Photonic network random lasing in the Anderson localized regime

Michele Gaio,1, ∗ Dhruv Saxena,1 Jacopo Bertolotti,2 Dario Pisignano,3, 4 Andrea Camposeo,3 and Riccardo Sapienza1

1The Blackett Laboratory, Department of Physics,Imperial College London, London SW7 2AZ, United Kingdom

2Physics and Astronomy Department, University of Exeter, Stocker Road, Exeter EX4 4QL,UK3NEST, Istituto Nanoscienze-CNR, Piazza San Silvestro 12, I-56127 Pisa (Italy)

4Dipartimento di Matematica e Fisica “Ennio De Giorgi”,Universita del Salento, via Arnesano, I-73100 Lecce (Italy)

Classical and quantum optical devices rely on the powerful light trapping and transporting archi-tectures offered by nanophotonic systems. These properties derive from the way the optical modesare sculpture by multiple scattering. Here we propose a graph approach to nanophotonics, and anetwork platform to design of light-matter interaction. We report a photonic network built from re-current scattering in a mesh of subwavelength waveguides, with Anderson-localized network modes.These modes are designed via the network connectivity and topology and can be modeled by a graphdescription of Maxwell’s equations. The photonic networks sustain random lasing action and exhibitlasing thresholds and optical properties which are determined by the network topology. Photonicnetwork lasers are promising new device architectures for sensitive biosensing and for developingon-chip tunable laser sources for future information processing.

Network science describes complex systems with a fo-cus on the interaction between the elementary units,looking beyond the microscopic details. This has provenfruitful in many different fields, to understand the way avirus spreads [1], quantum information is transmitted [2],or a power grid failure propagates [3]. In nanophotonics,within the coupled-dipole approximation [4], wave mul-tiple scattering can be naturally described as a networkof connections between the scattering elements. There-fore a network approach to the study of light propagationin complex media offers a simple framework to describeaccurately realistic-sized systems.

Furthermore, a network architecture can enhance light-matter interaction and promote coupling between embed-ded emitters, due to its low dimensionality. Such net-works have been proposed for optical routing [5, 6] andfor light localization [7]. A network decouples the pro-cesses of light scattering (at the nodes) and transport (inthe one-dimensional links), allowing to design the scat-tering strength at the nodes via the connectivity, i.e. thenumber of links, and the recurrent scattering loops viathe network topology, while minimizing the out-of-planescattering losses.

In particular, a network is ideal to promote lasing byenhancing the probability of stimulated emission. In adisordered material, the modes arising by multiple andrecurrent scattering boost the stimulated emission prob-ability, resulting in efficient random lasing action [8, 9]and in a wealth of interesting physical phenomena [10–12]. Initial random lasing architectures were based on 3Ddisordered semiconductor powders [13] or randomly fluc-tuating colloids in solution [14] with spectrally broad andomnidirectional lasing emission. Lacking a direct designof the optical modes, their optimization was only drivenby the increase in the scatter density to minimize thescattering lengths. Recently, a new generation of ran-

dom lasing systems, based on designed-disordered archi-tectures, with more robust and collective light-trappingschemes, frequency control [15], tailored spatial correla-tions [16, 17], and Anderson localization [12], which arebetter suited for lasing control, have emerged.

Random lasing has been studied in a perforated mem-brane with subwavelength network-like topology [16] al-beit without light confinement in the links. Very recentlya macroscopic laser based on 4 connected optical fibreswas demonstrated, as a first network laser operated inthe single scattering regime [18]. Modeling lasing with agraph theory approach [19] capable of reducing its com-plexity is still largely unexplored, and only recently anetwork approach has been proposed for weakly scatter-ing lasing systems [20].

Here, we introduce a planar photonic network archi-tecture fabricated from a mesh of nanoscale waveguides,forming a micron-scale photonic material comprising overa thousand nodes. We describe light propagation andinterference with a graph model valid in the multiplescattering regime capable of revealing Anderson locali-sation of light. We report experimental observation ofnetwork lasing exhibiting low lasing threshold and evi-dence of light localization in the photonic network, whichcompares well with our predictions.

