Photonic crystals in two-dimensions based on semiconductors: fabrication, physics and technology

Applied Surface Science 164 Ž2000. 205–218 www.elsevier.nlrlocaterapsusc

Photonic crystals in two-dimensions based on semiconductors: fabrication, physics and technology H. Benisty, C. Weisbuch, D. Labilloy ) , M. Rattier Laboratoire de phsyique de la Matiere ` Condensee, ´ UMR 7643 du CNRS, Ecole Polytechnique, 91128 Palaiseau cedex, France

Abstract Wavelength-scale periodically structured dielectrics in two or three dimensions, the so-called photonic crystals ŽPCs., may acquire outstanding electromagnetic properties, due to the appearance of a photonic gap and of the peculiar photon dispersion relations around such gaps. One may take advantage of these properties to elaborate novel devices based on microresonators, integrated mirrors, etc. In this paper, we start with a brief introduction to two-dimensional Ž2D. crystals and to defects in these crystals. We next discuss the physical and technological issues raised by some recent realisations. The incorporation of PCs into various devices is then examined, restricting ourselves to applications to light-emitters and integrated optics, a case for which radiation losses of PCs, are discussed. q 2000 Published by Elsevier Science B.V. PACS: 42.70Qs; 42.55Sa; 42.82.m; 42.50-p Keywords: Photonic crystals; Microcavities; Semiconductors; Light-emitting devices

1. Introduction The role of semiconductors in optoelectronics owes much to their electro-optic properties and the ease with which one obtains electroluminescence or even gain from electrical injection. However, their purely dielectric properties are also important, for example, the index contrast of optical waveguides or of multilayer mirrors w1–3x. It is tempting to elaborate more complex dielectric structures to seek for still more interesting properties. The pioneering paper by Yablonovitch w4x suggested along this line, the presence of photonic gaps in three-dimensional Ž3D. structures, able to inhibit spontaneous emission from emitting species embedded in such a photonic crystal

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Corresponding author. Present address: CERF, Corning SA, 7bis Ave de Valvins, F-77210 Avon, France.

ŽPC.. Since then, the field has flourished with various studies and realisations of PCs, using, not only semiconductors, but also colloids, opals, glass nanotubes, etc. w5–10x. In the semiconductor area, scaling down the microwave experiments w11–14x to the optics length scale proved a difficult task, even in 2D. Indirect probes, such as light scattering from a PC-based waveguide w15x, or diffraction from luminescence generated inside PC-based cavities w16–18x, or the measurement of the lasing characteristics of a gain medium with PC-based mirrors w19–22x, are often easier to implement than a proper transmission or reflection measurement. In this paper, we give an overview of this state of affair and evaluate the potential of PCs in various kinds of semiconductor device applications. In Section 2, we give a brief introduction to two-dimensional Ž2D. PCs and to defects in these

0169-4332r00r$ - see front matter q 2000 Published by Elsevier Science B.V. PII: S 0 1 6 9 - 4 3 3 2 Ž 0 0 . 0 0 3 3 9 - 1

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crystals. In Section 3, we discuss the physical and technological issues raised by recent realisations based on macroporous Si, on membranes and on more classical waveguides, the latter being studied in the group of the authors. The issue of radiation losses of PCs is given special care. In Section 4, the incorporation of PCs into various devices is examined, restricting ourselves to applications to light emitters and integrated optics, a case where we attempt to find the limit set by radiation losses. Section 5 gives the conclusion.

Fig. 1. Rays in a multilayer stack with high index contrast: rays above the critical angle generate evanescent waves that couple the high-index layers Žresulting in minibands.; Right: TM polarised light fails to experience Bragg periodicity around the Brewster angleŽs..

2. Basics of 2D PCs 2.1. Why more than one-dimensional (1D) periodicity? The optics of 1D periodical stacks Žsome of them often named distributed bragg reflectors, DBR. is relatively simple. For the simplest alternate stacks made of two materials, one finds, with a little mathematics w23x, ranges of frequencies Ždenoted as v . and angles Ždenoted as u . called ‘‘stop-bands’’, for which, light cannot propagate and decays when entering the stack. The appearance of these gaps Žf stop-bands. and windows Žf bands. for propagation is explained in solid-state textbooks for electrons and periodic ionic potentials. A basic consequence of periodicity is that a convenient basis of waves in a periodic medium is the one labelled by the wavevector k spanning the first Brillouin zone, and having the form c s u k Ž r .expŽ ik P r y v t . with u k Ž r . periodic, known as Bloch waves. The multivalued plot of v Ž k . is the band structure. Like plane waves, Bloch waves can be combined to form, e.g. stationary waves. Unlike them, the electromagnetic energy distribution of Bloch waves can be arbitrarily non-uniform through the periodic part u k Ž r ., which ‘‘maps’’ the dielectric structure Žthe potential., so as to fit the required photon energy " v Želectron energy.. Along this line, in 1D, one can understand the usual splitting of photon dispersion relation at k s pra Žfor periodicity a.: in v s kcrn, replace the index n by its weighed value - n )s- c Ž r .< nŽ r .< c Ž r . ) Žwith - c < c )s 1.. Writing v s kcr- n ) , the role of c , that may preferentially sample high or low

