Quasi-3D Light Confinement in Double Photonic Crystal Reflectors VCSELs for CMOS-Compatible

Quasi-3D Light Confinement in DoublePhotonic Crystal Reflectors VCSELs forCMOS-Compatible IntegrationCorrado Sciancalepore,Badhise Ben Bakir,Xavier Letartre,Jean-Marc Fedeli,Nicolas Olivier,Damien Bordel, Christian Seassal,Pedro Rojo-Romeo,Member,IEEE,Philippe Regreny,and Pierre ViktorovitchAbstract—A novel architecture of one-dimensional photonic crystal membrane(PCM)reflectors embodying a heterostructure is proposed as a robust design aimed at a3-D efficient confine-ment of light in single-mode polarization-controlled 1.55-m vertical-cavity surface-emitting laser(VCSEL)microsources for heterogeneous integration on complementary metal-oxide-semi-conductor(CMOS).On the basis of a theoretical approach,the paper focuses on the deep interweaving between the kinetics of light transport in the mirrors and the physical nature of the ex-ploited Fano resonances.An example of VCSEL design for optical pumping employing heterostructure-confined photonic crystal mirrors is presented.The predicted photons kinetics along with the considerable improvement in cavity modal features owing to the enhanced mirror architecture have been confirmed by per-forming rigorous three-dimensionalfinite-difference time-domain (3-D FDTD)calculations.Finally,experimental observations of photoluminescence(PL)emission performed onfirst-ever fabri-cated devices for optical pumping show striking agreement with theoretical considerations and ab initio modelling.Index Terms—Photonic crystals,photons,semiconductor lasers, slow Bloch mode,vertical-cavity surface-emitting lasers(VC-SELs).I.I NTRODUCTIONA CHIEVING a full control of photons in thereal—reciprocal space as well as in the frequency—time domain is decisive for the design of innovative optical components aimed at further breakthroughs in thefield of micro-nano-photonics.An efficient harnessing of light is made possible by confining photons within the tiniest spatial domain (in comparison to the wavelength)for the longest time possible (as compared to the oscillation period),while allowing to be efficiently collected(from)or addressed(to)the photonic structures where are meant to be confined.Manuscript received December22,2010;revised April19,2011,May05, 2011;accepted May10,2011.Date of publication May23,2011;date of current version June15,2011.This work was supported by the European Commission in the framework of the project HELIOS.C.Sciancalepore is with the Institut des Nanotechnologies,Ecole Centrale de Lyon,F-69134Ecully,France(e-mail:corrado.sciancalepore@ec-lyon.fr) and with the Commissariatàl’énergie Atomique et auxÉnergies Alternatives, Département Optronique,(CEA-LETI Minatec),F-38054Grenoble,France (e-mail:corrado.sciancalepore@cea.fr).X.Letartre,C.Seassal,P.Rojo-Romeo,P.Regreny,and P.Viktorovitch are with the Institut des Nanotechnologies,Ecole Centrale de Lyon,F-69134Ecully, France.B.Ben Bakir,J.-M.Fedeli,N.Olivier,and D.Bordel are with the Commis-sariatàl’énergie Atomique et auxÉnergies Alternatives,Département Optron-ique,(CEA-LETI Minatec),F-38054Grenoble,France.Digital Object Identifier10.1109/JLT.2011.2157303An effective control of light is highly desirable in the case of laser microcavities and devices for non-linear applications where the need for a stronger light-matter coupling is even more binding.This concept is particularly true in vertical-cavity sur-face-emitting lasers(VCSELs),where the coupling of the op-tical mode with the active material is crucial for low-threshold emitters and modal control constitutes an additional require-ment to be addressed especially for telecommunication-oriented applications.As widely proposed in the literature,while ver-tical confinement is achieved through diffractive phenomena provided by distributed Bragg reflectors(DBRs),the lateral op-tical waveguiding(or antiguiding)and modal behaviour in VC-SELs generally relies on a complex interplay of index-[1]and gain-guiding mechanisms[2].Specifically,in arsenide lasers,the enhancement of the pho-tons-matter coupling along with modal selection was accom-plished by introducing an optimized transverse optical and elec-trical confinement via oxide windows[3]–[7].However,the main drawback of such solutions lies with the ineludible de-sign trade-offs between single-mode operation,low threshold and optical output power[6].Using shallow surface reliefs[5] has only partly addressed the issue given the still considerable drop in emitted power;on the other hand,VCSELs’designs based on external cavity configurations and index antiguiding [7]for the suppression of higher order modes are characterized by rather complicated and possibly unstable processing.Con-cerning InP-based VCSELs,while efficient carrier funnelling has been obtained by means of structured[8]–[10]or proton-im-planted[11]tunnel junctions,this device class is still waiting for an efficient optical confinement owing to the lack of index guiding[11].Solutions based on the incorporation of2-D pho-tonic