The photonic network is composed of subwavelengthwaveguides (links) connected together at their cross-ing (nodes). Poly(methyl methacrylate) (PMMA) singlemode nanofibers of diameter in the range 200–500 nm,doped with Rhodamine-6G (1% in weight), are fabri-cated by electro-spinning and self-assembled on a cm-scale metal TEM grid (200x200 µm opening). The net-work (Fig. 1a) is formed by annealing the mesh in nitro-gen atmosphere to join the waveguides at the nodes (seeSupplementary Information); the white lines and the redcircles superimposed identify the links and nodes of the

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FIG. 1. Network random laser. Far-field images of (a) fluorescence and (b) lasing from a network of freestanding subwave-length electrospun polymer (PMMA) nanofibers embedded with dye (Rhodamine 6G). The white lines and red circles in thetop left of image (a) illustrate the links and nodes of the network, respectively. The network topology is characterized by itsnode degree distribution (c) and link length distribution (d). The lasing spectrum (e) collected along the vertical dashed line in(b) reveals multimodal behavior and narrow linewidth (inset: laser peak at threshold). (f) The plot of peak emission intensityas a function of pump power shows a clear transition to lasing at threshold T ' 200 nJ.

network, respectively.The fabricated structures are planar disordered net-

works with a partial mesh topology with degree D inthe range 4 to 6 (〈D〉 = 4.4) and average link length〈le〉 = 26.5 µm, as shown in Fig. 1c and d. Each subwave-length nanofiber guides light with a propagation lengthlarger than 100 µm, as we have previously measured [6].Although 3D bulk arrangements of polymer nanofibershave been demonstrated [21–25], these structures wereweakly connected together or consisted of self-connectedloops, which is very different from our connected net-work. Compared to 3D arrangements, planar architec-tures are better suited for on-chip integration with otherphotonic components. Moreover, the network topologyprovides a strong in-plane confinement of light in thelinks with minimal out-of-plane scattering at the nodes(a few %), as confirmed by finite-difference time-domain(FDTD) simulations (see Supplementary Information).Thus, our photonic network serves as an efficient randomlasing platform.

Efficient lasing action requires a long-lasting popula-tion inversion and low-loss optical modes that can bepopulated by stimulated emission. While the first prop-erty is guaranteed by the efficient laser dye (gain-length`g ' 10 µm), the second, i.e. the optical modes, are de-signed by the network topology. The photonic network isvery efficient to reach room-temperature lasing when ex-cited with a single 500 ps green laser pulse (λ = 532 nmfrom a TEEM Microchip), which illuminates a spot of di-ameter ∼ 160 µm perpendiculalyr to the network plane.Optical images of the fluorescence and lasing are shown inFig 1a,b. Under low illumination intensity (P = 20 nJ)the outcoupled fluorescence from the fibers is recorded(Fig. 1a) and its pattern follows the network shape. Notethat spontaneous emission couples to the fundamental

waveguide mode in these subwavelength fibers with an ef-ficiency as high as 30–50% [6]. When the illumination in-tensity is increased, particularly above the lasing thresh-old, bright spots corresponding to the network nodes areobserved (Fig. 1b, P = 2000 nJ). This is because stim-ulated emission populates the guided modes, which areoutcoupled at the nodes due to out-of-plane scattering.

Spectrally, lasing is characterized by highly multimodeemission with sharp peaks, as shown in Fig. 1e, which hasonly a minor pulse-to-pulse variation. The experimen-tal linewidth is limited by the spectrometer resolution(0.05 nm, from an holographic grating, 1800 gr/mm, onPrinceton Instruments Isoplane-320). The threshold is

around 200 nJ, i.e. ∼ 0.3 mJ/cm2, as extracted from

the emitted peak power versus pump intensity relationshown in Fig. 1f. It is worth noting that the data inFig. 1 is representative of the sample; in total, data wascollected from 16 different networks and the thresholdranged between 120–325 nJ.

Network random lasing can be well-described by agraph model, which we develop to study the key elementsof the photonic network beyond its microscopic details.We solve Maxwell’s equations (scalar wave equation) ona metric graph, analogous to quantum graphs [26]. Cen-tral to this model is the description of the photonic net-work: multiple scattering occurs at each node, while lightpropagation along the (single-mode) link from node i tonode j of length Lij is fully described by phase aquiredby the electric field E(xj) = E(xi)e

ikLij . We describethe multiple scattering process as a boundary-conditionproblem for the waves meeting at the nodes. We employNeumann boundary conditions (Ei = Ej ,

∑dEi/dx = 0

[26]), which ensures that energy is conserved at the scat-tering node. The links at the boundary of the network

3

are left open and represent the only loss channels in themodeled system. Within the graph model, the full spa-tial profile of the modes can be easily calculated with-out the need for spatial discretization. This enables ef-ficient modeling of large networks (See supplementary).A similar approach has been developed in Ref.[18], al-beit limited to the single scattering process. Instead,here we focus on multiple scattering in the optical meso-scopic regime, by considering complex networks with alarge number of nodes (200–600 nodes, Fig. 2). Note thatnumerical methods such as finite-difference time-domain(FDTD) would be impractical for modeling such largesystems and provide extended statistics.

