Fig. 1 Schematic and working principle of multi-periodic nanostructures. Multi-periodic nanostructures are constructed by superimposing multiple binary gratings using a logical disjunction operation. (a) The component gratings have periods Λ_i and ridge widths l_rd,Λi giving duty cycles of l_rd,Λi/Λ_i. (b),(c) The resulting multi-periodic nanostructure is again a binary grating with period Λ that is the least common multiple of the component gratings' periods. The component gratings’ periods determine the resonances’ emission wavelengths and angles. The relative resonances’ intensities are controlled by the duty cycles as shown schematically for the case of two different two-component gratings (b,c).

Fig. 2 Fourier spectrum of a two-periodic nanostructure. (a) Calculated power Fourier spectrum of a 350|450 (l_rd,350nm = 100 nm, l_rd,450nm = 100 nm) multi-periodic nanostructure (black) together with the involved single period component spectra (red and green). The multi-periodic nanostructure spectrum comprises the dominant Fourier orders of the component gratings. (b) Control of Fourier spectrum of the 350|450 multi-periodic nanostructure for varying l_rd,350nm and fixed l_rd,450nm = 100 nm. The ridge width l_rd,350nm of the 350-nm component grating provides wide control over the dominating Fourier orders 7 and 9.

Fig. 3 Characterization of the nickel nanostructures by scanning electron microscopy (SEM) and atomic force microscopy (AFM). Exemplarily shown is a multi-periodic grating of type 196|400. (a), SEM image. (b), AFM height profile.

Fig. 4 Background removal and normalization of the emission spectrum. (a) Angularly- and spectrally-resolved, TE-polarized photoluminescence measurement of a nanostructured area on the sample. (b) TE-polarized photoluminescence measurement of an unstructured area on the sample. Multiplied with a matching factor, this emission is assumed to be the background of (a). (c) Background removal: measured signal (blue line) and background-free signal (red line) (TE-polarization, θ₀ = 0°). The guided mode outcoupling peaks are approximately separated by subtracting the background. Note that the plotted intensity range has been limited. (d) Normalization. To obtain excitation-independent resonance peaks, the background-free emission spectrum is subsequently divided by the background spectrum at 0°.

Fig. 5 Photoluminescence measurements of resonant scattering of the TE₀ guided mode at 16 different multi-periodic nanostructures. Shown is the normalized emission intensity as a function of wavelength and emission angle perpendicular to the grating grooves (continuous background subtracted and normalized). Symmetric pairs of peaks arise due to the presence of forward- and backward-traveling modes. The numbers next to the strongest peaks indicate the scattering order m. Note how the introduction of an additional component grating in the lower two rows leads to an additional pair of peaks. The ridge widths of the component gratings are varied along the columns and rows of the figure and provide control over the intensity of the resonances.

Fig. 6 Control of the peak intensities by a 350|450 multi-periodic nanostructure. Relative peak intensities with varying ridge width l_rd,350nm of the 350-nm component. The ridge width of the 450-nm component is 100 nm and the wavelength is λ₀ = 550 nm.