SI

# Multi-mode lasing in supercell plasmonic nanoparticle arrays

Rebecca Heilmann  Department of Applied Physics, Aalto University School of
Science, P.O. Box 15100, Aalto, FI-00076, Finland  Kristian Arjas  Department
of Applied Physics, Aalto University School of Science, P.O. Box 15100, Aalto,
FI-00076, Finland  Tommi K. Hakala  Center for Photonics Sciences, University
of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland  Päivi Törmä
Department of Applied Physics, Aalto University School of Science, P.O. Box
15100, Aalto, FI-00076, Finland<EMAIL_ADDRESS>
###### Abstract

Multicolour light sources can be used in applications such as lighting and
multiplexing signals. In photonic and plasmonic systems, one way to achieve
multicolour light is via multi-mode lasing. To achieve this, plasmonic
nanoparticle arrays are typically arranged in superlattices that lead to
multiple dispersions of the single arrays coupled via the superlattice Bragg
modes. Here, we show an alternative way to enable multi-mode lasing in
plasmonic nanoparticle arrays. We design a supercell in a square lattice by
leaving part of the lattice sites empty. This results in multiple dispersive
branches caused by the supercell period and hence creates additional band
edges that can support lasing. We experimentally demonstrate multi-mode lasing
in such a supercell array. Furthermore, we identify the lasing modes by
calculating the dispersion by combining the structure factor of the array
design with an empty lattice approximation. We conclude that the lasing modes
are the 74th $\Gamma$\- and 106th $X$-point of the supercell. By tuning the
square lattice period with respect to the gain emission we can control the
modes that lase. Finally, we show that the lasing modes exhibit a combination
of transverse electric and transverse magnetic mode characteristics in
polarization resolved measurements.

keywords: plasmonics, nanophotonics, surface plasmon resonance, multi-mode
lasing

## Introduction

Plasmonic nanoparticle arrays support surface lattice resonances (SLRs) that
are dispersive plasmonic-photonic modes arising from a hybridization between
the plasmonic resonances of individual nanoparticles and the diffracted orders
governed by the lattice geometry. The spectral position of the SLRs can be
easily tailored by varying the lattice geometry and period while
simultaneously yielding high quality (Q-) factors [1, 2, 3, 4, 5, 6, 7].
Combined with emitters such as organic dye molecules, plasmonic nanoparticle
arrays are an effective system to study light-matter interaction such as
strong coupling or Bose-Einstein condensation [8, 9, 10, 11, 12, 13]. Lasing
in plasmonic nanoparticle arrays has been studied in various lattice
geometries such as square, rectangular, honeycomb or hexagonal lattices.
Typically the systems produce lasing at a band edge originating at high
symmetry points of the lattice, for instance at the $\Gamma$-, $K$\- or
$M$-points [14, 15, 16, 17, 18, 19]. Also bound states in continuum which have
extraordinary high Q-factors have been recently exploited for lasing in
plasmonic arrays [20, 21, 22].
For lighting applications and optical communication, multicolour light sources
are necessary. Ideally, such sources span red, blue and green wavelengths in
order to create white light or NIR regions for optical communication [23, 24,
25, 26]. In photonic systems, one way to achieve multicolour light sources is
via multi-mode lasing, i.e. simultaneous lasing at a set of different modes.
As lasing occurs in plasmonic systems at band edges, multiple band edges need
to be created to realize multi-mode lasing. The most straightforward approach
is by organizing individual arrays in a larger superlattice network, where the
SLRs of the individual arrays couple to the Bragg modes of the superlattice,
leading to multiple band edges at different energies and wavevectors [27].
Multi-mode lasing has been observed in such superlattice geometries [28, 29].
Another possibility to realize multimode lasing is by dividing a square array
into smaller patches which have slightly different periods. This creates
additional band edges at different energies which simultaneously lase under
optical pumping [30].
Another way of creating additional band edges for a zero wavevector, i.e. into
the direction normal to the array plane, is by introducing an effective second
lattice period which yields a second SLR. This can be done in bipartite arrays
[6, 31, 32] or by introducing periodic vacancies to the arrays [33]. By
removing particles at designated positions, deterministic aperiodic lattices
that yield more complicated band structures and hence additional band edges
have been realised and lasing in such lattices has been demonstrated [34].
However, multi-mode lasing has not been explicitly studied. Other systems in
which multi-mode lasing has been observed include low-symmetry arrays, where
two polarization dependent modes lase simultaneously [35], light-cone SLRs
overlapping with higher Brillouin Zone (BZ) edges enabling lasing from several
high symmetry points at once [36], and lasing in quasi-propagating modes that
span a continuum of energies [37]. In addition to the aforementioned plasmonic
structures, multi-mode lasing has been achieved in various photonic systems
such as hyperuniform structures [38], topological insulators [39, 40], bound
states in the continuum [41, 21] and as simultaneous lasing in the magnetic
and electric resonances of a dielectric nanoparticle array [42].
Here we study lasing in a plasmonic square lattice where we create a periodic
supercell by removing particles at designated positions. We perform lasing
experiments by combining the array with a solution of dye molecules. Under
optical pumping we observe multiple lasing peaks that emerge at different
energies and non-zero wavevectors. To understand the interplay between the two
periods in the system (underlying square and the supercell periods) we
calculate the Empty Lattice Approximation (ELA) from the geometric structure
factor and free photon dispersion [34]. We can see additional modes enabled by
the much longer supercell period which would otherwise be suppressed in the
original square lattice. These new modes can be seen to exist at high-symmetry
points of the Brillouin Zone (BZ) as defined by the supercell: 74th $\Gamma$\-
and 106th $X$-point if all possible modes are considered. While these modes
are expected to exist in any square lattice matching the period of the
supercell, by changing the positions of the particles in the unit-cell we can
exhibit some control over the modes. This type of supercell and theoretical
framework provides a new platform for designing multimode lasing systems.

