Quasi-BIC Mode Lasing in a Quadrumer Plasmonic Lattice

Plasmonic lattices of metal nanoparticles have emerged as an effective platform for strong light–matter coupling, lasing, and Bose–Einstein condensation. However, the full potential of complex unit cell structures has not been exploited. On the other hand, bound states in continuum (BICs) have attracted attention, as they provide topologically protected optical modes with diverging quality factors. Here, we show that quadrumer nanoparticle lattices enable lasing in a quasi-BIC mode with a highly out-of-plane character. By combining theory with polarization-resolved measurements of the emission, we show that the lasing mode has a topological charge. Our analysis reveals that the mode is primarily polarized out-of-plane as a result of the quadrumer structure. The quality factors of the out-of-plane BIC modes of the quadrumer array can be exceedingly high. Our results unveil the power of complex multiparticle unit cells in creating topologically protected high-Q modes in periodic nanostructures.

: Q-factor dependence on lattice plane momentum (a) for modes polarized mainly in the lattice plane, and (b) for those polarized mainly out of plane, as calculated from FEM simulations.
Topological charge of the BIC as obtained from FEM simu-12 lations 13 Figure S2: Winding of the polarization for the mode D 6 , calculated by using the angle of the polarization vector: φ(k) = arg[p(k) ·x + ip(k) ·ŷ], with p(k) = (x · u k (r, z) )x + (ŷ · u k (r, z) )ŷ, where u k is the electric field that is obtained from FEM simulations for a single unit cell with periodic boundary conditions, and · means the spatial average over a surface z = 5.4 µm away from the lattice plane. The winding direction is unaltered by the choice of the plane for the spatial average. The topological charge is then calculated upon discretization of the formula:  Figure S3: Schematic of the experimental setup used in the lasing experiment. The two CMOS cameras and the 2D CCD camera in the spectrometer allow for simultaneous measurements of angle-resolved spectra, real space images, and 2D k-space images; in this way one can confirm that the images correspond to the lasing mode (single narrow peak at the spectrometer). Here, ND, BS and LP 850 stand for neutral density, beam splitter and long pass filter with 850 nm cutoff length, respectively.
The transmitted or emitted light from the sample is collected by a 0.3 NA 36 objective with a tube lens. The optional polarization filters are placed behind the 37 tube lens to filter out single polarization states. In lasing measurements, an additional 38 2 850 nm long pass filter is used to filter out reflections of the pump beam. The back 39 focal plane of the objective is focused onto the entrance slit of the spectrometer.

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Each point on the slit corresponds to an emission angle θ y , which is related to the 41 in-plane wavevector by k y = k 0 sin (θ y ) with k 0 = 2π/λ 0 . Here, λ 0 is the free space 42 wavelength. Thus, the 2D CCD camera inside the spectrometer resolves the angle 43 of the light coming from the sample along one axis and the energy along the other.

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The real space images and 2D k-space images are collected by two separate CMOS 45 cameras.

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Calculation of the modes in the isolated quadrumer 47 We consider an isolated quadrumer for which the dipole moment on the i-th particle is Particles are coupled to their neighbours in a polarization-dependent 49 way, see Fig. 2(a), such that Ω L = P i · L i |Ĥ|P i+1 · L i , when the dipoles are 50 oriented longitudinally to the link connecting neighbouring particles i and i + 1, while 51 Ω T = P i ·T i |Ĥ|P i+1 ·T i for dipoles that are oriented transversely to the neighbouring 52 link 1;2 ; see notation in Fig. 2(b). In the basis (P x 1 , P y 1 , P x 2 , P y 2 , P x 3 , P y 3 , P x 4 , P y 4 ), the 53 coupling matrix reads: where ε is a zero-point energy. In our case Ω L ≈ 0 and Ω T is finite; in this case 55 we find two sets of four-fold degenerate modes, at energy ≈ ε ± Ω T . The spatial 56 structure of the eigenmodes is shown in Fig. 2(c), where arrows correspond to the 57 electric dipole moment orientation.
where α is an inverse-length parameter depending on the setup, and J 1 is the first  Figure S5: Vectorial in-plane electric field E, and E x , E y field components (in V/m) of all the modes considered in Fig. 2(c) of the main text. The arrows are scaled in proportion to the field intensity. Only the relative values of the field amplitude (colorscale bars next to the E x and E y panels) are meaningful, and they are shown for comparing the relative magnitude of the different field components. These plots are depicted at the lattice plane, corresponding to the experimental surface of the glass substrate on which the nanoparticles sit.  Fig. 2(c) of the main text for an infinite lattice. The mode D 6 has both negative and positive contributions close to the nanoparticles which may cancel radiation to far field, as expected for BICs. We carried out analogous calculations at a XY plane located 8 µm away from the lattice plane, and we found that the out-of-plane component of the Poynting vector for the modes B 1,2 outgrows in four orders of magnitude that of the modes D 3 − D 5 , and in seven orders of magnitude that of the modes D 1 , D 2 and D 6 . This is consistent with the results of Figs. 2(f),(g) of the main text.
In addition to the lattice presented in the main text, we studied lattices with different calculations with different system sizes" and "Experiments with different array sizes" 167 below.

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The interparticle distance of the particles in the quadrumer, i.e. β, has a large 169 effect on the lasing ability of the structure. Arrays with β > 0.58 did not lase, whereas   Figure S8: Threshold curves for different array sizes (a) and threshold curves for different array sizes normalized by array size (b), integrated lasing peak intensities at a pump fluence of 1.995 mJ/cm 2 (c), Q-factor at threshold for different array sizes (d). With decreasing array size, the peak intensity decreases and the threshold pump fluence increases. Arrays with an edge size below 80 µm do not show any lasing action. The Q-factor at threshold appears to be constant.
The lasing peak Q-factors for the arrays with edge lengths ranging from 95 to 212 120 µm are decreasing with the edge length. In Fig. 2 (b) of the main text it seems, 213 however, that the Q-factors for the arrays with edge lengths of 85 and 90 µm have 214 higher Q-factors. We attribute this to the behaviour of the spectrometer for low 215 intensity peaks: If the intensity is very low, the counts of the pixels around the 216 central peak are very low and comparable to the background. Hence, the peak 217 is detected only on very few pixels and the peak appears to be narrow. For high 218 intensity peaks however, the counts in the pixels around the central peak are much 219 higher and distinguishable from the background and hence, the peak appears wider.