Fig. 2: Achieving single mode lasing through non-uniform pumping (theory).

a-b NetSALT calculations for lasing on a Buffon graph with non-uniform pumping indicate various processes for mode selection. a Mode trajectories in the complex k plane (shown over large Re(k) range, with red dots identifying the passive modes and black dots identifying the modes that lase. The non-uniform pump profile used to obtain these trajectories is shown in d, right-hand panel inset. b Zoom of a over a small range of wavelengths with trajectories shown in colour, where the colour scale indicates the pump strength. Three modes (1, 99, and 134) are highlighted to exemplify the different processes involved in mode selection. Mode trajectories with uniform pumping are shown by solid grey lines. c Example mode profiles for two network modes (1 and 134) before pumping (passive, D₀ = 0) and at threshold for mode 1 (D₀ = 0.004) and at pump strength D₀ = 0.01 for mode 134. Minimal change in profile is observed for mode 1, whereas mode 134 reshapes significantly. d Modal intensities as a function of pump power for a uniform pump and a pump chosen to maximise lasing of mode 1 (see Methods), with pump profiles represented as green edges in insets. e Heatmap of the modal intensities of the first 50 modes (along each row) under 50 patterns optimised for each mode (each column). The computations correspond to the top 50 modes of the Buffon graph ordered in descending order of Q-factor. Optimisation of the pump profile leads to good mode selectivity.