## Experiments

Figure 1: Lasing experiment in a supercell nanoparticle array. a) SEM image of
the array studied. The array is based on a square array structure (period
$p=596$ nm) where specific lattice sites are kept empty. The giant unit cell
is highlighted ($q$ is the unit cell size). b) Measured transmission spectrum
and c) results of the lasing experiments. For the experiments in c), the array
is combined with organic dye molecules (IR 140, 10 mM). Multiple lasing peaks
are visible at different energies and $k_{y}$ (see central plot). The
threshold behaviour of the lasing peaks is shown as indicated by the arrows.

We study a system based on a nanoparticle lattice with a square symmetry,
however, part of the sites of the square array are left empty. This creates a
unit cell of 13 $\times$ 13 sites which is repeated over the whole array. An
SEM image of the array is shown in Fig. 1 a). Details of the sample
fabrication and the measurement setup are given in the Supporting Information,
including Fig. S1. The period of the square array is $p=596$ nm and hence, the
period of the supercell is $q=7748$ nm. The gold nanoparticles have a diameter
of 120 nm and a height of 50 nm. A transmission measurement of the array is
shown in Fig. 1 b), where the dispersive surface lattice resonances (SLRs) are
clearly visible. The SLRs correspond to the underlying square array with the
$\Gamma$-point located at $k_{y}=0$, $E$=1.37 eV. In transmission
measurements, the finer features caused by the supercell are scarcely visible
below the TM-branch.
We combine the nanoparticle array with a solution of organic dye molecules (IR
140, 10 mM) by sandwiching a droplet between the sample slide and a
superstrate, and pump the system optically with a femtosecond laser (1 kHz
repition rate, 800 nm central wavelength). With increasing pump fluence, a set
of narrow peaks emerge as shown in Figure 1 c). The corresponding threshold
curves, i.e. emission intensity versus pump fluence are shown in the plots
surrounding the spectrometer data recorded at a pump fluence of 0.1329 mJ/cm2.
At the highest energy of $1.428$ eV there are five peaks at
$k_{y}=-1.662\mu\mathrm{m}^{-1}$, $k_{y}=-0.852\mu\mathrm{m}^{-1}$,
$k_{y}=0\mu\mathrm{m}^{-1}$, $k_{y}=0.852\mu\mathrm{m}^{-1}$, and
$k_{y}=1.677\mu\mathrm{m}^{-1}$. There are four modes along the transverse
electric (TE) SLRs at energies of $E=1.419$ eV
($k_{y}=-0.387\mu\mathrm{m}^{-1}$ and $k_{y}=0.392\mu\mathrm{m}^{-1}$) and
$E=1.417$ eV ($k_{y}=-0.370\mu\mathrm{m}^{-1}$ and
$k_{y}=0.366\mu\mathrm{m}^{-1}$). For clarity, only the threshold curves for
the modes at the slightly higher energy are shown in Fig. 1 c), the other
modes are shown in the Supporting Information, Figure
LABEL:fig:threshold_lower_TESLR. At energies of $E=1.416$ eV are two peaks
visible at $k_{y}=-1.281\mu\mathrm{m}^{-1}$ and
$k_{y}=1.279\mu\mathrm{m}^{-1}$. Lastly, there are two lasing peaks at
$k_{y}=-2.046\mu\mathrm{m}^{-1}$ and $k_{y}=2.050\mu\mathrm{m}^{-1}$ at an
energy of $E=1.400$ eV. Note that only the four lasing peaks that lie on the
TE modes of the SLRs coindice with modes that can be seen in Fig. 1 b),
whereas none of the modes at other lasing peaks are visible in the
transmission measurement.
Figure S3 in the Supporting Information shows the dye emission with the lasing
mode energies indicated. All of these modes coincide with the emission
maximum. Interestingly, the $\Gamma$-point of the underlying square lattice
($E=1.37$ eV) is located at an energy where no lasing takes place. This
suggests that the modes originating from the supercell experience more gain
and/or have lower losses and are therefore more likely to lase. By changing
the square lattice period, the dispersions of the arrays can be conveniently
shifted with respect to the emission maximum of the dye. As a consequence, the
modes that exhibit lasing can be tuned as shown in Figure S4 in the Supporting
Information.

