  Anapole states associated with the resonant suppression of electric-dipole
scattering exhibit minimized extinction and maximized storage of
electromagnetic energy inside a particle. Using numerical simulations, optical
extinction spectroscopy and amplitude-phase near-field mapping of silicon
dielectric disks, we demonstrate high-order anapole states in the near-infrared
wavelength range (900-1700 nm). We develop the procedure for unambiguously
identifying anapole states by monitoring the normal component of the electric
near-field and experimentally detect the first two anapole states as verified
by far-field extinction spectroscopy and confirmed with the numerical
simulations. We demonstrate that higher order anapole states possess stronger
energy concentration and narrower resonances, a remarkable feature that is
advantageous for their applications in metasurfaces and nanophotonics
components, such as non-linear higher-harmonic generators and nanoscale lasers.
