Lasing at $K$ Points of a Honeycomb Plasmonic Lattice

R. Guo, M. Nečada, T. K. Hakala, A. I. Väkeväinen, and P. Törmä

Phys. Rev. Lett. 122, 013901 – Published 2 January 2019

Abstract

We study lasing at the high-symmetry points of the Brillouin zone in a honeycomb plasmonic lattice. We use symmetry arguments to define singlet and doublet modes at the $K$ points of the reciprocal space. We experimentally demonstrate lasing at the $K$ points that is based on plasmonic lattice modes and two-dimensional feedback. By comparing polarization properties to $T$ -matrix simulations, we identify the lasing mode as one of the singlets with an energy minimum at the $K$ point enabling feedback. Our results offer prospects for studies of topological lasing in radiatively coupled systems.

Received 23 May 2018

DOI:https://doi.org/10.1103/PhysRevLett.122.013901

Physics Subject Headings (PhySH)

Lasers Plasmonics

Atomic, Molecular & Optical

Authors & Affiliations

R. Guo¹, M. Nečada¹, T. K. Hakala^1,2, A. I. Väkeväinen¹, and P. Törmä^1,*

¹Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland
²Institute of Photonics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland

^*paivi.torma@aalto.fi

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 122, Iss. 1 — 11 January 2019

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(a) A measured angle-resolved extinction spectrum of a honeycomb lattice with particle separation of $p = 576 nm$ . Color scale shows the extinction which is defined as ( $1 - normalized transmission$ ). The SLR modes correspond to extinction maxima, closely following the diffracted orders (dashed lines). The left inset shows the measured dispersion around the $K$ point (the color scale is from 0 to 0.05). The right inset shows the dispersion obtained by $T$ -matrix simulations. (b) The lasing measurements. Nanoparticle samples combined with IR-792 molecules in solution are pumped with a femtosecond laser. The hexagonal geometry of the lattice (inset: scanning electron microscope image of the gold nanoparticles, scale bar 500 nm, with the A and B unit cell sites marked) enables lasing emission in six distinct off-normal angles, collected by a 0.6 NA objective and further analyzed. In the Fourier image, the six angles correspond to lasing at the six $K$ points of the first Brillouin zone, with distinct polarization directions (grey arrows) of the electric field $E$ .
Reuse & Permissions
Figure 2
Eigenmodes of the honeycomb plasmonic lattice at the $K$ point. (a) Real-space electric dipole polarizations of the nanoparticles (circles) corresponding to two singlet modes and a doublet mode, at a specific time. The dipole polarizations depicted by orange and magenta arrows evolve in time rotating clockwise and counterclockwise, respectively, for the $K$ mode, and in the opposite directions for the $K^{'}$ mode. (b) Fourier transform of the dipole polarizations in the corresponding eigenmodes. (c) Band structure of the empty lattice model, that is, as given by diffracted orders of a periodic structure without the effect of the localized plasmonic resonance of the nanoparticles, with the studied $K$ point highlighted. (d) The first Brillouin zone (green area) of the honeycomb reciprocal lattice and its high symmetry points, $a$ is the lattice constant. (e) Singular values (SV) of the symmetry-adapted scattering problem at the $K$ point, whose minima give the mode energies, as function of energy. The color shows the results of projection of the corresponding eigenmodes on the singlets and doublets obtained by group theory (the singlet $A_{1}^{'}$ that was found to lase experimentally is shown in orange, the other singlet $A_{2}^{'}$ in green, and the doublet $E^{'}$ in blue), the energies are marked by ticks.
Reuse & Permissions
Figure 3
(a) Measured emission spectra of a honeycomb lattice with $P = 1.38 P_{th}$ , where $P_{th} = 0.47 mJ / {cm}^{2}$ is the threshold pump fluence for the $K$ -point lasing mode (particle distance $p = 576 nm$ and diameter $d = 100 nm$ ). (b) The mode output power (squares) and the line width (circles) at the $K$ -point angle ( $35 ° \pm 0.4 °$ ) as a function of pump fluence. Note that due to low intensity, we cannot determine the line width at pump fluences below the threshold, for below threshold emission, see Supplemental Material [39]. (c) The emission intensity as a function of angle at the $K$ -point energy ( $\sim 1.426 eV$ ) with several pump fluences.
Reuse & Permissions
Figure 4
Lasing mode polarization. (a) Angle resolved emission of the sample without any polarizer in detection. All six $K$ points are clearly visible. (b) Polar emission intensities at each $K$ point in the presence of a linear polarizer. The angles refer to the polarizer angles and the radii refer to the measured intensities. The red dashed lines are the calculated intensity distributions for linearly polarized light (along the $Γ - K$ directions) passing through the polarizer at the corresponding angle.
Reuse & Permissions
Figure 5
Upper row: examples of real-space images for different polarization filters (no filter, horizontal, vertical) used for the analysis of the lasing mode. In each case, the expected positions of the nanoparticles (small yellow-cyan circles) and the dipole polarizations (arrows) of the singlet mode $A_{1}^{'}$ for the ideal (zero) $K - K^{'}$ relative phase are depicted, projected to the corresponding filter direction. Lower row: theoretical prediction for the intensities for the ideal $K - K^{'}$ relative phase. The scale bar length is $1 μ m$ . For images over larger areas, see Supplemental Material [39].
Reuse & Permissions

Physical Review Letters