ceramics

Article Coupled Photonic Nanocavities as a Tool to Tailor and Control Emission

Annamaria Gerardino 1,* , Giorgio Pettinari 1 , Niccolò Caselli 2,3,†, Silvia Vignolini 2,3,‡, Francesco Riboli 3,§, Francesco Biccari 2,3 , Marco Felici 4 , Antonio Polimeni 4 , Andrea Fiore 5, Massimo Gurioli 2 and Francesca Intonti 2,*

1 National Research Council, Institute for and (IFN-CNR), Via Cineto Romano 42, I-00156 Rome, Italy; [email protected] 2 Department of Physics and Astronomy, University of Florence, via Sansone 1, I-50019 Sesto Fiorentino, Italy; [email protected] (N.C.); [email protected] (S.V.); francesco.biccari@unifi.it (F.B.); gurioli@fi.infn.it (M.G.) 3 European Laboratory for Non-linear Spectroscopy, via Nello Carrara 1, I-50019 Sesto Fiorentino, Italy; [email protected]fi.it 4 Department of Physics, Sapienza University of Rome, P.le A. Moro 5, I-00185 Roma, Italy; [email protected] (M.F.); [email protected] (A.P.) 5 Dep. Applied Physics and Institute for Photonic Integration, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands; a.fi[email protected] * Correspondence: [email protected] (A.G.); [email protected]fi.it (F.I.); Tel.: +39-064-152-2242 (A.G.) † Present address: Instituto de Ciencia de Materiales de Madrid (ICMM), CSIC, Sor Juana Inés de la Cruz 3, 28049 Madrid, Spain. ‡ Present address: Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, UK. § Present address: National Research Council, Istituto Nazionale di Ottica (INO-CNR), via Nello Carrara 1, I-50019 Sesto Fiorentino, Italy.

 Received: 29 November 2018; Accepted: 27 December 2018; Published: 14 January 2019 

Abstract: In this review, we report on the design, fabrication, and characterization of photonic crystal arrays, made of two and three coupled nanocavities. The properties of the cavity modes depend directly on the shape of the nanocavities and on their geometrical arrangement. A non-negligible role is also played by the possible disorder because of the fabrication processes. The experimental results on the spatial distribution of the cavity modes and their physical characteristics, like and parity, are described and compared with the numerical simulations. Moreover, an innovative approach to deterministically couple the single emitters to the cavity modes is described. The possibility to image the mode spatial distribution, in single and coupled nanocavities, combined with the control of the emitter spatial position allows for a deterministic approach for the study of cavity quantum electrodynamics phenomena and for the development of new photonic-based applications.

Keywords: ; photonic crystal ; resonant coupling

1. Introduction The development of fabrication technologies that enable the capability of patterning materials on dimensions smaller than the of has opened the way to new classes of applied physics fields, like plasmonics, photonics, spintronics, and also to the experimental exploitation of quantum effects [1,2]. This is of particular interest in the field of . The interaction between optical waves (i.e., ) and materials is strongly dependent on the material structure, so nowadays, it is possible to control and tailor the light propagation and localization by designing the material arrangement. Within this framework, photonic (PCs) represent a perfect example of how, by

Ceramics 2019, 2, 34–55; doi:10.3390/ceramics2010004 www.mdpi.com/journal/ceramics Ceramics 2019, 2 35 nanostructuring a material, we are able to impart to it new physical properties. As it is well known, PCs have been introduced starting from the idea of Yablonovitch [3] and John [4], by Joannopoulos and co-workers [5]. The idea is based on the possibility to create a periodic variation in the of materials, in order to affect the properties of photons, in a similar way to how ordinary crystals affect the properties of . If the difference in the refractive index of the materials is high enough, and if the absorption of light in the materials is sufficiently low, the refractions and reflections of the light wave through all of the various interfaces can produce the same effects for photons as the ones produced by the atomic periodical potential for electrons (for an extensive description see [5], in particular Chapter 3). As in the crystal, the atomic periodic potential gives origin to the band gaps between the valence and conduction energy bands, and in the photonic crystals, the periodic modulation of the refractive index of the dielectric materials allow for the formation of photonic band gaps, that is, a frequency range in which light propagation is forbidden. Photonic band gaps can be designed and fabricated in dielectric materials with a different refractive index; the propagation of photons can be inhibited for certain designed frequencies and allowed for others. The periodic variation can be realized in 1D, 2D, or 3D, determining the characteristics of the PC band gaps; for example, 3D PCs could have a complete developed and the light cannot propagate in any direction for any wavelength within that gap. A typical 2D PC consists of an array of holes (i.e., air cylinders) realized in a dielectric material slab. The presence of the slab also ensures confinement in the vertical direction by total internal reflection. The symmetry of the lattice, the diameter of the holes (d), the array lattice spacing (a), and the slab thickness (t), are the parameters that control the band gap properties for in-plane propagation. High resolution e-beam lithography and high precision anisotropic etching are the top-down fabrication processes that enable the fabrication of PCs. The control of the fabrication process has a direct influence on the PC performances, and must be carefully evaluated, as we will discuss in the following. Moreover, defects in the PCs can be introduced on purpose, in order to create integrated photonic elements. Linear defects, a missing line of holes, gives rise to ; point defects, namely missing a single hole or a few holes, results in a nanocavity. A single PC nanocavity can be considered as the photonic analogue of a bounded electronic quantum system, and shows a discrete set of energy levels with a characteristic spectral width and spatial extent, even if the photonic is inherently an open system. By combining these basic elements and tuning the characteristic parameters of PCs, integrated photonic circuits and devices can be designed and fabricated to manipulate light at ranging from visible to microwaves, according to the resolution limit of the fabrication techniques. PC nanocavities can be exploited to realize advanced light sources when coupled to two levels emitters, for example quantum dots (QDs). A quantum dot is an inclusion of material, confined in 3D on a nanometric scale into a hosting material with a higher band gap energy, thus resulting in a discrete energy spectrum and in strong carrier correlation effects, making the energy of each electronic configuration (excitons, biexcitons, and multiexcitons) easily distinguishable. According to the density of the QDs grown in the PC nanocavity, several quantum electrodynamic effects can be studied, from the deterministic coupling between a nanocavity and an emitter (Purcell effect [6], Rabi oscillations [7], etc.) to the controlled production of single or entangled photons [8–10]. The Purcell effect, or weak coupling, is the modification of the spontaneous rate of emission when the emitter transition is tuned with one of the discrete cavity modes. A faster emission is desirable, because it allows for the realization of more deterministic and reliable single photon sources (photon-on-demand source). Furthermore, the cavity can channel the emitted photons into a well-defined spatial mode in order to increase the extraction efficiency. However, the most striking change of emission properties occurs when the conditions for strong coupling are fulfilled. In this case, there is a change from the irreversible to a reversible exchange of energy between the emitter and the cavity mode, leading to vacuum Rabi splitting [11]. This coherent coupling may provide a basis for future applications in quantum information processing or schemes for coherent control. To summarize, low-threshold lasers [12,13], single photon sources [8,9], add-drop filters [14], and the Ceramics 2019, 2 FOR PEER REVIEW 3 Ceramics 2019, 2 36 summarize, low-threshold lasers [12,13], single photon sources [8,9], add-drop filters [14], and the implementation of cavity quantum electrodynamics experiments [15–18] are just some examples of implementationthe possible applications of cavity quantumof PC nanocavities. electrodynamics experiments [15–18] are just some examples of the possibleIn this applications review, of we PC describe nanocavities. some of our studies on coupled photonic-crystal cavities, an interestingIn this platform review, we to describeexploit the some features of our of studies photonic on coupled molecules photonic-crystal [19]. Photonic cavities, molecules an interestinghave been platformemployed to to exploit realize the photonic features device of photonic like lasers molecules [20], waveguides [19]. Photonic [21], molecules or memories have been[22]. employedThe basic tophotonic realize structure photonic we device considered like lasers is a [20two], waveguidesdimensional [triangular21], or memories lattice with [22]. a Thefilling basic factor photonic of f = structure35%, and wethe consideredanalyzed single is a two cavity dimensional is formed triangular by four missing lattice with holes a filling(see the factor topography of f = 35%, image and and the analyzedscanning singleelectron cavity microscope is formed (SEM) by four image missing in Figure holes (see 1a,b)[23], the topography and diamond-like image and and scanning denominated microscopeD2. The mode (SEM) distribution image inof Figurethese D21a,b) coupled [ 23], PC and cavities diamond-like is particularly and denominated interesting and D2. makes The modethem distributiongood candidates of these to D2study coupled a coupled PC cavities PC system; is particularly a system interesting of two D2 and coupled makes themPC cavities good candidates has been todesigned, study a fabricated, coupled PC and system; characterized a system ofusing two D2 coupled PC cavities (PL), has and been near designed, field and fabricated, far field andspectroscopy characterized in order using to photoluminescence understand the cavity (PL), mode and near behavior. field and The far two-cavity field spectroscopy system has in order been toexploited understand to realize the cavityan analogue mode behavior.of the Young’s The two-cavitydouble slit systemexperiment has beento probe exploited the photonic to realize mode an analoguesymmetry of and the to Young’s give an double experimental slit experiment proof of to the probe possibility the photonic to control mode the symmetry ground andstate to bonding give an experimentaland antibonding proof nature of the in possibilitythe photonic to controlmolecules the using ground the state geometrical bonding arrangement. and antibonding Once nature the two in thecoupled photonic PC cavities molecules case using has been the geometrical analyzed, we arrangement. also considered Once a thesystem two coupledof three coupled PC cavities PC casecavities has beento investigate analyzed, the we photon also considered hopping between a system the of interacting three coupled resonators. PC cavities The control to investigate of the mode the photon nature hoppingand of the between related the effects interacting is crucial resonators. in the exploitation The control of of photonic the mode molecules nature and as of building the related blocks effects of islarge crucial scale in photonic the exploitation integrated of photonic circuits in molecules analogy as with building electric blocks circuits of large [2423]. scale photonic integrated circuits in analogy with electric circuits [24].