crystals(PhCs)in proton-implanted devices[11]are af-fected by significant drawbacks due to light leaking through the photonic crystal holes disrupting laser operation,even calling into question the actual feasibility of the fabrication process. In electro-thermally tunable micro-electro-mechanical sys-tems(MEMS)-VCSELs,curved micro-machined DBR mem-branes have been successfully employed in order to improve the transverse confinement of the fundamental mode while main-taining a good modal selection[8].Nonetheless,these mirrors are affected by several important disadvantages such as the lack of lateral(typical diameters100m)and longitudinal(cavity physical length15m)compactness,tight epitaxial and fab-rication constraints.Regarding polarization control,different strategies have been so far adopted in VCSELs:elasto-[12]and electro-optic[13], [14]induced birefringence,asymmetric cavity geometries[15]0733-8724/$26.00©2011IEEEFig.1.Left:Exemplifying sketch of2.5-D PCM-VCSEL structures for op-tical pumping.Materials,refractive indices,and physical lengths are reported in Table I.Right:Top view of Si/SiO one-dimensional photonic crystal mir-rors.and current injection[16],elliptical surface reliefs[17]as well as sub-wavelength gratings[18],[19].Nevertheless,although the latter solution provides a good polarization mode suppres-sion ratio(PMSR),a precise control over the grating etch depth and duty cycle is necessary to ensure a stable polarization. Since Yablonovitch’s paper in the late‘80s[20],photonic crystals were increasingly employed to control the spatial-tem-poral trajectory of photons through the diffractive confinement arising from the high-index-contrast periodical structuring of the optical medium.In photonic crystal membranes(PCMs) [21]–[24],the control of light by diffractive phenomena is com-bined with the refractive confinement arising from the index guiding provided by the strong contrast between the high-index slab core and the low-index cladding.In the past years,dif-ferent groups[24]–[30]suggested to shift from thicker,effi-ciency-limited,narrow-bandwidth DBRs to PCMs as wideband polarization-sensitive reflectors in VCSELs.Recently,high-nu-merical-aperture PCM mirrors enabling a double focusing for both reflected and transmitted waves have been proposed as building block for a new class of VCSELs and solid-state lasers [31].Although PhCs operating in the photonic bandgap regime have been already successfully employed in VCSELs[32],[33], however,PCMs working in the slow-light regime through the excitation of surface-addressable slow Bloch modes via low-Q Fano resonances(FRs)represent a very attractive solution as efficient compact mirrors.Fano resonances arise from the res-onant coupling between slow Bloch modes wave-guided in the PCM reflector with the background radiation continuum.Spec-tral properties of the resonances and coupling strength depend on the parameters defining the membrane design such as dielec-tric constant corrugation,slab thickness andfilling factor. Another interpretation of the physics governing light in PCMs lies in the destructive(or constructive)interference between the modes propagating through the membrane[along -axis,see Fig.1(right)],which is set by the design,deter-mining the strong reflective(transmissive)behavior of the membrane[34].This innovative photonic architecture can accommodate in-plane wave-guided modes which are deliberately opened to the third spatial dimension by accurately tailoring the cou-pling with the radiation continuum.Hence,the optical modes supported by a VCSEL cavity employing double photonic crystal reflectors are combination of a wave-guided(within the mirrors)and a radiated component(coupled to the active material in between the mirrors)giving origin to so-called hybrid modes.TABLE IM ATERIALS,R EFRACTIVE I NDICES,AND P HYSICAL L ENGTHS IN PCM-VCSELS Essentially,from2-D micro-nano-photonics,where only in-plane truly wave-guided modes are concerned,we are moving into the realm of2.5-D-photonics where a quasi-3D accurate light harnessing at the wavelength scale is made possible.As already demonstrated[26],one-dimensional broadband PCMs in VCSELs led to significant improvements in terms of optical mode lateral confinement,modal selection and polarization control.By exploiting two spectrally over-lapped low-Q Fano resonances,resulting in large stopbands up to150nm,such reflectors opened new perspectives also for the design of electro-statically tunable single-mode polariza-tion-controlled laser microsources,thus gradually substituting DBRs as alternative competing mirrors.Starting from a deeper theoretical understanding of photons kinetics in1-D PCMs,with the present paper we propose a novel PCM design embodying a photonic crystal heterostruc-ture aimed at a dramatic increase in the lateral confinement in long-wavelength VCSELs meant for the III-V/Si integration on complementary metal-oxide-semiconductor(CMOS).This is achieved by reducing the lateral escape rate of photons out of the mirrors by means