## Results and Discussion

In Figure 2 the real space pattern of the nanoparticle array is shown (a)
along with its geometric structure factor $S(\mathbf{k})$(b). The structure
factor describes the scattering properties of the lattice for any given wave-
vector $\mathbf{k}$ and can be interpreted as a measure of constructive
interference along that scattering direction. In a typical square lattice
$S(\mathbf{k})$ has peaks of equal magnitude at reciprocal lattice sites, i.e.
at the centres of the Brillouin Zone (BZ). Removing particles from the array
removes some of the destructive interference in the system allowing for new
scattering directions to occur. The amplitude of these new scatterings depends
on the number and positions of the particles removed. If particles are removed
periodically, the system becomes periodic with a supercell period and has a
new, smaller BZ. However, these new BZ:s are not made equal as their
properties are dependent on the way particles are removed. Let us denote the
initial square lattice period and the supercell period as $p$ and $q$
respectively, with associated reciprocal lattice vectors of magnitude $a$ and
$b$. For a periodic structure with a multi-particle unit cell, the normalized
structure factor is

$S(\mathbf{k})=\frac{1}{N^{2}}\sum_{ij}e^{i\mathbf{k}\cdot(\mathbf{r}_{i}-\mathbf{r}_{j})}=\frac{1}{N_{u}^{2}}\sum_{i^{\prime}j^{\prime}}e^{i\mathbf{k}\cdot(\mathbf{r}_{i^{\prime}}-\mathbf{r}_{j^{\prime}})}\cdot\underbrace{\frac{1}{N_{\alpha}^{2}}\sum_{\alpha\beta}e^{i\mathbf{k}\cdot(\mathbf{q_{\alpha}-q_{\beta}})}}_{\delta(\mathbf{k}-m_{1}\mathbf{b_{1}}-m_{2}\mathbf{b_{2}})},$
(1)

where $N_{u},N_{\alpha}$ are the number of particles in a unit cell and number
of unit cells, $i,j$ and $i^{\prime},j^{\prime}$ the indices of particles in
the lattice and in a unit cell, $\mathbf{q_{\alpha},q_{\beta}}$ the positions
of unit-cells, $\mathbf{b_{1}}$,$\mathbf{b_{2}}$ reciprocal lattice vectors
and $m_{1},m_{2}$ integers. As can be seen in Figure 2 b), the system retains
information from both periods. In addition to the original scattering peaks
(yellow dots at the $\Gamma$-points of the square lattice) a new set of
secondary peaks appears at the $\Gamma$-points of the supercell lattice, the
strongest of which exist near the corners of the initial BZ. The new system
retains information from the original period $p$ as the $S(\mathbf{k})$ can be
seen to repeat with a period of $a=2\pi/p$.