Figure 1. Single D2 nanocavity with lattice parameter a = 311 nm. ( (aa)) Topographic Topographic image by scanning b c near-fieldnear-field optical optical microscope microscope (SNOM) (SNOM) scan; scan; (b () SEM) SEM image; image; (c) (near-field) near-field spectrum spectrum averaged averaged on (2 on × (2 × 2) µm2;(d–f) photoluminescence (PL) intensity maps associated to M1–M3 modes, white circles 2) μm2; (d–f) photoluminescence (PL) intensity maps associated to M1–M3 modes, white circles are are the holes surrounding the D2 defect; (g–i) calculated electric field distribution of the M1–M3 modes the holes surrounding the D2 defect; (g–i) calculated electric field distribution of the M1–M3 modes 30 nm above the photonic membrane. (j–l) Spectral shift maps associated with the M1–M3 modes. 30 nm above the photonic membrane. (j–l) Spectral shift maps associated with the M1–M3 modes. In In particular, the maximum spectral shift in (j) and (k) is 0.2 nm, while in (l) is 0.5 nm; all of the maps particular, the maximum spectral shift in (j) and (k) is 0.2 nm, while in (l) is 0.5 nm; all of the maps in in (d–l) have the same spatial extension (1.3 × 1.6) µm2. Adapted with permission from the authors (d–l) have the same spatial extension (1.3 × 1.6) μm2. Adapted with permission from the authors of of [23]. Copyright 2009 AIP Publishing. [23]. Copyright 2009 AIP Publishing.

In order to exploit the coupling effects between the coupled PC cavity modes and the two-level emitters (QD), (QD), a aspatial spatial and and spectral spectral overlap overlap between between the PC the nanocavity PC nanocavity and the emitter and the is mandatory. emitter is Usually,mandatory. QDs Usually, are grown QDs by are epitaxial grown techniques by epitaxial (MBE, techniques MOCVD, (MBE, etc.). MOCVD, It follows etc.). that It QDs follows are randomly that QDs

Ceramics 2019, 2 37 positioned and the spatial matching is not easy to implement. Several techniques have been exploited to reach this goal (see, for example, [25,26]). A solution based on nanofabrication techniques and on the unique characteristics of dilute nitride materials, in particular GaAsN, has been proposed and implemented. The characteristic of GaAsN will be briefly described, together with the fabrication process, which allows for the post-growth creation of QDs in controlled positions. This last achievement together with the knowledge of the mode distribution in the coupled PC cavities is crucial in order to realize the spatial and spectral matching between the emitter and the nanocavity, a condition needed to create efficient photonic devices.

2. Material and Methods With regards the PC coupled systems, the investigated structures have been fabricated on a 320 nm-thick GaAs membrane with three layers of high-density InAs QDs grown using a molecular beam epitaxy embedded at the center of the membrane [27]. If properly excited, the InAs QDs at room temperature are characterized by a bright photoluminescence band centered at 1300 nm and (considering state filling excitation) about 60 nm broad, which acts as an internal light source for the photonic structures. The membrane is grown on a sacrificial layer of AlGaAs on a GaAs substrate. The fabrication process’s initial step is the pattern transfer on a 150-nm-thick SiO2, deposited on the GaAs membrane, by 100 kV electron beam lithography (positive resist ZEP 520A) and the CHF3 reactive ion etching of SiO2. SiO2 acts as a mask for the transfer on the GaAs layer by chlorine-based reactive ion etching. The AlGaAs sacrificial layer is then selectively etched in a diluted HF solution to locally release the GaAs membrane. All of the fabrication process steps ensure the dimension control at the tens of nanometers. As our internal light source is much narrower than the photonic bandgap, we perform the lithographic tuning of the PC bandgap [28] (i.e., we use different values of the lattice parameter a, to spectrally move the PC bandgap around our fixed light source). To investigate the photonic modes, both near field and far field measurements have been performed. To perform the near field analysis, a commercial scanning near-field optical microscope (SNOM) in illumination/collection geometry, was used. The spatially resolved images were recorded by scanning the probe tip over the sample at a fixed distance (few tens of nm). The sample is excited by a laser diode at 780 nm, coupled into the tip. The emitted PL signal, collected by the tip, is coupled to a spectrometer and is finally detected by a nitrogen cooled InGaAs array. At every tip position, the entire spectrum of the sample is collected with a spectral resolution of 0.1 nm. Figure1c shows a PL spectrum of a single D2 cavity collected at a fixed tip position. The spectrum is characterized by three peaks, which correspond to the main modes of the cavity (labelled M1, M2, and M3). Figure1d–f shows the spatial distribution of the PL intensity of the M1, M2, and M3 modes, respectively, and is characterized by a subwavelength spatial resolution of the order λ/5. The spatial resolution is defined as the full width at the half maximum of the profile of the smallest feature that can be resolved by our system. These maps clearly indicate the main elongation direction of the modes. However, these maps are not detailed enough for resolving the typical features of the local (LDOS) that have been numerically calculated with a finite difference time domain software, and are reported in Figure1g–i for the M1, M2, and M3 modes, respectively. To map the LDOS with a better spatial resolution, we implemented the tip-induced spectral shift imaging technique [29]. As the probe is made of dielectric material (silica with a refractive index of about 1.5) and the dimension of its apex is comparable with the dimension of the PC holes, it turns out that the tip itself locally perturbs the dielectric environment of the photonic cavity. In fact, we observe that as the tip approaches the center of the cavity, the modes slightly shift to the red. Moreover, Koenderink and coworker theoretically predicted that the tip induced perturbation is proportional to the local strength of the intensity of the electric field [30]. This prediction has been recently debated in view of the non-Hermitian nature of photonics [31]; still, in the limit of a relative high Q (>1000, to be on the safe side), the previous prediction is quite accurate. Therefore, by reporting on a map the Ceramics 2019, 2 38 strength of the tip induced spectral shift as a function of the tip position, it is possible to map with a Ceramics 2019, 2 FOR PEER REVIEW 5 high fidelity the electric field intensity of the cavity modes (see Figure2).

FigureFigure 2. 2.Schematic Schematicexplanation explanationof ofthe thetip-induced tip-inducedspectral spectralshift shiftimaging imagingtechnique. technique. TheThe twotwo spectra,spectra, reportedreported with with blue blue and and red red lines, lines, correspond correspond to theto the signal signal collected collected by theby SNOMthe SNOM tip in tip position in position A and A Cand on the C on D2 the nanocavity, D2 nanocavity, respectively respectively (see inset). (see The inset). spectrum The acquired spectrum inside acquired the photonic inside the crystal photonic (PC) nanocavitycrystal (PC) (position nanocavity C) has (position higher C) counts has higher and a longercounts wavelength and a longer peak wavelength position withpeak respectposition to with the spectrumrespect to collected the spectrum outside collected the cavity outside (position the cavity A). By (position analyzing A). the By numberanalyzing of the counts number collected of counts as a functioncollected of as the a function tip position, of the we tip retrieved position, the we intensity retrieved map the of intensity the mode map (see of Figure the mode1d–f). (see On Figure the other 1d– hand,f). On by the reporting other hand, the by peak reporting wavelength the peak as a wavelength function of theas a tip function position, of the we tip obtained position, a spectral we obtained shift mapa spectral that reflects shift themap electric that reflects local density the electric of states local (LDOS) density for the of states mode under(LDOS) investigation. for the mode under investigation. The spectral shift maps are reported in Figure1j,l for the M1, M2, and M3 modes, respectively. The comparisonThe spectral of shift these maps experimental are reported maps in with Figure the calculated1j,l for the LDOSM1, M2, ones, and obtained M3 modes, by a respectively. commercial three-dimensionalThe comparison finite-difference of these experimental time-domain maps (FDTD) with the code calculated (CrystalWave, LDOS photon), ones, showed obtained a clear by a directcommercial correspondence three-dimensional and a quite finite-difference striking experimental time-domain spatial resolution,(FDTD) code better (CrystalWave, than λ/15. Withphoton), the sameshowed technique, a clear direct it is also correspondence possible to map and the a broadening quite striking of theexperimental resonance peaks,spatial which resolution, gives better important than informationλ/15. With the on same the tip-induced technique, lossesit is also and possible on the to SNOM map the detection broadening mechanism of the resonance [29]. peaks, which givesTo important measure theinformation polarization on the of thetip-induced electromagnetic losses and fields on associatedthe SNOM with detection the PC mechanism optical modes, [29]. a SNOMTo measure with the the polarization polarization control of the has electromagnetic been exploited fields [32]. Theassociated intensity with maps the ofPC the optical PL signal modes, of thea SNOM optical with mode, the inpolarization the two orthogonal control has polarization been exploited channels, [32]. The supply intensity an image maps of of the the twoPL signal electric of fieldthe optical components mode, in in the the plane two onorthogonal the PCC. polarization channels, supply an image of the two electric fieldThe components PCCs were in characterized the plane on inthe far PCC. field by a micro-photoluminescence setup using a microscopy objectiveThe with PCCs a numerical were characterized aperture of in 0.7. far The field external by a cone micro-photoluminescence of view forms with the setup normal using to the a samplemicroscopy surface objective at a 45◦ withangle, a andnumerical the angular aperture resolution of 0.7. is The 8◦. Forexternal the excitation, cone of weview used forms a -state with the lasernormal emitting to the atsample 532 nm. surface The angularlyat a 45° angle, selected and PL the emission angular fromresolution the sample is 8°. For was the collected excitation, with we a multimodeused a solid-state optical laser fiber, emitting dispersed at 532 by anm. spectrometer, The angularly and selected detected PL by emission a cooled from InGaAs the sample array; was the spectralcollected resolution with a multimode is of the order optical of 0.1 fiber, nm. dispersed by a spectrometer, and detected by a cooled InGaAs array; the spectral resolution is of the order of 0.1 nm. 3. Results 3. Results 3.1. Photonic Crystal Molecules: Two Nanocavities Case 3.1. PhotonicTwo or moreCrystal interacting Molecules: PC Two cavities Nanocavities give raise Case to the hybridization of the single cavity modes (photonicTwo orbitals), or more asinteracting in real molecules, PC cavities the inter-atomicgive raise to coupling the hybridization gives rise toof the the molecular single cavity orbital. modes So, the(photonic coupled orbitals), PC cavities as in can real also molecules, be called photonicthe inter-atomic molecules coupling [23,33]. gives In our rise case, to the coupledmolecular photonic orbital. nanocavitiesSo, the coupled have beenPC cavities designed can in also two differentbe called configurations; photonic molecules we will [23,33]. refer to verticallyIn our case, (or horizontally)the coupled alignedphotonic D2 nanocavities cavities if the have coupling been line designed of the two in two adjacent different D2 cavities configurations; lies along thewe principalwill refer M to (or vertically K) axis of(or the horizontally) photonic crystal aligned structure. D2 cavities In the if idealthe coupling case of identical line of the cavities, two adjacent frequency D2matching, cavities lies and along spatial the overlap,principal the M coupling (or K) axis results of inthe an photonic energy splitting crystal structure.of the modes In theand ideal in the case formation of identical of delocalized cavities, frequency matching, and spatial overlap, the coupling results in an energy splitting of the modes and in the formation of delocalized “symmetric” and “antisymmetric” coupled modes. In real samples, we have to take into account the disorder induced by the fabrication process that could determine the presence of a double peak in the spectrum even in the absence of coupling. A given peak can be