of energy barriers.A photonic crystal heterostructure[35]–[40]consists in one photonic crystal(i.e.,the PCM reflectors of our VCSEL)en-closed laterally between two adjacent different crystals.The two crystals are chosen in such a way that photons can propagate in the central layer—the pass-band well—while encountering a forbidden bandgap region in the side layers which forms the so-called barriers.These can be obtained either by changing the refractive index profile or thefilling factor of the two con-stituting crystals or even also by approaching two PhC layers of different lattice period.Although the concept of photonic crystal heterostructures has already been employed in several applications,however,the inclusion of heterostructures in both photonic crystal reflectors of PCM-VCSELs represents to our knowledge an original and innovative solution for a quasi-3D light confinement in such devices.Here follows the manuscript organization.In Section2,a brief description of the structure design and the photon kinetics describing the vertical confinement in PCM-VCSELs is pro-vided,while in thefirst part of Section3the crucial concept of light transport kinetics within1-D PCMs is addressed by pre-senting the two-dimensional dispersion surfaces of slow Bloch modes excited in the mirror.In the second subsection the the-oretical description is validated by3-D FDTD simulations and the integration of photonic crystal heterostructures in PCMs is introduced and discussed.In the third and last part of Section3 the experimental evidence of previous theoretical arguments isSCIANCALEPORE et al.:QUASI-3D LIGHT CONFINEMENT2017Fig.2.(a)RCWA-computed TE-and TM-reflectivity spectra(inset)of1-D PCMs.(b)Dispersion characteristics in the neighbourhood of the-point of slow Bloch modes excited via the low-Q Fano resonances(labelled as“FR”) shown in the TE-reflectivity spectrum.presented;finally,Section4is dedicated to conclusions and per-spectives.II.PCM-VCSEL S TRUCTURE AND V ERTICAL C ONFINEMENT A schematic cross section of the optically pumped structures under study along with its top view is given in Fig.1.Briefly, an embedded InAsP-InGaAsP multiple-quantum-well active re-gion grown by molecular beam epitaxy(MBE),such to pro-vide gain around1.55m,is placed between a top and bottom Si/SiO1-D PCMs broadband reflectors(50%Sifilling factor, lattice period m)spaced by means of two sym-metric SiO gaps.It is worth to point out that the use of ad-vanced III-V—semiconductors/SiO wafer bonding technology [41],[42]as well as deep-UV(DUV)lithography is required for mirrors fabrication.The light kinetics that governs the vertical confinement in PCM-VCSELs can be described in rather good approximation by simply evaluating the quality factor of the exploited Fano resonance.According to the relation(strictly rig-orous only in the case of laterally infinite photonic crystal mem-branes),where and are,respectively,the frequency and the lifetime of photons in the membrane,a broad resonance cor-responds to a strong coupling rate of photons to the ra-diation continuum,which represents a condition to obtain wideband mirrors.A rigorous coupled-wave analysis(RCWA)computation of PCM reflectivity spectra for transverse electric(TE,electric field parallel to silicon slits)and transverse magnetic polar-ization(TM,electricfield perpendicular to slits)is shown in Fig.2(a),while the dispersion characteristics of slow Bloch modes corresponding to both Fano resonances are reported in Fig.2(b).The calculation of dispersion curves was carried out by tracking the resonant wavelength of the corresponding Fano resonance.This method,which may appear a qualitative approximation due to the small quality factors involved,proves instead to be enough reliable in determining the frequencies of slow Bloch modes located at band-edges,resulting in a very good agreement with3-D FDTD simulations of the PCM-VCSEL cavity shown later on in this work.Designed in order to operate at a band-edge obviously situ-ated above the light cone,the PCM shows a wide TE stopband (140nm)provided by two spectrally overlapped broad reso-nances situated at1.37m and1.55m respectively.It should be remarked that both slow Bloch modes can couple at the -point,given their symmetric distributions over the photonic crystal unit cell[43].Although a high power reflectivity is ensured over the whole PhC stopband,however we decided to operate near the redder Fano in order to maximize the cavity vertical confinement in the wavelength region of interest.The guidelines for a proper choice of the Fano resonance lie in its dispersion characteristic and we will comment further on it below.As said before,the optical mode supported by the cavity can be seen as a hybrid mode made up of two components:a wave-guided component in the mirrors as well as a radiated(com-monly called Fabry-Perot)component in between the mirrors. In detail,one can express the average relative time spent by pho-tons in the cavity during the overall lifetime of the hybrid reso-nance as follows[24]:(1)where,being