slow light in photonic crystal

Wideband slow light with ultralow dispersion in a W1photonic crystal waveguideJian Liang,1,2Li-Yong Ren,1,*Mao-Jin Yun,2and Xing-Jun Wang31State Key Laboratory of Transient Optics and Photonics,Xi’an Institute of Optics and Precision Mechanics,Chinese Academy of Sciences,Xi’an710119,China2College of Physics Science,Qingdao University,Qingdao266071,China3State Key Laboratory on Advanced Optical Communication Systems and Networks,Peking University,Beijing100871,China*Corresponding author:renliy@Received1July2011;revised21September2011;accepted21September2011;posted26September2011(Doc.ID150243);published20October2011A dispersion tailoring scheme for obtaining slow light in a silicon-on-insulator W1-type photonic crystalwaveguide,novel to our knowledge,is proposed in this paper.It is shown that,by simply shifting thefirst two rows of air holes adjacent to the waveguide to specific directions,slow light with large group-index,wideband,and low group-velocity dispersion can be realized.Defining a criterion of restricting thegroup-index variation within aÆ0:8%range as a flattened region,we obtain the ultraflat slow light withbandwidths over5.0,4.0,2.5,and1:0nm when keeping the group index at38.0,48.8,65.2,and100.4,respectively.Numerical simulations are performed utilizing the three-dimensional(3D)plane-wave ex-pansion method and the3D finite-difference time-domain method.©2011Optical Society of AmericaOCIS codes:230.3990,230.5298,230.7390,260.2030.1.IntroductionFor the past decade or more,slow light,meaning the group velocity v g of light is slowed down as compared with the vacuum velocity c,has attracted a lot of attention because of its many potential applications. These include tunable optical delay[1],spectral re-solution enhancement[2],all-optical signal buffer-ing,and regeneration[3].In particular,slow light in photonic crystals has recently become a hot topic because of its numerous merits,such as room-temperature operation and easy on-chip integration. One can realize any slow-light operating wavelength by suitably scaling the structural parameters of the photonic crystal.Utilizing the photonic bandgap of photonic crystals,the phenomenon of slow light can be easily observed close to the Brillouin zone edge in a photonic crystal waveguide(PCW)[4–6].But un-fortunately,the dispersion relationship around the slow-light operating point inherently shows a near-parabolic behavior,which means a large group-velocity dispersion(GVD)may always accompany the slow-light propagation.This disadvantage coun-ters most of the advantages of operating in the slow-light region and severely limits the bandwidth[7,8]. In2004,Baba and coworker reported that,by using a chirping PCW,it is possible to significantly depress the GVD while enlarging the bandwidth[9]. Since then,the GVD and the bandwidth in a slow-light PCW structure were found to strongly rely on its geometric parameters.Up to now,many theoreti-cal and experimental papers have been presented, based on the method of destroying the overall unifor-mity of the photonic crystal structure to tailor its dis-persion,that aim to realize wideband and low GVD slow light with high group index n g.These ways mainly include reducing the waveguide width[8,10], chirping the waveguide structure[9,11],perturbing the radius of air holes[12,13],introducing ring-shaped holes[14],simultaneously shifting the first0003-6935/11/310G98-06$15.00/0©2011Optical Society of AmericaG98APPLIED OPTICS/Vol.50,No.31/1November2011and second rows of air holes[15–17]adjacent to the waveguide,infiltrating microfluid into the air holes [18],embedding quantum dots[19],solely shifting the third row of air holes[20]as well as merging coupled cavities[21],etc.For a simpler fabrication process,shifting the location of the air holes is widely considered to be the most effective one[4].In pre-vious literature,authors usually determine the bandwidth of a flattened wideband slow-light region by restricting the variation of n g within aÆ10% range[10,14,17].However,in fact,such aÆ10%cri-terion may be not good enough to allow high-fidelity slow-light propagation for a short pulse,especially with a high n g.For a higher n g,there must be larger GVD and smaller accompanying bandwidth[9],so a slow-light PCW structure with a much stricter criter-ion of n g is needed.Here,we propose a W1-type slow-light PCW struc-ture with a line defect in the center.Such a PCW pos-sesses an acceptably wide and ultraflat band regionfor each given n g.In simulations,we focus on realiz-ing the transverse electriclike(TE)polarized evenmode slow light because a laterally odd mode canhardly couple with a monomode dielectric waveguidedue to the symmetry mismatch[8].Moreover,thelight–matter interaction of the even mode in thePCW is strong,which interacts with several rowsof air holes adjacent to the waveguide.This phenom-enon implies that shifting these air holes can drama-tically detune the