Ceramics 2019, 2 39

“symmetric” and “antisymmetric” coupled modes. In real samples, we have to take into account the disorder induced by the fabrication process that could determine the presence of a double peak in the spectrum even in the absence of coupling. A given peak can be surely identified as a coupled-cavity mode, only if its intensity distribution is delocalized over the two cavities. On the other hand, if the intensity Ceramics 2019, 2 FOR PEER REVIEW 6 distribution is localized over one of the two cavities, we observe an uncoupled status. Considering that the coupledsurely identified cavities systemas a coupled-cavity as two linearly mode, coupledonly if its oscillatorsintensity distribution with identical is delocalized losses (inover our the case, the modestwo of eachcavities. isolated On the cavity other )[hand,23], if the the formulaintensity distribution of the photonic is localized splitting over forone theof the coupled-cavity two cavities, mode we observe an uncoupledp status. Considering that the coupled cavities system as two linearly can be obtained, Ω = ∆2 + 4g2, where ∆ is the disorder induced energy detuning and g is the coupled oscillators with identical losses (in our case, the modes of each isolated cavity) [23], the coupling energy between the modes of each cavity [34]. Therefore, the photonic splitting Ω is a direct formula of the photonic splitting for the coupled-cavity mode can be obtained, Ω= Δ +4푔, where measurementΔ is the disorder of the couplinginduced energy energy detuningg only and when g is the∆ = coupling 0; in the energy case ∆between g, thethe modes system ofis each uncoupled. So, it iscavity crucial [34]. to Therefore, determine the thephotonic nature splitting of the Ω coupled is a direct PC measurement cavity modes of the in coupling order to energy verify g only the coupling energywhen and Δ the = 0; accuracy in the case of Δ≫ theg,fabrication the system is process. uncoupled. So, it is crucial to determine the nature of the Tocoupled better PC understand cavity modes the in analysis order to ofverify the the coupled coupling PC energy cavities, and it the is necessaryaccuracy of tothe start fabrication from the main propertiesprocess. of a single D2 cavity [23]. Figure1a reports the topographic map of a D2 cavity with a To better understand the analysis of the coupled PC cavities, it is necessary to start from the latticemain parameter propertiesa = of 311 a single nm and D2 cavity a x,y reference[23]. Figure system 1a reports in the the topographic plane, with map the of vertical a D2 cavityy axis with aligned to the principala lattice parameter diagonal a of = 311 the nm D2 and cavity. a x,y Inreference particular, system the in the M1 plane, mode with is elongated the vertical alongy axis aligned the y direction (Figureto1 j), the the principal M2 mode diagonal is mainly of the elongatedD2 cavity. In along particular, the x thedirection M1 mode (Figure is elongated1k), while along the the M3y mode is mainlydirection distributed (Figure 1j), at the the M2 vertexes mode is mainly of a square elongated (Figure along1 thel). x In direction the analysis (Figure of1k), the while two the coupled M3 PC systems,mode we is will mainly consider distributed only at the the modesvertexes M1 of a andsquare M2. (Figure 1l). In the analysis of the two coupled PC systems, we will consider only the modes M1 and M2. Now, we can start with the analysis of the two coupled PC cavities. As seen, the M1 and M2 Now, we can start with the analysis of the two coupled PC cavities. As seen, the M1 and M2 modesmodes of the of D2 the cavity D2 cavity have have a verya very distinct distinct LDOS LDOS and and are are expected expected to show to showdifferent different coupling coupling energy energy g in theg in horizontal the horizontal and and vertical vertical coupling coupling design.design. As As we we will will demonstrate demonstrate in the infollowing, the following, the nature the nature of theof coupled the coupled cavity cavity modes modes is is strictly strictly relatedrelated to to the the disorder disorder induced induced by the by fabrication the fabrication process and process and by theby geometry the geometry of the of the system. system. In In Figure Figure3 3,, thethe comparison comparison between between the experimental the experimental results for results the for the horizontallyhorizontally (along (alongx) coupled x) coupled PC PC cavities cavities and andthe the calculatedcalculated electric electric field field intensity intensity is reported. is reported. The The PL PL maps indicate that the mode labelled P1 is concentrated on the left cavity (Figure 3b), and the maps indicate that the mode labelled P1 is concentrated on the left cavity (Figure3b), and the mode mode labelled P2 is localized on the right cavity (Figure 3c). labelled P2 is localized on the right cavity (Figure3c).

FigureFigure 3. (a) 3. Near-field (a) Near-field spectrum spectrum of of the the horizontallyhorizontally coupled coupled D2 PC D2 cavities PC cavities averaged averaged on a region on aof region of 2 × 3 μm2. Inset: (I) topographic map as obtained during the SNOM scan. (b–e) PL intensity maps 2 × 3 µm2. Inset: (I) topographic map as obtained during the SNOM scan. (b–e) PL intensity maps associated to the peak P1–P4. (f–i) Spectral shift maps associated to the peak P1–P4; in particular, the associatedmaximum to the spectral peak P1–P4.shift in (f ()f –andi) Spectral (g) is 0.3 nm, shift in maps (h) is 0.5 associated nm, and in to (i the) is 0.2 peak nm. P1–P4; (j–k) Calculated in particular, the maximumelectric spectral field distribution shift in (off) P3 and and (g P4) is30 0.3nm nm,above in the (h photonic) is 0.5 nm, membrane. and in All (i) of is the 0.2 maps nm. have (j,k) the Calculated electricsame field spatial distribution extension of of P32.4 and× 2.0 P4μm2 30. Adapted nm above withthe permission photonic from membrane. the authors Allof [23]. of theCopyright maps have the same spatial2009 AIP extension Publishing. of 2.4 × 2.0 µm2. Adapted with permission from the authors of [23]. Copyright 2009 AIP Publishing.

Ceramics 2019, 2 40

The PL signal related to the modes labelled P3 and P4 is instead delocalized on both cavities (Figure3d,e). The analysis of the spectral shift maps allows for a better understanding of the nature of the different modes, and to correlate them with the modes of a single D2 cavity. The P1 and P2 modes are similar to the M1 mode of a single D2 cavity (Figure1d), and the coupling energy ( g) for the M1 modes of the two horizontally aligned D2 cavities is very small. The spectral splitting of 2.6 meV between P1 and P2 has to be ascribed to the energy detuning due to structural disorder in the cavity realization. In the case of the P3 and P4 modes, the situation is quite different. Their experimental maps strongly suggest that P3 and P4 are the two coupled cavity modes originating from the M2 modes of the two D2 cavities. This is clearly demonstrated by the comparison of the spectral shift map of P3 (Figure3h), with the corresponding simulated map of the LDOS, as reported in Figure3j. The spatial distribution of both the experimental and theoretical P3 LDOS is extended over the whole coupled system, and is very similar to the electric field distribution of two M2 modes, (Figure1h), each centered on one of the two D2 cavities, forming the horizontal coupled system. Analogous results are obtained for P4 (Figure3e,I,k). Therefore, the large spectral splitting between P3 and P4 is attributed to their electromagnetic coupling, but still, we have to consider that the two D2 cavities are not identical. From the statistical analysis of several single D2 cavities, we found that, even if the absolute position of the M1 and M2 modes shifts up to 9 meV because of a structural disorder, their energy separation varies only up to 1 meV. Therefore, we can assume that the 2.6 meV energy splitting between P1 and P2 is also an estimation of the cavity detuning for the two M2 modes. Therefore, by p using Ω = ∆2 + 4g2, we obtained a coupling parameter of g = 5.9 meV for the M2 modes in the horizontal (along x) geometry. In the case of the vertically (along y) coupled PC cavities, we found a different situation. In Figure4, the experimental results are shown. The modes labelled P1 and P2 are delocalized over the entire system (Figure4b,c; PL maps), and can be attributed to the two coupled-cavity modes originating from the M1 mode of a single D2 cavity (Figure4f,g; spectral shift maps) with an overall photonic splitting of Ω = 11.7 meV. This attribution is also validated by the simulated map of the LDOS for P1 (Figure4j). Similar results are obtained for P2. The modes labelled P3 and P4 show a very small splitting (0.8 meV), which cannot be attributed to a structural disorder. As seen before, the signature of the frequency induced mismatch because of the structural disorder is the localization of the modes on the single D2 cavities. Looking to the near field maps of the modes (Figure4d,e,h,i), it is clear that they are extended over the entire system and can be considered as the symmetric and anti-symmetric coupled cavity modes coming from the M2 mode of the two D2 single cavities. This demonstrates that in this case, the disorder induced by the fabrication process is smaller than the coupling energy, and the matching condition are satisfied. The measured detuning is ∆ = 0.8 meV and the calculated coupling energy is g = 5.9 meV for the M1 modes and g = 0.4 meV for the M2 modes. These results are confirmed by numerical simulations.