the cavity optical thickness and c the speed of light.In contrast with low-index contrast DBRs, the fast coupling rate to radiation continuum provided by broad Fano resonances promotes a higher electromagnetic density within the laser active region,thus favouring the design of low-threshold devices.III.A DDRESSING THE L ATERAL C ONFINEMENT INPCM-VCSEL SA.Light Transport Kinetics in1-D Photonic Crystal Mirrors Given thefinite width of the photonic crystal membrane re-flector,the reflectivity yield is limited by the lateral escape rate which is in turn related to the average group velocity ex-by photons when propagating in PCMs.Therefore, concerning the issue of lateral confinement,a second necessary condition to realize efficient mirrors is set by minimization of ,where and indicate,respectively,the lat-eral of the mirror and the Fano resonance dispersion char-acteristic curvature around the-point[23].In other words,the resonant coupling efficiency can be ex-pressed as:the reduction in lateral losses is thus accomplished by putting photons through a slow-light regime via the excitation of surface-addressable slow Bloch modes located at veryflat band-edges(low)of dispersion curves.The dispersion characteristic of the hybrid mode is entirely defined by its wave-guided and radiated components in the re-ciprocal space.The isotropic dispersion characteristic of the FP component of the hybrid mode in the vicinity of the-point can be described by the following simple analytical expression(2) where indicates the transverse wavevector component,being the frequency at the-point and the quadratic approx-imation of the dispersion characteristic curvature evaluated at the band-edge.In a simplified but still reliable one-dimensional approach(strictly rigorous in case of laterally infinite struc-tures),the hybrid mode band curvature is a linear combi-nation of the wave-guided and FP components2018JOURNAL OF LIGHTWA VE TECHNOLOGY,VOL.29,NO.13,JULY1,2011corresponding band curvatures weighted by the relative average lifetimes and[28](3) Owing to the isotropic dispersion characteristic of the Fabry–Perot component of the hybrid mode,it follows that the optical cavity modal features are mainly determined by the wave-guided component.Consequently,a deep study of photons behavior when exciting slow Bloch modes in PCMs is essential to the understanding of the physics describing the device.Given the one-dimensional nature of the membrane refractive index structuring,photons propagating in PCMs experience different dispersion curves depending on the wave vector,thus originating strongly anisotropic two-dimensional dispersion surfaces.Relying on purely one-dimensional dispersion characteristics as commonly done would end up with the unrealistic modelling of light transport in the mirrors.We need to shift to dispersion surfaces where the energies of corresponding photonic states are now depending on a generic in-plane wave vector of compo-nents.In this way,grasping theflow of light within photonic crystal membrane reflectors provides us with a powerful tool to design highly compact and innovative single-mode PCM-VCSELs. Such complex picture has been studied by RCWA calcula-tions in the neighbourhood of the-point,which,in our case, corresponds to the domain of the reciprocal space experienced by the wave-guided component of the fundamental mode.As il-lustrated in Fig.3the-space dispersion surfaces show,beyond the well-expected anisotropy,very different behaviors.The slow Bloch mode at1.55m is described by a saddle surface, while the mode at higher energy is characterized by a strongly anisotropic paraboloid.Although both resonances provide high vertical confinement[see Fig.2(a)],the latter is affected by stronger lateral losses due to a higher average group velocity notably along the direction perpendicular to the mirror slits and should therefore be considered less appropriate for an efficient confinement of optical modes.On the other hand,it can be noted that both dispersion surfaces are sharing aflat but slightly negative curvature mainly parallelly to the slits.The existence of directions within PCMs along which photons propagate with a negative group velocity allows confining the light towards the mirror center,resulting crucial for the improvement of light control in the device as further shown later on.According to(3),we may suppose to exploit slow Bloch modes characterized by a negative curvature of the dispersion characteristic at the band-edge in order to com-pensate for the positive band curvature of the Fabry–Perot mode(2),resulting in a minimization of the global hybrid mode curvature over a wide reciprocal space domain around the-point.The enhanced light slow-down would turn out in high-Q cavity modes.Nevertheless,the strong anisotropy of 2-D dispersion surfaces prevents to accomplish an overall com-pensation between and for every in-plane direction. In other words,(3)should be re-written as a function of the in-plane wave vector and not just of.As a re-sult,innovative