index region, explains the appearance of different v ’s at k s pra where c Ž r . is either cosŽ kx . or sinŽ kx . in 1D. In 1D structures, however ŽFig. 1., even for high index contrasts Ž n hi :n lo s 3:1., a stop-band cannot extend to the highest angles of the system. For the TM polarization, the Brewster angle suppresses reflections and, thus, repels gaps. For TE modes, the stop-band is broad and may extend up to the critical angle Ž ucrit s siny1 Ž n lorn hi .. if both layers are ‘‘quarter-wave stacks’’. But then, beyond the critical angle, waves are evanescent in the lower index medium and tend to be guided in the higher index medium ŽFig. 1.. This situation is that of coupled guides with Bloch waves being still propagative in the relatively broad ‘‘minibands’’ that form as for electrons in superlattices with tiny barriers. Hence, although the modes of a planar stack are not fully trivial, the presence at any frequency of a continuum of propagative modes at some angle Ž k value. in a periodic structure prevents light confinement. It also implies that the photon density-of-states ŽDOS. and, from Fermi’s golden rule, the spontaneous emission rate is not much affected with respect to a bulk medium, the strongest enhancements being typically the factors ; 1.5. 2.2. Gaps in 2D Since straight fences of 1D structures are unavoidably leaky, affecting radiation from an emitter requires at least true 2D periodic structures with omnidirectional gaps. Obtaining photon gaps in 2D amounts to require the overlap of gaps occurring

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along the Brillouin zone edge w6x. Since the central frequency of a gap at the edge of the Brillouin zone scales like the modulus of k, the preferred zone shape is as round as possible, which points to a hexagonal shape and, thus, a triangular lattice Žof period denoted as a. w24x. The TE and TM polarisations that separate in Maxwell’s equations are also called H and E, respectively, as they feature a single H and E component along the invariant direction. We concentrate here on the case of air holes Žair rods. in a dielectric matrix, which has the advantage of electrical continuity for device applications. However, hexagonal lattices of pillars may have quite similar properties w25,26x. An example of band structure v Ž k . with a TE gap and no TM gap is given in Fig. 2, for a system with dielectric constant ´ m s 11.7 for the matrix and air-fillling factor Žthe areal fraction of air in a section. f s 30%. Such large f ’s and index contrasts are needed to open sizeable gaps. The gap map of Fig. 3 represents the evolution of both TE and TM gaps in normalized frequency units of u s arl s v ar2p c as a function of f Žsee Refs. w6,27x for other examples.. Only very large holes Ž f ); 65%. result in an absolute gap for TE and TM. On the contrary, the TE gap alone appears as

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Fig. 3. Gap map for the same dielectric constants as Fig. 2, as a function of the air filling factor.

soon as f ) 10%. ŽIn 1D and one direction, the gap opening is proportional to f.. In Section 2.3, we examine some simple defects practiced in a 2D systems, as these defects form a good basis to manipulate light in a frequency- or direction-selective way. 2.3. Modes of selected defects in 2D crystals

Fig. 2. Photon dispersion relation ŽTE Žor H .: solid lines, TM Žor E .: dashed lines. for a triangular lattice of circular air holes in a dielectric matrix with ´ m s11.7, and with an air filling factor f s 30%, a case where a full gap arises in TE polarisation. The inset represents the first Brillouin zone. The denser crystal rows run along the normal to GK.

Remaining in 2D, and with the triangular lattice, we consider two classes of simple defects. In Fig. 4a, we depict the first class: ‘‘0D’’ defects, formed by missing rods arranged in a hexagonal shape. Such defects introduce modes with eigenfrequencies in the gap. Even for a single missing hole, there can be several such modes within the gap. The photon-in-abox mode-counting tells us that the number of states per unit relative bandwidth, d NrŽd vrv . scales like S´ b , with S as the defect area and ´ b as the matrix dielectric constant. The prefactors are such that, for e.g. the f s 30% crystal in typical semiconductor Ž ´ b ; 10., one finds about 0.3 state per missing hole and per 10% bandwidth Žabout the gap relative width.. Going further requires simulations. Such a simulation is shown in Fig. 4b using plane wave expansion

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in a supercell, whereby one finds the modes of an artificial photonic superlattice, the cell of which may comprise hundreds of elementary PC cells, each with or without a hole w28,29x. The mode pattern is reminiscent of the eigenmodes of a disc without ‘‘angular momentum’’, as the main oscillation is radial. Similar ‘‘radial’’ 2D modes were evidenced inside circular concentric Bragg mirrors w30x. Another canonical class of defects are the line defects formed along dense rows according to the schemes of Fig. 5a–b, whereby a spacer of thickness Lc is introduced. The two halves are PC-based mirrors Žpossibly diffractive w31x. so that, at normal incidence, it as basically a Fabry–Perot ŽFP. cavity w32x at k ´ s 0: owing to the periodicity, the modes may be classified using k ´ w28x. For infinitely thick mirrors, there is no fundamental distinction between the k ´ s 0 FP modes and modes with k ´ as high as allowed, in the sense that energy remains confined in the cavity for any frequency in the gap. For mirrors of finite PC thickness, say a few rows, the modes unavoidably experience a finite coupling to outside radiations, allowing, e.g. unit transmission at resonance for a symmetric configuration. We now turn to the experimental systems,