3.1.1. Mode Symmetry After the comprehension of the mode distribution and the demonstration that the transition from the localized to non-localized mode is controlled by the mode detuning, the symmetry of the coupled modes have been verified in details [35]. This property depends on the characteristics and it is not trivial to be measured in the near field; we demonstrated that it is possible to probe it using a far field photoluminescence analysis by considering the two coupled PC cavities in analogy with Young’s double slit experiment. Several phase sensitive techniques have been proposed in near field and far field [36,37], but the method we proposed is quite simple and straightforward. Ceramics 2019, 2 41 Ceramics 2019, 2 FOR PEER REVIEW 8

FigureFigure 4. 4. (a()a Near-field) Near-field spectrum spectrum of ofthe the vertically vertically coupled coupled D2 D2PC-cavities PC-cavities averaged averaged on a onregion a region of (1.5 of ×(1.5 3.5)× μm3.5)2. µInsetm2. Inset(i): topographic (i): topographic map map as obtained as obtained during during the the SNOM SNOM scan. scan. Inset Inset (ii): (ii): near-field near-field high high resolutionresolution spectrum spectrum that that resolves resolves the contributionscontributions ofof P3P3 andand P4. P4. (b(b––ee)) PL PL intensity intensity maps maps associated associated to tothe the peak peak P1–P4. P1–P4. (f– i()f– Spectrali) Spectral shift shift maps maps associated associated to peak to peak P1–P4; P1–P4; in particular, in particular, the maximum the maximum spectral spectralshift in fshift and in g isf and 0.15 g nm, is 0.15 in (h nm,) is 0.2in ( nm,h) is and 0.2 innm, (i) and is 0.1 in nm. (i) is (j ,0.1k) Calculatednm. (j–k) Calculated electric field electric distribution field distributionof P1 and P2 of at P1 30 and nm aboveP2 at 30 the nm photonic above membrane.the photonic All membrane. of the maps All have of the samemaps spatialhave the extension same spatial(1.5 × extension3.5) µm2. Adapted(1.5 × 3.5) with μm2 permission. Adapted with from permission the authors from of [23 the]. Copyright authors of 2009 [23]. AIP Copyright Publishing. 2009 AIP Publishing. In an ideal photonic , the mode coupling determines the frequency splitting of the 3.1.1.eigenvalues, Mode Symmetry originating in the delocalized symmetric and antisymmetric eigenvectors. The symmetric mode,AfterE+ [ther], arisescomprehension from two in-phaseof the mode single distribution cavity modes and and the a demonstration constructive interference that the transition along the fromnormal the direction,localized andto non-localized is expected to mode be observed is controlled (as in by the the original mode Young’s detuning, double the symmetry slits experiment). of the coupledThe asymmetric modes have mode, beenE −verified[r], arises in details from two [35]. out This of property phase single depends cavity on modes,the phase and characteristics a destructive andinterference it is not trivial along theto be normal measured direction in the is near expected. field; Thewe demonstrated coupled modes that can it beis describedpossible to as probe a linear it usingsuperimposition a far field photoluminescence of the single cavity analysis modes, by depending considering on thethe spatialtwo coupled distance PC between cavities thein analogy cavities. withThe symmetryYoung’s double of the slit single experiment. cavity modes Several has phase a very sensitive strong impact techniques on their have angular been proposed emission in pattern, near fieldas can and be far deduced field [36,37], by analyzing but the method the Fourier we proposed transform is ofquite the simple coupled and modes. straightforward. Starting from these considerations,In an ideal thephotonic far field molecule, patterns the of themode coupled coupling modes determines have been the calculated frequency on splitting the basis of of the the eigenvalues,numerically simulated originating far in field the patterns delocalized of the symmetricmodes of the and single antisymmetric cavity. As before, eigenvectors. we refer to The the symmetricsingle cavity mode, modes 퐸[푟 as], arises M1 and from M2, two and in-phase to the coupledsingle cavity cavity modes modes and as a constructive P1–P4. So, to interference exploit the alongeffects the of normal the Young’s direction, double and slit is interferenceexpected to tobe probe observed the mode(as in symmetrythe original of Young’s a photonic double molecule, slits experiment).we need to measureThe asymmetric the polarization mode, 퐸 resolved[푟], arises far from field two patterns out of ofphase the single cavity modes, modes and toa destructiveestimate the interference corresponding along near the field normal patterns direction using is FDTD expected. calculations The coupled (Figure modes5), which can be are described linked by asa Fouriera linear superimposition transform. of the single cavity modes, depending on the spatial distance between the cavities. The symmetry of the single cavity modes has a very strong impact on their angular emission pattern, as can be deduced by analyzing the Fourier transform of the coupled modes. Starting from these considerations, the far field patterns of the coupled modes have been calculated on the basis of the numerically simulated far field patterns of the modes of the single cavity. As before, we refer to

Ceramics 2019, 2 FOR PEER REVIEW 9

the single cavity modes as M1 and M2, and to the coupled cavity modes as P1–P4. So, to exploit the effects of the Young’s double slit interference to probe the mode symmetry of a photonic molecule, we need to measure the polarization resolved far field patterns of the single cavity modes and to

Ceramicsestimate2019 the, 2 corresponding near field patterns using FDTD calculations (Figure 5), which are linked42 by a Fourier transform.

FigureFigure 5.5. Single D2 cavity. ( (aa)) PL PL spectra spectra in in the the x x (black (black line) line) and and y y (red (red line) line) polarization polarization channels. channels. ((bb)) SEM SEM image. image. (c – (ec)– Electrice) Electric field field finite-difference finite-difference time-domain time-domain (FDTD) (FDTD)near field near maps: field (c) x maps: component (c) x ofcomponent M1, (d) x componentof M1, (d) ofx M2,component (e) y component of M2, (e of) M2.y component (f–h) Experimental of M2. (f PL–h) far Experimental field intensity PL k patterns:far field (f)intensity x polarization k patterns: of M1, (f) (xg )polarization x polarization of ofM1, M2, (g) (h x) polarization y polarization of of M2, M2. (h (i)– yk )polarization FDTD far field of M2. intensity (i–k) kFDTD pattern: far field (i) x intensity polarization k pattern: of M1, ( (ij)) x x polarization polarization of of M1, M2, (j ()k x) polarization y polarization of ofM2, M2. (k) SEMy polarization and near fieldof M2. images SEM and are 1:5nearµm field× 2:0 imagesµm. Inare the 1:5 near μm field× 2:0 μm. maps In red the (blue) near field maps indicates red (blue) a positive color (negative) indicates amplitude.a positive (negative) The far field amplitude. patterns The cover far thefield whole patterns external cover solid the whole angle andexternal the white solid circlesangle and are thethe experimentalwhite circles coneare the of view.experimental Adapted cone with permissionof view. Adapted from the with authors permission of [35]. Copyrightfrom the authors 2011 American of [35]. PhysicalCopyright Society. 2011 American Physical Society.

TheThe modemode M1M1 isis mainlymainly polarizedpolarized alongalong the xx direction,direction, whilewhile thethe modemode M2M2 isis characterizedcharacterized byby anan ellipticalelliptical polarizationpolarization [[32].32]. TheThe calculatedcalculated near near field field maps maps show show the the electric electric field field amplitude amplitude with with a scalea scale color color to to indicate indicate the the amplitude amplitude sign. sign. The The measured measured far far field field intensity intensityk patternsk patterns are are shown shown on on a bluea blue background background for for the the PL PL experimental experimental data data and and on on a a blackblack backgroundbackground forfor the FDTD simulations. WeWe usedused the the denomination denominationx -evenx-even (x -odd)(x-odd) for for an an even even (odd) (odd) mode mode with with respect respect to the to xtheinversion, x inversion, and similarlyand similarly for y for, to y describe, to describe the modethe mode parity parity synthetically. synthetically. With With regards regards the FDTDthe FDTD near near field field maps, maps, the modethe mode M1, M1, elongated elongated along along the y thedirection, y direction, is an isx-even an x-even and y -evenand y-even mode mode (Figure (Figure5c). The 5c). mode The M2mode is moreM2 is symmetricallymore symmetrically distributed distributed (with a (with slight a elongation slight elongation along x )along and the x) twoand polarizationsthe two polarizations have an oppositehave an opposite parity. The parity.x polarization The x polarization is x-even is and x-eveny-odd, and while y-odd, the whiley polarization the y polarization is x-odd is and x-oddy-even and (Figurey-even 5(Figured,e). The 5d,e). corresponding The corresponding PL far field PL intensityfar field intensityk patterns k arepatterns verydifferent are veryfor different the following for the threefollowing cases: three M1 cases: shows M1 a horizontal shows a horizontal stripe with stripe a maximum with a maximum at the center at the (Figure center5i). (Figure M2 shows 5i). M2 a darkshows central a dark region central that region is vertical that foris vertical the x polarization for the x polarization and horizontal and for horizontaly polarization for y (Figurepolarization5j,k). The(Figure experimental 5j,k). The experimental data show a very data good show agreement a very good with agreement the FDTD with simulations the FDTD (Figure simulations5f–h). The(Figure far field5f–h).k Thepatterns far field directly k patterns follow directly from follow the near from field the maps, near field and maps, the following and the following two points two help points for understanding them: (i) imposes that the far field pattern is elongated in the perpendicular direction with respect to the near field map (see M1). (ii) The x-odd (y-odd) modes destructively interfere in the far field along kx = 0 (ky = 0) (see the two polarizations of M2). Ceramics 2019, 2 FOR PEER REVIEW 10 help for understanding them: (i) Diffraction imposes that the far field pattern is elongated in the Ceramics 2019, 2 43 perpendicular direction with respect to the near field map (see M1). (ii) The x-odd (y-odd) modes destructively interfere in the far field along kx = 0 (ky = 0) (see the two polarizations of M2). WithWith all all of these considerations in in mind, mind, we we can can analyze analyze the the results results for for coupled coupled PC PC structures. structures. InIn the the case case of of the the vertically vertically aligned aligned photonic photonic molecule, molecule, we we limit limit our our analysis analysis to to the the P1 P1 and and P2 P2 modes modes arisingarising from from the the overlap overlap of of the two M1 modes (see Figure 66).). FollowingFollowing anan analysisanalysis similarsimilar toto thethe singlesingle cavity cavity modes, modes, comparing comparing the the PL PL spectra, spectra, the the experimental experimental PL PL kk patternspatterns of of P1 P1 and and P2, P2, and and the the Young’sYoung’s predictions (with red frames) for constructive and and destructive destructive interference, interference, respectively respectively (see (see FigureFigure 66),), wewe couldcould demonstratedemonstrate thatthat P1P1 isis thethe symmetricsymmetric coupledcoupled modemode andand P2P2 isis thethe antisymmetric coupledcoupled mode. mode. The The FDTD FDTD simulations simulations (shown (shown in in Figure Figure 66f,g)f,g) agree well with thethe datadata andand withwith thethe Young’sYoung’s model predictions.