confinement strategies arenecessary.To this purpose,2-D dispersion surfaces represent a highly predictive and reliable tool for the design of2.5-D laser cavities.Fig.3.Two-dimensional reciprocal-space dispersion surfaces of slow Bloch modes at1.37m(a)and1.55m(b)respectively.In fact,although the hybrid mode spans in a three-dimensional world,however,its wave-guided(and most meaningful)com-ponent can be considered as purely two-dimensional.Hence,the picture describing the light behavior in the mirrors is thus given by dispersion surfaces which depend exclusively on in-plane wave vectors.On the basis of the morphology of dispersion surfaces[see Fig.3(b)]as well as referring to the considerations contained in the previous paragraph,we can infer that photons travel-ling within PCMs will experience a lateral confinement or,vice versa,a higher lateral escape rate,depending on the wave vector. Specifically,photons propagating in the mirrors along directions characterized by a negative group velocity are to a certain ex-tent confined within the mirror.Simply,starting from the definition of group velocity for the th dispersion band(4) we can determine confinement and deconfinement domains in the reciprocal space by computing the gradient of the Fano res-onance dispersion surface at1.55m(mode“B”)as illustrated in Fig.4.By deriving the group velocity as a two-dimensional vectorialfield we obtain a clearer overview about the magnitude and preferential directions along which lateral losses rise sig-nificantly.We expect the out-coupling of light to be maximized along those directions(i.e.,wave vectors)indicated by the gra-dient.Thus,mirrors reveal to be particularly leaky across the slits,while showing a natural confinement along the slits.Con-cerning single-mode operation,gradient paths in Fig.4clearly explain the achievement of transverse modal selection in the de-vice:those modes characterized by larger transverse wavevector components are leaking out of the PCM faster,resulting in an in-trinsic modal discrimination determined by the curvature of dis-persion surfaces.The increased lateral losses suffered by higherSCIANCALEPORE et al.:QUASI-3D LIGHT CONFINEMENT2019Fig.4.(a)–(d)Group velocity corresponding to the slow Bloch mode at1.55 m[see.Fig.3(b)]calculated on different-space domains.order modes boost the threshold discrimination for the benefit of the fundamental mode.B.Heterostructure-Confined1-D PCMs for Ultimate Transverse Light ControlGiven the high vertical confinement obtained when operating nearby the Fano resonance wavelength,the minimization of photons lateral escape rate within the PCMs is necessary for low-loss PCM-VCSELs.The introduction of a photonic crystal heterostructure suits quite well with the control of anisotropic losses arising from the peculiar light transport dynamics taking place in1-D PCMs.In our devices,barriers are obtained by introducing a local vari-ation of thefilling factor(silicon content)and oriented per-pendicularly to those directions affected by a higher average group velocity,aiming at preventing photons to leak out and al-lowing a more efficient optical transverse confinement[35][see Fig.5(a)].Similarly to the formalism used in[38]we can express the local modification of thefilling factor in the heterostructures Fig.5.(a)Sketch-up of a heterostructure-confined1-D photonic crystal mirror. Barriers are introduced in a typical1-D PCM to enhance lateral confinement (and represent,respectively,the siliconfilling factor in the well and barriers).(b)Schematic illustration of the heterostructure band-gap;the tun-nelling and diffraction losses coupling rates are indicated.(c) dispersion characteristics in the well and barriers .indicates the confinement frequency along direction of transport.as a spatially slowly varying perturbation modulating the average refractive index of the crystal(5) where2L and are,respectively,the width of the well and the index contrast between the core and the cladding of the het-erostructure.In a naïve but realistic approach,the heterostructure can be seen as a resonator where barriers serve as mirrors,while the well constitutes the centre cavity.The resulting confinement wavelength range arises from the bandgap introduced with the heterostructure and can be treated as an energy barrier[in Fig.5(c)].The lateral reflectivity provided by the heterostruc-ture bandgap increases the lifetime of the wave-guided mode in the PCM,promoting the confinement of the global hybrid mode.In particular,in narrow barriers with a tiny heterostruc-ture bandgap the electricfield decays exponentially but still al-lowing photons to pass through as in quantum-mechanical tun-nelling.On the contrary,in the case of wider and energetic bar-riers,the decay will be complete so that all the energy will be reflected within the well.This mechanism is responsible for the enhancement of the photonic crystal mirror reflectivity yield. In dielectric waveguides optical confinement is obtained when the dielectric contrast between core and cladding