Grating-assisted superresolution of slow waves in Fourier space

We present a far-field optical technique allowing measurements of the dispersion relation of electromagnetic fields propagating under the light cone in photonic nanostructures. It relies on the use of a one-dimensional grating to probe the evanescent tail of the guided field in combination with a high-numerical-aperture Fourier-space imaging setup. A high-resolution spectroscopy of the far-field emission diagram allows us to accurately and efficiently determine the dispersion curve and the group-index dispersion of planar photonic crystal waveguides operating in the slow-light regime.

Figure 1

(a) Electron microscopy image of a SW1 waveguide (see text) with the linear probe gratings at the waveguide boundaries. (b) Illustration of the folding of the theoretical SW1 dispersion curve (dark line) into the light cone (dash line). Gray region: region below the light cone.

Figure 2

(Color online) (a) Schematic of the Fourier imaging principle. (b) Real space near-infrared image of a $500\mu \mathrm{m}$ SW1 waveguide $(\lambda =1514\mathrm{nm})$. (c) Corresponding far-field image. Red circle: Output pupil of the objective.

Figure 3

[(a) and (b)] Far-field spectra of the SW1 waveguide at different wavelengths in regimes I and II, respectively. Each spectrum is spaced by $2\mathrm{nm}$ wavelength step.

Figure 4

Left: Experimental dispersion curves of the reference W1 waveguide mode (dark squares), of the SW1 even mode (open circles), and of the dielectric band (open squares), with error bars corresponding to the linewidth of the modal spectrum. The gray dashed curves and the dark dotted curve are theoretical 3D plane-wave expansion calculation of the SW1 and of the reference W1, respectively. Right: Theoretical (gray) and experimental (black) group-index dispersion curves.