FigureFigure 6. VerticallyVertically aligned aligned photonic photonic molecule. molecule. (a (a) ) PL PL spectrum spectrum (red (red line) line) compared compared with with the the PL PL spectrumspectrum of the singlesingle D2D2 cavitycavity (blue(blue line), line), the the inset inset shows shows the the SEM SEM image. image. (b –(bc)– Experimentalc) Experimental PL PL far farfield field intensity intensity k patterns k patterns of of P1 P1 and and P2. P2. (d (–de–)e Young’s) Young’s predictions predictions of of “+” “+” and and ” ”−“−“ modes. ( ff––g)) FDTD FDTD farfar field field intensity intensity k k patterns patterns of of P1 P1 and and P2. P2. The The far far field field patterns patterns cover cover the the whole whole external external solid solid angle angle andand the the white white circles circles are are the experimental cone cone of of view. view. Adapted with with permission from from the the authors authors ofof [35]. [35]. Copyright Copyright 2011 2011 American American Physical Physical Society. Society.

In the case of the horizontally aligned D2 photonic molecule, the situation is more complex. We In the case of the horizontally aligned D2 photonic molecule, the situation is more complex. analyzed the P3 and P4 modes arising from the overlap of the two M2 modes of the single cavity, We analyzed the P3 and P4 modes arising from the overlap of the two M2 modes of the single cavity, which shows elliptical polarization. So, in this case, we have to separately analyze the far field which shows elliptical polarization. So, in this case, we have to separately analyze the far field patterns patterns for the x and y polarizations of P3 and P4. As for the vertically aligned case, the results are for the x and y polarizations of P3 and P4. As for the vertically aligned case, the results are summarized summarized in Figure 7. From the comparison of the numerical predictions and the experimental in Figure7. From the comparison of the numerical predictions and the experimental measurements, measurements, it is possible to affirm that, with regards to the x polarization, P3 and P4 are the it is possible to affirm that, with regards to the x polarization, P3 and P4 are the antisymmetric and antisymmetric and symmetric coupled modes, respectively. In the case of y polarization, the symmetric coupled modes, respectively. In the case of y polarization, the understanding of the far understanding of the far field k patterns of P3 and P4 is not straightforward. The fingerprint of the field k patterns of P3 and P4 is not straightforward. The fingerprint of the destructive interference destructive interference for P3 is the broadening of the dark region along the kx = 0 direction. Instead, for P3 is the broadening of the dark region along the k = 0 direction. Instead, the fingerprint of the the fingerprint of the constructive interference for P4 canx be identified in the two additional vertical constructive interference for P4 can be identified in the two additional vertical dark fringes around dark fringes around 40°. A detailed explanation can be found in the literature [35], together with the 40◦. A detailed explanation can be found in the literature [35], together with the analysis of the analysis of the FDTD near field maps of the electric field amplitudes, and it is really worth discussing FDTD near field maps of the electric field amplitudes, and it is really worth discussing these last ones, these last ones, because from them, we can retrieve the parity of the modes that does not always because from them, we can retrieve the parity of the modes that does not always coincide with the coincide with the symmetry/asymmetry of the modes. For vertical coupling, P1 is y-even and P2 is y- symmetry/asymmetry of the modes. For vertical coupling, P1 is y-even and P2 is y-odd, as expected. odd, as expected. Instead, for the horizontal coupling, the situation is again more complicated. In this Instead, for the horizontal coupling, the situation is again more complicated. In this case, it is the x-symmetry that defines the Young interference. The P3 mode is x-odd for the x polarization and CeramicsCeramics 20192019,, 22 FOR PEER REVIEW 44 11

case, it is the x-symmetry that defines the Young interference. The P3 mode is x-odd for the x xpolarization-even for the and y polarization. x-even for the Instead, y polarization. the two polarizations Instead, the two of the polarizations P4 mode have of the an P4 opposite mode xhave-parity an withopposite respect x-parity to P3. with These respect puzzling to P3. parity These properties puzzling ofparity P3 and properties P4 can be of understoodP3 and P4 can by be simply understood noting thatby simply the “symmetric” noting that mode, the “symmetric”E+[r], is x-even mode, (x-odd) 퐸[푟] whenever, is x-even the (x-odd) electric whenever field mode the ofelectric the single field cavitymode isof xthe-even single (x-odd). cavity On is thex-even contrary, (x-odd). the “antisymmetric”On the contrary, modethe ‘‘antisymmetric’’E−[r] is x-odd ( xmode-even) 퐸 whenever[푟] is x- theodd electric (x-even) field whenever mode of the the electric single field cavity mode is x-even of the (x single-odd). cavity This means is x-even that (x the-odd). parity This (with means respect that tothe inversion) parity (with and respect symmetry to inversion) with respect and symmetry to the mode with building respect do to notthe alwaysmode building coincide, do and not that always the orientationcoincide, and of that the coupledthe orientation PC cavities of the determines coupled PC the cavities mode properties.determines the mode properties.

Figure 7.7. HorizontallyHorizontally aligned photonic molecule. ( a) PL spectrum (red line) compared with the PL spectrum of the singlesingle D2D2 cavitycavity (blue(blue line),line), thethe insetinset showsshows thethe SEMSEM image.image. ((b––ee)) ExperimentalExperimental PLPL farfar fieldfield intensity k k patterns: patterns: (b (b) )and and (c ()c x) xpolarization polarization of ofP3 P3 and and P4; P4; (d– (ed), ey) polarization y polarization of P3 of and P3 and P4. P4.(f–i) ( fYoung’s–i) Young’s predictions: predictions: (f–g ()f x,g polarization) x polarization of “−“ of “and−“ and“+” modes; “+” modes; (h–i) ( hy, ipolarization) y polarization of “−“ of “and−“ and“+” “+”modes. modes. (j–m ()j – FDTDm) FDTD far field far field intensity intensity k patterns. k patterns. (j–k (j), k x) polarizationx polarization of of P3 P3 and and P4; P4; ( (ll–,m) y polarization of P3 and P4.P4. The far fieldfield patternspatterns cover the wholewhole external solid angle and the whitewhite circles are are the the experimental experimental cone cone of view. of view. Adapted Adapted with permission with permission from the from authors the of authors [35]. Copyright of [35]. 2011Copyright American 2011 PhysicalAmerican Society. Physical Society.

3.1.2. Controlling Bonding and Antibonding GroundGround StateState After the demonstration of the possibility to control the symmetry and parity of the coupled PC cavities modes by the geometrical arrangement, aa further step forward was done. Theoretical studies on two coupled PC cavities showed that the ground state in the photonic crystals may actually change fromfrom aa bondingbonding toto antibondingantibonding charactercharacter depending on the spatial alignmentalignment or thethe distancedistance of thethe twotwo isolatedisolated cavitiescavities [[38,39].38,39]. In this case, the analogy with the quantum mechanics is not observed; whenwhen aa degenerate electron groundground energyenergy levellevel ofof twotwo identical atoms splits into two states through electronic couplings,couplings, the lower (higher) energy statestate always has an even (odd) parity independent of the atomic distance. By using the same Young’s type experiment, it has been possible to demonstrate

Ceramics 2019, 2 45 the atomic distance. By using the same Young’s type experiment, it has been possible to demonstrate the different behavior of the photonic molecules and the capability of tailoring the mode nature by the coupled PC cavity design [40]. Again, we investigated the system of two D2 PC cavities, coupled in a vertical (principal axis of the photonic crystal, M) and horizontal (K axis) configuration. By means of the FDTD solver package, numerical simulation of the x component of the electric field of ground (G) and the first excited (E) states of the M and K photonic molecules have been carried out. Both G and E derive from the coupling of the ground modes of the two D2 coupled PC cavities, which are even along both the M and K axes. For the M coupling (vertical), a large mode overlap is found (energy splitting Ω =12 meV) and the mode parity must be considered with respect to the spatial inversion to K axis (horizontal) and the predicted G mode parity is even (E is odd), in accordance with the quantum mechanics analogy. Instead, in the case of K coupling, the observed overlap is small (Ω = 0.4 meV) and the parity is considered for the spatial inversion with respect to the M axis. In this case, the predicted G mode is odd (E is even). So, in this case, the G mode should have an antibonding nature. To experimentally prove these numerical previsions, we had to keep in mind the analysis of the energy splitting Ω for the two coupled PC cavities. When the Ω value is small, we have to verify that this is not because of the possible disorder induced by the fabrication process. Therefore, in the M alignment, because of the large value of the coupling energy, we can neglect the disorder component; in the K coupling, instead, we have to be sure to eliminate the possible detuning coming from the disorder. To do this, many techniques are available, like the local change of the refractive index by liquid infiltration [41] or nano-oxidation [42]. Here, we choose to tune the mode cavity by laser assisted local oxidation [43,44]; the laser exposure allows for photoinduced oxidation and produces a blue shift on the cavity photonic modes. The spatially resolved PL has been used to detect and monitor the spatial localization of the photonic modes of the cavity. The red-shifted mode of the K coupled system has been exposed at steps of about half an hour, at a power of 1 mW. An anticrossing curve has been obtained and the initial experimental splitting of Ω = 1.6 meV could be reduced to a minimum value of 0.9 meV after a total exposure of 150 min. In Figure8, the comparison between the FDTD calculation and the experimental data for the G (a) and E (b) state in both the M and K configuration are reported. As we know, the far field intensity patterns give direct insight on the mode parity. Even the modes originating from the in-phase single cavity modes and the odd modes come from the out of phase single cavity modes. Following the Young’s analogy, in the first case, we expect constructive interference, and in the second one, a destructive one, along the axis perpendicular to the alignment direction. The far field pattern of the G mode of a single D2 cavity is a bright horizontal band (see Figure5i). So, the bright horizontal line present in the far field pattern of the M (vertical) coupled PC cavities in Figure8a, is evidence of the evenness of the G mode, and the dark vertical line in far field patterns of the K mode indicates that its G mode is odd. The numerical results confirm the experimental outcomes. These results mean that the G state is bonding for the M coupling and antibonding for the K coupling, demonstrating that in photonic molecules, the G state can have both a bonding and antibonding nature, and that this property can be controlled by the cavity geometrical design. With regards to the E states, the results are shown in Figure8b. Also, in this case, the agreement between the numerical previsions and experimental results is very good. The presence of a dark horizontal band in the middle of the far field pattern of the M coupling mode for the excited case, is the fingerprint of the oddness of E mode and of its antibonding nature. Instead, the far field pattern of the E mode in the K coupling shows a bright central spot deriving from the product of the horizontal band of the single D2 cavity by the vertical constructive bright fringe because of the evenness of the E mode in K coupling. Ceramics 2019, 2 46 Ceramics 2019, 2 FOR PEER REVIEW 13