is posi-tive.In photonic crystal heterostructures,the diffractive-based confinement of waveguided modes depends on the sign of the th band curvature in the well at the band-edge[38](i.e., the photon effective mass2020JOURNAL OF LIGHTWA VE TECHNOLOGY,VOL.29,NO.13,JULY1,2011evaluated along the direction of perpendicular transport.The positive band curvature along implies that optical confine-ment can be obtained if the average refractive index in the well is greater than in barriers,which implies the use of a higher siliconfilling in the On the contrary,usingin presence of a positively-(negatively-) curved dispersion band in the well would introduce antiguiding at the expense of modal confinement in the VCSEL. Regarding modal selection between two competing trans-verse modes,the single-mode behavior of a laser microcavity is usually assessed through the modal stability parameter(6) where the subscripts and label,respectively,vertical and hor-izontal mode orders and represents the threshold gain the corresponding mode needs for lasing.Alternatively,modal se-lection can be appreciated to a larger extent by estimating the proportion between the quality factor of competing modes,since it provides the ratio between the energy stored in the cavity and the dissipated power for each cavity mode.In order to lay out an accurate comparison between heterostructure-confined devices and ordinary PCM-VCSELs in terms of their modal properties, rigorous3-D FDTD simulations(without considering the gain medium)were performed.In Table II resonant wavelengths and corresponding quality factors of the fundamental andfirst order transverse modes are reported for a standard15-m-wide PCM-VCSEL with and without the use of lateral barriers. Results confirm the expected increase in lateral confinement of cavity modes provided by heterostructure-confined photonic crystal mirrors respect to standard1-D PCMs.This can be fully appreciated by comparing the cavity modes nearfields reported in Fig.6(a)–(d).First of all,lateral losses are strongly anisotropic and this perfectly matches with the description provided by dispersion surfaces:photons loss rate across the mirror slits is sensibly higher respect to the parallel direction [see Fig.6(a),(b)].Moreover,the same behavior is reproduced in the MQW active region,implying that the modal properties of the hybrid mode are shaped by its wave-guided component in the mirror.In second place,the heterostructure-confined mirrors[see Fig.6(c),(d)]suppress lateral loss almost com-pletely,resulting in a considerable quality factor enhancement originated by the maximization of the PCMs reflectivity yield. The observed red-shift of cavity modes in structures including photonic heterostructures is simply due to the variation of the transverse boundary.Referring to Table II,we observe that the rise in the funda-mental mode quality factor does not come at the expense of the modal selection,but,on the contrary,strengthens the modal dis-crimination of the cavity.The reason for that can be found out by introducing in our descriptive model two additional parame-ters[see Fig.5(b)]:1)the photons tunnelling rate through the barriers,which expresses the efficacy of the heterostructure in con-fining the light laterally;2)a secondterm related to diffraction loss affecting cavity modes,which is induced by the perturbation modu-lating the refractive index profile.Fig.6.nearfields within the top PCM(a)and the MQW active region(b)in standard PCM-VCSELs compared to a heterostructure-confined device(c),(d).Mirror dimensions are defined by the superimposed white square(co-ordinates are reported inm),while slits orientation is illustrated schematically in the left part of theTABLE IIM ODAL B EHAVIOR C OMPARISON Consequently,we can define the quality factors of transversemodes as follows:(7) where refers to out-coupling vertical losses which mainly depend on modal wavelengths.The increase in lateral reflectivity yield provided by het-erostructure is strictly linked to the width of the bandgap originated by the perturbation.In particular,the exponential decay of the electricfield characterized by penetration depth within barriers is related to the size of the heterostructure band-gap.Residual lateral losses are thus mainly con-trolled by the barriers physical extension as well as the width of the confinement frequency range.Modal selection is related to diffraction losses stem-ming from the different spatial overlap of transverse modes with barriers.In detail,the coupling to radiation continuum via diffractive scattering due to barriers is proportional to the overlap between the optical mode and the perturbation to the dielectric constant introduced in the photonic crystal mem-brane.The mode overlap with barriers affects also the photons tunnelling rate.In fact,a major extension of the cavity mode profile within barriers gives rise to a drop in reflectivity yield due to a diminished efficacy in the confinement.It follows that barriers position governs the selective lateral confinement。