Fig. 5. Scheme of Ža. a basic 2D PC and Žb. a line defect of thickness Lc ; Žc. micrograph of the corresponding cavity Žcrystal period: as 240 nm.. This system sustains both ‘‘FP’’ modes and ‘‘guided’’ modes, as shown.

for which the third dimension is finite or even quantized. 3. 2D PC in the real world 3.1. Macroporous Si The system closest to infinitely thick 2D PCs has been realized using photoelectrochemical etching of Si to produce so-called macroporous Si. Columns of aspect ratio ) 100 are etched at locations defined by a first photolithographic step, producing the starting pits from which anisotropic etching then proceeds w26,33x. This system allows high air-filling factors, as has been examplified by the pioneering demonstration of a complete gap ŽTE and TM. around l s 5 mm by Gruning et al. w34x, using a triangular ¨ lattice with 2.3-mm pitch, leaving only 0.17-mm thick veins between the holes. It is known, however, that Si is a poor light-emitting material so that only detector applications should be sought. 3.2. Membranes

Fig. 4. Magnetic field amplitude for TE polarised eigenmode of a hexagonal cavity of side 5a in a 2D PC lattice of period a and parameters close to those of Fig. 2.

Self-supported membranes are the other extreme in terms of height. One dissolves a sacrificial layer beneath the membrane by selective etching, a process best mastered in InP-based alloys w17,35x. A

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virgin membrane always sustains at least one guided mode. But given the high index contrast, the minimum mode thickness corresponds to membrane thicknesses of the order of lr2 n, hence, 0.2 mm for near-infrared applications. Can a periodically perforated membrane still sustain truly confined, lossless guided modes? This is indeed possible for PCs which are infinitely extended in-plane, although it has never been experimentally clearly demonstrated. If the system is infinite, modes are again properly labelled by k ´ , the in-plane wavevector, now restricted to the first Brillouin zone. Radiation modes have a real k z component in the third direction so that v s c k 52 q k z2 ) ck 5 . If a mode lies below the vacuum light line v s ck ´ of the dispersion diagram ŽFig. 6., it cannot couple to radiation modes for which v ) ck ´ . Due to periodicity, the modes in-plane Fourier components Ž k ´ , k ´ q G 1 , k ´ q G 2 , . . . , 0, G 1 , G 2 , . . . being the in-plane reciprocal lattice vectors. all feature larger k’s and automatically obey v - ck. Upper dispersion branches lying in the region v ) ck ´ correspond in principle to delocalized modes, filling the full space, but modulated spatially by the presence of the PC w36x. These modes tend to form a continuum Ž k z is now continuous., but some localization in the membrane is still possible, corresponding to the case of guided modes a relatively well-defined k z value that have a modest coupling to the outside. This gives rise to the anomalies in transmission or reflection of impinging plane waves at a characteristic angle, a phenomenon known in lamellar gratings, and which

(

Fig. 6. A typical dispersion relation of a periodic structure Ždotted curves. and the various light lines Ždashed lines.; k-components of a localized defect mode are also pictured as a function of k ´ .

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can be traced back to the Wood anomaly of metallic gratings Žsee Ref. w37x for the case of PCs.. Close but distinct from the 2D PC geometry is the air-bridge geometry probed by Villeneuve et al. w38,39x with 1D, rather than 2D, periodicity and few periods. In these studies, however, the emphasis has been more on gaps and defect modes, than on dispersion relations and existence of true guided modes. Experiments on membrane-based PCs have been rather devoted to 0D defects, such as micro-hexagons w16,17x. Any laterally localized mode in such a system has nonzero k-components, with a continuous k-spectrum, allowing k-components above the light line Ži.e. closer to k s 0 at given v .. Electromagnetic energy fed into such a defect mode eventually radiates into 3D modes and the cavity field behaves as a damped oscillator w40,41x. In terms of light-emitting device, this corresponds to a perfect extraction efficiency, whereby light generated inside a dielectric is fully radiated outside w42x. This is in contrast with the fate of light generated in a simpler macroscopic shape, where indefinite total internal reflections ŽTIR. occur. This issue is well documented for LEDs w43,44x and leads to extraction efficiency h far from 100%: one has typically h s 2–20% depending on substrate and superlayer absorption, use of lateral light, etc. However, using a membrane into a mass-production device raises several issues Žsee Section 4.. Refs. w16,17x evidenced the confined in-plane resonances of hexagonal cavities by means of the radiation losses towards air, using 1.5–2 mm spontaneous emission Žphotoluminescence.. In Ref. w16x, spatial and spectral resolution are poor, the cavity is still rather large Ž; 15 mm. and the modes are not individually resolved. In Ref. w17x, the cavity is smaller Ža few microns., excitation is localized, and individual modes are apparently resolved. Self-supported membranes are thus probably excellent laboratory systems, but one may prefer more robust ones. An easy way to strengthen such membranes is to bond them to a glass substrate Ž n s f 1.46. and lift them off their original substrate. This has the clear drawback to lower the light line down to v s ck ´rn s , meaning that at a given v , higher k ´ are able to radiate into substrate modes and damp possible defect modes. A route more adapted to GaAs-based heterostructures is to selectively steam-