Figure 8. (a) Comparison between the FDTD calculations (left) and experimental data (right) for the G state of both M- and K-coupledPC cavities. (b) Comparison between the FDTD calculations (left) andFigure experimental 8. (a) Comparison data (right) between for the the E FDTD state of calculations both M- and (left) K-coupledPC and experimental cavities. data The (right) white circlesfor the ◦ ◦ indicateG state of the both aperture M- and angles K-coupledPC of 30 and cavities. 60 , respectively. (b) Comparison Adapted between with permission the FDTD fromcalculations the authors (left) ofand [40 experimental]. Copyright 2012data American(right) for Physicalthe E state Society. of both M- and K-coupledPC cavities. The white circles indicate the aperture angles of 30° and 60°, respectively. Adapted with permission from the authors Theof [40]. physical Copyright origin 2012 of American this behavior Physical can Society. be ascribed to the role played by the optical evanescent waves. In an atom, the amplitude of the electron wave-function decays exponentially with the distance fromThe the nucleus.physical origin In the photonicof this behavior crystal can case, be instead, ascribed the to the localization role played of theby the photons optical is evanescent due to the strongwaves. multiple In an atom, scattering, the amplitude and the field of ofthe localized electron photons wave-function oscillates decays with an exponentially decayingwith the envelope.distance from In the the case nucleus. of the In coupled the photonic PC cavities, crystal the case, interference instead, the between localizati theon oscillatory of the photons evanescent is due electromagneticto the strong multiple waves determinesscattering, theand number the field of of nodes localized in the photons total waves oscillates and depends with an on exponentially the distance betweendecaying the envelope. sources [In45 ].the Therefore, case of the it is coupled this oscillating PC cavities, nature the of theinterference evanescent between waves inthe the oscillatory photonic bandevanescent gap that electromagnetic originates the bondingwaves determines or antibonding the number nature of of nodes the G statein the of total the coupledwaves and PC depends system. on the distance between the sources [45]. Therefore, it is this oscillating nature of the evanescent 3.2.waves Photonic in the Crystal photonic Molecules: band gap Three that Nanocavities originates Casethe bonding or antibonding nature of the G state of the coupledTo continue PC system. the study on the photonic molecules, we also designed, fabricated, and characterized the photonic systems made of three D2 coupled cavities along the two principal axes (x and y) [46]. The3.2. Photonic systems Crystal of identical Molecules: coupled Three PC Nanocavities cavities, alsoCase named photonic arrays, are characterized by spatiallyTo delocalizedcontinue the optical study modes on andthe spectralphotonic minibands, molecules, which we allow also fordesigned, photon hoppingfabricated, between and thecharacterized adjacent resonator the photonic (in analogy systems with made coupled of three resonators) D2 coupled [47 cavities,48]. In along this section, the two we principal will briefly axes describe(x and y) their [46]. main The properties. systems of In thisidentical case, eithercoupled the PC nearest cavities, neighbor, also andnamed the nextphotonic nearest arrays, coupling are mustcharacterized be taken intoby spatially account. delocalized As in the two modes molecules, and spectral the geometry minibands, plays which a crucial allow role; for the photon mode splittinghopping andbetween spatial the distribution adjacent resonator depend strongly(in analogy on thewith cavity coupled arrangement. resonators) To [47,48]. study the In this behavior section, of thewe three-cavitywill briefly describe photonic their molecule, main properties. we considered In this the case, coupling either between the nearest the neighbor, lower energy and modethe next of eachnearest single coupling D2 cavity. must As be we taken know, into this account. mode, namedAs in the M1, two is elongatedcavity molecules, along the the M geometry axis (y direction) plays a ofcrucial the photonic role; the crystal.mode splitting The consequence and spatial of distribu this characteristiction depend is strongly that in theon Mthe coupled cavity arrangement. cavities, the modeTo study overlap the behavior is large, similar of the tothree-cavity what we have photonic described molecule, for the we two considered coupled PCthe system, coupling and between so it is thethe nearest-neighborlower energy mode interaction. of each single Instead, D2 in cavity. the case As of we the know, K alignment, this mode, the modenamed overlap M1, is iselongated smaller, determiningalong the M axis a non-intuitive (y direction) mode of th behavior.e photonic The crystal. theoretical The consequence studies of thisof this system characteristic have been is widely that in describedthe M coupled in the literaturecavities, the [46 ].mode Here, overlap we report is onlarge, two-dimensional similar to what plane we wavehave expansiondescribed calculationsfor the two usedcoupled to reconstruct PC system, the and system so it is the nearest-neighb curve, and moreor interaction. extensively onInstead, the experimental in the case outcomes, of the K whosealignment, results the validated mode overlap the theoretical is smaller, predictions. determining a non-intuitive mode behavior. The theoretical studiesFigure of this9 shows system the have mode been dispersion widely describe relationd of in a the two-dimensional literature [46]. infiniteHere, we array report of theon two- D2 coupleddimensional PC cavities plane wave plotted expansion in the energy calculations range relativeused to reconstruct to the fundamental the system mode dispersion of a single curve, D2, and for bothmore kinds extensively of M and on the K alignment. experimental The outcomes, finite width whose of bothresults of validated the dispersion the theoretical curves demonstrates predictions. the realFigure photonic 9 shows hopping the mode between dispersion the coupled relation PC of cavities.a two-dimensional Clear minibands infinite appeararray of for the both D2 configurations,coupled PC cavities but with plotted different in the behavior.energy range In the rela Mtive alignment, to the fundamental the energy mode of the of photonic a singleguided D2, for both kinds of M and K alignment. The finite width of both of the dispersion curves demonstrates the

Ceramics 2019, 2 47 Ceramics 2019, 2 FOR PEER REVIEW 14 modemode increasesincreases monotonicallymonotonically asas aa functionfunction ofof k,k, correspondingcorresponding toto aa modemode withwith aa positivepositive groupgroup velocity;velocity; inin the K K alignment, alignment, the the dispersion dispersion of of the the photonic photonic mode mode is small is small and and non-monotonic. non-monotonic. The Thegroup group velocity velocity changes changes its sign its at sign kxD/2π at kx =D/2 0.25,π =and 0.25, the and light the propagation light propagation behavior behavior is not trivial. is not In trivial.this case, In thisthe interaction case, the interaction between next-nearest-neighbors between next-nearest-neighbors or distant orcavities distant must cavities also mustbe taken also into be takenaccount. into To account. calculate To calculatethe neighbor the neighborcoupling couplingterms, the terms, tight thebinding tight approximation binding approximation has been has applied. been applied.In Figure In 10, Figure the near10, the field near PL field spectra PL spectra averaged averaged on the on whole the whole array arraystructure structure are reported are reported for both for boththe M- the and M- andK-alignment. K-alignment. Three Three peaks peaks are detected are detected (denominated (denominated T1, T1,T2, T2,and and T3) T3)for fordecreasing decreasing the thewavelength. wavelength. In Figure In Figure 10b, 10 b,the the peak peak positions positions of of T1, T1, T2, T2, and and T3 T3 for for the M alignedaligned cavitiescavities withwith aa nominalnominal identicalidentical designdesign areare shown. The peaks peaks are are almost almost equally equally spaced spaced at at 10 10 nm. nm. It Itis isimportant important to tonote note that that we we observed observed a afew few nanometers nanometers spread spread in in the peak positions, butbut thethe separationseparation betweenbetween T1–T2T1–T2 andand T2–T3T2–T3 isis almostalmost thethe same,same, andand soso thethe spreadspread isis betweenbetween thethe T1–T3T1–T3 modesmodes (about(about 2020 nm).nm). ThisThis means means that that the the detuning detuning due to due the to fabrication the fabrication process process and the possible and the variation possible of variation the membrane of the thicknessmembrane is smallthickness enough is small to not enough impact onto not the modeimpact coupling on the mode in the coupling case of M in aligned the case cavities. of M Foraligned the Kcavities. alignment For geometry,the K alignment the result geometry, is shown the in result Figure is 10shownd and in the Figure peak 10d positions and the are peak summarized positions are in Figuresummarized 10e the in for Figure nominally 10e identicalthe for nominally structures. identical The three structures. mode peaks The for three each structuremode peaks are withinfor each a 5structure nm range, are smaller within than a 5 nm the Mrange, case, smaller denoting than a weaker the M case, coupling. denoting The peaka weaker position coupling. spread The instead peak isposition much higher,spread soinstead in this is case, much the higher, fabrication so in disorderthis case, is the significant fabrication and disorder comparable is significant to the mode and coupling.comparable The to array the mode #1 is the coupling. one in whichThe array the disorder#1 is the isone less in important; which the itsdisorder PL spectrum is less isimportant; reported inits FigurePL spectrum 10d, where is reported an almost in degeneration Figure 10d, inwhere the mode an almost T1 and degeneration T2 is observed. in Inthe our mode sample, T1 and the arrayT2 is #1observed. represents In theour fabricatedsample, the structure array #1 nearest represents to the the theoretical fabricated system structure of the nearest three identicalto the theoretical coupled PCsystem cavities, of the as it three is possible identical to verify coupled by PCcomparing cavities, the as experimental it is possible PL to spectrum verify by with comparing the FDTD the calculatedexperimental one (Figure PL spectrum 10f). We with also the mapped FDTD the calculated PL intensity one at (Figure the peak 10f). wavelength We also for mapped each resonant the PL modeintensity as a at function the peak of thewavelength SNOM tip for position, each resonant in order mode to study as a the function electromagnetic of the SNOM field distributiontip position,of in theorder modes, to study and, tothe understand electromagnetic the mode field behavior, distribution a minimal of the model modes, based and, on ato three understand coupled resonatorthe mode hasbehavior, been developed. a minimal model In the Mbased aligned on a structures,three coupled the resonator T1 and T3 has modes been aredeveloped. mainly localizedIn the M aligned on the centralstructures, cavity. the TheT1 and T2mode T3 modes instead are ismainly localized localized on the on external the central coupled cavity. PC The cavities. T2 mode These instead results is confirmedlocalized on the the FDTD external simulations coupled PC and cavities. the fact These that all results of the confirmed modes are the roughly FDTD simulations formed by a and linear the combinationfact that all of of the the modes single cavityare roughly modes. formed So, the by M a alignment linear combination case can be of reproduced the single cavity considering modes. only So, thethe nearest-neighbor M alignment case coupling. can be reproduced considering only the nearest-neighbor coupling.