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FIG. 2. Properties of the network modes. (a) Givena network topology realization (143 nodes, 293 links, D = 6,〈le〉 = 6.5 µm), the optical modes spatial profile and theircomplex k can be computed. (b) Modes are characterizedby the lasing threshold (-Im(k)) and a degree of localizationdescribed by IPR. The white dashed line is a prediction formodes localized at the center of the network. (c) The natureof the modes (intensity in red) in different area of the plot(indicated by the corresponding symbol) is confirmed by in-specting their spatial profile: ((c), circle) delocalized modesoccupying a significant system area; (square, (d)) lossy modesconfined close to the network boundary; ((e), triangle) modeslocalized in the center of the system, which are those with thelowest threshold.

The networks modes are labeled with complexwavevector k (Im(k). Under the assumption of linear andundepleted gain we can calculate the mode threshold byevaluating −Im(k) which is the amount of gain requiredto bring the mode to net amplification. This approach iscommon to many lasing models [27, 28],and is valid forpump energies not far from the lasing threshold, whichis the common assumption for the first lasing modes.

A direct correlation between the degree of localizationand lasing threshold can be evidenced by plotting thelasing threshold, i.e. -Im(k), against the inverse partic-ipation ratio (IPR) as shown in Fig. 2b. The IPR isdefined as IPR = L

∫dx|E|4 /(

∫dx|E|2)2 and it is nor-

malized to range between 1 (fully uniformly delocalizedmodes) and ∼ N (where N is the total number of links

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FIG. 3. Mode spacing statistics. (a) Normalized modespacing statistics obtained from experimentally measured las-ing spectra from different parts of the sample (inset shows aselected spectrum). The distribution is well fitted by an expo-nential curve (dashed line), up to the experimental spectralresolution indicated by the gray band. The Wigner-Dysondistribution is also shown for comparison (dotted line). (b)The level spacing statistics obtained from the model complexk’s (inset shows a typical map), for networks with 150 nodes,D = 6, 〈le〉 = 6.5 µm. The distribution follows mostly anexponential trend with limited mode repulsion.

in the network) for modes predominantly confined to afew links. The modes localized at the center of the net-work, with a radial spatial profile following exp(−r/ξ),are expected to lie along the dashed white line shownin Fig. 2b (IPR' (R/2ξ)2, −Im(k) ∝ R exp (−R/ξ)/2ξ2,where R is the radius of the network and ξ is the lo-calization length (see Supplementary Information). InFig. 2b two main branches are visible, one follows theabove predicted trend, with lower threshold modes be-ing characterized by stronger confinement, while anotherbranch seems to have the opposite behavior with largerthreshold for the most localized modes. Upon inspectionof their spatial profiles, these are interpreted as modeslocalized close to the network boundary, as in Fig. 2cmiddle plot, therefore more subjected to losses, but stillcapable of lasing, albeit with larger threshold. This istherefore simply a finite-size effect. Interestingly, theAnderson localized modes localized in the middle of thenetwork, as in Fig. 2e, i.e. with the lowest in-plane out-coupling losses, are the one with the lowest threshold.

In order to experimentally address the nature of thelasing modes of the network we investigate the statisticsof the nearest-neighbor level spacing, which also offersa direct comparison between our model and the experi-ments. The level spacing is a robust experimentally ac-cessible quantity that relates the modes with the trans-port regime [29]. Diffusive and localized systems are

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FIG. 4. Degree and threshold relation. (a) Thresh-old (-Im(k)) distributions for networks with average degreeD = 3, 4, 5, 6 and constant number and density of nodes(N = 600). (b) The lowest lasing threshold estimated fromthe model (averaged over 1000 different realizations) as a func-tion of the average degree.

known to follow universal nearest-neighbor level spac-ing statistics [30] regardless of the details of the system,as predicted for instance in the random matrix frame-work [31]. Broad spatial overlap and mode coupling ofdelocalized modes induces mode repulsion [32] resultingin the Wigner-Dyson distribution of the nearest-neighborlevel spacing, while localized modes lack spatial over-lapping and therefore feature a random spectrum withno mode repulsion are well described by a Poissoniannearest-neighbor level spacing distribution [33, 34].