dispersion[12,22].In this paper,we discover that the flattened wideband slow lightwith large n g can be generated by shifting the firstrow of air holes(above and below the waveguide)hor-izontally and the second row vertically.By using sucha novel lattice-shifting method,slow light with an ul-traflat band is easily achieved even for n g up to100.4;i.e.,our method permits us to achieve a simulta-neously wideband slow-light PCW with ultralow dis-persion at large n ing the three-dimensional(3D)plane-wave expansion(PWE)method and the3Dfinite-difference time-domain(FDTD)method,thedetailed numerical simulations are performed.2.Theoretical SimulationFigure1depicts the schematic diagram of the PCWstructure we propose,where the silica substrate isremoved to see clearly.The PCW above the substrateis a triangular lattice photonic crystal slab with aline defect in the center,usually called a W1-typePCW.The backbone material is silicon(n Si¼3:4). We denote the period of the lattice,the diameter ofthe air holes,and the thickness of the PCW by a,D,and h,respectively.Here h just indicates the thick-ness of the slab.In simulations,the thickness of thesilica substrate(n¼1:45)is set to1μm.The shifting method can be described as follows.Inaccordance with the definition of the x and z axesshown in Fig.1,we useΔx andΔz to denote theshifted distances of the air holes along the directionsof the x and z axes,respectively.For the lower tworows of air holes adjacent to the waveguide,the first row is shifted along the x direction and the second one along the z direction.The upper ones are shifted just opposite to the lower ones.We use the3D PWE method to calculate the dispersion of this slow-light waveguide[23].In the supercell calculation,we se-lect the plane waves’numbers of each axis as32and the eigenvalue tolerance as10−8for a sufficient cal-culation precision and a moderate time consumption and memory demand.The relationship between the dispersion and n g can be described asn g¼n effþωdn effdω;ð1Þwhere n eff is the effective index andωis the central angle frequency of the incident pulse.Here n eff andωcan be substituted by the propagation constantβ(β¼2πλn eff,whereλis the operating wavelength)and the normalized frequency u(u¼ωa2πc,where c is the light speed in vacuum),respectively.Considering the condition n g≫n eff in the case of slow light, Eq.(1)can be transformed ton g¼a2πdβdu:ð2ÞObviously,low-dispersion(equivalently,nearly con-stant n g)slow light can be obtained in specific fre-quency regions where u varies linearly withβ.Our purpose in this paper is to optimize the parametersΔx andΔz in the W1PCW to obtain a wideband and ultraflat slow light at large n g simul-taneously.In the numerical simulations,we fix D¼0:62a and h¼0:5a.Note that these parameters are referred theoretically and experimentally to most publications.Figure2(a)shows the relation uðβÞof the TE even mode whenΔx andΔz are varied for op-timization.As mentioned above,the linear region of each curve corresponds to a flatband slow-light re-gion with a specific n g(thusυg)determined byits Fig.1.(Color online)3D model diagram of the W1PCW structure we propose.It is a slab with air holes in silicon,which lies on a substrate made of silica.The two rows of air holes adjacent to the waveguide are shifted along the x and z arrows.a is the period of the lattice.D and h are the diameter and thickness of the slab, respectively.1November2011/Vol.50,No.31/APPLIED OPTICS G99slope.Such a linear region has been labeled in Fig.2(a).Figure 2(b)shows the calculated n g as a function of u .In order to evaluate the low-dispersion bandwidth of the slow light,some previous papers usually select a bandwidth criterion where n g varies within a Æ10%range.Bandwidths of 9.0,6.7,4.7,and 2:7nm are obtained for nominal group indices of 38.0,48.8,65.2,and 100.4,respectively .It should be noted that the dispersion relation-ships plotted in Fig.2remain unchanged provided that one enlarges or reduces the PCW structure pro-portionally (i.e.,by changing the lattice period a while maintaining the relative geometry).It is fasci-nating that one can obtain arbitrary operating wave-length of interest by only adjusting the value of a without renewing the simulation.This relationship can be expressed bya ¼u 0λ0;ð3Þwhere u 0denotes the specific normalized frequency for each given n g with a flat band [as shown in Fig.2(b)]and λ0denotes the specific operating wavelength.To further illustrate the dispersion property of the ultraflat band in Fig.2(b),we introduce the GVD parameter D defined as [14]D ¼1c ∂n g∂λ:ð4ÞAccording to Eq.(4),if the slope of n g ðλÞis near zero,the PCW structure must have a near-zero GVD.Figure 3shows dispersion curves as a function of wavelength for different n g .In Fig.3,we have fixed all the nominal operating wavelengths to be around 1550nm by setting a to different values according to Eq.