FigureFigure 9. 9.Mode Mode dispersion dispersion relation relation oftwo-dimensional of two-dimensional infinite infinite array arrayof D2 coupled of D2 coupled PC cavities PC plotted cavities inplotted the energy in the range energy relative range to relative the fundamental to the fundamental mode of a single mode D2, of fora single configurations D2, for configurations of alignment ofof bothalignment M, (a), of and both K, (M,b). ( Adapteda), and K, with (b). permissionAdapted with from permission the authors from of [ 46the]. Copyrightauthors of ©[46].2015 Copyright American © Chemical2015 American Society. Chemical Society.

Ceramics 2019 2 Ceramics 2019,, 2 FOR PEER REVIEW 48 15

FigureFigure 10.10. ((aa)) ExperimentalExperimental PLPL nearnear fieldfield spectrumspectrum averagedaveraged overover thethe wholewhole arrayarray structurestructure ofof threethree M-coupledM-coupled D2 nanocavitiesnanocavities (see thethe SEMSEM imageimage inin thethe inset).inset). ThreeThree resonancesresonances (labeled(labeled T1,T1, T2,T2, andand T3)T3) areare clearlyclearly observed.observed. ((bb)) TheThe experimentalexperimental resonantresonant modesmodes wavelengthwavelength valuesvalues forfor fivefive nominallynominally identicalidentical M-coupledM-coupled arrays, arrays, evaluated evaluated by by fitting fitting every every peak peak with with a Lorentziana Lorentzian function. function. The The array array #2 in#2 (inb) (b) corresponds corresponds to to the the case case reported reported in in (a ().a). (c ()c Theoretical) Theoretical spectrum spectrum obtained obtained by by three-dimensionalthree-dimensional FDTDFDTD calculations,calculations, averagedaveraged over thethe M-coupledM-coupled arrayarray structurestructure reportedreported inin thethe inset.inset. ((dd––ff)) SameSame analysisanalysis ofof ((aa––cc)) concerningconcerning thethe K-axisK-axis alignedaligned arrayarray ofof threethree D2D2 nanocavities.nanocavities. InIn ((dd)) thethe modesmodes T1T1 andand T2T2 areare almostalmost degeneratedegenerate asas forfor thethe nominalnominal designdesign structurestructure calculatedcalculated byby FDTDFDTD thatthat isis reportedreported inin ((ff).). ((ee)) ResonantResonant modesmodes wavelengthwavelength valuesvalues forfor sixsix nominallynominally identicalidentical K-coupledK-coupled arrays.arrays. TheThe arrayarray #1#1 inin ((ee)) correspondscorresponds toto thethe casecase shownshown inin ((dd).). TheThe scalescale barbar inin allall ofof thethe insetsinsets isis 600600 nm.nm. AdaptedAdapted withwith permissionpermission fromfrom thethe authorsauthors ofof [[46].46]. CopyrightCopyright 20152015 AmericanAmerican PhysicalPhysical Society.Society.

InIn thethe KK aligned structures, wewe observedobserved aa complexcomplex behavior.behavior. TheThe T3T3 modemode isis mainly delocalized onon thethe twotwo externalexternal coupledcoupled PCPC cavities.cavities. TheThe T1T1 andand T2T2 modesmodes areare mainlymainly distributeddistributed overover bothboth thethe centralcentral andand leftleft sideside coupledcoupled PCPC cavities. The comparison ofof the FDTD calculations showsshows thatthat aa better agreementagreement wouldwould bebe obtainedobtained byby exchangingexchanging thethe T1T1 andand T2T2 distribution.distribution. ThisThis couldcould happenhappen becausebecause ofof the detuning due due to to the the fabrication fabrication disorder disorder and and because because the the two two modes modes are are almost almost degenerate. degenerate. In Inthe the K-alignment K-alignment model, model, the the contribution contribution of of the the next-nearest-neighbor next-nearest-neighbor has has been been included togethertogether withwith thethe nearestnearest neighbor.neighbor. InIn thisthis case,case, thethe comparisoncomparison betweenbetween thethe experimentalexperimental andand modellingmodelling showsshows somesome discrepancy,discrepancy, andand thethe realreal systemsystem isis provenproven toto bebe moremore complexcomplex thanthan aa threethree coupledcoupled resonatorresonator system.system. Part Part of of the the complexity is is surely due to the disorder induced energy detuning; nevertheless,nevertheless, this lastlast oneone cancan bebe compensatedcompensated andand tunedtuned byby postpost fabricationfabrication techniquestechniques asas wewe havehave seenseen inin SectionSection 3.1.23.1.2.. The dielectricdielectric environment of the coupled PC cavitiescavities cancan bebe finelyfinely modifiedmodified andand inin thisthis way resonances betweenbetween thethe coupledcoupled PCPC cavities,cavities, andand soso thethe photon hopping betweenbetween thethe adjacentadjacent resonatorresonator cancan bebe controlledcontrolled onon demand.demand. TheThe casecase wherewhere thethe photonphoton hoppinghopping raterate isis thethe samesame forfor allall ofof thethe resonatorsresonators representsrepresents thethe startingstarting pointpoint forfor studyingstudying quantumquantum many-bodymany-body phenomenaphenomena withwith lightlight [[49,50].49,50]. TheThe possibilitypossibility ofof controllingcontrolling thethe resonanceresonance betweenbetween thethe adjacentadjacent resonatorsresonators usingusing postpost fabricationfabrication tuningtuning methodsmethods andand introducingintroducing aa uniformuniform andand controlledcontrolled gradientgradient inin thethe photonphoton hoppinghopping rate rate between between the the different different cavities cavities allows allows for engineering for engineering photonic photonic arrays suitable arrays for suitable quantum for informationquantum information processing processing and optical and communication. optical communication.

3.3. All Lithographic Approach to PCC and QD Coupled Systems As detailed in the previous sections, the ability to fabricate photonic crystal structures with a high spatial resolution might open the way to the realization of novel optical elements, which are

Ceramics 2019, 2 49

3.3. All Lithographic Approach to PCC and QD Coupled Systems As detailed in the previous sections, the ability to fabricate photonic crystal structures with a high spatial resolution might open the way to the realization of novel optical elements, which are useful for the manipulation and routing of light in nanophotonic circuits operating at the sub-wavelength level. Equally important, within this context, would be the possibility to integrate one or more light emitters at prescribed points of such photonic circuits. Ideally, the spatial and spectral position of these emitters should be controllable with a precision of a few nm; moreover, they should be able to generate non-classical light states “on demand”, that is, exactly one photon [17] or one entangled photon pair [18] should be produced for each excitation pulse. In recent years, semiconductor quantum dots (QDs) have emerged as particularly promising candidates for the realization of non-classical light emitters [16,51], owing to their inherent integrability with optoelectronic devices, as well as to continuous improvements in terms of their single-photon purity/indistinguishability and degree of entanglement [22,52]. The ability to control the position and emission energy of QDs has also made great strides in the past decade, because of the joint efforts of several research teams [26,53]. Recently, a novel and versatile approach for the post-growth fabrication of site-controlled QDs has been developed based on the spatially selective incorporation or removal of atoms in dilute nitride structures [54,55]. Hydrogen incorporation in GaAsN results, indeed, in the formation of N–H complexes, which neutralize all of the effects of N on GaAs, including the N-induced large reduction of the bandgap energy [56,57]. Therefore, by engineering the spatial incorporation and/or removal of hydrogen in dilute nitrides, it is possible to attain a spatially controlled modulation of the bandgap energy in the growth plane and, eventually, to tailor the carrier-confining potential down to a nm scale, resulting in the fabrication of site-controlled QDs that are able to emit single photons on demand [54,55]. Such a novel fabrication approach has made it possible to obtain a fully lithographic approach for the deterministic integration of a site-controlled QD in a PC system [58]. The fabrication process is sketched in Figure 11a. The sample is first provided with a series of Cr/Au alignment markers by means of a standard lift-off process. Then, an ordered array of GaAsN QDs is realized aligned with the metallic markers, by making use of the spatially selective hydrogenation approach described in the literature [54,58]. After the fabrication of the QDs, a photonic crystal cavity can be realized around each QD. The PC cavity fabrication follows a standard procedure that consists of covering the sample with a thin layer of positive-tone electron-beam resist (ZEP520A), on which the desired PC pattern is realized by electron beam lithography (EBL). The resist acts as a mask during the transfer of the PC design onto the sample via the Cl-based dry etching of the GaAsN/GaAs layer. Finally, the mask resist is removed by a wet etching in hot anisole, and the membrane containing the PC structure integrating the GaAsN QD is released by the wet etching of the AlGaAs sacrificial layer with a 5% solution of hydrofluoric acid, in order to provide vertical optical isolation. It is worth mentioning that the presence of the alignment markers on the sample—together with the spatial accuracy in defining the QD and PC cavity both given by the EBL—guarantees a spatial coupling between the QD and the cavity of about 20 nm (limited only by the realignment precision of the EBL system). Further details on the fabrication process can be found in the literature [58]. As summarized in Figure 11b–e, integrated QD-PC systems have already been successfully realized using the fully lithographic approach, and their properties have been optically characterized [57,58]. The energy of the fundamental cavity mode of a series of L3 photonic defects (wherein the microcavity is obtained by removing three holes from an otherwise perfectly periodic photonic lattice [59]) was lithographically tuned into resonance with the QD emission by adjusting the PC lattice pitch, a [28] (see Figure 11d). After achieving a coarse spectral matching between the cavity mode and the QD exciton (X) for a = 255 nm, the system was progressively tuned into the resonance by varying the sample temperature, T, as displayed in Figure 11b. This was made possible by the much stronger T dependence of the energy of the X transition, which follows the band gap reduction of GaAsN with T [60], with respect to the cavity mode, which linearly redshifts (at a rate of ~20 meV/K, consistent with the literature [61]) because of the variation of the refractive index of GaAs with T. An interesting outcome of the progressive reduction of the QD-cavity mode Ceramics 2019, 2 50 energy detuning with T is reported in Figure 11c, which displays the temperature dependence of the micro-PLCeramics 2019 intensity, 2 FOR PEER of the REVIEW QD and cavity mode peaks. As T is increased from 10 K to 50 K, the PL signal 17 shows the intensity drop-off that is usually expected in semiconducting samples, chiefly because of the thermalof the non-radiative activation of therecombination non-radiative channels recombination [62]. For channelsT > 50 K, [however,62]. For T a> large 50 K, increase however, in a the large PL increaseintensity in can the PL be intensity observed can as be the observed X line as is the moved X line into is moved resonance into resonance with the with cavity the mode. cavity This mode. is Thisconsistent is consistent with the with ~10-fold the ~10-fold enhancement enhancement of the ofradiative the radiative recombination recombination rate (i.e., rate the (i.e., Purcell the Purcell effect effect[6]) measured [6]) measured for this for system this system [63]. [63].