The level spacing obtained from experimental lasingspectra is shown in Fig. 3a and follows an exponen-tial trend indicating Anderson localization. The dip atsmaller spacing is due to the finite resolution of our spec-trometer (0.05 nm). For comparison, we also overlaythe Wigner-Dyson distribution (dotted lines). The sameanalysis can be performed for the modes obtained numer-ically from the model (Fig. 3b). Their distribution alsofollows an exponential trend. However, a limited moderepulsion is still visible, which comes from the finite-sizeof the simulated network, chosen to match the exper-imental one. If we simulate larger networks, the levelspacing distribution converges towards an exponential(see Supplementary Information). Thus, the model ade-quately describes the level-spacing distribution observedin experiments.

We further use our model to investigate the role of net-work topology on the lasing action. Specifically, we con-sider effect of the network degree on the lasing threshold,for degree D = 3, 4, 5, 6. For a given number and densityof scattering points, changing the connectivity of the sys-tem has two main effects: i) it affects the scattering at thenodes, as light is distributed among different number oflinks, and ii) it modifies the linear length of the network,thereby changing the total gain available. The simulation

in Fig. 4a shows how the network degree impacts the las-ing threshold. A rapid decrease of the lasing threshold forincreased connectivity is visible, with a drop of thresholdby an order of magnitude between D = 3 and D = 6(TD=3/TD=6 = 10 ± 0.2), as shown in Fig. 4b. Thesevalues are obtained from the threshold distributions bycomputing all the modes in a given wavelength window(∆λ = 1 nm centered at λ0 = 600 nm) and calculatingthe average of the lowest threshold modes over 1000 dif-ferent realizations. The increase in network length withincreasing degree and therefore the increased total gain(by a factor of ∼ 2 when comparing D = 3 and D = 6),cannot explain the threshold decreases alone.

In conclusion, we have proposed a graph approach tolight transport and localisation in a network, which isfast and correlates well with experimental results. Wehave shown experimental evidence of random lasing froma photonic network of connected subwavelength waveg-uides. The planar network architecture promotes lasingof the most confined optical modes, which are shown tobe Anderson localized. Network random lasers can beeasily extended to networks connected in three dimen-sions or other kind of networks, and the light localiza-tion length and threshold can be tuned with the net-work topology. This family of lasers holds potential forsensitive sensors due to the high surface area, narrowlinewidth and strong mode competition, as well as foron-chip laser sources outcoupled to external waveguides.

The authors would like to thank M. Moffa for samplepreparation. The research leading to these results hasreceived funding from the Engineering and Physical Sci-ences Research Council (EPSRC), the Leverhulme Trust,the Royal Society, the European Research Council underthe European Union’s Seventh Framework Programme(FP/2007-2013)/ERC Grant Agreement n. 306357 (ERCStarting Grant NANO-JETS). The data is publicly avail-able in Figshare [35].

∗ m.gaio@imperial.ac.uk[1] Albert-Lszl Barabsi, Gulbahcem Natali, and Loscalzo

Joseph, “Network medicine: a network-based approachto human disease,” Nature Reviews Genetics 12, 56–68(2011).

[2] S. Perseguers, M. Lewenstein, A. Acın, and J. I. Cirac,“Quantum random networks,” Nature Physics 6, 539543(2010).

[3] A. Caz, R. Pierrat, and R. Carminati, “Voltage col-lapse in complex power grids,” Nature Communication7, 10790 (2016).

[4] A. Caz, R. Pierrat, and R. Carminati, “Spatial Co-herence in Complex Photonic and Plasmonic Systems,”Phys. Rev. Lett. 110, 063903 (2013).

[5] Eyal Feigenbaum and Harry A Atwater, “ResonantGuided Wave Networks,” Physical Review Letters 104,147402 (2010).

5

[6] Michele Gaio, Maria Moffa, Marta Castro-Lopez, DarioPisignano, Andrea Camposeo, and Riccardo Sapienza,“Modal Coupling of Single Photon Emitters WithinNanofiber Waveguides,” ACS Nano 10, 6125–6130(2016).