(3).Taking n g ¼38:0as an example,the GVD stays within Æ1ps =ðmm ·nm Þover 3nm,which has been considered a low-dispersion region in [14].Simi-lar to the GVD feature under a small value of n g ,the GVD still keeps very low as expected even for a high n g ,which was quite difficult to obtain in previous models published [14,16].We can obtain this ultra-low dispersion region over 0:3nm in a high value of n g ¼100:4.Note that the dispersion region de-creases constantly as the value of n g increases,so the discussion of an even higher value of n g gradually becomes meaningless.If one considers n g to be prac-tically constant when it varies less than Æ0:8%,in fact,one proposes a much stricter criterion for flatness of the slow-light band region.In the sense of this criterion,we obtained an ultraflat bandwidth over 5.0,4.0,2.5,and 1:0nm for n g ¼38:0,48.8,65.2,and 100.4,respectively.The simulation results are listed in Table 1.It implies that,no matter whether this structure is used at a small n g or at a large n g ,an optical pulse will suffer almost no distortion because of the ultralow GVD.We also simulate the pulse propagation in the PCW using a 3D FDTD method with the perfectly matched layer as absorbing boundary conditions [24,25].Note that we have chosen 32unit cells per axis in our simulations,which has been verified to be good enough to ensure a reflectionless transmis-sion [26].We suppose that a sinusoidallymodulatedFig.2.(a)Normalized frequency as a function of the propaga-tion constant,four groups of selected Δx and Δz included with a ¼420nm and a diameter of air holes of D ¼0:62a ,h ¼0:5a .(b)Group index as a function of normalized frequency accord-ing to(a).Fig.3.GVD as a function of wavelength when the selected group indices are deemed to be constant with n g ¼100:4,65.2,48.8,and 38.0,respectively .We keep the nominal operating wavelength region at 1550nm by changing a .G100APPLIED OPTICS /Vol.50,No.31/1November 2011Gaussian pulse impinges upon the PCW.The 1=e half width of the input pulse was set at τ¼0:067ps.It should be pointed out that this choice of τ,as an ex-ample,corresponds to c τ¼20μm.The temporal pulse profiles in terms of the normalized field ampli-tude are then obtained as shown in Fig.4,corre-sponding to the two selected detection points at a and 10a ,respectively ,behind the light source.The normalized central frequency of the pulse is chosen to be u ¼0:269,which corresponds to the approxi-mately constant n g of 38.0as shown in Fig.2(b).Figure 4indicates that the shape of the pulse keeps almost unchanged even after a propagating distance of 9a [≈3:76μm].The 1=e half width of the pulse at 10a is 0:0672ps,meaning that the pulse experiences a broadening of only 0.3%.The value of n g can be cal-culated based on the relationship between the propa-gation time t and the propagation distance l ,i.e.,t ¼l =v g ¼ðn g =c Þl .Therefore,we obtain n g ¼36:6from Fig.4.Note that the value of n g ¼38:0we ob-tained above is determined by the 3D PWE method.We think that the slight discrepancy of n g is tolerant as far as the discretization in FDTD calculations and the certain number of plane waves adopted in PWE calculations are concerned.The nearly coinciding re-sult means that the dispersion relationship (thus n g )obtained with the 3D PWE method is correct.Hence,we can conclude that,in our proposed PCW struc-ture,an ultranarrow pulse can propagate with a slow group velocity without obvious distortion owing to its broad bandwidth and near-zero GVD.3.Results and DiscussionThe mechanism of the slow-light effect in the PCW can be classified into two different aspects,namely,coherently backscattering and omnidirectional re-flection [5].For an air-hole-based PCW ,the former one is usually the main mechanism.However,the op-posite shifting method along the x direction brings the latter one into the PCW due to the opposite change in refractive index.This mechanism avoids interference between the signal and the backscatter-ing pulse,so the GVD can be significantly depressed.We further discuss the independent impact of Δx and Δz on this model.Figure 5(a)indicates that,for a given Δz ,the group-index curve,as a function of the normalized frequency,has a left shift and changes from monotonous to nonmonotonous behavior with increasing Δx .For a given Δx ,similar characteristics happen with the increase of Δz as shown in Fig.5(b),except that the curve has a right shift.Besides,the impact of Δx is somewhat stronger.Therefore,by appropriately combining Δx and Δz ,it is possible to realize a nearly constant n g even in a broad frequency region.This is the reason why we can ob-tain wideband,ultraflattened slow light with a large group index at different normalizedfrequencyFig.5.n g as a function of normalized frequency .(a)Δx varied from 0:124a to 0:144a with a step of 0:01a ,where Δz is fixed at 0:015a .