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oxidize a high Al-content GaAlAs layer into ‘‘Alox’’, of index n s f 1.6. The Alox system has been used for air bridges w38,39x, but not for true 2D PCs Žexcept as a mask in Ref. w19x.. The glass substrate has been used recently by Baba et al. w15x, to produce interesting qualitative experiments on guides in 2D PCs. 3.3. Systems with standard substrates Can one preserve the advantages of freezing the third dimension into a guided mode but retain a more classical guiding heterostructure with moderate index contrast, Ž D n s 0.2–0.6.? The primary penalty is to lower the light line down to v s Ž crn clad . k ´ and to subsequently allow much more k-components of a given mode to radiate away from the guide. To minimize the extent of this phenomenon, one can play first with the strength of the basic mechanism, as discussed below in Section 4. Let us simply say here that the hole diameter and the index contrast are the essential parameters to tune this strength. Another way to handle this is to minimize the field itself in the holes where leakage takes place. Following the first measurements by Krauss et al. w45x, our group has acquired extensive experience of this kind of 2D PCs for filling factors f in the range 20–30%, where, as suggested originally by Krauss et al., out-of-plane radiation losses are still affordable compared to the case f ) 60%, where they tend to wash out all the expected photonic features of the 2D crystal. A quite complete set of measurements of transmission, reflection and diffraction by 2D PCs of this kind can be found in the following references, using photoluminescence as an internal and very practical light source w18,30–32,46–49x. This source can also be made relatively broadband Ž) 10%. using quantum dot layers ŽQDs. instead of the quantum wells, so that combined to ‘‘lithographic tuning’’ with 10% steps of the PC period a, continuous spectra can be obtained. Typical periods to obtain a gap around 1 mm are a s 200–240 nm, and nominal hole diameters 130–160 nm. A sample transmission measurement of crystals of variable periods a Žfrom 180 to 330 nm. is presented on Fig. 7a–e on a log scale, for PCs with a variable number of rows N, showing the build-up of the gap in the TE polarization and the GK orientation of the

Fig. 7. Ža–e. Near infrared transmission, on a log scale, of 2D PCs with N s 2, 5, 8, 12 and 15 rows, along the GK direction and for the TE polarisation, as a function of the normalized frequency. The gap build-up is clearly visible. Note in Že. the data added for the GM orientation, showing how the partial gaps overlap; Žf. the N s15 GK transmission spectrum on a linear scale, showing the ‘‘fine structure’’ oscillations around the gap and the smoother high energy band edge.

impinging wave w48x. For N s 15, the GM orientation Žat 308 from GK. is also shown and the full gap can be seen to extend from u s 0.21 to u s 0.245, over a 15% relative frequency range. Fig. 7f presents a transmission spectrum along GK on a linear scale, on which the oscillations around the gap are clearly visible. As discussed in Refs. w29,48x, the presence of these oscillations, with an amplitude that compares

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well with pure 2D theory, demonstrates that a Bloch wave launched at one crystal side may be little attenuated upon a round-trip through 30 rows of cylinders, about 60 semiconductorrair interfaces. The fact that the pure 2D picture provides a satisfactory first-order description of the present PC properties shows that the various perturbations Žfabrication fluctuations and mostly, the presence of the waveguide. are still modest. The damping of the oscillation amplitude on the high energy side of the gap is also interesting, as it reveals large losses when the Bloch wave samples the holes rather than the dielectric. It suggests that one handle regarding losses is the relative location of the working frequency in the gap, which dictates the local field distribution and, thus, the associated losses. The two classes of cavity modes introduced above, the micro-hexagon and the linear defect, have also been experimentally demonstrated using the same waveguide-based PCs. Micro-hexagons with side lengths from 4 a to 7a were fabricated and probed using the internal PL of embedded QDs photoexcited inside the cavity ŽFig. 8a.. Detection was made at the cavity boundary Žalthough the cavity dimensions are not much larger than the setup resolution, about 1 mm. in order to collect light radiated by the decaying guided mode field into air and to remove the background of QDs directly emitting towards air Žthe luminescence of InAs QDs in the PCs is quenched at room temperature by etching-induced nonradiative recombination. w18,30x. A typical spectrum is shown on Fig. 8b for a cavity of side 5a. Narrow peaks with Q ; 1000 clearly appear. They are the signatures of in-plane cavity modes. However, the number of such peaks is smaller than the ; 20 modes expected for the 10% present relative bandwidth and cavity size Ž61 missing holes.. We attribute the absence of some modes to the fact that their nontrivial field pattern prevents far-field radiation towards the collecting lens. Nevertheless, the sharp spectral signatures are a wonderful fabrication reproducibility test w29x. To obtain physical information in the simplest way, we found that 1D defects ŽFP cavities bounded by PCs. were most interesting w32x as the analysis of the FP resonance is a well documented matter that gives easy insight into, e.g. the mirror losses. For PCs of N s 4 rows only, periods of 200–260 nm,