The first approximation for the band-structure is given by the empty-lattice
approximation (ELA) which is obtained by taking the convolution of geometric
structure factor $S(\mathbf{k})$ with the in-plane free-photon dispersion
$|\mathbf{k}|=\sqrt{k_{x}^{2}+k_{y}^{2}}=nE/(\hbar c)$ [34]: $\int
d\mathbf{q}S(\mathbf{q})\delta(|\mathbf{q-k}|-nE/(\hbar c))$.This corresponds
to placing light cones at the peaks of the structure factor (in case of square
(super) lattices the $\Gamma$-points of each Brillouin zone) and weighing them
by the value of the structure-factor. The results are shown in Figure 2 c). As
$S(\mathbf{k})$ indicates the amount of constructive interference, the weight
of the light-cone correlates with the strength of the mode. In addition to the
typical ELA-dispersions of a square lattice, additional weaker dispersions
emerge from the $\Gamma$-points of the supercell. The experimentally measured
dispersion overlayed with the calculated ELA is shown in Figure S5 in the
Supporting Information showing a good agreement with the theoretical model and
measurements. By comparing the measurements to the ELA-dispersions, we find
that the lasing peaks exist slightly below crossings of two or more bands at
the high-symmetry points of the new BZ as is shown in figure 2 d). Due to the
large size of the supercell, the new BZ is small and can be repeated multiple
times in the measured range. The modes at equal energy are separated by a
multiple of $k_{y}=b$, so when folded back to the first BZ they can be seen to
correspond to the same high-symmetry points. It is unclear whether in the
experiments there exist even more lasing peaks at the same energy as we are
limited in $k_{y}$ by the optics, i.e. the numerical aperture of the
objective, (see Supporting Information Figure S1). Interestingly, some of the
bands in the band-structure in Figures 2 c) and d) have different slopes than
the TE and TM modes originating from the underlying square lattice (the lines
with the highest weight). These modes with different slopes cannot be
categorized as purely transverse electric/magnetic (TE/TM) and the modes
observed in the measurements coincide with such bands.This implies that the
lasing modes are not purely TE or TM polarized.

Figure 2:  a) Schematic picture of the lattice with the unit cell (orange) and
superlattice period highlighted. b) Structure factor calculated with equation
1 with dashed lines marking the reciprocal lattice as defined by the
underlying square lattice. For clarity the values shown are limited to
$[0,0.2]$. A close-up is shown in Figure S6. c) Weighed empty-lattice
approximation calculated from the structure factor. Line color correlates with
the ELA weight. d) Close-up of the dispersion relation close to the
experimentally measured region (shaded area in c)). The weights have been
increased to make the fine-structure more visible. Measured lasing peaks
(green dots) are found to exist at crossings of at least two bands. When
folded into the first BZ as defined by the superlattice (orange dots) period
we find that lasing peaks at the same energy actually correspond to the same
high-symmetry point. For visual clarity the $\Gamma$ and $X$-points of the
supercell BZ have been highlighted in cyan and black dashed lines.

To verify the hybrid TE/TM nature of the lasing states, we experimentally
studied an array with the same lattice parameters ($p=596$ nm and $q=13p$),
however, the edge length is now $240$ $\mu$m. We combined the nanoparticle
array with a reservoir of dye molecules with the same concentration of 10 mM
as in the previous measurements, leading to increased total amount of
molecules. The larger array as well as the increased gain caused by the higher
amount of molecules lead to a stronger signal. This is needed as we added a
polarizer into the detection path which decreases the amount of measurable
light. We collected angle-resolved spectra as well as full momentum space
images as is shown in Figure 3. Here, a vertical orientation refers to the
axis of the polarizer oriented parallel $k_{x}=0^{\circ}$ and a horizontal to
the axis of the polarizer oriented parallel to $k_{y}=0^{\circ}$.
Changing the size of the array changes the lasing spectrum. In total, five
lasing peaks are clearly observable in Figure 3 a) with additional four peaks
with less intensity. The peaks at the higher energy ($E=1.403$ eV) are at
wavevectors $k_{y}=-1.680,-0.859,0,0.845,$ and $1.661$ $\mu$m-6 which
correspond directly to the wavevectors of the highest energy mode ($E=1.428$
eV) of the smaller array shown in Figure 1 c), and based on the weights of the
ELA we conclude that the lasing mode is now the 71st $\Gamma$-point.The peaks
at the lower energy in Figure 3 a) ($E=1.393$ eV) occur at wavevectors
$-2.047,-0.443,0.431,$ and $2.024$ $\mu$m-6, where the larger $k_{y}$ directly
correspond to the $X$-point lasing peaks in Figure 1 c). The peaks at the
smaller $k_{y}$ are the $X$-points closest to $k_{y}$ in Figure 2 d). The
shift in energy is most likely caused by the increased amount of dye molecules
that leads to a shift in the refractive index.
The majority of these peaks are visible in the angle-resolved spectra with
polarization filters applied, albeit with varying intensity. Further, the
peaks at $k_{y}=\pm 0.8$ $\mu$m-1 and $E=1.403$ eV do not appear in the case
where a horizontally oriented polarizer is applied (Figure 3) b), implying
that these modes are TM polarized.
The full momentum space images in Figure 3, bottom row, show strong features
along $\theta_{y/x}=0^{\circ}$ if a horizontally/vertically oriented polarizer
is applied. These images are in logarithmic scale to make weaker features more
visible. And indeed, although a polarizer is applied, the peaks along
$\theta_{x/y}=0^{\circ}$ can still be distinguished with a horizontal/vertical
polarization filter applied. Nevertheless, these peaks are much weaker in
intensity as the others and although this implies the hybrid TE/TM nature of
the modes, the modes are mainly TE or TM polarized.

