the middle of the comb.

III. APPLICATIONS

In this section, we shall numerically illustrate the aboanalytic results for comblike waveguides composed of idtical constituents, i.e.,«15«2, with the same characteristiclengthsd15d2 and the boundary conditionE50. In Fig. 2,we present the band structures and transmission coefficof the waveguide with and without defects. The top and btom panels, respectively, describe the situations wherenumber of grafted side branches at every node isN851 andN854. Figure 2~a! shows the six lowest dispersion curvesthe photonic band structure forN→` andN851. The plotsare given in terms of the reduced frequencyV5vd1A«1/cversus the reduced wave vectorkd1. In this perfectly peri-odic system, one notes the presence of pass bands sepaby gaps and in particular the existence of a cutoff frequeni.e., a forbidden band that commences at zero frequency.flat bands at reduced frequencies equal top, 2p, 3p, . . .correspond to modes whose eigenfunctions vanish at e

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13 448 PRB 59J. O. VASSEURet al.

FIG. 2. Top panel: ~a! Dispersion curves for the one-dimensional structure depicted in Fig. 1, withN→`, N851 and the boundaryconditionE50. The other parameters ared15d2 and«15«2. ~b! Transmission coefficient through the same structure with the side brangrafted atN55 nodes.~c! Transmission coefficient through a comb containing one defect of lengthd350.7d1 («35«15«2) located on themiddle node (N2851 andN3851). The other parameters are the same as in~b!. ~d! Transmission coefficient through the same comb as in~c!but the defect is located on the second or on the fourth node.Bottom panel: Same as in top panel, but forN854 side branches grafted aevery node. In~g! @respectively~h!#, N2854 resonators grafted at the middle~respectively second or fourth! node have been replaced bN3851 defective branch (d350.7d1 «35«1).

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node of the comb structure. This means that these modeeigenfunctions of the independent chains, constituting bthe resonators and the backbone, without any interactiontween the different chains. In Fig. 2~b!, we have plotted thevariations of the transmission coefficientT versus the re-duced frequency for a finite number of nodes, namelyN55. Despite the finite number of resonators, one can nothat T approaches zero in regions corresponding to theserved gaps in the electromagnetic band structure of2~a!. The flat bands in Fig. 2~a! do not contribute to thetransmission since the corresponding eigenfunctions vaat every node of the comb structure. We have shown13,14thatincreasing the numberN of nodes to 10 and more, does nmodify significantly the edges of the forbidden bands. In F2~c!, the resonator located at the middle node, namelyN53, has been replaced by a defective branch of lengthd350.7d1 and of relative dielectric constant«35«1. In thefrequency range displayed in Fig. 2~c!, there are five peaksassociated with the localized states; those falling inmiddle of a gap being much narrower than those situatethe vicinity of a band edge. The transmission inside a pband is also significantly affected by the presence ofdefect as can be observed in the second pass band of2~c! where T is totally depressed. Figure 2~d!, where thedefect is located on the second or on the fourth node, hlights the influence of the location of the defect site ontransmission coefficient and in particular on the intensitiesthe peaks associated with the localized modes. The bopanel of Fig. 2 contains similar results to those of thepanel when several (N854) side branches are grafted at eery node. This panel illustrates the narrowing~widening! ofthe pass bands~forbidden bands! when increasingN8. Figure2~g! @respectively Fig. 2~h!# shows the variation of the transmission coefficient when the four resonators located onmiddle ~respectively second or fourth! node have been re

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placed by one defective branch (d350.7d1 and«35«1). Inthis case, some of the peaks associated with the defect mare more clearly detached from the band edge, than in2~c!, especially those around the second pass band. Oneobserves that these peaks are much narrower in Fig.~g!than in Fig. 2~c!. This is clearly illustrated in Fig. 3 wherethe peaks appearing aroundV'9 in Figs. 2~c! and 2~g! havebeen plotted on the same graph at the same scale. Wealso investigated the influence of the number of graftedfects on the localized modes. The top panel of Fig. 4 shothe variations of the transmission coefficient through a ndefective comb whose parameters areN55, N851. Thethree other panels present the transmission when the restor grafted on the middle node has been replaced byN3851, 2, and 4 defective branches of lengthd350.7d1. Thetransmission factor in the pass bands is more and morepressed asN38 increases. The localized states are gettcloser to the middle of the gaps when increasingN38 . Thesestates become more and more confined, i.e., the qualitytor of the peaks defined as the ratio between the central

FIG. 3. Comparison between the shape of the peaks assocwith the localized state appearing aroundV'9 in Figs. 2~c!~dashed line! and 2~g! ~solid line!. One observes that increasingN8produces a decrease in the width of the peak.