[7] Z Q Zhang, C C Wong, K K Fung, Y L Ho, W L Chan,S C Kan, T L Chan, and N Cheung, “Observation of lo-calized electromagnetic waves in three-dimensional net-works of waveguides,” Physical Review Letters 81, 5540–5543 (1998).

[8] Diederik S. Wiersma, “The physics and applications ofrandom lasers,” Nat. Phys. 4, 359–367 (2008).

[9] H Cao, “Review on latest developments in random laserswith coherent feedback,” J. Phys. A. Math. Gen. 38,10497 (2005).

[10] Ravitej Uppu and Sushil Mujumdar, “ExponentiallyTempered Levy Sums in Random Lasers,” Physical Re-view Letters 114, 183903 (2015).

[11] N. Ghofraniha, I. Viola, F. Di Maria, G. Barbarella,G. Gigli, L. Leuzzi, and C. Conti, “Experimental evi-dence of replica symmetry breaking in random lasers,”Nature Communications 6, 6058 (2015).

[12] J Liu, P D Garcia, S Ek, N Gregersen, T Suhr, M Schu-bert, J Mørk, S Stobbe, and P Lodahl, “Randomnanolasing in the Anderson localized regime,” Nat Nano9, 285–289 (2014).

[13] C Gouedard, D Husson, and C Sauteret, “Genera-tion of spatially incoherent short pulses in laser-pumpedneodymium stoichiometric crystals and powders,” Jour-nal of the Optical Society of America B 10, 2358–2363(1993).

[14] N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes,and E. Sauvain, “Laser action in strongly scattering me-dia,” Nature 368, 436–438 (1994).

[15] Stefano Gottardo, Riccardo Sapienza, Pedro D. Garcıa,Alvaro Blanco, Diederik S. Wiersma, and Cefe Lopez,“Resonance-driven random lasing,” Nature Photonics 2,429–432 (2008).

[16] Heeso Noh, Jin-Kyu Yang, Seng Fatt Liew, Michael JRooks, Glenn S Solomon, and Hui Cao, “Photonic net-work laser,” Optics Letters 36, 3560–3562 (2011).

[17] Jin-Kyu Yang, Svetlana V. Boriskina, Heeso Noh,Michael J. Rooks, Glenn S. Solomon, Luca Dal Negro,and Hui Cao, “Demonstration of laser action in a pseu-dorandom medium,” Applied Physics Letters 97, 223101(2010).

[18] Stefano Lepri, Cosimo Trono, and Giovanni Giacomelli,“Complex active optical networks as a new laser con-cept,” Physical Review Letters 118, 123901 (2017).

[19] R Albert and AL Barabasi, “Statistical mechanics ofcomplex networks,” Reviews of Modern Physics 74, 47–97 (2002).

[20] M. Hofner, H. J. Wunsche, and F. Henneberger, “Arandom laser as a dynamical network,” New Journal ofPhysics 16, 033002 (2014).

[21] Dengfeng Huang, Tingshuai Li, Shenye Liu, Tao Yi,Chuanke Wang, Jin Li, Xiyun Liu, and Min Xu, “Ran-dom lasing action from electrospun nanofibers dopedwith laser dyes,” Laser Physics 27, 035802 (2017).

[22] Luana Persano, Andrea Camposeo, and Dario Pisignano,“Active polymer nanofibers for photonics, electronics, en-ergy generation and micromechanics,” Progress in Poly-mer Science 43, 48–95 (2015).

[23] Sarah Krammer, Christoph Vannahme, Cameron L C

Smith, Tobias Grossmann, Michael Jenne, StefanSchierle, Lars Jørgensen, Ioannis S. Chronakis, AndersKristensen, and Heinz Kalt, “Random-cavity lasing fromelectrospun polymer fiber networks,” Advanced Materi-als 26, 8096–8100 (2014).

[24] R. Dhanker, A. N. Brigeman, A. V. Larsen, R. J. Stew-art, J. B. Asbury, and N. C. Giebink, “Random lasingin organo-lead halide perovskite microcrystal networks,”Applied Physics Letters 105, 151112 (2014).

[25] R. Zhang, S. Knitter, S. F. Liew, F. G. Omenetto,B. M. Reinhard, H. Cao, and L. Dal Negro, “Plasmon-enhanced random lasing in bio-compatible networks ofcellulose nanofibers,” Appl. Phys. Lett. 108 (2016),10.1063/1.4939263.

