A nanophotonic laser on a graph

Conventional nanophotonic schemes minimise multiple scattering to realise a miniaturised version of beam-splitters, interferometers and optical cavities for light propagation and lasing. Here instead, we introduce a nanophotonic network built from multiple paths and interference, to control and enhance light-matter interaction via light localisation. The network is built from a mesh of subwavelength waveguides, and can sustain localised modes and mirror-less light trapping stemming from interference over hundreds of nodes. With optical gain, these modes can easily lase, reaching ~100 pm linewidths. We introduce a graph solution to the Maxwell’s equation which describes light on the network, and predicts lasing action. In this framework, the network optical modes can be designed via the network connectivity and topology, and lasing can be tailored and enhanced by the network shape. Nanophotonic networks pave the way for new laser device architectures, which can be used for sensitive biosensing and on-chip optical information processing.


Supplementary Note 1: Estimation of scattering loss at a node
Finite-difference time-domain (FDTD) simulations (using Lumerical Solutions) were performed to estimate the scattering loss at nodes of the photonic network lasers. Since the most common degree (D) in the networks was 4, we modelled a node with an X-crossing (schematic shown in Supplementary Figure 1). To calculate the scattering loss, a guided mode was injected along a branch towards the node and the total power transmitted (or reflected) was measured through all branches.
Supplementary Figure 1 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 fibre diameter was 500 nm, index was 1.5 and the mode wavelength was 600 nm. Loss was calculated for the configuration where the two fibres were just touching at the node, and when the two fibres were merged together. As shown in Supplementary Figure  The electric field profiles of these modes is shown in the inset. The data was calculated from 3D FDTD simulations, in which the diameter of the fibres was 500 nm, index was 1.5 and the mode wavelength was 600 nm. The fibres were either merged or not merged, as indicated.

Supplementary Note 2: 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 wave equation: where v = c/n is the speed of light. On each edge r of the graph the solutions of the wave equation in the Helmholtz form: with k = ω/v, are of the form: A graph with N edges is described by N of such equations. The 2N coefficients X 1..2N = {B + 1..N , B − 1..N } are defined by the behaviour at the nodes. As boundary conditions we impose the continuity of the solution at the node position x i : for all edges r, s which are connected to the node i, giving overall 2N − n equations, and for all connected edges r at each node (n equations), with the positive direction of x being consistently in-going or out-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 diagonal matrix of the graph.
Given a certain k, such conditions might be put in a 2N × 2N sparse matrix M (k), such that: The eigenvalues of the system are given by the condition: det(M (k)) = 0 (7) Given the high sparsity of the matrix M , the eigenvalues can be efficiently computed even for very large number of nodes. Numerically, the condition number C of M (k) is estimated (condest function in MATLAB) on a grid of the complex k region of interest to identify k (maxima of C) for which the matrix is singular. Subsequently, each located guess for k is optimized (simplex search) until convergence.

Supplementary Note 3: Estimate of threshold and inverse participation ratio
The inverse participation ratio (IPR) is defined as: with L the total length of the network. From the definition of Q value: with u the energy density and S the pointing vector: We compute these quantities assuming that a mode electric fields E(x, y) is exponentially decaying radially from the centre of the network which occupy a disc or radius R: with ξ the localization length. This integrals are computed in the limit of the radius of the network R ξ (localized regime), and assuming uniform density of the networks nodes, so that the integrals over the network (dl) can be computed over the plane in which it is embedded (dl = αdxdy), with α = L 2πR 2 , while the summation is computed defining the density of output channels over the perimeter β = #(output channles) 2πR , so that: from which: The parameter α and β are obtained directly from the network structure and depend on the density of nodes and network nodes degree (β/α 0.5).

Supplementary Note 4: Spectral evolution when increasing the pump energy
Supplementary Figure 2 shows the evolution of the lasing spectrum as a function of the pump energy. Above the lasing threshold many sharp spikes develop. When the pump energy is increased even further more modes start to lase while the already present lasing peaks simply grow in intensity with minimal spectral shift. This is consistent with lasing from localized modes, which are expected to be stable both spatially and spectrally.

Supplementary Note 7: Gain length calculation
The gain length of Rh6G (L g ) is estimated from its density in polymer (1% by weight) and its stimulated emission cross-section (σ = 3 × 10 −20 m 2 ). The density of the polymer is ∼ 1 g cm −3 and of Rh6G is 479 g mol −1 , so the density of Rh6G in polymer (Π) is: Π = 0.01 × 1 g cm −3 479 g mol −1 × (6 × 10 23 molecules mol −1 ) = 1.25 × 10 25 molecules m −3 Gain length is thus: However, this estimate is a best case scenario as it neglects losses and assumes perfect confinement of the mode inside the fibre. By taking into account the overlap of the fundamental mode with the fibre, which varies from 0.4 -0.8 for fibre diameters between 200-500 nm, the gain length is estimated to be 5-10 µm.