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PRB 59 13 449DEFECT MODES IN ONE-DIMENSIONAL COMBLIKE . . .

quency and the full width at half maximum increases. Fig5 focus on the peak associated with the localized statepearing aroundV'9 in Figs. 4~b!, 4~c!, and 4~d!. This fig-ure shows together the shifting in frequency of this peakwell as the increase of its quality factor with increasingN38 .From Figs. 2, 4, and 5, one can conclude that an increasN8 or N38 in the comb gives rise to narrower peaks.

In Fig. 6~a!, we present the reduced frequencies oflocalized modes as a function of the ratiod3 /d1 for a fixed

FIG. 4. Transmission coefficients versus the reduced frequethrough four combs without defect@~a!# and with defects@~b!, ~c!,and ~d!#. The results are illustrated forN55, N851, «1

5«2 , d15d2 with the boundary conditionE50. In ~b!, ~c!, and~d!, the resonator grafted at the middle node has been replaceN3851, 2, and 4 defective branches (d350.7d1 and«15«3).

FIG. 5. The shape of the peak associated with the localized sappearing in Figs. 4~b!, 4~c!, and 4~d! aroundV'9 is illustrated ata larger scale to show the increase in the quality factor with increing N38 .

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value of«3 when one resonator in an infinite comb has bereplaced by one defective branch. The hatched areas cspond to the pass bands of the perfect comb withN851. Oneobserves that the localized modes emerge from the pbands, decrease in frequency by increasingd3 and finallymerge into a lower pass band. At each frequency, thereperiodical repetition of the localized state as a function ofd3.The same observations may be done in Fig. 6~b! where thelengthd3 of the defect has been fixed and its relative dieletric constant«3 varies. These figures clearly show that tfrequencies of the localized states inside the forbidden bastrongly depend on the physical characteristics of the detive side branch. For example, one observes in Fig. 6~a! thatthe localized states occur in the middle of the second forbden band ford3 /d1'0.5,1,1.5 . . . . Figures 6~c! and 6~d!present similar results for an infinite comb withN854 andthe same other parameters. The defect modes were obtaby replacing four resonators in one node by a defectbranch of lengthd3 or made of a material of relative dielectric constant«3.

The results presented in this paper were obtained withboundary conditionE50 at the free ends of the resonatorQualitatively similar results are also obtained by usingboundary conditionH50 at the free extremities of the sidbranches, although the gaps are narrower in this caseour choice ofd15d2 and«15«2.13,14

In order to check the validity of the above theoreticpredictions in a frequency range up to 500 MHz, we haperformed experiments where both the one-dimensiowaveguide and the side branches are constituted by stan50V coaxial cables. The transmission measurements hbeen realized by using a tracking generator coupled tspectrum analyzer. We have compared the experimemeasurements with the theoretical predictions for comwithout and with defects. As predicted by the theoreticresults, an increase ofN8 or N38 in the comb, gives rise to anarrowing of the peaks associated with the localized staFor comb-like waveguides withN8 and N38>4, the resolu-tion of our experimental setup does not allow to detect thvery sharp peaks. Then we limited ourself to study expementally the propagation of electromagnetic waves in deftive combs withN8 and N38<2. The upper panel of Fig. 7shows the theoretical and experimental variations oftransmission coefficient through a perfect comb whoserameters areN55, N851, «15«252.3, andd15d251m;the boundary condition beingE50 at the free ends of thegrafted resonators. The width and the frequency domainthe pass and stop bands are quite similar in both spectra.attenuation of the amplitude in the experimental signal wgrowing frequency, observed in Fig. 7~b!, is well-known inthis kind of coaxial cables. The variations ofT when thecoaxial cable of lengthd251m grafted on the middle node ireplaced by an other cable of lengthd351.175m are pre-sented in the middle panel. The localized modes appeathe experimental spectrum as peaks whose frequencies aquite well with the theoretical prediction. The wideningthese peaks with growing frequency is also observed in bspectra. Because of the resolution of the experimental sethe amplitude of the peaks appears weaker in the experimtal spectrum than in the theoretical spectrum. The prese

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13 450 PRB 59J. O. VASSEURet al.