FigureFigure 11. 11. ((aa)) Sketch Sketch of of the the processing processing steps steps leading leading to to the the deterministic deterministic integration integration of of a a single single Ga(AsN)Ga(AsN) quantum quantum dotdot (QD)(QD) withwith a PC cavity. First, an an array array of of H-opaque H-opaque masks masks is is aligned aligned to to a aset set of ofmetallic metallic (chromium–gold) (chromium–gold) markers markers previously previously realized realized on on the the sample sample surface. surface. Both Both the the masks masks and and the themarkers markers are are defined defined by byelectron electron beam beam lithography. lithography. Second, Second, H irradiation H irradiation results results in the in theformation formation of a ofsite-controlled a site-controlled GaAsN GaAsN QD underneath QD underneath each eachH-opaque H-opaque mask. mask. Finally, Finally, a PC cavity a PC cavityis fabricated is fabricated around aroundeach QD. each The QD. reference The reference system system defined defined by bythe the metallic metallic markers markers ensures ensures a a near near perfect (~20 (~20 nm nm accuracy)accuracy) alignment alignment between between the the QD QD and and the the PC PC cavity. cavity. (b) ( Micro-photoluminescenceb) Micro-photoluminescence (PL) (PL) spectra spectra of an of integratedan integrated QD-PC QD-PC cavity cavity system, system, showing showing the temperature-dependent the temperature-dependent of the cavity of the mode-QD cavity detuning.mode-QD Thedetuning. exciton The transition exciton of transition the QD is of labeled the QD as X.is labeled(c) Temperature as X. (c) dependence Temperature of dependence the integrated of PL the intensityintegrated of thePL cavityintensity mode of the (black cavity dots) mode and of(black the X dots) peak and (blue of dots). the X The peak intensity (blue dots). increase The observed intensity forincrease temperatures observed above for temperatures ~50 K is a result above of an~50 increased K is a result QD-PC of an cavity increased coupling QD-PC (i.e., cavity of the coupling Purcell effect),(i.e., of because the Purcell of the effect), QD coming because into of resonance the QD coming with the into cavity resonance mode. ( withd) Lithographic the cavity tuning mode. of (d) theLithographic cavity mode tuning energy of ofthe a cavity PC L3 defectmode energy cavity byof varyinga PC L3 thedefect value cavity of the by pitchvarying (a) ofthe the value photonic of the lattice.pitch ( Thea) ofr /thea ratio photonic (where lattice.r is the The radius r/a ratio of each (where PC hole)r is the is radius kept constant of each ( rPC/a hole)= 0.29). is Askept expected, constant the(r/a dependence = 0.29). As expected, of the cavity the dependence mode energy of on thea cavityis roughly mode linear, energy with on aa dEis roughlyCM/da ~3.5 linear, meV/nm. with a (edE) Second-orderCM/da ~3.5 meV/nm. autocorrelation (e) Second-order of the X transitionautocorrelation of the of site-controlled the X transition QD of integrated the site-controlled in a L3 cavity. QD Noteintegratedg(2)(0)

WithWith regards regards to to the the properties properties of of the the quantum quantum emitter emitter integrated integrated in in the the cavity, cavity, it it is is important important to to stressstress that that in in this this condition, condition, thethe site-controlledsite-controlled QDQD isis alsoalso ableable toto emitemit atat thethe singlesingle photonphoton regime,regime, whichwhich isis a a crucial crucial property property forfor the the successful successful employmentemployment ofof thesethese systemssystems inin futurefuture applications.applications. EvidenceEvidence of of such such a a non-classical non-classical behavior behavior of of light light is is given given by by the the observationobservation of of a a strongstrong antibunching antibunching forfor the the near-zero near-zero time time delay delay in the in autocorrelation the autocorrelation histogram histogram of the QD of exciton the QD emission exciton line emission (reported line in(reported Figure 11 ine). Figure 11e). Finally, we want to stress here that a deterministic integration of QDs in PC systems might also be achieved using the spatially selective hydrogen-removal approach presented in [54,57]. In this case, indeed, the SNOM ability to “see” the of a PC cavity as we have seen before [29,64] can be used to map the field distribution of the fundamental cavity mode of a fully hydrogenated GaAsN/GaAs PC cavity, and, eventually, using near field illumination, to fabricate a

Ceramics 2019, 2 51

Finally, we want to stress here that a deterministic integration of QDs in PC systems might also be achieved using the spatially selective hydrogen-removal approach presented in [54,57]. In this case, indeed, the SNOM ability to “see” the electromagnetic field of a PC cavity as we have seen before [29,64] can be used to map the field distribution of the fundamental cavity mode of a fully hydrogenated GaAsN/GaAs PC cavity, and, eventually, using near field illumination, to fabricate a quantum emitter coupled with the cavity mode. Although, within this further approach the spatial accuracy is worse than that achievable with the EBL-based method presented above (only a spatial precision of ~100 nm can be reached by this fabrication approach [55]), the overall process flexibility is improved by the possibility to tailor the QD emission energy to that of the cavity mode, simply by varying the QD fabrication parameters.

4. Conclusions In this review, we described the characteristics of photonic molecules based on D2 (diamond-like) coupled PC cavities. The shape of this photonic crystal nanocavity, which is elongated along the M axis (vertical) of the photonic crystal, has a direct influence on the mode distribution. The M1 mode is also elongated along the M axis, while the M2 mode is mainly elongated along the K axis (horizontal). In the two D2 coupled PC cavities, these characteristics are reflected in the fact that the PC cavity coupling behaves differently accordingly to the geometrical arrangement of the coupled cavities. In general, mode delocalization requires that 2 g > ∆, then the disorder induced detuning ∆ is expected to be similar for any geometry, while the coupling strength (g) can largely vary for the different cavity arrangement. In the vertical arrangement investigated, all of the modes of the coupled system (P1–P4) are distributed over both of the coupled PC cavities; this spatial delocalization, similar to the molecular orbitals, is a fingerprint of the effective resonance between the modes (M1 and M2) of the single D2 cavities, and the photons can tunnel among the two cavities (that is, 2 g > ∆). In the case of the horizontal arrangement investigated, only the coupled modes (P3 and P4) originating from the M2 modes are really coupled and spatially delocalized on both cavities (that is, 2 g > ∆). The P1 and P2 modes, coming from the resonance of the M1 mode of the two D2 cavities, are not really coupled and are spatially distributed on the left cavity and on the right one, respectively (that is, 2 g < ∆). Their spectral shift can be attributed to the disorder induced by the fabrication process, that in this way can be evaluated. Therefore, the presence of a fabrication induced disorder can hamper the coupling strength between the adjacent cavities. Nevertheless, this effect can be compensated by applying the post fabrication control of both ∆ and g, thus tailoring the tunneling/coupling performances of the studied structure [65]. The coupled modes have been extensively analyzed, investigating the symmetry properties and their bonding and antibonding nature. All of these characteristics are strictly linked to the geometrical features and to the disorder presence, and can be controlled by the system design and/or by post fabrication tuning. Also, the three-cavity system has been studied together with the possibility of realizing the photon hopping among the cavity array. All of the examined photonic molecules have been realized in samples in which high-density quantum dots play the role of internal emitters. The possibility of integrating single emitters at precise spatial locations would path the way to the precise manipulation of single photons. To this aim, an all lithographic approach to create single QDs coupled with PC cavities has been developed by the exploitation of the diluted nitride properties and the high-resolution capabilities of e-beam lithography. With this technique, the deterministic integration of single QD and PC cavities have been carried out and single photon emission is achieved. The natural perspective of the work described in this paper is the realization of the photonic molecules on the dilute nitride material, so including single photon emitters accordingly to the mode spatial distribution. The versatility of the fabrication technique will allow for the engineering of unique platforms to control the photon coupling, to be applied in quantum information processing and communications. Ceramics 2019, 2 52

Author Contributions: As a review article, the paper describes the work of a long lasting collaboration between the authors. The IFN-CNR group (A.G. and G.P.) contributed to the design, fabrication, optical characterization and data discussion; The University of Florence and European Laboratory for Non-linear Spectroscopy group (N.C., S.V., F.R., F.B., M.G., F.I.) contributed to the design, optical characterization, data analysis, validation and discussion; The Sapienza University of Rome group (M.F. and A.P.) contributed to the design, optical characterization, data analysis, validation and discussion as regards the work described in Section 3.3. The Eindhoven University group (A.F.) contributed to sample growth, fabrication and data discussion as regards the work discussed in Sections 3.1 and 3.2. Writing-original draft preparation, A.G.; writing-review and editing, F.I., G.P., M.G. and all authors. Funding: M.F. and A.P. acknowledge support by Sapienza Università di Roma under the “Ricerche Ateneo” Grants 2016 and 2017. F.B., M.F. and G.P. also acknowledge support and funding from the Italian Ministry for Education, University and Research within the Futuro in Ricerca (FIRB) program (project DeLIGHTeD, Prot. RBFR12RS1W). The work of F.B. was partially supported by Fondazione Ente Cassa di Risparmio di Firenze within the project PERBACCO (no. 2016.1084). A.P and M.F. have also received funding from the LazioInnova project “SINFONIA”, (Prot. n. 85-2017-15200). Conflicts of Interest: The authors declare no conflict of interest.

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