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Fig. 8. Ža. Experimental configuration used to detect in-plane cavity resonances by means of scattered photoluminescence of embedded QDs; Žb. typical spectrum obtained for a hexagon of side 5a surrounded by 2C PC Ž as 240 nm..

and still the same kind of air-filling factors 20–30%, we introduced spacer L c ranging from 40 to 360 nm and found perfectly defined FP transmission peaks with quality factors Q from 50 to 200 and peak transmissions from 15% to 45%. We could infer that the mirror reflectivities are in the range of R s 90%, with transmission T and losses L Žpresumably radiation losses. sharing each about 5% of the remaining 10% power ŽT q R q L s 1.. These performances compare well with those of membranes, the Q’s obtained in similar microhexagons being apparently of the same order. Both systems could hence be competing in terms of reflectivity. Mirrors with 90–95% reflectivities are already interesting enough for many applications. To go beyond, e.g. to R s 99%, one has to understand the origin of these limits, as discussed below. We now discuss the etching issues for these various systems. 3.4. Etching issues Etching submicron arrays of holes in typical semiconductors is the basic task for fabricating PCs. Practical dimensions for near-infrared applications

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are in the range d s 100–300 nm for hole diameters and a s 200–600 nm for the period. Applications to mid-infrared emitters Ž l s 10 mm. is also envisioned with d s 1 mm and a s 2 mm, or even with pillars instead of holes. The main advantage of membranes over substrate approaches is the limited etch depth required, typically 200–400 nm, whereas the full profile of the guided mode has to be overlapped by the hole in the substrate approach, which leads to depths of the order of 800 nm in the GaAlAs system around l s 1 mm, and possibly up to 2 mm in the InP system around l s 1.6 mm, in which the smaller index steps increases the mode tail. Attaining aspect ratio of 6 to 10 is thus advisable for the substrate approach, instead of 1 to 2 in the membrane approach. This demand translates into a technological quest of a mask, withstanding the etching process with minimal erosion. The resist itself Žsay PMMA. cannot be used in general. One has to transfer the pattern to another more robust mask, silica w50x or nitride, or to bilayer masks. The etching chemistry and parameters Žkind of plasma, etc.. will not be discussed here but have important effects on the shape and properties of the etched surface. Nonradiative recombination is strongly systemdependent, being very large in the GaAlAs system, but remaining moderate in the InP system. Only in this latter can QW luminescence be maintained at room temperatures close to interfaces, say, within less than a micron. QD luminescent layers are much less sensitive to nonradiative recombination because dots Žquantum or not. localize the carriers w51,52x. This trapping is, however, far from perfect for InAs dots in GaAs around room temperature, for which an important nonradiative quenching of the PL occurs, e.g. for hexagonal cavities with diameter less than 1 mm. Below 150 K, trapping is efficient and carriers trapped in the dots experience negligible re-evaporation in the GaAs waveguide before recombination in the dot takes place. InAs dots embedded in InP have similar properties, but the numbers are different because dots are larger and band discontinuities are different. They are certainly interesting for the investigation of PCs around the 1.55 mm region, at least because they again provide larger PL spectra.

The uniformity and accuracy of etching are also of some importance. Their degradation would induce some broadening of the spectral features. From our experience, based on a Leica e-beam exposure system operated at 5 nm pixel size and followed by RIE, disorder and fluctuations were not sufficient to broaden the measured features, in particular, the sharp low energy band edge of relative width ; 1‰. Taking the 1D FP cavities bounded by 2D PC as another example, we could fit the eigenfrequencies of cavities with spacers Lc in steps of 10 or 20 nm, without any adjustable parameter, just using the guide effective index in a pure 2D model w53x. Similarly, we show in Ref. w29x that for two nearby microhexagons, the spectra with sharp peaks of Fig. 8 are accurately reproduced Žone of them had some modifications, but far enough from the cavity to have a very small impact on the mode frequencies.. At the present point, our opinion based on the study of various in-plane microresonators is thus that the obtainment of sharp, reproducible photonic features is not limited by the various e-beam and etching fluctuation issues within a given run, but rather, by the intrinsic performances of the PCs themselves. The reproducibility from run to run is, of course, delicate to assess in the context of academic facilities, and would certainly be improved in the industrial realm. These conclusions should be revisited for larger resonators Že.g. for PCs used as laser end facets with, say, 100 mm spacing., but the progresses in technology are such that, in our view, no fundamental bottleneck can be foreseen on this matter. 3.5. Radiation losses The best documented system from the experimental viewpoint is the substrate one. Namely, we found that PCs with the smallest periods and holes Ž a s 200 and mostly, a s 180 nm, with holes of 100 nm diameter or less. had much weaker transmission than expected, whereas periods a s 220 nm and above were more in agreement with the theoretical expectation of a 2D PC. As the etching technology reaches its limits around diameters of 100 nm, the weaker transmission is a clear effect of the finite etch depth, thus, owing to etching limitations and not intrinsic to the substrate approach. The recovery of acceptable transmissions for PCs with a s 220 nm and above