FIG. 6. Top Panel: Reduced frequencies othe localized modes associated withN3851 defec-tive side branch inserted in an infinite comb (N→`). The characteristics of the comb areN851, «15«2 , d15d2 and the boundary condition is E50. ~a! The lengthd3 of the defectivebranch varies and its relative dielectric permittiity «3 is fixed and equal to«1. ~b! The relativedielectric permittivity«3 of the defective branchvaries and its lengthd3 is fixed and equal tod1.Bottom Panel: Same as in top panel, but forN854 side branches grafted at every node. In~c!and ~d!, N2854 resonators grafted at one nodethe infinite comb have been replaced byN3851defective branch (d3 ,«3).

FIG. 7. Comparison between the theoretic@~a!, ~c!, and~e!# and experimental@~b!, ~d!, and~f!# transmission coefficients through~a! and~b!:a perfect comb of characteristicsN55, N851, «15«252.3, d25d151m with the bound-ary condition E50 (N285N3851). ~c! and ~d!:the same comb containing one defect of lengd351.175m («35«152.3) located on themiddle node.~e! and~f!: the same comb but withthe defect located on the fourth node.

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PRB 59 13 451DEFECT MODES IN ONE-DIMENSIONAL COMBLIKE . . .

of the defective branch also affects the transmission inthird, fourth and fifth pass bands. Finally the bottom pashows the behavior ofT when the same defect branchlocated on the fourth node. This behavior is similar to tprevious one except for the strongest attenuation of the trmission in the fourth and fifth passbands as well asweaker amplitude of the peaks associated with the localstates.

Other experiments have been performed on the abodescribed comb structure with onedefect coaxial cablegrafted on the middle node. Transmission has been measfor various values ofd3. The frequencies of the localizestates associated with the defective resonator were measusing the spectrum analyzer. For each value ofd3, we havecompared the experimental frequencies measured on acomb with the theoretical frequencies calculated onequivalentinfinite comb. The results are displayed in Fig.The theoretical and experimental values are in reasonagreement. The shift in frequency observed for a few pomay be attributed to the finite number of resonators tainto account in the experimental setup. We have obtai

FIG. 8. Comparison between the theoretical and the experimtal frequencies of the localized states associated with the presof a defect of lengthd3 in a comb (N285N3851) thin line: calcu-lated frequencies in an infinite comb (N→`). filled dots: measuredfrequencies in a finite comb withN55. The other parameters arN851, d15d251m, «15«25«352.3 and the boundary condition is E50. The flat lines correspond to the edges of the pabands.

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similar results with star waveguides possessing two graresonators at each node. The forbidden bands are sligwider in this case.

IV. CONCLUSION

We have investigated the propagation of electromagnwaves in comblike structures composed of multiple danglside branches grafted on an infinite waveguide. The theoical model assumed that the cross sections of the wavegand of the side branches are small compared to their lindimensions, that is, they may be considered as odimensional media. This retains the validity of the modelmonomode propagation of electromagnetic waves. Thcomblike structures may present large stop bands andgood candidates for PBG materials. We have shown thatpresence of defective branches in the comb gives rise tocalized states inside the forbidden bands. These defect mappear as very narrow peaks of strong amplitude in the tramission spectrum. The localized states are very sensitivthe length of the defects, to their location in the comb as was to their number. Experimental measurements usingaxial cables have confirmed the theoretical predictionsfrequencies up to 500 MHz. However, our theoretical mois, in principle, universal and then also valid in other frquency domains of the electromagnetic spectrum. Indeedfrequency domains where the PBG occur only dependsthe periodicity of the comblike structure and the length of tgrafted branches. The manufacturing of such waveguicould be very usefull in making, for instance, filtering,multiplexing devices. It would even be more interesting fintegrated structures working at optical frequencies. Omust observe that for optical frequencies, the periodicity athe length of the side branches must be of the order of mnitude of the micrometer. Recents works show clearly tthe manufacturing of such comblike structures at a submicmetric scale becomes realizable with the new technologdevelopments using high-resolution electron belithography.12,17

ACKNOWLEDGMENTS

We thank S. Randoux, Laboratoire de SpectroscoHertzienne, Centre d’Etudes et de Recherches Lasers etplications, Universite´ de Lille I, for his assistance in the experimental measurements. This work was partially suppoby le Conseil Re´gional Nord-Pas de Calais.

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