[26] Peter Kuchment, “Quantum graphs: I. Some basic struc-tures,” Waves in Random Media 14, S107–S128 (2004).

[27] H. E. Tureci, H. G. L. Schwefel, Ph. Jacquod, andA. Douglas Stone, “Chapter 2 - Modes of wave-chaotic di-electric resonators,” in Progress in Optics, Vol. 47, editedby E. Wolf (Elsevier, 2005) pp. 75–137.

[28] Takahisa Harayama, Peter Davis, and Kensuke S Ikeda,“Stable Oscillations of a Spatially Chaotic Wave Func-tion in a Microstadium Laser,” Physical Review Letters90, 063901 (2003).

[29] Jing Wang and Azriel Z Genack, “Transport throughmodes in random media,” Nature 471, 345–348 (2011).

[30] Stefan Rotter and Sylvain Gigan, “Light fields in com-plex media : mesoscopic scattering meets wave control,”Reviews of Modern Physics 89, 015005 (2015).

[31] C. W. J. Beenakker, “Random-matrix theory of quantumtransport,” Rev. Mod. Phys. 69, 731–808 (1997).

[32] B.L. Al’tshuler, I. Kh. Zharekeshev, S.A. Kotochigova,and B.I. Shklovskii, “Repulsion between energy levels andthe metal-insulator transition,” Journal of Experimentaland Theoretical Physics 67, 625 (1988).

[33] M. P. Sorensen and T. Schneider, “Level-spacing statis-tics for the Anderson model in one and two dimensions,”Zeitschrift fur Physik B Condensed Matter 82, 115–119(1991).

[34] S. Sorathia, F. M. Izrailev, V. G. Zelevinsky, and G. L.Celardo, “From closed to open one-dimensional Ander-son model: Transport versus spectral statistics,” Physi-cal Review E 86, 011142 (2012).

[35] Michele Gaio, Dhruv Saxena, Jacopo Bertolotti, DarioPisignano, Andrea Camposeo, and Riccardo Sapienza,“Photonic network random lasing in the Anderson local-ized regime,” Figshare (2017).

6

Supplementary information

Photonic network fabrication

Poly(methyl methacrylate) (PMMA) was dissolved in a mixture of chloroform and dimethyl sulfoxide (DMSO)(volume ratio 9:2), while Rhodamine-6G was added to the solution with a concentration of 1% weight:weight relativeto the polymer matrix. The addition of DMSO allows fibers with sub-micron diameters to be collected [S1,S2].The solution was mixed by mechanical stirring and loaded in a 1 mL syringe, tipped by a stainless needle. Theelectrospinning was performed by applying a bias (10-15 kV, EL60R0.6-22, Glassman High Voltage) between theneedle and a 10×10 cm2 Cu plate, positioned 10 cm away from the needle, while the solution was injected at constantflow rate (0.5-1 ml h-1) using a syringe pump (Harvard Apparatus). Free-standing fiber networks were depositedon TEM grids (TAAB Laboratories Equipment Ltd.), used as collectors for the fiber networks. After deposition,electrospun fiber networks were stored in a glovebox (Jacomex, GP[Concept]) and annealed at 80 ◦C for 5 minutes innitrogen atmosphere to favor the formation of fiber joints at the nodes of the network, without degrading the emissionproperties of the embedded chromophores.

Estimation of scattering loss at a node

Finite-difference time-domain (FDTD) simulations (using Lumerical Solutions) were performed to estimate thescattering loss at nodes of the photonic network lasers. Since the most common degree (D) in the networks was 4,we modeled a node with an X-crossing (schematic shown in Fig. 5). To calculate the scattering loss, a guided modewas injected along a branch towards the node and the total power transmitted (or reflected) was measured throughall branches.

Fig. 5 shows the loss in a X-branch node as a function of angle, for the two lowest order modes. In these simulations,the structure was free-standing in air, and the fiber diameter was 500 nm, index was 1.5 and the mode wavelengthwas 600 nm. Loss was calculated for the configuration where the two fibers were just touching at the node, and whenthe two fibers were merged together. As shown in Fig. 5, the average loss is below 0.15 when the fibers are mergedand below 0.04 when the fibers are touching.

FIG. 5. Scattering loss in a X-branch. Loss in a X-branch is calculated for the two lowest order modes (HE11 modes ofdifferent polarizations) as a function of angle. The electric field profiles of these modes is shown in the inset. The data wascalculated from 3D FDTD simulations, in which the diameter of the fibers was 500 nm, index was 1.5 and the mode wavelengthwas 600 nm. The fibers were either merged or not merged, as indicated.