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leads us to consider that we only have intrinsic effects, such as radiation losses for most of our PC samples. Remind that the smoothing of the high energy edge of the gap in Fig. 7f, accompanied by a damping of the ‘‘fine structure’’ oscillations when approaching this very edge, has lead us to consider holes as the source of losses. More quantitative effects of radiation losses are seen in the 1D FP cavities in the form of limited Q factors Žpure 2D theory predicts Q’s about twice higher. and relatively poor transmissions ŽT ; 0.25.. In Ref. w54x, based on the ideas also found in Labilloy’s thesis, we attempted to build up a simple scattering treatment for radiation losses. We start from the fact that a separable solution of the field, EŽ x, y, z . s c Ž x, y . z Ž z ., taken as scalar for simplicity, holds for the case in which ´ Ž x, y, z . is the sum of a ‘‘horizontal’’ and a ‘‘ vertical’’ term. This would hold if the index contrast of the original waveguide, D ´ , taken as symmetric for simplicity, were also existing along the crystal rods, with, say, air Ž ´ s 1. at the cladding level and a filling, such that ´ core s 1 q D ´ at the level of the holes in a volume V s wp r 2 defined by the core waveguide thickness w and the hole area p r 2 . Conversely, the actual system Ž ´ s 1 all along the crystal rods. can then be seen as a perturbation to the separable case differing by a term VD ´ . The perturbed polarization standing in volume V is then basically P Ž x, y, z . s EŽ x, y, z . D ´ and under simplifying assumptions, one can view radiation losses as originating from the power dissipated at each hole by a dipole p s PV s EŽ x, y, z . D ´ V. This power obviously scales like p 2 s E 2D ´ 2 V 2 . The E 2 factor is met in any dissipative process, such as Joule heating. Using this analogy, an effective imaginary dielectric constant could be defined in the holes and was found to be of the order of ´ Y s 0.1–0.2 in our 2D PCs. This data will be used in Section 4. The V 2 factor is quite interesting, as it tells that small holes are quadratically better than large ones. But the D ´ 2 factor is still more exciting, as it suggests that losses are minimized for the weakest waveguide confinement! This is when the light-line is intrinsically very close to the guided mode relation dipersion, since the slopes in the diagram are given by ´y1 r2 . This is not so paradoxical for two reasons: Ži. firstly, we did not consider in which direction the

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dipole was allowed to radiate its power p 2 . It is obvious that, since the dipole p radiates in a waveguide, only a fraction of its power may be extracted outside the guide, a large fraction for small index contrast, but a small one for large contrast. So the D ´ 2 trend is much offset by the ‘‘power extraction’’ effect. Also, coherence ŽBragg. effects have not been discussed, as they depend whether a propagative mode or a defect mode is considered, opening many possibilities for the Bragg summation in 3D space directions. Precautions are thus required to extrapolate the present reasoning to resonators in membranes. The other reason is that Žii. approaching the light-line just tells about the selection rule of a transition, allowed or not, not about the strength of the mechanism. What happens as D ´ vanishes is that the guided mode then mimics a plane wave and feels less the disturbance caused by the holes. It is well known that for shallow index profiles used, for example on silica waveguides, interrupted guides behave surprisingly well. In some cases, ‘‘segmented’’ waveguides with many interrupts can be of great interest for mode transformation w55x. However, in this kind of limit, the guide becomes large Žmany l,so that aspect ratio of holes lie again in the range 20–100., and has a vanishing critical angle Žsay, that of glass optical fibers. and is unable to concentrate spontaneous emission of inner species. Hence, the kind of guides usually met in semiconductor heterostructures, with D ´ ; 1, critical angles around 40–508 and a thickness ; lr2–0.2 mm, could well be not far from the best compromise between low losses, good capture of spontaneous emission and limited etch depth. The task of experimentalists and theoriticians is now to assess as well the possible radiation losses of membrane- and of substrate-based 2D PCs, in order to know the limits of each system. 4. Devices and 2D PCs 4.1. 2D PCs compatibility Before examining possible PC-based devices, we address in this subsection various issues of interest for applications. Ø Heat sinking: For the membrane approach, heat sinking takes place horizontally only. Heating was

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seen to be a major issue upon photopumping at relatively moderate densities. Temperature variation of the order of 100 K was suggested w16x. Ø Current: Bringing current laterally through a 2D PC is possible in the air rod case Žnot in the pillar case!., with only a small impedance penalty. The more challenging issue is to bring both electron and holes horizontally to a common area Ždefect. enclosed by PCs. We found that this required well-defined lateral current apertures, such as those used in VCSELs and based on steam oxidised AlAs. However, an additional complication is that these current apertures have to go around the PC holes. Ø Tunability: In the substrate approach, tuning has to rely on the same electro-optic Žor nonlinear optical. coefficients, as is usual elsewhere in optoelectronics, with typical index variations of only 1%. For a given index variation, the largest potential variations in PC optical behaviour are expected around band edges. But one should be aware that a band edge is also basically very sensitive to fabrication parameters: in the center of a gap, the phase of the amplitude, reflected by a PC is as stationary as possible, whereas at a band edge, this phase experiences rapid variations. As an example, using the slow-down of group velocity near band edges is probably a delicate experimental matter Žthe same holds for the ‘‘superprism’’ effects w56–58x.. In the case of membranes, mechanical effects could be of great interest to provide ‘‘slow’’ tunability Ža few microseconds, see Ref. w35x and the literature on MOEMs.. For example, electrostatically approaching a defect in a membrane from the substrate would certainly provide a large tuning of the defect frequency. Radiation losses would also be tuned in this geometry, so that solutions able to tune only the frequency or only the quality factor are to be sought. 4.2. Light-emitting diodes (LEDs) LEDs are a first class of application of PCs. The basic issue in LEDs is to raise the external efficiency, since the internal efficiency of industrially grown epitaxial active layers is often above 90%, whereas the light extraction efficiency ranges from 2% to 20%. Three approaches can be pursued. In the most feasible, one aims at re-absorbing the guided light propagating horizontally in the structure, which can amount to 40% of the generated light, or