(b)Δz varied from 0:005a to 0:025a with a step of 0:01a ,where Δx is fixed at 0:134a.Fig.4.Temporal pulse propagation at two selected detection points at a and 10a ,respectively ,behind the light source.n g ¼38:0;the relevant parameters are the same as those in Table 1.Table 1.Group Index,Bandwidth under Different Optimized Shifting ParametersAdjusted Parameters n g ΔλðÆ10%;nm ÞΔλðÆ0:8%;nm ÞΔx Δz 0:134a 0:015a 38.09.0 5.00:120a 0:045a 48.8 6.7 4.00:103a 0:060a 65.2 4.7 2.50:084a0:070a100.42.71.01November 2011/Vol.50,No.31/APPLIED OPTICSG101regions,as shown in Fig.2,by purposely changing Δx and Δz in opposite directions by simply shifting the related air holes.Shifting the locations of the first two rows of air holes adjacent to the waveguide using our method has been verified a functional modification to detune the dispersion in the PCW as discussed in Section 2.However,there is another geometric parameter that exists that may be very sensitive to the slow-light ef-fect and is usually neglected in previous PCW simu-lations.It is the thickness of the waveguide slab,h .In the simulation above,we set this value as a constant to 0:5a ,which seems not important in the simulation.Here we take n g ¼38:0as an example for which the relevant parameters are listed in Table 1.Figure 6shows the dependence of n g on the wavelength as h varies from 200to 220nm by a step of 10nm,where a is set to 420nm.Note that,for the convenience of discussing,the abscissa was adopted to the wave-length instead of the normalized frequency.We can see from Fig.6that the thickness of the slab plays an important influence on the slow light in the PCW.With the increase of h ,the flattened region in-dicates a considerable redshift,while the shape of n g stays almost unchanged.Such a performance can be used to excellently explain the redshift between the simulative result and the experimental one in [27],where the shift is 20nm and the transmittance curve (there the transmittance curve is directly decided by n g )stays almost unchanged.Although shifting air holes is considered the sim-plest way in fabrication,there will also be some un-avoidable errors.So in the following,the acceptable fabrication tolerance of the model is discussed.We define δx and δz as the fabrication errors relative to the shifts Δx and Δz of interest,respectively.The signs of δx and δz are the same as that of Δx and Δz ,as shown in Fig.1.We take n g ¼38:0as an example with the optimized parameters being listed in Table 1.The results are shown in Fig.7.The solid curve denotes the ideal model without any fabrica-tion error.For simplicity ,we suppose that each airhole in the two first rows adjacent to the waveguide has the same error δx and that each one in the 2s rows has the same error δz .Of course this is an ex-treme scenario.In principle,if one of them has smal-ler absolute error,the fabrication error of the other parameter can be relaxed accordingly.We can see from Fig.7that the curve remains almost unchanged when δx is positive,while the curve becomes worse when δx is negative.This is easy to understand if one realizes that a positive (or negative)δx means a consistent (or reverse)shifting direction of air holes owing to δx with respect to that owing to Δx .Figure 7also indicates that the curve is more sensitive to δx than δz .Therefore,this wideband low-dispersion property can be easily reserved provided that we keep a positive δx .This is true for a positive δx even up to 1nm as shown in Fig.7.4.ConclusionBy simply shifting the four rows of air holes adjacent to the waveguide,a W1PCW structure,novel to our knowledge,is proposed to realize wideband slow light with ultralow GVD.This scheme includes a hor-izontal-opposite shift to the first upper and lower rows of air holes adjacent to the waveguide,as well as a vertical-opposite shift to the second two rows accordingly .Compared with some other schemes,such as introducing ring-shaped air holes in [14],this scheme is more convenient to fabricate.The most im-portant thing is that we realize ultraflat group-index curves varying only in a range of Æ0:8%even for the group index up to 100.4.This work was partially supported by the West Light Foundation of the Chinese Academy of Sciences (CAS)(2009LH01),the Innovation Founda-tion of CAS (CXJJ-11-M22),the National Natural Science Foundation of China (NSFC,60778020),the Open Research Fund of State Key Laboratory of Transient Optics and Photonics of CAS,and the Scientific Research Foundation for theReturnedFig.6.n g as a function of normalized frequency with h varied from 200to 220nm with a step of 10nm.n g ¼38:0;the relevant parameters are the same as those in Table 1.Fig.7.n g as a function of normalized frequency with several δx and δz .δx and δz denote fabrication errors relative to Δx and Δz ,respectively .n g ¼38:0,the relevant parameters are the same as those in Table 1.G102APPLIED OPTICS /Vol.50,No.31/1November 2011Overseas Chinese Scholars,State Education Minis-try.The authors gratefully acknowledge helpful discussions with Prof.Romano A.Rupp. References1.A.Talneau,“Slowing down the 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