Figure 3: Polarization characteristics of the lasing modes in a large
supercell array. The square lattice period is $p=596$ nm and the edge length
of the array is 240 $\mu$m. Angle-resolved spectra for a) no polarizer
applied, b) horizontally, and c) vertically oriented polarizer. Full momentum
space images in logarithmic scale for d) no polarizer applied, e)
horizontally, and f) vertically oriented polarizer.

For comparison, we simulated a superlattice-type version of our structure and
found the results of such calculation to be in agreement with [28], as is
shown in Supporting Information S7. Since both structures have the same
supercell periodicity, band crossings happen at same values of $\mathbf{k}$
and $E$. However, due to the differences in structure factors, these crossings
have different weights, and thus different bands are expected to be
responsible for lasing. The strongest secondary bands in the supercell lattice
exist in the vicinity of the original square lattice modes while in our case
the strongest secondary bands are found closer to the $M$-points. In this
case, the observed crossings are not the results of these stronger bands as is
evident in Figure 2 c) and d). Instead, we find the crossings to come from
multiple colliding weak modes. While these modes exist for the superlattice
structure as well, the relative weights of the modes can be estimated to be
different, as is shown by the different weight of the lines in Figure 2 and
S7. In fact, the relative strengths of these crossings can be seen to
correlate with the mode brightness.

## Conclusion

We demonstrated multi-mode lasing in a plasmonic supercell lattice. We showed
that by introducing a periodic supercell in a square lattice geometry,
additional band edges are formed near high symmetry points of the supercell.
These band edges enable lasing and we observed lasing in multiple modes. By
calculating the empty lattice approximation based on the structure factor of
the lattice design, we identified the lasing modes to be the 74th $\Gamma$\-
and 106th $X$-point of the supercell. Due to their higher order nature these
lasing modes are not purely TE or TM polarized. By tuning the square lattice
period with respect to the emission maximum of the gain medium we were able to
select the lasing modes. One advantage of the supercell approach compared to
the superlattice approach is the relative size of the structure. In previous
superlattices in which multimode lasing has been achieved, the individual
arrays of the size of 10s of $\mu$m were arranged on a centimeter square
($10^{-4}$ m) scale [28]. The supercell arrays presented in this work on the
other hand provide multimode in arrays with a size of 115 $\mu$m x 115 $\mu$m
($10^{-8}$ m) and are therefore significantly smaller. The supercell design
approach provides new possibilities to engineer band edges for multimode
lasing for instance by distributing the particles within the supercell
differently or by changing the supercell period.

## Acknowledgements

We thank Grazia Salerno for valuable discussions on the calculation of the
band structure based on the structure factor.
Funding: This work was supported by the Academy of Finland under Project No.
318937 (PROFI), 322002, the Academy of Finland Flagship Programme, Photonics
Research and Innovation (PREIN), Project No. 320166 and 320167, and the Vilho,
Yrjö and Kalle Väisälä Foundation. Part of the research was performed at the
OtaNano Nanofab cleanroom (Micronova Nanofabrication Centre), supported by
Aalto University. We acknowledge the computational resources provided by the
Aalto Science-IT project. R.H. acknowledges financial support by the Finnish
Foundation for Technology Promotion.
Author contributions: R.H. initiated the project and P.T. supervised it. R.H.
fabricated the samples and performed the measurements. K.A. calculated the
band structures. All authors discussed the results. R.H. and T.K.H. wrote the
manuscript with input from all coauthors.
Competing interests: There are no competing financial interests.

## Supporting Information

The Supporting Information is available free of charge. Experimental Methods;
Threshold curves of lower TE SLR lasing modes; Dye emission; Lasing
experiments with different lattice periods; Measured dispersion and calculated
ELA; Close-up of the structure factor; ELA of a superlattice.

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