even more in substrate-less structures, and is useless at the user’s surface. This re-absorption, followed by re-emission is called photon recycling. It readily exists to some extent in ordinary LEDs, but it is not a very profitable scheme for various reasons. For the specific case of fiber-coupled LEDs, the brightness of the front LED chip is the key issue. It is increased if enough recycling takes place in front of the fiber core. This can be achieved by enclosing the emitting area, say, 10 mm in diameter, by PC-based omnidirectional mirrors Žwith respect to guided modes.. However, controlling the other modes Žnot guided and not extracted directly, the so-called leaky modes for example. is advisable at the same time. See Ref. w59x for the interplay of vertical microcavity effects and photon recycling. The second way to use PCs is to take advantage of their radiation losses, much as surface emitting second-order Bragg grating w1x. PCs are then located around the emitting area w60x. However, one need not be in the PC gap, but rather, in conduction band modes to favor radiation losses. Being quantitative about this approach is not obvious, either from the point of view of theory Žwhere a detailed angular and spectral knowledge of radiation losses is needed. or experiment w60x. In the third approach, more ambitious, one builds a cavity with few modes in which electroluminescence takes place. One privileged mode is well-coupled to the outside, as what occurs for planar microcavity LEDs w44x. It thus takes away a large fraction of the emission. In the present case of 2D PC, this could partly apply to guided light generated in the 1D cavities discussed above, which may sustain in adequate instances a single mode around mid-gap, around 70 nm thickness for our parameters. Emission taking place inside such a cavity would be channeled by the single mode to an open end of the cavity and a large brightness would be achieved on a very small spot. However, obtainment of acceptable internal efficiencies inside such a tiny cavity is a challenge, even by using QDs. The goal of electroluminescence is not easier, but the forced carrier drift may somewhat alleviate diffusion issues. In conclusion, one should probably consider first larger cavities making use of either recycling or direct extraction, both effects still requiring precise estimations.

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4.3. Micro-lasers and the Purcell effect In fully confined cavities Žsuch as 0D defects in a 2D PC in a waveguide, or micropillars etched through planar microcavity, or also mircodiscs ‘‘whispering gallery modes’’ w61,62x or ‘‘quasi-radial mode’’ w30x., the DOS is basically a series of Dirac peaks, broadened by the quality factor Q. If an emitter has a linewidth smaller than D v cav s vrQ, it may experience the peak DOS, scaling as 1rQ. Applying the Fermi golden rule, it has been shown since Purcell, that this can lead to an enhancement of the spontaneous emission rate of the mode by a factor proportional to Q l3rV ŽPurcell factor., where V is the modal volume w51x. This enhancement has been experimentally evidenced for InAs QDs in AlGaAs-based micropillars with Q’s in the range 1000–4000 and diameters around 1 mm w51,63x, with a radiative lifetime divided by a factor of five for the best pillars. For light emitters, this phenomenon could apply to spontaneous emission, as well as stimulated emission. The basic idea is that the competition between the previously indistinct modes is now biased in favor of the resonant mode by the larger Purcell factor. For example, if 10 modes compete for a single emitter, emission in the single mode with a Purcell factor of 10 can be one half of the total instead of one tenth. For spontaneous emission, the aim is to beat the speed limitations of present LEDs, operating not above ; 1 GHz at high injection. Speeding up LEDs is presently achieved to the expense of efficiency by introducing a large density of nonradiative centers. The ‘‘Purcell effect’’ offers a more satisfying alternative, whereby carriers trapped in those QDs resonant with the high Q mode directly experience an enhanced radiative rate. For micro-lasers, the interest is in changing the ‘‘spontaneous emission coupling factor’’ denoted as b , giving the fraction of photons emitted into the lasing mode. b is only 10y4 –10y5 in edge-emitting laser diodes, and peaks around 0.1 in whispering-gallery modes of microdiscs. A larger b means that generated photons are not spoiled in useless modes. It has been suggested that in the b 1 limit, the threshold would disappear from the laser L-I curve. It is clear that a mode with a large Purcell factor also