7

Wave equation on a graph

A graph is a collection of nodes i = 1..n, some pairs connected by edges r = 1..N . We solve the 1D scalar waveequation:

∂2E(x, t)

∂t2− v2 ∂

2E(x, t)

∂x2= 0

where v = c/n is the speed of light. On each edge r of the graph the solutions of the wave equation in the Helmholtzform:

∇2E(x) + k2E(x) = 0,

with k = ω/v, are of the form:

Er(x, k) = B+r exp(ıkx) +B−

r exp(−ıkx).

A graph with N edges is described by N of such equations. The 2N coefficients X1..2N = {B+1..N , B

−1..N} are defined

by the behavior at the nodes. As boundary conditions we impose the continuity of the solution at the node positionxi:

Er(xi) = Es(xi),

for all edges r, s which are connected to the node i, giving overall 2N − n equations, and∑r

dE(x)

dx

∣∣∣∣x=xi

= 0,

for all connected edges r at each node (n equations), with the positive direction of x being consistently in-going orout-going for all terms. This boundary conditions are equivalent to those of the graph laplacian operator defined as∇2 = A − D, where A is is the adjacency matrix describing the graph and D is the connectivity degree diagonalmatrix of the graph. Given a certain k, such conditions might be put in a 2N × 2N sparse matrix M(k), such that:

M(k) ·X = 0

The eigenvalues of the system are given by the condition:

det(M(k)) = 0.

Given the high sparsity of the matrix M , the eigenvalues can be efficiently computed even for very large number ofnodes. Numerically, the condition number C of M(k) is estimated (condest function in MATLAB) on a grid of thecomplex k region of interest to identify k (maxima of C) for which the matrix is singular. Subsequently, each locatedguess for k is optimized (simplex search) until convergence.

Threshold and IPR estimation for localized modes on the network

The inverse participation ratio IPR is defined as:

IPR = L

∫|E|4dl(∫|E|2dl

)2 , (1)

with L the total length of the network. From the definition of Q value:

Q =Re(k)

2|Im(k)|= ω · STORED ENERGY

LOSSES PER CICLE=

∫u(l)dl∑

perimeter S, (2)

with u the energy density and S the pointing vector:

u = ε0|E|2, S = cε0|E|2.

8

We compute these quantities assuming that a mode electric fields E(x, y) is exponentially decaying radially from thecenter of the network which occupy a disc or radius R:

|E|2(r) ' e−r/ξ,

with ξ the localization length. This integrals are computed in the limit of the radius of the network R� ξ (localizedregime), and assuming uniform density of the networks nodes, so that the integrals over the network (dl) can becomputed over the plane in which it is embedded (dl = αdxdy), with α = L

2πR2 , while the summation is computed

defining the density of output channels over the perimeter β = #(output channles)2πR , so that:

IPR =

(R

2ξ

)2

(3)

Q =Re(k)ξ2α

βRe−R/ξ(4)

from which:

− Im(k) =βRe−R/l

2ξ2α. (5)

The parameter α and β are obtained directly from the network structure and depend on the density of nodes andnetwork nodes degree (β/α ' 0.5).

Mode spacing for different system size

The modes spacing distribution for networks of increasing number of nodes is computed. For small network,well below the localization length, the mode spacing resembles the Wigner-Dyson distribution (dash-dot line). Forincreased number of nodes, the distribution converge towards an exponential curve (dashed line), expected for random(Poissonian) distributed modes.

0 1 2 3 4 5

normalized spacing

0.0

0.2

0.4

0.6

0.8

1.0PoissonianWigner-DysonN=15N=150N=600

FIG. 6. Mode spacing for network of different size. Modes spacing distribution for networks of increasing number ofnodes (N = 15, 150, 600), for a fixed degree D = 6 and average link length 〈le〉 = 6.5 µm.

References

S1 : Zhong, W.; Li, F.; Chen, L.; Chen, Y.; Wie, Y. J. Mater. Chem. 2012, 22, 55235530.

S2 : A. Camposeo, R. D. Pensack, M. Moffa, V. Fasano, D. Altamura, C. Giannini, D. Pisignano, G. D. Scholes,J. Am. Chem. Soc. 138, 1549715505 (2016).