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has a large b , but a full realistic model of such micro-lasers should be considered to predict what can be gained. A first step for the Purcell factor would be to compare the performances of the micro-pillars and of the micro-hexagons. Both systems have demonstrated rather comparable Q’s Ž1000 or more. and normalized volumes Vrl3 are similar, albeit with a different shape. Micro-discs can also be serious contenders, with a simpler mode structure, but their high Q modes are also the most difficult to couple outside. In PC-based systems, more tunability is offered, as the mirror is a periodic structure with many ‘‘handles’’. To conclude this subsection on ambitious matters, one should keep in mind that we are dealing with very small cavity volumes and very sharp resonances, both factors allowing only a few QDs to be on resonance. Even if 10 10 recombination events take place every second in the proper mode, the generated power would be only 1 nW! For fiberoptics applications, the existing infrastructure requires mW power levels, a figure, which has no reason to diminish soon. However, the realization of ‘‘single-photon’’ sources, with a potential for quantum cryptography and secured information exchange could still make single faint emitters Žor arrays of them. interesting. 4.4. Integrated optics Integrated optics is devoted to routing optical signals. Semiconductor-based integrated optics requires expensive substrates such as InP, but allows the integration of various electro-optic functions and of light sources, amplifiers, etc. However, to date, only a few functions can be cascaded on a single chip, the size of which is often closer to 1 cm2 than 1 mm2 w1x. The radius of curvatures of the guides are in the 100’s of mm range. Guides with acceptable scattering are typically strip-loaded or buried ones, with a weak lateral contrast in terms of effective index. Thus, the basic TIR mechanism of a straight guide is rapidly degraded in a curve of modest curvature. PCs have been suggested as novel guide boundaries allowing virtually 908 turns within a wave-

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length or so w6,14,64–67x. This novel lateral guide implementation has, of course, a large potential to shrink the size of integrated optics circuits. We can even envision more than shrinking the dimensions, as the optical properties of PCs are so rich: combining different functions presently requires many regrowth steps to implement the proper layers, different for grating Žoptical feedback. regions, electrooptic sections Žtunable phase., etc. The versatility of PCs could be used to pattern, in a single-step, different complex functions on the same wafer. Some of the optical effects presently relying on different materials could be made to rely on different PC shapes and parameters. Although the detail of these ideas is still fuzzy, there is no doubt that novel routes are available for the design of integrated optics circuits. One could wonder why PC-based waveguides are so much better than narrow stripe Ž; l. waveguides, with a strong semiconductorrair contrast. The answer is that an irregularity at the surface of the latter gives rise to scattering into propagative waves, whereas no such waves exist in the gap of a PC. Hence, irregularities cause only scattering to evanescent modes, which do not carry any power away. Since the intrinsic radiation losses of PCs are left as the dominant scattering phenomenon, we estimate here their order-of-magnitude for a PC-based straight guide, using the kind of 2D PCs we explored Ž f s 20–30%.. We assume a width of the order of 600 nm, which is laterally a multimode guide. For the fundamental bell-shaped mode profile, a realistic overlap factor with the PC boundaries as a whole is G PC s 0.1. For the losses, the overlap G holes with the holes should be minimized. For f s 30%, G holes s f = G PC s 0.03 is an average value, but a proper choice of the side of the gap, close to the low energy edge, could reduce this factor to, say, G holes s 0.015. Next, we use the value ´ Y s 0.1 in the holes that can be expected from an adequate vertical heterostructure, elaborating on the discussion above, so that an effective value for attenuation can be deY duced from ´eff s ´ YG holes s 1.5 = 10y3 : propagation occurs, as for a plane wave propagating in a Y medium with ´ ) s ´ X y i ´eff s 9–1.5 = 10y3 i. Ž ´ X s 9 is the squared effective index., or n ) s 3–2.5 = 10y4 i s nX y inY . Elementary optics tells that this corresponds to an attenuation length lr2p nY ; 1 mm, or, for a s 330 nm PC period, 3000 holes. This

is enough to implement many functions with quite some margin. Even in an ultimate laterally-monomode PCwaveguide of width - 100 nm, and thus, G PC ; 0.5 Žhalf or more of the mode is in the PC boundary region., this would still leave about 500 holes to play with. Hence, integrated optics could soon profit by the impact of 2D PCs to drastically increase its packing density.

5. Conclusion PCs with bandgaps were presented as a novel ‘‘optical material’’, with a number of striking properties and many promises. We have seen that 2D PCs are not only an ideal system for theory, but can also be implemented in practice in semiconductors, in particular in III–V’s. Freezing the third dimension by means of a standard heterostructure waveguide or even a membrane achieves a nearly 2D situation. Each of the two approaches has its pros and cons: simplicity and compatibility for the substrate approach, shallower etching and more tunability for the membrane approach. Radiation losses arise in these systems as soon as some light ‘‘molding’’ w6x is required, e.g. when a 0D defect mode, such as a micro-hexagon is introduced. We could give a quantitative insight on these losses in the substrate realisation and, thus, attempted to extrapolate the possible performances of straight waveguides bounded by 2D PCs. We also discussed the prospects for lightemitters. In both cases, we have only given a modest account of technological issues. Although the technological task is certainly a difficult one, with the interplay of many factors Želectrical, optical, thermal, surface-related, etc.., the new prospects raised by these systems and the understanding that has already been reached are sufficient incentives to overcome these difficulties.

Acknowledgements Many of the results shown in this paper are the result of fruitful collaborations. Concerning the samples, T.F. Krauss, C.J.M. Smith and R.M. De La Rue in Glasgow University have performed etching of

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